second

Edition
Reviews ofthe First Edition

“The processes are always

explained in the simplest
terms, while maintaining full su
Usingfundamental physics, the theory of stellar
mathematical rigor... requires
structure and evolution can predict how stars are born,
only basic undergraduate physics
how their complex internal structure changes, what
and mathematics and no prior
nuclear fuel they burn, and their ultimate fate. This knowledge of astronomy...”
textbook is a stimulating introduction for students of Orion, societe Astronomique
astronomy, physics, and applied mathematics, taking a
fD
de Suisse
course on the physics of stars, it uniquely emphasizes the
basic physical principles governing stellar structure and “... a first-class textbook... The
evolution. host of student exercises... ensure —
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that any dedicated physics or
o
This second edition contains two new chapters on mass mathematics undergraduate can,
loss from stars and interacting binary stars, and new with some effort, understand
exercises. Clear and methodical, it explains the processes
what is going on.”
David Hugh£$New Scientist
in simple terms, while maintaining mathematical
rigour, starting from general principles, this textbook
”... a book that I can strongly
leads students step-by-step to a global, comprehensive
recommend as a suitable textbook
understanding ofthe subject. Fifty exercises and full
to anyone teaching a course in
solutions allow students to test their understanding. stellar structure, at advanced
No prior knowledge of astronomy is required, and only undergraduate or beginning m
a basic background in undergraduate physics and graduate level... An excellent
mathematics is necessary. book, which certainly deserves to
become a classic.”
Dina Prialnik is Professor of Planetary Physics at Tel Aviv Robert connon smith,
university. Her research interests lie in stellar evolution; The obseruatpry
the structure and evolution of cataclysmic variables;
comet nuclei and other small solar system bodies, and
the evolution of planets.

cover designed by’hil Treble

Cambridge
cover illustration: light echo around V878 wonocerotis. UNIVERSITY PRESS
Courtesy of NASA, f SA. and h. Bond (STSO). www.cambridge.org

o5R
 second Edition

An introduction to the

Theory of
stellar structure
'• and
Evolution

Dina Prialnik

Cambridge

I
 An Introduction to the Theory of
Stellar Structure and Evolution

Second Edition

Using fundamental physics, the theory of stellar structure and evolution can predict
how stars are born, how their complex internal structure changes, what nuclear
fuel they burn, and their ultimate fate. This textbook is a stimulating introduction
for students of astronomy, physics and applied mathematics, taking a course on the
physics of stars. It uniquely emphasizes the basic physical principles governing
stellar structure and evolution.
This second edition contains two new chapters on mass loss from stars and
interacting binary stars, and new exercises. Clear and methodical, it explains
the processes in simple terms, while maintaining mathematical rigour. Starting
from general principles, this textbook leads students step-by-step to a global,
comprehensive understanding of the subject. Fifty exercises and full solutions
allow students to test their understanding. No prior knowledge of astronomy is
required, and only a basic background in undergraduate physics and mathematics
is necessary.

Dina Prialnik is a Professor of Planetary Physics at Tel Aviv University. Her

research interests lie in stellar evolution; the structure and evolution of cataclysmic
variables; comet nuclei and other small solar system bodies; and the evolution of
planets.
An Introduction to
the Theory of
Stellar Structure
and Evolution
Second Edition

Dina Prialnik
Tel Aviv University

Cambridge
UNIVERSITY PRESS
 Cambridge
UNIVERSITY PRESS
University Printing House. Cambridge CB2 8BS. United Kingdom

Cambridge University Press is part of the University of Cambridge.

It furthers the University’s mission by disseminating knowledge in the pursuit of
education, learning and research at the highest international levels of excellence.

www.cambridge.org
Information on this title: www.cambridge.org/97805218ft6040

First edition © Cambridge University Press 2000

Second edition © D. Prialnik 2010

This publication is in copyright. Subject to statutory exception

and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of D. Prialnik.

First published 2000

Reprinted 2004. 2005. 2006. 2007, 2008
Second edition printed 2010
Reprinted 2010 (with corrections)
8th printing 2015

Printed in the United Kingdom by TJ International Ltd. Padstow, Cornwall

A catalog record for this publication is available from the British Library

Library of Congress Cataloging in Publication data

Prialnik. Dina.
An introduction to the theory of stellar structure and evolution / Dina Prialnik. 2nd cd.
p. cm.
ISBN 978-0-521 -86604-0 (hardback)
I. Stars - Structure. 2. Stars - Evolution. I. Title.
QB808.P75 2009
523.8'8 dc22 2009034267

ISBN 978 0 521 86604 0 Hardback

Cambridge University Press has no responsibility for the persistence or

accuracy of URLs for external or third-party Internet websites referred to
in this publication, and does not guarantee that any content on such
websites is. or will remain, accurate or appropriate.
To my son
 Contents

Preface to the second edition page xi

Preface to the first edition xiii

1 Observational background and basic assumptions I

I. I What is a star? I
1.2 What can we learn from observations? 2
1.3 Basic assumptions 6
1.4 The H-R diagram: a tool for testing stellar evolution 9

2 The equations of stellar evolution 15

2.1 Local thermodynamic equilibrium 16
2.2 The energy equation 17
2.3 The equation of motion 19
2.4 The virial theorem 21
2.5 The total energy of a star 23
2.6 The equations governing composition changes 25
2.7 The set of evolution equations 28
2.8 The characteristic timescales of stellar evolution 29

3 Elementary physics of gas and radiation in stellar interiors 34

3.1 The equation of state 35
3.2 The ion pressure 37
3.3 The electron pressure 38
3.4 The radiation pressure 42
3.5 The internal energy of gas and radiation 43

vii
viii Contents

3.6 The adiabatic exponent 44

3.7 Radiative transfer 46

4 Nuclear processes that take place in stars 51

4.1 The binding energy of the atomic nucleus 51
4.2 Nuclear reaction rates 53
4.3 Hydrogen burning 1: the p — p chain 57
4.4 Hydrogen burning II: the CNO bi-cycle 59
4.5 Helium burning: the triple-a reaction 61
4.6 Carbon and oxygen burning 63
4.7 Silicon burning: nuclear statistical equilibrium 65
4.8 Creation of heavy elements: the .v- and /--processes 66
4.9 Pair production 67
4.10 Iron photodisintegration 68

5 Equilibrium stellar configurations - simple models 70

5.1 The stellar structure equations 70
5.2 What is a simple stellar model? 71
5.3 Polytropic models 72
5.4 The Chandrasekhar mass 77
5.5 The Eddington luminosity 78
5.6 The standard model 80
5.7 The point-source model 83

6 The stability of stars 87

6.1 Secular thermal stability 88
6.2 Cases of thermal instability 89
6.3 Dynamical stability 92
6.4 Cases of dynamical instability 94
6.5 Convection 96
6.6 Cases of convective instability 98
6.7 Conclusion 103

7 The evolution of stars - a schematic picture 104

7.1 Characterization of the (log T. log p) plane 105
7.2 The evolutionary path of the central point of a star in the
(log T, log p) plane 110
7.3 The evolution of a star, as viewed from its centre 113
 Contents ix

7.4 The theory of the main sequence 116

7.5 Outline of the structure of stars in late evolutionary stages 122
7.6 Shortcomings of the simple stellar evolution picture 126

8 Mass loss from stars 130

8.1 Observational evidence of mass loss 130
8.2 The mass loss equations 131
8.3 Solutions to the wind equations - the isothermal case 136
8.4 Mass loss estimates 139
8.5 Empirical solutions 142

9 The evolution of stars - a detailed picture 144

9.1 The Hayashi zone and the pre-main-sequence phase 145
9.2 The main-sequence phase 151
9.3 Solar neutrinos 155
9.4 The red giant phase 160
9.5 Helium burning in the core 165
9.6 Thermal pulses and the asymptotic giant branch 168
9.7 The superwind and the planetary nebula phase 173
9.8 White dwarfs: the final state of nonmassive stars 177
9.9 The evolution of massive stars 182
9.10 The H-R diagram - Epilogue 186

10 Exotic stars: supernovae, pulsars and black holes 189

10.1 What is a supernova? 189
10.2 Iron-disintegration supernovae: Type II - the fate of
massive stars 193
10.3 Nucleosynthesis during Type 11 supernova explosions 197
10.4 Supernova progenies: neutron stars - pulsars 200
10.5 Carbon-detonation supernovae: Type la 204
10.6 Pair-production supernovae and black holes - the fate of very
massive stars 205

I I Interacting binary stars 208

11.1 What is a binary star? 208
11.2 The general effects of stellar binarity 211
1 1.3 The mechanics of mass transfer between stars 216
1 1.4 Conservative mass transfer 219
X Contents

11.5 Accretion discs 220

11.6 Cataclysmic phenomena: Nova outbursts 223

12 The stellar life cycle 231

12.1 The interstellar medium 231
12.2 Star formation 232
12.3 Stars, brown dwarfs and planets 236
12.4 The initial mass function 239
12.5 The global stellar evolution cycle 243

Appendix A — The equation of radiative transfer 251

Appendix B - The equation of state for degenerate electrons 259
Appendix C — Solutions to all the exercises 270
Appendix D - Physical and astronomical constants and conversion
factors 300
Bibliography 303
Index 308
 Preface to the second edition

It is now a decade since the publication of the first edition of this book. Despite
the large number of research papers devoted to the subject during this period of
time, the basic principles and their applications that are addressed in the book
remain valid and hence the original text has been mostly left unchanged. And
yet a major development did occur soon after the book first appeared in print:
the ‘solar neutrino problem' that had puzzled physicists and astrophysicists for
almost four decades finally found its solution, which indeed necessitated new
phy sics. However, the new physics belongs to the theory of elementary particles,
which must now account for neutrino masses, rather than to the theory of stars.
Also worth mentioning is a major recent discovery that finally provides support
to the theory proposed about four decades ago regarding the end of very massive
stars in powerful supernova explosions triggered by pair-production instability:
SN2006gy, the first observed candidate for such a mechanism. Thus the section
on solar neutrinos is now complete and that on supernovae expanded.
Stellar evolution calculations have made great progress in recent years, fol­
lowing the rapid development of computational means: increasingly faster CPUs
and greater memory volumes. Nevertheless. I have made use of new results only
when they provide better illustration for points raised in text. For the most part, old
results are still valid and this long-term validity is worth emphasizing: the theory
of stellar structure and evolution, with all its complexity, is a well-established
physical theory.
The text was expanded to include two new chapters on topics that were
not addressed in the first edition: mass loss and interacting binary stars. Both
are complicated subjects, some aspects of which are still not well understood,
similarly to star formation. Although this may justify their exclusion from a basic-
textbook on stellar structure and evolution theory, an exposition of the theory
would not be complete without some reference to them. Each one deserves a full
textbook by itself, and in fact books have been devoted to each in the last decade,
not to mention older texts dealing with these subjects. In the new chapters I have

xi
xii Preface to the second edition

touched upon them briefly enough to adapt the treatment to the general level and
scope of this book, but also in sufficient detail to arouse interest and enable a
basic understanding of where the problems lie.
I have also added an appendix that explains and develops more rigorously the
concept of degeneracy pressure in an attempt to dispel some confusion related
to the applicability of complete degeneracy, which was the only form developed
in the early edition: is the omission of temperature an assumption or a justified
result? Another, minor, addition is a concise discussion of the mixing-length
treatment of convection. Finally, I have included a few more exercises, which
are mostly of the same nature and serve the same purpose as the older ones: to
elucidate points made in the text or provide additional information.
While I am still grateful to those who have helped, supported and encouraged
me during the writing of the original version of this book, it is with new pleasure
and gratitude that I thank those who have commented on it since, who have used
the book in their classes and have helped to improve it. Among them are Nuria
Calvct. Aparna Vcnkatesan, Allan Walstad. Werner Dappen. Nicolay Samus. Bill
Herbst, Phil Armitage, Silvia Rossi and Barry Davids, and my long-time friends
Mike Shara, Mario Livio and Oded Regev. Special thanks are due to Robert Smith
for pointing out a number of inaccuracies and for making important suggestions.
 Preface to the first edition

For over ten years 1 have been teaching an introductory course in astrophysics
for undergraduate students in their second or third year of physics or planetary
sciences studies. In each of these classes, I have witnessed the growing interest
and enthusiasm building up from the beginning of the course toward its end.
It is not surprising that astrophysics is considered interesting; the field is
continually gaining in popularity and acclaim due to the development of very
sophisticated telescopes and to the frequent space missions, which seem to bring
the universe closer and make it more accessible. But students of physics have an
additional reason of their own for this interest. The first years of undergraduate
studies create the impression that physics is made up of several distinct disciplines,
which appear to have little in common: mechanics, electromagnetism, thermo­
dynamics and atomic physics, each dealing with a separate class of phenomena.
Astrophysics - in its narrowest sense, as the physics of stars - presents a unique
opportunity for teachers to demonstrate and for students to discover that complex
structures and processes do occur in Nature, for the understanding of w'hich all
the different branches of physics must be invoked and combined. Therefore, a
course devoted to the physics of stars should perhaps be compulsory, rather than
elective, during the second or third year of physics undergraduate studies. The
present book may serve as a guide or textbook for such a course.
Books on astrophysics fall mostly into two categories: on the one hand,
extensive introductions to the field covering all its branches, from planets to
galaxies and cosmology, quite often including an introduction to the main fields
of physics as well; and on the other hand, specialized books, often including up
to date results of ongoing studies. The former are aimed at readers who have not
yet received any real training in physics, the latter, at graduate students who are
specializing in astronomy or astrophysics. The present book is aimed at students
who fall between these extremes: undergraduates who have acquired a basic
mathematical background and have been introduced to the basic law's of physics

xiii
xiv Preface to the first edition

during the first two or three semesters of studies, but have no prior knowledge of
astronomy.
The purpose of this book is to satisfy the eagerness to comprehend the realm
of stars, by focusing on fundamental principles. The students arc made to under­
stand, rather than become familiar with, the different types of stars and their
evolutionary trends. As far as possible, I have refrained from burdening the reader
with astronomical concepts and details, in an attempt to make the text suitable
for students of physics who do not necessarily intend to pursue astrophysics any
further. Thus, odd as it may seem, there is no mention of concepts that are so famil­
iar to astronomers, such as magnitude, colour index, spectral class and so forth.
Equally odd may appear the use of SI units, which is still alien to astrophysics, but
has become common, in fact mandatory, in physics studies. I have complied with
this demand, despite my conviction that, perhaps surprisingly, astrophysicists still
think in terms of cgs units. (One hardly comes across stellar opacities expressed
in square metres per kilogram, or densities in kilograms per metre cubed.) As is
customary in textbooks, exercises are scattered throughout the book and solutions
are provided in an appendix.
The theory of stellar evolution is developed in a methodical manner. The
student is led step by step from the formulation of the problem to its solution
on a path that appears very natural, even obvious at times. I have tried to avoid
the widely adopted alternative of following the progress of a star’s evolution,
enumerating the different phases with their inherent physical aspects. I find the
logical, rather than the chronological, method the best way of presenting this
theory, the way any other established theory is usually presented. When each
chapter of a scientific book relies on the preceding one and leads to the next,
there is hope of arousing in the reader sufficient curiosity for reading on. The
fascinating history of the theory of stellar structure and evolution is sometimes
alluded to in ‘Notes’ and quotations.
The first chapter introduces the subject of stellar evolution, as it arises from
observations: the problem is defined and the basic assumptions (axioms) are laid
down. The following six chapters are essentially theoretical: the second formulates
the problem mathematically by introducing the equations of stellar evolution;
the third summarizes briefly the basic physical laws involved in the study of
stellar structure, serving for reference later on. Chapters 4, 5 and 6 - dealing with
nucleosynthesis in stars, simple stellar models and stabi lity - build up to Chapter 7,
which is the heart of this book. Combining the material of Chapters 3-6, it presents
a general, almost schematic picture of the evolution of stars in all its aspects.
From my experience, this picture remains imprinted in the students’ minds long
after the details have faded away. Chapter 8 is, in a way, a recapitulation of the
previous chapter from a different angle: the story of stellar evolution is retold,
filling in many details, as it emerges from numerical computations. Emphasis
is now put on comparison with observations, thereby closing the circuit opened
 Preface to the first edition xv

in Chapter 1. The next chapter deals with special objects: stipernovae and their
remnants, pulsars, black holes (very briefly) and other radiation sources. Finally,
Chapter 10 touches on the global picture of the stellar evolution cycle, from the
galactic point of view.
I have tried to give proper credit where it was due. but occasionally I may
have failed or erred. I apologize for any such failure or error, my only defence
being that it was not intentional. 1 have refrained from referring to original papers
in the text, in order not to interfere with fluency. A selection of references (by no
means complete) is given in the bibliography.
Enthusiasm toward a subject of study is instigated not only by the subject
itself, but quite often by the teacher. In this respect I was lucky to have been
introduced to astrophysics by Giora Shaviv and I hope to have earned on some of
his passion to my own students. Computing and numerical modeling, on which
the subject matter of this book relies, are not merely a skill but a true art of unique
beauty and elegance. For having introduced me to this art long ago and for having
been a constant source of encouragement and advice during the writing of this
book. I am grateful to my husband (and former teacher) Allay Kovetz. I would
like to express my gratitude and appreciation to Leon Mcstcl for a very careful
and thorough reading of the original manuscript. This book has tremendously
benefited from his countless observations, comments and suggestions. Special
thanks are due to Michal Semo and her team at the Desktop Publishing unit of Tel
Aviv University for their skilful and painstaking graphics work, not to mention
their endless patience and cheerfulness. Above all, I am grateful to my son Ely
for gracefully bearing with a busy and preoccupied mother during the rather
demanding years of adolescence.
CHOOSE SOMETHING LIKE A STAR
by Robert Frost

0 Star, (the fairest one in sight),

We grant your loftiness the right
To some obscurity of cloud -
It would not do to say of night,
Since dark is what brings out your light.
Some mystery becomes the proud.
But to be wholly taciturn
In your reserve is not allowed.
Say something to us we can learn
By heart and when alone repeat.
Say something! And it says T burn’.
But say with what degree of heat.
Talk Fahrenheit, talk Centigrade.
Use language we can comprehend.
Tell us what elements you blend.
It gives us strangely little aid,
But does tell something in the end.
 I

Observational background and

basic assumptions

I. I What is a star?

A star can be defined as a body that satisfies two conditions: (a) it is bound by
self-gravity: (b) it radiates energy supplied by an internal source. From the first
condition it follows that the shape of such a body must be spherical, for gravity
is a spherically symmetric force field. Or. it might be spheroidal, if axisymmetric
forces are also present. The source of radiation is usually nuclear energy released
by fusion reactions that take place in stellar interiors, and sometimes gravita­
tional potential energy released in contraction or collapse. By this definition,
a planet, for example, is not a star, in spite of its stellar appearance, because
it shines (mostly) by reflection of solar radiation. Nor can a comet be consid­
ered a star, although in early Chinese and Japanese records comets belonged
with the ‘guest stars' - those stars that appeared suddenly in the sky where
none had previously been observed. Comets, like planets, shine by reflection of
solar radiation and. moreover, their masses are too small for self-gravity to be of
importance.
A direct implication ofthe definition is that stars must evolve: as they release
energy produced internally, changes necessarily occur in their structure or com­
position. or both. This is precisely the meaning of evolution. From the above
definition we may also infer that the death of a star can occur in two ways:
violation of the first condition - self-gravity - meaning breakup of the star and
scattering of its material into interstellar space, or violation of the second condi­
tion - internally supplied radiation of energy - that could result from exhaustion
ofthe nuclear fuel. In the latter case, the star fades slowly away, while it gradually
cools off. radiating the energy accumulated during earlier phases of evolution.
Eventually, it will become extinct, disappearing from the field of view of even
the most powerful telescopes. This is what we call a dead star. We shall see
that most stars end their lives by a combination of these two processes: partial
breakup (or shedding of matter) and extinction. As to the birth of a star, this is
2 I Observations and assumptions

a complex process, which presents many problems that arc still under intensive
investigation. We shall deal with this phase only briefly, mainly by pointing out
the circumstances under which it is expected to occur.
We shall therefore start pursuing the evolution of a star from the earliest
time when both conditions of the definition have been fulfilled, and we shall stop
when at least one condition has ceased to be satisfied, completely and irreversibly.
Finally, we shall consider the life cycle of stellar populations and the effect of
stellar evolution on the evolution of galaxies within which stars reside. Galaxies
are large systems of stars (up to 10" or so), which also contain interstellar clouds
of gas and dust. Many of the stars in a galaxy are aggregated in clusters, the
largest among them containing more than 1 (P stars. The object of reference in
stellar physics is, naturally, the Sun, and in galactic physics, the Galaxy to which
it belongs, also known as the Milky Way galaxy.

1.2 What can we learn from observations?

Astrophysics (the physics of stars) does not lend itself to experimental study, as do
the other fields of physical science. We cannot devise and conduct experiments in
order to test and validate theories or hypotheses. Validation of a theory is achieved
by accumulating observational evidence that supports it and its predictions or
inferences. The evidence is derived from events that have occurred in the past and
are completely beyond our control. The task is rather similar to that of a detective.
As a rule of thumb, a theory is accepted as valid (or at least highly probable) if
it withstands two radically different and independent observational tests, and of
course, so long as no contradictory evidence has been found.
The information we can gather from an individual star is quite restricted. The
primary characteristic that can be measured is the apparent brightness, which
is the amount of radiation from the star falling per unit time on unit area of a
collector (usually, a telescope). This radiation flux, which we shall denote ZObS
is not, however, an intrinsic property of the observed star, for it depends on the
distance of the star from the observer. The stellar property is the luminosity L,
defined as the amount of energy radiated per unit time - the power of the stellar
engine. Since L is also the amount of energy crossing, per unit time, a spherical
surface area at the distance d of the observer from the star, the measured apparent
brightness is

L
/obs ~ 4^' (1.1)

and L may be inferred from ZObS if d is known. The luminosity of a star is usually
expressed relative to that of the Sun, the solar luminosity Lq — 3.85 x 1026 J s-1.
Stellar luminosities range between less than l()-5Leand over 105LQ.
 1.2 What can we learn from observations? 3

to distant stars

Figure I. I Sketch of the parallax method for measuring distances to stars.

Note: The only direct method of determining distances to stars (and other celestial
bodies) is based on the old concept of parallax - the angle between the lines of sight
of a star from two different positions of the observer. The lines of sight and the line
connecting the observer’s positions form a triangle, with the star at the apex, as shown
in Figure 1.1. The larger the distance to the object, the wider the baseline required for
obtaining a discernible parallax: for objects within the solar system distant points on Earth
suffice; for stars, a much larger baseline is needed. This is provided by the Earth’s orbit
around the Sun, yielding a maximal baseline of ~3 x 1 O’1 m, twice the Earth-Sun distance
a(— 1 AU). Thus, the stellar parallax is obtained by determining a star's position relative
to very distant, fixed stars, at an interval of half a year. Even so, the triangle obtained
is very nearly isosceles, with almost right base angles, while the parallax p, defined as
half the apex angle, is less than 1" (the largest known stellar parallax is that of Proximo
Centauri-the star closest to our Sun, p = 0".76). Consequently, to a good approximation,
d a/p. Based on this method, distances of up to about 500 light-years may be directly
measured. (One light-year, 9.46 x 1015 m, is the distance travelled in one year at the
speed of light.) A common astronomical unit for measuring distances, called parsec, is
based on the parallax method: as its name indicates, it is the distance corresponding to a
parallax of 1", amounting to about 3 light-years. Recently, the number of stars for which
we have accurate distances has grown a hundredfold as a result of the activity of the
satellite specially designed for this task, Hipparcos (High Precision Parallax Collecting
Satellite), named after the greatest astronomer of antiquity. Hipparchus of Nicea (second
century bc), who measured the celestial positions and brightnesses of almost a thousand
stars and produced the first star catalogue. The satellite Hipparcos. which operated during
1989-93, gathered data on more than a million nearby stars. But on the astronomical scale,
distances that can be directly measured are quite small and hence indirect methods have
to be devised, some of which are based on the theory of stellar structure and evolution, as
we shall see in Chapter 9.
4 I Observations and assumptions

The surface temperature of a star may be obtained from the general shape of
its spectrum, the continuum, which is very similar to that of a blackbody. The
effective temperature of a star Tctf is thus defined as the temperature of a blackbody
that would emit the same radiation flux. Il provides a good approximation to the
temperature of the star’s outermost layer, called the photosphere, where the bulk
of the emitted radiation originates. If R is the stellar radius, the surface flux is
L/4tt Rz, and hence:

4 L
Cr'H = ------ - (],2)
c" 4tiRz
where n is the Stefan-Boltzmann constant. Thus

L =47r/?2a7^. (1.3)

The surface temperatures of stars range between a few thousand to a few hundred
thousand degrees Kelvin (K). the wavelength of maximum radiation Xmax shifting,
according to Wien’s law

AmaxT = constant, (1.4)

from infra-red to soft X-rays. The effective temperature of the Sun is 5780 K. We
should bear in mind, however, that conclusions regarding internal temperatures
cannot be drawn from surface temperatures without a theory.
The chemical composition, too, can be inferred from the spectrum. Each
chemical element has its characteristic set of spectral lines. These lines can
be observed in the light received from stars, superimposed upon the continuous
spectrum, either as emission lines, when the intensity is enhanced, or as absorption
lines, when it is diminished. The elements that make up the photosphere of a
star, which emits the observed radiation, may thus be identified in the stellar
spectrum. But since the photosphere is very thin, the deduced composition is
not representative of the bulk, opaque interior of the star. Most of the chemical
elements were found to be present in the solar spectrum. In fact the existence
of the element helium was first suggested by spectral lines from the Sun (in the
1860s); its name is derived from ‘helios’, the Greek word for Sun.
Under certain conditions, the mass of a star that is a member of a binary
system can be calculated, based on spectral line shifts, as we shall show in
Chapter 11. Very seldom, in eclipsing binary systems, may the radius of a star
be directly derived; it can. however, be estimated from the independently derived
luminosity (when possible) and effective temperature using Equation (1.3). Stellar
masses and radii are measured in units of the solar mass, MQ = 1.99 x IO30 kg,
and the solar radius, Rq — 6.96 x 10s m. The mass range is quite narrow -
between ~0. IA/O and a few tens stellar radii vary typically between less
than 0.01 Rq to more than 1000/?©. Much more compact stars exist, though, with
radii of a few tens of kilometres.
 1.2 What can we learn from observations? 5

Besides being sparse, the information one can gather is confined to a very
brief moment in a star’s life, even if observations are carried on for hours or
years, or, hypothetically, hundreds of years. To illustrate this point, let us compare
the life span of a star to that of a human being: uninterrupted observation of
a star since, say. the discovery of the telescope some 400 years ago. would be
tantamount to watching a person for about 3 minutes! Obviously, it would be
impossible to learn anything (directly) about the evolution of the star from such
a fleeting observation. The body of data available to the astrophysicist consists of
accumulated momentary information on a very large number of stars, at different
evolutionary stages. From these data, the astrophysicist is required to form a
scenario describing the evolution of a single star.
Imagine, for comparison, an explorer who has never seen human beings,
trying to figure out the nature and evolutionary course of these creatures, based
solely on a large sample of photographs of many different humans chosen at
random. The explorer will find that humans differ in many properties, such as
height, colour of skin, etc., and will note, for example, that the height of the
majority varies within a narrow range around a mean of, say, 1.75 m, and only
the height of a small minority is significantly below this mean. These findings
may be interpreted in two ways: (a) humans are intrinsically different, the tall
ones being more numerous than the short ones; (b) humans are similar to one
another, but their properties change in the course of their lives, their height
either increasing or decreasing with age (one would not be able to tell which).
In the latter case, based on the hypothesis that humans evolve, it may also be
inferred that individual human beings arc tall for a longer part of their lives
than they are short. It might even be possible to calculate the rate of change
of the human height from the relative number of individuals in different height
ranges.
In a similar manner, if we find that a certain property is common to a great
number of stars, we may infer - on the basis of the evolution hypothesis - that
such a property prevails in stars for long periods of time. By the same token, rarely
observed phenomena might not be rare events, but simply short-lived ones. At
the same time, the possibility of actually rare phenomena cannot be entirely ruled
out. This is a sample of the problems one would have to face if the understanding
of stars and their evolution were to rest entirely on observation.
As the information available for any given star is so limited, the theory of
stellar evolution is not meant to describe in detail the structure and expected
evolutionary course of any individual star (with the exception of the Sun).
Its purpose is rather to construct a general model that explains the large var­
iety of stellar types, as well as the relations between different stellar proper­
ties revealed by observations (such as the correlation between luminosity and
surface temperature, or between luminosity and mass, which we shall shortly
encounter).
6 I Observations and assumptions

1.3 Basic assumptions

Guided by the observational evidence, we may add several fundamental assump­

tions (or axioms) to the general definition of a star, on which to base the theory
of stellar structure and evolution.

Isolation

Regarding its structure and evolution, a single star may be considered isolated in
empty space, although it is invariably a member of a large group - a galaxy -
or even a denser group within a galaxy - a stellar cluster. (We exclude from the
present discussion binary stars - a pair of stars that form a bound system that we
shall address in Chapter 11.) Consequently, the initial conditions will exclusively
determine the course of a star's evolution. Thus the evolutionary process of a star
(metaphorically termed life) differs from that of live creatures, the latter being
influenced to a large extent by interaction with their environment. To better grasp
the isolation of stars, consider the star closest to our Sun (Proximo Centauri),
which is at a distance of 4.3 light-years. This distance is larger than the solar
diameter by a factor of 3 x IO7. Such a situation would be similar to nearest
neighbours on Earth being separated by a distance 3 x IO7 times their height,
which roughly amounts to 50000 km. This is four times the Earth diameter or
one seventh of the distance to the Moon. We would call this isolation! Both the
gravitational field and the radiation flux, which vary in proportion to 1 /d2, are
diminished by a factor of at least 1 /(3 x IO7)2 ~ 10 from one star to another.

Uniform initial composition

A star is born with a given mass and a given, presumably homogeneous, com­
position. The latter depends on the time of formation and on the location within
the galaxy where the star is formed. The composition of stars has been a question
of intense debate for a long time. It turned out, finally, that most of the material
of a newly formed star, about 70% of its mass, consists of hydrogen. The second
most important element is helium, amounting to 25-30% of the mass, and there
are traces of heavier elements, of which the most abundant are oxygen, carbon
and nitrogen (in that order), known collectively as the CNO group. In the Sun, for
example, for every 10000 hydrogen atoms, there are about 1000 helium atoms,
8 oxygen atoms, almost 4 carbon atoms, one atom of nitrogen, one of neon and
less than one atom of each of the other species. The composition of stellar material
is usually described by the mass fractions of different elements, the mass of each
element per unit mass of material. It is common to denote the mass fraction of
hydrogen by X. that of helium by Y. and the total mass fraction of all the other
elements by Z, so that X + Y + Z = I.
 1.3 Basic assumptions 7

Exercise 1.1: Calculate the mass fractions of hydrogen, helium, carbon, oxygen,
nitrogen and neon in the Sun.

Thus, since both hydrogen and helium, the predominant stellar components,
are found in the gas phase unless the temperature is extremely low or the density
(pressure) extremely high, we may quite safely deduce that stars are made of
gas. We shall return to this point later on, when we gain more insight into stellar
interiors.
With very few exceptions, the abundances of the chemical elements, as derived
from stellar spectra, are remarkably similar. Moreover, they are very similar to
those prevailing in the interstellar medium. As stars are born in interstellar clouds,
and the composition of their surface layers is expected to be the least affected by
evolutionary processes, it may be concluded that there is little difference in the
initial composition of stars. The largest differences occur for the abundances of
the heavy elements, which vary among different stars between less than 0.001 to a
few per cent of the entire stellar mass. But differences in the initial abundances of
these elements are of secondary importance to stellar evolution. For simplicity, we
shall ignore differences in the initial composition of stars. In numerical examples
we shall generally adopt the solar composition. The fate of a star will then be
solely dependent upon its initial mass M.

Historical Note: The first to show that the Sun’s atmosphere is dominated by hydro­
gen was Cecilia Payne in her doctoral dissertation completed in 1925. Not only did she
show that the most abundant elements were hydrogen and helium, but she also suggested
that the relative abundances of the heavier elements were roughly constant throughout the
galaxy, thus indicating the homogeneity of the universe. These findings followed from
Saha’s equation (see Section 3.6), then new, according to which, the strength of spectral
lines depends on physical conditions as well as on elemental abundances. These conclu­
sions, very much opposed to the common wisdom of the time, were largely ignored. It
was only a few years later, when, corroborated by further evidence, the prevalence of
hydrogen and helium in the Sun’s atmosphere was convincingly argued by Henry Norris
Russell, whose fame will become apparent shortly.
A doctoral degree awarded to a woman was extremely unusual in those days. In her
autobiography, Cecilia Payne-Gaposchkin writes ‘One serious obstacle existed: there was
no advanced degree in astronomy, and I should have to be accepted as a candidate by
the Department of Physics. The redoubtable Chairman of that department was Theodore
Lyman, and Shapley [Harlow Shapley, her mentor] reported to me that he refused to
accept a woman candidate.’ In the end she became the first person to earn a doctorate in
astronomy from Harvard University.
8 I Observations and assumptions

Spherical symmetry

Departure from spherical symmetry may be caused by rotation or by the star’s

own magnetic field (since by assuming isolation, we have excluded all possible
external force fields). In the overwhelming majority of cases, the energy associated
with these factors is much smaller than the gravitational binding energy. We know,
for example, that the period of revolution ofthe Sun around its axis is about 27
days, so that its angular velocity is co — 2.5 x IO-6 s_|. The spin velocity of more
distant stars can be deduced from the broadening of spectral lines caused by the
Doppler effect. The kinetic energy of rotation relative to the gravitational binding
energy is of the order:

Mco2R2 co2R2 s
-------— -------------- 2 x KT5,
GM-/R GM
where G is the constant of gravitation. (This is also the ratio of the centrifugal
acceleration to the gravitational acceleration at the equator.)
The magnetic fields of stars similar to the Sun range from a few thousandths
to a few tenths of a tesla. The larger ones may be directly deduced from split
spectral lines caused by the Zeeman effect, whose separation can be measured.
The energy density associated with a magnetic field B is B2/Zp.Q, while the
gravitational energy density is of the order of GM2/7?4; for the Sun, even taking
B = 0.1 T (typical of sunspots, but larger than the average magnetic field), we
have
g2/Mo _ b~ r2 |0_„
GM-/R* noGM2
Compact stars tend to have higher magnetic fields, but their small radii (large
binding energies) compensate for them. Hence, magnetic effects on the structure
of a star can usually be ignored.
Neglecting deviations from spherical symmetry, the physical properties within
a star change only with the radial distance r from the centre and are uniform over
a spherical surface of radius r. The spatial variable r may be replaced by the mass
m enclosed in a sphere of radius r, as shown in Figure 1.2. The transformation
between these variables is given in terms of the density p\

or, in differential form,

dm = pĄnr2dr. (1.5)

The advantage of using m instead of r in calculations of the changing stellar

structure is that its range of variation is bounded, 0 < m < M, whereas the radius
may change by several orders of magnitude in the course of evolution of a star.
 1.4 The H-R diagram 9

Figure 1.2 The relationship between space variables r and m in spherical symmetry.

Exercise 1.2: In a star of mass M. the density decreases from the centre to the
surface as a function of radial distance r, according to

P = Pc 1

where pc is a given constant and R is the star’s radius, (a) Find m(r). (b) Derive
the relation between M and R. (c) Show that the average density of the star (total
mass divided by total volume) is 0.4pc.

1.4 The H-R diagram: a tool for testing stellar evolution

As we have seen, the two most fundamental properties of a star that can be inferred
from observation are the luminosity L and the effective temperature 7~ctl. It is only
natural that a possible correlation between them be sought. This was initiated
independently by two astronomers at about the same time: Ejnar Hertzsprung in
1911 and Henry Norris Russell in 1913. Hence the diagram whose axes are the
(decreasing) surface temperature (or related properties) and the luminosity (or
related properties) bears their names, being known as the H-R diagram. Each
observed star is represented by a point in such a diagram, an example of which
is given in Figure 1.3. The results depend to some extent on the criterion used
for choosing the sample of stars, for example, stars within a limited volume in
the solar neighbourhood, or members of a given star cluster, or stars of apparent
brightness greater than a prescribed limit, etc. The question we are interested in
is whether something can be learned from this diagram regarding the evolution
of stars.
It is immediately obvious from the examination of any H-R diagram that only
certain combinations of L and Tcn values are possible (a priori there is nothing
to impose such a constraint): most points are found to lie along a thin strip that
10 I Observations and assumptions

Figure 1.3 The H-R diagram of stars in the neighbourhood of the Sun.

runs diagonally through the (log Tetf, log L) plane. This strip is called the main
sequence and the corresponding stars are known as main-sequence stars.
Another populated area of the diagram is found to the right and above the
main sequence: it represents stars that arc brighter than main-sequence stars of
same Tcif, or of lower Teff for the same L. meaning that their spectrum is shifted
toward longer wavelengths and their colour is reddish. A higher L and lower Teff
implies, according to Equation (1.3), a large radius. Such stars are therefore called
red giants. Their radii may attain several hundred solar radii and even more. If the
Sun were to become a red giant, it would engulf the Earth and reach beyond Mars.
Another region of the (log logL) plane that is relatively rich in points
is located at the lower left corner: low luminosities and high effective tempera­
tures. Stars that fall in this region have a small radius and a bluish-white colour;
accordingly, they arc named white dwarfs. White dwarf radii are of the order of
the Earth's, although their masses are close to the Sun’s. The typical densities of
such stars arc therefore tremendous; one cubic centimetre of w hite dwarf material
would weigh more than a ton on Earth.
There are points outside these three main regions and there are conspicuously
empty spots within densely populated areas of the diagram, but we shall ignore
them for the moment and concentrate on the three main ones. What, if anything,
can wc learn from them? We recall that, in view of our basic assumptions, stars
 1.4 The H-R diagram II

may differ from one another only in their initial mass and their age. We can
therefore interpret the H-R diagram in two different ways:

1. The scatter of points is due to the different ages of the stars. The implied
assumption in this case is that the stars were formed at different times, and
hence there are ‘old- stars and ‘young’ stars. According to this hypothesis
the evolution of a star can be traced in the H-R diagram by some line,
with the time elapsed from the formation of the star being the changing
parameter along it. Looking at a large sample of stars, each one is caught
at a different age - hence the scatter of points in the diagram.
2. The properties of a star, in particular its luminosity and surface temper­
ature, depend strongly upon its mass, the only distinguishing parameter
at birth. Thus, different points in the diagram represent different stellar
masses.

This is the same dilemma our earlier explorer of the human race was faced
with: are the observed differences inherent or evolutionary? The explorer would
have been able to choose the correct explanation if sets of snapshots of humans of
the same age, for example, pupils of different school grades, were supplied. The
explorer would have immediately concluded that height is determined by age,
whereas skin-colour is an innate property. Similarly, the astrophysicist is aided
by H-R diagrams of star clusters. Stars within a cluster are formed more or less
simultaneously, by fragmentation of a large gas cloud (as will be explained in
Section 12.2). Images of star clusters are shown in Figure 1.4. Examples of H-R
diagrams of such clusters are given in Figure 1.5. We note that the main sequence
ends at different luminosities for each cluster: in one case it extends up to very
high luminosities; in another case it is shorter, but at the same time there appear
some red giants, which were absent in the first cluster; in yet another one. the
main sequence is shorter still, and red giants are numerous.
Generally, as the main sequence is depleted, the red giant and white dwarf
branches are enriched. The lower part of the main sequence is always present
and equally populated, allowing for observational constraints. We may therefore
conclude that being on or outside the main sequence is determined by age, w hereas
the location of a star along the main sequence is determined by its initial mass.
We are still unable to trace the evolutionary trajectory, whether toward or
away from the main sequence, so long as the cluster ages are not determined;
but the second inference can be tested. We may choose main-sequence stars with
known masses and look for a correlation between their masses and luminosities.
This is shown in Figure 1.6, which demonstrates that indeed there is a power-law
dependence of a main-sequence star’s luminosity upon its mass:

L oc M",
12 I Observations and assumptions

Figure 1.4 Stellar clusters: (a) the young open (amorphous shape) Pleiades cluster; (b) the
old globular cluster 47 Tucanae (copyright Anglo-Australian Observatory/Royal Obser­
vatory Edinburgh, photographs by D. Malin).

with v ranging between ~3 and ~5 over most of the mass range. As the Sun is a
main-sequence star, the relation can be calibrated to read

L M
(1.6)
Lq

Do stars leave the main sequence to become red giants? Do they later turn
into white dwarfs? Or, do some stars become red giants and others white dwarfs?
Why are there always - in all H-R diagrams - lower main-sequence stars? Why
 1.4 The H-R diagram 13

Log(Te(f)

Figure 1.5 The H-R diagram of star clusters: (a) the Pleiades cluster (adapted from
H. L. Johnson & W. W. Morgan (1953), Astrophys. J.. 117); (b) the Hyades cluster
(adapted from H. L. Johnson (1952), Astrophys. J., 116); (c) the globular cluster M3
(adapted from H. L. Johnson & A. R. Sandage (1956), Astrophys. J., 124).

are some changes in the stellar structure so rapid as to leave a blatant gap in the
H-R diagram? Observation alone is incapable of providing answers to all these
questions. We must resort to theory, and use the observations that have guided
us so far, in particular the H-R diagram, as a test. Here Martin Schwarzschild’s
words come to mind:
14 I Observations and assumptions

Log(M/Mo)

Figure 1.6 The mass-luminosity relation for main-sequence stars. Data from O. Yu.
Malkov (2007). Mon. Not. Roy. Astron. Soc., 382, based on detached main-sequence
eclipsing binaries (triangles). E. A. Vitrichenko, D. K. Nadyozhin and T. L. Razinkova
(2007), Astron. Lett., 33 (squares) and from the compilation by O. Yu. Malkov, A. E.
Piskunov and D. A. Shpil’kina (1997), Astron. Astrophys., 320 (dots).

If simple perfect laws uniquely rule the universe, should not pure thought be
capable of uncovering this perfect set of laws without having to lean on the
crutches of tediously assembled observations? True, the laws to be discovered
may be perfect, but the human brain is not. Left on its own, it is prone to stray, as
many past examples sadly prove. In fact, we have missed few chances to err until
new data freshly gleaned from nature set us right again for the next steps. Thus
pillars rather than crutches arc the observations on which we base our theories;
and for the theory of stellar evolution these pillars must be there before we can
get far on the right track.
Martin Schwarzschild: Structure and Evolution of the Stars, 1958

Our aim throughout most of the following chapters w ill be to develop a theory of
the stellar structure and evolution based on the laws of physics. This should ulti­
mately lead to a theoretical H-R diagram, to be confronted with the observational
one.
 2

The equations of stellar evolution

We have learned a star to be a radiating gaseous sphere, made predominantly of

hydrogen and helium. Radiation may be regarded as a photon gas, each ‘particle’
carrying a quantum of energy h v, proportional to the frequency u of the associated
electromagnetic wave, and a momentum hv/c, where h is Planck's constant and
c is the speed of light. This mixture of gases that makes up a star is governed by
frequent collisions between its particles, ions, electrons and photons alike. This
is how Sir Arthur Eddington describes The Inside of a Star.

... Try to picture the tumult! Dishevelled atoms tear along at 50 miles a second
with only a few tatters left of their elaborate cloaks of electrons torn from them in
the scrimmage. The lost electrons are speeding a hundred times faster to find new
resting-places. Look out! there is nearly a collision as an electron approaches
an atomic nucleus; but putting on speed it sweeps round it in a sharp curve. A
thousand narrow shaves happen to the electron in 10“10 of a second; sometimes
there is a slide-slip at the curve, but the electron still goes on with increased
or decreased energy. Then comes a worse slip than usual; the electron is fairly
caught and attached to an atom, and its career of freedom is at an end. But only
for an instant. Barely has the atom arranged the new scalp on its girdle when a
quantum of aether waves (photon) runs into it. With a great explosion the electron
is off again for further adventures. Elsewhere two of the atoms arc meeting full
tilt and rebounding, with further disaster to their scanty remains of vesture....
And what is the result of all this bustle? Very little. Unless we have in mind
an extremely long stretch of time the general state of the star remains steady.
Sir Arthur S. Eddington: The Internal Constitution ofthe Stars, 1926

Frequent collisions lead to a state of thermodynamic equilibrium, which is char­

acterized by a temperature, indicative of the energy distribution of the particles.
For example, a free ideal gas in thermodynamic equilibrium is described by a
Maxwellian velocity (kinetic energy) distribution.

IS
16 2 The equations of stellar evolution

2.1 Local thermodynamic equilibrium

When the average distance travelled by particles between collisions - the mean
free path - is much smaller than the dimensions of the system, thermodynamic
equilibrium is achieved locally, and the system may assume different temperatures
at different points. It is thus described by a temperature distribution. If, moreover,
the time elapsed between collisions - the mean free time - is much shorter than the
timescale for change of macroscopic properties, then thermodynamic equilibrium
is secured, but the temperature distribution may change with time. Such is the
situation in stars.
Equilibrium between matter and radiation can be achieved as well, by ‘col­
lisions’ (interactions) between mass particles and photons. In this case the radia­
tion becomes a blackbody radiation, where the energy distribution of the photons
is described by the Planck function, and the temperatures of gas and radiation are
the same. As we shall see in more detail in the next chapter, the average mean
free path of photons in stellar interiors is many orders of magnitude smaller than
typical stellar dimensions. Needless to say, the corresponding mean free time of
photons is vanishingly small. Consequently, the gas and the radiation may be
assumed in thermodynamic equilibrium locally, that is, the gas temperature is the
same as the radiation temperature at each point (although the temperature of a star
is neither uniform nor constant). This means that the radiation in stellar interiors is
very nearly blackbody radiation, described by the Planck function corresponding
to the local unique temperature. Such a state is known as local thermodynamic
equilibrium (LTE). It should be stressed that radiation and matter are not always in
a state of equilibrium. For example, the solar radiation passing through the Earth’s
atmosphere does not reach equilibrium with the gas: the radiation temperature is
the effective temperature of the Sun, about 6000 K, while the gas temperature,
around 300 K, is more than 20 times lower. Similar situations occur in gaseous
nebulae that are illuminated by stars embedded in them. There are also mixtures
involving more than two temperatures; for example, in an ionized gas, the tem­
peratures of the electrons, the ions and the photons may all differ from each other.
Such is the situation in the solar wind - the flux of particles, mainly protons and
electrons, emanating from the Sun. In this case, the characteristic temperatures
of the two gases - about IO6 K for the protons, and almost twice as much for the
electrons - arc higher than that of the radiation (6000 K).
The assumption of LTE constitutes a great simplification, for it enables the
calculation of all thermodynamic properties in terms of the temperature, the
density and the composition, as they change from the stellar centre to the surface.
Thus the structure of a star of given mass M is uniquely determined at any given
time t, if the density p, the temperature T and the composition - the mass fractions
of all the constituents - are known at each point within it. By ‘point’ we mean
any value of the independent space variable chosen (r or m), which refers to a
spherical surface around the centre. The temperature, density and composition
 2.2 The energy equation 17

Figure 2.1 Spherical shell within a star and the heat flow into and out of it.

change not only w ith distance from the centre of the star, but also with lime. Hence
the evolution of a star composed of n different elements is described by the n + 2
functions, p(m. t), T(m, t) and the mass fractions X,(m. t), w'here 1 < i < n,
of two independent variables, time and space. A set of n + 2 equations is thus
required, of w'hich these functions are the solutions.
We thus invoke the basic conservation law's that apply to any physical system:
conservation of mass, momentum, angular momentum and energy. As we have
assumed a star to be a nonrotating system, the angular momentum is uniformly
zero at all times. (Nevertheless, the global conservation of angular momentum
will be invoked later on to explain special features of peculiar stars.) Conservation
of mass is implicitly included in the relation between dm and dr. Only two
conservation law s remain to be applied, for energy and momentum, which together
with the equations for the rate of change of abundance for each species w ill form
the set of equations of stellar evolution.

2.2 The energy equation

The first law of thermodynamics, or the principle of conservation of energy, states

that the internal ehergy of a system may be changed by two forms of energy trans­
fer: heat and work. Heat may be added or extracted, and work may be done on the
system, or performed by the system, and involves a change in its volume - expan­
sion or contraction. Consider a small element of mass dm within a star, over w hich
the temperature, density and composition may be taken as approximately con­
stant. In view of the spherical symmetry assumed, such an element may be chosen
as a thin spherical shell between radii r and r + dr - as shown in Figure 2.1 -
so that its volume is dV = Ąnr~dr and

dm — pdV — pĄnr^dr (2.1)

(see Equation (1.5)). Let u be the internal energy per unit mass and P the pressure.
18 2 The equations of stellar evolution

We denote by 8f a change that occurs in the value of any quantity f within the
mass element over a small period of time 8t (a Lagrangian rather than Eulerian
change). Then, if 8Q is the amount of heat absorbed (5 Q > 0) or emitted (8 Q < 0)
by the mass element and 8W is the work done on it during the time interval 8t,
the change in the internal energy, according to the first law, is given by:

8(udm) = dm8u — 8Q + 8W, (2.2)

where we have used the conservation of mass in assuming dm to be constant. The

work may be expressed as

dV / • \
8W = — P8dV = —P8 —dm = — P8 I — I dm. (2.3)
dm \P/
We note that compression means shrinking of the element’s volume, or 8d V < 0,
and hence entails an addition of energy, while expansion (8dV > 0) is achieved
at the expense of the element's own energy.
The sources of heat of the mass element are: (a) the release of nuclear energy,
if available, and (b) the balance of the heat fluxes streaming into the element and
out of it. The rate of nuclear energy release per unit mass is denoted by q and the
heat flowing perpendicularly through a spherical surface by F(m). Thus F has the
dimension of power (not to be confused with the strict definition of a heat flux -
power per unit area), and obviously F(M) = L. Accordingly,

8Q — q dm8t + F(m)8t — F(m + dm)8t.

But F(m + dm) — F(m) + (8F/'dm)dm and hence

dF\
8Q = q-------- dm8t. (2.4)
dm /

Substituting Equations (2.3) and (2.4) into Equation (2.2), we may write the
latter as

dF\
drnSu 4- P8 q--------dm8t. (2.5)
dm /

and in the limit 8t —> 0 we obtain

IV dF
- = <7 - 7— ■ (2.6)
p/ dm

where we have used the notation f for the temporal (partial) derivative df/dt of
a function f (the notation introduced by Newton).
In thermal equilibrium, when temporal derivatives vanish, we have

dF
(2.7)
dm
 2.3 The equation of motion 19

P{r+dr)[ |

Figure 2.2 Cylindrical mass element within a star.

Integrating over the mass,

M pM
qdm = dF = L, (2.8)
Jo
for the heat flow must vanish at the centre to avoid singularity. The left-hand side
is the total power supplied in the star by nuclear processes, which is commonly
denoted by Lnuc, the nuclear luminosity,

Tnuc = / qdm, (2.9)

Jo
and thus thermal equilibrium implies that energy is radiated away by the star at
the same rate as it is produced in its interior. L — Lnuc.

2.3 The equation of motion

Newton’s second law of mechanics, or the equation of motion, states that the net
force acting on a body of fixed mass imparts to it an acceleration that is equal
to the force divided by the mass. This is the momentum conservation law for a
body of fixed mass. Consider a small cylindrical volume element within a star,
with an axis of length dr in the radial direction, between radii r and r + dr, and
a cross-sectional area dS. as shown in Figure 2.2. If the (approximately uniform)
density within the element is p. its mass Ahi is given by

^m — pdrdS. (2.10)

The forces acting on this element are of two kinds: (a) the gravitational force,
exerted by the mass of the sphere interior to r (the net gravitational force exerted
by the spherical mass shell exterior to r vanishes) and (b) forces resulting from
the pressure exerted by the gas surrounding the element. The gravitational force
20 2 The equations of stellar evolution

is radial and directed toward the centre of the star. Due to the spherical symmetry
assumed, the pressure forces acting perpendicularly to the side of the cylindrical
element are balanced and only the pressure forces acting perpendicularly to its
top and bottom remain to be considered. Denoting by r the acceleration d2r/dt2
of the element, we may write the equation of motion in the form

GmAm
rAm —-------- -— + P(r)dS — P(r + dr)dS. (2.11)
r-
But P(r + dr) = P(r) + (dP/dr)dr and hence

GmAm d P Am
rAm =-------- z-----------------------------------,
r- dr p

where we have substituted Am from Equation (2.10). We may now divide by Am

to obtain
Gm 1 dP
(2.12)
r2 p dr

If m is chosen as the independent space variable rather than r and the transfor­
mation dr — dm/\4nr2p) is used. Equation (2.12) becomes

Gm -,'dP
r ------ ------4ttr—. (2.13)
r- dm
When accelerations are negligible, Equations (2.12) and (2.13) describe a state of
hydrostatic equilibrium, with gravitational and pressure forces exactly in balance:

dP Gm
(2.14)

or
dP _ Gm
(2.15)
dm 4ttr4
As the right-hand side of Equation (2.14) or (2.15) is always negative, hydrostatic
equilibrium implies that the pressure decreases outward. The pressure gradient
vanishes at the centre, since on the right-hand side of Equation (2.14) mJ r2 tends
to zero with r.
We may estimate the pressure at the centre of a star in hydrostatic equilibrium
by integrating Equation (2.15) from the centre to the surface of the star,

M Gm dm
P(M) — P(0) = - , , d . (2.16)
Jo 4trr4

On the left-hand side we are left with the central pressure Pc = P(0), since at the
surface the pressure practically vanishes. P(M) ~ 0. On the right-hand side we
may replace r by the stellar radius R > r, to obtain a lower limit for the central
 2.4 The virial theorem 21

pressure:

rM Gindin Gm dm
(2.17)
Jo 4,t/?4 ’

yielding

GM2
N m"2. (2.18)
8.t/?4

The pressure at the centre of the Sun exceeds 450 million atmospheres!

Exercise 2.1: For a star of mass M and radius R, find the central pressure and
check the validity of inequality (2.18) for the following cases: (a) a uniform
density and (b) a density profile as in Exercise 1.2.

Exercise 2.2: Suppose that the greatest density in a star is pc at the centre and
let Pc be the corresponding pressure. Show that

Pc < (4rr)l/30.347GM2/3pc4/3.

2.4 The virial theorem

An important consequence of hydrostatic equilibrium is a link that it establishes

between gravitational potential energy and internal energy (or kinetic energy in
a system of free particles). Multiplying the equation of hydrostatic equilibrium
(2.15) by the volume V = and integrating over the whole star, we obtain

i /' w Gm dm
VdP = \/o ~ (2.19)

The integral on the right-hand side of Equation (2.19) is none other than the
gravitational potential energy of the star, that is. the energy required to assemble
the star by bringing matter from infinity.

rw Gm dm
Q= - ---------- (2.20)
Jo ''
The left-hand side of Equation (2.19) can be integrated by parts,

/J(W)
VdP = [PV]$ PdV. (2.21)
22 2 The equations of stellar evolution

The first term on the right-hand side vanishes, since at the centre V = 0 and at
the surface P = 0. Combining Equations (2.19)—(2.21), we finally obtain
/•V(K)

-3/ Pr/V = Q, (2.22)

Jo
or, since dV — dm/p,
rM p
— 31 —dm — Q. (2.23)
Jo P
This is the general, global form ofthe virial theorem, which will prove extremely
valuable in many later discussions. A similar relation, applicable to part of the
star, may be obtained by carrying the integration of Equation (2.19) up to a radius
P> < P:
p
PSV,- — dm = {Qi. (2.24)
Jo P
Here. is the gravitational potential energy of the sphere whose boundary is at
Ps. which is unaffected by the shell outside it (between Ps and P). while Ps is the
pressure at Rs, exerted by the weight of the overlying shell.
Consider the particular case of an ideal gas of density p and temperature T
(to be treated in more detail in the next chapter): let the mass of a gas particle be
mg. The gas pressure is then given by P = (p/ms)kT, where k is the Boltzmann
constant. The kinetic energy per particle is *kT and since for an ideal gas the
internal energy is the kinetic energy of its particles, the internal energy per unit
mass is

Combining Equation (2.25) with the virial theorem (2.23), we have u dm =

—|Q. The integral on the left-hand side is simply the total internal energy U and
hence

U = -|Q. (2.26)

We can use this result to estimate the average internal temperature of a star
(assuming that stellar material behaves as an ideal gas - an assumption that will
be justified later on). The gravitational potential energy. Equation (2.20). of a star
of mass M and radius P is given by

GM-
= (2.27)

where a is a constant of the order of unity, determined by the distribution of

matter within the star, that is, by the density profile. By the virial theorem, we
 2.5 The total energy of a star 23

have on the one hand U = \aGM~/R-, on the other hand, from Equation (2.25),

fw 3kT 3 k -
U = /------ dm =------ TM, (2.28)
Jo 2 2 mg

where T is the temperature averaged over the stellar mass. Combining the two
results, we obtain

a m„G M
(2.29)
3~lT~R'

Substituting the average density p — 3M/4n R3 in Equation (2.29), we obtain

T or M~13 p}'3, meaning that between two stars of the same mass, the denser one
is also the hotter.

Exercise 2.3: For a star of mass M and radius R. find the value of a in the
expression for the gravitational potential energy for two cases: (a) a uniform
density and (b) a density profile as in Exercise 1.2.

Taking a = = and assuming the gas to be atomic hydrogen, we find that the
average temperature of a star is

4 x IO6 (2.30)

We note that T is much higher than the surface temperature Ten (as obtained
from observations), implying that internal temperatures must reach still higher
values. At temperatures of millions of degrees Kelvin, hydrogen and helium
are completely ionized, and even heavier elements are found in gaseous, highly
ionized form. Stellar material is therefore a plasma, a mixture of ions - nuclei
stripped of almost all their electrons - and free electrons.

2.5 The total energy of a star

Wc start by integrating the energy equation (2.6) over the entire star:

Since the variables t and m arc independent, the order of differentiation and
integration may be interchanged. Hence the first term on the left-hand side is

f" d fw
/ = — / udm = U. (2.32)
Jo dt Jo
24 2 The equations of stellar evolution

Now

(2.33)

and

V — 47tr'r. (2.34)

Integrating by parts the second term on the left-hand side of Equation (2.31), we
obtain
fw DE f'w <) P
/ P—dm = [PV]^—l 4jrr2r—dm. (2.35)
J„ Jo dm
and since al the centre V vanishes with r and al the surface P vanishes, we finally
have
’w , <) P
U - 4nr2r—dm = Lnuc _ T. (2.36)
o dm
We turn now to the equation of motion and integrate it, too, over the entire
star, after multiplying by r:
rM pM
M ->.dP
4nr r—dm. (2.37)
Jo Jo r* Jo
As the total kinetic energy of the star is given by
pM
IC = I ^r2dm. (2.38)
o
the integral on the left-hand side of Equation (2.37) is
rw . fM d , d . 2, .
/ r 'r dm = I —(J;r~)dm = — / Gdm — K,. (2.39)
Jo Jo dt - dt Jo
The first term on the right-hand side of Equation (2.37) is
r Zl\- d fM Gm dm
— I Gm—dm — I C/n[ - ) dm — — / ----------- = —Q. (2.40)
Jo /- Jo Yd J dt Jq r
Thus Equation (2.37) reads
rw dp
)C + Q = - / 47rr2/-—dm. (2.41)
Jo dm
Combining Equations (2.36) and (2.41), we have

U T K- + Q = Cnuc — E, (2.42)

where the left-hand side is the rate of change of the total stellar energy, that is,
E = U + /C T £2,

E = L nuc - L. (2.43)
 2.6 The equations governing composition changes 25

If a star is in thermal equilibrium, it follows that E = 0 and the energy is

constant. If, in addition, the star is in hydrostatic equilibrium. JC vanishes. In this
case U and Q arc related by the virial theorem, and hence either of them determines
the total energy of the star. Consequently, each one of them is conserved, not only
their sum. For example, a star in thermal and hydrostatic equilibrium cannot cool
throughout and expand (although cooling decreases the energy and expansion
increases it); it must conserve the internal (thermal) energy and the gravitational
potential energy separately. Another, apparently puzzling, conclusion is that a star
in hydrostatic equilibrium has a negative heat capacity, meaning that it becomes
hotter upon losing energy! This follows again from the virial theorem: for an ideal
gas. we have

E = U + Q= = -U (2.44)

and in the general case the right-hand side is multiplied by a constant ofthe order
of unity. Hence if E < 0, then (7, and with it (by Equation (2.28)) the average
temperature of the star, must increase. At the same time, the star must contract.
(We shall see shortly that contraction docs not necessarily imply violation of
hydrostatic equilibrium, so that the last argument is not contradictory.) In fact,
the gravitational potential energy released in contraction supplies both the energy
that is lost (radiated) and the thermal energy that causes the temperature to rise -
in equal amounts in the case of an ideal gas.

Exercise 2.4: Assuming that a star of mass M is devoid of nuclear energy sources,
find the rate of contraction of its radius, if it maintains a constant luminosity L.

2.6 The equations governing composition changes

As we have seen that stellar material is composed of free electrons and (almost)
entirely bare, chemically unbound nuclei, composition changes - if any - cannot
be of a chemical nature. The only possible changes in the abundance of the
constituents can occur by transformations of one element into another, that is, by
nuclear reactions - interactions between nuclei.
The atomic nucleus is made of protons and neutrons, collectively called
nucleons, which belong to the class of heavy particles named baryons (‘baryon’
meaning ‘heavy one’ in Greek). The proton has a positive electric charge of
unity (in units of the elementary charge e); the neutron has zero charge. These
particles may therefore be characterized by two numbers, baryon number A and
charge Z. (1, +1) for the proton and (1.0) for the neutron. Electrons, like protons,
are charged particles. Their baryon number is 0, meaning that electrons are not
heavy particles (indeed, the electron mass is almost 2000 times smaller than the
26 2 The equations of stellar evolution

proton mass). Associated with the electrons arc the neutrinos, of baryon number
0 (the neutrino mass is still controversial, although it is no longer thought to be
zero) and charge 0. Electrons and neutrinos belong to a class of particles called
leptons (‘light ones’ in Greek). For each particle, relativistic quantum mechanics
postulates the existence of an antiparticle, for which the signs of baryon or lepton
number and charge are reversed. The best known antiparticle is the positron, the
electron’s antiparticle.
Protons and neutrons are bound together in the atomic nucleus by a force of
attraction called the strong nuclear force. This is a short-range force, independent
of charge, that surpasses the repulsive Coulomb force between protons at nuclear
length scales - a few fertnis (1 fermi = 10-15 m). Another force that can act on
protons and neutrons is the weak force, whose range is estimated to be still shorter
(<10_|7m). The weak interaction is responsible for the conversion of protons
into neutrons (or vice versa). In a nuclear reaction (interaction by means of the
strong or the weak force) the charge as well as the baryon and lepton numbers arc
conserved. Hence in a weak interaction, an electron ora positron must be involved
in order to conserve charge, and a neutrino or antincutrino, so as to conserve the
lepton number. Conservation of lepton number means equal numbers of leptons
and antileptons: hence a positron (antilepton) will be accompanied by a neutrino
and an electron by an antineutrino.
If the bulk density in some part of a star is p and the partial density of the /th
nuclear species is p,-, the mass fraction of this species is given by
Pi
Xi = — (2.45)
P
and the number density - number of nuclei per unit volume - is given by the
partial density divided by the mass of one nucleus. The mass of an atomic nucleus
is slightly less than the sum of masses of its constituent protons and neutrons as
free particles. However, to a good approximation we may write

where mH is the atomic mass unit representing the mass of a (bound) nucleon,
usually defined as a twelfth of a carbon nucleus mass. Despite the notation. ///h
is slightly different both from the proton mass and from the mass of a hydrogen
atom. Combining Equations (2.45) and (2.46), we obtain the relations

p Xj Ai
n, =------ — and X,>= n,—n/H. (2.47)
/?/H A, p
The number of nuclei in a given volume may change as a result of nuclear
reactions that create it and others that destroy it. The creation or destruction of a
nucleus takes place by fusion of lighter nuclei or by breakup of a heavier nucleus
and may involve capture and release of light particles, such as positrons and
 2.6 The equations governing composition changes 27

electrons, neutrinos (and antineutrinos) and energetic photons. Specific nuclear

reactions will be considered in Chapter 4. A general way of describing a nuclear
reaction is by two different nuclei combining to produce two other nuclei. Since
the nucleus of any element is uniquely defined by the two integers X,- and Z,, we
denote the reactants by the symbols Z(X,-, Z,) and J(Aj. Z,), and the products
by the symbols K(Ak.Zk) and L(Ak. Zt). A nuclear reaction can proceed in
either direction (similarly to chemical reactions and ionization-recombination
processes), depending on the temperature (kinetic energy) and density of the
particles, and can therefore be described symbolically by

/(X,. Z,) + J(Aj, Zj) K(Ak, Zk) 4- L(Ah Zt), (2.48)

subject to two conservation laws:

A,- + Aj — Ak 4- X/, (2.49)

Zj + Zj = Zk 4- Z\. (2.50)

If positrons or electrons are also involved, they must be taken into account as
well: for them X — 0 and Z = ±1 and conservation of lepton number must be
obeyed. Therefore, any three ofthe four nuclei involved in the reaction uniquely
determine the fourth. The reaction rate, say from left to right, can be identified by
three indices: two for the reactants and one for one of the products.
Let us now attempt to evaluate the rate at which nuclei of type / are destroyed
by reactions of type (2.48) with the help of a simplified picture. Consider a unit vol­
ume and assume that each 1 nucleus within it has a cross-sectional area g, meaning
that any J nucleus striking this area will cause a reaction to occur. Assume further
that the relative velocity of / nuclei with respect to J nuclei is v. so that I nuclei
may be considered as targets at rest, while J nuclei flow toward them at velocity
v. The effective target area is therefore n,g\the number of particles crossing a unit
area per unit time is n, v. Hence the number of reactions that occur per unit time in
this unit volume is jgv or n,ny Rjjk, where /?,jk = gv - having the dimension
of volume divided by time - is called the reaction rate. In the case of particles
of one kind, say /, interacting with each other, the product Ujiij is replaced by
|n?. The velocity of gas particles in the star is the thermal velocity - this is
why nuclear reactions occurring in stars are called thermonuclear reactions -
and the cross-section depends on the properties of the reacting nuclei (such as
their charges) as well as on the properties of the products.
We may now write the rate of change ofthe (th element's abundance resulting
from all possible nuclear reactions, both destructive and constructive, in the form

n, = -u, £(i + M—J—Rlik + £ (2.51)

1 +tr1+
28 2 The equations of stellar evolution

where <5^ has its usual meaning: for j /, Sq = 0, while for j = i, 8,, = 1 (the
Kronecker delta). We note that two particles of type i are destroyed when j = i,
hence the factor (1 + <5(J ) in the first term on the right-hand side of Equation (2.51).
Using Equation (2.47), we obtain the rate of change of the mass fraction:

_p_ X./ Rijk X,Xk Rlki

Aj 1 + 8j j + E A/ Ak 1 4- 8ik
(2.52)
A,
l.k

and similar equations for the other mass fractions. For simplicity, we may define
a composition vector by X = (Xi.........X„) so that the set of equations describ­
ing composition changes may be symbolically written as one equation with n
components, one for each clement.

X = ftp. T, X). (2.53)

In nuclear equilibrium, when X, — 0, the mass fractions X, are readily obtained

from the set of equations f, = 0.

2.7 The set of evolution equations

The set of non-linear partial differential equations describing the evolutionary

course of the internal structure of a star is:

Cm ,dP
r =----- -- -4nr- —.
r- dm
71V OF (2.54)
"+ P - =?-7->
\p / dm
X = f(p. T.X),

together with the time-independent relation (1.5). As it stands, the set is not
complete; besides the structure functions - p(m. /). T(m, t), and Xj(jn, t) - that
form the set of unknowns, it contains additional functions: (a) P and u, (b) F
and (c) q and f, which have to be supplied in terms of the unknowns. To this
purpose, we shall have to invoke different branches of physics: thermodynamics
and statistical mechanics in case (a); atomic physics and the theory of radiation
transfer in case (b); and nuclear and elementary particle physics in case (c).
This is, in fact, what distinguishes astrophysics from other physical disciplines.
Astrophysics does not deal with a special, distinct class of effects and processes, as
do the basic fields of physics. Nuclear physics, for example, deals exclusively with
the atomic nucleus: there are many ramifications to this field of research, such as
nuclear forces, nuclear structure and nuclear reactions, but they are all intimately
connected. Nuclear physics has very little to do, say. with hydrodynamics, the
 2.8 The characteristic timescales of stellar evolution 29

study of the motion of continuous media. By contrast, astrophysics deals with

complex phenomena, which involve processes of many different kinds. It has to
lean, therefore, on all the branches of physics, and this makes for its special beauty.
The theory of the structure and evolution of stars presents a unique opportunity
to bring separate, seemingly unconnected physical theories under one roof. In the
next chapter we shall interrupt our pursuit of the evolutionary course of stars, in
order to extract from different physical theories the information that will enable
us to resume it.
Finally, in order for the set of differential equations to be solved, boundary
and initial conditions have to be supplied. The two space derivatives require
two boundary conditions and the n + 3 time derivatives require n -I- 3 initial
distributions of physical properties. The boundary conditions are straightforward:
P(M. t) — 0 and. in order to avoid a singularity at the centre. /•'(0. t) = 0. The
initial conditions could be p(in.O), T(m.O), i'(m.O) and X,(w,0), or related
functions. Here, it seems, we run into serious difficulties: as mentioned before,
star formation is still a subject of study. The initial state of a star is, therefore,
rather obscure. Fortunately, as we shall see shortly, this problem can be overcome,
or more precisely, avoided.

2.8 The characteristic timescales of stellar evolution

The evolution of a star is described by the three time-dependent Equations (2.54),

each dealing with a different type of change: the first involves dynamical or
structural changes, the second describes thermal changes and the third deals with
nuclear processes leading to changes in composition (and in the rest-mass energy).
Each change, or process, has its characteristic timescale r, which can be defined
as the ratio of the quantity (or physical property) </> that is changed by the process
and the rate of change of this quantity:

The simplest example would be of a motion, whose duration - or characteristic

timescale - is given by distance divided by velocity. Obviously, a rapid process
(large </>) has a short timescale and vice versa. It is instructive to estimate and
compare the timescales of the different processes that occur in stars.

The dynamical timescale

We can envisage a considerable change in the structure of a spherically symmetric

star as a change in its characteristic dimension, the radius R; hence in this case
we may take <f) = R. As gravity is the binding force of a star, the typical rate of
30 2 The equations of stellar evolution

change of R would be the characteristic velocity in a gravitational field: the free­

fall or escape velocity vesc = -J2GM / R\ hence </> = J2GM / R. The dynamical
timescale may therefore be estimated by

R _ /
Tdyn (2.56)
vZ ~ V 2GM'

or. in terms ofthe average density p = 3M/4ttR \ neglecting factors ofthe order
of unity.

Tiyn (2.57)
x/G^‘
There are many ways to obtain the dynamical timescale, but they all lead to the
same result, within factors of the order of unity. The dynamical timescale of the
Sun is about 1000 s (roughly a quarter of an hour), and generally:

(2.58)

The dynamical timescale is extremely short, many orders of magnitude shorter

than typical stellar ages. The estimated age of the Sun. for example, is 4.6 billion
years, or ~1.5 x I()l7s, about 1014rdyn. What is the meaning of this result? A
dynamical process occurs in a star whenever the gravitational force is not balanced
by the pressure forces (see Equation (2.12)). Such a situation can develop cither
into contraction, if there is insufficient pressure to counteract gravity, or into
expansion, if the pressure is too high. It can end either in a catastrophic event -
collapse or explosion - or in restoration of hydrostatic equilibrium, when the
forces are again in balance. Either of these end states will be achieved within
a period of time of the order of the dynamical timescale. This leads us to the
following conclusions:

1. If a star cannot recover from a dynamical process (by restoring hydrostatic

equilibrium), the ensuing collapse - or explosion - should be observable
in its entirety. Indeed, such events have been known to occur: they are
called supernovas. We shall return to them in Chapter 10.
2. Rapid changes that are sometimes observed in stars may indicate that
dynamical processes arc taking place, but on a smaller scale, not involving
the entire star. From the timescale of such changes - usually oscillations
with a characteristic period - we may roughly estimate the average density
ofthe star. The Sun has been observed to oscillate with a period of minutes.
Oscillations with periods of a few tens of seconds indicate that the star
should be a compact one, such as a white dwarf.
3. As a rule, stars may be assumed to be in a state of hydrostatic equilibrium
throughout. Any perturbation of this state is immediately quenched. This
 2.8 The characteristic timescales of stellar evolution 31

does not mean, of course, that stars are static during their entire life
span, but rather that they evolve quasi-statically, constantly adjusting their
internal structure so as to maintain dynamical balance. Consequently, the
left-hand side of Equation (2.12) may be assumed to vanish, and the virial
theorem (Section 2.4) may be assumed to hold at all times. This means
that the gravitational potential energy and the thermal energy of the star
each follows the behaviour of the total energy.

The thermal timescale

Thermal processes affect the internal energy of the star; hence in this case we
may take <j) — U. By the virial theorem (which, as we have seen, is applicable),
U % GM2 / R. The characteristic rate of change of U is the rate at which energy
is radiated away by the star; thus we may set <p = L. The thermal timescale may
be therefore estimated by

_ U GM2
T'h~ T. ~~ RL ’ (2.59)

For the Sun, z,], % 1015 s, or about 30 million years, and generally

(2.60)

The thermal timescale is many orders of magnitude longer than the dynamical
timescale, but it still constitutes only a small fraction - about 1% or less - of the
lifespan of a star. Thus, although we would not be able to observe the development
of a thermal process in a star (in fact, we have no way of knowing whether any
observed star is in thermal equilibrium or not), we may assume that throughout
most of its life a star is in a state of thermal equilibrium. If a star maintains
both thermal and hydrostatic equilibrium during an evolutionary phase, its total
energy is conserved (or changes very slowly) during that phase, and by the virial
theorem, the gravitational potential energy and the thermal energy, separately, are
conserved. Thus, if contraction occurs (quasi-statically) in some part of the star,
it follows that other parts should expand so as to conserve Q. Similarly, if the
temperature rises in some place, it should decrease in another, so as to keep U
constant. Later on we shall make use of such arguments.
The thermal timescale may be interpreted as the time it would take a star to
emit its entire reserve of thermal energy upon contracting (as we have shown in
Section 2.5), provided it maintains a constant luminosity. This was, in fact, the way
William Thomson (better known as Lord Kelvin) and, independently, Hermann
von Helmholtz estimated the Sun’s age more than a century ago, and for this
reason, the thermal timescale is often called the Kelvin-Helmholtz timescale.
32 2 The equations of stellar evolution

Historical Note: Kelvin's (1862) estimate imposed an upper limit on the age of the
Earth, which was in marked conflict with the new theory put forward by Charles Darwin
(in 1859). This theory required that geological time be much longer, so as to account for
the slow evolution of countless species of plants and animals (living and fossil) by natural
selection. A long and intense debate ensued between the two eminent scientists. To the
end Darwin remained convinced that, in time, physicists would change their minds. Harsh
criticism of Kelvin’s estimate came toward the end of the nineteenth century from the
geologist Thomas C. Chamberlin:

Is present knowledge relative to the behaviour of matter under such extraordinary

conditions as obtain in the interior of the sun sufficiently exhaustive to warrant
the assertion that no unrecognized sources of heat reside there? What the internal
constituents of the atoms may be is yet an open question. It is not improbable
that they are complex organizations and the seats of enormous energies.
T. C. Chamberlin: Annual Report of the Smithsonian Institution, 1899

And two decades later Eddington, addressing the same issue, predicted that the source of
energy in stars should be ‘subatomic’:

Only the inertia of tradition keeps the contraction hypothesis alive - or rather,
not alive, but an unburied corpse ...
A star is drawing on some vast reservoir of energy by means unknown to
us. This reservoir can scarcely be other than the subatomic energy which, it
is known, exists abundantly in all matter ... There is sufficient in the Sun to
maintain its output of heat for 15 billion years...
If, indeed, the subatomic energy in the stars is being freely used to maintain
their great furnaces, it seems to bring a little nearer to fulfillment our dream of
controlling this latent power for the well-being of the human race - or for its
suicide.
Sir Arthurs. Eddington: Observatory 43. 1920

Finally, after about ten more years, the controversy was settled (in Darwin’s favour!) by
quantum and nuclear physics, which solved the puzzle of the energy source of stars.

The nuclear timescale

The quantity that is changed by nuclear processes, besides abundances, is a (small)

fraction of the rest-mass energy given by Einstein’s famous relation E = me2.
This fraction, which may be turned into other forms of energy, constitutes the
nuclear potential energy. Hence we may take </> = eMc2. where e can be estimated
by the typical binding energy of a nucleon divided by the nucleon’s rest-mass
energy, which amounts to a few 10-3. The rate of change of the nuclear potential
energy is, obviously, the nuclear luminosity Enuc, and since we are allowed to
 2.8 The characteristic timescales of stellar evolution 33

assume thermal equilibrium, we may take (j> = Tnuc = L. Hence

eMc1
Aiuc (2.61)

or. using solar units,

'in ( M \
Tnuc ~ £4.5 X 10“° I ---- I ( —- | S. (2.62)

For the Sun, this is many times its age; in fact, rnuc is larger than the estimated
age of the universe. An immediate conclusion that emerges is that stars seem to
have actually consumed only a small fraction of their available nuclear energy,
meaning that only a fraction ofthe stellar mass has changed its initial composition.
Another is that, generally, nuclear equilibrium is not to be expected.
To summarize our results,

Tlyn Ah Tiuc- (2.63)

Consequently, it is the rates of nuclear processes that determine the pace of

stellar evolution, throughout which the star may be assumed to maintain thermal
and hydrostatic equilibrium at each stage. The set of evolution equations (2.54)
reduces to

(IP Gm
dm 4jrr4
dF (2.64)
~r
am
=cf
X = f(p,T.X).

This is a considerable simplification of the original set. In particular, we need

not know the initial structure of the star in order to be able to trace its evolution.
All we need to know is the initial composition, which we shall assume to be
homogeneously distributed throughout the star, and this is precisely what we do
know reasonably well. Our task of investigating the evolution of stars now divides
into two different parts, or two main questions: (a) What is the sequence of nuclear
processes that take place in stellar interiors? (b) Given the composition, what is
the structure - distribution of temperature and density - of a star in hydrostatic
and thermal equilibrium? Clearly, these questions cannot be separated: nuclear
processes depend on temperature and density, and the structure of a star depends
on its composition. But they can be answered in turn and the answers may then
be combined into a comprehensive picture of stellar evolution.
We shall address the first question in Chapter 4 and the second in Chapters 5
and 6. The next chapter, dealing in a general manner with the physics of stellar
interiors, may be skipped by readers who are familiar with the physics of gaseous
systems and of radiation.
 3

Elementary physics of gas and

radiation in stellar interiors

As a star consists of a mixture of ions, electrons and photons, the physics of stellar
interiors must deal with (a) the properties of gaseous systems, (b) radiation and
(c) the interaction between gas and radiation. The latter may take many different
forms: absorption, resulting in excitation or ionization; emission, resulting in
de-excitation or recombination; and scattering. In order not to stray too far from
our main theme, we shall only consider processes and properties that are simple
enough to understand without requiring an extended physical background, and
yet sufficient for providing some insight into the general behaviour of stars.
The full-scale processes are incorporated in calculations of stellar structure and
evolution, performed on powerful computers by means of extended numerical
codes that include enormous amounts of information. These, however, should be
regarded as computational laboratories, meant to reproduce, or simulate, rather
than explain, the behaviour of stars. Our purpose is to outline the basic principles
of stellar evolution and we arc therefore entitled to some simplification. Eddington
defends this right quite forcefully:

I conceive that the chief aim of the physicist in discussing a theoretical problem
is to obtain ‘insight’ - to see which of the numerous factors are particularly
concerned in any effect and how they work together to give it. For this purpose
a legitimate approximation is not just an unavoidable evil; it is a discernment
that certain factors - certain complications of the problem - do not contribute
appreciably to the result. We satisfy ourselves that they may be left aside; and
the mechanism stands out more clearly, freed from these irrelevancies. This
discernment is only a continuation of a task begun by the physicist before the
mathematical premises of the problem could be stated; for in any natural problem
the actual conditions are of extreme complexity and the first step is to select those
which have an essential influence on the result - in short, to get hold of the right
end of the stick. The correct use of this insight, whether before or after the
mathematical problem has been formulated, is a faculty to be cultivated, not a

34
 3.1 The equation of state 35

vicious propensity to be hidden from the public eye. Needless to say the physicist
must if challenged be prepared to defend the use of his discernment; but unless
the defence involves some subtle point of difficulty it may well be left until the
challenge is made.
Sir Arthur S. Eddington: The Internal Constitution of the Stars. 1926

3.1 The equation of state

A relation between the pressure exerted by a system of particles of known com­

position and the ambient temperature and density, P - P(p, T, X), is called an
equation of state.
In previous sections we have repeatedly assumed the stellar gas to be an ideal
gas, which implies a mixture of free, noninteracting particles (a perfect gas).
The time has come to justify this assumption. At the temperatures prevailing in
stars, gases are ionized, and Coulomb interactions can be expected to occur. We
shall show that the energy involved in such interactions is small compared with
the kinetic (thermal) energy of the particles. For an average density p and a gas
particle mass .A/wh, the mean interparticle distance is

3/W ) (3.1)
p /
where p has been expressed in terms of stellar mass M and radius R. If the particle
charge is Ze (e denoting the electron charge), the typical Coulomb energy per
particle may be estimated as
I Z-e1
(3.2)
d
The kinetic energy (per particle) is of the order kT and hence, after substituting
T from Equation (2.29),

kT 4.t£0 A^rn^GM2^'

ignoring factors of the order of unity. For Z = 1, A = I and M =MO, this

ratio is about 1% and, to this accuracy, Coulomb interactions may therefore be
neglected. For higher Z, we have A 2Z, and the ratio (3.3), which varies as
Z2/3, remains well below unity even for a composition of pure iron. We should
note, however, that ec/kT A I for mass M A 1()-3M3. Although stars do not
belong to this dangerous zone, planets do: the mass of Jupiter is roughly 103M3.
Consequently, the structure of planets cannot be described by a mixture of free
gases; more complicated equations of state must be invoked. Since ec/kT 3> I
characterizes solids, we may conclude that the smaller a planet's mass, the closer
to a solid is its structure.
36 3 Elementary physics of gas and radiation

Figure 3.1 Beam of particles impinging on a hypothetical surface, making an angle 0

with the normal to the surface.

A general theorem enables the calculation of the pressure of a free particle

system by means of an integral known as the pressure integral:
/'OC
P = | / v pn(p)dp. (3.4)
Jo
where v is the particle velocity, p its momentum, and n(p)dp is the number
of particles per unit volume with momenta within the interval (p, p + dp). The
proof of this theorem is as follows: Consider a surface (real or imaginary) within
the system of particles. The pressure on this surface results from the momentum
imparted by particles colliding elastically with it. The momentum transferred by
an incident particle is twice the momentum component normal to the surface

A/? — 2pcers0. (3.5)

Consider a beam of particles impinging on the surface at a velocity v making an

angle 0 with the normal to the surface (as shown in Figure 3.1). Let n(0, p) d0 dp
be the number density of particles with momenta in the range (p, p + dp) and
directions within a cone (0, 0 + d0). Since the particle distribution is isotropic,
the number of particles within any solid angle da) is proportional to the solid
angle, or

n(0. p)d0 dp da> 2tt sin 0 d0

= | sin 0 d0. (3.6)
n(p)dp 4rr 4rr

The number of particles from this beam striking the surface in a time interval
81 is given by the number density multiplied by the volume v8l dS cos#, where
dS is the area of incidence of the beam on the surface. Hence the momentum
transferred to the surface by these particles is given by

8p<) — n(0, p) d0 dp v 81 dS cos 0 Ap. (3.7)

 3.2 The ion pressure 37

The contribution of these particles to the pressure - the momentum transferred per
unit time per unit surface area - is therefore given by Equation (3.7), combined
with Equations (3.5) and (3.6), after dividing by 8tdS'.

dP = I sin0r/0/?(/?)r//> v cos(? 2pcos0 = vpn(p) dp cos2 0 sin0 d0 (3.8)

and the total pressure is obtained by integrating over all angles of incidence
(0 < 0 < n/2) and all momenta. Since
rir/2 /■ I
/ cos2 0 sin (9 d0 = / cos2 0 d cos 0 = |, (3.9)
Jo ./()

the proof is completed. Obviously, the pressure of a mixture of free, noninteracting

particles of different species will be given by the sum of the pressures exerted
by each species separately. This will include the radiation pressure, which is the
pressure exerted by photons. Consequently, in stars, P will be the sum of three
different terms: P\ for the ions, Pc for the electrons and Prac| for the photons, the
first two constituting the total gas pressure,

P — P\ + Pc + /’rad — /gas + /rad- (3.10)

It is customary to define a parameter as the fraction of the pressure contributed

by the gas; thus

P^ = fiP (3.11)

/>rad = (l -p)P. (3.12)

3.2 The ion pressure

The equation of state for an ideal ion gas is the well-known relation

Pi T, (3.13)

where nj is the number of ions per unit volume. This relation is obtained by apply­
ing the theorem just proven to a free particle gas in thermodynamic equilibrium,
which is characterized by a Maxwellian velocity distribution:

n\4np2dp P
n(p)dp = ------- —^e (3.14)

Using relations (2.47), we obtain the total number of ions in a unit volume by
summing overall the ion species i:

= = <3J5)
/»H Ai
I I
38 3 Elementary physics of gas and radiation

The mean atomic mass of stellar material is defined by

(3.16)

so that

— » (3.17)
Mi'«h
and /Z| may be approximated by
1 Y 1 - X - Y
*X + 4 + M) (3.18)

where (A) is the average atomic mass of the heavy elements (elements other than
hydrogen and helium, sometimes referred to as metals). For the Sun, for example,
X = 0.707. Y = 0.274, and (A) ~ 20: whence /zj = 1.29. The ratio k/mu is
usually known as the ideal gas constant

H = —. (3.19)
"'ll

Substituting Equations (3.17) and (3.19) into Equation (3.13), we finally obtain
Tl
P\ = —pT. (3.20)
Ml

3.3 The electron pressure

If the electrons constitute an ideal gas, the equation of state is, as Equation (3.13)
above.

/<■ = M'7', (3.21)

where ne is the number of (free) electrons per unit volume. Here we shall make
a simplifying assumption by taking the atoms to be completely ionized. This
is certainly correct for the main stellar constituents, hydrogen and helium, at
temperatures exceeding 106 K. The assumption is obviously incorrect for stellar
photospheres, but we are mainly concerned with the interior. With this assumption,
the total number of electrons per unit volume is

— Vx — (3.22)
mH ' A/

We define /ze 1 as the average number of free electrons per nucleon.

(3.23)
 3.3 The electron pressure 39

leading to

ne = (3.24)
Me'»H

In terms of the mass fractions X and Y, wc have

— = X4-|r + (l-X-X)(4k (3.25)

Me ' AI

where (4j) is the average value for metals, which may be approximated reasonably
well by 1. Hence

— *|(1+X), (3.26)
Me

which for the Sun amounts to Me 1-17 and for hydrogen-depleted stars, to
/jLe % 2. The electron pressure is thus given by

R.
Pe = — pT. (3.27)
Me

Combining Equations (3.20) and (3.27), we obtain the total gas pressure

P^ = Pi + Pc = (- + —}npT = -pT, (3.28)

\Mi Me/ M

where

- = -+ (3.29)
M Ml Me

yielding p = 0.61 for the solar composition. Note that for hydrogen, the contribu­
tions of ions and electrons to the gas pressure are equal; for all heavier elements,
the electron pressure is higher than the ion pressure (twice as high for helium, for
example).
The assumptions explicitly made so far were (a) lack of interactions between
gas particles, and (b) complete ionization. Other assumptions were, however,
implicitly included when adopting the classical physics approach, ignoring quan­
tum and relativistic effects. But the conditions of stellar interiors arc such that
these effects cannot always be neglected.
According to quantum mechanics, the simultaneous position and momentum
of an electron (or any other particle) cannot be known more precisely than allowed
by the Heisenberg uncertainty principle. More specifically, if a particle’s location
is known to be within a volume element AV and its momentum is within an
element A3p in the three-dimensional momentum space, then AV and A3p are
constrained by the condition

AVA3/? > h\ (3.30)

40 3 Elementary physics of gas and radiation

Consider now an ideal electron gas of temperature T; the temperature deter­

mines the distribution of momenta according to Equation (3.14). In particular,
the average momentum (or velocity) is uniquely determined by T. Imagine now
that the gas is compressed. The volume occupied by each particle, AV oc p~},
decreases. As long as the temperature is sufficiently high (and. with it, the aver­
age velocity, or momentum), so that compression (decrease of AV) does not
lead to the violation of Heisenberg’s principle, we should be allowed to ignore
quantum effects. Eventually, however, the density may become high (AV low)
enough for the range of momenta dictated by the uncertainty principle to exceed
the momentum corresponding to the gas temperature. Practically, this means that
the electron pressure must be higher than that inferred from the temperature. In
order to estimate the electron pressure under these conditions, we have to take
account of another quantum mechanics principle, the Pauli exclusion principle,
which postulates that no two electrons can occupy the same quantum state, that
is. have the same momentum and the same spin. Since an electron can have two
spin states (up and down), this means that each element of phase space - location
and momentum space - can be occupied by two electrons at most. The pressure
generated by electrons that arc forced into higher momentum states as their den­
sity increases is called degeneracy pressure. A state of complete degeneracy is
obtained when all the available momentum states are occupied up to a maximum
momentum value. In this case AV A3/? is minimal and condition (3.30) becomes
an equality. Such an ideal situation can only be achieved at zero temperature,
but it constitutes a good approximation to states of high degeneracy and has the
advantage of enabling a straightforward calculation of the pressure. Therefore,
although the transition from a Maxwellian to a completely degenerate momen­
tum distribution occurs gradually, we shall discuss only the extreme situations.
A more detailed treatment of the degenerate equation of state may be found in
Appendix B.
Applying the Heisenberg and Pauli principles to a completely degenerate
isotropic electron gas yields the momentum distribution - the number of electrons
with momenta in the interval (p. p + dp) per unit volume:

2 2 ,
ne(p)dp = —— = —Ąttp-dp, p < p(}. (3.31)
AV IP

The maximal momentum />() can be obtained by integrating, nc = nc(p)dp,

and reversing the relation between ne and p<y.

We may now use Theorem (3.4), substituting Equation (3.31), taking v = p/mc,
where me is the electron mass, and carrying the integral up to po, to obtain the
 3.3 The electron pressure 41

degeneracy pressure of an electron gas

5_ h> /3\2/3 1 /p\5/3

15/ne/?3/{) 20me \tt/ w3/? \Me/ (3.33)

where we have used relation (3.24) for /ie. We note that the degeneracy pressure
is inversely proportional to the particle (electron) mass. Hence, although the argu­
ments presented for electrons could be equally applied to protons and neutrons, as
they are nearly 2000 times more massive than electrons, quantum effects become
important in their case under much more extreme conditions (much higher den­
sities for a given temperature, and much lower temperatures for a given density),
and may usually be ignored. We also note that, in spite of the high densities
characteristic of degenerate matter, the particles may be still considered free,
since the particle energy, of the order of /?„/2/??c, is still higher than the Coulomb
energy eC-

Exercise 3.1: Find the condition that the electron number density ne must satisfy,
for a degenerate electron gas to be considered perfect.

Inserting the numerical values of constants in Equation (3.33), we obtain

/ \5/3
/’e.deg = J ( — ) , (3.34)

where K\ = 1.00 x 107m4kg-2/3s“211.00 x 10l3cm4g-2/3s"2]. Foracompo-

sition devoid of hydrogen (and not very rich in extremely heavy elements), pc & 2
and hence the degeneracy pressure (3.33) is simply given by

Pe,deg = K1P5/3, (3.35)

where K\ is a constant. This relation will be often used in future discussions.

If the electron density is increased further, the maximal momentum in a
completely degenerate electron gas grows larger. Eventually, a density is reached
such that the velocity po/mc approaches the speed of light. The electrons now
constitute a relativistic degenerate gas. for which the simple relation between
momentum and velocity p = mv no longer holds and has to be replaced by the
relativistic kinematics relation. Here again, for simplicity, we shall only consider
the extreme case in which the velocity is very close to c. Replacing v by c in the
pressure integral (3.4), we obtain by the same procedure as before

_^mi/3j_/M4/3 (3.36)
\*V \Mc/

for the pressure exerted by a completely degenerate relativistic electron gas in

the limit v c. The transition between relations (3.33) and (3.36) is a smooth
42 3 Elementary physics of gas and radiation

function of v/c or p/m<.c, which we shall not address here (but see Appendix B).
Inserting the numerical values of constants in Equation (3.36), we obtain
/ \4/3
/’e.r-deg = ~ I , (3-37)
\Me/

where K', = 1.24 x IO10 m3 kg“1/3 s-1 [1.24 x 1015 cm3g-'/3 s"1] and finally,
for a fixed value of /ic,

Pe.r-deg = W/3, (3.38)

where AS is another constant.

We should keep in mind that relations (3.35) and (3.38) for the pressure were
obtained on the assumption of vanishing temperature and hence, naturally, the
pressure is only a function of density (for a given composition). It is true, however,
that even in the case of incomplete - or partial - degeneracy, the temperature
plays a far smaller role than in the case of an ideal (nondegenerate) gas. Thus as
a crude approximation, the degeneracy pressure may be considered as insensitive
to temperature. The approximation is good provided kT is only a fraction of the
kinetic energy of a particle with the highest momentum /?o(«e)- A more accurate
treatment of degeneracy pressure may be found in Appendix B.

3.4 The radiation pressure

Radiation pressure is due to photons that transfer momentum to gas particles

whenever they are absorbed or scattered. In thermodynamic equilibrium the pho­
ton distribution is isotropic and the number of photons with frequencies in the
range (v, v + dv) is given by the Planck (blackbody distribution) function
8.7 v2 dv
n(v)dv = —-------------- . (3.39)
C QIT — 1
The pressure is then readily obtained from Equation (3.4):

^rad = | [ c—n(v)dv = \aT4, (3.40)

Jo c
where a is the radiation constant

Although the expression for radiation pressure was easily derived from the
pressure integral, the concept deserves further (intuitive) explanation. Imagine a
collimated beam of photons striking an atom. Each photon is absorbed, thereby
exciting the atom, which consequently returns to its original state by emitting
a photon. The direction of the emitted photon is random, the initial direction
 3.5 The internal energy of gas and radiation 43

of the absorbed one having been ‘forgotten’. Each such interaction involves an
exchange of momentum. By absorbing the photon, the atom gains momentum in
the direction of the photon beam. When it emits a photon, the atom recoils in
the direction opposite to that of the emitted photon. After a long series of such
interactions, the random changes of momentum due to emission cancel out and
the net change in the atom’s momentum is in the direction of the photon beam, as
if material pressure has been exerted on it in that direction.

3.5 The internal energy of gas and radiation

The specific energy (energy per unit mass) of a perfect gas, which is due to the
kinetic energy e of the motion of the individual particles, is generally given by

u = (3.42)

where the integral represents the energy density (energy per unit volume). For a
classical gas, e = p2/2m^, for a relativistic gas

P~
e - /ngc- (3.43)
m^c2

which tends to p2/2ms in the limit p m^c. Performing the integral for a simple
classical ideal gas, we obtain for the energy density the well-known result *nkT,
which is equivalent to . The specific energy is therefore

_ 3 /T,
"gas — . (3.44)
2 p

For a classical completely degenerate electron gas we obtain the specific energy
by integrating Equation (3.42) up to the highest momentum p$ and the result is
identical with that obtained for the classical ideal gas - Equation (3.44). For the
relativistic completely degenerate case we obtain by the same procedure
Pgas
"gas — 3 (3.45)
P
The energy density of radiation is given by

/ hvn(v) dv — aT\ (3.46)

Jo
where the integral is the same as in Equation (3.40), the specific energy being
«7’4 -/rad
"rad = ------- = 3------ . (3.47)
P P
44 3 Elementary physics of gas and radiation

Exercise 3.2: Assuming a uniform value of fl throughout the star and defining
U = f(ugas + uni)dm. show that the virial theorem (2.23) leads to

for a classical (nonrelativistic) gas. Note in particular the limits -> 1 and
fl -+ 0. If the star contracts, maintaining the same uniform fl, which fraction of
the gravitational potential energy released is radiated away and which fraction is
turned into heat?

3.6 The adiabatic exponent

Thermodynamic processes of a special kind, which will be of interest in later

discussions, arc those occurring in a system without exchange of heat with the
environment. Such processes are called adiabatic. From the first law of thermo­
dynamics (mentioned in Section 2.2) it follows that adiabatic processes satisfy
the condition
du + Pd ( - ) = 0. (3.48)
\P/
In the previous section we have seen - at least for simple systems - that the
specific energy u is always proportional to P/p. We may therefore write

u = <p—, (3.49)
P
which, by differentiating and substituting into Equation (3.48). leads to

<j)Pd(- | +</>-dP + Pd(~} =((/> + l)Pjf-) + (fl-dP =0. (3.50)

\pj P \P) \P) P
Accordingly, the dependence of the pressure on density is described by a power
law
0+1
P oc p . (3.51)

The power (d In P/d In p) is called the adiabatic exponent, denoted ya; the pro­
portionality factor (to be denoted A"a) is determined by the properties of the
system (it is a direct function of the entropy). In conclusion, adiabatic processes
are characterized by the law

P = KApY\ (3.52)

It is easily seen that for the systems we have considered, the value of ya is 5/3
in the case of a nonrelativistic ideal gas or a completely degenerate electron gas.
and 4/3 in the case of a relativistic degenerate electron gas or of pure radiation.
 3.6 The adiabatic exponent 45

Intermediate values will obtain for mixtures, such as gas and radiation, and for
nonextreme cases, such as a moderately relativistic degenerate electron gas.
So far we have considered gases of a fixed number of particles: either (almost)
fully ionized, as in the deep stellar interior, or (almost) fully recombined, as in
the outer layers of a cool stellar atmosphere. When ionization takes place and
the number of particles changes with the other physical properties, the adiabatic
exponent changes too. Since this will prove to be of particular importance to the
stability of stars, it deserves some discussion. We shall only consider the very-
simple case of a singly ionized pure gas (rather than a mixture of gases), say,
hydrogen. Hence we have to deal with three different types of particles: neutral
atoms, whose number density we denote by n0, ions of number density n+, and
free electrons of number density ne (obviously, ne = n+). The pressure exerted
by the gas is proportional to /t0 + «+ + ne, while the mass density is proportional
to »o + n+- The degree of ionization is defined by
n+
x =----- —. (3.53)
no + »+
The densities of ions and neutrals are related by Saha’s equation (after Meghnad
Saha, who derived it in 1920)

= 4(27rwe^7')3/2e"zAr, (3.54)
»o h-
where g is a constant and / is the ionization potential (the energy required to
create an ion by removing an electron from an atom). In terms of the degree of
ionization, we have

P = (1 + -r)(«0 + n+)kT = (I + xYRpT. (3.55)

and Saha’s equation becomes

In the case of a partially ionized gas, the specific energy has an additional term,
Xn+/P = x7?+/[(/!o + n+)mH] = Xx/mH- which is due to the available potential
energy of ionization. Thus

IP X
u zz—I------ * (3.57)
2 p mu
replaces Equation (3.49). Using Equations (3.55) and (3.56) to express the degree
of ionization as a function of pressure and density x — x(P, p), differentiating
Equation (3.57), and substituting into Equation (3.48) yields

3 / 1 \ 3 / 1 \ X dx X dx ( 1 \
- - ]dP + -Pd - + — — dP + —------ dp + Pd ( - = 0.
2\p/ 2 \p J m\\dP wh dp \p /
(3.58)
46 3 Elementary physics of gas and radiation

Figure 3.2 Radiation flux passing through a slab.

Multiplying by p/P and assembling terms, we have

3 X ( P \ / dx \ (IP F5 x / p \ /dx \ j dp
2 + 17 \ I +x/ \dP/fi ’y~|_2-k7\i+.xj W7/J ~P

(3.59)

from which, after not inconsiderable manipulation, ya(x) lliay be calculated:

In the limit x — 0 or x - I. we obtain ya — 5/3, as before; the minimum value

is obtained for x =0.5; it is 1.63 for x/^T = I. for example, and 1.21 for
X/kT = 10.

3.7 Radiative transfer

Consider a slab of thickness dr and density p between parallel surfaces A and

B of unit area, as shown in Figure 3.2. A radiation flux 11 (energy per unit area
per unit time) incident on A emerges at B after losing an amount dH, which
has been absorbed by the slab (in reality, the slab may also emit radiation, which
would have to be taken into account). Obviously, the amount of absorbed radiation
should be proportional to the incident flux and to the amount of material, that
is, to the density of absorbing (scattering) particles and to the length of the path
travelled by the photons, and hence to their product lipdr. We may therefore
write:

dH — —KHpdr. (3.61)
 3.7 Radiative transfer 47

where the minus sign indicates that the flux has been diminished, and k is the
proportionality factor, called the opacity coefficient, determined by the properties
of the materia] the slab is made of, such as composition, density and temperature.
A dimensionless quantity, r, may be defined by dr = —Kpdr', called optical
depth, it is a measure of the transparency of a medium to radiation. The minus sign
appears because radial distance is measured positively outwards, while depth is
measured positively inwards. An opaque medium has a large optical depth, which
may be due to a large physical depth, a high opacity, or a high density, or to
a combination of these factors. A transparent medium, which lets through most
of the radiation crossing it, has a low optical depth. Integrating Equation (3.61)
inwards, we thus obtain

H(r) = (3.62)

for the radiation flux at a distance r from a source Ho. By definition, tq > f(r),
so that H(r) < //(>. The characteristic absorption length, (/rp)-1, may be regarded
as the mean free path of a photon.
In a star, the concept of optical depth serves to define the photosphere. Being
a gaseous sphere, a star does not have a well-defined surface; the stellar radius is,
by definition, the radius of the surface where T = Teff. To find this surface, we
recall that the bulk of stellar radiation is emitted from the region lying above R,
which is the photosphere; hence the optical depth of the photosphere, f^ —Kpdr,
must be of the order of unity. The condition Kpdr 1 may be regarded as a
definition of R. the exact value of the photospheric optical depth being determined
by a detailed treatment of radiative transfer and its inherent assumptions and
approximations.

Exercise 3.3: Show that the equation of hydrostatic equilibrium may be written
as
dP _ g
dr k'
where g is the local gravitational acceleration. (This form is useful in models of
stellar atmospheres.) Use this form to evaluate the pressure at R and estimate the
ratio between the centre and surface pressures of a star. For simplicity, assume
an average constant opacity for the photosphere.

As photons of different frequencies interact differently with matter, the opacity

coefficient is also a function of the frequency (or wavelength) of the radiation and
the foregoing discussion applies strictly only to monochromatic radiation. It is,
however, possible to define an average opacity, independent of wavelength (see
Appendix A). The most important interactions between stellar matter (of high
48 3 Elementary physics of gas and radiation

temperature) and radiation are those involving electrons (rather than the much
heavier nuclei). These are of several types.

1. Electron scattering - the scattering of a photon by a free electron. In

the classical case, known as Thomson scattering (after J. J. Thomson),
the photon’s energy (that is, frequency) remains unchanged. In the less
common, relativistic case, known as Compton scattering, the photon’s
energy changes.
2. Free-free absorption - the absorption of a photon by 'A free electron, which
makes a transition to a higher energy state by briefly interacting with a
nucleus or an ion. The inverse process, leading to the emission of a photon,
is known as bremsstrahlung.
3. Bound-free absorption - which is another name for photoionization - the
removal of an electron from an atom (or ion) caused by the absorption of
a photon. The inverse process is radiative recombination.
4. Bound-bound absorption - the excitation of an atom due to the transition
of a bound electron to a higher energy state by the absorption of a photon.
The atom is then de-excited either spontaneously or by collision with
another particle, whereby a photon is emitted.

In the deep stellar interiors, where temperatures are very high, the first two
processes are dominant, simply because there are very few bound electrons,
the material being almost completely ionized. Furthermore, the energy of most
photons in the Planck distribution is of the order of keV, whereas the separation
energy of atomic levels is only a few tens eV. Hence most photons interacting w ith
bound electrons would set them free. Thus bound-bound (and even bound-free)
transitions have extremely low probabilities, interactions occurring predominantly
between photons and free electrons.
Opacity coefficients may be measured or - for conditions typical of stellar
interiors - calculated, taking into account all the possible interactions between
different elements and photons of different frequencies. This is a tedious task
that requires an enormous amount of calculation. When it has been performed,
the results arc usually approximated by relatively simple formulae in the form of
power laws in density and temperature for a given composition:

k = KapaTh. (3.63)

The opacity resulting from electron scattering is temperature and density inde­
pendent (a = b = 0); it is given by

^es = — «Ues.0(l+X), (3.64)

Me
 3.7 Radiative transfer 49

Log T(K)

Figure 3.3 Opacity coefficients (in units of cnrg-1) for a solar composition as a func­
tion of temperature for different density values; the numbers beside each curve are
logp(gcm-3) (data from C. A. Iglesias & F. J. Rogers (1996), Astrophys. J. 464).

where /ces,o = 0.04 m2 kg 1 (0.4 cm2 g_|). The opacity resulting from free-free
absorption, first derived by Hendrik A. Kramers, is well approximated by a power
law of the form (3.63). with a = 1 and b = -7/2. known as the Kramers opacity­
law,

K(f = —fypT-7/2 * |/Cff.o(l + X)l^-\pT~1/2, (3.65)

pe ' A I \AI

where complete ionization is assumed and (^) = X,^-. The constant «o

has the value 7.5 x IO18 m5 kg"2 K7/2 (7.5 x IO22 cm5 g~2K7/2) with an accuracy
of about 20%. Electron scattering and free-free opacities arc both due to the free
electrons; both coefficients, Kes and Ka, are thus proportional to the electron
number density and hence to //“’ (see Equation (3.24)). Opacity coefficients for
solar composition material are given, as an example, in Figure 3.3; note that above
a few 104 K they may indeed be quite accurately represented by power laws.
The average opacity of stellar material (of solar composition) is of the order
of 0.1 nr kg 1 (1 cm2 g-1), and since the average density is of the order of 1000
kg m-3 (1 g cm-3), the mean free path of photons in the interior of a star is about
0.01 m (1 cm). The temperature drop over such a radial distance within a star is
about 0.001 K (estimated as T/R). This is why the radiation in stellar interiors is
so close to that of a blackbody. But blackbody radiation is also isotropic: what,
then, is the meaning of a radiation flux from the interior of the star to the surface
(the function F in Equation (2.6))? It turns out that the minute deviation from
isotropy is sufficient for the transfer of energy that results in the stellar luminosity.
50 3 Elementary physics of gas and radiation

To calculate the radiative flux we shall adopt a simple approach due to Edding­
ton. The absorption of radiation energy by the slab just considered also involves
a corresponding amount of momentum: the momentum absorbed by the slab per
unit time is \dH\/c. The rate of increase of the momentum must be equal to the net
force applied to the slab by the radiation field (Newton’s second law). This force
is simply the difference of the radiation pressures exerted on the surfaces A. say
at r. and B, at r + dr (see Figure 3.2): Pra(j(/') — Praii(r + dr) — —(dPta<i/dr) dr.
Consequently,
Hkp _ dPrMi
(3.66)
c dr
and since the radiation may be assumed to be blackbody radiation, the pressure
is given by Equation (3.40) and
H _ 4acT3 dT
(3.67)
3i<p dr
A rigorous derivation of this equation, leading to the correct evaluation of the
average opacity, is given in Appendix A. To obtain the total flux F crossing a
spherical surface of radius r, we multiply H by the surface area 4?rr2:

4acT3 dT
F = —4.t/-2 (3.68)
3kp dr
We may invert this relation to obtain the temperature gradient in terms of the flux:
dT 3 Kp F
(3.69)
dr 4acT3 47tr2
or. using m as the independent space variable,
dT _ 3 k F
(3.70)
dm 4a c T2 (4jrr2)2

We have now gathered sufficient information on the physics of stellar interiors

to allow us to pursue the investigation of stellar evolution.
 4

Nuclear processes that take

place in stars

The evolution - continuous change - of stars is due to their sustained emission of

radiation originating from an internal source. The energy source that supplies the
luminosity of stars during most of their lifetimes is nuclear fusion, which turns a
small fraction of the rest mass into energy. Although this was only realized at the
beginning ofthe twentieth century, with Einstein’s formula E = me2, the concept
of conversion of matter into light dates back to Newton, at the beginning of the
eighteenth century.

The changing of bodies into light, and light into bodies, is very conformable to
the course of Nature, which seems delighted with transmutations.
Isaac Newton: Opticks, 1704

The formalism by which nuclear reactions arc incorporated into the stellar evo­
lution theory was given in Section 2.6. The purpose of the present chapter is to
examine in more detail the nuclear processes that are bound to take place in stars
and the energy each of them can supply.

4.1 The binding energy of the atomic nucleus

The energy released or absorbed in a nuclear reaction being a fraction of the

rest-mass energy of the particles involved, mass is not strictly conserved: the
total mass of the products differs slightly from the total mass of the reactants,
the difference depending on the binding energies of the interacting nuclei. As we
have seen in Section 2.6, the general description of a nuclear reaction is of the
form (2.48),

KA,. Zi) + J(Ar Zj) K(Ak. Zk) + L(Ah Zt).

51
52 4 Nuclear processes that take place in stars

Denoting by Qijk the amount of energy released in this reaction, and by .M, the
mass of a nucleus of type /. we have

(4.1)

neglecting the small masses of possible light particles that may be involved. Using
the mass unit mu, we may write Equation (4.1) as

Qijk — l(-M, — A/»h) + (-M j — Ajinw) — (Mk — Akm\\) — (Mi — X//»h)]c2

+ (A, + Aj — Ak — v4/)nzi|C", (4.2)

where the second term on the right-hand side vanishes, by conservation of baryon
number, Equation (2.49). The difference

^M(l) = (Mi -Aimn)c2 (4.3)

(whether positive or negative) is called mass excess, despite its being a measure
of energy. Mass excesses are listed in tables of nuclear data in units of MeV
(I MeV = 106 eV = 1.6021772 x 10"13 J). We note that mass-excess values
depend on the atomic mass unit employed, but Qqk, which involves differences
of mass excesses, is independent of the (arbitrary) mass unit.
We may now calculate the total rate of energy release at a given point in a star:
since the number of reactions of type (2.48) that occur per unit volume per unit
time is n,n jR^ (if I = J. should be replaced by |n?). the energy released
by such reactions per unit volume per unit time is nai, R,jkQi,k- Summing over
all nuclear reactions that can occur at that point, and dividing by p to obtain the
rate of energy released per unit mass, we have
p V-. 1 Xi Xj
(-4)

'''II ijk 1 T °'J -I

for the term that appears on the right-hand side of the energy equation (2.6). In
reality, the available energy (to be turned into heat) may be less. If neutrinos are
produced by the nuclear reactions (or by other processes), their energy is lost to
the star, which is transparent to neutrinos. These ‘particles’ leave the star without
undergoing collisions and sharing their energy with the medium. Therefore, the
net rate of energy release is <?nuc — qv, where r/nuc is given by Equation (4.4) and
qv is the neutrino energy lost per unit mass per unit time.
Neutrinos are not only produced in nuclear reactions involving electrons and
positrons, but also in interactions similar to those of an electron with the radiation
field (photons), in which a change in the electron’s momentum occurs. Simply,
the emerging photon is sometimes - usually extremely seldom - replaced by a
neutrino-antincutrino pair. Thus photoneutrinos are produced when a photon is
scattered by an electron, replacing the outgoing photon. The annihilation of an
electron-positron pair (discussed in Section 4.9). which normally results in the
 4.2 Nuclear reaction rates 53

creation of two photons, may produce a neutrino-antineutrino pair instead, with

a probability of about IO-19. Bremsstrahlung photons, emitted when an electron
is decelerated by the Coulomb field of a nucleus or an ion (see Section 3.7),
may also be replaced by a neutrino-antineutrino pair. Finally, a photon may itself
decay into a neutrino-antineutrino pair, when the radiation field is affected by the
electromagnetic field of the stellar plasma. The role of the electron is replaced
in this case by a virtual particle called plasmon - essentially, a quantized plasma
wave. All these processes become important either at very high densities or at
very high temperatures (or both). Then, due to the transparency of stellar material
to neutrinos (the mean free path of stellar neutrinos is about l()9/?o!), they may
cause efficient local cooling.
The energy released in a nuclear reaction Qa^ is a measure of the difference
between the binding energies of the reactants and the products. The total binding
energy of a nucleus is, of course, a function of the number of nucleons; but
even the binding energy per nucleon differs from element to element, or among
isotopes of the same chemical element (nuclei with the same Z, but different A).
This implies that some nuclear structures are more stable than others. There are
also unstable nuclei that spontaneously decay by emitting light particles such
as electrons (fl decay) or positrons (fl+ decay) - these are called radioactive
isotopes; and there are excited nuclei, in a high-energy state, that emit energetic
photons, thereby becoming more strongly bound. The nuclear-shell model, similar
in many respects to the electron-shell model of the atom, explains these properties.
For our purposes, the important result is the variation of binding energy
per nucleon with baryon number A, shown in Figure 4.1, relative to the free
proton - the hydrogen nucleus. The general trend is an increase of the binding
energy per nucleon with atomic mass up to iron (A = 56) and a slow monotonic
decline beyond iron. The steep rise of the binding energy from hydrogen, through
deuterium and 'He. to 4He implies that fusion of hydrogen into helium should
release a large amount of energy per nucleon (unit mass), considerably larger
than that released in. say. fusion of helium into carbon. Energy may be gained
by fusion of light nuclei into heavier ones up to iron and. to a lesser extent, by
breakup, or fission of heavy nuclei into lighter ones down to iron. In a different
context, we recognize the first process as the basic mechanism of the H-bomb,
and the second, as the mechanism of the A-bomb. Another important fact related
to the binding energy of atomic nuclei is that there arc no stable configurations for
>1 = 5 and for A = 8; 4He is more lightly bound than its immediate neighbours.

4.2 Nuclear reaction rates

In Section 2.6 we have seen that the rate of a nuclear reaction is essentially
the product of the cross-sectional area of a (target) nucleus and the relative
54 4 Nuclear processes that take place in stars

Figure 4.1 Variation of the binding energy per nucleon with baryon number.

velocity of the interacting gas particles. For the latter, we may simply assume a
Maxwellian velocity distribution (Equation (3.14)): this means that the probability
of the velocity of a particle of mass mg being within an interval dv around a
velocity v would be proportional to exp(— mgv2/2kT), decreasing with increasing
v. Since, in reality, the target nuclei are not at rest (as assumed, for simplicity, in
Section 2.6) and as v is the relative velocity of the interacting particles / and J ,mg
is their reduced mass [mg,,mg.;/(?Mg.,- + ffig,/)|. The cross-sectional term ę in the
product presents a more difficult problem: in order to induce a nuclear reaction,
nuclei have to come within a distance comparable to the range of the strong force.
Since they are positively charged, to do so they must overcome the Coulomb
repulsive force, which tends to separate them. This force imposes an effective
barrier at a separation distance d, where the kinetic energy of the particles equals
the electric potential energy,

47Tf() |fflgV2

For average stellar temperatures (as derived in Section 2.4), the thermal velocities
are such that the Coulomb barrier is set at a distance which is almost three orders
 4.2 Nuclear reaction rates 55

Figure 4.2 Schematic representation of the Coulomb barrier - the repulsive potential
encountered by nucleus in motion relative to another - and the short-range negative
potential well that is due to the nuclear force. The height of the barrier and the depth of
the well depend on the nuclear charge (atomic number).

of magnitude larger than the typical range of the strong nuclear force! This
is illustrated schematically in Figure 4.2. In other words, the kinetic (thermal)
energy of the gas in stellar interiors is of the order of keV. while the height of the
Coulomb barrier at nuclear distances is of the order of MeV.
We can now understand why, during the first quarter of the twentieth century,
it was thought impossible for such interactions to lake place in stars: simply, stars
appeared not to be sufficiently hot. The solution to this puzzle was provided by
quantum mechanics. A rigorous explanation is beyond the scope of this text; suf­
fice it to say that, according to quantum mechanics, there is a finite (nonvanishing)
probability for a particle to penetrate the Coulomb barrier, as if a ‘tunnel’ existed
to carry it through. This quantum effect, discovered by George Gamow in 1928
in connection with radioactivity, is indeed called ‘tunnelling’. It was applied to
energy generation in stars by Robert Atkinson and Fritz Houtermans in 1929,
soon after its discovery.
The penetration probability, as calculated by Gamow, and with it the nuclear
cross-section, is proportional to exp(— rr Z-,Zje2/e^hv), thus increasing with u. In
conclusion, the product cxp(-:rZ/Z;<r/£o/!v)cxp(-mgL'2/2F/'), where the first
56 4 Nuclear processes that take place in stars

exponent increases and the second decreases with increasing v. has a maximum
known as the Gamow peak. To calculate the reaction rate, we would have to
integrate the product over all velocity values. It can be shown that the value of
the integral, and with it the reaction rate, is proportional to the maximum of the
product, which occurs for

v = {yr ZiZje2kT. (4.6)

Hence the reaction rate

3 /7iZ,Zje2\-' /tns\}
ęv oc (kT{ 2 ■’exp
2 \ eQh J

increases with increasing temperature and decreases with increasing charges of the
interacting particles. Fusion of heavier and heavier nuclei would therefore require
higher and higher temperatures. Reactions of a special type, called resonant
reactions, interfere with this monotonic trend. They occur when the energy of
the interacting particles corresponds to an energy level of the compound nucleus
(/ + J), which is formed for a very brief period of time, before decaying into
the reaction products K and L. In this case the reaction cross-section has a very
sharp peak at the resonant energy, several orders of magnitude higher than the
cross-sections at neighbouring energies.
The typical timescale of a nuclear reaction is inversely proportional to the
reaction rate. For example, the characteristic time of destruction of type / nuclei
by collisions with type J nuclei, leading to reactions of the form (2.48), would
be given by

r,- = (njRijky '■

The extremely high sensitivity of nuclear reaction rates to temperature leads

to the concept of ‘ignition’ of a nuclear fuel: each reaction (or nuclear process)
has a typical narrow temperature range over which its rate increases by orders
of magnitude, from negligible values to very significant ones. Around this range,
the temperature dependence of the reaction rate may be well approximated by a
power law (with a high power) and an ignition or threshold temperature may be
defined. Hence, by Equation (4.4), we should characteristically have q oc pT", or

c/ = qopTn. (4.7)

The process of creation of new nuclear species by fusion reactions is called

nucleosynthesis. And since the kinetic energy of particles is that of their thermal
motion, the reactions between them are called thermonuclear, as mentioned in
Section 2.6. The simple chain of arguments presented here may be misleading;
nuclear reaction rates involve quite complicated calculations, taking into account
the particular structure (energy states) of the interacting nuclei. A detailed account
 4.3 Hydrogen burning I: the p - p chain 57

ofthe physics of nuclear reactions may be found in Donald Clayton’s classic book.
Principles of Stellar Evolution and Nucleosynthesis, first published in 1968.

4.3 Hydrogen burning I: the p - p chain

The most abundant element in newly born stars is hydrogen, with Z — 1.

Fusion of hydrogen into the next element, helium, with Z = 2. would require
an encounter of three or four protons - hydrogen nuclei - within a distance of
the order of fermis. The probability of such a multiple encounter is vanishingly
small. Thus the process by which hydrogen is, eventually, turned into helium does
not happen at once but gradually, through a chain of reactions, each involving the
close encounter of only two particles. The first link of this chain should obviously
be fusion of two protons (by the nuclear force - the strong interaction). The result­
ing particle would be. however, unstable and it would immediately disintegrate
back into two separate protons. The way out of this impasse was found by Hans
Bethe in 1939: during the close encounter of two protons, the weak interaction
may convert one proton into a neutron, thus forming a heavier, stable isotope of
hydrogen, deuterium:

p+p 2D + e ' + v.

We note that all three conservation laws are obeyed - baryon number, lepton
number and charge. Then deuterium captures a proton to form the lighter helium
isotope. 'He:

2D + p -> 3He + y,

where y indicates the emission of an energetic photon, which will soon be

absorbed, and whose energy will be shared by neighbouring particles. The chain
now ramifies: one branch following the encounter of two 'He isotopes, and the
other, the encounter of a 'He isotope with a 4He one;

?He + 3He -> 4He + 2p.

or

3He + 4He 7Be + y.

The first branch marks the end of a chain - called the p - pl chain - that
turns six protons into a 4He nucleus (also known as an a particle), returning two
protons, as illustrated in Figure 4.3. The second branch ramifies again, defining the
p - p 11 and the p - p 111 chains shown in Figure 4.3. The p - p II chain proceeds
with the capture of an electron by the beryllium nucleus, accompanied by the
58 4 Nuclear processes that take place in stars

Hydrogen burning

Figure 4.3 The nuclear reactions of the p - p 1, II and 111 chains.

emission of a neutrino:

'Be + e~ -+ 7Li + v,

and the subsequent capture of another proton, to form two 4He nuclei:

Li + p —> 2 4He.

The p - pill chain results from the capture by 'Be of a proton, instead of an
electron:

7Be + p —> 8B + y.

The radioactive boron isotope SB decays into 8Bc, which is highly unstable and
immediately breaks into two 4He nuclei:

8B 8Be + e+ +v

8Be -> 2 4He.

This completes the p - p chain, whose three branches operate simultaneously.

The relative importance of these chains, that is, the branching ratios, depend
upon the conditions of hydrogen burning: temperature, density and abundances
 4.4 Hydrogen burning II: the CNO bi-cycle 59

of the elements involved. For example, for X = Y. the transition from p - p 1 to

p - p II occurs gradually between temperatures of 1.3 x IO7 K and 2 x 107 K;
above 3 x IO7 K the p - p III chain dominates. However, at such high temper­
atures, a different hydrogen-burning process may favourably compete with the
p - p chains, as we shall see shortly.
The energy released in the formation of an a particle by fusion of four protons
is essentially given by the difference of the mass excesses of four protons and one
a particle,

Qp_p = 4A,M('H) - A.M(4He) = 26.73 MeV,

according to the atomic mass table. Since any reaction chain that accomplishes
this task must also turn two protons into neutrons, two neutrinos are emitted,
which carry energy away from the reaction site. (In fact, it is these neutrinos that
bear direct testimony to the occurrence of nuclear reactions in the interiors of
stars, which would be otherwise unobservable. We shall return to this point in
Section 9.3, when we discuss solar neutrinos.) The amounts of energy carried
by the neutrinos vary for the different reaction chains: from 0.26 MeV for the
creation of deuterium, to 7.2 MeV for the boron decay. Since the p - p III chain,
which includes the boron decay, has a small probability (branching ratio), 26 MeV
are liberated on the average for each helium nucleus assembled, which, translated
into energy per unit mass, yields 6 x 1014 J kg 1 (6 x 10ls erg g-1).
Finally, the rate of energy release is determined by the slowest reaction in the
chain, which is the first one. with a typical timescale of almost IO10 yr. Il may be
approximated by a power law in temperature with an exponent ranging from less
than 4 and up to ~6. Roughly, we may assume

qp_pC<.pT\ (4.8)

Not only does the p - p chain require the lowest temperature among fusion
processes, but it also exhibits the weakest temperature sensitivity.

4.4 Hydrogen burning II: the CNO bi-cycle

We have seen in Chapter 1 that a small percentage of the initial composition of

any star consists of carbon, nitrogen and oxygen (CNO) nuclei. These nuclei may
induce a chain of reactions that transform hydrogen into helium, in which they
themselves act similarly to catalysts in chemical reactions: they are destroyed
and reformed in a cyclic process. The process, which is accordingly named
the CNO cycle, was suggested by Bethe and, independently, by Carl-Friedrich
von Weizsacker, in 1938. The reactions involved are shown schematically in
Figure 4.4. We note that here, too, as in the case of the p - p chain, the process
may ramify, with the two different branches forming a bi-cycle. Each of the
60 4 Nuclear processes that take place in stars

Hydrogen burning

Figure 4.4 The nuclear reactions of the CNO bi-cycle.

two closed chains that form the CNO bi-cycle involves six reactions resulting
in the production of one 4Hc nucleus: four proton captures and two fl ' decays
accompanied by the emission of neutrinos per chain. They are listed below, in
parallel.

I2C + ’ll -> l3N + y i4N + 'H l5O + y

13N -> ,3C + e+ + v l5O -» l5N + e+ + v

l3C + 'H -> l4N + y l5N + 'H-> l6O + y

i4N + 1H -> l5O + y l6O + 'H -+ l7F + y

l5O -> i5N + e+ + v l7F -> l7O + e+ + v

15N + 'H l2C

13+ 4He l7O + 'H 14N + 4He

We note that the number (total abundance) of CNO (and F) nuclei taking
part in the process is constant in time: the relative abundances of the species
depend upon the conditions of burning, mainly the prevailing temperature. The
burning rate - as in any chain of reactions - is determined by the slowest reaction
 4.5 Helium burning: the triple-a reaction 61

in the chain. In this context, it is important to note that, while fl decays are
independent of external conditions, capture reactions arc extremely sensitive to
temperature. Hence a very wide range of burning rates is to be expected, but only
so long as capture reactions proceed more slowly than decays. At the extremely
high temperatures for which the situation is reversed, fl decays would act as a
bottleneck to the nuclear reaction sequence, regardless of temperature. This may
occur in explosive hydrogen burning (sec Section 11.6).
The energy released in the formation of a 4He nucleus by the CNO cycle is
~25 MeV, after subtracting the energy carried away by the neutrinos. The temper­
ature dependence of the energy generation rate q may be roughly approximated
by a steep power law

r/CNoapT16. (4.9)

Thus, both processes of hydrogen burning - the main source of stellar

energy - were brought to light at about the same time, and Bcthc played a crucial
role in both. Many years later, in 1967. he was awarded the Nobel Prize for
Physics for his contribution to the understanding of energy production in stars.

4.5 Helium burning: the triple-cr reaction

As in the case of hydrogen burning, the simplest and most obvious nuclear
reaction in a helium gas should be fusion of two helium nuclei (a particles).
But we have seen in Section 4.1 that there exists no stable nuclear configuration
with >1 = 8 (regardless of Z). Two helium nuclei may be fused into a beryllium
isotope

4He+ 4He 8Be,

but the 8Bc lifetime is only 2.6 x 10-16 s! The solution to this new problem was
provided by Edwin Salpeter in 1952. Short as the 8Be lifetime may seem, it is
nevertheless longer than the mean collision (scattering) time of a particles at
temperatures of the order of 108 K. Therefore, even at the seemingly negligi­
ble sBc abundance of one in IO9 particles, there is a nonvanishing probability
that an a particle will collide with a 8Be nucleus before it decays, to produce
carbon:

8Be+ 4He -> l2C.

Fred Hoyle realized shortly afterwards that the small probability of an a capture
by a beryllium nucleus would be greatly enhanced if the carbon nucleus had an
energy level close to the combined energies of the reacting 8Be and 4He nuclei.
The reaction would then be a relatively fast resonant reaction. Remarkably, such a
resonant energy level of l2C (at 7.65 MeV) was subsequently found experimentally
in the Kellogg Radiation Laboratory at the California Institute of Technology. The
bl 4 Nuclear processes that take place in stars

excited l2C nucleus decays back into three a particles with high probability, but
with a nonnegligiblc probability it decays to its ground state, emitting an energetic
photon (y-decay).
Thus helium burning proceeds in a two-stage reaction that leads to the fusion
of three helium nuclei into l2C; hence the name of this reaction: triple-a (or ,3a).
The energy released in such a reaction is easily calculated:

Q3a = 3A,M(4He) - A,M(I2C) = 7.275 MeV.

and, correspondingly, the energy generated per unit mass is 5.8 x 1013 J kg-1
(5.8 x 1017 erg g-1). This is about one tenth of the energy generated by fusion
of hydrogen into helium! The rate of this process is determined by the second
reaction in the chain (which itself has two stages: a-capture and /-decay). It
is thus proportional to the xBc abundance, which itself varies as the square of
the helium abundance. Consequently, the energy generation rate depends on the
square of the density. Its temperature sensitivity is quite astounding:

ocp2T40. (4.10)

When a sufficient number of carbon nuclei have accumulated by 3a reactions, it

seems reasonable that a captures by these nuclei, and possibly by their products,
could lead to the formation of heavier and heavier particles. In reality, it turns out
that the increasing Coulomb barrier renders the probability of such captures very
low compared with that of the 3a reaction, at least until the helium abundance
becomes small. Hence the only significant a capture reaction that takes place is

l2C+ 4He -> l6O.

The energy released by this reaction is 7.162 MeV, amounting to 4.3 x 1013 J kg-1.
To summarize, the products of helium burning are carbon and oxygen, in relative
abundances which depend on temperature. The process is shown schematically
in Figure 4.5.

Note: It was the competition between the 12C + 4He and the 8Be + 4He reactions that
led Hoyle to the prediction of the resonant energy level in the carbon nucleus. Already in
1946, with remarkable foresight, Hoyle had postulated that all nuclei (not only helium)
build up from lighter nuclei by fusion reactions that take place in the interior of stars.
Pursuing this idea, he considered the synthesis of elements from carbon to nickel, in
1953-1954, with the Salpeter process as starting point. He then showed that the observed
cosmic abundance ratios He: C: O could be made to fit the yields calculated for the
above reactions, if the 8Be +4He reaction had a resonance corresponding to a level at
~7.7 MeV in the l2C nucleus. Otherwise, the inferred cosmic carbon abundance would
be too low. Eager to test this prediction. Hoyle even collaborated in the first attempts to
detect such a level experimentally.
 4.6 Carbon and oxygen burning 63

Helium burning

T=108K —$ n(8Be):n(4He) = 1:109

p= 105gcm"3

Figure 4.5 The triple-a process.

Exercise 4.1: Calculate the energy generated per unit mass, if helium burning
produces equal amounts (mass fractions) of carbon and oxygen.

4.6 Carbon and oxygen burning

Carbon burning - fusion of two carbon nuclei - requires temperatures above

5 x IO8 K and oxygen burning, having to overcome a still higherCoulomb barrier,
occurs only at temperatures in excess of IO9 K. Interactions of carbon and oxygen
nuclei need not be considered, for at the intermediate temperature required by the
intermediate Coulomb barrier, carbon nuclei are quickly exhausted by interacting
with themselves.
The processes of carbon and of oxygen burning are very similar: in both cases
a compound nucleus is produced, at an excited energy level, and it subsequently
decays. Several decay options are open, with different, temperature-dependent
64 4 Nuclear processes that take place in stars

Carbon burning

Figure 4.6 The nuclear reactions involved in carbon and in oxygen burning.

probabilities (branching ratios).

I2C + ,2C -» 24Mg +y l6O+l6O 32S + y

—> 23Mg + n 3lS + n

-> 23Na + p ->3lP + p

20Ne + a —> 28Si + a

16O + 2a 24Mg + 2a

The possible reaction channels are shown schematically in Figure 4.6.

On the average, 13 MeV are released for each l2C + l2C reaction and
about 16 MeV for each lftO + l6O reaction, amounting to ~5.2 x IO13*16J kg 1 and
~4.8 x 1013 Jkg-1, respectively. These reactions entail production of light
particles, such as protons and helium nuclei, which are immediately captured
by the heavy nuclei present, because of the relatively low Coulomb barriers. Thus
 4.7 Silicon burning: nuclear statistical equilibrium 65

Table 4.1 Major nuclear-burning processes

7 threshold Energy per

Nuclearfuel Process (IO6 K) Products nucleon (MeV)

H P~P ~4 He 6.55
H CNO 15 He 6.25
He 3<z too C.O 0.61
C C+C 600 O. Ne, Na. Mg 0.54
0 0+0 1000 Mg, S, P. Si ~0.3
Si Nuc. eq. 3000 Co, Fe. Ni <0.18

many different isotopes are created by secondary reactions, besides those primar­
ily produced by fusion of carbon or oxygen. The major nucleus formed by oxygen
burning is silicon (28Si). although other elements are also significantly abundant.

4.7 Silicon burning: nuclear statistical equilibrium

In principle, we may now assume by analogy that two silicon nuclei could fuse
to create iron, the most stable element - the end-product of the nuclear fusion
chain. In reality, however, the Coulomb barrier has become prohibitively large. At
temperatures above the oxygen burning range, but way below those that would be
required for silicon fusion, another type of nuclear process takes place. It involves
the interaction of massive particles with energetic photons, which are capable
of disintegrating nuclei, much as less energetic photons are capable of breaking
up atoms by tearing electrons away. The process, called photodisintegration, is
similar in many respects to photoionization of atoms, except that the binding force
is nuclear, instead of electric, and the emitted particles are light nuclei, instead of
electrons. As in the case of ionization, reactions can proceed both ways and equi­
librium may be achieved, with relative abundances depending on the prevailing
physical conditions. The reaction l6O + a 20Ne + y, for example, produces
neon at temperatures around 109 K. but reverses direction above 1.5 x 109 K. The
energy absorbed in the inverse reaction (photodisintegration) is supplied by the
radiation field.
Silicon disintegration occurs around 3 x 109K; the light particles emitted
are recaptured by other silicon nuclei, building up an entire network of nuclear
reactions, with light particles exchanged between heavy nuclei. Although the
nuclear reactions tend to equilibrium, where direct and inverse reactions occur at
(almost) the same rate, the resulting slate of nuclear statistical equilibrium is not
perfect: a leakage occurs toward the stable iron group nuclei (Fe, Co, Ni), which
resist photodisintegration until the temperature reaches ~7 x IO9 K.
The major nuclear-burning processes that we have encountered and their
main characteristics are summarized in Table 4.1. Their common feature is the
66 4 Nuclear processes that take place in stars

release of energy upon consumption of nuclear fuel. The amounts and the rates
of energy release vary, however, enormously. But nuclear processes that absorb
energy (from the radiation field) are also possible under conditions expected to
occur in stellar interiors. Their consequences may range from mild to catastrophic,
depending on the amount of absorbed energy and, especially, on the rate of energy
absorption. Such are the processes discussed in the following sections.

Exercise 4.2: Estimate the minimal stellar mass required for the central ignition of
the different nuclear fuels, according to the threshold temperatures of Table 4.1,
by assuming (a) a density profile as in Exercise 1.2; (b) solar composition; (c)
nondegeneracy.

4.8 Creation of heavy elements: the s- and / -processes

So far we have considered charged particle interactions, their rates being con­
trolled by the height of the Coulomb barrier and interactions of nuclei with
photons, which become efficient at high temperatures. Another type of interac­
tion becomes possible in the presence of free neutrons, which are produced during
carbon, oxygen and silicon burning. Neutron capture by relatively heavy nuclei
is not limited by the Coulomb barrier and can therefore proceed at relatively low
temperatures. The only obstacle in the way of neutron-capture reactions is the
scarcity of free neutrons.
Suppose a sufficient number density of neutrons is available. A chain of
reactions would then be triggered, with nuclei capturing more and more neutrons,
thus creating heavier and heavier isotopes of the same element:

/(A, 2) + // /|G4+ 1,2),

I\ (>4 + 1.2) + n —> Tl-d + 2. 2).

Af-d + 2, 2) + n —> fs(-d -I- 3, 2), etc.

So long as /,v is stable, the chain of neutron captures may continue, but eventually
a radioactive isotope should be formed. Such an isotope would subsequently
decay by emitting an electron (and an antineutrino), thus creating a new element

7,v(A + N, 2) -+ J (A + N. 2 + I) + e~ + v.

If the new element is stable, it will resume the chain of neutron captures. Other­
wise, it may undergo a series of fi~ decays:

J(A + N. 2 + 1) K(A + N,2 + 2) + e~ + v,

K(A + N, 2 + 2) —> L(A + N, 2 + 3) + e + v, etc.

 4.9 Pair production 67

until a stable nucleus of mass A + A' and atomic number Z + M, say, is produced.
Either way, increasingly heavier elements and their stable isotopes are thereby
created.
In the process just described two types of reactions - neutron captures and
/3~ decays - and two types of nuclei - stable and unstable - are involved. Stable
nuclei may, of course, undergo only neutron captures: for unstable ones both
tracks are open and the outcome depends on the timescales of the two processes.
The timescales of /J- decays (or half-life times of ^-unstable isotopes) are con­
stants - independent of prevailing physical conditions. Those of neutron captures
may change according to temperature and density. Hence neutron-capture reac­
tions may proceed more slowly or more rapidly than the competing fl decays.
The resulting chains of reactions and products will be different in the former
case, called the s-process and in the latter, called the r-process (terms coined by
Margaret and Geoffrey Burbidge, William Fowler and Fred Hoyle in their seminal
paper of 1957). This is illustrated schematically in Figure 4.7, where the s-process
products are labelled .v, those of the r-process are labelled r, and those which may
be produced by both are labelled s,r.
In the course of the main burning processes, the a- and r-processes operate
as secondary reactions and a wealth of nuclear species results, although the
abundances of elements heavier than iron are relatively small.

4.9 Pair production

We have already seen in this chapter many examples of transmutations of mass

into ‘light’: all the major burning stages release energy at the expense of a small
fraction (less than 1%) of the mass. But the reverse transmutation - of light into
mass - is also possible.
During the interaction with a nucleus, a photon may turn into an electron­
positron particle pair, provided its energy hv exceeds the rest-mass energy of the
particles, hv > 2mec2. The presence of the nucleus is required for the simultane­
ous conservation of momentum and energy. The typical temperature at which the
condition for pair production is satisfied may be estimated by kT hv 2mcc2,
yielding 7' ~ 1.2 x 10l0K. However, even at temperatures T > 109K, a large
number of photons - at the tail of the Planck distribution function (3.39) - are
already sufficiently energetic to produce electron-positron pairs. At the same time,
the inverse reaction - annihilation of electron-positron pairs into photon pairs -
tends to destroy the newly created positrons. As a result, the number of positrons
reaches equilibrium. Pair production, as photodisintegration, bears similarity to
the ionization process: an increase in temperature leads to an increase in the num­
ber of particles, at the expense of the photon energy; an increase in density has the
opposite effect. Thus, at a few times 109 K (depending on the electron density)
68 4 Nuclear processes that take place in stars

Creation of heavy elements

s-process /■■process

r)
rapid

Figure 4.7 Schematic representation of the s-proccss and the r-process, showing reac­
tion chains that involve neutron captures and f decays, leading to the formation of
stable isotopes. Nuclei marked v, r or s, r are formed by one of the processes (respec­
tively), or by both (adapted from D. Clayton (1983), Principles of Stellar Evolution and
Nucleosynthesis, University of Chicago Press).

the number of positrons becomes a significant fraction of the number of electrons.

We note that having a lot of pairs at a temperature of a few IO9 K (much less than
12 x 109 K) is similar to having a considerable fraction of, say, ionized hydrogen
at a few 104 K (much less than ~15 x 104 K. corresponding to / = 13.6eV).

4.10 Iron photodisintegration

If sufficiently high temperatures are achieved, even the stable iron nuclei do not
survive photodisintegration. They break into a particles and neutrons.

56Fe -> 134He + 4n,

 4.10 Iron photodisintegration 69

thus reversing almost entirely the nucleosynthesis process. Each reaction of this
kind absorbs about 124 MeV of energy.
Al temperatures above ~7 x 1O9 K. helium becomes more abundant than iron.
Al still higher temperatures, helium itself is disintegrated by the energetic photons
into protons and neutrons. In conclusion, bound nuclei require temperatures above
a few 106 K in order to be created and below a few 1O9 K so as not to be destroyed.
This, as we shall see, is precisely the range of temperatures characteristic of stellar
interiors.
 5

Equilibrium stellar
configurations - simple models

5.1 The stellar structure equations

The main conclusion of Chapter 2 was that the evolution of a star may be perceived
as a quasi-static process, in which the composition changes slowly, allowing the
star to maintain hydrostatic equilibrium and. generally, thermal equilibrium as
well. The chain of processes through which the composition gradually changes
has been described in the previous chapter. Our present task is to describe the
equilibrium structure of a star of a given composition (this chapter) and to find
whether the equilibrium is stable (next chapter). The (static) structure of a star is
obtained from the solution of the set of differential equations known as the stellar
structure equations, formulated in terms of either of the previously encountered
space variables, r or m:

dP Gm dP Gm
---- — ~P T" — = <5J)
dr----------- r2 dm 4rrr4
dm dr 1
— = 4,T/-p ~r = <5-2>
dr dm 4rtrLp
dT _ 3 tep F dT 3 k F
— =------------------------ (5.3)
dr 4ac T* 4nr2 dm 4ac T2 (4rrr2)2
dF , dF
— = 4nr-pq ~T = q (5.4)
dr dm

where the first is the hydrostatic-equilibrium equation, the second is the con­
tinuity equation, the third is the radiative-transfer equation (provided radiative
diffusion constitutes the only means of energy transfer) and the fourth is the

70
 5.2 What is a simple stellar model? 71

thermal-equilibrium equation, supplemented by the relations

—pT + Pe + \aT* (5.5)

F\
K =KOp“Tb (5.6)

q = qOpmT". (5.7)

Integration of these differential equations provides the profiles of four functions

throughout the star: T, p, in (or r) and /•', from which any other function of
interest may be derived. Four boundary conditions (integration constants) have to
be supplied. The three straightforward ones are at m — 0, r = 0 and F = 0; at
m = M (or at r = /?), P = 0. A more complicated condition relates the emitted
radiation L — F(R)~ or, equivalently, the effective temperature - to the temper­
ature obtained at some depth below the surface. Although this set of equations is
simpler by far than the set of evolution equations derived in Chapter 2, it does not
lend itself to simple, analytic solutions. The reason is threefold: first, the equa­
tions are highly nonlinear, particularly in view of the power-law relations (5.5) to
(5.7); secondly, they are coupled and have to be solved simultaneously; thirdly,
they constitute a two-point boundary value problem, which requires iterations
for its solution. And yet, a great deal of our understanding of stellar structure
dates back to the early decades of the twentieth century, when fast computers
were not only unavailable, but inconceivable. Eddington’s book, which we have
already mentioned, saw light in 1926; Subramanyan Chandrasekhar’s book An
Introduction to the Study of Stellar Structure, a cornerstone in the study of stars,
was first published in 1939.

Exercise 5.1: Derive the behaviour of m(r), P(r), F(r) and T(r) near the centre
of a star by Taylor expansion for given composition and physical properties at
r = 0: pc, Pc and Tc.

In what follows we shall see that insight into the structure of stars may be
gained both by analysing the equations, without actually solving them, and by
seeking simple solutions based on additional simplifying assumptions.

5.2 What is a simple stellar model?

A fundamental principle that enables a simple solution of the structure equations

is finding a property that changes moderately enough from the stellar centre to
the surface to allow us to regard it as uniform (independent of r or m). At first
sight, this demand appears rather strange, keeping in mind that the temperature,
72 5 Equilibrium - simple models

for example, is expected to change throughout a star by more than three orders of
magnitude (according to simple estimates) and the pressure by more than eleven!
However, properties can be found that do not change significantly with radial
distance. Many models, for example, assume the composition to be uniform. Is
such an assumption justified? It would be for a star which is thoroughly mixed
by convection (a process that we shall address shortly), or for a star composed
mainly of elements heavier than hydrogen, where the gas pressure is dominated
by electrons and hence depends on /xe, which is very nearly 2 regardless of the
detailed abundances. A homogeneous composition is also typical of young stars,
since the initial stellar composition is uniform.
Another principle that enables an analytic investigation of the behaviour of
stars is the representation of a star by its two extreme points - the centre and the
surface (the surface is, of course, not a point in the strict sense of the word, but all
points on the surface are identical by the spherical symmetry assumption). The
hidden implication is that properties change monotonically between these two
points. This is certainly correct for the pressure, from Equation (5.1), and also for
the temperature, by Equation (5.3), since from Equation (5.4), F > 0. The latter
condition is not necessarily correct in the case of strong neutrino emission, which
may turn the net q negative and may eventually lead to a temperature inversion.
But we shall disregard such complications.
As a further simplification, we may represent a star by only one ofthe extreme
points; the centre, for example. Assuming that both P and T decrease outward
(and so must p. otherwise we would encounter the unstable situation in which
heavy material lies on top of light material, resulting in a turnover), the centre
of a star is the hottest and densest place. There, therefore, the nuclear reactions
are fastest and since nuclear processes dictate the evolutionary pace, the centre
would be the most evolved part ofthe star. We should be able to learn a great deal
about the evolution of a star by considering its central point alone. This will be
the subject of Chapter 7. The surface ofthe star (the global stellar characteristics)
is important from an entirely different point of view - it is the only ‘point’ whose
model-derived properties can be directly compared with observations. In some
cases, global quantities and relations between them may be obtained, as we shall
see in Chapter 7, without solving the set of structure equations.
For now, we shall consider several simple models based on the principle of a
uniform property.

5.3 Polytropic models

The first pair of stellar structure equations, (5.1)—(5.2), is linked to the second
pair, (5.3)—(5.4), by the dependence of pressure on temperature. If the pressure
were only a function of density (and composition, of course), the first pair would
 5.3 Polytropic models 73

be independent and eould be solved separately, meaning that the hydrostatic

configuration would be independent of the flow of heat through it. Analytic
solutions of this form are now more than a century old.
Multiplying Equation (5.1) by r2/p and differentiating with respect to r we
have

d rj_cLP dm
= —G—. (5.8)
Tr p dr dr

Substituting Equation (5.2) on the right-hand side, we obtain

1 d fr-dP
= —47rGp. (5.9)
r2 dr \ p dr

We now-' consider equations of state of the form

P = KpY, (5.10)

where K and y are constants, known as polytropic equations of state. It is cus­

tomary to define the corresponding polytropic index, denoted by n, as

(5.11)

Thus the equation of state of a completely degenerate electron gas is polytropic,

with an index of 1.5 (y = 5/3) in the nonrelativistic case and 3 (y = 4/3) in the
extreme relativistic limit. An ideal gas, too, may be described by a polytropic
equation of state under certain conditions; we shall encounter such cases later
on. Substituting Equations (5.10)—(5.11) into (5.9), we obtain a second-order
differential equation:

(n + \)K I d r2 dp
= -p- (5.12)
4xGn r2dr ^Th-

The solution p(r) for 0 < r < R, called a polytrope, requires two boundary con­
ditions. These arc p = 0 at the surface (r = R), which follows from P(R) = 0,
and dp/dr — 0 at the centre (r = 0), since hydrostatic equilibrium implies
dP/dr — 0 there (see Section 2.3). Hence a polytrope is uniquely defined by
three parameters: K. n and R, and it enables the calculation of additional quan­
tities as functions of radius, such as the pressure, the mass or the gravitational
acceleration.
It is convenient to define a dimensionless variable 0 in the range 0 < 0 < I
by

P = PcO". (5.13)
74 5 Equilibrium - simple models

Figure 5.1 Normalized polytropes for n = 1.5 and n = 3.

to obtain Equation (5.12) in a simpler form.

’(« + I )K I 1 d / 2d0\
------------ -- r-- =-0". (5.14)
L4ttG7V J '-(lr V ^7

Obviously, the coefficient in square brackets on the left-hand side of Equation

(5.14) is a constant having the dimension of length squared,

(n+l)/cl ,
---- — =a-- (5.15)
_47rGpc" _
which can be used in order to replace r by a dimensionless variable £,

r — a£. (5.16)

Substituting Equation (5.16) into Equation (5.14), we now obtain the well-known
Lane-Emden equation of index n.

(5.17)
d$ V /
subject to the boundary conditions: 0 — 1 and dd/d^ = 0 at £ = 0. Equation
(5.17) can be integrated starting at $ = ();forn < 5, the solutions (?(£) are found to
decrease monotonically and have a zero at a finite value £ = £j, which corresponds
to the stellar radius,

(5.18)

Examples of solutions (p/pc as a function of r/R), for n = 1.5 and n = 3, are

given in Figure 5.1. As shown, the structure of a polytrope depends only on n. A
polytrope of index 3 describes a star in which the mass is strongly concentrated
 5.3 Polytropic models 75

Table 5.1 Polytropic constants

1.0 3.290 3.14 3.14 0.233

1.5 5.991 2.71 3.65 0.206
2.0 11.40 2.41 4.35 0.185
2.5 23.41 2.19 5.36 0.170
3.0 54.18 2.02 6.90 0.157
3.5 152.9 1.89 9.54 0.145

at the centre, whereas a polytrope of index 1.5 describes a more even mass
distribution.
The total mass M of a polytropic star is given by

CR , f£|
M = I 4ytr2pdr = 4na3pc I %20"d%. (5.19)
Jo Jo

From Equation (5.17) we have

, f*1 d / ,de\ ,
M = -4naypc — I £2 — ) d% = -4Tra2p^ I — I . (5.20)
Jo d$ \ d£J

Exercise 5.2: Solve the Lane-Emden equation analytically for (a) n = 0 and
(b) n = 1 and find |i and M(R) in each case.

In later discussions we shall often resort to general relations between stellar

properties resulting from a polytropic equation of state. These follow easily from
Equation (5.20). Eliminating ct between Equations (5.18) and (5.20), we obtain a
linear relation between the central density and the average density p,

M
Pc — p — D„ , (5.21)

which is generally valid. Only the constant D„ derives from the solution of
Equation (5.17) and depends on the value of n;

3 /de\
Dn = (5.22)

Values of D„ for various n can be found in Table 5.1.

Using Equation (5.20) again, but now eliminating pc with the aid of Equation
(5.15) and substituting a from Equation (5.18), we obtain a relation between the
stellar mass and radius, which may be expressed in terms of two constants. Mn
76 5 Equilibrium - simple models

and R„, in the form

[(/?+ l)/fj"
(5.23)
4,tG

The values of the constants M„ = and R„ = vary with the

polytropic index n in the range from 1 to 10. as listed in Table 5.1. We note that
n = 3 is a special case: the mass becomes independent of radius and is uniquely
determined by K,

M = 4.T/W3 (5.24)

Thus for a given K, there is only one possible value for the mass of a star that
will satisfy hydrostatic equilibrium. Another special case is n — I, for which the
radius is independent of mass and is uniquely determined by R:

R = R} (5.25)

Between these limiting values of n, 1 < 11 < 3. we have from Equation (5.23)

R-~" oc (5.26)
M"-''
meaning that the radius decreases with increasing mass: the more massive the
star, the smaller (and hence denser) it becomes.
A final important relation is obtained between the central pressure and
the central density by substituting K from the mass-radius relation (5.23) in
i+-
Eqnation (5.10). Pc = Kpc ", whence

(4;rG)« /GM\^ / R 2±i

7j4-i \ Pc (5.27)

Eliminating R between Equations (5.27) and (5.21), and assembling all n-

dependent coefficients into one constant Bn, reduces Equation (5.27) to

Pc = (47r)1/35,,GM2/3pc4/3. (5.28)

The remarkable property of this relation is that it depends on the polytropic

equation of state only through the value of B„, which, as we see from Table 5.1.
varies very slowly with n. Il therefore constitutes an almost universal relation,
and as such it will be used in Chapter 7. Note that expression (5.28) for is
consistent with the upper limit derived in Exercise 2.2 (Section 2.3).

Exercise 5.3: For a given mass M and central pressure Pc, which poly trope yields
a bigger star: that of index 1.5 or that of index 3?
 5.4 The Chandrasekhar mass 77

Exercise 5.4: Capella is a binary star discovered in 1899. with a known orbital
period, which enables the determination of the mass and radius of the brighter
component: M = 8.3 x 1030kg and R — 9.55 x 109m. Assuming that the star
can be described by a polytrope of index 3, find the central pressure and the
central density. Check whether the central pressure satisfies inequality (2.18).

5.4 The Chandrasekhar mass

Stars that arc so dense as to be dominated by the degeneracy pressure of the

electrons (discussed in Chapter 3) would be accurately described by a polytrope
of index n — 1.5, with K = K\ of Equation (3.35). We know from observations
that such compact stars exist - they are the white dwarfs mentioned in Chapter 1,
which have masses comparable to the Sun's, and radii not much larger than the
Earth's. Their average density is thus higher than l()8 kg m-3 (IO3 gem-3), about
five orders of magnitude higher than the average density of the Sun. We might
learn some more about these stars by investigating the properties of this particular
polytrope. From Equation (5.23), the relation between mass and radius becomes
R oc M~]/3. (5.29)

The density, therefore, increases as the square of the mass,

p ex M R~3 ex M2. (5.30)

Imagine now a series of such degenerate gaseous spheres with higher and higher
masses. The radii will decrease along the series and the density will increase in
proportion to M2. Eventually, the density will become so high that the degenerate
electron gas will turn to be relativistic, departing from the simple n = 1.5 poly­
trope. As the density increases (the radius tending to zero), the correct equation
of state will approach the form (3.38). still a polytrope, but of index n — 3. with
K — Ki. We have seen, however, that in such a case there is only one possible
solution for M. uniquely determined by K. Hence our scries of degenerate gaseous
spheres in hydrostatic equilibrium ends at this limiting mass. The existence of
an upper limit to the mass of degenerate stars was first found by Chandrasekhar
in 1931 and hence the upper limit bears his name. A/Ch- About half a century
later, this work earned Chandrasekhar the 1983 Nobel Prize for Physics, which
he shared with Fowler (for their contributions to the understanding of stellar
evolution).
Substituting in Equation (5.24), we have

he
= ——
~ 4/3
(5.31)
4,t c'”h
78 5 Equilibrium - simple models

Figure 5.2 The mass-radius relation for white dwarfs (/ze = 2).

Inserting the values of constants, we obtain

MCh = 5.83/z;2/W (5.32)

which yields for /2C = 2 a limiting mass of 1,46A/O. The mass-radius relation for
white dwarfs is shown in Figure 5.2 for /zc = 2 (He, C, O, ...). For /ze = 2.15
(Fe), the limiting mass is 1.26A/©. In conclusion, hydrogen-poor compact stars,
where the pressure is supplied predominantly by the degenerate electron gas, can
have masses only up to the critical mass of 1.46M©. Indeed, no white dwarf is
known with a mass exceeding this value.

Exercise 5.5: Calculate the critical mass using relation (5.28) between central
pressure and central density; show that the numerical coefficient in Equation
(5.31) is equivalent to Bj3/2>/L5/32^.

5.5 The Eddington luminosity

So far we have dealt with the first two of the structure equations. We shall now
add the third, thus taking into account the temperature and the radiation pressure.
Substituting the radiation pressure Pra(j = in Equation (5.3) and dividing
Equation (5.3) by Equation (5.1), we obtain
dPnii = kF
(5.33)
dP 4ttcGih
The result of this manipulation will be the derivation of an upper limit for the
stellar luminosity. Since P — Ps&s 4- PrM)- and both Pgas and PrMj decrease outward
 5.5 The Eddington luminosity 79

(provided q > 0). it follows that sgn[<7/Jraci ]=sgn[<7 /Jgas | and we obviously have
dPn<\/dP < 1, implying

kF < 4rr cGm. (5.34)

This inequality may be violated either in the case of a very large heat flux, which
may result from intense nuclear burning, or in the case of a very high opacity,
as encountered at the ionization temperatures of hydrogen or helium. In such
cases Equations (5.1) and (5.3) cannot simultaneously hold, and if we require
hydrostatic equilibrium, then heat transport must be described by a different
equation; that is, it must occur by a means other than radiative diffusion, which
has become inefficient. We know from everyday experience that near a strong
heat source, such as a stove, convective motions develop in the surrounding air,
which carry the heat efficiently and distribute it throughout the room. If the stove
is not very hot. it spreads heat by thermal radiation alone. The same phenomenon
occurs in stars - on appropriately larger scales. Stars transfer energy by radiation
alone under moderate conditions, in which case they are said to be in radiative
equilibrium and inequality (5.34) is satisfied, or by convection, under more severe
conditions, when the rate of heat generation becomes too rapid for radiation to
carry, or when ionization interferes too much with the radiative transfer. It may
also happen that some regions of a star arc in radiative equilibrium and others arc
not; the former are called radiative regions or zones, and the latter, convective
ones.

Exercise 5.6: Find the expression for the gas pressure gradient, assuming radiative
equilibrium, and its relation to inequality (5.34).

Near the centre of a star. Equation (5.4) and F(0) = 0 yield F/m -> qc as m -> 0.
where qc = q(m = 0); hence inequality (5.34) imposes a universal upper limit on
the central energy generation rate that can be accommodated by radiative energy
transfer:
4ttcG
(5.35)
K

The surface layer of a star is always radiative; applying inequality (5.34) for
m = M, we have

4?r cGM
L < (5.36)
K

Violation of this condition then implies violation of hydrostatic equilibrium: mass

motions arise leading to a stellar wind. As pointed out by Eddington, the right­
hand side of inequality (5.36) represents a critical luminosity Tcrit that cannot be
80 5 Equilibrium - simple models

surpassed; it is, therefore, also known as the Eddington luminosity Ecdd^

(5.37)

where the opacity is expressed relative to the electron-scattering opacity kcs, which
is a constant (see Equation (3.64)). To summarize, radiative equilibrium requires

7, < Lndd-

To show the possible implications of this result, we may indulge in some spec­
ulation. If we assume k & /ces to be a reasonable approximation, E^dd becomes
uniquely determined by M. We have seen in Chapter 1 that for a certain type of
stars, those of the main sequence, a correlation exists between the luminosity and
the mass. If the outer layers of such stars are in hydrostatic and radiative equilib­
rium, restriction (5.36) combined with the mass-luminosity relation imposes an
upper limit on the mass of main-sequence stars. We should then expect the main
sequence to have an upper end.

5.6 The standard model

After this brief digression, we proceed to derive the so-called standard model,
which is due to Eddington and is therefore also known as Eddington's model.
We define a function q by

F _ L
(5.38)
m “ ''a?
and insert it into Equation (5.33). which becomes

d Prad P
= ----------- KI). (5.39)
dP---- 4ncGM
At the surface, q = 1, and for stars that burn nuclear fuel mostly in a (small)
central core, thus maintaining an almost constant flux outside the core, q increases
inward, as m decreases. The opacity, on the other hand, usually increases from
the centre outward. If. from the centre outward, the increase in k is approximately
compensated by the decrease in q, we may take their product to be constant.
This is the uniform property of the Eddington model (a controversial assumption,
which has been subject to severe criticism over the years). With

Kq = constant = ks, (5.40)

where ks is the surface opacity, we have by integrating Equation (5.39)

k\L
P<^ = r^P' (5.41)
4:rcGM
 5.6 The standard model 81

since the total pressure and the radiation pressure tend to zero at the surface. Thus
the constancy of K)] implies a constant ratio of radiation pressure to total pressure
throughout the star: in other words, a constant /J (see Equation (3.12)). We also
obtain
4ttcGM
L — ----------- (1 — /?) = ^Edd(l ~ $), (5.42)

meaning that the luminosity approaches the limiting value as the radiation pressure
becomes dominant (/> 0). Assuming the gas pressure to be given by the ideal
gas law. Equation (3.28), we have

Combining the extreme left and extreme right expressions, we get

and the equation of state may be written as

Since K is a constant, we have obtained a polytropic equation of state of index

3. which implies a unique relation between K and M, Equation (5.24) derived in
the previous section. Rearranging terms and inserting the values of constants, we
obtain

M
I — yfj = 0.003 (5.46)

a fourth order equation for known as the Eddington quartic equation,

whose solution is given in Figure 5.3. The quartic equation is valid for a hypothet­
ically wide range of masses. We note, however, that only for a rather restricted
range does differ significantly from unity (pure gas pressure) or from zero (pure
radiation pressure) and this range more or less coincides w ith the range of stellar
masses, as derived from observations. In Eddington ’s own words:

We can imagine a physicist on a cloud-bound planet who has never heard tell
of the stars calculating the ratio of radiation pressure to gas pressure for a series
of globes of gas of various sizes, starting, say, with a globe of mass 10 gm.,
then 100gm., 1000gm., and so on, so that his zrth globe contains 10" gm. [.. .J
Regarded as a tussle between matter and aether (gas pressure and radiation
pressure) the contest is overwhelmingly one-sided except between Nos. 33-35,
where we may expect something interesting to happen.
82 5 Equilibrium - simple models

Figure 5.3 Solution of the Eddington quartic equation.

What ‘happens’ is the stars.

We draw aside the veil of cloud beneath which our physicist has been
working and let him look up at the sky. There he will find a thousand million
globes of gas nearly all of mass between his 33rd and 35th globes - that is to
say, between | and 50 times the sun’s mass.
Sir Arthur S. Eddington: The Internal Constitution of the Stars, 1926

Exercise 5.7: The quartic equation may be written in terms of a mass that
is a combination of natural constants, (a) Find the expression for this mass and
calculate it. (b) Express the Chandrasekhar mass in terms of Mt.

What can one learn about the evolution of stars based on this simple model?

1. For stars of given composition (fixed /j.). fi decreases as M increases,

meaning that radiation pressure becomes particularly important in massive
stars.
2. Inserting Equation (5.42) into (5.46), we obtain

4ttcG M& ..
------------- 0.003m 0 (M, m) (5.47)

This is close to a power-law relation between the luminosity and the

mass, similar to that obtained from observations of main-sequence stars.
In fact, the mass-luminosity relation derived by Eddington was at the
time a theoretical prediction, to be confirmed only later by observations.
Differences in composition, and hence in the value of g, may explain the
observed scatter of points in the (Af. L) relation.
 5.7 The point-source model 83

3. For a given M and changing p - as along the evolutionary course of a star -

fl decreases with increasing /z. Since nuclear reactions cause a gradual
increase in /z, we should expect radiation pressure to play an increasingly
greater role, as a star gets older. With it. by Equation (5.42). the luminosity
should approach its limiting value. Could this mean that a star should lose
(eject) some of its mass in its late stages of evolution, compensating for
the rise in /z, so as to prevent fl from dropping too low? Wc may consider
this a first hint to the existence of stellar winds, which should intensify as
the luminosity approaches Z-Edd-

We have formulated these conclusions very cautiously, for they derive from such
a simple model. So formulated, they are acceptable and they provide important
and easy to understand clues to the complex structure and evolution of stars.

Historical Note: The theoretical mass-luminosity relation (5.47) has two parameters:
the mean molecular weight and the opacity. Assuming one of them, one may derive the
other by comparing the relation with its observational counterpart. Eddington started by
assuming stars to be made of iron (or terrestrial material), which implied a value of /z
slightly in excess of 2, considering the highly ionized state of stellar interiors. This led to
the estimate of an ‘astronomical opacity coefficient’, which exceeded by about a factor
of 10 the ’theoretical opacity coefficient’ that had been calculated following the Kramers
theory. Although he was aware that including a considerable proportion of hydrogen in
the chemical composition of stars would resolve the discrepancy, this solution seemed
improbable at first, both to him and to others, and the alternative of seeking a correction to
the opacity coefficient was pursued for a time. However, around 1930 it became established
that in the atmospheres of the Sun, and the stars in general, hydrogen amounts to about
half the mass (see Section 1.3). The possibility of hydrogen floating to the surface of the
star was discarded; Eddington had already shown that diffusion in stars should proceed
negligibly slowly. Thus, in 1932, the prevalence of hydrogen in stellar interiors was finally
recognized by Eddington, and independently advocated by Bengt Stromgren. on the basis
of the mass-luminosity relation.

5.7 The point-source model

Models presented so far have considered the first three structure equations. This
chapter would not be complete without mentioning another group of relatively
simple models, which take account of the fourth equation. Equation (5.4), and
assume a power law for the opacity in the form (5.6). If the nuclear-energy source
of a star is confined to a very small central region, it may be considered a point­
source, so that <7 = 0 for r > 0. In this case, the equation of thermal equilibrium
84 5 Equilibrium - simple models

implies a constant energy flow (energy per unit time) throughout the star. Thus

F = constant = F(R) = L (5.48)

constitutes the basic assumption of the point-source models. Such models were
first investigated by Thomas Cowling, in 1930, and hence they arc also known
as Cowling models. It is reasonable to assume a homogeneous composition for
point-source models. Expressing the opacity in terms of p and T as in Equation
(3.63), k — K^paTh, the set of equations to be solved reduces to

dP Gmp
(5.49)
~dr ~ ~~r2~
dPriĄ _ k0L pa+'Tb
(5.50)
dr c 4rrr2
dm A 2
— = 4,7 f-p. (5.51)
dr
together with an equation of state for the gas. say Pgas = ('R./p)pT. This is by
no means a very simple or transparent model, but Equations (5.49)—(5.51) can
be integrated numerically for a given opacity law. A somewhat simpler and more
elegant version of the point-source model may be obtained if one further assumes
the opacity to be constant (« = b = 0). Equations (5.49) and (5.50) may then be
written as
d(Psas + Prad) _ Cmp
(5.52)
dr r2
(I Prad _ KLp
(5.53)
dr 4ncr2'
and dividing them (as we did to obtain Equation (5.33)), we get

r/Pcas 4ncG
------- = -------- m — I. (5.54)
dPrM\ kL

We now differentiate Equation (5.54) to obtain

d~ PSiis dPTad _ 4ttcG dm

(5.55)
dP2d dr kL dr

Inserting Equations (5.53) on the left-hand side and (5.51) on the right-hand side,
and rearranging terms, we finally have

d2P«as /'647t3c2G\ ,
(5.56)

We may express p in terms of Pgas and Praj as follows:

p = ^(a\^P
(5.57)
72. V 3 / gas d
 5.7 The point-source model 85

and cast Equation (5.53) in the form

</ 47tcK/3\l/4 , IA
_____ ( _ p~] p1 (5.58)
PkL \a) gas rad

Thus the original set of four equations has been reduced to a pair of differential
equations, (5.56) and (5.58), in three variables: Pgas, r and Ąa(j (as the independent
one). With the introduction of appropriate dimensionless variables - x for PXiV\, y
for Tgas and Z for r 1 - the equations to be solved become

^~-v _ _,-4 _ —i 1/4

dx- " dx y (5.59)

The solutions y(x) and z(x) may be inverted to obtain Pea!i(r) and Prad(/), and
with them the temperature and density variation throughout the star. Even the
solution of this pair of equations is far from being straightforward (a detailed
analysis may be found in Chandrasekhar's book. An introduction to the study
of stellar structure, 1939). It is interesting to note that the point-source model
yields a mass-luminosity relation whose slope, on logarithmic scales, is much
less steep for large masses than for small ones, which is in qualitative agreement
with observations (Figure 1.6). A tendency toward a steeper slope for low mass
stars is exhibited by the relation resulting from the standard model (Equation
(5.47)) as well. (The domain of very low-mass stars of Figure 1.6, where the
slope changes again, is not relevant here, for these stars cannot be treated by the
equations considered, as we shall see later.)

Note: We stress the mass-luminosity relation in particular, because of the primary role
it played at the early stage of the stellar-evolution theory, when a great deal of confusion
regarding the nuclear reactions responsible for energy generation still prevailed:

Our discussion has been based on the relation generally called the mass­
luminosity-relation. [... ] The relation however contains several unknowns, and
without certain assumptions with regard to some of them no definite results can
be reached. So far it is the only relation between the unknowns in question, which
it has been possible to establish. When our knowledge of the energy-generation in
the stars advances so that one more relation can be established, we shall probably
be able to give definite answers to the questions raised by the discussion.
Bengt Strbmgren: On the Interpretation of the Hertzxprung-Russell-Diagrain
in Zeitschriftfur Astrophysik, 1933

and, a few years later.

The researches of the last two decades into the constitution of the stars have
resulted in considerable advance in the understanding of the physical processes
in stellar interiors. The chief success of the investigations is the establishing of
86 5 Equilibrium - simple models

a mass-luminosity relation. This relation has been obtained without reference to

the actual nuclear reactions that are the source of stellar energy, merely from
consideration of the mechanical and thermodynamical equilibrium of the star.
It follows therefore that exact knowledge of the rate of generation of subatomic
energy cannot overthrow the mass-luminosity relation, but may serve only to
place some restriction on the range of magnitude [luminosity] corresponding to
a given mass.
Fred Hoyle and Raymond A. Lyttleton: The Evolution ofthe Stars in
Proceedings of the Cambridge Philosophical Society, 1939

When nuclear energy generation in stars finally became understood. Cowling wrote in
retrospection:

With the advent of this new physical information, complete data for constructing
stellar models were for the first time available; it was like supplying the fourth
leg of a chair which so far had had only one back leg.
Thomas G. Cowling: The Development ofthe Theory of Stellar
Structure in Quarterly Journal ofthe Royal Astronomical Society, 1966

Models such as those briefly mentioned here were developed more than
70 years ago, long before computers became available. Since the advent of com­
puters they became rather obsolete, for the relatively simple computation involved
ceased to be a real advantage. They have been described here mainly for one pur­
pose: to demonstrate how complicated is the solution of the apparently simple set
of structure equations, even under the most extreme simplifying assumptions that
are still (barely) consistent with physical reality.
 6

The stability of stars

In the previous chapter we have dealt with models of the stellar structure under
conditions of thermal and hydrostatic equilibrium. But in order to accomplish our
first task toward understanding the process of stellar evolution - the investigation
of equilibrium configurations - we must test the equilibrium configurations for
stability. The difference between stable and unstable equilibrium is illustrated
in Figure 6.1 by two balls: one on top of a dome and the other at the bottom
of a bowl. Obviously, the former is in an unstable equilibrium state, while the
latter is in a stable one. The way to prove (or test) this statement is also obvious
and it is generally applicable; it involves a small perturbation of the equilibrium
state. Imagine the ball to be slightly perturbed from its position, resulting in a
slight imbalance of the forces acting on it. In the first case, this would cause
the ball to slide down, running away from its original position. In the second
case, on the other hand, the perturbation will lead to small oscillations around
the equilibrium position, which friction will eventually dampen, the ball thus
returning to its original point. The small imbalance led to the restoration of equi­
librium by opposing the tendency of the perturbation. Thus a stable equilibrium
may be maintained indefinitely, while an unstable one must end in a runaway,
for random small perturbations arc always to be expected in realistic physical
systems.
As stars preserve their properties for very long periods of time, we may guess
that their state of equilibrium is stable. But is it always? What is the mechanism
that renders it stable? Is this mechanism always operating? If not. what are the
conditions required for it to operate? We shall presently address these questions
for each of the two types of equilibrium of stellar configurations: thermal and
hydrostatic.

87
88 6 The stability of stars

Figure 6.1 Illustration of stable (left) and unstable (right) equilibrium states.

6.1 Secular thermal stability

The total energy of a star in hydrostatic equilibrium is given by the sum of the
internal energy U and the gravitational potential energy Q, as we have seen in
Chapter 2. These arc related by the virial theorem. Equation (2.23):
/•.W p
3 / —dm — —Q.
Jo P
In the case of an ideal gas with negligible radiation pressure, we have by Equation
(3.44)

ć/ = -|S2 (6.1)

and consequently.

E = = -U. (6.2)

For an ideal gas and a nonnegligible radiation pressure, we have from Equations
(3.28). (3.40). (3.44) and (3.47)

E _ ^gas F’raj _ 7?. ft T'4 , ,

— —---------- 1----------— — / + "Z------ = aWgas + ^Mrad-
P P P P 3p
Applying the virial theorem, we obtain

L^gas = — 1(^2 + (Jrati), (6.3)

where t/gas is the total internal energy of the gas and f7rad is the total radiation
energy, whence

E = 1(Q + (7rad) = -Cgas. (6.4)

The effect of radiation is to reduce the gravitational attraction; Q + C/rad may thus
be regarded as an effective gravitational potential energy. In both cases the star
heats up upon contraction (| Q | increases and with it (Jgas and hence the average
temperature) and cools upon expansion.
We have also seen that the rate of change of the energy is given by the
difference between the rate of nuclear energy production and the rate of emission
 6.2 Cases of thermal instability 89

of radiation:

E = Lmc-L. (6.5)

A state of thermal equilibrium is obtained when these terms are in balance,

£nuc = E, and hence the energy is constant (£ = 0). Suppose now that a small
perturbation causes a slight imbalance, so that £nuc exceeds L. By Equation (6.5),
the total energy will increase (£ > 0), and since £ is negative, it means that
its absolute value will become smaller. Therefore, by Equation (6.2) or (6.4),
the average temperature will decrease. At the same time the star will expand,
the average density thus decreasing. As a result, the (average) rate of nuclear
reactions, which is proportional to positive powers of p and 7'. will slow down
and Tnuc will drop. The perturbation will be reversed and thermal equilibrium
will eventually be restored. This thermostat, provided by the virial theorem, is
the stabilizing mechanism by which stars are capable of maintaining thermal
equilibrium for such long times. Stars are said to be in a state of secular stability.

6.2 Cases of thermal instability

The crucial link in the chain of arguments leading to the conclusion of secular
stability was the dependence of the internal energy of the star on temperature,
more precisely, the negative heat capacity of stars. Only if a change in internal
energy involves a change in temperature that, in turn, affects the energy supply,
is the thermal stability secured.

Thermal instability of degenerate gases

We have seen that when the pressure is due mainly to the degenerate electron gas,
it is insensitive to temperature. The same applies to the internal energy of the gas
(as shown in Section 3.5). and hence although Equation (6.2) still holds (for a
nonrelativistic gas), a decrease in internal energy - resulting from a perturbation
£nuc > £ - will lead to expansion, but it will not entail a drop in temperature.
Since nuclear energy production is far more sensitive to temperature than to
density, the nuclear energy output will not diminish. Instead of a restoration of
thermal equilibrium, a runaway from equilibrium will ensue: the temperature will
continue to rise due to the enhanced nuclear energy release, this will cause the
nuclear energy generation to escalate, and so forth. Such an instability is called a
thermonuclear runaway. It is encountered whenever nuclear reactions ignite in a
degenerate gas, and it may result in an explosion. A catastrophic outcome may.
however, be avoided: the gas may, eventually, become sufficiently hot and diluted
to behave as an ideal gas, for which the stabilizing mechanism operates. We say
in this case that (he degeneracy has been lifted. An entire class of stellar outbursts.
90 6 The stability of stars

known as novae, that we shall encounter in Chapter 11, constitute well known
examples of such thermonuclear runaways, which develop into explosions on the
surfaces of white dwarfs and are subsequently quenched.
The secular instability caused by temperature-sensitive nuclear reactions in
degenerate matter was first studied by Tsung-Dao Lee (in 1950) and Leon Mestel
(in 1952). In 1958 Evry Schatzman proposed that unstable burning on the surfaces
of white dwarfs may lead to recurrent ejection of gaseous shells. This avant-garde
suggestion, then barely supported by observations, became in time the very model
of nova outbursts (see Chapter 11).
To better understand the application of the stability criterion, consider a star
which burns nuclear fuel at its centre. In hydrostatic equilibrium, the central
pressure and density are related by
dPc _ 4dpc
(6.6)

(see Equation (5.28)). The pressure, density and temperature are linked by the
equation of state, which may be written in general form as
dPc dpc , dTc
----- = a-------- f b-----, (6.7)
Pe Pc Tc
where a and b are positive coefficients. Combining Equations (6.6) and (6.7), we
obtain
4 \ dpc b^
3 J Pc (6.8)
Tc
So long as a < 4/3, sgn[Jpc/pc] = sgn|ć/Tc/Tc], and hence contraction (caused
by energy loss) is accompanied by heating, while expansion (caused by energy
gain) is accompanied by cooling, as required for stability. This is the case for
ideal gases, where a — b — I. For degenerate material, on the other hand,« > 4/3
and 0 < b 1. Thus dpc/pc and dTc/Tc have opposite signs. This means that
expansion, which would result from an increase in internal energy, would be
accompanied by a (small) rise in temperature, that in turn would lead to a further
enhancement of Lnuc. Such a situation is obviously unstable. But since the tem­
perature rises as the gas expands, the gas may gradually become ideal (in terms of
the coefficients a and /?, the former decreasing and the latter increasing), in which
case stability will be restored. We note, in passing, that, generalizing Equation
(6.8), a degenerate star that loses energy is expected to contract and cool, unlike
an ideal gas one.

The thin shell instability

Consider a thin shell of mass A/n, temperature T, and density p within a star
of radius R. between a fixed inner boundary r0 and an outer boundary r, so that
 6.2 Cases of thermal instability 91

its thickness is £ = r - r0 << R. Assume nuclear reactions to take place in this

shell. If the shell is in thermal equilibrium, the rate of nuclear energy generation
is equal to the net rate of heat flowing out of the shell (see Equation (5.4)). If
the rate of energy generation exceeds the rate of heat flow, the shell will expand,
thus pushing outward the layers above it. Lifting of these layers will result in
a diminished pressure. In hydrostatic equilibrium, the pressure within the shell,
determined by the weight of the layers above it, varies as r ~4 (see Equation (5.1));
hence
dP
(6.9)
P
The shell’s mass is given by Am = 4nTj£p and therefore the density varies with
the shell’s thickness as
dp dt dr dr r
(6.10)
p t t r~t

Substituting dr/r from Equation (6.9), we obtain a relation between the changes
in density and pressure of the form

dP _^dp
(6.11)
P r p
To obtain the resulting change in temperature, we use the equation of state in the
general form (6.7), leading to
' £ dp dT
4- - a — = b— (6.12)

For thermal stability we require the expansion of the shell to result in a drop in
temperature, or, since b > 0,
£
4->a. (6.13)
r
Obviously, for a sufficiently thin shell (£/r -»■ 0) the stability condition would
eventually be violated. If the shell is too thin, its temperature increases upon
expansion (even if the gas within it is an ideal gas), and this may lead to a
runaway. Thus with respect to nuclear energy generation, a thin shell behaves
much in the same way as a degenerate gas. The thermal instability of thin shells
was first pointed out by Martin Schwarzschild and Richard Harm in 1965.
Before we leave the subject of thermal stability (or lack thereof) a word of
caution would be in order. In all our foregoing discussions we have neglected
the possibility that a change in temperature might affect not only the energy
generation rate, but also the heat flux. Thus an increase in temperature, resulting
from Lnuc < L. may not only lead to a higher Lnuc, but also to a higher Ł, and the
outcome Łnuc > L - interpreted as thermal stability - might not be guaranteed.
Similarly, if a surplus of heat in a thin shell causes the temperature to rise, in spite
92 6 The stability of stars

of the shell’s expansion, this might enhance the rate of heat flow out of the shell
so as to prevent a runaway, even if the rate of nuclear energy generation increases.
As it happens, the heat flux is far less sensitive to temperature than is the rate of
nuclear energy generation. Hence, generally, changes in L or dF/dm may safely
be neglected compared with changes in L„uc or q that are due to a temperature
perturbation, and our foregoing arguments remain valid.

6.3 Dynamical stability

Dynamical stability is related to motions of mass parcels in the star, that is, to
macroscopic motions; on the microscopic scale, the gas particles are always in
random, local motion. In hydrostatic equilibrium, no macroscopic motions occur;
more precisely, they occur imperceptibly slowly. In order to test the stability of
this equilibrium, we have to consider the response to small perturbations of the
balance between the gravitational attraction and the outward force exerted by the
pressure gradient. Since we deal with a spherically symmetric configuration, we
shall consider radial perturbations: compression or expansion. The basic question
is whether a temporary contraction will result in expansion toward the original
state or in further contraction, escalating in a runaway.
A rigorous treatment of dynamical perturbations within a star is far from
simple. But in order to illustrate the basic principles involved, a highly simplified
example should suffice. Consider a gaseous sphere of mass A/, in hydrostatic
equilibrium. The pressure at any point r(m) is equal to the weight per unit area of
the layers between m and M. as obtained by integrating the equation of hydrostatic
equilibrium (5.1) and taking P(M) — 0:

The density at r(m) is given by Equation (5.2):

Consider now a small, uniform, radial compression, so that the new radii are
everywhere obtained from the original ones by a small perturbation:

= r — Er = /-(1 — £). (6.16)

If e 1, the binomial approximation

(1 ±(?)" % | ±n£ (6A7)

 6.3 Dynamical stability 93

can be used. The new densities will be

1 dm dr
(6.18)
1 4ttt2(1 - s)2 dr dr' (I— £)•'

Assuming furthermore the contraction to be adiabatic (and neglecting radiation

pressure), we find that the new gas pressure will relate to the initial one as (p'/p)Y\
where ya is the adiabatic exponent (introduced in Section 3.6):

P'gas = /,(l+3£)y’«/>(l+3eya). (6.19)

Similarly, by Equation (6.14), the new hydrostatic pressure will relate to the initial
one as
/*w Gm dm
P'h = -7---------- r % P( I + 4e). (6.20)
47rr4(l-£)4

It is to be expected that after this perturbation the gaseous sphere will no longer be
in hydrostatic equilibrium, that is, Pc'as P^. The condition required for restoring
equilibrium is in our case

P'gas > P'h, (6.21)

so as to cause the sphere to expand back to its original state. Substituting Equations
(6.19) and (6.20) into Equation (6.21), we thus require

P(1 +3ya£) > P( 1 +4e). (6.22)

Hence the condition for stable hydrostatic equilibrium, or in other words, the
condition for dynamical stability is
4
Ta > f6-23)

The same result obtains in the case of expansion, when e < 0 and condition (6.21)
is reversed.
It can be shown rigorously that a star in which ya > 4/3 everywhere is dynam­
ically stable (and neutrally stable, if y.d = 4/3 everywhere). The case in which
Ta < 4/3 somewhere requires further examination. Global dynamical instability
is obtained if the integral f (y-a — 4/3)^dm over the entire star is negative. Thus,
if Ea < 4/3 in a sufficiently large core, where P/p is high, the star will become
unstable. If, however, ya < 4/3 in the outer layers, where P/p is small, the star
as a whole need not become unstable.

Exercise 6.1: The equation of state for solid, self-gravitating bodies, such as
planets, must allow a finite density at the surface, where the pressure vanishes.
Neglecting effects of temperature, which are generally small, such equations of
94 6 The stability of stars

state are usually cast in the form

r/pV1 7V1
p(p) = k r _ r ,
. X Po / X Po ) .
where K is a constant, py is the surface density and y\ > yj. Pass to normalized,
dimensionless variables y = P/Pcandx = p/pc, where PQ and pc denote central
pressure and density, respectively, and show that

(a) Dynamical stability requires that either yj, y2 > 4/3, or yi, y? < 4/3;
(b) In the latter case, the allowed ratio of central to surface density is limited.

6.4 Cases of dynamical instability

The question we have to ask is, ‘What are the stellar configurations that may
lead to violation of the stability criterion, that is, to ya < 4/3?’ We have already
encountered such cases in Section 3.5.

Relativistic-degenerate electron gas

For a relativistic-degenerate electron gas, ya tends to 4/3. The instability expected

in the limit ya = 4/3 results in this case in the Chandrasekhar limiting mass
(derived in Section 5.4): the degeneracy pressure can sustain the gravitational
attraction only if the stellar mass is smaller than this limit. For a higher mass,
contraction will end in collapse.

Dominant radiation pressure

The second case for which ya tends to 4/3 (as shown in Section 3.6) is that of
a dominant radiation pressure, or in terms of the parameter /? - introduced in
Section 3.1 — > 0. In the limit fl = I (ideal gas without radiation), ya = 5/3,
and hence an ideal gas would be dynamically stable under its own gravitational
field. As fl decreases, ya decreases as well, tending to 4/3 in the limit fl = 0 (pure
photon gas). Another way of showing that a radiation pressure dominated gas
tends to be dynamically unstable is by using the virial theorem. For pure radiation
PIp — Mrad/3 and hence by Equation (2.23)
pM p
— Q =3 / —dm = t/rad, (6.24)
Jo P
meaning that the total energy of a star E — Q + U vanishes; that is, the star
becomes unbound. We see, therefore, that the consequences of dynamical insta­
bility may differ.
 6.4 Cases of dynamical instability 95

Exercise 6.2: Show that for an adiabatic process, stable hydrostatic equilibrium
corresponds to a minimum state of the total energy (E — U + Q).

Ionization-type processes

Dynamical instability, or ya < 4/3, is also prone to occur in any system of parti­
cles in which the number of particles is not conserved, but changes with changing
physical conditions. Ionization (Section 3.6) provides a typical example: a single
atom may produce two particles, an ion and an electron, by absorbing the right
amount of energy from a collision with another particle or with a photon. At the
same time the reverse reaction - recombination - occurs, which tends to dimin­
ish the number of particles. When the system is compressed, recombination is
enhanced, whereas if the volume is increased, ionization is favoured. Therefore,
the number of particles changes in inverse proportion to the density (Le Chate-
lier’s principle). The following simple argument is meant to provide an intuitive
explanation to the effect of this property on the value of ya. Consider two systems
of particles of volume V and pressure P: in one the number of particles N is con­
served, in the other it may change due to ionization-type reactions. We assume
an ideal gas and recall that the pressure is proportional to the number of particles
(regardless of their nature) and inversely proportional to the volume. Suppose now
that the volume is slightly compressed to V < V. In the first system the pressure
would obviously increase, since N/ V > N /V. In the second system, however,
N would change as well, say, to N' < N. Hence the new ratio N'/V would
be smaller than in the first system, N'/ V < N/V. Consequently, the pressure
would increase to a lesser extent, meaning that the dependence of the pressure on
volume (and hence, density) is weaker in the second system. This should translate
into a smaller value of ya, possibly smaller than 4/3. For a pure, singly ionized
gas. for example, according to Equation (3.60), ya < 4/3 between 5% and 95%
ionization (for y/kT 10). Hence in a cool atmosphere, only an almost entirely
neutral or a completely ionized gas would be dynamically stable.

Exercise 6.3: Show that there is a critical temperature above which partially
ionized hydrogen will always be dynamically stable, and find this temperature.

Since in stellar interiors temperatures are sufficiently high to ensure total

ionization - at least for the major components, hydrogen and helium - ionization
in itself is of no great consequence regarding the global stability of stars, for
ya < 4/3 only in restricted zones, where P/p is small. We have encountered,
however, two other ionization-type processes, which occur at high temperatures:
iron photodisintegration (Section 4.10) and pair production (Section 4.9). We
96 6 The stability of stars

Figure 6.2 Illustration of Schwarzschild’s criterion for stability against convection by a

mass element dm moving radially from point 1 to point 2 within a star.

shall see in the next chapter that both these processes drastically affect the course
of stellar evolution.

6.5 Convection

We have seen in Section 5.5 that the radiative energy flux through a star in
hydrostatic equilibrium is limited by the requirement

kF < 4ticGm,

which may be violated in cases of intense nuclear burning, when F — f qdm

becomes exceedingly high, or when the opacity is very high. By Equation (5.3),
a high flux or a high opacity leads to a steep temperature gradient. However, the
temperature gradient may only increase up to a limit beyond which convection
occurs, involving cyclic macroscopic mass motion (but not a net mass flux) that
carries the bulk of the energy flux. When the total flux satisfies Equation (5.4),
(convective) thermal equilibrium is achieved. Thus convection may be regarded as
a type of dynamical instability, although it does not have disruptive consequences.
In fact, in spite of being a dynamic process, convection affects the structure of a
star only as an effective heat carrier and as a mixing mechanism. The condition
for the onset of convection (or the limiting temperature gradient) is determined
by a simple criterion, as was shown by Karl Schwarzschild in 1906.
The Schwarzschild criterion for stability against convection derives from
the following argument: Consider a mass element Am at some point within a
star, as shown in Figure 6.2. Denoting this point by 1, let the local values of
 6.S Convection 97

Figure 6.3 Schematic density-pressure diagram, leading to the mathematical formulation

of Schwarzschild’s criterion for stability against convection.

density and pressure at 1 be p\ and P\, respectively. Suppose the element moves
a small distance outward in the radial direction to point 2, where the density
is p2 and the pressure is P2. Since the pressure in a star decreases outward.
P2 < P\‘, that is, the surrounding pressure at point 2 will be lower than the
pressure within the mass element. The element will therefore expand until the
internal and external pressures are in balance. In view of the great difference
between dynamical and thermal timescales, it is reasonable to assume that no
heat exchange with the environment occurs while the mass element expands.
Hence the element undergoes an adiabatic change leading to a final density p*,
which is not necessarily equal to the density of its surroundings. If p, > pj,
the mass element will descend back toward its initial position. We regard such
a situation as stable, for any mass motion that may accidentally arise will be
damped. If, on the other hand, p* < p2, the element will continue its upward
motion (by the Archimedes buoyancy law). In this case, the system is unstable
against convection; that is, convective motion is prone to develop. The extent of
the convectively unstable region may be found by applying the same criterion for
increasingly more distant points. It is possible that a star be fully convective, all
the way from the centre to the photosphere.
To obtain a mathematical formulation of the convective stability condition, we
resort to the (p. P) diagram of Figure 6.3, where the starting point I — (pi. Pil­
is marked. As shown in Section 3.6, the dependence of pressure on density in
an adiabatic process is given by P = KApY'. The curve labelled A represents
the adiabatic (P. p) relation passing through point 1, obtained from the physical
characteristics of the gas at that point. The curves labelled S and S' represent
hypothetical stellar configurations: possible variations of the pressure with density
in the star in the neighbourhood of point I. The slope of S is steeper and that of S'
98 6 The stability of stars

is shallower than the slope of A. The horizontal line P = Pi intersects each of the
curves A, S’ and S': the intersection with A corresponds to the density p* within the
mass element, while the intersections with S’ and S' correspond to the density of
the surroundings in each case. If the stellar configuration is described by S’, then
p2 > p*, meaning instability, whereas if S' describes the stellar configuration,
p'2 < /?*, indicating stability against convection. In conclusion, the condition for
stability is

(6.25)

and multiplying both sides by p/P, we have

p /dP \
< Ta- (6.26)
P \ dp / slar
It is noteworthy that the general validity of this simple criterion was not proved by
rigorous mathematical methods until six decades after it came into use, in 1967,
by Shmuel Kaniel and Attay Kovetz.
For an ideal gas and negligible radiation pressure, the pressure is proportional
both to temperature and to density, whence
dP dp dT
------— — 4~ — (6.27)
P p T
for a given composition. Combining Equations (6.27) and (6.26), we obtain the
condition for convective stability in the form

P(dT\ < Ta - 1
'/'V^Atar< Ta (6.28)

which may also be written as

(6.29)

recalling that the temperature and the pressure gradients are negative. We have thus
obtained the upper limit for the magnitude of the temperature gradient allowed
before convection sets in.

6.6 Cases of convective instability

The criterion for convective stability that we have just derived is very general;
it may be equally applied to stellar interiors and, for example, to the Earth’s
atmosphere. But can we be more specific about the conditions that may lead to
convective instability in stars? In particular, how is restriction (5.34) connected
to the criterion of convective stability?
 6.6 Cases of convective instability 99

We have seen in Section 3.6 that during ionization the adiabatic exponent is
lowered. Therefore, in regions of the star where the gas is partially ionized, the
condition for convective stability is more difficult to satisfy. At the same time
these regions may become dynamically unstable, if ya < 4/3.
Condition (6.28) may be generalized to include the effect of radiation pressure,
in which case ya = ya(^), but the adiabatic exponents that appear in conditions
(6.26) and (6.28) become different functions of /J. Both adiabatic exponents tend
to 5/3 for fl -> I and to 4/3 for fl 0.

Exercise 6.4: Following the procedure of Section 3.6, derive the adiabatic expo­
nents in conditions (6.26) and (6.28) for an ideal gas and radiation, as functions
of fl. Calculate their values for /) = 0, | and 1.

If we now use the radiative diffusion equation (5.3) for the temperature gradient
and the hydrostatic equation (5.1) for the pressure gradient, we obtain the condition
for convective stability (6.29) in the form

kF <c 4ncGm 4( Ta Ya- 1 )(1 ~fl) (6.30)

This is similar to condition (5.34), which imposed an upper limit on the product
kF, above which radiative equilibrium could no longer hold. We note that con­
dition (6.30) is stronger, since the term in square brackets on the right-hand side
is smaller than unity. Therefore convection arises before the upper limit for kF
(condition (5.34)) is reached. The two conditions converge as fl tends to zero.
In the case of ionization, the high opacity and low adiabatic exponent combine
to induce convection. This effect is particularly important in the outer regions of
stars, where temperatures are sufficiently low for helium and hydrogen to be
only partially ionized. In stellar interiors, especially in zones of high temperature
where the opacity is constant, the dominant factor that may induce convection is
a high energy flux. Such a flux is expected to result from intense nuclear burning.
Assuming that the nuclear energy generation rate may be expressed as a power
law of the form (5.7), q — q^p"'T" with n 3> m, it should be possible to translate
the condition on the intensity of nuclear burning into a limiting value for n. This
is by no means a simple task: it cannot be accomplished analytically but requires
solution of the stellar structure equations. The question of a limiting value for
n has already been addressed in the 1930s; for example, using relatively simple
models (more elaborate variants of the models described in Chapter 5). Cowling
obtained the following conditions: for a constant opacity, no convectively stable
configuration is possible if n exceeds a number lying between 3 and 4. while for
a Kramers opacity law' (Section 3.7), no such solution is possible for n in excess
of about 8. In the early 1950s the problem was pursued and elaborated by Roger
100 6 The stability of stars

Tayler, with similar results. Generally, a high temperature sensitivity of the energy
generation rate is bound to trigger convection.
Condition (6.30) indicates that convection is more likely to occur when
besides kF being high, f is not too far from unity, meaning that gas pressure is
dominant. We have seen in Section 5.6 that, based on the simple standard model,
f is strongly related to the stellar mass, increasing with decreasing M. Hence,
we should not be surprised to find that convection is dominant in low-mass stars
burning nuclear fuel. When low-mass stars arc sufficiently cold and dense for
degeneracy pressure to dominate, stability against convection is regained. This is
for two reasons: first, degenerate matter is highly conductive; that is, its effec­
tive opacity is very low, and second, no stable nuclear burning is possible under
degenerate conditions (Section 6.2 above), implying that such stars must be inert.
In conclusion, we should not expect convection to develop in the interiors of white
dwarfs.

Convective energy transport - the mixing-length method

When convective energy transport takes place, Equation (5.3) is no longer valid in
the sense that the flux appearing on the right-hand side is the radiation flux, which
now differs from the total dux F (of Equation (5.4)), amounting to only a small,
unknown fraction of F. Hence Equation (5.3) must be supplemented or replaced
by another one that takes account of convection. Since convective motions are
clearly not entirely radial, there are only approximate ways of estimating the con­
vective flux for spherical, one-dimensional stellar models. The most commonly
adopted method was first proposed by Ludwig Biermann in the 1930s, based on
the concept of mixing-length, which had been introduced by Ludwig Prandtl a
few years earlier, as the distance traversed by a mass element while conserving
its properties, before it blends with its surroundings. The arguments for esti­
mating the convective flux constitute what is known as the mixing-length theory
of convection, an approximate method for calculating convective transfer by an
appropriate parametrization. Since the mixing length, which we shall denote by
€c, cannot be determined from first principles, it is taken as a free, adjustable
parameter.
In the case of convective transfer, the energy is transmitted by turbulent mass
motions. Consider a rising mass element at a radial distance r (mass m), where
the temperature is T and the density p. The basic assumption is that the mass
element travels a distance £c adiabatically, at some velocity vc, until it reaches
pressure equilibrium with its surroundings and releases its surplus heat. We now
estimate the differences (8) between properties of the element and those of its
surroundings at the equilibrium stage: first, 8P — 0. Next, the temperature surplus
is given by the difference between the change in temperature that has occurred
in the surroundings over the small distance and the corresponding adiabatic
 6.6 Cases of convective instability 101

change that has occurred within the mass element itself,

8T = fl\ —
dr 1 1 star
— 1 £c,
dr la/
(6.31)

where
dT T (d\nT\ (IP
(6.32)
dr 7 \ JlnP A, ~dr

Using a similar form for the temperature gradient within the star, we have

d In T Jln7'\ 1 dP
8T = T ---------------------- T (6.33)
Jin P dlnP I, P dr
star ' aJ
where the term P/(dP/dr) on the right-hand side, with a minus sign, is the
pressure scale-height, which constitutes the characteristic local lengh-scale. Mea­
suring the mixing length in units of this length scale, a dimensionless parameter
a is defined, known as the mixing-length parameter,

’ P/(-dP/dr) (P/gpY

which is the sole parameter of the model, typically a small number. Finally,
from Equation (6.27), the density deficit is related to the temperature surplus by
\8p/p\ = \8T/T\.
The mass element releases heat at constant pressure, hence the amount of
heat released per unit mass is cy8T. where cp is the heat capacity, a function
of temperature and density, and pc?8T is the heat released per unit volume.
Multiplying by the velocity, we obtain the average rate of heating per unit area,
or the convective heat flux

Hc = pvccp8T. (6.35)

The convective velocity may be estimated by uc — y/g'£c, where g' is the accel­
eration, which, by the Archimedes buoyancy principle, is the local gravitational
acceleration g, reduced by the factor \8p/p\. Hence

Vc = Jg\8p/p\tc = Jg\ST/T\te = JedP/p)\8T/T\, (6.36)

where we have used Equation (6.34) to eliminate gtc. We note that JP/p is the
thermal speed (which is of the order of the sound speed). Sometimes half this
velocity is assumed, but this is unimportant, since the uncertain factor may be
lumped into the free parameter a.
Combining Equations (6.35), (6.36), (6.33) and (6.34), we obtain

d In T d In T
//c = pcpT^/P/p a2. (6.37)
Jin P star
Jin P
102 6 The stability of stars

The total energy transported per unit time /•' is the sum of the radiative part, given
by Equation (3.68), and the convective part 4ttA’2/7c. Thus in a convective zone,
Equation (5.3) that relates the temperature gradient to the energy flux is replaced
by a more complicated one.

Adiabaticity

We shall now show that within convective zones in stellar interiors the departure
from adiabaticity is very small. To obtain an order of magnitude estimate for the
superadiabaticity ST/T, we replace Hc by L/R-, p by M/R\ cPT by U/M,
and P/p by GM/R. and take a = 1 in Equation (6.37), which yields, with
Equation (6.33),

3T\ 3/2 L 1 (6.38)

U JGM/IP'

We recognize the first term ofthe product on the right-hand side as the reciprocal
of the thermal (Kelvin-Helmholtz) timescale (Equation (2.60)), of the order of
1015 s, and the second, as the dynamical timescale (Equation (2.56)), ofthe order
of 103 s. In conclusion

ST_
(6.39)
T

which clearly shows that in a convective zone the temperature gradient must be
very nearly adiabatic (y ya). Thus, instead of Equation (5.3), the temperature
gradient may be replaced by the adiabatic one in deep convective zones, to a
very good approximation. This approximation is not valid close to the stellar
surface, where P/p GM/R. We note, however, that assuming an adiabatic
temperature gradient leaves F undetermined throughout the convective zone. It is
precisely the tiny extent of superadiabaticity that determines the rate of convective
energy transport. This is reminiscent of the fact that it is the slight departure from
homogeneity in the radiation field that drives the radiative energy transport in
stellar interiors (see Appendix A).

Exercise 6.5: Assuming that a star of uniform k (opacity) and fi has a convective
core, and no nuclear energy generation outside the core, show that the mass
fraction of this core is given by , ,,.

The assumption of adiabaticity leads to a polytropic equation of state. Now,

since dynamical stability requires ya > 4/3, and since ya is at most 5/3, a star in
 6.7 Conclusion 103

hydrostatic equilibrium must satisfy

4 5
- < Ta < (6.40)

which means that if the configuration of a star is to be approximately described

by a polytropc, the index n may only vary between 1.5 and 3.

6.7 Conclusion

To summarize the question of stability of equilibrium in a star (whether radiative

or convective) raised at the beginning of this chapter, let us say that stability
depends on the ability of the gas - particles and photons - at any given point to
sustain the weight of the overlying layers by means of the pressure it exerts, so as
to maintain exact balance despite possible perturbations. To this end, the pressure
must depend strongly enough both on temperature and on density. Sensitivity
to temperature is required in order to prevent thermonuclear runaways, and to
density - so as to avoid collapse. There are additional cases of instability in stars,
essentially resulting from violation of this principle, but since they occur in the
course of evolution when special interior configurations develop, we shall deal
with them as they arise.
 7

The evolution of stars - a

schematic picture

Having answered the two basic questions posed at the end of Chapter 2, our
present task is to combine the knowledge acquired so far into a general pic­
ture of the evolution of stars. We recall that the timescale of stellar evolution
is set by the (slow) rate of consumption of the nuclear fuel. Now, the rate of
nuclear burning increases with density and rises steeply with temperature, and
the structure equations of a star show that both the temperature and the density
decrease from the centre outward. We may therefore conclude that the evolu­
tion of a star will be led by the central region (the stellar core), with the outer
parts lagging behind it. Changes in composition first occur in the core, and as
the core is gradually depleted of each nuclear fuel, the evolution of the star
progresses.
Thus insight may be gained into the evolutionary course of a star by con­
sidering the changes that occur at its centre. To obtain a simplified picture of
stellar evolution, we shall characterize a star by its central conditions and follow
the change of these conditions with time. We have seen that besides the com­
position, the temperature and density are the only properties required in order
to determine any other physical quantity. If we denote the centra) temperature
by Tc and the central density by pc, the state of a star is defined at any given
time t by a pair of values: Tc(t) and pc(t). Consider now a diagram whose axes
are temperature and density. The pair [77(r), pc(t)] corresponds to a point in
such a diagram, and the evolution of a star is therefore described by a series
of points, [?;.(?]), pc(fi)], [Tc(f2), pAti)\. |Tt(/3), pc(t3)], [Tc(t4). , for
times t] < r2 < ?3 < H,..., which forms a parametric line [Tc(t), pc(f)]. Since
the only property that distinguishes the evolutionary track of a star from that
of any other star of the same composition is its mass, we may expect to obtain
different lines in the (T, p) plane for different masses.

104
 7.1 Characterization of the (log T, log p) plane 105

Note: A study of the late stages of stellar evolution, based on homogeneous and
isentropic (uniform entropy or adiabatic structure) models, was performed by Gideon
Rakavy and Giora Shaviv in 1968. The progress in time was simulated by decreasing the
entropy s, and thus parametric lines [Tc(s), pc(i)] were obtained for different values of
the stellar mass. A beautiful general picture of the end states of stars emerged, for which
a qualitative explanation (and, in a sense, validation) was offered a year later by Kovetz.
The following discussion, which will eventually lead to a more comprehensive picture
arising from very simple arguments, was inspired by these works.

All the processes that are bound to occur in a star have characteristic temperature
and density ranges, and hence different combinations of temperature and density
will determine the prevailing state of the stellar material and the dominant physical
processes that should be expected to occur. Thus the (T, p) plane may be divided
into zones, representing different physical states or processes. Our first step will be
to get acquainted with the terrain through which the evolution paths ofthe stellar
centre arc winding; the second step will be to identify the track corresponding to
each stellar mass: finally, by following each track through this terrain, we shall
be able to trace the chain of processes that make up the evolution of a star.

7.1 Characterization of the (log T, log p) plane

The (T, p) plane will be divided into zones dominated by different equations of
state and different nuclear processes. Of particular interest will be those regions
where the conditions are bound to lead to dynamical instability. As the ranges of
density and temperature typical of stellar interiors span many orders of magnitude,
logarithmic scales will be used for both.

Zones of the equation of state

The following arguments are based on the material of Chapter 3 and lead to
Figure 7.1. The most common state ofthe ionized stellar gas is that of an ideal gas
for both components: ions and electrons. Hence the common equation of state is
of the form

P = -pT = K()pT. (7.1)

P
where is a constant (sec Equation (3.28)). At high densities and relatively low
temperatures, the electrons become degenerate, and since their contribution to the
pressure is dominant, the equation of state may then be approximated by

P = K\p5/i (7.2)
106 7 The evolution of stars - a schematic picture

Figure 7.1 Mapping of the temperature-density diagram according to the equation of

state.

(see Equation (3.35)), which replaces Equation (7.1). The transition from one state
to the other is, of course, gradual with the change in density and temperature,
but an approximate boundary may be traced in the (log T, log p) plane on one
side of which the effect of degeneracy is clearly important, while on the other
side an ideal-gas law prevails. This boundary may be defined by the requirement
that the pressure obtained from Equation (7.1) be equal to that obtained from
Equation (7.2),

log p — 1.5 log T + constant. (7.3)

which is a straight line with a slope of 1.5, as shown in Figure 7.1. The electron­
degeneracy zone, labelled 11 in Figure 7.1, where

K]P5^ > KnpT,

lies above (to the left of) this line. The ideal-gas zone, labelled I, lies below it.
Note that the transition refers to the total pressure of the stellar gas meant to
represent two different states of evolution, not just to the electron pressure (in the
degenerate case, the contribution of the ion pressure is negligible), since we are
interested in the behaviour of a star, rather than that of an hypothetical electron
gas. Thus the constants also involve different compositions.
 7.1 Characterization of the (log T. log p) plane 107

For still higher densities, when relativistic effects play an important role, the
equation of state changes to the form

P = K2p4/3 (7.4)

(see Equation (3.38)). The boundary between the ideal-gas zone and the
relativistic-degeneracy zone may be obtained, as before, from the requirement

W/3 = ^opT.

which defines a straight line

log p = 3 log T + constant (7.5)

with a slope of 3. Thus the boundary between the ideal-gas zone and the electron­
degeneracy zone changes slope, becoming steeper as the density increases.
Within the electron-degeneracy region, the transition from nonrelativistic to
relativistic degeneracy occurs when the rise in pressure with increasing density
becomes constrained by the limiting velocity c. Hence relativistic degeneracy
should be considered when

KlP5/3 » K2P4/3.

or

that is. above a high density level (a horizontal line in the (log T. log p) plane).
This is roughly indicated in Figure 7.1, where the relativistic-degeneracy zone is
labelled III.
In zone I radiation pressure has been neglected. Its contribution to the total
pressure becomes important, however, at high temperatures and low densities
(the lower right corner of the diagram) and should be added to that of the gas.
Eventually, radiation pressure would become dominant, with the equation of state
changing to

P = jaT4 (7.7)

(see Equation (3.40)). Taking the gas pressure to be negligible for, say, Pra(j =
10Peas. we obtain an approximate boundary for the zone of dominance of radiation
pressure (labelled IV in Figure 7.1) in the form

log p — 3 log T + constant (7.8)

again a straight line of slope 3 (with a constant different from that in Equation
(7.5), of course).
108 7 The evolution of stars - a schematic picture

Zones of nuclear burning

The following arguments are based on the material of Chapter 4. A nuclear

burning process of any kind becomes important in a star whenever the rate of
energy release by this process constitutes a significant fraction of the rate at
which energy is radiated away, that is, of the stellar luminosity. Although stellar
luminosities vary within a wide range, the variation in the conditions prevailing
in burning zones is quite restricted, due to the high sensitivity of nuclear reaction
rates to temperature. Hence a narrow threshold may be defined for each nuclear
process that takes place in stars. On one side of the threshold the rate of nuclear
burning may be assumed negligible, and on the other side, considerable. The
threshold for each process constitutes a line in the (log T, log p) plane, defined
by the requirement that the rate of nuclear energy generation q exceed a certain
prescribed limit z/min, say, 0.1 J kg-1 s-1 (lO’crgg-1 s-1). Since for each process,
q may be approximated by a power law of the form

q =qap"'T\ (IS))

the threshold given by q = r/tnjn is

logp = — — log? + — log f Y (7.10)

m \ r/o /

The exoergic transformation of hydrogen into the iron group elements com­
prises five major stages: hydrogen burning into helium either by the p — p chain
or by the CNO cycle, helium burning into carbon by the 3a reaction, carbon
burning, oxygen burning and silicon burning. The five corresponding thresholds
are plotted in Figure 7.2.
In most of the cases m — 1 and n >> 1, and hence the (negative) slope in
Equation (7.10) is so steep that the thresholds are almost vertical lines. Strictly,
the threshold defined by Equation (7.10) should be a straight line; in reality, the
values of the powers in Equation (7.9) change slightly for different temperature
ranges; this is the reason why the lines in Figure 7.2 are not perfectly straight.
For hydrogen burning, the slope is milder at low temperatures, corresponding to
the p — p chain (w % 4), and becomes steeper at higher temperatures, where the
CNO cycle (n «« 16) becomes dominant.
Nucleosynthesis by energy releasing fusion of lighter elements into heavier
ones ends with iron. Iron nuclei heated to very high temperatures are disintegrated
by energetic photons into helium nuclei. This energy absorbing process reaches
equilibrium (called, as in the case of silicon burning, nuclear statistical equilib­
rium), with the relative abundance of iron to helium nuclei determined by the
values of temperature and density. A threshold may be defined for the process of
iron photodisintegration, as a strip in the (log T, log p) plane, by the requirement
 7.1 Characterization of the (log T. logp) plane 109

Log[T(K)]

Figure 7.2 Mapping of the temperature-density diagram according to nuclear processes.

that the number of helium and iron nuclei be approximately equal. This threshold
is shown in Figure 7.2.

Zones of instability

The following arguments are based on the material of Chapter 6. The condition of
dynamical stability is ya > 4/3 (Section 6.3). We thus expect stellar configurations
to become dynamically unstable in those regions of the (loglog p) plane where
ya is reduced to 4/3 or less. Such regions are the far extremes of the relativistic-
degeneracy zone III and of the radiation-pressure-dominated zone IV. where
ya tends asymptotically to 4/3. Another is the iron-photodisintegration zone,
where ya < 4/3. As we arc dealing with the centre of stars, restricted regions
of instability caused by the ionization of hydrogen and helium lie outside the
ranges of temperature and density that we consider. Pair production, which is
an ‘ionization’-type process as well, defines an additional unstable zone, with
Ya < ^/^- as shown in Figure 7.3. With all these unstable zones marked, the
stable part of the (log T. log p) plane becomes completely bounded on two sides:
at high densities and at high temperatures. Hence severe constraints are imposed
on the possible evolutionary tracks of stars. We finally recall that nuclear burning
is thermally unstable in degenerate gases, whether relativistic or not. Hence the
110 7 The evolution of stars - a schematic picture

10

8 10
Log [T(K)J

Figure 7.3 Outline of the stable and unstable zones in the temperature-density diagram.

nuclear burning thresholds of Figure 7.2 have been discontinued after crossing
the boundary into the degeneracy zone II.

7.2 The evolutionary path of the central point of a star in the

(log T. log p) plane

Having become acquainted with the (log T, log p) plane, the question we now­
ask is whether the centre of a star of given mass M may assume any combi­
nation of temperature and density values, that is, may be found anywhere in
this plane, or whether these values are in some way constrained by M. We now
regard the (log T. logp) plane as a (log 7^, logpc) plane, referring to the stel­
lar centre. Assuming a polytropic configuration (Equation (5.10)) for a star in
hydrostatic equilibrium, the central density is related to the central pressure by
Equation (5.28),

Pc = (47r)l/3B„GM2/3pc4/3. (7.11)

This relation is only weakly dependent on the polytropic index n, especially for
stable configurations, for which n varies between 1.5 and 3 (see Section 6.6), and
the coefficient Hn. between 0.157 and 0.206 (see Table 5.1), and it is independent
of K. It is valid whether K is determined by processes on the microscopic scale,
such as electron degeneracy (Section 3.3), or on the macroscopic scale, such
as convection (Section 6.6). Although a star in hydrostatic equilibrium is not
 7.2 The evolutionary path of the central point of a star III

7 8 10
Log[T(K)]

Figure 7.4 Relation of central density to central temperature for stars of different masses
within the stable ideal gas and degenerate gas zones.

a perfect polytrope (even if its composition is homogeneous), relation (7.11)

provides a good approximation to hydrostatic equilibrium for any configuration.
Note that simply by dimensional analysis of the hydrostatic equation, the central
pressure must be proportional to GM2!)p\
In addition, the central pressure is related to the central density and temper­
ature by the equation of state. Within the different zones of the (logTL-, logpc)
plane we have different equations of state. Combining each of them with
Equation (7.11), we may eliminate Pc, to obtain a relation between pc and Tc.
Consider a star of mass M. whose central point is found in the ideal gas zone
I, where Equation (7.1) holds. The relation between 7'c and pc in this case is of
the form
J'S
o = 0 (7 12)

meaning that for a star of given mass, the central density varies as the central
temperature cubed. For stars of different masses but the same central temperature,
the central density decreases as the mass squared increases. On logarithmic scales,
relation (7.12) becomes a straight line with a slope of 3. Thus different masses
define different parallel lines, which intersect the temperature axis at intervals
proportional to log M. The lines corresponding to M = 0.1. 1, 10 and 100AY© are
plotted in Figure 7.4; these masses being successive powers of IOAf©, the intervals
between the lines are equal.
112 7 The evolution of stars - a schematic picture

If at the centre of a star the electrons are strongly but nonrelativistically degen­
erate, the central point is found in zone II and Equation (7.2) holds. Substituting
Ą. from Equation (7.11) with n = 1.5, we obtain

/ B | s G \'
pc = 4?r I —J A/2, (7.13)

which replaces the ideal gas relation (7.12). Here pc is independent of 7C, and
the corresponding line in the (log Tc. logpc) plane is horizontal at a height that
increases with mass M, as plotted in Figure 7.4. Strictly, from Equation (7.13) the
central density should vary as the mass squared, but relativistic effects increase
the power. Zones I and II are the only stable regions in the (log T, log p) plane
and hence we need not consider the others.
For relatively low masses, relations (7.12) and (7.13) will merge at the bound­
ary between zones I and II, as shown by the dashed segments in Figure 7.4,
resulting in a continuous bending-path characteristic of each mass. We have seen
in Section 5.4 that the density of degenerate stars tends to infinity as the stellar
mass approaches the critical Chandrasekhar limit Afch - the highest mass that can
be sustained in hydrostatic equilibrium by electron-degeneracy pressure. Thus the
paths corresponding to increasing masses will bend at higher and higher density
values in the (log Tc. log pc) plane, deeper into the region of relativistic degeneracy.
It is easy to see that the limiting case will be represented by a straight line, which
will also mark the division between paths that bend into the degeneracy zone II
and those which remain in zone I. We recall that the boundary between the ideal­
gas zone and the degenerate-electron-gas zone, close to its relativistic part, has a
slope of 3. Since the (log 7’c, log pc) curves in the ideal-gas zone have a slope of 3
as well, there exists a value of M that coincides with the relativistic-degeneracy
boundary. This mass is AfCh. which was obtained by equating the right-hand sides
of Equations (7.4) and (7.11), while the boundary between zones I and III was
obtained by equating the right-hand sides of Equations (7.11) and (7.1). Hence
the boundary between zones 1 and III merges with the path corresponding to A/ch
in the (log Tc, log pc) plane.
In conclusion, a star of fixed mass has its own distinct track in the
(log Tc, log pc) plane, which we shall refer to in the following text as 'I'm- There
are two characteristic shapes of : straight lines for M > A7ch and knee-shaped
ones for M < Meh- In general terms, we may understand the relationship between
tracks corresponding to different masses as follows. With increasing stellar mass,
the gravitational pull toward the centre becomes stronger. Hence a higher pres­
sure is required to counterbalance gravity. This may be achieved in an ideal gas
by a higher density or a higher temperature. A higher density implies, however,
smaller distances between material particles, which further enhance the gravi­
tational field. In fact, since the hydrostatic pressure is proportional to a higher
power of the density than is the gas pressure (4/3 as compared to 1), a higher
 7.3 The evolution of a star, as viewed from its centre 113

CO
E

o
o

8
Log [T(K)1

Figure 7.5 Schematic illustration of the evolution of stars according to their central
temperature-density tracks.

density would only worsen the imbalance. Thus a lower density or a higher tem­
perature are required for equilibrium, if the stellar mass is increased. In the case
of a degenerate electron gas, the temperature plays a far less important role. But
now the hydrostatic pressure is proportional to a lower power of the density than
is the gas pressure (4/3 as compared to ~5/3), so that a higher density is needed
for equilibrium in a more massive star.
The question we now have to answer is, ‘Where docs the evolutionary course
of a star lead the central point along a track?’

7.3 The evolution of a star, as viewed from its centre

Combining Figures 7.1 to 7.4 into one picture, we obtain a full, albeit schematic,
view of stellar evolution, as shown in Figure 7.5. We may now choose a mass
M. identify its path (marked in Figure 7.5 by the value of M). and follow the
journey of the (log Tc. log pc) point along it, to discover what it encounters on its
way. Stars form in gaseous clouds, where densities and temperatures are much
lower than those prevailing in stellar interiors; therefore, the starting point is on
the lower part of the path. At the beginning, a star radiates energy without an
internal energy source, which means that it contracts and heats up (as we have
114 7 The evolution of stars - a schematic picture

seen in Chapter 2 and again in Chapter 6). Hence in the (logTc, logpc) plane
the central point ascends along - which we recall to be a straight line of
slope 3 - toward higher temperatures and densities. Eventually, it will cross the
first nuclear burning threshold. At this point in the evolution of a star hydrogen is
ignited at the centre and the star adjusts into thermal equilibrium with Lnuc and L in
balance. The journey of the central point along T.v comes to a very long pause. Wc
note that for low masses crosses the threshold on the upper part, corresponding
to the p — p chain, whereas for high masses the threshold is crossed on the lower
part, corresponding to the CNO cycle. We should therefore expect stars to burn
hydrogen by different processes according to their masses.
It was shown in the previous section that the boundary between the ideal-gas
zone and the radiation-pressure-instability zone has a slope of 3 (regardless of
the criterion adopted for its definition), the same as the slopes of the curves.
Hence as the mass increases. inevitably approaches this boundary. This means
that in massive stars radiation pressure becomes progressively more important and
eventually dominates gas pressure. Since a star dominated by radiation pressure
is dynamically unstable (becoming unbound), an upper limit thus emerges for the
stellar mass, roughly near (or somewhat above) l()()M,3, as marked by the curve
,wmax
in Figure
©
7.5.
A lower limit for the stellar mass range may also be inferred from the
(log Tc, log pc) diagram. The hydrogen-burning threshold does not extend to tem­
peratures below a few times 106K (see Figure 7.2). The highest value of M
for which still touches this threshold, before bending into the degeneracy­
pressure zone, may be regarded as the lower stellar mass limit, marked T,,Wmin in
Figure 7.5. Objects of mass below this limit will never ignite hydrogen nor any
other nuclear fuel; they will first contract and heat up and will then contract more
slowly while cooling off. Such objects do not fit into our definition of stars (sec
Chapter 1). Based on Figure 7.5, the lower mass limit fora star is somewhat below
0.1 Mq.
When the hydrogen supply in the stellar core is finally exhausted, the star
loses energy again, and the core contracts and heats up. The central point resumes
its climb up the path. For low mass stars, will soon cross the degeneracy­
pressure boundary and bend to the left into a horizontal line. The pressure exerted
by the degenerate electron gas has become sufficient for counteracting gravity.
The contraction slows down and the star cools while radiating the accumulated
thermal energy, tending to a constant density (and radius), determined by M. The
higher the mass, the higher will be the final density and the lower the final radius.
For higher M, 'P.w will cross the next nuclear burning threshold. Helium
now ignites in the core and another phase of thermal equilibrium is established,
marking the beginning of another pause in the journey of the central point. We
note that among the paths that cross the helium burning threshold, those
corresponding to low masses do so very close to the degeneracy boundary. Wc
 7.3 The evolution of a star, as viewed from its centre 115

may expect to encounter some form of thermonuclear instability in stars of such

mass.
The story repeats itself after the exhaustion of helium: the lower-mass stars
among those which have burnt helium contract, develop electron-degenerate cores,
and start cooling. Contraction stops when the final density (and radius) is reached,
as determined by the mass M. We have thus identified two classes of compact,
cooling stars: one including stars of very low mass, made predominantly of
helium, and another including more massive stars composed (at least partly) of
helium-burning products, carbon and oxygen.
The dividing line between stars that eventually become degenerate and cool
off as compact objects and those which remain in the ideal gas state due to their
high temperatures, even when reaching high densities, corresponds, as we have
seen, to 4/.wCh. In principle, for M — Meh contraction may go on indefinitely on
the borderline of dynamical stability. However. crosses the carbon-burning
threshold very near the degeneracy zone. This indicates that carbon ignites in
a highly degenerate material. Nuclear burning is thermally unstable under such
circumstances (sec Section 6.2), and should result in a thermonuclear runaway -
or carbon detonation - that is bound to have cataclysmic consequences. Such a
fate is. however, only hypothetical for the single isolated stars of fixed mass that
we are considering, since the probability of a star’s having been born with a mass
of (or almost) Meh is negligibly small. In reality, stellar collapse due to carbon
detonation is prone to occur as a result of mass exchange in a close binary* system.
For M above the critical value of ~ 1.46 the central point will continue
its journey up 4';W, stopping temporarily when a nuclear burning threshold is
crossed. Note that all T.w paths with M > Ma terminate at the unstable iron-
photodisintegration boundary. Thus massive stars undergo contraction and heat­
ing phases alternating with thermal-equilibrium burning of heavier and heavier
nuclear fuels until their cores consist of iron. Further heating ofthe iron inevitably
leads to its photodisintegration, which is a highly unstable process. We therefore
expect the life of these stars to end in a catastrophe!
For very large M. the paths enter the pair-production-instability zone
before crossing the burning thresholds of heavy elements. Thus very* massive stars
arc expected to be extremely short-lived, developing pair-production instability
that should result in a catastrophic event at early stages of their lives. In conclusion,
two main types of catastrophic events are expected to terminate the lives of
relatively massive stars: carbon detonation and iron photodisintegration (and,
possibly, a third - caused by pair production). We may note, in passing, that
the paths leading to carbon detonation and iron photodisintegration meet at the
upper right corner of the (log Tc, logpc) diagram (since collapse caused by the
latter implies an almost vertical ascent of Tv within the instability strip). Thus
the outcomes of the two different types of instability should have a great deal in
common. We shall return to this speculative point later on.
116 7 The evolution of stars - a schematic picture

To summarize, stellar masses are confined to a range spanning about three

orders of magnitude, between A/injn ~ 0.1 MG and Mmax ~ 100A/Q. All stars
undergo hydrogen burning at their centres and since hydrogen is the most potent
of the nuclear fuels, we expect central hydrogen burning to be the most common
and long-lived state of stars in general. Evolution following hydrogen exhaustion
proceeds differently for stars of different masses. Those under the critical mass
Meh ~ 1 -46 contract and cool off cither after the completion of hydrogen burn­
ing or after the completion of helium burning. In stars near the critical mass, carbon
detonation leads to thermonuclear instability that should end in collapse. Stars
above the critical mass undergo all the nuclear burning processes, ending with
iron synthesis. Subsequent heating of the iron core develops into a highly unstable
state, expected to end in a catastrophic collapse or explosion (or both). Stars of
very high mass may reach dynamical instability sooner, due to pair production.
How docs this picture relate to the realm of observed stars? The most com­
mon among observed stars are the main-sequence ones, mentioned in Chapter 1.
May we deduce that these are stars burning hydrogen in their cores? To prove
this inference, we have to show that for hydrogen-burning stars a correlation
exists between luminosity and effective temperature, of the kind that defines the
main sequence in the H-R diagram. This will be done in the next section. The
identification of the compact cooling stars, corresponding to the horizontal part
of Tv/ tracks, with the observed white dwarfs is quite straightforward. It will be
pursued in more detail in Chapter 9. Indeed, according to observations, one dis­
tinguishes between two types of white dwarfs: low-mass ones and more massive
ones; also, the two types differ in composition, although the connection between
the observed surface composition and that of the interior is debatable. Where do
red giants fit into this picture? May we guess that they should be associated with
that phase of evolution following hydrogen exhaustion, when the core contracts
toward the next core burning episode? This puzzling question will be addressed
shortly (Section 7.5).
Finally, observed stellar explosions - supernovae - arc of two distinct types,
termed Type 1 and Type IT with possible subdivisions (Type la, lb, etc.). One may
be tempted, even at this early stage in our understanding of stellar evolution, to
associate one type with carbon detonation and the other with iron photodisinte­
gration. By analysing in more detail the properties of each, observationally as
well as theoretically, we shall show in Chapter 10 that this, indeed, is the case.

7.4 The theory of the main sequence

Observationally, the main sequence is defined by an empirical relation between

the luminosity and the effective temperature of a group of stars called, accordingly.
 7.4 The theory of the main sequence 117

main-sequence stars. This relation has the form

log L = a log Teff + constant. (7.14)

where the slope a is shallower at the lower end (low L) and becomes steeper
at large L. Another property of main-sequence stars is an apparent correlation
between mass and luminosity, also in the form of a power law:

LocMv (7.15)

(sec Figure (1.6) in Chapter 1). Our hypothesis based on theory is that main-
sequence stars are those stars that burn hydrogen in their cores, their centres lying
along the hydrogen-burning threshold in Figure 7.5, where the paths intersect
this threshold. Wc therefore have to prove that for such stars a correlation of
the type (7.14) exists and an additional one between mass and luminosity, like
correlation (7.15).
Consider stars that have begun burning hydrogen at the centre and are in
thermal and hydrostatic equilibrium. We may take their composition to be uniform
throughout, equal to the initial composition that we have already assumed to be
shared by all stars (sec Chapter 1). Provided they are in radiative equilibrium,
their structure is described by Equations (5.1 )—(5.7). With the further assumptions
of (a) negligible radiation pressure, and (b) an analytic opacity law (for the sake
of simplicity, wc shall adopt a constant opacity), these equations become

dP Gm
(7.16)
dm 4irr4

dr 1
(7.17)
dm 4?rr2/>

di 3 k F
(7.18)
dm 4ac T2 (47tr2)2

— qr>pT" (7.19)

P = —pT. (7.20)
P

to be solved for r(m), P(m), p(m), T(m) and F(m) in the range 0 < m < M.
for any value of the mass M, which is the only free parameter. Is it possible
to learn something about the characteristics of these solutions without actually
solving this complicated set of nonlinear differential equations? As in other cases
of complex physical systems, a great deal may be learned from the dimensional
analysis of the equations.
118 7 The evolution of stars - a schematic picture

First, we define a dimensionless variable x, the fractional mass:

m
x = —. (7.21)
M
The functions r(/n), P(m\ p(m), T(m) and F(m) may then be replaced by dimen­
sionless functions of x - /i(x), f2(x) and so forth - by the following definitions:

r = /,(x)/?. (7.22)

P = f2MPt (7.23)

P = fjUlp. (7.24)

T = ./4(-r)7; (7.25)

F = /5(x)F„ (7.26)

where the starred coefficients have the dimensions of the original functions,
respectively.
Next, by substituting relations (7.21) to (7.23) into Equation (7.16), we obtain

P„df2 _ GMx
(7.27)
M dx ~ MtftRf

In a physical equation the dimensions on the two sides must match, and hence in
(7.27), wherex, f\ and are dimensionless, P* must be proportional to GM1/Rf
Adopting (without loss of generality) a proportionality constant of unity, we may
separate Equation (7.27) into

dfr x GM1
— —---------- pt —------- (7.28)
dx Rj

and repeating the procedure for Equation (7.17), then Equations (7.20), (7.18)
and (7.19),

df\ _ 1___ _ M_
(7.29)
dx Ry.

fl = hfx T-=^
(7.30)
Jlp.
dfx _ 3/5 F _ac TfRt
(7.31)
dx 4/43(47r/,2)2 ’ k M

C£=flfZ F^qop.T/'M.
(7.32)

On the left of Equations (7.28)—(7.32) we have a set of nonlinear differential

equations, now independent of M. for the variable functions f\s that have been
defined in the range 0 < x < 1 by Equations (7.22)—(7.26). The dimensional
coefficients that appear on the right-hand side of Equations (7.22)-(7.26) are
 7A The theory of the main sequence 119

obtained as functions of the stellar mass M by solving the set of algebraic

equations on the right of Equations (7.28)—(7.32). Combining the solutions of the
differential equations and the algebraic equations, we may obtain from Equations
(7.22)-(7.26) the profiles of any physical characteristic (temperature, density,
pressure, etc.) for any value of M. The important conclusion is that the shape
of the profiles as a function of the fractional mass is the same in all stars, the
profiles differing only by a constant factor determined by the mass. This similarity
property is called homology.
By solving only the simple set of relations between the starred quantities, we
may therefore derive the dependence of physical properties on the stellar mass,
without actually solving the differential equations. Substituting Equations (7.28)
and (7.29) into Equation (7.30), we obtain

and inserting this relation, in turn, into Equation (7.31), we have

ac /iiG \4 ,
F. = - P— (7.34)
K \ K /

Thus fluxes at a given fractional mass in stars of different masses relate as the
cube of the mass ratio. For example, the radiative flux across a spherical surface
enclosing, say, half the total mass will be a thousand times larger in a star of 1(WQ
than in a star of 1 Mq. The same applies to any other value of x. In particular, the
surface (x = 1) flux, or the luminosity, will be proportional to the mass cubed,

L a (7.35)

This is the desired relation between luminosity and mass, to be compared with
that derived observationally for main-sequence stars (see below). We recall that a
similar relation emerged from the simple standard model discussed in Chapter 5.
If we retain the dependence on p as well, then relation (7.34) implies L oc A/ja4.
Combining Equations (7.34) and (7.32) and substituting Equations (7.28)-(7.30)
yields the dependence of R„ on the mass M in the form
R. ocA/H, (7.36)

which relates radii corresponding to a given fractional mass in stars of different

masses. This holds, in particular, for x — 1, that is, for the stellar radius R. Hence,
for a large n, such as n 16 corresponding to CNO-cycle hydrogen burning, the
radius will be almost proportional to the mass. For/? = 4 that approximates hydro­
gen burning by the p — p chain, the dependence is weaker. R a . We note
that in all cases the radius increases with increasing mass, in contrast to compact,
degenerate stars (white dwarfs), where the radius is inversely proportional to
some power of the mass. The power in relation (7.36) is always smaller than unity
120 7 The evolution of stars - a schematic picture

(tending to I. in principle, as n oo). Inserting relation (7.36) into Equation

(7.29), we obtain the variation of density with mass M:

p. oc M2^. (7.37)

Since n > 3. the density decreases with increasing stellar mass. Thus stars of low
mass are denser than massive stars at any x, again in contrast to degenerate stars.
That this holds for the stellar centre (x = 0) is obvious from Figure 7.4.

Exercise 7.1: Derive the dependence of the pressure P„ and of the temperature
T. on M (a) in general form; (b) for n = 4 and n = 16.

We are now ready for the crucial test of our hypothesis that Equations (7.16)—
(7.20) may be taken to describe main-sequence stars. In the relation between
luminosity and effective temperature L — 4?r R2ct T^(( (Equation (1.3)) the radius
R may be eliminated, using relations (7.35) and (7.36), to obtain

a (7.38)

Taking logarithms on both sides, we have for n — 4

log L = 5.6 log Teff + constant, (7.39)

while for n = 16

log L — 8.4 log Tcff + constant. (7.40)

These are the calculated slopes for the lower part (low L and M) and for the
upper part (high L and M) of the main sequence in the (log Tcff, log L) diagram,
shallow on the lower part and much steeper on the upper part, as those derived
observationally.
Other characteristics of the main sequence are also readily explained. The
nuclear energy reservoirof a star is, obviously, proportional to its mass. In thermal
equilibrium the rate of consumption of the nuclear fuel is equal to the rate of energy
release L. Hence the duration of the main sequence (hydrogen-burning) stage,
Ims should roughly satisfy

^ms oc — oc M , (7.41)

where we have used relation (7.35). The larger the stellar mass, the shorter the time
spent by the star on the main sequence (burning hydrogen). This explains why in
an ensemble of stars born at the same time, the more massive among them leave
the main sequence earlier. With the passage of time, the main sequence of this
ensemble becomes gradually shorter, as stars that are less and less massive leave
it. This is the reason for the different extent (or upper end) of main sequences
 7.4 The theory of the main sequence 121

corresponding to stellar clusters of different ages, as we have encountered in

Chapter 1.
We have concluded on the basis of the schematic picture of stellar evolution
(Figure 7.5) that there should be a minimal mass for stars capable of igniting
hydrogen. We may now attempt to calculate it more accurately. According to
Equation (7.33) and relation (7.36). the temperature within stars of different
masses varies as M/R, oc A/4/("+3), that is, as M to a positive power. This holds,
in particular, for the central temperature (the highest temperature within a main
sequence star), which thus decreases with decreasing M.

Tc oc M4/7, (7.42)

where we have substituted n = 4. appropriate to low stellar masses (low tem­

peratures). The lowest temperature required for hydrogen burning into helium
is Tmin ~ 4 x IO6 K, applying to the p — p chain. We know that the Sun is a
main sequence star burning hydrogen predominantly via the p — p chain, from
its location in the H-R diagram and from detailed studies of its interior. We may
therefore calibrate relation (7.42):

A (7.43)
Tc.o '

The condition for hydrogen ignition

7c > TTnin

may thus be translated into a condition on the mass

M
(7.44)

yielding A/,nin % 0.1Mofor the estimated Tc,o ~ 1.5 x 10 K. The luminosity

corresponding to this mass may be calculated from relation (7.35) after calibrating
it with the aid of Lo and Mq,

7-min MminY |q-3

(7.45)
7-0 Mo /
thus defining the lower end of the main sequence.

Exercise 7.2: Calculate the effective temperature corresponding to the lower end
of the main sequence.

Exercise 7.3: Using the condition L < Egad (with Z.E(Jd given by Equation (5.37)),
derive an upper limit for the mass and the luminosity of main sequence stars.
Estimate the effective temperature at the upper end of the main sequence.
122 7 The evolution of stars - a schematic picture

Exercise 7.4: Find the relation between L and M and the slope of the main
sequence, assuming an opacity law k — K^pT-112 (the Kramers opacity law)
and n = 4.

We now return briefly to the mass-luminosity relation (7.35). Generally, the

power depends on the adopted opacity law as well as on n, although for the
constant opacity we have assumed - appropriate for electron scattering, which
dominates at high temperatures - the power of 3 is independent of n. For the
Kramers opacity law, appropriate for relatively low temperatures, the power is
Y.'s' (Exercise 7.4), which is close to 5 for it > 4. As we have seen that the
temperature scales with the mass, this explains the changing slope of the observed
mass-luminosity relation (Figure 1.6) from ~5 on the lower part to 3 on the upper
part. The slopes of the main sequence, as derived earlier, would also change to
some extent for a different opacity law, but the upper part would still remain much
steeper than the lower one.
In conclusion, we have succeeded in explaining most features of the observed
main sequence. Furthermore, if we take into account that the initial composition
is not strictly the same for all stars, we also understand why the main sequence
is a strip rather than a line. The hypothesis that main-sequence stars are those
stars that burn hydrogen in a relatively small core has thus turned into the theory
ofthe main sequence. Strictly, the theory applies to the zero-age main sequence,
when the composition is truly homogeneous. And we should bear in mind that it
ignores convection. But why can the simple and quite general procedure employed
be applied only to hydrogen burning? The reason why it cannot be applied to
other types of stars (other stages of evolution) is that the crucial homogeneity
assumption - both of composition and of physical state - is no longer valid at
advanced evolutionary stages.

Exercise 7.5: Repeat the dimensional analysis using the Kramers opacity law
and n = 4, but taking into account the dependence of temperature on the mean
molecular weight p. Derive the scaling laws of stellar properties on the main
sequence with respect to p as well as M.

7.5 Outline of the structure of stars in late evolutionary stages

The same basic diagram that was used to describe the evolution of stars may also
serve to describe the structure of a star at a given evolutionary stage. Consider a
star of mass M\ for any given point m within it we have the value of the local
temperature Tim) and the value of the local density p(m), which define a point
in the (log T, logp) plane. Joining the points corresponding to different values
 7.5 Structure of stars in late evolutionary stages 123

Figure 7.6 Schematic illustration of the stellar configuration in different evolutionary

phases for a 10/W3 star (A, B, C, D, E) and a white dwarf (WD).

of in between 0 and M. wc obtain a parametric line that traces the structure of

the star in the (log T. log p) plane. One end of the line - the central point - lies
on the 'I'm curve; the other - the surface - is characterized by a temperature
considerably below 106K and by a very low density. Hence structure lines run
across the (log?, logp) plane toward the lower left corner. The exact shape of
these lines may be complicated (only polytropes would be described by straight
lines on logarithmic scales) and may change with time - as the central point moves
along Nevertheless, they will invariably lead (more or less) monotonically
from the central point toward low temperatures and densities.
An example is given, schematically, in Figure 7.6 for a star of 1(WG by
a series of structure lines labelled A, B, C,... with origins at a chronological
series of points (labelled O) lying along the evolutionary track T110 of Figure 7.5.
These lines may be taken to roughly represent the evolving structure of the star.
We recognize line A as outlining the main-sequence structure. The next line,
B, describes the star at a later stage, when hydrogen has been depleted in the
core. We note that the conditions for hydrogen burning are now fulfilled at some
point outside the core, where A intersects the hydrogen-burning threshold. Thus
hydrogen burning continues in a shell outside the helium core. The relatively cool
region beyond the burning shell constitutes a chemically homogeneous envelope.
The core itself, now devoid of energy sources, is contracting and heating up.
124 7 The evolution of stars - a schematic picture

Regarding this particular evolutionary phase. Martin Schwarzschild wrote in

his book:

... It would thus clearly be safer if we stopped our discussion of stellar evolution
here and waited for the results from the big computers, which we may expect in
the nearest future. But for those whose curiosity is stronger than their wish for
safety we shall go on - fully aware of the risk.
Martin Schwarzschild: Structure and Evolution ofthe Stars, 1958

So we. too, shall take the risk and go on, the results of numerical calculations
awaiting us in Chapter 9.
Assuming the contraction of the core to occur quasi-statically. on a timescale
which is much longer than the dynamical timescale, the virial theorem may be
assumed to hold. Provided the amount of energy gained (or lost) during this phase
is negligible with respect to the total stellar energy (that is, thermal equilibrium
is maintained), the latter may be assumed to remain constant. As discussed in
Section 2.8, under such circumstances the gravitational potential energy and the
thermal energy are each conserved. Consequently, contraction of the core must
be accompanied by expansion of the envelope, so as to conserve the gravitational
potential energy. At the same time, heating of the core must result in cooling of
the envelope, for the thermal energy to be conserved. In particular, the surface
(effective) temperature drops, the blackbody radiation thus shifting to the red.
The star assumes the appearance of a red giant (RG).
In order to get a rough idea ofthe amount of expansion that might take place,
we may do a very simple exercise: consider two equal mass elements /Sni\ and
Azzzi at a distance r<) from the centre of a star and regard zzz(z-0) as a point mass.
Suppose that one element moves toward the centre, to a distance zq, and the
other outward, to a distance z^. so that the gravitational energy of the system is
conserved. It is easily verified that the distances measured in units of z-o, zq = zq /z*o
and zĄ — ’’i/ro, are related by ŻS = (2 — rj" )“*. We find that when one element
moves inward ~ 10% of z() (zq = 0.9). the other moves outward by about the same
amount (ŻS = 1.13). When the inward displacement is 20% (rq = 0.8). however,
the outward one is more than 30% (rq = 1.33), and the difference increases, r2
tending to infinity as zq approaches half the original distance. This exercise should
not be taken too literally: the gravitational energy is conserved globally, not by
separate mass elements; the motion occurs within the mass of the star and not
outside it, and so forth. Nevertheless, the general conclusion that it was meant to
emphasize - that a moderate amount ofcorc contraction may entail a significant
expansion of the envelope - is true.
If the total energy does not remain constant as assumed, but rather increases
(L nuc > L on the average), then it is easy to sec that the effect of envelope
expansion upon core contraction will be all the more considerable. Therefore
 7.5 Structure of stars in late evolutionary stages 125

the giant dimensions that red giants may reach should not surprise us. We note,
however, that if the total energy of the star decreases while the core contracts
(Lnuc < L), we cannot draw any definite conclusion: the envelope may then either
expand, or remain unchanged, or even contract too. depending on the difference
between the energy drop resulting from core contraction and the overall energy
drop (Lnuc — L). Only detailed stellar-evolution computations can provide the
answer as to the departure from thermal equilibrium (its trend and extent). But
provided Lnue > L, we may safely claim that a star with a contracting core should
evolve into a red giant.

Note: The first detailed calculations of evolving inhomogeneous stellar models,

carried out by Allan Sandage and Martin Schwarzschild in 1952, indeed showed this
effect. The models consisted of a contracting core and an envelope, separated by a
hydrogen burning shell. It was found that *... as the cores contract, the envelopes greatly
expand. Thus from the initial configuration, which is near the main sequence, the stars
evolve rapidly to the right in the H-R diagram....’
It is interesting to note that these calculations were done during the brief period of
time between Salpeter’s solution for the 3a process and Hoyle’s prediction of its resonant
character (see Section 4.5). Thus, at the time, the estimated threshold temperature for
helium ignition was ~2 x 108 K. Sandage and Schwarzschild found, to their disappoint­
ment, that while the cores contract and heat up toward helium ignition, the envelopes
expand way beyond the observed red giant branch. They concluded, or rather specu­
lated. that when the central temperature reaches 1.1 x 108 K. '.. .a physical process not
included in the present computations should start to play an essential role... so as to
halt the contraction of cores and expansion of envelopes. This could have been a sec­
ond, independent argument for postulating a resonant energy level in the carbon nucleus.
Indeed, this level reduces the threshold temperature for the 3a process to about 108 K!

When the helium-ignition temperature is finally reached at the centre, core

contraction stops. The structure of the star is described by line C in Figure 7.6.
Two energy sources are now exploited: the main one, helium burning in the core,
and a secondary one, hydrogen burning in a shell around it. When helium is
exhausted in the core, another phase of core contraction and envelope expansion
sets in. Since the core is now more condensed, envelope expansion is even more
pronounced, turning the star into a supergiant. The structure of the star at this point
is described by line D in Figure 7.6: outside the carbon-oxygen core resulting from
helium burning, we find two burning shells, where D intersects the helium burning
threshold and the hydrogen burning threshold, respectively. The composition
of the star is stratified: enveloping the carbon-oxygon core is a helium layer,
with the helium burning shell between them. The outer boundary of the helium
layer is defined by the hydrogen-burning shell, which separates the helium layer
126 7 The evolution of stars - a schematic picture

from the hydrogen-rich envelope. The hydrogen-burning shell feeds fresh fuel
to the helium-burning one, and so both advance outward. The process is quite
complicated in detail, as we shall see in Chapter 9 (where symbols HB - horizontal
branch - and AGB - asymptotic giant branch - will be explained).
Finally, when all the nuclear processes are over in the stellar core, the structure
of the star, line E in Figure 7.6, is layered like an onion, each layer having
a different composition, with lighter elements lying above heavier ones. The
supergiant (SG) is now a supernova progenitor. The spectacular albeit brief
remainder of its evolutionary course will be discussed in Chapter 10.
A similar chain of arguments may be applied to stars of other masses.

7.6 Shortcomings of the simple stellar evolution picture

It is noteworthy that the very first rough sketch of the global evolution of stars
was outlined by Bethe in 1939(1); this is how Bcthe ended his treatise on energy
production in stars, which paved the way to the modern theory of stellar evolution:

... It is very interesting to ask what will happen to a star when its hydrogen is
almost exhausted. Then, obviously, the energy production can no longer keep
pace with the requirements of equilibrium so that the star will begin to contract.
Gravitational attraction will then supply a large part of the energy. The contraction
will continue until a new equilibrium is reached. For ‘light’ stars of mass less
than 6/1 2 sun masses, the electron gas in the star will become degenerate and a
white-dwarf will result. In the white dwarf state, the necessary energy production
is extremely small so that such a star will have an almost unlimited life....
For heavy stars, it seems that the contraction can only stop when a neutron
core is formed. The difficulties encountered with such a core may not be insuper­
able in our case because most of the hydrogen has already been transformed into
heavier and more stable elements so that the energy evolution at the surface of
the core will be by gravitation rather than by nuclear reactions. However, these
questions obviously require much further investigation.
Hans A. Bcthe: Physical Review, 1939

In the present chapter, we have built a frame for the theme of stellar evolution
and we have outlined a more elaborate sketch (along the same basic lines!), but
the picture is still far from being complete. In order to fill in the details, we shall
have to rely on numerical computations of stellar evolution - the computational
laboratories of stars. This will be the subject of Chapter 9, but in order to assess
the authenticity of our sketch, the results of complex numerical calculations for
the evolution of stars of various masses, as they appear in the (log 7^, logpc)
plane, are shown in Figure 7.7. The general trend is remarkably similar to that of
 7.6 Shortcomings of the simple stellar evolution picture 127

Log [Tc (K)]

Figure 7.7 Relation of central density to central temperature obtained from complex
numerical calculations of the evolution of stars of various masses, as marked (adapted
from A. Kovetz, O. Yaron and D. Prialnik (2009). Mon. Not. Roy. Astron. Soc., 395).

Figure 7.5 obtained on the basis of simple arguments. Leaving aside the deviations
associated with the ignition of a nuclear fuel (in particular the expected explosive
helium ignition at the centre of the 1 Mq star), we may be surprised to discover
that stars as massive as 8M3, and perhaps up to IOMq, end their lives as white
dwarfs. We have expected this to happen only below Afch!
This points out the fallacy of our assumption concerning the conservation
of the stellar mass during evolution. We should have suspected this assumption
to be wrong, especially for massive stars, from the conclusions of the standard
model (Section 5.6). Observational considerations, too, suggest that mass loss
must occur. Several low-mass white dwarfs in the solar neighbourhood, with
accurately determined masses (0.4A/o or less) are long known. If stars conserved
their masses, it would follow that the Galaxy is old enough for stars of 0.4MQ or
less to have evolved off the main sequence. We should then expect to encounter
at least some star clusters with main sequences ending at luminosities below that
corresponding to a mass of 0.4Mo, but no such clusters are known. In fact, the
main sequences of all known clusters extend to considerably higher luminosities
128 7 The evolution of stars - a schematic picture

(corresponding to masses above 0.7 A/o), indicating a younger age. It would be

very difficult to explain why all star clusters should be much younger than the
Galaxy within which they reside. It is far more natural to assume that the Galaxy
is about as old as its oldest clusters, which forces us to conclude that stars lose
mass, particularly after leaving the main sequence.

... We are forced to accept the short time scales for most clusters and look for
processes by which a massive evolved star is able to lose a large fraction of its
mass, so it can settle down into a cooling white dwarf; thus, we link the problem
of the origin of white dwarfs with that of the ultimate fate of stars well above the
Chandrasekhar limit.
Leon Mestel: The Theory of White Dwarfs, 1965

Nowadays, when modern telescopes are able to detect white dwarfs in dense
globular clusters, this argument is even stronger: the white dw'arfs have lower
masses than main-sequence stars of the same cluster.
As it turns out, stars lose a significant fraction of their masses by a stellar wind,
such as that emanating from the Sun, only much more substantially in the case of
massive stars, where radiation pressure is considerable. Hence the evolutionary
paths U'm in Figure 7.5, should have increasingly steeper slopes, as the initial mass
M increases. This means that stars initially more massive than Afci, may become
white dwarfs, their T paths shifting quickly toward paths corresponding to lower
and lower masses, the evolutionary course being very similar to that described
earlier for a mass of about 1M3. Therefore the general picture remains valid,
except that the dividing mass between stars that will end up as white dwarfs and
stars that will become supernovae is, in reality, higher than 1.46Afo. To determine
how high, a model of mass loss is required. This will be the subject of the next
chapter.

Exercise 7.6: Consider the hypothetical evolution of a star of initial mass Mq. The
star’s core grows in mass as a result of nuclear burning. The nuclear processes
release an amount of energy Q per gram of burnt material. The star loses mass
(by means of a stellar wind) at a rate proportional to its constant luminosity
L: M = —aL. (a) Find the mass of the core as a function of time, A/C(r), assuming
that Afc(0) = 0. (b) Find the mass of the envelope as a function of time. Me(t),
noting that M(,(0) = Mq. (c) What is the core mass when the envelope mass
vanishes? (d) Calculate the upper limit of Mo, for which the star will become
a white dwarf, given Q = 5 x 1014 J kg-1 (from turning solar composition into
carbon and oxygen) and a = 10_|4kgJ_|.

Another process that has been neglected is neutrino emission in dense cores,
which has a marked cooling effect. As the rise in temperature between late
 7.6 Shortcomings of the simple stellar evolution picture 129

burning stages is impeded by neutrino cooling, the slopes of the curves

should become somewhat steeper than 3. However, this effect docs not alter any
of our conclusions.
The main shortcomings of the simple picture arc (a) the total lack of time
spans for the different processes and (b) the ignorance of the outward appearance
of the star at each stage. Both factors render a comparison with observations
impossible (statistically as well as individually). Since our main purpose is to
reproduce as accurately as possible the observed stellar characteristics - not only
their trends - we must resort to detailed stellar models. Having acquired a basic
understanding of the principles involved, we may expect a smooth sail through
the ocean of evolutionary computations that we shall reach in Chapter 9.
 8

Mass loss from stars

8.1 Observational evidence of mass loss

It is an acknowledged fact that stars lose mass. In addition to the outflow of

photons, there usually is an outflow of material particles. But unlike the flow of
radiation, which is supplied by energy generation in the interior, the flow of mass
is not replenished. As a result, the stellar mass decreases at a rate that is usually
measured in solar masses per year and denoted by M, where the negative sign
is omitted. Shedding of mass may take two forms: a sudden ejection of a mass
shell, usually following an explosion, or a continuous flow, usually referred to as
a wind. We shall deal with explosive mass ejection in Chapter 10, and devote the
present discussion to stellar winds.
Indirect evidence for mass loss was brought in the previous chapter and
theoretical indication for its probable occurrence was mentioned in Chapter 5.
There is, however, direct observational evidence for continuous rapid expansion
of the outer layers of stars beyond the stellar photosphere that marks the outer
edge, and into the interstellar medium. The most common is exhibited by a
characteristic shape of spectral lines, known as P-Cygni lines, named after the
star P Cygni - one of the brightest in our Galaxy, discovered in 1600 as a new star
(sec upcoming Chapters 10 and 11) - where they are prominent. A P-Cygni line
profile consists of a blue-shifted absorption component and a red-shifted emission
component.
To understand this peculiar profile, imagine a spherically symmetric outflow
from a star. Assume the star emits radiation at some wavelength Ao (at rest).
This radiation is scattered by the outflowing gas and since the gas velocities with
respect to an observer range from —v to +v, where v is the expansion velocity,
the emission line will appear symmetrically broadened - red-shifted and blue-
shifted - on both sides of Xq, as a result of the Doppler effect. The region along
the line of sight of the star, where the gas velocity is positive (directed toward
the observer) will scatter the radiation out, while the star is occulting the region

130
 8.2 The mass loss equations 131

Figure 8.1 Line spectrum of P Cygni, where P-Cygni profiles are apparent.

where velocities are negative. This will result in depletion of only blue-shifted
radiation, that is, blue-shifted absorption superposed on the broadened emission.
Examples of P-Cygni lines are shown in Figure 8.1. where a portion of the line
spectrum of the star P Cygni is shown; the typical profiles are apparent for three
lines.
In a complicated way, the analysis of the detailed shape of a P-Cygni line and
its intensity enables the derivation of the gas density, as well as its velocity and
radial distance from the star. As we shall show below, these lead to the estimation
of the mass-loss rate. Mass-loss rate estimates are also possible based on other
kinds of spectral lines. We shall refrain from spectral-line analysis here, and only
mention the significant result that measured mass-loss rates vary over a very wide
range of values: from ~10-14 to ~10-4 MQ yr-1, depending on stellar mass and
evolutionary stage.

8.2 The mass loss equations

The outward flow of mass is generated in the outermost layers of the star, usually
referred to as the stellar atmosphere, while the bulk of the star maintains hydro­
static equilibrium and retains its size. This is beautifully apparent, for example,
during a full solar eclipse, when the disc of the Sun is occulted and the corona
132 8 Mass loss from stars

Figure 8.2 Total solar eclipse of August 11, 1999. (Photograph by Fred Espcnak).

becomes visible, as shown in Figure 8.2. Thus at the base of the wind region the
velocity must be vanishingly small.
Consider mass outflow in the outer layers of a star of mass M, under the
assumption that the mass of these layers is negligible compared with the total
mass of the star. The basic assumption of spherical symmetry introduced in
Chapter 1 still holds, hence the flow is radial. The mass enclosed in a sphere of
radius r can no longer serve as space variable, since it is allowed to How, and
the conservation laws applied to fixed mass elements in Chapter 2 will have to
be adapted to mass flow. In this case, it is sometimes convenient to adopt the
volume V enclosed in a sphere of radius r as the independent space variable.
The equations of mass, momentum and energy conservation that we derived in
Chapter 2 now have to be reformulated.

Mass conservation - the equation of continuity

Consider a small volume element dV between radii r and r + dr, or volumes

V and V + dV (dV = 4?ir2dr), over which physical properties may be taken
as uniform. The mass contained in this volume is pdV. Over a small period
of time 8t, the change in mass will be 8(pdV) — dV8p. This change will be
caused by mass flowing into and out of the volume element. Let J be the
amount of mass that crosses a spherical surface of radius r per unit time.
 8.2 The mass loss equations 133

Then J = Mrr2pv, where u is the flow velocity, taking the positive direction
outward. Thus

dV8p = J(V)8t - J(V+dV)8t (8.1)

and passing to the limit <5r —> 0 and r/V -> 0. we obtain the equation of mass
conservation, also known as the continuity equation

where the second term on the left-hand side is. in fact, the divergence of the mass
flux, (l/r2)d(r2pv)/3r.
In what follows wc shall consider steadyflows, where the local density remains
constant in time. Thus

— = 0, (8.3)
dt
which implies that J is constant, that is, does not change with radial distance,
and is thus equal everywhere to the amount of mass lost by the star per unit time.

J = 4td-2 pv = M = constant. (8.4)

Momentum conservation - Euler’s equation

Conservation of momentum, as derived from Newton’s second law (see

Section 2.3). requires the calculation of the rate of change of the velocity of
a given mass element (the Lagrangian derivative). Since now mass elements are
no longer fixed in space, whereas the temporal derivative has to be expressed in
terms of quantities at a given point in space (the Eulerian derivative), the rate of
change of a property f of a moving mass element is given by

;)/ + V dr (8.5)

Used for the velocity v, it leads to Euler’s equation (first obtained by Leonhard
Euler in 1755),
du dv 1 dP GM
— H- v — —---------- — —“— (8.6)
dt dr p dr r1
For steady flows all properties are constant in time at any given point (radial
distance), although they may change with r, thus df/dt — 0 for any function f.
In particular, dv/dr = 0 and hence
dv 1 dP GM
v--- =------------------ — (8.7)
dr p dr r2
134 8 Mass loss from stars

Energy conservation

The change in internal energy is given by the first law of thermodynamics,

Equation (2.6) derived in Section 2.2 and involves temporal derivatives for a
fixed mass. Using again relation (8.5), assuming that there is no nuclear energy
generation within the volume and recalling that F is the heat flowing per unit
time across a spherical surface of radius r, we obtain the equation of energy
conservation in the form
i)w 'du d 1 dF
+ v—+ + VT-
di dt dr 4nr2p dr

For a steady flow, the first and third terms on the left-hand side vanish and hence

du d /1\ 1 dF
VT = ~PvT (~ _ 7—2—r- (8 8)
dr dr \p / Mrr~p dr
Multiplying Equation (8.7) by v and adding Equations (8.7) and (8.8), which will
now have the same dimension, we obtain
/ dv du\ [\dP GM d /1 \"| 1 dF
v I v~j—F — I — —v —-—I-------—F P — I - I — -—— —. (8.9)
\ dr dr / [_p dr rz dr \p) J Mir-p dr
Noting that

GM_ d (GM\
r2 dr \ r )

1 dP d ( \\ d ( P\
------- + P— - =— -
p dr------- dr \ p / dr \p /

and multiplying Equation (8.9) by 4?rr2p, we finally obtain

.2^/12 P GM\ dF n
Mtr~pv— I |tr + u -I---------------- ) + —— = 0. (8.10)
dr p r J dr

Since 4?rr2pv = M is constant. Equation (8.10) may be integrated to yield

• /, , P GM\
M I + m 4---------------- + F = constant (8.11)
\ P r )
and F may be expressed using the radiative-transfer equation (5.3), assuming
there is no convective flux in these layers.

The full set of equations

It is reasonable to assume that the gas is ideal and the composition homogeneous
in the outer layers of a star, and hence, using Equations (3.44). (3.47) and (5.3),
 8.2 The mass loss equations 135

we may substitute

U = 1^+3^S and />,,„ = l<,r> (8.12)

2 P P
H/Prad \dPTAidT kF
= = -. ( 0. 1 3)
p dr------ p dT dr--------- 4ncr2
In summary, the full set of equations to be solved for v(r), p(r). T’(r) and F(r) is

4nr2pv = M = constant (8.14)

dv \dP^ GM / kF \
V--- =--------------------- Z— 1--------------- I (8.15)
dr p dr r1 \ 4ncGM J
■ (\ -> 5 Poas 4a T2 GM\
M I |tr H--------—F ----------------- I + F — constant (8.16)
\ 2 p 3 p r /

— =—(8.17)
dr 4acTy 4rrr2
Besides the two constants, two more conditions, or boundary values, are required
for the integration of the two differential equations. We note that the gravity
term in the momentum equation is diminished by a factor equal to the radiation
Dux divided by the Eddington limiting flux. We should mention that besides the
gravitational force and the force exerted by the radiation pressure, there may be
other forces impeding or driving the mass flow, such as friction, proportional to
the velocity, or acceleration caused by photons in a particular line or band that
are not part of the continuum (black body) radiation.

Historical Note: The concept of corpuscular radiation, or particles streaming out of

the Sun, was proposed by Ludwig Biermann in a paper published in 1951, to explain the
well-known fact that, whether a comet moves towards or away from the Sun, its tail always
points in the antisolar direction. A few years later, Eugene Parker pursued this idea and
developed the first model of what he termed the solar wind. It may be interesting to note that
the novel idea met with strong opposition, so much so that the paper submitted by Parker to
the Astrophysical Journal in 1958, was rejected by two reviewers. It took the intervention
of the editor, Chandrasekhar, whom we have already encountered, to get it published. It
is also noteworthy that this model was not entirely new. It is true that the model considers
only positive velocities describing outward flows. However, the equations depend on
v2 and the same treatment and solutions would apply to inward flows corresponding to
negative velocities. Indeed, Hermann Bondi, studying the problem of spherical accretion
of mass by a star, arrived at the same equations and solutions in 1951, but was apparently
unaware of their implication to the newly postulated effect of mass ejection.

It will not come as a surprise that there is no simple solution for the set of
Equations (8.14)—(8.17). We may go one step further, however. It is easily seen
136 8 Mass loss from stars

that the mass-conservation equation (8.14) may be written as

2 1 dp \ dv
- + --T + --T = ° (8.18)
r p dr v dr
and for an ideal gas, the derivative of the gas pressure as
I dP„as 1 dT 1 dp
---------- = + - —. (8.19)
Pgas dr T dr------ p dr

Eliminating 2 between these equations and substituting the resulting expression

for in the momentum equation (8.15), wc obtain

( 2 2Pgas Po3idT GM / kF \
\ p / dr L pr pT dr r2 \ 4?tcGM '_

This equation has a singularity, that is, dv/dr is undefined when the flow velocity
v is equal to the isothermal sound speed of the gas

(8.21)

Since at the point where v = vs, the left-hand side of Equation (8.20) vanishes,
and since obviously v ± 0 there, the term in square brackets on the right-hand
side must vanish as well. This condition serves to determine the point rc, where
v = vs, known as the critical point, which in our case defines a spherical surface
of radius rc.

Exercise 8.1: Assuming the temperature to be uniform in the flow region (denoted
by Tq), find the critical radius rc. Find the relation between the sound speed and
the escape velocity at rc.

8.3 Solutions to the wind equations - the isothermal case

In order to gain some insight into the properties of stellar winds, avoid­
ing at the same time numerical complications, we now consider the simplest
form of the wind equations for which the solutions can be relatively easily
described. The conclusions will be qualitatively valid for more complicated
cases.
We assume a uniform temperature for an outer region of a star starting at
a radius r0, thus describing an isothermal stellar wind. In this case, strictly, the
radiation flux vanishes, by Equation (8.17). For an ideal gas.
2 JIT
v/ =----- = constant
 8.3 Solutions to the wind equations - the isothermal case 137

and we may express T in terms of vs. The set of wind equations (8.14)—(8.17)
reduces to

4,Tr2pt' = M = constant (8.22)

(8.23)

to be solved for v(r) and p(r). where

GM
(8.24)

is the point at which the right-hand side of Equation (8.23) vanishes.

Equation (8.23) is independent of the other equation of the set and may be solved
for v(r) given a boundary condition, say, v(/'o) = Vo- The full solution requires
an additional constant, say, p(r0) = Po- We shall now show that this solution is
unique, that is, there is only one value of Vo, or equivalently. M, for which a
physical solution exists.
It is convenient to define dimensionless variables
/ \ 2
x = - and v=(-) (8.25)
G \vj
in terms of which Equation (8.23) becomes

dy / 1X4/ 1 \ „_
— (8.26)
dx \ y/ x \ x)

The family of solutions is determined by the initial values of y; the various classes
of solutions are shown in Figure 8.3. We assume that the outer layer of the star,
where the mass flow occurs (or the wind is generated), is tightly bound to the
star, which means that at the base ro of this region vcsc vs uq and therefore
/'o < rc. Hence we start the integration of Equation (8.26) at some x < 1 and
y < 1. The derivative dy/dx is thus positive, and y increases. If y reaches unity
before x does, then x(y) has a maximum at y = 1 and thus y(x) is a multi­
valued function - shown in region 2 of Figure 8.3 - which is unphysical. Equally
unphysical arc the solutions in region 4 of the figure. If, on the other hand, x
reaches unity before y does, then y(x) has a maximum at x = 1. which means
that the velocity will reach a maximum value and decrease thereafter, as shown
by the solutions in region 1 of Figure 8.3. For this to be physically possible,
an inward directed pressure must be exerted on the flowing mass, which may
be shown to surpass the interstellar gas pressure by many orders of magnitude.
Hence this solution is unphysical as well. Solutions appearing in Figure 8.3 that
correspond to infinitely large velocities at the base of the wind, as those of region
3, arc in conflict with the requirement of vanishing velocities there. The only
viable solution, therefore, is that for which uq is such that x and y, starting at
138 8 Mass loss from stars

Figure 8.3 The normalized isothermal wind solutions. The circle marks the sonic point.
The unique viable solution is highlighted in bold.

low values, reach unity simultaneously, which means that u(r) passes through the
point (rc, us). Thus the wind is transonic: subsonic below rc and supersonic above
it.
Integrating Equation (8.26) and applying this condition, we obtain the unique
solution for v(r) as the root of
4
v - In v = 4 In x 4------- 3, (8.27)
x
which defines the only acceptable velocity at the base ro of the isothermal-flow
region for a given temperature of this region and a given stellar mass. With it, the
mass-loss rate is obtained as M — 4?rr^PoVq(T, M). The resulting density profile
is
p(r) r0\2 Vo(7 • M)
(8.28)
Po r/ r(r)
and since v(r) increases with r, the density decreases with radial distance more
steeply than r~2.
In this simple example, the energy conservation requirement (8.16) was dis­
carded. being replaced by the condition 7'(r) — constant, and F = 0 was obtained
from Equation (8.17). Alternatively, we may discard Equation (8.17) and obtain
 8.4 Mass loss estimates 139

the heat flux as the solution of Equation (8.16), which may be written as

M[|(t’(r)2 - i>esc(r)2) + constant] + F = constant.

or, in differential form, as

^ = -|A/-^[u(r)2-L'esc(r)2]. (8.29)
dr - dr
At the base of the flow, vesc vg and the term in brackets is negative. This means
that in order to maintain an isothermal wind, energy must be pumped into the
flow along the way. As r increases, v increases while uesc decreases, hence this
term will monotonically increase (and will eventually become positive). Thus
F / 0, and if, as expected, F = —f(T)dT/dr, where f is some function of T,
the isothermal case is not realistic, unless a driving force operates that pushes the
wind by doing work on it. Nevertheless, detailed numerical and analytical studies
of the wind problem show that it is not so far from reality.

Exercise 8.2: A polytropic wind is defined by the assumption

P = KpY,

where A" is a constant. Hence

T _
n
(a) Show that the solution to the wind velocity equation has the same form as
that obtained for an isothermal wind, with the isothermal sound speed replaced
by

(b) Show that for y = 5/3, one obtains F — constant, which means that the flow
is adiabatic.

8.4 Mass loss estimates

Based on the isothermal wind solution developed in the previous section we may
attempt to evaluate the resulting mass-loss rates. Our free variables are the stellar
mass and the uniform temperature in the wind region. At a given stage ofcvolution,
the stellar mass determines the stellar radius as well. On the main sequence, for
example, relationship (7.36) may be adopted, normalized by the solar values. For
a red giant, an order of magnitude estimate for the radius may be obtained using
Equation (1.3), where we may take the Eddington critical luminosity (5.37) as
an estimate for L, and - the radiation peaking in the red part of the black body
140 8 Mass loss from stars

spectrum - we may use Wien's law (1.4) to obtain an estimate for the effective
temperature, say, 4000 K.
With known M. R and T. the sound speed vs is known and the critical radius
rc may be calculated by Equation (8.24). We thus have the solution v(r) by
denormalizing the dimensionless relation y(.v), the solution of Equation (8.27).
Since M = 4nr2pv is constant, it may be evaluated at any point of the flow. To
do so. however, we need to know the density at that point.
In Section 3.7 we have encountered the concept of optical depth and its
relation to the stellar radius, the radius of the photosphere, where most of the
stellar radiation is emitted into space. Thus, the relation
/•OO
r = / Kpdr — 1
Jr

used as the definition of the photospheric radius R. will supply p(R). The density
profile is given by Equation (8.28) in the form p = Cr-S, where C is a constant,
and 2 < .v < 3 for the supersonic region (for most of the subsonic region. 2 <
s < 4). The opacity k is a function of temperature and density, given generally by
a power law of the form (3.63) and since the temperature is constant and known.
k becomes a function of density only. We thus obtain

/»OO
K{}Th [Cr-'\“+'dr = 1. (8.30)
Jr

where a = h = 0 for electron scattering, and a = 1, b = —7/2 for the Kramers

opacity law (3.65). The integral in Equation (8.30) may be performed to yield

k\T. p(R)]p(R)R * 1

(up to a factor of order unity), from which p(R) is derived for given T. The
velocity at R is given by v = vsS/y(x) for.t — R/rc and thus M may be evaluated
at R.
Examples of such simple evaluations are shown in Figure 8.4 for several
stellar configurations: a main-sequence star of solar mass, a much more massive
main-sequence star, and a red giant of solar mass. We note that the actual measured
values of the solar wind. M = 2 — 3 x 10_|4A/Oyr_| and T % 1.5 x 1()6K. are
not very far from the curve of possible solutions, considering the very wide range
of variation of stellar mass-loss rates. The significant result is that during its
lifetime as a main-sequence star, the sun should lose an insignificant amount of
material particles, amounting to less than a thousandth of its mass. In fact, the
wind mass-loss rate during the main-sequence stage is lower than the rate of
mass loss due to conversion of mass into the energy radiated away by the Sun.
Loc~2 = 6.7 x I0-14A/Oyr-'.
 8.4 Mass loss estimates 141

Red giant wind

-5

■s
Wind from a 30 A40
05
O main-sequence star

-10

Solar
wind
o

4 5 6
Log [T(K)]

Figure 8.4 Estimated mass-loss rates for isothermal winds. Note the point corresponding
to the measured values of the solar wind.

Exercise 8.3: Estimate the rate of mass loss from the Sun. if at Earth the measured
velocity of the solar wind is ~400kms_| and the proton density in the wind is
roughly 7 particles per cm3. Assume spherical symmetry for the wind expansion
away from the Sun.

This is not the case for much more massive stars. For them, not only is the
wind rate orders of magnitude higher than the rate of conversion of mass into
radiated energy, but during the time spent on the main sequence, the star is bound
to lose a significant fraction of its mass. For the example of Figure 8.4. the time
spent by a 30 MQ star on the main sequence is shorter than the Sun’s by a factor
of 3O2, according to relation (7.41), but the mass-loss rate is higher by a factor
~107. Thus the total mass lost during the main-sequence phase will be IO4 times
higher, hence of the order of solar masses. Similarly, a red giant - including the
future Sun - is bound to lose an appreciable amount of mass during the red-giant
stage, even if this stage of evolution lasts only a few' thermal timescales.
Our evaluations arc extremely crude, and the simple solutions obtained may
mislead us into underestimating the problem. Although the multiple solution
142 8 Mass loss from stars

classes, the uniqueness of the viable wind solution and the transonic prop­
erty of the velocity variation with distance are common to a much wider and
less restrictive range of conditions, the mass-loss problem is far from being
solved. Complex cases and solutions are treated in books devoted solely to stellar
winds in general and the solar wind in particular; they are beyond the scope of
this text.

8.5 Empirical solutions

The simple cases that we have addressed have avoided the radiation flow. We have
found that energy must be continually supplied in order to maintain or accelerate
the wind out ofthe gravitational potential well of the star. This is the crux ofthe
problem.
Equation (8.15) shows that the velocity would easily increase with distance,
were the second term on the right-hand side vanishingly small. This term van­
ishes when the radiation flux approaches the Eddington limiting flux, that is,
when the radiation pressure becomes dominant, or /?-»■() in Equation (5.42).
Thus, although the stellar-wind phenomenon is not yet fully solved theoretically,
it is well recognized that mass loss is driven by the increasingly dominant radiation
pressure, as the stellar luminosity approaches the Eddington limit. The importance
of radiation pressure for the ejection of matter by novae was first acknowledged
by William McCrea, as early as 1937 in the context of nova outbursts, which
we shall encounter in Chapter 11. The idea was later pursued in a vast number
of analytical and numerical studies of steady winds. However, it is not always
necessary for the bulk luminosity to approach critical value and thus disrupt
hydrostatic equilibrium (sec Section 5.5). Radiation pressure is capable of accel­
erating material out of the stellar gravitational potential well, even for an overall
state of hydrostatic equilibrium, because material particles vary widely in their
ability to absorb radiation. While the interaction of a particle with a gravitational
field depends solely on the particle’s mass, its interaction with a radiation field
depends on its composition, structure, size and density, as well as on the radiation
wavelength. Thus, if in the outer layers of a star there are such particles that are
exceptionally absorbent at the leading wavelength ofthe photons - as determined
by the temperature - then for these particles the radiation pressure might just
overcome gravity. In other words, the high opacity rather than the photon flux
would cause k F/4ttcGM in Equation (8.15) to reach or exceed unity. The result
would be an outward acceleration leading to a mass outflow of such particles, and
others entrained by them.
By a heuristic argument, at a mass-loss rate M driven by radiation pressure, the
mass Mfit ejected during a time interval fit acquires escape velocity by absorbing a
 8.5 Empirical solutions 143

fraction, say 0', of the momentum carried by the radiation (L/c)8t. Consequently,

MSt vesc -
c
and substituting vjsc = 1GM/R. and (/> = we may write

M =(</>—(8.31)
\ c / GM
Thus the mass-loss rate must have the dimension of LR/GM and this is the
key to empirical formulae used to express M. The transfer of momentum from
the radiation field to mass may be very complicated, involving turbulence, shock
waves or acoustic energy. The dimensionless coefficient in parentheses is diffi­
cult to calculate theoretically, but it may be obtained from the observed global
properties of stars whose mass-loss rates can be measured. We shall return to this
point in the next chapter, where we consider numerical stellar models in the light
of observations.

Exercise 8.4: (a) Estimate the mass-loss timescale, rm_|, and compare it with
the thermal timescale of a star, (b) Show that the rate of energy supply required
for mass loss at a rate M is a very small fraction of L. (c) Find the relation
between the mass-loss timescale and the nuclear timescale of the star and show
that, usually, rm_] < rnuc.

Exercise 8.5: Assuming that the mass-loss rate may be parametrized as in

Equation (8.31): M <x LR/GM, show that for main-sequence stars M oc La
and evaluate a.
 9

The evolution of stars - a

detailed picture

This chapter differs from previous ones by being descriptive rather than analytical.
An account will be given of the evolution of stars as it emerges from full-scale
numerical calculations - solutions of the set of equations (2.54), with accurate
equations of state, opacity coefficients and nuclear reaction rates. Such numerical
studies of stellar evolution date back to the early 1960s. when the first computer
codes for this task were developed. The first to program the evolution of stellar
models on an electronic computer were Brian Haselgrove and Hoyle in 1956.
They adopted a method of direct numerical integration of the equations and
fitting to outer boundary conditions. A much better suited numerical procedure
for the two-boundary value nature of the stellar structure equations (essentially
a relaxation method) was soon proposed by Louis Hcnycy; it is often referred to
as the Henyey method and it has been adopted by most stellar-evolution codes to
this day. Among the numerous calculations performed by many astrophysicists
all over the world since the early 1960s. the lion’s share belongs to Icko Iben Jr.
The detailed results of such computations cannot always be anticipated on the
basis of fundamental principles, and simple, intuitive explanations cannot always
be offered. We must accept the fact that, being highly nonlinear, the evolution
equations may be expected to have quite complicated solutions.
As the complete solutions ofthe evolution equations provide, in particular, the
observable surface properties of stars, we shall focus in this chapter, more than we
have previously done, on the comparison of theoretical results with observations.
The ultimate test to the stellar-evolution theory is the understanding of the H-R
diagram in all its aspects (described briefly in Chapter 1). We thus expect to find
stars in the H-R diagram where theoretical models predict them to be. Moreover,
the basic statistical principle mentioned in Chapter 1 should apply: the longer
an evolutionary phase of an individual star, the larger the number of stars to
be observed in that particular phase. A detailed comparison between theoretical
predictions and observations is thus possible for long evolutionary phases, such as
core-hydrogen burning and. to a lesser extent, core-helium burning. Proceeding

144
 9.1 The Hayashi zone and the pre-main-sequence phase 145

to advanced evolutionary stages, neutrino emission from the dense, hot cores of
massive stars, acting as an efficient energy-removing agent, accelerates the rate
of evolution by requiring an enhanced rate of nuclear energy supply. Hence the
weak nuclear fuel (from carbon to iron, the amount of energy release per unit
mass of burnt material is relatively small) is quickly consumed. Consequently,
the probability of detecting stars during these brief evolutionary phases is low.
Cooling, following the completion of nuclear burning in relatively low-mass stars,
is again a slow process, but cooling stars - white dwarfs - become gradually fainter
and more difficult to detect.

9.1 The Hayashi zone and the pre-main-sequence phase

Chapter 7 dealt with the evolution of stars by following the path of the stellar
centre in the (logT. log p) plane. The present chapter, focusing on the stellar
surface, follows evolutionary tracks in the (log Teff. log L) plane, the theorists’
H-R diagram. In the (log 7”, log p) plane we found zones of instability, which
have constrained the evolutionary paths of stars. We shall now' show that the
(log Tcff. log L) plane has its own ‘forbidden zone". It is known as the Hayashi
forbidden zone and its boundary as the Hayashi track, after Chushiro Hayashi,
w ho was the first to point out and study this type of instability in the early 1960s.
The forbidden zone’s boundary is determined by the hypothetical evolution of a
fully convective star.
Consider a fully convective star of mass M, where convection reaches out
to the stellar photosphere. In Section 6.6 we showed that in a convective zone
the temperature gradient is very closely adiabatic. On the one hand, even a slight
superadiabaticity gives rise to high heat fluxes which reduce the temperature
gradient. On the other hand, subadiabaticity quenches convection and reduces the
heat flux; as a result, the temperature gradient steepens. Therefore, if convection
persists, the temperature gradient remains very close to the adiabatic. Neglecting
the mass and thickness of the photosphere with respect to the stellar mass M
and radius R, we may adopt a very simple description for the interior of a fully
convective star as a polytrope of index n = (ya — I)-1,

P = Kp{+" (9.1)

(see Section 5.3). The coefficient K is related to M and R by Equation (5.23):

K" = C„G"Mn-'Ri-n, (9.2)

the constant C„ depending on n, C„ = yI • We have one free parameter, the

value of R. which will be determined by joining the fully convective interior to the
radiative photosphere at the boundary r — R. The ability of the photosphere to
radiate the energy flux crossing this boundary will depend on the change in
146 9 The evolution of stars - a detailed picture

density, temperature and pressure across it. Hydrostatic equilibrium requires

dP GM
77 * ~P7? (9.3)

and integrating from /?, where the pressure is Pr. to the point where the pressure
vanishes, or, for simplicity, to infinity, we obtain

GM f (X)
= ^L pdr- (9.4)

The temperature at R is the effective temperature of the star, satisfying L =

R-aT7if. The optical depth of the photosphere is of the order of unity (see
Section 3.7), the exact value depending on the type of solution of the radiative
transfer equation. Thus JR Kpdr - k JR pdr = 1, where k is the opacity aver­
aged over the photosphere. Taking k to be the opacity at R and expressing it as
a power law in density pr and temperature TCff of the form (3.63), we have, as a
crude approximation,

KopRTdf ( pdr=\.
(9.5)
JK

Combining Equations (9.4) and (9.5) wc obtain

Pr = r\.Pr e,f' (9.6)

A further relation among pressure, density and temperature at R is provided

by the equation of state, for which we adopt the simplest case of an ideal gas
and negligible radiation pressure, Equation (3.28), Pr = (7?./p.)pRTeff. Wc thus
arrive at a set of four equations, all in the form of products of powers of physical
quantities, which are easily solved when turned into linear logarithmic equations:

log Pr = log M — 2 log R — a log pr — b log Teff + constant (9.7)

n log Pr = (n — 1) log M + (3 — n) log R + (n + 1) log pR + constant (9.8)

log PR — log pR + log Tcff + constant (9.9)

log L = 2 log R + 4 log Teff + constant. (9.10)

By eliminating log/?. logp« and logP«. we obtain a relation between log A.

log Teff and log M. in the form

log L — A log Tcff + B log M + constant. (9.11)

(7 - n)(a + 1) - 4 - a + b (/j - 1)(« + 1) + 1

A =-------------------------------------- . B —--------------------------------- .
0.5(3 — n)(a + 1) — 1 0.5(3 - w)(a + 1) - 1
(9.12)
 9.1 The Hayashi zone and the pre-main-sequence phase 147

which traces a line in the (log7'eff. logL) diagram, the Hayashi track, for each
value of M. These tracks play a similar role to that of the tracks in the
(log T. log p) plane, but they cannot be taken to represent evolutionary paths, as
their counterparts, because the assumptions on which they were derived arc
not generally valid. They represent asymptotes to evolutionary' paths, as we shall
show shortly.
To simplify the discussion, we assume a = 1, which is a reasonable approxi­
mation. The power b, however, may assume a wide range of values, mostly positive
(as seen from Figure 3.3), because photospheric temperatures are relatively low.
The coefficients (9.12) thus reduce to

9 - 2/1 + b 2n — 1
A = —-----------, B = -~------- . (9.13)
2—n 2 — 71

In addition, we recall that the polytropic index is constrained by dynamical sta­

bility to n < 3 (Section 6.3), and hence overall 1.5 < n <3. The first conclusion
to be drawn from Equations (9.11) and (9.13) is that the slope of the Hayashi
track is extremely steep. For b — 4 and n = 1.5, typical of low temperatures, we
obtain A = 20 - an almost vertical line. Consequently, tracks corresponding to
different stellar masses are close to each other, with larger values of M shifting
the lines always to the left. This is because sgn| B | = —sgn[A], and hence for
A > 0 a higher M lowers the line, while for A < 0, a higher M raises it. (We
recall that in the H-R diagram, the effective temperature increases leftward.) We
further note that the slope changes sign for n > 2. Thus, for example, the slope
will be differently inclined for a photosphere of atomic hydrogen (n = 1.5) and
one of molecular hydrogen (n = 2.5). Accurate calculations of convective stellar
models show that the Hayashi track corresponding to a given M changes slope,
bending slightly to the left at low L and shifting to the right at large L.
In order to understand the significance of Hayashi tracks, we characterize a
star by a unique value y, obtained by averaging y — over the entire star.
Similarly, ya denotes the average adiabatic exponent. For a fully convective star,
we obviously have y = ya. If the star has radiative zones, then in some regions
y < ya, and hence y < ya. The corresponding average polytropic index n for such
stars satisfies n > na, where na is taken to denote the adiabatic polytropic index
that defines the Hayashi track. Therefore, y > ya, or n < na, can only arise from
superadiabaticity, which is unstable and hence ‘forbidden’. Now, consider a star
of mass M and luminosity L, whose configuration can be described, as above, by a
polytropc overlaid by a photosphere. How would its effective temperature change
with the polytropic index zi? To answer this question, we have to reconstruct
Equations (9.7)-(9.10), taking account of the dependence of the constants (on
the right-hand sides) on the polytropic index, in order to obtain the function
log Teff(n). When this tedious task is accomplished, the result is d log Teff/dn > 0.
148 9 The evolution of stars - a detailed picture

Consequently, the forbidden zone corresponding to n < na lies to the right of the
Hayashi tracks in the H-R diagram.
The role of Hayashi tracks and forbidden zone is best illustrated by the pre-
main-sequence evolution of stars. The very beginning of a star’s life is marked
by a rapid collapse of an unstable gaseous cloud. The initiation of such a collapse
being a galactic, rather than a stellar process, will be discussed in Chapter 12. At
first, the material is transparent, but as it condenses and its temperature rises, it
eventually becomes opaque. The interior is now shielded and the boundary layer
from which radiation escapes defines a discernible object which will become a
star. This occurs at densities of about 10 l0-10-9kgm-3 and temperatures of
a few hundred degrees Kelvin. Under such conditions hydrogen is in molecular
(H2) form. The gas is too cold to resist the gravitational force and contraction
proceeds, essentially as radial free fall, on the dynamical timescale (of the order
of \/s/Gp, Equation (2.57)). We note that this timescale is considerably longer
than that typical of mature stars, densities being so much lower. The rising gas
temperature becomes, eventually, high enough for dissociation of the hydrogen
molecules to take place, then for ionization of the hydrogen atoms and, finally,
for ionization of the helium atoms. All these processes absorb a vast amount of
energy, which is supplied by the gravitational energy released in contraction. The
gas temperature is now prevented from further increase, much in the same way as
the temperature of boiling water remains constant, although energy is continually
supplied to it to keep it boiling. Thus free fall continues throughout these stages.
When ionization of hydrogen and helium is almost complete, the gas temperature
increases again due to the release of gravitational energy. There comes a time
when it generates sufficient pressure to oppose the gravitational pull and a state of
hydrostatic equilibrium is established. The gaseous condensation has now become
a protostar.
A rough estimate of protostellar characteristics may be obtained by assuming
that all the gravitational energy released in collapse to the protostellar radius /?ps
practically from infinity,aGM-/R[n. was absorbed in dissociation of molecular
hydrogen and ionization of hydrogen and helium, although in reality a fraction was
emitted as radiation. Denoting by xh2 the dissociation potential of H2 (4.5 eV), by
Xh the ionization potential of hydrogen (13.6 eV) and by xne the total ionization
potential of helium (79 eV — 24.6 eV + 54.4 eV), we have

GM2 M (X Y \
%— —Xh. + *Xh + -7 XHe (9.14)
Rps wH \ 2 4 /

and taking Y 1 — X and a we obtain

ftps 50 M
— ------------------- . (9.15)
Rq 1 - 0.2X ’
 9.1 The Hayashi zone and the pre-main-sequence phase 149

The protostar being in hydrostatic equilibrium, an average internal temperature

may be estimated from the virial theorem, as in Section 2.4 (Equation (2.29)).
Using approximation (9.15) with X 0.7, we obtain

a n GMmH
6 x IO4K, (9.16)
3 I /?ps

independent of the stellar mass. At this temperature the opacity is still very high
(see Figure 3.3), the flow of radiation is hindered, and hence the protostar is
fully convective. This is the starting point of the Hayashi evolutionary phase. In
the (log Tefi. log L) diagram the star descends along its Hayashi track at almost
constant effective temperature, its radius decreasing steadily and its luminosity
decreasing, roughly as R2. In time, as the internal temperature continues to rise,
ionization is completed and the opacity drops. The convective zone recedes from
the centre and the star moves away from the Hayashi track toward higher effective
temperatures. The increasing core temperatures cause nuclear reactions to start,
slowly at first, far from thermal equilibrium, but gaining in intensity. This causes
the stellar luminosity to reverse its trend and start rising. The evolution toward
thermal equilibrium is complicated by the gradual ignition of different reactions
of the hydrogen-burning chains. This is illustrated in Figure 9.1 by the winding
paths traced by stars of various masses in the (log Tcn. log L) diagram, obtained
from detailed evolutionary calculations. The corresponding time intervals are
listed in Table 9.1.
The relevant timescale throughout the protostellar stages is the relatively short
Kelvin-Helmholtz (thermal) timescale given by Equation (2.59). Stars in the pre-
main-sequence evolutionary phase are hard to detect not only because they are
scarce, this phase being relatively short, but also because they are still shrouded in
the remains of the cloud out of which they were formed. The less massive among
them, which evolve more slowly, appear as highly variable mass ejecting objects,
known as T Tauri stars. They are surrounded by circumstellar discs, probable sites
of planet formation, which are estimated to dissipate on timescales of up to IO7 yr.
An example of jets of material ejected by a young star hidden in a nebula of gas
and dust is shown in Figure 9.2.
Only on the main sequence will the evolutionary timescale finally shift to
the nuclear one and will stars become numerous. Contraction toward the main
sequence takes up less than 1% of a star’s life; in contrast, the star will spend
about 80% of its life as a main-sequence star. For example, a star of 1 MQ spends
3 x 107 yr contracting prior to hydrogen ignition, in contrast to the IO10yr it
spends burning hydrogen in the core. For initially more massive stars the time­
scales shrink significantly: thus for a 9Mq star the contraction phase takes only
about 105 yr, and the main-sequence phase 2 x IO7 yr.
 Table 9.1 Evolutionary lifetimes (years)

1-2 2-3 3-4 4-5

15 6.7(2) 2.6(4) 1.3(4) 6.0(3)

9 1.4(3) 7.8(4) 2.3(4) 1.8(4)
5 2.9(4) 2.8(5) 7.4(4) 6.8(4)
3 2.1(5) 1.0(6) 2.2(5) 2.8(5)
2.25 5.9(5) 2.2(6) 5.0(5) 6.7(5)
1.5 2.4(6) 6.3(6) 1.8(6) 3.0(6)
1.25 4.0(6) 1.0(7) 3.5(6) 1.0(7)
1.0 8.9(6) 1.6(7) 8.9(6) 1.6(7)
0.5 1.6(8)

Note: powers of 10 are given in parentheses.

Log(Teff)

Figure 9.1 Evolutionary paths in the H-R diagram for stars of different initial mass (as
marked) during the pre-main-sequence phase. The shade of segments is indicative of the
time spent in each phase, ranging from less than IO3 yr (light) to more than 107 yr (dark),
as given in Table 9.1 (adapted from I. Iben Jr. (1965), Astrophys. J., 141).
 9.2 The main-sequence phase 151

Figure 9.2 Jet of gas, one-half ly long, ejected by a young star, bursting out of a dark
cloud of gas and dust that hides the star (photograph by J. Morse (STScI), with NASA’s
Hubble Space Telescope).

9.2 The main-sequence phase

All stars undergo the main-sequence phase, characterized by corc-hydrogcn burn­

ing. The major product is helium, but other important isotopes are synthesized
during the main-sequence phase as well, such as nitrogen, resulting from proton
capture on carbon nuclei in the stellar core, and the rarer isotopes 3He and i3C,
produced in cooler regions, outside the core. In the course of this long phase
there is ample time for the stellar configuration to achieve both hydrostatic and
thermal equilibrium and ‘forget’ its former structure and evolutionary phases.
Thus the main sequence may be regarded as the starting point of stellar evolution;
fortunately so, for the earlier stages of evolution are less well understood, partly
perhaps because the guidance provided by observations is restricted by the short
duration of these phases.
The nuclear energy generated in the hydrogen-burning core is transported
outward by radiation or by convection. In low-mass stars (M ~ 0.3Mo) the main
means of energy transfer is convection, these stars being fully convective (except
for the photosphere, of course), as we have anticipated in Section 6.6. In the
H-R diagram they are found in the region where the Hayashi track meets the
main sequence. More massive stars have smaller and smaller outer convective
zones; in the Sun, for example, the convective zone extends over only about
152 9 The evolution of stars - a detailed picture

Figure 9.3 Correlation between luminosity and effective temperature obtained from
model calculations of hydrogen-burning stars of solar composition and various masses
and the resulting main sequence in the H-R diagram (adapted from R. Kippenhahn and
A. Weigert (1990). Stellar Structure and Evolution, Springer-Verlag).

Log

Figure 9.4 The extent of convective zones (shaded areas) in main-sequence star models
as a function ofthe stellar mass (adapted from R. Kippenhahn and A. Weigert (1990).
Stellar Structure and Evolution, Springer-Verlag).

2% of the solar mass below the photosphere. Stars more massive than the Sun.
which burn hydrogen predominantly by the temperature-sensitive CNO cycle,
develop convective cores, while the envelopes are in radiative equilibrium. The
main sequence emerging from complex stellar-model calculations is shown in
Figure 9.3. with masses marked along it. The extent of convective zones is shown
in Figure 9.4.
It is important to stress that the composition throughout a convective zone is
uniform, as a result of continual mixing, even if the nuclear reaction rates are not.
As a result, hydrogen-burning products migrate to cooler regions ofthe star, where
they could not have been found otherwise. And there they remain, even when the
central convective zone shrinks or disappears altogether. Later on. when the stellar
 9.2 The main-sequence phase 153

envelope will, eventually, become convective, its inner boundary overlapping the
outer boundary of the formerly convective core, hydrogen-burning products will
make their way to the surface of the star, where they can be observed in the
spectrum. This effect of consecutive, overlapping convecting zones. leading to the
dredge-up of processed material to the surface, enables us to infer the occurrence
of nuclear burning processes in the shielded stellar cores. Thus the detection in
spectra of evolved stars of heavy elements and isotopic ratios that are different
from those of young stars constitutes another crucial test, as well as guide, to the
theory of stellar evolution. Although there is ample observational evidence that
indirectly validates the theory of nuclear energy generation in stellar interiors,
great efforts arc devoted to testing it directly. Experiments aiming to test the very
hydrogen-burning process taking place in the core of the Sun will be described in
the next section.
One of the salient features of stellar evolution is mass loss. As wc have seen in
the previous chapter, stars lose mass at all evolutionary stages, including the main
sequence, and the rates of mass loss vary over a very wide range. On the lower
main sequence the mass-loss rate is so slow as to have no discernible effect on the
stellar mass. As shown in Section 8.4. the solar wind, for example, removes mass
from the Sun at a rate of a few ~10“l'lAf.3 yr ’, which will amount to less than
1/1000 of the Sun’s mass at the completion of its main-sequence phase. As we
go from low-mass to massive stars, the wind becomes more intense. As a result,
although the main-sequence lifespan decreases rapidly with increasing stellar
mass, the evolution pace of massive stars, which shed a considerable fraction of
their mass by the wind, slows down compared with the evolutionary rate of these
stars had they conserved their mass. To illustrate this effect we define a parameter
a by

•og(rMS/TMS,0)
a = --------------------- .
log(M/Mo)

We have seen in Chapter 7 that evolutionary timescales are determined by the

stellar mass, as they are inversely proportional to the square (or a higher power)
of the mass. Solar models are calibrated by requiring them to reproduce the solar
radius and luminosity at the present age of the Sun, which is independently known
from geologically based estimates of the age of the Earth. This calibration is used
for computing the main-sequence phase of stars over a wide range of masses.
The main-sequence lifetimes of stars within the initial mass range 0. 1Mq<
M < 25Maare listed in Table 9.2 and are marked onto the main sequence in
Figure 9.5. along with the corresponding stellar masses. The values of a arc listed
in the third column of Table 9.2: |or| is not constant; for relatively large M. it
decreases with increasing M. We note the large span of main-sequence ages: at
the lower end of the main sequence they exceed by far the age of the universe;
by contrast, at the upper end. they become shorter than the thermal timescale
of the Sun.
154 9 The evolution of stars - a detailed picture

Table 9.2 Main-sequence

lifetimes

Muss (Mq) Time (yr) a

0.1 6 x I012 -2.8

0.5 7 x 1010 -2.8
1.0 1 x 10'°
1.25 4 x 109 -4.1
1.5 2 x 109 -4.0
3.0 2 x 108 -3.6
5.0 7 x 107 -3.1
9.0 2 x 107 -2.8
15 1 x 107 -2.6
25 6 x 106 -2.3

Figure 9.5 The main-sequence lifespan for stars of different masses marked along the
main sequence in the H-R diagram (see Figure 9.3), which may be used to determine
stellar-cluster ages according to the main-sequence ‘turnoff point’.

Consider now a stellar cluster, which is, essentially, a large group of stars born
at the same time, more or less. The age of the cluster will show on its H-R diagram
as the upper end of the main sequence, or the turnoff{joint: stars within the cluster,
with masses corresponding to main-sequence lifetimes shorter than the cluster’s
age, would have already left the main sequence toward the red giant branch. In
other words, such stars would have consumed the hydrogen supply of their cores,
the cores contracting toward the next burning stage (or toward becoming white
dwarfs). Clearly, stars with main-sequence lifetimes longer than the cluster’s age
will still dwell on the main sequence of the H-R diagram. We are thus provided
with a reliable tool (a clock) for measuring the age of star clusters, as illustrated
in Figure 9.5, where superimposed on the clock are the H-R diagrams of different
clusters. The oldest cluster provides a lower limit to the age of the galaxy within
which it resides and to the age of the universe itself.
 9.3 Solar neutrinos 155

Note: The main sequence of stellar clusters serves not only as a time instrument but
also as an instrument for distance determination. The H-R diagram obtained directly from
observations has the measured apparent brightness, defined by Equation (1.1), instead of
the luminosity as ordinate. Since, on the logarithmic scale, this translates into a uniform
vertical shift of magnitude log(47rd2), matching such a diagram with the calibrated one
enables the determination of the shift, and hence of the distance to the cluster. As the
lower main sequence is the most populated region of the H-R diagram of any cluster,
the matching procedure relies mainly on the main sequence and thus this method of
distance determination (which, in reality, is complicated by such factors as metallicity
and interstellar absorption. Chapter 12) is called main-sequence fitting.

Stars of all masses partake in the main-sequence phase, but subsequent evo­
lution differs for stars of different masses. In what follows we shall distinguish
between stars whose main-sequence lifespan exceeds the present age of the uni­
verse (according to latest estimates, ~l.4x IO10 yr) and stars that could have
evolved off the main sequence, were they born early enough. Stellar models
yield the upper mass limit for stars that are still on the main sequence (even if
they are as old as the universe) at 0.7 Afo. Due to their low surface temperature
and thus reddish colour, these stars are also known as red dwarfs. Stars of mass
M > 0.7Mo may be divided in turn into two subgroups according to their mass:
those with initial masses below 9-1 ()M3, and the rest, with the former ending their
lives as white dwarfs (after shedding a considerable fraction of the initial mass)
and the latter undergoing supernova explosions. The former fall into low-mass
stars (0.7 < M < 2MQ) and intermediate-mass stars (2 < Af < 9-IOA/q), and the
latter (>1()A/O) are known as massive stars. The distinction between low-mass
and intermediate-mass stars is based on the way of helium ignition in the core,
that is, whether or not it occurs under degenerate conditions.
We now make a short digression from the course of stellar evolution to address
the crucial issue of solar neutrinos.

9.3 Solar neutrinos

Since the mean free path of photons in stars is barely 1 cm, stellar cores, where
nuclear reactions take place, cannot be directly observed. We infer the occurrence
of nuclear reactions from the fact that stars shine and that their luminosities
are well predicted by the theory of stellar evolution based on nuclear energy
generation, and also from the variety of surface abundances and isotopic ratios.
However, a direct test of the theory would be possible by devising means of
capturing neutrinos that are expelled in nuclear reactions. For them, the mean free
path exceeds stellar dimensions by about ten orders of magnitude. But if matter
156 9 The evolution of stars - a detailed picture

Table 9.3 Properties of solar neutrinos

Flux at Earth Energy Average

Source (m~2 s~[) (MeV) (MeV)

p + p —> 2D + e+ + v 6.0 x IO14 <0.42 0.263

7Be + <>--> 7Li + v 4.9 x 10” 0.86(90%), 0.38(10%) 0.80
8B -> 8Be + <?+ + v 5.7 x 10"’ <15 7.2

is so utterly transparent to these elusive particles, how can we expect to capture

them? It turns out that a tiny fraction of the immense neutrino flux from the Sun
(the nearest source) that sweeps the Earth can be captured by very ingenious
experimental devices.
The energy generation process put to the test is thus the fusion of four protons
into a helium nucleus, emitting two positrons and two neutrinos, and liberating
thermal energy

4/7 4He + 2e++ 2v + 0,

where Q & 25 MeV after subtracting the average energy removed by the neu­
trinos. The number of neutrinos emanating from the Sun per second can be easily
derived: ILq/Q %2x I03Ss-1. In the Sun. hydrogen burning proceeds mainly
through the p — p reaction chain (described in Section 4.3), which is, in fact, the
most common energy-generating process in stars. The chain, as we know, has
three branches, involving three neutrino-emitting reactions. Due to the branching
ratios of the p — p chain, the neutrinos emitted in each case have widely different
fluxes, and also different energies, as given in Table 9.3. The branching ratios
are directly related to the core-temperature profile in the Sun (and vary from star
to star). Now, the probability of absorption of a neutrino - small as it may be -
increases with increasing energy; hence the easiest to detect would be the XB
neutrinos, which have the highest energies.

Exercise 9.1: Using the data of Table 9.3, calculate the following: (a) the branch­
ing ratios of the p — p chain; (b) the neutrino luminosity of the Sun; and (c) the
range of neutrino emission (particles per second) that would be expected, if the
branching ratios of the p — p chain were not known.

Indeed, the SB neutrinos were the main target of the first neutrino experiment,
started in the early 1960s by Raymond Davis with the support of John Bahcall on
the theoretical side. The experiment was turned on in 1967 and ran continuously
for almost thirty years. The basic principle is the capability of 37C1 (a rare chlorine
isotope), to absorb a high-energy neutrino and produce 37Ar, a radioactive isotope
 9.3 Solar neutrinos 157

of argon,

v + 37C1 —-> e~ + 37 Ar,

which subsequently decays, having a half-life of 35 days. Eventually, if the chlo­

rine is exposed for a sufficiently long time, equilibrium is achieved between the
production and destruction of argon. The equilibrium abundance of37 Ar isotopes
can be used to derive the flux of highly energetic neutrinos. As the threshold
energy of the reaction is 0.8 MeV, these are mostly SB neutrinos; only a small
fraction (~1 /6) may result from Be. To grasp how formidable this experiment
was, we should mention that it involved a huge tank containing about 600tons
of C2CI4 fluid (an ordinary cleaning fluid) placed in the abandoned !500m-deep
Homestakc gold mine in South Dakota, shown in Figure 9.6. (The experiment had
to be conducted deep underground in order to avoid background noise caused by
cosmic ray particles.) The equilibrium abundance of argon atoms was no more
than a few tens among the ~2 x IO30 chlorine atoms in the tank! Nevertheless,
the radioactive argon atoms could be extracted and, by means of a Geiger counter,
counted. The inferred neutrino flux being about a factor of 2 to 3 lower than
predicted by solar models for the same energy range constituted what has been
known for many years as ‘the solar neutrino problem’.
The persisting puzzle prompted further investigation, always on a grand scale,
as demanded by the difficulty of neutrino detection. Thus other experiments
aiming at the 8B neutrinos followed: the Kamiokande II in the mid 1980’s, and
its successor, the Super Kamiokande, about ten years later, and still operating.
The latter consists of a big tank - 40 m in diameter and 40 m high - containing
50000 m3 of very pure water, of which about half is used for the experiment
itself, with the other half surrounding and shielding it. The tank is placed in the
Kamioka Mozumi mine in Japan, at a depth of more than 1000 m. Its walls are
lined with some 11 000 very sensitive light detectors - photomultiplier tubes -
capable of detecting single photons. (The earlier version used a smaller amount
of water and yielded somewhat less accurate results.) The experiment is based on
neutrino-electron scattering reactions,

v + e~ —> v' + e~',

which produce electrons moving with a speed that surpasses the speed of light in
water (but is less than the speed of light in vacuum). Such electrons radiate energy,
known as Cherenkov radiation, an effect that resembles a shock wave produced
by an aircraft moving at supersonic speed. This radiation hits the detectors on the
tank walls. The threshold energy of this experiment is close to 7 MeV, and hence it
is only sensitive to the more energetic among 8B neutrinos. The number of events
per day is less than 20. In principle, if one knows the neutrino flux detected by the
Kamiokande experiment, one can predict from the energy distribution of the XB
neutrinos the flux that should be detected by Davis’s chlorine experiment. This
158 9 The evolution of stars - a detailed picture

was found to be higher than the flux actually detected, which further complicated
‘the solar neutrino problem’.
The remarkable achievement of the Kamiokande experiments is to have estab­
lished that the detected neutrinos do indeed come from the Sun. The observed
directions of the scattered electrons, which recoil in the direction of the scattering
neutrinos, are found to trace out accurately the position of the Sun in the sky.
The disadvantage of both the chlorine and the water-Cherenkov experiments
was that they tested a rather insignificant branch of the p — p chain. The bulk of
solar neutrinos are the low energy ones produced by the fusion of two protons
into deuterium. As it turns out, such neutrinos can interact with gallium

v + 7lGa —> e~ + 7lGe,

the threshold energy being only 0.23 MeV, to produce radioactive germanium,
which has a half-life of 11.4 days and decays back to gallium. Two experiments
based on this reaction were soon designed: one named SAGE - a Russian (for­
merly Soviet)-American collaboration - in an underground excavation in the
Caucasus region of Russia, and another named GALLEX - a primarily European
collaboration - in an underground laboratory in Gran Sasso, Italy. Both operated
between 1990 and 2006. SAGE used 60 tons of metallic gallium (more than the
amount produced worldwide in a year!); GALLEX used half this amount in an
aqueous solution. Similar to the method used in the chlorine experiment, the
way of detecting the neutrinos was to collect and count the radioactive atoms
in the target. More than half the neutrinos detected in these experiments came
from the p + p reaction, providing for the first time unambiguous confirmation
of hydrogen fusion at the centre of the Sun. The comparison with theoretical pre­
dictions was significantly improved. The experimental results now came within
~65% of the solar model predictions, and the discrepancy diminished as more
data accumulated and more refined effects were included in these models (such
as diffusion, improved methods of dealing with convection, better opacities).
This was the situation close to the turn of the millennium: solar neutrinos
had been observed in five different experiments, with the expected energies and
roughly, but not quite, the expected fluxes. Moreover, it was unequivocally con­
firmed that their source was the Sun. We might safely claim that the main goal
of the neutrino-detection experiments - the validation of the theory concerning
the nuclear engine that powers the Sun and stars - had been attained. But the
discrepancy between detected and predicted neutrino fluxes, even if smaller than
at the onset of the Davis experiment, was still nagging. The quest for a solution
thus continued.
The solution to the solar neutrino problem and with it, strong indication for
‘new physics’, was soon to be found, provided by new experiments, even more
sophisticated and sensitive than those just described. The first experiment of this
kind, named SNO (for Sudbury Neutrino Observatory), was designed to detect
 9.3 Solar neutrinos 159

(a) (b)

Figure 9.6 The Homestake-mine experiment (a) (photograph by courtesy of the

Brookhaven National Laboratory) and the SNO detector (b) (photograph by courtesy
of Ernest Orlando Lawrence Berkeley National Laboratory).

solar neutrinos through their interaction with deuterium nuclei and electrons
present in salt heavy water, based again on Cherenkov radiation. The spherical
detector, shown in Figure 9.6. was placed 2000 m below ground in the Creighton
Mine located in Ontario, Canada, and filled with 1000 tons of salt heavy water.
It was surrounded by clean water that served as shield and by almost 10000
photomultiplier tubes. The special advantage of this experiment was its sensitivity
to different kinds of neutrinos, as besides the common electron neutrino, there are
two more types associated with two other kinds of leptons, the muon and the tau.
In 2001-2002 it provided the first evidence of transitions between the different
kinds (neutrino oscillations), which not only paved the way to the solution of the
solar neutrino problem, but also indicated that neutrinos have mass, and placed
constraints on its magnitude. Very briefly, while the earlier experiments were
sensitive to electron neutrinos only - the kind that is released in the nuclear
reactions of the p - p chain - a fraction of these neutrinos decayed to other
types before reaching the detectors and thus escaped detection, which explains
the apparent deficiency in the neutrino flux. For these transitions to be possible,
neutrinos must possess mass.
The SNO experiment was terminated in 2006 and soon after that, in May of
2007. another experiment was launched at the same underground laboratory in
160 9 The evolution of stars - a detailed picture

Gran Sasso that had housed the GALLEX experiment. It is called BOREXINO
and its goal is to measure the 'Be neutrinos from the Sun not targeted in the other
experiments, which makes it a low-energy experiment (see Table 9.3). The detector
core is a transparent spherical vessel 8.5 m in diameter, filled with 300 tons of a
liquid scintillator and surrounded by 1000 tons of a high-purity buffer liquid. The
photomultipliers are supported by a stainless steel sphere, which also separates
the inner part of the detector from the external shielding, provided by 2400 tons
of pure water.
Several other neutrino experiments are currently operating, among them the
KamLAND in Kamioka, and MINOS - at the Fermilab of the University of
Chicago; still others are being designed and planned. But they are now drifting
away from the ‘solar neutrino problem’, heading towards new particle-physics
theories, imposed by the neutrino mass.

Note: The 30-years long Homestake Experiment earned Raymond Davis the Nobel
Prize for physics for 2002, which he shared with Masatoshi Koshiba, who worked on
the Kamiokande and Super-Kamiokande experiments, and with Riccardo Giacconi. The
prize was awarded for pioneering contributions to astrophysics, Davis and Koshiba for the
detection of cosmic neutrinos, and Giacconi for having led to the discovery of cosmic X-
ray sources. Davis was almost 88 years old at the time, making him the oldest ever recipient
of a Nobel Prize. His collaborator on the solar neutrino experiment, John Bahcall, was
awarded the Dan David prize for cosmology and astronomy the next year, and summed
up the extraordinary achievement of the persistent and unrelenting effort devoted to the
neutrino experiments with the words: ‘1 am amazed that flashes of light in a mine, the
temperature of the Sun, and the properties of neutrinos can be linked in such a beautiful
way.’

9.4 The red giant phase

As the main-sequence phase advances, a hydrogen-depleted core grows gradually

in mass. Hydrogen burning proceeds in a shell surrounding the core, which
separates it from the envelope. Since the core is now devoid of energy sources,
the heat flow through it falls to zero (from Equation (5.4), F = f q din ->> 0) and
with it the temperature gradient decreases (Equation (5.3)). Thus, as the burning
shell moves outward, the core becomes isothermal while its mass increases. We
shall now show that an isothermal core of ideal gas cannot have an arbitrarily
large mass: given the stellar mass M, an upper limit exists for the core mass
Mc, beyond which the pressure within the core is incapable of sustaining the
weight of the overlying envelope. Mario Schonberg and Chandrasekhar were
the first to point out and derive the limiting mass from model calculations in
 9.4 The red giant phase 161

1942, and this type of dynamical instability is therefore known as the Schonberg-
Chandrasekhar instability. It can be easily understood on the basis of the virial
theorem (Section 2.4) - as McCrea showed 15 years later, although in a completely
different context (McCrea was studying star formation by gravitational collapse).
Denoting the core radius by /?t, its volume by Vc. the mean molecular weight
within it by /ze, and the temperature by Tc, we have from Equation (2.24)

C / G M~ = P'Vc + |a----- (9.17)

PdV
J()

where Ps is the pressure at the core’s boundary. Now. for an ideal isothermal gas,

fVc TZ f TZ
/ PdV = — Tc / pdV = — TCMC. (9.18)
Jo Me J Me

Substituting relation (9.18) and Vc = 4tt/?^/3 into Equation (9.17), we obtain

3 1ZTCMC aG M- n n
P,(RC) =--------- £ ------------ 7. (9.19)
4:r Me R 4tt R4

For a given core mass, the pressure at the core boundary increases with the core
radius from Ps = 0 at

aG
Ro =-------- — (9.20)
371 Tc

to a maximum value Ps.max at

4aG Mcp
l\ I —-------------- (9.21)
9TZ Tc

obtained by setting dPJdRc = 0. A core of radius < /?o would collapse under
its own gravity, without any external pressure. For a core of radius /?c > 7?,, the
pressure at its boundary would be smaller than Ps,max- Thus the maximal pressure
that can be attained at the core boundary as a function of the core mass is

7',4
/’s.maxCMJ = constant —7-7. (9.22)
^cMc

This pressure, exerted by the gas within the core, must balance the pressure Pcnv
exerted by the envelope. To estimate the latter, we may assume the core to be a
point mass (/?c R) and make use of inequality (2.18) obtained in Section 2.3:
/’em > GM2/8tr/?4. Obviously, if A, max < GM2/%nR4, no equilibrium con­
figuration would be possible. Hence the stability condition for an isothermal core
is
T4 GM2
/’s.maxfK) = constant (9.23)
M2iĄ ~ 8tt /?4 ’
162 9 The evolution of stars - a detailed picture

Still regarding the core as a point mass, we may use homology relation (7.33) of
Section 7.4, with /zenv denoting the mean molecular weight of the envelope,

to eliminate Tc and R in condition (9.23). The stability condition thus becomes

Me < /Menv\ ...

— Z constant ----- . (9.24)
M \ Me /

Schonberg and Chandrasekhar arrived at this result with the dimensionless con­
stant of 0.37. Assuming a solar composition for the envelope and a mostly
helium composition for the core, we have by Equations (3.29), (3.26) and (3.18)
/zenv ~ 0.6 and gc ~ 1. leading to Mc/M Z 0.13. When the mass of the hydrogen-
depleted core reaches this limit, the core starts contracting rapidly.
Main-sequence stars more massive than about have homogeneous con­
vective cores surpassing the critical limit, as shown in Figure 9.4. Once hydrogen is
exhausted in such a core, energy generation subsides, convection is quenched, and
the core becomes isothermal. Since its mass is already greater than the Schonberg-
Chandrasekhar limit, the dynamically unstable core starts collapsing. In time, it
acquires the temperature gradient necessary for balancing gravity. The temper­
ature gradient causes loss of heat and hence core contraction and the increase
in temperature that goes with it continue, but on a thermal (Kelvin-Helmholtz)
timescale.
When hydrogen burning in the core ceases, thermal equilibrium is destroyed
and for a brief period of time the stellar energy decreases (L > Lnuc). How­
ever, as hydrogen burning shifts from the core to a shell surrounding it, and as
the temperature in this shell rises with the rising core temperature, the nuclear
energy generation rate soon increases again. But since hydrogen is burnt by the
CNO cycle, whose rate varies as a very high power of the temperature (see
Section 4.4), the energy-production rate is accelerated beyond thermal equilib­
rium and during most of the core-contraction phase the stellar energy increases
(Lnuc > L). This is illustrated in Figure 9.7(a), where the change with time of
the total energy of a 7M© star model is plotted, beginning at the end of the
main sequence and ending on the red giant branch. Core contraction is thus
necessarily accompanied by expansion of the envelope (see Section 7.5), as illus­
trated in Figure 9.7(b), and the star becomes a red giant, moving to the right
in the H-R diagram, as shown in Figure 9.7(d). Overall, the transition from a
main-sequence to a red-giant configuration is characteristically of short dura­
tion, and hence the probability of detecting stars undergoing this transition is
vanishingly small. This is the reason for the conspicuous gap between the main
sequence and the red giant branch in the H-R diagram, known as the Hertzsprung
gap.
 9.4 The red giant phase 163

Time (years) Log(Teff)

(b) (0

Figure 9.7 Evolution of an intermediate-mass star (7MS) during the crossing of the
Hertzsprung gap: (a) total energy as a function of time (the time is arbitrarily set to zero
at the onset of core contraction); (b) central density and average density (3M/4ttR3) as a
function of time; (c): evolutionary track in the H-R diagram (where lines of equal radius
are marked); (d): changing of central temperature with effective temperature.

Note: The question ‘How does a star become a red giant?’ constitutes a long-standing
puzzle. But the puzzle is connected not so much with the physics of red giants as with
our perception of understanding a phenomenon. We may claim to understand a physical
process in the following cases: (a) if we can lay down the physical principles governing it;
(b) if we can write down the equations describing it and solve them; (c) if we can explain
the process in simple terms, step by step. Of course, if all three conditions are fulfilled,
the process may be considered well understood. But in fact, condition (b) alone suffices.
This is the case with red giants: all numerical computations of the evolutionary phase
following hydrogen exhaustion in the core obtain red giant configurations as solutions of
the stellar-evolution equations. Moreover, the simple explanation offered in Section 7.5
points out the virial theorem as the basic principle involved, given the contraction of the
core - thus satisfying condition (a). Nevertheless, we feel uncomfortable in accepting
these solutions so long as condition (c) is not satisfied. We would like to be able to
identify the precise mechanism that drives a star to become a red giant. However, this
last condition is not always considered imperative for understanding a physical process.
For example, we understand and explain the outcome of a collision of two rigid balls on
a smooth surface in terms of conservation of momentum and energy, without bothering
about the exact manner in which momentum is transferred from one ball to another during
their brief contact. And yet we still worry about red giants ...
164 9 The evolution of stars - a detailed picture

As the helium core grows in mass by hydrogen burning in the shell outside it,
it continues to contract, liberating gravitational energy. Consequently, the temper­
atures in the core and shell go on rising (see Figure 9.7(d)), accelerating further
the rate of hydrogen burning and core growth. Finally, thermal equilibrium is
restored and the luminosity, which is proportional to the rate of core growth,
increases. The need to transfer an increasing energy flux on the one hand, and the
increasing opacity of the cool envelope on the other hand, cause the envelope to
become convectively unstable. Hence red giants develop convective envelopes,
extending from just outside the hydrogen-burning shell all the way to the surface.
The base of the convective envelopes reaches layers where nuclear processes have
taken place earlier and thus hydrogen-burning ashes make their way to the sur­
face. This is the first occurrence of ‘dredge-up’ (explained in Section 9.2) that is
observationally detected. In the (log Teff, log T) plane these stars are said to climb
up the slanted red giant branch (very close to their Hayashi tracks) toward higher
luminosities and slightly lower effective temperatures. The red giant branch in the
H-R diagram roughly coincides with the boundary of the Hayashi forbidden zone.
Eventually, the core temperature becomes sufficiently high for helium to ignite.
The Schonberg-Chandrasekhar instability applies, however, only to ideal
gases. A cold and dense gas, in which the degenerate electrons supply most
of the pressure, is capable of building up a sufficient degeneracy pressure to sup­
port the weight of the envelope, even in a relatively massive core. The appropriate
conditions for electron degeneracy.

/ \5/3

A.max(Mc)</f, ,
\ 3 /

using Equation (3.35), are found to develop in the helium cores of stars with
masses below about 2MO. The core-contraction phase of these stars is slow and
gradual. The temperature rises throughout the contracting core and the burning
shell outside it. As a result, the nuclear energy generation rate increases and,
with it, the stellar luminosity. At the same time, the envelope expands and the
temperature decreases throughout it, as well as at the stellar surface. The star
assumes gradually the appearance of a red giant. Indeed, the ascent toward the
red giant branch is clearly seen at the lower part of the main sequence, in particular
in H-R diagrams of old globular clusters. However, these low-mass stars, evolving
quietly toward higher core temperatures, are bound to encounter a different type
of instability. Heating of the core as a result of contraction is impeded by neutrino
emission, which acts as an energy sink. Hence the core material becomes strongly
degenerate before helium burning sets in. We have seen in Section 6.2 that nuclear
burning in degenerate material is thermally unstable, leading to a runaway. Thus in
these relatively low-mass stars, when the temperature finally reaches the helium-
ignition threshold, helium ignites in an explosion, known as the helium flash.
This occurs when the core mass has grown to about 0.5Afo, regardless of the
 9.5 Helium burning in the core 165

total stellar mass. During a few seconds, the temperature rises steeply at almost
constant density, the local nuclear power reaching 1011 Lq (roughly, the luminosity
of an entire galaxy). Nevertheless, an outsider would not be aware of the intense
central explosion, which is almost entirely quenched by the energy-absorbing
stellar envelope. Thus, there is no apparent clue in the H-R diagram to the helium
flash. Soon, the core temperature becomes sufficiently high for the degeneracy to
be lifted, the core expands, and helium burning becomes stable.
A fraction of the red giants, however, do not attain helium ignition. This is
due to the effect of mass loss that characterizes red giants. During the red-giant
phase, when the stellar envelope is considerably less bound gravitationally than in
the main-sequence phase, the stellar wind intensifies. Hence low-mass red giants
lose their small envelopes before the core has a chance to reach sufficiently high
temperatures for helium ignition. The degenerate helium cores continue their
contraction, leaving the red giant branch to become helium white dwarfs.

Exercise 9.2: Assume a star of mass M and radius R has a core of mass M, and
radius R]. Let the density distribution be given by
/ r \2
Pc - (Pc - Pi) I — ) for 0 < r <
\ «l /

f> \ r ) \ R )
P\ —------------ —5----- for R\ < r < R,
• - (-Y
\ RJ
where pc is the central density and = p(/?i). Find the dependence of the
ratio R/R] on .v, pc/p\ and yt = M/M\. Calculate the ratio for.v, = 10 and
yi — 7.5 (consistent with condition(9.24)).

9.5 Helium burning in the core

The phase of stable helium burning in the stellar core is significantly shorter
than the main-sequence phase of core hydrogen burning. The reason is twofold:
first, the fusion of helium - into carbon and oxygen - supplies only about one
tenth of the energy per unit mass supplied by hydrogen fusion (as we have seen
in Chapter 4), and secondly, the stellar luminosity is higher by more than an order
of magnitude compared with the main-sequence luminosity of the same star. In
fact, helium burning would have been still shorter, were it not for the additional
energy source provided by hydrogen burning in the shell outside the core.
In low-mass stars (0.7-2Ms), which undergo the helium flash, the subsequent
rapid expansion of the core has an effect on the star’s structure similar to the
contraction of the core at the end of the main sequence, only in reverse. As the
core expands and cools, the envelope contracts and its temperature rises to some
166 9 The evolution of stars - a detailed picture

extent. As a result of core expansion and cooling, the temperature in the hydrogen­
burning shell decreases and the nuclear energy supply diminishes. This, combined
with the diminished stellar radius, cause the luminosity to drop and the star is
said to descend from the red giant branch. Since the effective temperature has
increased, the star moves to the left in the H-R diagram. The locus of low-mass
core-helium-burning stars in the H-R diagram forms the horizontal branch, a
roughly horizontal strip stretching between the main sequence and the red giant
branch, corresponding to luminosities of the order of 50-100Lq. There they dwell
for about 108 yr. All these stars have equally massive cores at the end of the red-
giant phase; hence their different positions along the horizontal branch must be
determined by another factor. For stars of similar Z (heavy element content), this
factor is found to be the envelope mass, a function of the initial stellar mass and
the rate of mass loss up to this stage, itself possibly a function of the rotation
rate of the star. The highest envelope masses are found at the red (low effective
temperature) end ofthe branch, where the hydrogen shell contributes most ofthe
energy and the convective envelope’s structure is similar to that of red giants.
Proceeding toward the blue end, we find smaller envelope masses and weaker
hydrogen-burning shells. The envelopes are now radiative rather than convective.
Stars in this region of the horizontal branch arc found to go through a phase of
dynamical instability in their envelopes, in the regions of hydrogen and helium
ionization (sec Section 6.4). This instability manifests itself by pulsations, causing
a cyclic variability ofthe luminosity with periods of a few hours. Such pulsating
stars on the horizontal branch arc indeed observed; they arc known as RR Lyrae
variables. At the blue end of the branch the hydrogen-rich envelopes are small -
in both mass and radius - and inert.
Intermediate-mass stars (2-10A/o) ignite helium quietly when the central
temperature reaches IO8 K. Subsequently the rate of energy supply by the helium-
burning core steadily increases, while the rate of energy supply by the hydrogen­
burning shell decreases. As the temperature in the burning shell at the base of the
envelope drops, the envelope cools too and. eventually, it starts contracting; this
occurs when the contributions from the two energy sources become roughly equal.
At this point the stars leave the red giant branch in the H-R diagram by looping
toward higher effective temperatures, the higher the mass, the more extended the
loop. As luminosity increases with mass, these stars form a helium main sequence,
with a slope similar to that of the (hydrogen) main sequence, but closer to the
red giant branch. In fact, observationally, the helium main sequence is hardly
discernible from the thick red giant branch. Intermediate-mass helium-burning
stars, too, go through a phase of envelope instability resulting in pulsations,
but the pulsation periods are longer, ranging from days to several months. Such
pulsating luminous stars are known as Cepheid variables, or simply Cepheids.
Their importance to astronomy warrants another digression from the pursuit of
stellar evolution.
 9.5 Helium burning in the core 167

Log [P(days»

Figure 9.8 The period-luminosity correlation for Cepheids derived from observations
(from A. Sandage and G. A. Tammann (1968), Astrophys. J. 151).

It turns out that a well-defined correlation exists between the (average) lumi­
nosity of a Cepheid star Łceph and its pulsation period Pceph, as shown in Fig­
ure 9.8. The correlation emerges from observations of Cepheids with well-known
distances, for which accurate luminosities can be derived, and thus Tceph(^’ceph)
is established. Imagine now that a pulsating star is detected in a distant cluster of
stars or a distant galaxy, with a period PObS characteristic of Cepheids. If the star
is identified as a Cepheid (based also on its spectral characteristics), its apparent
brightness 70bS and its pulsation period can be used to derive its distance cl, which
is also the distance to the cluster or galaxy within which it resides

Tęcph( T’obs)
(9.25)
4?r /obs

Cepheids constitute what are called in astronomy standard candles - and are the
most accurate and reliable among them. The period-luminosity relationship was
first discovered in 1912 by Henrietta Swan Leavitt, for the Cepheids in the nearby
galaxy called the Small Magellanic Cloud (or SMC), and these stars immediately
rose to fame. A year after the discovery, the period-luminosity relation had already
been used by Hertzsprung and other famous astronomers to determine distances
to galaxies.

Note: Within our own Galaxy the relative brightness of stars in a given volume is
largely affected by their different distances. Stars of a distant galaxy, however, are all
equally distant from an observer on Earth, because the distance to a galaxy is far larger
than its size. Consequently the ratio of apparent brightnesses is equal to the ratio of
intrinsic luminosities for these stars. Hence statistical analyses are far more reliable for
stars in the nearby galaxies, such as the Magellanic Clouds.
168 9 The evolution of stars - a detailed picture

Table 9.4 Evolutionary lifetimes (years)

1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10

15 1.0(7) 2.3(5) <— 7.6(4) —> 7.2(5) 6.2(5) 1.9(5) 3.5(4)

9 2.1(7) 6.1(5) 9.1(4) 1.5(5) 6.6(4) 4.9(5) 9.5(4) 3.3(6) 1.6(5)
5 6.5(7) 2.2(6) 1.4(6) 7.5(5) 4.9(5) 6.1(6) 1.0(6) 9.0(6) 9.3(5)
3 2.2(8) 1.0(7) 1.0(7) 4.5(6) 4.2(6) <— 6.6(7) —> 6.0(6)
2.25 4.8(8) 1.6(7) 3.7(7) 1.3(7) 3.8(7)
1.5 1.6(9) 8.1(7) 3.5(8) 1.0(8) >2(8)
1.25 2.8(9) 1.8(8) 1.0(9) 1.5(8) >4(8)
1.0 7.0(9) 2.0(9) 1.2(9) 1.6(9) >1(9)

Note: powers of 10 are given in parentheses.

Due to its high temperature sensitivity, helium burning occurs in a convective

core (as does hydrogen burning by means of the CNO cycle). This is the inner
part of the larger helium core, which grows as a result of hydrogen burning in
the shell surrounding it. So long as the inner core is convective, its composition
is constantly mixed and turns gradually from helium to carbon and oxygen,
although helium burning, which varies with temperature as ~7'40, is confined
to its very centre. When the inner convective core becomes depleted of helium
and hence burning within it subsides, convection is quenched as well. The star
consists of a carbon-oxygen core, surrounded by a helium layer - remnant of
the original helium core - which, in turn, is separated from the hydrogen-rich
envelope by the hydrogen-burning shell. Helium burning continues in a shell at
the C-0 core boundary. Evolutionary paths in the H-R diagram up to helium
burning in a shell are traced in Figure 9.9 for stars of different masses, and
the characteristic time intervals are listed in Table 9.4. A comparison between
theory and observations is made possible by varying the shade of the paths in
Figure 9.9 in proportion to the length of time spent - and hence to the number of
stars expected to be observed - in each phase. Among core-hclium-burning stars,
those observed are predominantly the low-mass ones populating the horizontal
branch.

9.6 Thermal pulses and the asymptotic giant branch

The carbon-oxygen core’s evolutionary course and its consequences are similar
to the helium core’s. Devoid of energy sources, the core contracts and heats up; as
a result, the envelope expands and cools, and convection sets in again throughout
it. As the inner boundary of the convective envelope overlaps the earlier outer
boundary of the now extinguished hydrogen-burning shell, processed material,
mainly helium and nitrogen, is once more dredged up and mixed into the envelope.
 9.6 Thermal pulses and the asymptotic giant branch 169

Log (Teft)

Figure 9.9 Evolutionary paths in the H-R diagram for stars of different initial masses
(as marked) up to the stage of helium burning in a shell. The shade of the segments is
indicative of the time spent in each phase, ranging from less than 10? yr (light) to more
than IO9 yr (dark), as given in Table 9.4. The different phases, indicated by numbers,
are: 1-2, main sequence; 2-3, overall contraction; 3-5, hydrogen burning in thick shell;
5-6, shell narrowing; 6-7, red giant branch; 7-10, core-helium burning; 8-9. envelope
contraction (adapted from I. Iben Jr. (1967), Ann. Rev. Astron. Astrophys., 5).

The signature of these elements appears in the star’s spectrum, again bearing
witness to the processes taking place in its deep interior. The expanding star
becomes redder and resumes its climb on the giant branch in the H-R diagram,
which has been interrupted by the core-helium-burning episode. This part of the
giant branch, populated by stars with C-0 cores, is called the asymptotic giant
branch (AGB); it is an extension ofthe red giant branch toward higher luminosities
and lower effective temperatures at the boundary of the Hayashi forbidden zone.
170 9 The evolution of stars - a detailed picture

Figure 9.10 Evolution of the interior structure of a 6Mq model star from the main
sequence to the AGB phase. Dark areas indicate nuclear burning and shaded ones, con­
vective zones (adapted from D. Prialnik and G. Shaviv (1980), Astron. Astrophys., 88).
Note the occasional changes in timescale.

Hence stars in this phase of evolution (known as AGB stars) are even bigger than
the former red giants - they are now becoming supergiants. Cooling of the layers
above the C-0 core extinguishes the hydrogen-burning shell temporarily; it will
reignite later on, after envelope expansion will come to a halt. Contraction of the
core raises the density up to the point when electrons become degenerate, and since
degenerate matter is an efficient heat conductor, the core becomes isothermal. An
illustration of the internal evolution of a 6Mq star from the main sequence up to
the onset of the asymptotic giant phase is given in Figure 9.10: changing burning
zones, convective zones and boundaries between regions of different composition
are marked. The remarkably different evolutionary timescales are particularly
noteworthy.
There are three outstanding characteristics for AGB stars:

1. Nuclear burning takes place in two shells - a thermally-unstable configu­

ration - leading to a long scries of thermal pulses.
2. The luminosity is uniquely determined by the core mass, independently of
the total mass of the star.
3. A strong stellar wind develops as a result of the high radiation pressure in
the envelope, the star thus losing a significant fraction of its mass.
 9.6 Thermal pulses and the asymptotic giant branch 171

We shall now address each characteristic in more detail. The two burning shells
that supply energy during the asymptotic giant phase are separated by a helium
layer. The external shell, at the bottom of the hydrogen-rich envelope, burns
hydrogen, thus increasing the helium layer’s mass. The internal shell, on top the
C-C) core, burns helium, thus eating into the helium layer and building up the
C-0 core. In principle, a steady state could be achieved, with the two burning
fronts advancing outward at the same rate. However, the great differences between
the two nuclear burning processes do not allow such a steady state to develop.
As it happens, the two shells do not supply energy concomitantly, but in turn,
in a cyclic process, and the mass of the helium layer separating them changes
periodically.
During most ofthe cycle’s duration hydrogen is burnt in the external shell,
while the inner shell is extinct. As a result, the helium layer separating the shells
grows in mass. With no energy supply, this layer contracts and heats up until the
temperature at its base becomes sufficiently high for helium to ignite. Helium
burning in this thin shell is thermally unstable, as explained in Section 6.2; it
resembles the helium flash that takes place in the electron-degenerate cores of
low-mass stars at the tip of the giant branch. At the peak of the short-lived flash
the nuclear energy generation rate reaches 108Le. The energy is absorbed by the
overlying layers, which expand and cool. As these layers contain the hydrogen­
burning shell, the rate of hydrogen burning quickly declines. During an ensuing
short period of time, the helium-burning front advances through the helium shell,
turning helium into carbon and oxygen, until it catches up with the now extinct
hydrogen shell.
The high temperatures attained in the helium-burning shell lead to a chain of
reactions that produces neutrons. Capture of these neutrons by traces of heavy
elements that are present in the shell leads to the creation of trans-iron isotopes
by the .s-proccss explained in Section 4.8.
The proximity of the hot helium-burning front causes the hydrogen to reignite.
Due to its lesser sensitivity to temperature, hydrogen burning in a shell is stable.
The temperature and density adjust into thermal equilibrium. At the same time,
helium burning is quenched as a result of the relatively low temperature now
prevailing in the hydrogen-burning shell and its vicinity. Thus a new cycle begins.
The evolution throughout a thermal-pulse cycle, known also as a shell flash, is
shown schematically in Figure 9.11. Particularly noteworthy is the dredge-up of
processed material into the convective envelope by the moving inner boundary of
the convective zone.
Although thermal pulses entail periodic changes of the stellar luminosity,
these cannot be observed because the periods vary between hundreds to thousands
of years. The lasting result of each cycle is the growth of the carbon-oxygen core.
This brings us to the second characteristic of AGB stars.
Evolutionary calculations show that the luminosity of an AGB star with a core
mass > 0.5 Mq (recalling that the star is now well past the onset of helium
172 9 The evolution of stars - a detailed picture

Figure 9.11 Sketch of the progress of a thermal-pulse cycle through its different stages
(not in scale). Hydrogen is burning during stages a and d, while helium is burning during
stages b and c. When, in stage c. the outer convective zone extends inward beyond the
helium-burning shell’s boundary, hydrogen- and helium-burning products are mixed into
the envelope and dredged up to the surface. Stage a' is the same as a, except that the
carbon-oxygen core has grown at the expense of the envelope.

burning) is quite accurately represented by the following relation:

— = 6 x IO4 — - 0.5 , (9.26)

\ /

regardless of the stellar mass, as was first pointed out by Bohdan Paczyński in
1971. We note that this luminosity is of the order of the Eddington luminosity
L/Lq — 3.2 x \()4M/Mo (see Section 5.5). Thus stars of the same core mass
are found at the same height on the asymptotic giant branch in the H-R diagram,
regardless of their envelope mass. Stars reach the asymptotic giant branch at
different points, depending on the mass of the core at the end of the central
helium-burning stage. They climb up the branch, as the core continues to grow
during the thermal-pulses stage. At the same time the envelope mass decreases,
not only at the expense of core growth, but mainly because of mass loss at the
surface. Hence the point at which a star leaves the asymptotic giant branch is
determined by the mass of the envelope at the end of core-helium burning and by
the intensity of the stellar wind. This brings us to the third characteristic of AGB
stars.
The outer layers of giant and supergiant stars are sufficiently cool for atoms to
coalesce into molecules and molecules into tiny dust particles. It is these particles
that are accelerated by the radiation pressure that drives the stellar wind (see
Section 8.5). However, the nature of such particles and the interactions involved
are extremely difficult to calculate and the resulting mass-loss rate difficult to
assess. At this point, the stellar evolution theory has to rely on observations in
 9.7 The superwind and the planetary nebula phase 173

order to continue its pursuit of the changing structure of stars. Observations of red
giants and supergiants reveal that these stars lose mass at rates ranging from 1(T9
to 10~4 M& per year. Mass loss is generally classified into two types of winds:

1. A stellar wind that may be described by an empirical formula due to Dieter

Reimers, linking the stellar mass, radius and luminosity, by a relation of
the form (8.31). the constant being determined from observations over a
wide range of stellar parameters:

,, L R Mo
M 10“ 3------------- - Mo yr’1. (9.27)
Z.0 Rq m

Typical wind rates are of the order of IO-6 Mq yr-1, which for character­
istic M, L and R values, imply 0 ~ 1 in Equation (8.31), that is, a high
efficiency in momentum transfer.
2. A superwind, essentially a stronger wind, leading to a concentration of the
stellar ejecta in an observable shell surrounding the central star.

9.7 The superwind and the planetary nebula phase

The existence of the superwind is imposed by two different and independent

observations: first, the high density within the observed shells formed by the
stellar ejecta (a slow wind would have given rise to more diffuse shells), and
secondly, the relative paucity of very bright stars on the asymptotic giant branch
of H-R diagrams of statistically significant stellar samples.
The inference of a strong wind based on the second observation is not straight­
forward. but it can be understood in terms of simple arguments. The number of
stars expected to reside on the asymptotic giant branch in the H-R diagram is
proportional to the time spent by stars in the double-shell-burning evolutionary
phase. An upper limit to this time span may be obtained by considering only the
evolution of the core. Essentially. AGB stars turn hydrogen into carbon and oxy­
gen. growing C-O cores at the expense of envelope material. If Q is the amount
of nuclear energy released in the process per unit mass, roughly 5 x 1()14 J kg-1,
thermal equilibrium implies

M = - % 1.2 x IO-11— Moyr“', (9.28)

Q

where L is the luminosity averaged over a thermal-pulse cycle. Substituting L/Lq

from Equation (9.26), we obtain

dMc
= 7.2 x Ur1 yrdt. (9.29)
M. - 0.5 Mo
174 9 The evolution of stars - a detailed picture

At the beginning of the asymptotic giant phase the core has some mass Mc$
(>0.5Mo). Assuming that as a result of contraction the electrons become degen­
erate. the maximal mass the core could reach is the Chandrasekhar mass MCh.
Integrating between Mc — Mc_o and Mc — we obtain an upper limit to the
duration ofthe asymptotic giant phase,

, /MCh-0.5M3\
rAGB < 1.4 x IO6 In Jh
\ Mc.o - 0.5My /
yr. (9.30)

Evolutionary calculations show that a relation exists between the initial core mass
and the initial mass of the star Mo, of the form

Mc,o a + /?M0, (9.31)

where a and b are constants. Hence Tagb is, essentially, a function of the initial
stellar mass. Therefore, if the distribution of initial stellar masses is known, the
number of stars on the asymptotic giant branch can be computed for any given
population of stars. As it turns out, the expected number of AGB stars exceeds
by far the actual number of observed AGB stars, with the discrepancy being as
large as a factor of 10. This means that stars are prevented by some process from
completing their sojourn on the asymptotic giant branch, while losing mass at the
moderate rate dictated by Reimers’s formula. This process is the superwind, which
consumes the envelope mass before the core has grown to its maximal possible
size. In fact, mass loss must be so intense as to allow the core to grow by only
about 0.1 Mq while the entire envelope is ejected. It should be mentioned that, in
addition to the indirect indications, the hypothesis of a superwind is confirmed by
observations of stars which eject mass at rates of the order of 10~4MG yr-1. In
some AGB stars, those believed to descend from relatively low-mass progenitors,
the high mass-loss rate is associated with a pulsation instability in the envelope,
similar to that of RR Lyrae stars and Cepheids that we have encountered earlier.
These stars, known as Miras, or long-period variables, pulsate with periods of the
order of a year.
As a consequence of the superwind, stars of initial mass in the range 1MQ<
M < 9MO shed their envelopes and are left with C-0 cores of mass between
0.6Mo and 1. 1MO, a higher final core mass corresponding to a higher initial total
mass. These cores will subsequently develop into white dwarfs. Since, as we
shall see shortly, low-mass stars are far more numerous than massive ones, we
expect most white dwarfs to have masses near 0.6Mo. This conclusion is verified
by observations. Thus white dwarfs originating from AGB stars have masses
considerably smaller than the Chandrasekhar critical mass, and hence, although
degenerate, these stars arc in no danger of a catastrophic denouement (contrary to
some early theories). But they do undergo a short episode of particular brilliance
before fading into cooling, inert white dwarfs.
 9.7 The superwind and the planetary nebula phase 175

The cores of stars at the end of the asymptotic giant phase are surrounded by an
extended shell, a more or less spherical nebula formed by the ejected material. The
inner part of this shell - resulting from the superwind - is relatively dense. When
mass loss finally ceases, the core, freed from the burden of a massive envelope,
expands slightly and as a result, the small envelope remnant contracts. This causes
a distinct separation, a void, between the star and its ejecta. Subsequently, as the
central star contracts, the effective temperature rises considerably. When it reaches
~30 000 K, the radiated photons become energetic enough to ionize the atoms in
the nebula and cause them to shine by fluorescence (the same mechanism that
is responsible for fluorescent lamps). A shining nebula of this kind is called a
planetary nebula; it appears as a bright circular ring surrounding a point-source
of light, although many appear twisted or elongated. An example is given in
Figure 9.12.

Historical note: Despite the name, planetary nebulae have nothing to do with
planets. The first planetary nebula ever detected was the Dumbbell Nebula, which was
discovered by Charles Messier in 1764. The comparison to a fading planet followed, about
15 years later, with the discovery of the second such object, the famous Ring Nebula by
Antoine Darquier. It was Sir Frederick William Herschel who eventually coined the name
‘planetary' nebula’ for these objects in his classification of nebulae in the 1780s, because
he found them to resemble the planet Uranus that he had newly discovered, although
earlier he and others thought them to be unresolved clusters of objects. A few years
later, Herschel found a planetary nebula with a very bright central star; thus he became
convinced that planetary nebulae were nebulous material (gas or dust) associated with a
central star. Recently, however, and quite ironically, it has been suggested that the peculiar
shapes of some of these nebulae may be due to the presence of giant planets orbiting the
central star and interfering with the flow of material it emanates. So planetary nebulae
may have something to do with planets after all!

Although the ring around the central star may appear like a disc, if this were
the case, then obviously, at least some planetary' nebulae should have appeared
flattened, due to the inclination of the disc with respect to the line of sight to the
observer. The fact that all planetary nebulae appear almost circular indicates that
what we see is the projection of a spherical shell. As the line of sight through
the nebula is much longer near the edges than at the centre, the material appears
opaque toward the edge and transparent at the centre, making it possible to see
the hot central star, as illustrated in Figure 9.13. This explains the ring shape. The
central source is called the planetary nebula nucleus.
The path that planetary nebulae trace in the H-R diagram is a horseshoe­
shaped track, first leftward, toward higher surface temperatures, meaning that the
nucleus preserves its luminosity during the transition, and then downward and
176 9 The evolution of stars - a detailed picture

Figure 9.12 (a) The Helix nebula, the nearest (450 ly away) and largest observed planetary
nebula (copyright Anglo-Australian Observatory; photograph by D. Malin), (b) detail of
the Helix nebula captured by NASA's Hubble Space Telescope, showing knots of gas.
Each gaseous head is at least twice the size of our solar system and each tail stretches to
about 1000 AU (photograph by C. R. O’Dell, Rice University).

to the right. The energy is provided by nuclear burning in the thin shell still left
on top of the C-0 core. When the mass of this shell decreases below a critical
size, of the order of 10 '-I0”4Mq, the shell can no longer maintain the high
temperature required for nuclear burning. The energy source becomes extinct,
the luminosity of the central star drops and its ionizing power diminishes. At the
same time, the nebula, which expands at a rate of a few 10 km s_|, grows in size
and gradually disperses. Thus a planetary nebula fades away and disappears after
some 104—105 yr. We now turn to the evolution of the remnant central star into a
cool white dwarf.
 9.8 White dwarfs: the final state of nonmassive stars 177

Figure 9.13 Sketch of a planetary nebula and its nucleus (PNN).

9.8 White dwarfs: the final state of nonmassive stars

Most white dwarfs - compact stars of high surface temperature - descend from
AGB stars, which develop C-0 electron-degenerate cores. As we have seen,
these stars lose mass by a strong stellar wind, while undergoing thermal pulses
caused by the alternate burning of hydrogen and helium in thin shells. The end
of mass loss, brought about by the dissipation of the entire envelope, occurs at a
random phase of a thermal pulse. If it occurs during the hydrogen-burning phase,
the star will be left with a thin coating of hydrogen-rich material, a vestige of
the lost envelope. If it occurs during helium burning, which takes place at the
bottom of a helium layer, the outer envelope will be composed predominantly of
helium. Since helium burning takes up only a small fraction of the pulse cycle,
the probability of a star ending the asymptotic-giant stage with a helium, rather
than a hydrogen-rich, envelope is proportionally smaller.
Soon after the end of mass loss, nuclear burning comes to an end as well.
During the intervening, short-lived planetary-nebula phase, the final stage of
nuclear burning supplies the energy that lights up the ejecta of the former AGB star.
The planetary nebula nucleus - the degenerate core of the former AGB star, with
the remnant thin envelope - becomes a white dwarf. We should therefore expect
to encounter two types of white dwarf spectra: a prevalent one, show ing hydrogen
lines, and a rarer type, with no evidence of hydrogen. Indeed, observations confirm
this expectation: about 25% of white dwarfs have no hydrogen lines in their
spectra.
Another source of white dwarfs is low-mass stars in a narrow initial-mass
range: 0.7 < M < \MQ. These stars do not reach high enough temperatures to
ignite helium, simply because they do not grow sufficiently massive helium cores.
Following the main-sequence phase, they turn into red giants and lose most of
178 9 The evolution of stars - a detailed picture

their envelopes, while the cores grow - by shell-hydrogen burning - to only

0.4MG, or less. Skipping the core-hclium-burning phase, the asymptotic-giant
phase and the planetary-nebula phase, these low-mass stars become even lower-
mass white dwarfs, composed mainly of helium. Indeed, the mass distribution
of white dwarfs derived from observations shows two peaks. The main peak, to
which most white dwarfs belong, corresponds to an average mass of ~0.6A/3.
The secondary, smaller, peak is found between 0.2Mo and 0.4A7o, confirming the
prediction of two distinct sources of white dwarfs.
How do white dwarfs evolve, as they must, since they radiate? This question
was considered by Mcstel in 1952. The structure of stars in the white dwarf stage
is characterized by two basic properties:

1. The internal pressure is supplied predominantly by degenerate electrons.

2. The internal energy source responsible for the radiation emitted at the
surface is the thermal energy stored by the ions (as the heat capacity of
a degenerate electron gas is negligible). The star has no nuclear energy
sources. If it had, nuclear burning would have been unstable (as seen in
Section 6.2), and something would have happened to cither stop it or
disrupt the star.

Note: In fact, as a white dwarf cools, it does contract slightly, releasing some
gravitational energy. At the same time, however, the higher density raises the internal
energy of the degenerate electrons (for which u oc p2/3 - see Sections 3.3 and 3.5) and
also the electrostatic potential energy. Mestel and Malvin Ruderman showed (in 1967) that,
to first order, the release of gravitational energy compensates for the rise in degeneracy and
electrostatic energy. Thus they vindicated the long-standing assumption that the energy
source of white dwarfs is the thermal energy of the ions, as if the white dwarf were rigid.

A degenerate electron gas behaves much like a metal, conducting heat very effi­
ciently. Since, by Equation (5.3), a very low opacity value implies a very small
temperature gradient, the internal temperature of a white dwarf is very nearly
uniform. The white dwarf structure - a homogeneous, isothermal gas, with neg­
ligible radiation pressure and no nuclear reactions - appears simple enough to be
described by analytical models with reasonable accuracy. (Elaborate numerical
models are nevertheless required for supplying the finer structural details.)
A simple model for the evolution of a white dwarf is obtained following
Mestel. A typical white dwarf may be described by an isothermal electron­
degenerate core comprising most of the star's mass M. As the density decreases
(tending to zero) toward the surface, an outer layer exists, however, where the
electrons cease to be degenerate and behave as an ideal gas. Across this surface
 9.8 White dwarfs: the final state of nonmassive stars 179

Figure 9.14 Sketch of the configuration of a cooling white dwarf.

layer the temperature drops as well and radiative equilibrium may be assumed,
with the temperature gradient determining the luminosity. The configuration bears
similarity to that of a fully convective star, discussed in Section 9.1: there, too, the
luminosity was determined by the conditions prevailing in a thin, radiative outer
layer (the photosphere). Obviously, the transition from a degenerate state to an
ideal-gas state is gradual, but, for simplicity, we shall assume a sharp transition
across a surface boundary between the degenerate core and the ideal-gas outer
layer, defined as the point where the physical conditions arc such that equal values
result for the ideal-gas pressure and for the degenerate-gas pressure. Let the radius
of this boundary be rb, as shown in Figure 9.14. For r < rh the temperature is
constant and equal to the central value Tc. For r > rb the luminosity is constant;
in addition, m(r > rb) ~ M. The structure equations for the outer layer reduce,
therefore, to

dP GM
= -P—T (9.32)
dr

dT 3 Kp L
(9.33)
dr 4ac T3 Mtr-

The first is derived from Equation (5.1) with m = M. and the second is derived
from Equation (5.3) with F = L. For the opacity we shall assume a power-law
dependence on temperature and density, the Kramers opacity law (3.65),

' = npT-^ = ^PT-W, (9.34)

where p has been replaced by P, using the ideal-gas equation of state (3.28).
Substituting Equation (9.34) into Equation (9.33) and dividing Equation (9.32)
by Equation (9.33), we obtain a relation between the pressure and the temperature
of the form

acR.G M K/,
PdP =------------------- T'il2dT. (9.35)
3/co/z L
180 9 The evolution of stars - a detailed picture

Integrating from the surface, where P = T = 0. inward, we have

64,toc7?.G \ 1/2 /M\i/2
____________ | I __ I y1 I 7/4
P(T) = (9.36)
51/fo/i / \ L /
This relation, which may be applied to the outer fringe (atmosphere) of stars in
general, is known as the radiative zero solution.
Reverting to the density by means of the ideal-gas equation of state, a relation
is obtained between the density and the temperature
MztacGn \ 1 2 / M \ 1 2 ? 13/4
p(T) = (9.37)
517?X(> / \ L /
which holds down to n,. Since the ions constitute an ideal gas on both sides of r^,
it follows that is the point where the ideal-electron pressure. Equation (3.27),
and the degenerale-electron pressure. Equation (3.34), are the same. This leads to
a second relation between density and temperature at r^.

TIT — (9.38)
MeJb

Clearly, we must have 7], = Tc in order to prevent a jump in temperature, which

would result in an infinite heat flux. Eliminating p between Equations (9.37) and
(9.38), we finally obtain
L 64?r acG K\3 /z 7,,
__ _ __________ I ' ? / / 2 (9.39)
M ~ 5171%^

which relates the luminosity emitted at the surface to the core temperature of the
white dwarf. Inserting the values of constants in Equation (9.39) for a typical
white-dwarf composition (say. half carbon and half oxygen), we have

L/Lq
6.8 x IO"3 (9.40)
M/Mq
or

(9.41)

Exercise 9.3: (a) Show that the temperature profile throughout the outer layer of
a white dwarf of mass M and radius R is given by

<9-42)

(b) Show that the layer’s thickness £ = R - rb <<c R.

(c) Calculate the relative change in thickness, G/^2, for a drop in luminosity
from L\ — 10~2Lo to L2 = 10-4i© (neglecting the small change in R).
 9.8 White dwarfs: the final state of nonmassive stars 181

As we have already mentioned, the energy source of a white dwarf is the

thermal energy of the ions in the isothermal core (the outer layer's contribution
being negligible):
3 71
Ui =----- MTC. (9.43)
2 /a.
Hence the rate of energy emission L must equal the rate of thermal-energy
depletion:

dU\ 3'JI dTc 3'JZ TcdL

—- =-------- M — =--------- M — —. (9.44)
dt 2 gi dt 7 gi L dt
where we have used the TC(L) relation (9.39). It is easily shown that this implies
r/L ,
-------- ex M T6, (9.45)
dt
which means that the rate of change of the luminosity (or. equivalently, the cooling
rate) decreases sharply with decreasing temperature. Thus the evolutionary pace
of a white dwarf slows down gradually, and a white dwarf of low mass evolves
more slowly than a massive one. To estimate the time it would take a white dwarf
of mass M to cool from an initial temperature TJ (and corresponding luminosity
L') to a temperature Tc (luminosity L), we integrate Equation (9.44).

7£
Tcoo! = 0.6— M (9.46)
ID
If Tc' 3> 7'c, then by Equation (9.39). TJL' Tc/L and the time required for
a white dwarf to cool to a temperature Tc (from a much higher temperature) or
decline to a luminosity L (from a much higher luminosity) is given by

Awl * 2.5 x 106 yr. (9.47)

\ L/Lq )

For example, about 2x 109yr would be required for the luminosity of a

\Mq white dwarf to drop to 1()-4Lq. For comparison, only ~1()7 yr would bring
the luminosity down to 0.1 Z,3 from, say, the typical planetary nebula luminosity,
of the order of 1 04Lq.
In reality, when a white dwarf reaches very low temperatures (luminosities),
the cooling rate no longer follows the simple relation (9.44). This is because the
ion gas ceases to be perfect: Coulomb interactions increase in importance until,
eventually, they become dominant. As the ratio (c/kT (discussed in Section 3.1)
approaches and then surpasses unity, the ion gas crystallizes into a periodic lattice.
At first, the corresponding heat capacity per ion increases due to the additional
vibrational degrees of freedom (from |A- to 3k). However, below a critical tem­
perature (the Debye temperature), typically a few million degrees Kelvin, the
heat capacity falls rapidly with temperature, following a T ' law. This means that
182 9 The evolution of stars - a detailed picture

Figure 9.15 White-dwarf luminosity function: number density of white dwarfs within a
logarithmic luminosity interval corresponding to a factor of IO2'5 2.5 against luminosity
(data from D. E. Winget et al. (1987). Astmphys../.. 315).

for a given amount of radiated energy, the drop in temperature is far larger than in
the free-gas regime. Thus the cooling of while dwarfs is accelerated considerably.
If k-ooi oc La, then a — —5/7 (Equation (9.47)) holds down to ~10^?L3, with
a increasing to small positive values below ~ 1CT4Lq. The number density of
observed white dwarfs as function of their luminosity - shown in Figure 9.15 -
bears witness to this effect.
The density distribution of a white dwarf, quite accurately described by an
n = 1.5 polytropc for M < l.2AfQ(see Section 5.4), remains almost constant
during the long cooling phase and hence so does the radius R. Therefore, the
cooling track in the H-R diagram is essentially a R - constant (straight) line

log L = 4 log 7'eff + constant, (9.48)

the effective temperature decreasing with the luminosity (and almost linearly with
the core temperature). Since R — R( M). the evolution of white dwarfs of different
masses corresponds to a strip in the H-R diagram, as show n in Figure 9.16. The
lower part of this strip should be much more heavily populated than the upper
part because of the rapidly decreasing cooling rate. White dwarfs spend far more
time at low luminosities than at high ones. These conclusions are confirmed by
observations. Unfortunately, however, as white dwarfs grow still fainter, they also
become more difficult to detect (and. besides, their number per luminosity interval
drops due to the rapid cooling). In the end (hey will turn into practically invisible
black dwarfs.

9.9 The evolution of massive stars

The evolution of massive stars (M() > 10Ms) has the following general charac­
teristics:
 9.9 The evolution of massive stars 183

Log(Teff)

Figure 9.16 White dwarf's in the H-R diagram. Lines of constant radius (mass) are marked
(data from M. A. Sweeney (1976), Astron. & Astrophys., 49).

1. The electrons in their cores do not become degenerate until the final
burning stages, when the core consists of iron.
2. Mass loss plays an important role along the entire course of evolution,
including the main-sequence phase (since the mass-loss rate of these stars
is still uncertain, this is also the reason for the poorer understanding of
their evolution).
3. The luminosity, which is already close to the Eddington critical limit
on the main sequence, remains almost constant, in spite of internal changes.
The evolutionary track in the H-R diagram is therefore horizontal, shifting
back and forth between low and high effective temperatures. Such tran­
sitions are slow during episodes of nuclear burning in the core and rapid
during intervening phases, when the core contracts and heats up, while the
envelope expands.

Stars of initial mass exceeding 30A/o have so powerful stellar winds as to result
in mass-loss timescales M/M shorter than main-sequence timescales MQ/L.
Consequently, their main-sequence evolutionary paths converge toward that of a
30A/Ostar. In particular, the extent of the helium core at the end of the main-
sequence phase is similar, and hence so are the ensuing evolutionary stages. The
intense mass loss that occurs during the main-sequence phase leads to configu­
rations composed mainly of helium, with hydrogen-poor envelopes (X % 0.1) or
no hydrogen at all. Such stars - luminous, depleted of hydrogen, and losing mass
at a high rate - are indeed observed, being known as Wolf-Rayet stars. They have
relatively low average masses, between 5 and 10/Wo, and are considered as the
bare cores of stars initially more massive than 3(WQ. There are different types of
Wolf-Rayet stars, distinguished according to their surface composition. Element
abundances in the sequence of types correspond to a progression in peeling off of
184 9 The evolution of stars - a detailed picture

Figure 9.17 Mass ejection by massive stars captured by NASA’s I lubblc Space Telescope,
(a) Eta Carinae, one of the brightest and most massive mass-losing stars. Its luminosity
is estimated at about 5 x 106£q, and its present mass at roughly IOO.Mq. Two lobes
of ejected stellar material are located very near the star, moving outward at a velocity
of ~600kms_| (photograph by J. Morse, University of Colorado), (b) a massive, hot
Wolf-Rayct star embedded in the nebula created by its intense wind. The blobs result
from instabilities in the wind which make it clumpy. The expansion velocity is about
40kms_| and the nebula is estimated to be no older than IO4yr (photograph by Y.
Grosdidier, University of Montreal and Obscrvatoire de Strasbourg; A. Moffat, University
of Montreal; G. Joncas, Laval University; and A. Acker, Observatoire de Strasbourg).

the outer layers of evolving massive stars; thus, some show the undiluted burning
products of the CNO cycle - helium and nitrogen, while others show the products
of 3a and other helium-burning reactions, mostly carbon and oxygen. A well-
known example of vigorous mass loss is provided by the peculiar star Eta Carinae,
shown in panel (a) of Figure 9.17. The nebula is considerably enriched in nitrogen,
and generally the observed abundances are consistent with those obtained from
model calculations for the supergiant phase of an initial 120AfQ star evolving with
 9.9 The evolution of massive stars 185

miM

Figure 9.18 Composition profiles in the inner 8MQof a 25MG star prior to supernova
collapse. Burning shells are marked (adapted from S. E. Woosley & T. A. Weaver (1986),
Ann. Rev. Astron. Astrophys., 24).

H Burning

He Burning

C burning

0 burning

Si burning

Figure 9.19 Schematic structure of a supernova progenitor star.

mass loss. A recent image of mass ejection by a typical Wolf-Rayet star is shown
in panel (b) of the figure.
In all massive stars, helium burning in the core is succeeded by carbon burning.
At this stage the core temperature is so high as to cause significant energy losses
due to neutrino emission. Thus the nuclear energy source has to compensate for
these losses, as well as supply the high luminosity radiated at the surface. As fusion
of heavy elements releases far less energy per unit mass of burnt material than
fusion of light elements (see Chapter 4), nuclear fuels are very rapidly consumed.
All the major burning stages pass in rapid succession, until an inner core made of
iron group elements is formed. Surrounding this core are shells of different compo­
sitions - silicon, oxygen, neon, carbon, helium - and, finally, the envelope, which
for Mo < 30AfQ retains most of the original composition and contains most of the
stellar mass. The inevitable contraction ofthe iron core will lead the star toward
collapse in a supernova explosion. The structure of a massive star and its schematic
configuration in the supernova-progenitor stage arc shown in Figures 9.18 and
9.19. The final stages of evolution will be described in the next chapter.
186 9 The evolution of stars - a detailed picture

He -> C+O Core Collapse. Supernova II

c+c
5 He

25Me Thermal Pulses Begin

To White Dwarf (0.85Me) \ AGB

Second Dredge-Up Begins PN

4 To White Dwarf (0.6M,:.) Ejection"

Thermal
t
Fluorescence
He -> C+O Pulses
Begin
Surrounding PN
3 Begins Here > H-> He
Core
5Mq Helium
First Dredge-Up Begins
Flash

2 -
Horizontal Branch

He

RGB

Dredge-Up
Begins

-1
4.5 4.3 4.1 3.9 3.7 3.5 3.3

Log (Teff)

Figure 9.20 Evolutionary tracks of 1 Ma,5Ma and 25MQ star models in the H-R diagram.
Thick segments of the line denote long, nuclear-burning evolutionary phases. The turnoff
points from the AGB are determined empirically (from I. Iben Jr. (1985), Q. J. Roy. Astron.
Soo., 26).

9.10 The H-R diagram - Epilogue

We have come to the end of our discussion on the H-R diagram and its
theoretical counterpart, the (log T^f, logL) diagram, thus completing the task
that we set out to accomplish at the end of Chapter 1. The success of the stellar
evolution theory in explaining the many different, often puzzling, characteristics
of stars, as exhibited by the H-R diagram, is remarkable: it explains the preva­
lence of main-sequence stars, the main-sequence turnoff point in star clusters,
the red giant, the supergiant, and the horizontal branches, the planetary neb­
ula and the white-dwarf regions, the gap between the main sequence and the
giant branch and many other, subtler properties of stars. To conclude this dis­
cussion. we show two more figures. In Figure 9.20 full evolutionary tracks in
the H-R diagram are given for a low-mass star, an intermediate-mass star and a
massive star.
 9.10 The H-R diagram - Epilogue 187

Figure 9.21 Evolutionary calculations for stars of different masses forming a hypothetical
cluster result in an evolving H-R diagram, shown at four ages. The number of stars and
their mass distribution is arbitrary. The dashed lines are lines of constant radius. The dotted
lines mark the main-sequence slopes. We note that at 107 years (a), the low-mass stars
are not yet settled on the main sequence, while the very massive ones have already left it:
the open triangles show the main sequence of massive stars at a much earlier epoch, 105
years. The Hertzsprung gap is conspicuous at 108 years (b) resembling the Hyades-cluster
H-R diagram shown in Figure 1.5. By contrast, the continuously-populated track toward
the red giant branch is clearly seen at later epochs (c and d). when low-mass stars leave
the main sequence.

Finally, crowning the stellar evolution theory, Figure 9.21 presents the evolv­
ing H-R diagram of a hypothetical star cluster, based on evolutionary calculations
of a large number of star models of different masses. Disregarding the number
of stars, the various populations are hardly distinguishable from those of actual
H-R diagrams of stellar clusters of different ages, as shown in Chapter 1.
188 9 The evolution of stars - a detailed picture

Nevertheless, the picture of stellar evolution is not yet complete, although

it is far more elaborate, detailed, and clear than the rough sketch traced in
Chapter 7. On close inspection, there are still fuzzy spots, especially where mass
loss or convection are concerned. Eddington ended his famous 1926 book with
the following: ‘... but it is reasonable to hope that in a not too distant future we
shall be competent to understand so simple a thing as a star.’
'We now do understand a great deal about stars; in particular, we understand
that they are not all that simple.
 10

Exotic stars: supernovae, pulsars

and black holes

Stars of the types considered in this chapter differ from those discussed so far,
inasmuch as. for various reasons, they do not (or cannot) appear on the H-R
diagram. As before, we shall rely on stellar evolution calculations to describe
them. Whenever possible, we shall confront the results and predictions of the
theory with observations, either directly or based on statistical considerations. We
shall find that, as we approach the frontiers of modern astrophysics, theory and
observation go more closely hand in hand.

10.1 What is a supernova?

We should start by making acquaintance with the astronomical concept of a

supernova, as we did with main-sequence stars, red giants and white dwarfs in
Chapter 1. Stars undergoing a tremendous explosion (sudden brightening), during
which their luminosity becomes comparable to that of an entire galaxy (some
10" stars!), are called supernovae. Historically, nova was the name used for an
apparently new' star; eventually it turned out to be a misnomer, novae being (faint)
stars that brighten suddenly by many orders of magnitude. So are supernovae,
but on a much larger scale. Not until the 1930s were supernovae recognized as
a separate class of objects within novae in general. They were so called by Fritz
Zwicky, after Edwin Hubble had estimated the distance to the Andromeda galaxy
(with the aid of Cepheids) and had thus been able to appreciate the unequalled
luminosity of the nova discovered in that galaxy in 1885, amounting to about one
sixth ofthe luminosity of the galaxy itself.
Since supernova outbursts last for very short periods of time (several months
to a few years), the chances of detecting them are small, even if an appreciable
fraction of stars go through this stage. Thus in a large stellar population, such as
a galaxy, supernova explosions are detected once in a few decades. Fortunately,
they become bright enough to be observable at very large, cosmic distances, and

189
190 10 Exotic stars: supernovae, pulsars and black holes

Table 10.1 Historical supernovae

Galaxy: Distance
Name Year x 3000ly

Milky Way:
Lupus 1006 1.4
Crab 1054 2.4
3C58 118K?) 2.6
Tycho 1572 2.5
Kepler 1604 4.2
Cas A 1658±3 2.8

Andromeda 1885 700

LMC: SN1987A 1987 50

a) 1937 Aug.23. Exposure 2Om. Maximum brightness

b! 1938 Nov.24. •• 45m. Faint.
c) 1942 Jon 19. ■■ 85m. Too faint to observe.

Figure 10.1 Supernova in the galaxy IC4182. At maximum brightness (a), it completely
obscures the galaxy; 5 years later (c), it becomes too faint to observe and the parent
galaxy appears in the picture. (Mt. Wilson 100-in. telescope photographs from the Hale
Observatories).

hence hundreds of such events have been recorded and studied. An example is
given in Figure 10.1. where a supernova in outburst outshines the galaxy within
which it resides by about a factor of 100. The most famous supernovae are those
which occurred and were observed in our own Galaxy - the historical supernovae,
listed in Table 10.1. These, however, represent only a fraction of all supernova
explosions that must have occurred in our Galaxy, say, in the last millennium,
 10.1 What is a supernova? 191

because most regions of our Galaxy are obscured by its radiation-absorbing central
bulge. (It is much easier to detect lights turning on in a neighbouring building
than in one’s own.)

Historical Note: The close occurrence of the supernovae of 1572 and 1604 led to a
philosophical revolution, by shattering the Aristotelian conception of the universe, which
had prevailed for almost two millennia. Aristotle’s universe consisted of a set of concentric
spheres, with the Earth at the centre. Each of the planets known at the time revolved in
its own sphere, while the outermost sphere contained the fixed stars. The lowest sphere
contained the Moon and marked the boundary between the imperfect, changeable world
below it and the perfect and eternal universe above. This is the reason why comets, of
unpredictable and transient apparition, were considered atmospheric phenomena. The
supernova of 1572 was intensively observed and studied by the Danish astronomer Tycho
Brahe, who devoted a book (De Nova Stella} to the new star. He paid particular attention
to its distance and concluded that it must reside within the fixed stars, far above the
Moon, showing that changes could take place in what had been considered the immutable
universe. But he chose to explain the new star as an immutable object that had so far
been concealed from the human eye. It took one more (soon to follow) supernova, another
great astronomer - Johannes Kepler, Tycho’s former assistant - and one more publication
(bearing a similar title, De Stella Nova) to overthrow the conception of the immutability
of the heavens. Kepler observed the 1604 supernova and concluded that, like Tycho’s
supernova, it, too, was among the fixed stars. Aristotle’s model had failed again and was
soon to be abandoned, although reluctantly at first, in favour of the Copernican heliocentric
theory and Kepler’s famous laws of planetary motion.

The nebulae ejected in supernova explosions, the so-called supernova rem­

nants, survive for much longer periods of time and are regarded as some of the
most spectacular astronomical objects. Expansion velocities being very high, up to
10000km s“' (0.03c!), these nebulae extend quite rapidly to remarkable dimen­
sions and remain visible for thousands of years, even when they become so dilute
as to be almost transparent. The Crab nebula, a remnant of the 1054 supernova, is
shown in Figure 10.2. A much more diluted, older supernova remnant is shown
in Figure 10.3.
According to the stellar evolution theory’ that wc have outlined in Chapter 7,
a catastrophic end to the life of a star is bound to arise from two types of quite
different circumstances, which lead to a state of dynamical instability. One is the
collapse of the iron cores of massive stars. The other is the collapse of white
dwarfs that have reached the Chandrasekhar limiting mass. As we have seen in
the previous chapter, the masses of degenerate cores of intermediate-mass stars,
which turn eventually into white dwarfs, are considerably lower than the critical
mass. Hence single stars are spared the catastrophic fate of collapse. This fate
192 10 Exotic stars: supernovae, pulsars and black holes

Figure 10.2 Crab nebula: the expanding remnant of the supernova that exploded in 1054
(from plates taken in 1956 with the Hale 5-m telescope, copyright D. Malin & J. Pasachoff,
Caltech).

Figure 10.3 Remnant of a supernova (N132D) that exploded some 3000 years ago in the
Large Magellanic Cloud. The progenitor star, which was located slightly below and left
of centre in the image, is estimated to have had a mass of 25MQ (photograph by J. A.
Morse, Space Telescope Science Institute, taken with NASA’s Hubble Space Telescope).
 10.2 Iron-disintegration supernovae 193

awaits, however, white dwarfs evolving in binary systems, which may interact
with their companion stars and reach A/ch-
Indeed, supernova explosions arc classified into two types according to their
observed properties: the so-called Type 1 and Type II supernovae. The main dis­
tinguishing characteristic is the presence of hydrogen lines in the spectrum of the
latter and their absence in the former. Each type has its own characteristic light
curve, although a wide variety of deviations from the general shape is detected,
resulting from individual properties, and subclasses have been defined (which we
shall ignore). Type II supernovae are not observed in old stellar populations (such
as elliptical galaxies), but mostly in the gas and dust rich arms of spiral galaxies,
where star formation is going on and young stars are abundant. Type I supernovae,
by contrast, are found in all types of galaxies.
It is the Type II supernovae that are associated with the collapse of the iron
cores of massive stars. These stars have large hydrogen-rich envelopes: hence the
evidence of hydrogen in the spectrum. As massive stars evolve much more rapidly
than low mass stars, old stellar populations, where no star formation occurs, have
outgrown the Type II supernova stage. Type I supernovae - more precisely, the
predominant Type la subclass members - arc those believed to arise from the
collapse of white dwarfs that have reached the limiting Chandrasekhar mass,
presumably by accretion or coalescence. Since in a given stellar population white
dwarfs form at all times, there is nothing to prevent the occurrence of Type la
supernovac in old populations as in young.

10.2 Iron-disintegration supernovae: Type II - the fate

of massive stars

To summarize Section 9.9, stars of initial mass exceeding ~10AfG undergo all
the major burning stages, ending with a growing iron core surrounded by layers
of different compositions. These are separated by burning fronts, which turn
the lighter nuclear species of the overlying layer into the heavier species of the
underlying one. Anticipating the imminent collapse, we have called such stars
supernova progenitors.
At the beginning, the iron core contracts - as all inert stellar cores do - simply
because no nuclear burning is taking place and. eventually, the electrons become
a degenerate gas. When the degenerate core’s mass surpasses the Chandrasekhar
limit (which, for iron, is somewhat lower than 1.46MO), the degenerate electron
pressure is incapable of opposing self-gravity and the core goes on contracting
rapidly. Two types of instability soon develop. First, electron capture by the heavy
nuclei deprives the core of its main pressure source and thus accelerates the infall.
Secondly, due to the high degeneracy of the gas - and hence its low sensitivity to
194 10 Exotic stars: supernovae, pulsars and black holes

temperature - the temperature rises unrestrained. In time, it becomes sufficiently

high for the photodisintegration of iron nuclei (sec Section 4.10):

56Fe —> 134He + 4n - 124 MeV.

This reaction is highly endothermic, absorbing ~2MeV per nucleon (just as the
reverse transition of helium into iron releases ~2 MeV per nucleon). The loss of
energy is so severe as to turn the collapse into an almost free fall. The continued
contraction is followed by a further rise in temperature. The pressure increases
too, but not sufficiently to arrest the process (ya < 4/3). The infall continues until
the photons become energetic enough to break the helium nuclei into protons and
neutrons. As this reaction entails an even greater energy absorption, about 6 MeV
per nucleon, the core contracts still further. Eventually, the density becomes high
enough for the free protons to capture the free electrons and turn into neutrons. Not
only docs this process absorb energy, but it also reduces the number of particles.
Hence the pressure drops and core collapse continues. Finally, the neutron gas,
which is in many ways similar to an electron gas, becomes degenerate. This
occurs at a density of about 1018kg m-3 (10l5gcm-3) and generates sufficient
pressure to halt the collapse. A neutron core is thus created, of a density similar
to that of an atomic nucleus - one single huge nucleus, about 40 km in diameter.
It was Hoyle who, as early as 1946, suggested the instability associated with
the photodisintegration of iron to be the triggering mechanism for supernova
explosions.

Exercise 10.1: Assuming a (free-fall) collapsing core to maintain a uniform

density (this is called homologous contraction), show that the solution of the
equation of motion tends to |v| ex r.

Exercise 10.2: Show that the free-fall collapse of a stellar core (of uniform initial
density) is homologous.

What happens to the outer layers of the star during and following the few
hundred milliseconds of core collapse? To answer this question, we consider the
energy budget of the star. Clearly, the energy source of a supernova explosion is
gravitational: the collapse of a core of mass Mc(~1.5Afo) from an initial white
dwarf radius Rc ~0.01 /?©to the final radius Rnc ~20km (<g Rc) of the neutron
core releases an amount of gravitational energy of the order of

, / I I \ GM; ,,
A£„rav % - GM:-------------- %--------- £- % 3 x 1046 J. (10.1)
e ’ C l n n I n
\ nc <*nc/ J'nc

The energy absorbed in nuclear processes amounts to

A£nuc % 7 Mev— % 2x IO45 J, (10.2)

"'h
 10.2 Iron-disintegration supernovae 195

about one tenth of AEgrav. There remains ample energy for ejecting all the material
outside the core, for imparting to it enormous velocities, and for producing the
huge luminosities observed. The radiated energy may be estimated by assuming
a typical luminosity Z>sn of IO3'Js-1 (3 x 1()IOLS) for a typical period rSN of
one year:

A Erad * ESNTSN * 3 X l()44J. (10.3)

Although this is an overestimate, it is still only a few percent of the released

energy. A similar amount would be required for the ejection of the (for the most
part) loosely bound envelope.

assuming a total stellar mass M ~ I ()MG. and a comparable amount would suffice
for supplying the high expansion velocities of the ejecta:

AEkin ~ l(M-MJv;xp * 1045J, (10.5)

adopting vexp ~ 10 000 km s_|, as derived from observations.

Two questions immediately arise: first, if such a small fraction of the released
energy is sufficient for powering a supernova explosion, where does the bulk ofthe
energy go? Second - the question that has puzzled astrophysicists for decades -
what is the mechanism that deposits the required energy in the envelope? The
answers to these questions are linked and involve one of the major factors affecting
the entire supernova process that we have yet to mention. These are the neutrinos,
which take part in any weak interaction so that the lepton number be conserved
(see Section 2.6).
As the iron core turns, essentially, into a neutron core, all the protons that
have been locked up in the iron nuclei undergo a weak interaction. Hence as
many as 1057 neutrinos are released, which can easily remove ~1046 J of energy,
given that their masses are very small (see Section 9.3), so that they move with
velocities very close to the speed of light. The second question is then merely how
to transfer a small fraction of the neutrino energy to the envelope surrounding
the collapsing core. Keeping in mind that matter is normally highly transparent
to neutrinos, this has proved to be a very puzzling question. Given, however,
the enormous neutrino flux and the unusually high densities involved, it turns
out that a nonncgligible neutrino opacity builds up. Some of the neutrino energy
is absorbed by the envelope layers that bounce off the stiffened neutron core
and are thus precipitated outward. The release of gravitational energy as the pri­
mary energy source in supernova explosions as well as the transfer of energy
to the envelope by neutrinos were first proposed and studied by Stirling Col­
gate and Richard White in 1966. Recent numerical simulations - which include
extensive, often multi-dimensional calculations performed on the most efficient
196 10 Exotic stars: supernovae, pulsars and black holes

Time (days)

Figure 10.4 Light curves resulting from calculated models of a 15 supernova com­
pared with observations of SN 19691. The models differ in magnitude of the explosion
energy: 1.3 x 10” erg (solid line) and 3.3 x 10” erg (dashed line) (adapted from T. A.
Weaver and S. E. Woosley (1980), Ann. NY Acad. Sci. 336).

computers - are quite successfully accounting for the observed characteristics of

supernova explosions.
The flare-up of the supernova begins when the shock wave propagating from
the collapsed core boundary breaks out through the surface of the hydrogen-rich
envelope. At first, the temperature is so high that most of the energy is radiated in
the UV, but very soon the envelope expands and the temperature drops sufficiently
for the object to become visible. A typical light curve of a Type II supernova,
where calculated models are superimposed on observational data points, is shown
in Figure 10.4.
A unique opportunity to test the core-collapse-neutrino-generating theory
was provided by the supernova that exploded in February of 1987 (known as
SN 1987A) in the Large Magellanic Cloud (LMC), a nearby galaxy, about 170 000
light-years away. A few of the neutrinos produced by that supernova (170000
years ago) - to be exact, 20 out of an estimated 1013 per m2 - were intercepted
by the neutrino detecting devices Kamiokande (described in Section 9.3) and
a similar one, named IMB. located in a 1570-m deep salt mine in Ohio. The
first detections of the two widely separated devices were simultaneous to within
the accuracy of time determination and the entire neutrino capture event lasted
about 12 s. It is noteworthy that since the detectors are located in the Northern
Hemisphere, the neutrinos from the LMC traversed the Earth before hitting the
detectors from below. All this occurred several hours before the supernova became
visible, as the theory would have it, for some time must elapse following the
collapse until the envelope expands enough to produce the typical supernova
luminosity. Besides providing the first detected neutrinos associated with core
collapse. SN1987A was unique in another, quite different sense: it was the first
(and so far only) supernova whose progenitor had been identified and its location
in the H-R diagram established (log Ten - 4.11-4.20, log L = 5.04). A mass
 10.3 Nucleosynthesis during Type II supernova explosions 197

Figure 10.5 (a) SN1987A in the LMC, before and after outburst, (b) SN1987A in the
LMC photographed in March 1987, about a month after discovery. Overlaid on the picture
is the negative image taken a few years before. The image of the supernova progenitor
is confused with two other stars in the same line of sight and thus appears noncircular
(copyright Anglo-Australian Observatory: photographs by D. Malin).

of ~18Mq was inferred (having probably evolved from a star of initial mass
somewhat above 20Mo), in good agreement with the outburst and postoutburst
characteristics. The supernova near maximum brightness is shown in Figure 10.5;
superimposed is the negative of the progenitor star. All other supernovae we
have known were either too distant or too old for their progenitors to have been
distinguishable.

10.3 Nucleosynthesis during Type II supernova explosions

Perhaps the most important and long-lasting outcome of supernova explosions

is the production of heavy elements (heavier than helium) and their dispersion
198 10 Exotic stars: supernovae, pulsars and black holes

Table 10.2 Characteristic masses of supernova models (in A/©)

Initial mass Helium core Iron core Neutron core Ejected (Z > 6)

15 4.2 1.33 1.31 1.24

25 8.5 2.05 1.96 4.31

throughout the interstellar medium. These elements are produced both during
the stages preceding the explosion, in the layers surrounding the iron core, and
during the explosion itself, as a result of the shock wave that sweeps the mantle.
Most of the shock-wave energy turns into heat, which raises the temperature
to peak values attaining 5 x IO9 K; at such high temperatures nuclear statistical
equilibrium is achieved (see Section 4.7) on a timescale of seconds (the dynamical
timescale). The main product is 56Ni, rather than iron, which is obtained at
lower temperatures, when nuclear reactions are slower. The reason is that the
nuclear fuel has ZfA I, and since time is too short for fl decays to occur and
change the ratio of protons to neutrons, the product must also have Z/A =
as 56Ni does, whereas for 56Fe, Z/A — || < |. As the shock wave moves out,
it loses energy and its temperature declines. When the temperature falls below
~2 x 1O9 K, which occurs when the wave has reached the neon-oxygen layer,
explosive nucleosynthesis ceases. Thus elements heavier than magnesium are
produced during the supernova explosion, while lighter elements are produced
during the stages preceding it.
Typical values for the estimated ejected mass, as well as other characteristic
masses of supernova models, are given in Table 10.2. The supernova ejecta mix
with the pre-existing interstellar clouds made predominantly of primordial hydro­
gen and helium and thus determine the evolving galactic (cosmic) abundances of
the elements. We shall return to this point in Chapter 12. We only note for now
that the agreement between the calculated ejecta abundance pattern and the solar
system abundance pattern is striking, all the more so when one considers the span
of seven orders of magnitude among the different species.
The production of ?6Ni, which is radioactive with a half-life of 6.1 days, has
a marked effect on the supernova light curve and can therefore be verified by
observations. The product of 56Ni decay is 56Co, itself radioactive with a half­
life of 77.1 days, decaying into 56Fe. These /J decays release the energy (3.0 x
IO12 J kg-1 for 56Ni and 6.4 x 10l2Jkg_| for 56Co) that powers the supernova
light curve after the initial decline from maximum. As the rate of decay and energy
release decline exponentially on the appropriate timescales, it can be compared
with the rate of decline of the light curve. A perfect match is obtained, as shown
in Figure 10.6 for SNI987A. If the distance to the supernova is known, as it is in
the case of SN 1987 A, the amount of 56Ni produced can be inferred (0.075Mo for
SN1987A).
 10.3 Nucleosynthesis during Type II supernova explosions 199

Time (days)

Figure 10.6 Light curve of SN1987A. Points correspond to observational data obtained
at the Cerro Tololo Inter-American Observatory (CTIO) and the South African Astro­
nomical Observatory (SAAO). The dashed line is obtained from a model assuming decay
of ().075Afo of 56Ni and later. 56Co (from D. Arnett et al. (1989), Ann. Rev. Astron.
Astrophys., 27).

Exercise 10.3: Derive the expression for the light curve L(t) of a supernova
powered by the decay of 56Ni and 56Co, assuming that 1MQ of 56Ni was initially
expelled in the explosion.

A rather recent observation that bears direct witness to ongoing nucleosyn­

thesis and to the continual dispersion of nuclear ashes throughout the Galaxy is
the detection of 26A1 - a radioactive isotope with a half-life of 7.2 x 105yr -
in the interstellar medium. Nuclei, like atoms, have their distinct spectra, due to
their discrete energy levels, only the energies involved are more than three orders
of magnitude higher. Excited nuclei, like excited atoms, emit photons whose
energies, corresponding to the differences between nuclear energy levels, are
measured in MeV, rather than eV (or keV), as in the case of atoms. The radioac­
tive aluminium isotope is produced by nuclear reactions at high temperatures in
an excited state. Its subsequent decay to 26Mg releases a characteristic 1.8 MeV
photon, appearing as a line in the y-ray spectrum. This line was detected by sev­
eral satellites equipped with a variety of y-ray detectors: COMPTEL (COMPton
TELescope on board the Compton Gamma Ray Observatory) launched in 1991,
RHESSI (Ramaty High Energy Solar Spectroscopic Imager) and INTEGRAL
(INTEmational Gamma Ray Astrophysics Laboratory), both launched in 2002,
to name only a few. The detection of this line indicates that26Al has been pro­
duced in quite significant amounts no earlier than 106 yr ago, a very short time
on the stellar evolution scale. Moreover, traces of the daughter product have been
detected in meteorites. This means that the same process of nucleosynthesis and
200 10 Exotic stars: supernovae, pulsars and black holes

heavy element dispersion was taking place at the time when the solar system
formed, 4.6 x 109 yr ago.

10.4 Supernova progenies: neutron stars - pulsars

With the expulsion of the envelope in a supernova explosion, the neutron core
becomes a neutron star. The existence of such exotic objects as neutron stars
was first postulated by Lev Landau, as early as 1932 (more precisely, Landau
mentioned the possible formation of ‘one gigantic nucleus’, when atomic nuclei
come in close contact in stars exceeding the critical mass). Their resulting from
supernovae was soon suggested by Walter Baade and Fritz Zwicky. in 1934. and
the first physical model was offered by Robert Oppenheimer and George Volkoff
in 1939. The governing equation of state is similar to that appropriate to a degen­
erate electron gas, a n = 1.5 polytrope (see Section 5.4) leading to a relation
between mass and radius R oc Af_| so long as relativistic effects arc negligible.
For example, a 1.5MS neutron star would have a 15-km radius. Thus, whereas
a white dwarf is similar in size to the Earth, the diameter of a neutron star is
no bigger than that of a large city. As in the case of degenerate electrons, the
relativistic limit of the equation of state for the degenerate neutron gas imposes an
upper limit on the neutron star's mass (equivalent to the Chandrasekhar limiting
mass for white dwarfs derived in Section 5.4). Above this critical mass, a neutron
star would not be able to generate enough pressure for balancing self-gravity
and collapse would ensue. In the case of neutrons, however, this limiting mass
is far more difficult to estimate. The value of 5.83 AW, that would result from
Equation (5.32), by taking /zn = I for neutrons, rather than /j.e = 2 for electrons,
is incorrect for two reasons. First, in a relativistic neutron gas the kinetic energy
of the particles is comparable to the rest-mass energy, and hence the Newtonian
gravitational theory is no longer valid and Einstein’s General Theory of Rela­
tivity (1915) must be used instead. Secondly, the gas is imperfect and particles
can no longer be considered free (noninteracting) at the high neutron star den­
sities. Interparticle distances are of the order of the strong force range. Hence
nuclear forces have to be taken into account and the equation of state becomes
more difficult to calculate. Although the first correction lowers the upper limit to
about 0.7 A7O, the second correction raises it. Thus, depending on the equation of
state used, the upper limit to the neutron star mass is estimated to lie between
2A7© and 3M3. Fortunately, this limit does not impose serious constraints, since
the iron cores of massive stars do not appear to exceed 2/W3 by much (see
Table 10.2). And yet, in principle at least, a third end state would be possible
for extremely massive stars - the collapse of a too massive (neutron) core into a
black hole.
 10.4 Supernova progenies: neutron stars - pulsars 201

In all cases considered so far. the different types of astronomical objects

had been observed long before they were understood. Such are main-sequence
stars, red giants, white dwarfs, planetary nebulae, different types of variable stars,
novae, supernovae — many misnomers bearing witness to that. Neutron stars,
on the other hand, first emerged as theoretical, hypothetical objects. Then, in
1967. an important discovery was made, quite accidentally. Jocelyn Bell Burnell,
a doctoral student working under the supervision of Anthony Hewish at a new
radio telescope in Cambridge, detected variable radio sources of extremely high
and very regular frequencies, the first such source having a period of barely more
than I s. They were called pulsars, short for /w/sating stars, and by analysing
the pulses more closely, it was very soon realized that pulsars must be very
compact galactic objects, smaller and denser than white dwarfs. To stress the
impact of this discovery, we note that the 1974 Nobel Prize for Physics was
awarded to Hewish for ‘his decisive role in the discovery ofpulsars' (the prize was
shared with Martin Ryle, for pioneering work in radio astronomy). Although the
explanation for these strange objects was soon to follow, it was not soon enough to
prevent another misnomer. Pulsars are not pulsating stars, but rather - as suggested
by Thomas Gold in 1968 - rotating neutron stars that formed in supernova
explosions.
The association of pulsars with supernovae - suggested by Hoyle as soon as
pulsars were made known - became widely accepted with the discovery, still in
1968, ofthe famous Crab Pulsar, with a period of only 0.033 s, at the centre of the
nebula bearing the same name, which had already been identified as the remnant
of the 1054 supernova. Another supernova remnant pulsar is the Vela Pulsar, with
a frequency of 0.089 s, discovered soon afterwards within the dispersed nebula of
a supernova that occurred some 10 000 years ago.
Having identified pulsars as the neutron star survivors of supernova explo­
sions, we now turn to the pulsar mechanism. This invokes two main factors:
rotation and a magnetic field, both having been tremendously intensified in the
course of the supernova core collapse. Taking into account that all stars rotate,
even if very slowly and insignificantly, we may estimate the expected rotation
rate of a neutron star by applying the law of angular momentum conservation
to the collapsing core. Since the collapsing core is weakly coupled to the enve­
lope, the transfer of angular momentum between them (if any) is negligible. We
know that the Sun rotates with a period of 27 days. Let us assume this period
Pq to be typical of a I Rq object of mass similar to the Sun’s, such as the pre­
supernova core of mass Mc. The corresponding angular momentum is of the
order of

2rtMcR^
(10.6)
202 10 Exotic stars: supernovae, pulsars and black holes

Figure 10.7 Sketch of the lighthouse model for pulsars.

If the angular momentum is conserved while the core turns into a neutron star of
radius Rns ~ 20 km. the rotation period of the neutron star will be

= *2x10"3s, (10.7)
\ "0 /

of the same order as the pulse period of the fastest pulsars.

But rotation alone is not sufficient for sending out pulsed radiation; a beam
would be required that would be periodically directed toward the observer, sim­
ilarly to a lighthouse beam. Here the magnetic field, which has been enhanced
by the collapse up to 10x T - nine orders of magnitude higher than the Sun’s and
more than four orders of magnitude higher than a white dwarf’s - comes into play.
Charged particles arc accelerated by a magnetic field, and accelerated charged
particles emit radiation. Radiation of this kind, characteristic of strong magnetic
fields, such as those produced in particle accelerators called synchrotrons, is
accordingly called synchrotron radiation. It is mainly produced by electrons and
it is strongest along the direction of the magnetic poles, where the magnetic field
lines are most concentrated. Thus the radiation emitted by the neutron star comes
from two radiating cones around the poles of the magnetic (dipole) axis, as shown
in Figure 10.7. If the rotation axis and the magnetic field axis were to coincide,
the neutron star, visible to observers lying within the radiation cones, would have
appeared as a constant source. There is no reason, however, for the axes to be
aligned - on Earth, for example, they are not. If they are not. we have our rotating
beam that sweeps the sky at the frequency of the stellar spin. It should be noted
that, since pulsars radiate in a preferred direction, we miss many such objects,
even if they are close by.
 10.4 Supernova progenies: neutron stars - pulsars 203

What is the energy source of pulsars? The perhaps surprising answer is kinetic
energy - the kinetic energy of rotation, amounting initially to

zl *■
A Erot % | Mc * 5 x 1045 J. (10.8)

to use the values of the former example. This energy, whose source is the gravi­
tational energy of collapse, should have been added, in fact, to the energy budget
of the supernova - Equation (10.5) above - but would not have changed any of
the conclusions. Indeed, now that over a thousand pulsars have been detected
in our Galaxy, and the oldest have been observed for a relatively long period
of time, it has been established that the pulsars’ periods increase with time (as
Gold had predicted), implying that the spinning rate slows down. The rate of
rotational energy loss Erot derived from the observed slow-down of the spinning
rate, — Erot oc P/P3 exceeds by many orders of magnitude the rate of emission
of pulsed radiation (a factor of ~107 for the Crab pulsar), so most ofthe energy is
emitted by a different mechanism. In simple terms, the rapidly changing magnetic
field of a spinning magnetic dipole with unaligned spin and dipole axes generates
a strong electric field and emits electromagnetic radiation at the spin frequency,
known as magnetic dipole radiation.
The magnetic-dipole-radiation mechanism for pulsars helps to solve a long­
standing enigma related to the very source of energy that powers the Crab nebula
and, in particular, to that part of the radiation emitted by the nebula which is due
to relativistic electrons. Although such electrons were produced in the supernova
explosion, they should have radiated away their energy a long time ago. Moreover,
an initial magnetic field that might have permeated the supernova ejecta, and
could have accelerated these electrons, should have weakened considerably with
the expansion of the nebula. As it turns out. for the Crab pulsar, — Erot ~ 105Co,
very close to the power required to explain the radiation and expansion of the Crab
nebula. If the rotation slowdown is due to emission of magnetic dipole radiation,
the relativistic electrons arc a by-product of the huge electric field associated with
the rapidly changing magnetic field of the spinning pulsar. The pulsar radiation
and the transfer of energy from the pulsar to the nebula are not yet well understood,
but, quite remarkably, John Archibald Wheeler and Franco Pacini had suggested
that the Crab nebula might be powered by the magnetic dipole radiation of a
rotating neutron star a short time before pulsars were discovered.
From the known number of pulsars and their estimated lifetimes it is possible
to derive an average rate of pulsar formation: this turns out to be about one every
few decades, very close to the rate of supernova explosions. This observation
provides an indirect, but independent, corroboration for the association of pulsars
with supernovae.
Finally, as their energy source wears out (after some 1 (P-106 yr), pulsars, too,
arc destined to 20 into oblivion.
204 10 Exotic stars: supernovae, pulsars and black holes

10.5 Carbon-detonation supernovae: Type la

Type la supernovae are by far the brightest standard candles, or distance indica­
tors, and thus play an important role in cosmology. The typical correlation that
serves for distance determination in this case is an empirical relationship between
the peak luminosity and the light-curve width. Type la supernovae have been
instrumental in the relatively recent revolutionary discovery of the acceleration
in the expansion of the Universe. And yet. the circumstances that lead to a Type
la supernova explosion are still controversial and subject to investigation, both
observational and theoretical.
As we have already mentioned in Section 10.1, there is general agreement
that the explosion is induced by a thermonuclear runaway in a carbon-oxygen
white dwarf whose mass has exceeded the Chandrasekhar mass. The reasons for
such a configuration to explode are twofold: first, dynamical instability, caused
by the inability of the degeneracy pressure to balance gravitational attraction
(see Section 6.4); second, thermal instability of degenerate matter, caused by
the insensitivity of the degeneracy pressure to temperature (see Section 6.2).
The former leads to rapid contraction and to ignition of carbon in an electron­
degenerate core - carbon detonation. Nuclear burning raises the temperature but
not the pressure and hence the temperature keeps rising, escalating into runaway.
The very high temperature causes carbon and oxygen to turn into iron-peak
elements on a dynamical time scale throughout a large fraction of the white
dwarf. The resulting enormous nuclear power blows off the entire star.
As explained in Section 10.3, the main product of the explosive nucleosyn­
thesis is 56Ni, which has the same ratio of neutrons to protons as carbon and
oxygen (the nuclear fuel), since there is no time for /(-decays to occur. Later on,
56Ni decays to 56Co, and finally 56Co to stable 56Fe, and these decays are reflected
in the light curve of supemovae of both types. The effect is more conspicuous
in the light curves of Type 1 supernovae, where a much larger fraction of the
mass, almost the entire progenitor star, turns into 56Ni. The decay of 56Ni and
56Co dominates the light curve in this case. A composite Type I supernova light
curve is shown in Figure 10.8. illustrating the striking uniformity of these gigantic
explosions.
The energy that powers the Type la supernova explosion is thus nuclear, in
contrast to the Type II case, where the energy source is gravitational contraction.
The amount may be easily estimated: the mass excess per nucleon in the progeni­
tor, assuming equal mass fractions of carbon and oxygen, is —0.1480 MeV while
the mass excess of56Ni is —0.9625 MeV/nucleon. Hence 0.8145 MeV are released
per nucleon and the number of nucleons is A/ch/mn % 1.75 x 1057, which yields
a total energy of ~2.3 x 1044 J. Most of this energy goes to the disruption of the
white dwarf, whose binding energy is of the same order.
 10.6 Pair-production supernovae and black holes 205

0
Type I Supernovae

-1
X
ra
E

-3

0 50 100 150 200 250 300

Days from maximum

Figure 10.8 Composite light curve of 38 Type I supernovae (superimposed so that the
maxima coincide) (from R. Barbon et al. (1973). Astron. Astrophys. 25).

So far the process is well understood. What is still uncertain is the evolution
that produces the Chandrasekhar-mass white dwarfs, which necessarily involves
stellar interaction, since single white dwarfs are produced with lower masses. We
shall address this problem in the next chapter.

10.6 Pair-production supernovae and black holes - the fate of very

massive stars

Stars of mass M > 60 (or M > 80Mo, according to some estimates) encounter
a different type of instability in their evolutionary course. The brief hydrogen-
burning phase is followed, as usual, by helium burning. Helium burning in these
stars produces mostly oxygen (due to the high core temperatures attained) and
hence oxygen rather than carbon constitutes the next nuclear fuel. Oxygen ignites
in a core exceeding 30M© at a temperature of ~2 x 109 K. At this temperature the
photon energy is sufficiently high for spontaneous electron-positron pair creation
(see Section 4.9). Pair production, much as ionization or photodisintegration,
reduces the adiabatic exponent below the stability limit of 4/3, leading to a
dynamical instability, as discussed in Section 6.4. The core (or part of it) - whose
mass exceeds the limiting neutron star mass - collapses and a black hole is formed
on a dynamical timescale.
The description of a black hole,, in fact the very concept of such an object,
is entirely based on the theory of general relativity, which is beyond the scope
of this text. Suffice it to say that even simple arguments indicate that something
odd must occur when the radius of a star of given mass becomes so small that the
escape velocity approaches the speed of light. This limiting radius, known as the
206 10 Exotic stars: supernovae, pulsars and black holes

Figure 10.9 Light curve of supernova SN2006gy, compared with typical light curves of
Type II and Type la supernovae and with SN 1987A. (Adapted from NASA/CXC/UC
Berkeley/N. Smith et al.)

Schwarzschild radius (after Karl Schwarzschild), is given by

2GM M
/<sch = —— - 3——km. (10.9)
c~
With remarkable foresight, Laplace pointed out the difficulty in 1796 (in his
‘Exposition du systeme du monde’), about a century after Newton’s classical
theory of gravitation, but also about a century' before Einstein’s relativistic one. (In
fact, Laplace was anticipated by John Michell, a less well known English geologist
and clergyman, who expressed similar ideas in a letter to Henry Cavendish in
1784.) Classical mechanics cannot account for such an enormous gravitational
field as obtains at r < 2?sch to prevent photons from escaping it, since photons are
massless, but lacking more advanced knowledge, this is the intuitive explanation
for the blackness of a black hole.
While the core collapses, the outer layers of the star are ejected as in a super­
nova explosion. Whether the explosion resembles a Type 1 or a Type II supernova
or whether it is different from both is uncertain. The results of evolutionary cal­
culations depend on as yet undetermined parameters, such as the mass-loss rate
during early evolution. It is unclear, for example, whether such a star would lose
all. or only part of, its hydrogen-rich envelope prior to explosion. Nor can we
rely on observations to guide us, for, as we shall see shortly, very massive stars
are rarely born, and moreover, they are extremely short-lived, aging and dying,
as it were, almost as soon as they are born. The remnant black hole, as the name
 10.6 Pair-production supernovae and black holes 207

indicates, would be elusive to observation as well. We have come to an impasse:

we can never be certain of an explanation to a phenomenon that we do not see,
even if we can explain why we do not see it. A theory which cannot be tested or
confirmed by observation seems unworthy of pursuit.
And yet, a candidate for the pair-production driven supernova was discovered
a few years ago, the first so far! Dubbed SN2006gy. it is the brightest supernova
known to date, about a hundred times more luminous at its peak than the brightest
Type II supernovac. as shown in Figure 10.9. It occurred in a very distant galaxy,
about 240 million light-years away, and hence about 240 million years ago. The
progenitor is believed to have been a very massive star, its mass exceeding 100Afo,
and perhaps close to the upper mass limit of stars. The high luminosity may have
resulted from the decay of 50-100 times more 56Ni than a regular supernova, about
10Afo. Interestingly, the presumed structure of its progenitor bears similarity to the
star Eta Carinae of our own Galaxy. This discovery casts doubts on the hypothesis
that the demise of a very massive star must be associated with the creation of
a black hole. Nevertheless, in some cases there is compelling evidence for the
presence of black holes (of stellar mass), inferred from phenomena related to the
strong gravitational fields these objects generate around them and the resulting
interaction with their surroundings. This brings us to the subject of the next
chapter.
 Interacting binary stars

11.1 What is a binary star?

As it turns out, the majority of stars are members of binary systems, or even
multiple systems. The term binary in the stellar context was coined in 1802 by
William Herschel only a few years after he introduced the term planetary nebula,
as mentioned in Section 9.7. The first telescopic discovery of a double star. Mizar,
is attributed to Giambattista Riccioli in 1650. just 41 years after Galileo’s first
telescope. Other stellar pairs were found by the mid-eighteenth century, but little
effort was devoted al the lime to their study.
A binary system consists of two stars revolving around their common centre
of mass, as shown in Figure 11.1, and is defined by three parameters: the masses
of its member stars and the distance d between their centres. The distance is not
necessarily constant in time; it may vary periodically or change secularly. The
masses, too, may change in the course of time. So perhaps a better characteri­
zation should be: initial masses and separation, and current age. Each parameter
spans a wide range of values and their combinations are innumerable. In most
cases, however, the members are so far apart that their individual structures and
evolutionary courses are barely affected; they arc thus no different from single
stars, except that their dynamics as point masses is more complicated.
Binary stars are born together as a bound system; in principle, a star may
capture another, in the presence of a third body, into a bound (negative energy)
state, but the chances for that to happen are small even in a dense star cluster.
Born at the same time and having different masses, binaries may be expected to
consist of stars in widely different evolutionary stages, since evolutionary time
scales - as we have learned - depend strongly on stellar mass. Interactions may
thus result in a wealth of phenomena. For binary stars to be interacting, their
mutual distance must be relatively small, when measured in units of the larger of
their radii. How small the distance may be, what form the interaction takes, and
to what consequences, are the questions that will interest us in this chapter.

208
 11.1 What is a binary star? 209

Figure 11.1 Orbits of two masses about their common centre of mass, for an elliptic orbit
of eccentricity 0.8 and for a circular orbit. The mass ratio is 2 in both cases, with the
massive star labelled I and the less massive one labelled 2.

Orbital motion

Consider a system of two stars of masses M\ and M2, isolated in space. In the
absence of external forces, the centre of mass of the system is at rest, and we may
adopt it as reference point. The stars may be considered point masses moving
with respect to the centre of mass under their mutual gravitational forces, which
are equal in magnitude and opposite in direction, according to Newton’s third
law. Let d| and d2 be the distances of M\ and M2, respectively, from the centre
of mass. The distance between the masses d is then

d = d, - d2, (11.1)

where all three vectors d| (/). d2(t) and d(r) are colinear, their magnitudes satis­
fying the relations:

M\d\ — M2d2 and d\ + r/2 = d. (H.2)

The equations of motion for the two stars are

GM\M^
Mid, M2A2 — ~ (H.3)

and combining them, we obtain

G(M} + M->) j
d = - d. (11.4)
d3
210 11 Interacting binary stars

The well-known solution d(Z) of this equation is periodic, described by an ellipse,

if we discard the possibility of unbound states, where stars would move away from
each other on hyperbolic or parabolic trajectories. Thus the distance between the
stars traces an ellipse in the orbital plane, the stars moving apart and coming
closer together periodically in time. This means that each star moves in an elliptic
orbit with respect to the other. An elliptic orbit is defined by two parameters: the
semi-major axis a and the eccentricity e, and in polar coordinates the equation
that describes it is

2)
a( I - eI.
d(6) =------------- (11.5)
1 + e cos 0

where 6 is the angle between the vector d and the major axis with the origin
at a focal point. It may be easily shown that not only the mutual distance, but
also the trajectory of each one of the stars in a rest frame of reference traces an
elliptic orbit, each around the common centre of mass. The periods of revolution
are obviously the same. This is shown in the upper part of Figure 11.1.

Exercise 11.1: Show that in the rest frame of reference, the equation of motion of
each member of a binary system has the form (11.4), whose solution is an elliptic
orbit, with the centre of mass at a focal point. Show further that the ellipses have
the same eccentricity.

The simplest form of an ellipse is a circle, an ellipse of zero eccentricity,

where the focal points and the centre coincide, the distance between the stars is
constant in time. d(t) = a. and so are the velocities of each. Most binaries that are
close enough to be interacting have circular orbits. This property is of particular
importance, for it enables the determination of the masses of the binary members,
providing the only way of measuring stellar masses.

Observational classification

Only seldom arc both members of a binary system visible; in such cases the
system is known as a visual binary, the most famous example of which is Sirius,
consisting of a main-sequence star and a white dwarf. When only one member is
observed, which is usually the case, evidence for the existence of a companion is
provided in one or more of the following manifestations (signs):

I. Astrometric binary: the star is wobbling periodically with respect to a

fixed point, the result of the projected orbital motion on the celestial
sphere perpendicular to the line of sight.
 11.2 The general effects of stellar binarity 211

Figure 11.2 Light curve of Algol-type (see below) eclipsing binary stars. Data was
obtained by the International Gamma-Ray Astrophysical Laboratory (INTEGRAL) (from
J. M. Mas-Hesse et al. (2003), Astronomy and Astrophysics, 411).

2. Spectroscopic binary: the spectral lines of the star show a periodic variation
of their Doppler shift, blue-ward and red-ward alternate, as the revolving
star moves towards and away from the observer.
3. Eclipsing binary: the star’s luminosity varies periodically, as a result of
eclipses of one star by the other. For this to be possible the inclination of
the orbital plane with respect to the line of sight must be small: the angle
between the normal to the plane and the direction of the observer, close to
90 degrees.

Exercise 11.2: Consider a system of two stars that revolve about their centre of
mass in circular orbits, for which it is possible to separate the spectral lines of
the two components (a spectroscopic binary system). As a result of the Doppler
effect, the lines shift periodically about a mean to shorter and longer wavelengths,
as each star moves toward or away from the observer. From these shifts it is
possible to determine the orbital period "P and the velocity components along the
line of sight, vo,[ and uo,2, respectively. Denoting the angle of inclination of the
orbital plane with respect to the observer by i, find the masses of the two stars,
M\ and Mz. in terms of the observables and sini.

I 1.2 The general effects of stellar binarity

Generally, the structure of each member of a binary system may be affected in

two different ways: the star may be irradiated by its companion’s luminosity, or
distorted by its gravitational field. The resulting effect is complicated to calculate
accurately, mainly due to the lack of symmetry; nevertheless, we shall try to obtain
rough estimates.
Consider irradiation of a star by another’s luminosity: the additional energy
will be absorbed in a thin outer layer, which will cause the effective temperature
212 11 Interacting binary stars

to rise, so that the surplus is re-emitted together with the energy flux flowing from
within. The radiation energy absorbed by the irradiated star is given by the flux
emanating from the radiating star at a distance J, given by Li/Ąird2 multiplied
by the cross-section of the former, tr R\. Denoting the intrinsic luminosity of the
first star by L| and assuming the radius of this star to remain unchanged, its total
luminosity is

ttRr
L = L\ + At-—jy (H.6)
4ttć/2
and the resulting effective temperature is

Ttff — Teff.l (11.7)

where T^j j would be the effective temperature of the star, if undisturbed. Usually,
the correction term on the right-hand side of Equation (11.7) will be small.
To get an estimate of the length-scale affected by the incoming radiation,
we use the same principle that has led to timescale estimates in Section 2.8: we
divide the quantity that is bound to change by the process - in our case, the
temperature - by its rate of change with distance within the star (its gradient),
irrespective of sign. Adopting m as space variable and using Equation (5.3), we
have for the absorbing mass scale that we denote by /,
T = 4aeT^rr^
\dT/dm\ 3k F
As a rough approximation, near the stellar surface we may substitute on the
right-hand side: F L. T Tetf. r =» R\. With a = 4cr/c and L = 4tt R]a7^,
Equation (11.8) reduces to
16 4tt/??
(11.9)
T K

Normalizing / by the stellar mass M\ — (4n/3)pR\ we obtain

X 16 £
(11.10)
KpR\ p
where p is the density near the surface of the star, which is orders of magni­
tude lower than the average density, while KpR\ is a crude approximation for
the photospheric optical depth, hence of order unity. Thus the outer layer of
the star where most of the incoming radiation is absorbed has negligible mass
in comparison with the stellar mass, the bulk of which remains unaffected. It
is worth noting that whenever heat diffusion is involved, the thickness of the
zone that absorbs most of the inflowing heat is known as the skin depth. A negli­
gible mass docs not necessarily imply a negligible thickness - since the density in
 11.2 The general effects of stellar binarity 213

the outer layers of a star may be very low - but even an extended skin of negligible
mass will not affect the stellar interior and its evolution.
To get an idea about the extent of distortion caused by the presence of a
companion star, imagine two stars, I and 2. of masses M| and M2. respectively,
separated by a distance d measured between their centres. The gravitational force
per unit mass exerted by star 2 on star 1 as point masses is GM2/d2. The force
per unit mass exerted by star 2 on a small mass element at a distance r from
the centre of star I along the line of centres is GM2/(d — r)2. The difference
between the two is the force that will distort the spherical shape of the star, known
as tided force, which pulls the mass element outwards, toward the companion
star.

GM2 gm2 gm2 2GM2r

JtideO') —
~d^ * ~d^ (11.11)
(d - r)1

where we have assumed r/d < R\/d 1 and used the binomial approximation.
An order of magnitude estimate for the tidal pressure along the line of centres,
obtained by integrating /,j(je(r)p(r)Jr, yields

3GM2M}
Ptide (11.12)

This pressure is opposed by the hydrostatic pressure, which increases steeply with
depth. By Equation (5.1). at the bottom of a surface layer of mass x, we have

GMlX
/’h(X) (11.13)
4?rPf ’

Hence the mass / of the outer zone that is bound to be affected by tidal forces
may be roughly estimated by requiring PMe ~ Ph, which yields

(H.14)

usually a small fraction of the total stellar mass.

Thus, irradiation and tidal distortion, which act on a star al a distance, have
relatively small effect. The main effect of stellar interaction becomes apparent
when the stars come into direct contact by transferring material between them, or
even sharing material. Observational indication for the possibility of mass transfer
between stars was provided by the Algol paradox: an Algol binary is a binary­
system where the lower-mass member is a giant, while the more massive one is
still on the main sequence (Fig. 11.2). This contradicts w hat we have learned so
far: that the more massive a star, the sooner it leaves the main-sequence phase,
evolving rapidly towards its final state.
214 11 Interacting binary stars

Historical Note: Many thousands of Algol binaries are now known. The prototype
of Algol star’s is the star itself called Algol (or fl Persei). Algol was first recorded as a
variable star in 1667 by Geminiano Montanari. Only more than a hundred years later
was a mechanism proposed for the variability of this star by a young deaf-mute amateur
astronomer, John Goodricke, who was the first to establish the periodic nature of these
variations. He published his findings in the Royal Society’s Philosophical Transactions
in 1783, suggesting that the periodic variability was caused by a dark large body passing
in front of the star (or else that the star itself had a darker spot that was periodically turned
toward the Earth) and was awarded the Copley Medal for his report. Another hundred years
later, in 1881, Edward Pickering presented evidence that Algol was indeed an eclipsing
binary. This was confirmed a few years later, in 1889, when Hermann Carl Vogel found
periodic Doppler shifts in the spectrum of Algol, inferring variations in the radial velocity
of this binary system. Thus Algol became not only the first detected eclipsing binary, but
also one of the first known spectroscopic binaries.

As we shall see shortly, the solution to the Algol puzzle is found in the change of
mass ofthe binary members in the course of evolution, with the initially massive
star losing and the low-mass star gaining mass.
Having already encountered the outcome of mass loss in the evolution of
single stars, we should not be surprised to find that the opposite effect of mass
accretion has its own significant consequences. We start by devoting some thought
to a simple scenario where a star is embedded in a medium of low-density material
rather than a void, as we have so far assumed, disregarding for now the source
of this material. Obviously, the star will accrete some ofthe surrounding material
and we may assume spherically symmetric accretion.
When a star of mass M and radius R accretes an amount of mass 8m corning
from infinity, its (negative) gravitational potential energy decreases by an amount

._ GM8m
^'grav = ~ •

If the material is accreted over a time interval St, the average rate of gravitational
energy release Egrav — 8Egtay//8t is proportional to the average accretion rate,
M — 8m /St:

■ _ GMM
E*w ~ ~R~ (11.15)

If the star is to maintain thermal equilibrium, this energy surplus must be radiated
away. Thus an accretion luminosity may be defined in relation to the accretion
process:

GMM
I
^acc — ^grav
— F (11.16)
R
 11.2 The general effects of stellar binarity 215

Obviously, the larger the gravitational field of a star, the larger would be its
accretion luminosity. Thus, for example, if a star of 1 Afowere to double its mass,
say. during 10l0yr (comparable to the age of the universe), it should accrete at
an average rate M = 1O~ioA7q yr-1. A main-sequence star, would thus produce
a luminosity of GMqM/Rq % 3x 10~3Lg, entirely negligible compared with
the natural luminosity of such a star. For a white dwarf, the resulting luminosity
would be about a hundred times higher, a few tenths LQ, significantly higher
than the typical luminosity of white dwarfs. For a neutron star, it would reach
1 OOLq, while for a black hole (assuming the accretion radius to be /?sch) it would
approach l000Ło.
The accretion rate is limited by the requirement that the resulting luminosity
be lower than the Eddington critical luminosity. Otherwise, the radiation pressure
exerted on the infalling material would push it back and prevent it from accumu­
lating. We recall that the luminosity approaches the critical limit as the radiation
pressure becomes dominant, and the binding energy of the star tends to zero. The
requirement Lacc < Z^dd leads, according to Equation (5.37), to

which for electron scattering opacity translates into

M < 10-3/?//?© Af© yr-1. (11.18)

We note that the upper limit of the accretion rate depends solely on the stellar
radius, regardless of the mass.
In conclusion, even a moderate accretion rate (far below the upper limit) may
induce the three types of compact stars to emit a significant luminosity. What
kind of radiation would we expect in such instances? In order to maintain thermal
equilibrium by emitting the surplus gravitational energy, a star must adjust its
surface temperature. The gravitational energy of the infalling matter is absorbed
by a surface boundary layer, which acquires a temperature Tj, and re-emits the
energy as blackbody radiation. Since compact objects are stiff, the radius is barely
affected. Hence Tj, can be estimated by

An upper limit is obtained by substituting the critical value of M (11.18) on

the right-hand side, which provides a reasonable estimate in view of the weak
dependence of 7), on M (a power of 1 /4). For a white dwarf we obtain Tj, % 106 K,
for a neutron star. 7'b % 1.5 x 10' K. and for a black hole. Tj, ~ 3 x 107 K. These
would appear as bright UV, X-ray, and even y-ray sources. Indeed, such sources
are quite abundant in our Galaxy (and beyond), as satellite-mounted modern
detectors reveal.
216 11 Interacting binary stars

Thus extinct compact stars, which would otherwise escape observation, may
be rejuvenated by accretion. In fact, accretion leads to a wide variety of fascinating
phenomena - an entire zoo of exotic objects - but the simple principles of stellar
evolution that we have encountered remain the same and can be applied to explain
the evolution of binary stars, as they have explained the more straightforward
evolution of single ones. In order to understand these complex phenomena, we
must first consider more carefully the mechanism of mass exchange between
stars.

11.3 The mechanics of mass transfer between stars

A single star, viewed as an isolated point mass, generates a spherically symmet­

ric gravitational field that may be described by spherical equipotential surfaces
centered on it. The meaning of such a surface is that a test particle may move on
it freely, without requiring additional energy (or work done on it by an external
force), nor releasing energy. Moving from one equipotential surface to another,
on the other hand, entails either energy gain or energy loss: the former in order to
move away from the star (rise to a higher potential value), the latter for moving
closer to it (dropping to a lower potential state).
When a binary star, viewed as two point masses isolated in space, is involved,
the equipotential surfaces - retaining their meaning - adopt more complicated
geometrical shapes. Very close to each star, that is, at distances which arc small
compared with the separation between the stars, as well as at very large distances
by the same standard, we may expect the equipotential surfaces to be almost
spherical: closed, separate surfaces around each star in the former case, and one
surface corresponding to a point mass equal to the sum of masses, in the latter.
These extreme cases are. however, not ‘interesting’; of interest is the intermediate
domain. In particular, we may already guess that in the transition from two
separate equipotential surfaces to a single one, a critical point - a case of special
significance - will arise. It is this case and its consequences that we now wish to
pursue.
Consider stars labelled I and 2 as point masses M\ and Mi, respectively,
where M\ > M2, revolving in circular orbits around their common centre of mass
at constant angular velocity co. It is easy to see from Equation (11.4) that the
angular velocity is

co2 = G(M\ + M2)/a\ (11.20)

where a is the (constant) distance between the stars (symbol cl is reserved

for a periodically changing separation; a. for a fixed one). Since co — 2n/V,
where V is the rotation period, Equation (11.20) expresses Kepler’s third law,
 11.3 The mechanics of mass transfer between stars 217

which states V2 oc The axis of rotation is perpendicular to the line of

centres.
In a Cartesian coordinate system (,r, y. z) co-moving with the stars with the
origin at star 1, the masses are located at (0.0,0) and (a, 0,0), respectively,
and the centre of mass at (%cm> 0- 0), where %cm — Mi_a/(M\ + Mi). The grav­
itational potential experienced by a test particle at any point in space will be
given by
G M\ GMi
U2 + y2 + z2)l/2 [Or -a)2 + y2 + z2]*/2

Mi_a
(11.21)
M | + Mi

where the first two terms are the gravitational potentials of the point masses,
while the third is the rotational potential resulting from the fictitious centrifu­
gal force in the rotating system. Defining a new parameter q as the mass ratio
q = Mi/M\, and measuring distances in units of a, we may normalize the
potential:

(1 + </)(x2 + y2 + z2)1/2 + (1 +c/)lU- I)2 + y2 + z2]l/2

(11.22)

where

G( M \ + Mi)
(11.23)

Thus the normalizing coefficient, which depends on all three independent param­
eters of any binary system, is a scaling factor, while the normalized potential
depends solely on one parameter, the mass ratio q. Equipotential surfaces gen­
erated by <f>' - C - where C is a constant - arc known as Roche equipotentials,
after the French mathematician of the nineteenth century. Edouard Albert Roche,
who was the first to study this problem of celestial mechanics.
If C is large, the equipotential surfaces will be closed, separate, elongated
spheroids around each point mass; the larger C. the more spherical the surfaces.
With decreasing value of C, the closed surfaces become more distorted, especially
towards the centre of mass, along the line of centres. Eventually, for a critical
value of C. the surfaces surrounding each point mass will touch at one point on the
line of centres, creating a dumb-bell shaped configuration, known as the Roche
limit surface. The point of contact is known as the inner Lagrangian point L\,
and the volumes enclosed by the limit equipotential surface are known as Roche
218 11 Interacting binary stars

lobes. The significance of these lobes is that they delimit the volume within which
material is gravitationally bound to only one of the stars. For q — 1, the lobes are
identical in size and shape; as q diminishes, the lobe around the more massive
component, of mass M\, expands, while that of the less massive one, of mass
Mi, shrinks. For still smaller values of C, the lobes open up into one continuous
surface, with a narrow neck close to L\. As C decreases further, the surface
becomes more and more spherical around the two stars.
The region of interest for stellar interactions is that within the Roche lobes.
The reason is that, although the formalism of Roche equipotentials is strictly valid
only when point masses are involved, its application may be extended - at least
approximately - to more realistic cases, where the stars occupy some volume
within their respective lobes. To make things simpler, a Roche radius rL is defined
as the radius of a sphere that has the same volume as the respective Roche lobe.
A very good approximation for the Roche radius was provided by Peter Eggleton
in the form

a 0.6^2/3+ ln(l +r/'A3)’

corresponding to the lobe of Mo, while for that of M\,q must be replaced by q~}.
A less accurate but more versatile approximation is
/ \ i/3
— ^0.5 |—^—l , (11.25)
a \ I +q)

which yields for the ratio of Roche lobe radii: q}^.

The crucial question now is whether or not one or both binary members
overflow their respective Roche lobes. The answer to this question serves to
distinguish among different classes of close binaries, as proposed by Zdenek
Kopal about half a century ago:

1. Detached binary: the radii of both stars are smaller than their respec­
tive Roche radii, thus the stellar photospheres lie within their Roche
lobes.
2. Semidetached binary : the radius of one of the stars exceeds its correspond­
ing Roche lobe. Material may thus pass from the Roche-lobe-filling star
to its companion.
3. Contact binary: the radii of both stars are larger than the respective Roche
lobes. A common envelope thus forms, surrounding the Roche limit sur­
face, with both stars buried in it and hidden from individual view.

It is the semidetached binary that leads to stellar interaction and opens up a realm
of phenomena related to mass transfer between stars, and therefore we shall pursue
this case further.
 11.4 Conservative mass transfer 219

11.4 Conservative mass transfer

The simplest case of mass transfer between stars is the conservative one, where
both the mass and the angular momentum of the system arc conserved. Consid­
ering the case of circular orbits, the total angular momentum of a binary system
is given by

,/ = M\a\a> + M2a;a> + I\co\ + Ąoh, (11.26)

where the first two terms relate to the orbital motion, and the last two relate
to stellar spin. The spin angular momenta /|.2CU|,2 usually constitute small cor­
rections, since stars are centrally condensed and hence have small moments of
inertia /|,2. We shall therefore neglect them in our following discussion. Using
Equations (11.2), we may substitute

<71 = a------------- and «2 = a------------- (11.27)

M\ + M2 M\ + M2
to obtain
M\M2 ■>
J =------------- a~co. (11.28)
M | + iW 2
Substituting co(a) from Equation (I 1.20), we obtain
/ G \'/2
./ =------------- M|M2a1/2. (11.29)
\ + M2 J

Conservative mass transfer implies

d
— (M\ + M2) — 0 > M\ = —M2 (11.30)

meaning that all the mass lost by one binary member is gained by the other, and

dJ M\ M2 1d
— =0 (11.31)
dt M\ M2 2a
Combining Equations (1 1.30) and (11.31), we have

d _ 2A/,(A/, - M2)
(11.32)
a M i M2
The conclusion is that the separation between the stars - and with it, the period
of revolution - changes at a rate which is proportional to the mass-transfer rate.
Whether it increases or decreases is determined by the direction of mass transfer:
from M2 to M] or vice versa. If it is the massive star that loses mass to its
companion, then M\ < 0, hence d < 0, and thus the orbital size of the system
shrinks. This is bound to enhance the rate of mass transfer, which in turn will bring
the stars still closer together. The process may escalate into runaway. It is this
process that accounts for the reversal of the initial mass ratio in a binary system.
220 11 Interacting binary stars

which solves the Algol paradox; for massive stars evolve faster than low-mass
ones, and hence they are the first to expand and eventually overflow their Roche
lobes and transfer mass to their less massive companions. This unstable state will
only cease when the mass ratio is reversed.
If, on the other hand, it is the less massive star that fills its Roche lobe and
transfers mass to its companion, the distance between the stars will increase. At
the same time, the ratio of Roche lobe radii will decrease with decreasing q, so it
may still be possible for the mass-losing star to fill its Roche lobe. In this case, a
stable state of mass transfer may result. As it turns out, this configuration of slow
and stable mass accretion onto the massive and more compact binary component
gives rise to a wide range of eruptive phenomena, lumped together under the
general name of cataclysmic variables.

Exercise 11.3: Consider a binary system in circular orbit, with M2 < Mi,where
M\ > OandMj = —M\. (a) Find the condition^ must satisfy for the Roche lobe
of M2 to shrink, (b) Assuming the donor star to expand slightly upon losing mass,
find the condition q must satisfy to ensure Roche-lobe overflow (use a relation
of the form R oc. for (he mass-radius dependence).

11.5 Accretion discs

With cataclysmic variables in mind, consider a binary system where the massive
member is a compact star. say. a white dwarf that accretes material from its
companion, say. a low-mass main-sequence star. As customary for such systems,
we shall refer to the massive, mass-accreting component as primary’, and to the
less massive, mass-losing one as secondary. The critical point of the configuration,
as we have seen, is the contact point L\ of the Roche lobes.
What is the meaning of the inner Lagrangian point Z.,? As it lies at the
intersection between surfaces that belong either to one star or to the other, a test
particle at this point belongs to both or to none. This means that the compounded
forces acting on it by the two stars must exactly provide the centripetal force
required for rotation around the centre of mass, so that the particle will remain
in equilibrium. Clearly. L\ will be more distant from M\ (the origin at the larger
mass star) than the centre of mass located at

a I +q’
for the force exerted by M\ on a test particle at xCm exceeds the force exerted by
M2. unless, of course, the masses are equal. This excess will supply the required
centripetal force towards the axis of rotation that passes through the centre of
mass, when the particle is removed from xcm- Hence, to reach equilibrium, the
 11.5 Accretion discs 221

test particle can only be moved towards x > Xcm- Thus, X/., > Xcm- Since on the
line of centres in the co-moving frame y — z = 0, the value of xi,t is obtained as
the solution of the equation

GM\ GM2
(11.34)
x2 (a — x)2

which is equivalent to the condition d<t>(x. 0, (Y)/dx = 0, corresponding to

extremum points of the potential. Using the same normalization as for <!>', we
obtain after some algebra

(1 + q)x> - (2 + 3q)x4 + (1 + 3<?)x3 - x2 + 2x - 1 = 0, (11.35)

where distances are now measured in units of a, so that the equation is dimen­
sionless. For q = 1, the only real root of Equation (11.35) is x^ = 5 = xcm- As
the mass ratio decreases, (/ —> 0, we get xcm -+ 0, w'hile X/., -» 1.
At A i, the potential <t> has a maximum, located between the gravitational
potential wells of the two stars. Another way of interpreting the inner Lagrangian
point is to imagine a test particle delicately balanced at the potential top, and
prone to fall into either of the wells at the smallest perturbation. Two additional
maxima - solutions of Equation (11.35) for 0 < q < 1 - exist, L2 on the far side
of M2 and on the far side of M\, in both cases the forces of the two stars acting
in the same direction towards xcm- Through these points a test particle may fall
into the binary system or else altogether escape from it.
The projection of several Roche equipotentials on the orbital plane (z = 0) is
shown in Figure 11.3. where all three Lagrangian points are marked. The closed
Roche lobes can be clearly seen. The complicated multi-valued function gives
one an idea about the even more complex structure of the equipotential in three-
dimensional space. One should bear in mind, however, that this elegant-looking
geometry is based on greatly simplified physical assumptions and hence not all
of it is relevant to real systems.
The mass passing through L i from the Roche lobe of the secondary to that
of the primary cannot just fall directly onto the primary. Consider a test particle
at L| that has just acquired a small velocity in the direction of M\. This would be
of the order of the thermal velocity typical of the temperature in the atmosphere
of the secondary star, and hence small compared with its rotational velocity.
Nevertheless, it will be sufficient for displacing it from the unstable equilibrium
position at L\. But the particle cannot fall directly onto the primary star, for it
possess angular momentum, as it revolves around the centre of mass of the system
(x/_, > xcm). Actually, viewed from M\, the particle is seen as moving almost
perpendicular to the direction of L\. The trajectory of the particle is complicated
to compute, but it will eventually settle into a nearly Keplerian orbit around the
primary. This will be the fate of all particles passing through L\ towards the
primary, and since the initial velocity that causes the fall towards Mi is small,
it will have little effect on the trajectories, which will thus be almost identical.
Ill II Interacting binary stars

Figure 11.3 Projection of Roche equipotentials on the orbital plane (z = 0) for mass ratio
q = 0.6. The centres of the stars, the centre of mass (CM) and Lagrangian points L|, Lt
and Lt, are marked along the line of centres.

The ring of material that will form around the primary star in the orbital plane
of the binary system will slowly evolve into a disc, as particles will lose angular
momentum due to friction and will spiral in towards the primary. Eventually, they
will accumulate on the equator of the compact star, and the strong gravitational
field will spread them evenly over the entire surface. This disc, typical of accretion
in binary systems, is known as an accretion disc.
An estimate of the radial extension of the disc q may be obtained on the
following somewhat simplifying assumptions: (a) within the primary's Roche
lobe the effect ofthe secondary may be neglected, and (b) the angular momentum
is conserved, that is, the specific angular momentum at L| is equal to that of the
Keplerian orbit around M\. both taken with respect to an axis passing through
the centre of M\ perpendicularly to the orbital plane in a stationary (nonrotating)
frame of reference. The specific angular momentum at L\ with respect to M\ is
1/2
2 2 (j(M\ + Ml )
71 = XLW = XLi (11.36)

We shall further assume that x£| ~ a - rL,2 and adopt the relatively simple
approximation (11.25) for rt.2, to obtain after some algebra

Ji = VGMjad + r/)1/2 1 - 0.5 (11.37)

 11.6 Cataclysmic phenomena: Nova outbursts 223

A particle in circular orbit around the primary at a distance from its centre has
velocity -/(jMy/rj and hence specific angular momentum

h = y/GM\rj. (1 1.38)

The equality j\ = fa thus yields

q \
- = (1 +q) -0.5 \+q' (11.39)
a

and relative to the primary’s Roche radius, zi.i = 0.5«/(l + q), using again
approximation (11.25),

— = 2[(1 + <7)l/3 - 0.5<?1/3]4, (11.40)

/"L.l

which varies between 0.8 for q = 0.1 to 0.65 for q = 0.9. Thus, allowing for
the approximations employed, the conclusion is that the accretion disc stretches
to a significant distance within the primary’s Roche radius, and is only weakly
dependent upon the binary mass ratio. The large area of the disc enables it to
radiate a significant luminosity.
Observational evidence for the existence of an accretion disc in an interacting
binary system is also provided by the area where the stream of particles from the
donor impinges on the rim of the disc at supersonic speeds (speeds that surpass
the thermal velocity), resulting in shock-heating. (We have encountered a similar
phenomenon of shock-heating, albeit on different scale, in supernova explosions.)
This area, known as the hot spot may radiate copious amounts of energy, often
more than the energy emitted by both stars and the accretion disc combined.

11.6 Cataclysmic phenomena: Nova outbursts

Sustained mass transfer at relatively low rates in close semidetached binary sys­
tems gives rise to periodic outbursts of enhanced luminosity separated by peri­
ods of quiescence. These variable systems are known collectively as cataclysmic
variables, although the processes involved, their timescales and luminosity ampli­
tudes, differ considerably. Of all these cataclysmic variables, novae are the earliest
discovered, the best known and the most spectacular. We therefore choose them
to illustrate the salient points of the cataclysmic process, which will crown and
conclude our brief discussion of interacting binary stars.
A nova, short for nova Stella (new star) is a star that brightens suddenly several
hundred- to a million-fold, remains bright for a few days to several months and
then returns to its former, low luminosity, as shown in Figure 11.4.
224 11 Interacting binary stars

Figure 11.4 Examples of nova light curves for different novae that evolve on somewhat
different timescales. Axes are arbitrary, with markings at intervals of 10 days on the
abscissa, and 0.4 log L on the ordinate. (Adapted from C. Payne-Gaposhkin (1957), The
Galactic Novae, Amsterdam: North Holland Publishing, and D. B. McLaughlin (1960),
Stellar Atmospheres, University of Chicago Press.)

Historical Note: In ancient times, novae were classed with the guest stars, which
also included supernovae, as well as comets, all transient objects. Early observations
of such objects were made mostly in the Far East - China, Japan and Korea - where
professional astronomers (astrologers, in fact) were employed by rulers to constantly
watch the sky for signs of impending dangers. Meanwhile, ancient and even medieval
Europe showed little interest in these temporary stars, which were in marked conflict with
the dominant Aristotelian concept of a perfect, immutable celestial sphere that we have
already encountered in the previous chapter. Detailed records from China go back to about
200 BC; in Korea and Japan, regular observations began around AD 800. While comets
were quite early suspected and then recognized as being a separate class of objects, the
distinction of novae from supernovae was made, as we know, only in the 1930s, when it
 11.6 Cataclysmic phenomena: Nova outbursts 225

was realized that the two differed in maximal brightness by about six orders of magnitude.
Around the turn of the twentieth century, the number of nova discoveries rose considerably,
and then settled at an average of ~4 galactic novae per year.
By the early 1960s ample observational evidence had accumulated, mainly through
the work of Robert Kraft, indicating that novae were invariably members of close binary
systems. The nova companion was found to be a low-mass main-sequence star. Obser­
vations of novae after eruption, and in a few cases, prior to eruption, showed them to be
hot compact stars. Mass estimates, albeit scarce and uncertain, suggested that the erupt­
ing stars were white dwarfs. This led to the hypothesis that the red-dwarf companion is
extended enough to fill its Roche-lobe and allow mass transfer to the hotter star through
the inner Lagrangian point. Indeed, in some cases, a rapidly rotating accretion disc was
detected around the hot star.
Thus, novae appear to be hot white-dwarf members of close binary systems, which
accrete matter from a cool red-dwarf companion. This sets the scene for the theory that
explains the outburst mechanism, its many distinctive features and its consequences.

Considering the galactic rate of nova outbursts on the one hand and the
restrictive requirements for a system to undergo a nova outburst on the other, one
arrives at the inevitable conclusion that nova outbursts must recur in the same
system a great many times, as was realized already in the late 1930s. Most of
the time, however, is spent in quiescence, while the white dwarf accretes mass
from its companion. As a result, old novae are difficult to detect; the oldest nova
that has been recovered was discovered in 1670 in the constellation Vulpecula.
Although the eruption is recurrent, for most novae the time elapsed between
outbursts is thousands to tens of thousands of years, and hence only one outburst
is recorded. These are often referred to as classical novae, to be distinguished
from recurrent novae that erupt at intervals of tens of years, so that a number
of such outbursts have been recorded for each. Nova outbursts are accompanied
by mass ejection and the formation of nova shells, which slowly disperse into
the interstellar medium. Although the term nova refers to the variable star that
undergoes temporary explosive eruptions, it is sometimes used to designate the
outburst itself (as in the case of supernovae. Chapter 10).

The outburst mechanism

The material gradually accumulating on the white dwarf’s surface becomes com­
pressed and the electrons at the bottom of the accreted envelope become degener­
ate. At the same time the temperature at the bottom of the hydrogen-rich envelope
rises. When it reaches ~2 xl()7 K, hydrogen is ignited in a thin shell by the
CNO nuclear reaction cycle. The energy released raises the temperature further,
but since the degeneracy pressure is insensitive to temperature, no expansion
and cooling results and the temperature keeps rising exponentially, boosting the
lib 11 Interacting binary stars

nuclear reaction rates in a runaway process, as explained in Section 6.2. However,

above about 1O8 K, the CNO cycle rate is limited by the decay rates of /3+ unstable
nuclei, which are temperature independent, as we have mentioned in Section 4.4.
In addition, the temperature becomes sufficiently high for the degeneracy to be
lifted, turning the gas into an ideal one. The runaway is thus quenched: the shell
expands, while the burning temperature starts dropping after having reached a
few 108 K. Following ignition, a convective region forms just above the thin shell
source and extends towards the surface. Convection mixes the /^unstable nuclei
throughout the envelope and brings fresh CNO nuclei into the burning shell, until
the unstable isotopes prevail over the extent of the envelope. Energy generation
continues, supplied by the decay of the fi ~ unstable nuclei. Heat absorption now
results in rapid expansion and cooling of the accreted envelope.
During the runaway, the white dwarf’s 1 uminosity rises until it attains - or even
surpasses briefly - the Eddington critical luminosity (given by expression (5.37)).
When the luminosity reaches maximum, the star’s radius is still relatively small
and its surface still hot; hence it radiates mostly in the UV or even extreme
UV. The rapid expansion of the envelope proceeds now at constant, close to
critical, luminosity. Hydrostatic equilibrium cannot be achieved; instead, mass is
driven out by radiation pressure in an optically thick wind, as we have already
encountered in Chapter 8.
Maximum luminosity in the visible part of the spectrum is obtained when
the maximal photospheric radius is reached, of order l()0/?o, corresponding to
effective temperatures of several thousand Kelvin. Thereafter, when the envelope
becomes highly diluted, and the opacity drops, the photosphere recedes through
the expanding mass; its radius decreases, while the effective temperature rises.
When most of the envelope has been ejected, mass loss comes to an end.
The small remnant shell on the white dwarf’s surface contracts and then starts
cooling slowly. Hydrogen burning continues until almost all of the hydrogen in
the remnant shell has turned into helium. Then nuclear burning ceases and the
white dwarf returns to its preoutburst state. The decline takes roughly one to
several years and the white dwarf remains almost unaffected by the outburst that
has taken place. Accretion resumes towards the next outburst and a new nova
cycle begins.

Exercise 11.4: In the long run, a white dwarf that undergoes repeated nova
outbursts loses mass. Assuming a constant average rate of mass loss, show that
the central density of the white dwarf will increase at first and then steadily
decrease. To this purpose, use Equation (B.37) derived in Appendix B. which
gives the second approximation to the electron-degeneracy pressure that includes
a temperature-dependent term.
 11.6 Cataclysmic phenomena: Nova outbursts 227

General characteristics

Despite the complexity of nova outbursts, some simple relations between the basic
properties that characterize the development of such outbursts may be obtained
from simple considerations. We have seen in Section 6.2 that nuclear burning
in a degenerate electron gas is bound to trigger a thermonuclear runaway, once
a nuclear fuel is ignited. Therefore, the temperature must exceed the ignition
threshold 7ign. As the temperature decreases outwards, ignition will start at the
deepest point where hydrogen is present, that is, at the bottom ofthe accreted layer
of mass A/h (or slightly deeper, if some mixing has taken place between accreted
material and white dwarf material), at some radius rb. Denoting 7}, = T(rb), we
thus require:

TbZTls„. (11.41)

We may express the condition of electron degeneracy by demanding that ideal

and degenerate electron gas pressures be comparable,
/ \ / \ 5/3
P\ , / P \
TZTb 1 — 1 % K\J— , (11.42)
\ Mc / h \ Me / b

which yields {p/p.e)h in terms of Tb. Setting T/, = 7i„n, we obtain from the equation
of state an estimate for the critical pressure above which a thermonuclear runaway
is bound to develop,

C^Tisn)5/2
^,3/2 /’erit^------- (11.43)

Since the accreted material is hydrogen-rich, and since strong temperature depen­
dence of the burning process will accelerate the runaway, the ignition temperature
is expected to be that of the CNO cycle (see Section 4.4), roughly 1.5 x IO7 K.
This yields %2x 10l7N m-2 (2 x IOl8dyncm-2).
We may now estimate the amount of material above rb required to balance this
pressure hydrostatically. Assuming its thickness to be negligible, that is, assuming
r/, ~ R, we have

For a white dwarf the radius and mass are correlated, as we have seen in
Section 5.4. With the simple relation (5.29), /? ot A7_|/3, Equation (11.44)
leads to

Am a M~7/\ (11.45)

Thus massive white dwarfs require considerably smaller accreted envelopes in

order to erupt. This conclusion has further consequences. The outburst recurrence
228 11 Interacting binary stars

time is given by V = Am/M and, as the mass accretion rate M is independent

of M. outbursts are more frequent on massive white dwarfs than on low-mass
ones. In addition, since the companion’s mass is another independent parameter,
massive white dwarfs undergo, on average, a larger number of outbursts than
low-mass ones. Consequently, the probability of discovering novae with massive
progenitors is higher, although they arc not necessarily more numerous, as was
believed at one time.
The luminosity at outburst is close to LWl), which is proportional to M
(sec Equation (5.37)). In quiescence, the luminosity is that of accretion Lacc,
given by Equation (11.16). We may thus estimate the amplitude of the out­
burst, that we define as the ratio of peak relative to quiescence luminosity,
A = log(LEdd/Lacc),

/4ttc7?(M)\
A % log -------- ■.---- - , (11.46)
\ kM J
which yields, roughly, the observed range of 3 to 6 orders of magnitude.

Note: Novae are extremely luminous at optical wavelengths, brighter than Cepheids
and surpassed only by supernovae, and they are about a hundred times more frequent
than supemovae. They are therefore easy to detect in external galaxies. In particular, the
constant luminosity maintained during part of the outburst can be used as a standard candle.
Perhaps the best known property of nova outbursts is the apparent correlation between the
maximum magnitude attained at outburst and the rate of decline, a relationship that was
already pointed out by Fritz Zwicky in 1936, and was first calibrated by Dean McLaughlin
in 1945. Since then, great effort has been devoted to the absolute calibration of this relation,
which is considered a reliable distance indicator, along with Cepheids and supemovae.

The ejected mass mC) may be estimated by assuming that it is supported

against gravity at the star’s surface - at the extended radius R - by the radiation
pressure PrMj = jaT4 (sec Equation (3.40)). Thus, ntejg = 4nR2PrMs, where g is
the gravitational acceleration at the surface. Substituting the effective temperature
for T and the Eddington luminosity for L,

i i AttcG Mwd
4.t R-a T4ff = L =----------- —. (11.47)
K
leads to
I 6tt R2
—----- • (11.48)
3/c
The range obtained - between ~ 10 5 to a few I () 4 M& - overlaps with that result­
ing from the independent estimate (11.45) and agrees with masses determined
observationally. We note that the expression for is similar to (1 1.10), which
 11.6 Cataclysmic phenomena: Nova outbursts 229

is not surprising, for in both cases the mass scale of interaction between matter
and radiation is considered.
The energy required to power a nova outburst may be estimated as follows:
the radiated energy is roughly the Eddington luminosity multiplied by the dura­
tion of an outburst, typically several weeks, which yields ~5 x 103 J; and the
kinetic energy of the expanding shell is ~2.5 x 1037 J, assuming a shell mass of
10 "4Mk, and an average velocity of 500 km s-1. But both are negligible compared
with the energy required to remove the shell from the deep gravitational potential
well of the white dwarf, which is of order IO39 J. This energy is supplied by
nuclear burning of only a small fraction, about 5%, of the accreted hydrogen-rich
envelope mass.

Exercise 11.5: Consider explosive hydrogen burning at the bottom of a thin

hydrogen-rich layer on the surface of a white dwarf that leads to the expulsion
of this layer, (a) For a white dwarf of mass M = MQ, which has a radius R =»
0.01 Ro, calculate the fraction f of the layer’s mass that has to be transformed
into helium in order to supply the energy necessary for expulsion, assuming
the layer to be of solar composition, (b) Derive the dependence of f on M for
M < Meh-

The problem of Type la supernova progenitors

For a long time it was thought that essentially the same configuration that leads
to nova outbursts - a close binary system, where a white dwarf accretes mass
from its companion - may result in accumulation of sufficient mass for the white
dwarf to approach, eventually. Meh- The problem is that nova outbursts occur on
the way and then the white-dwarf mass may still grow only if the mass accreted
between outbursts is larger than the mass ejected at outburst.
Observations as well as theoretical studies point, however, to the opposite:
the mass of a white dwarf that undergoes nova outbursts is gradually eroded. The
observational evidence is provided by the composition of nova shells, which is
enriched, sometimes strongly, in heavy elements that are typical of white dwarfs
(C, O. Ne, Mg) and the mass donor cannot supply these elements. Nor could these
peculiar abundances be produced during the nova eruption, for the energy required
to power it is readily supplied by burning a tiny fraction of the accreted mass (see
Exercise 11.5). Indeed, theory shows - by numerical evolutionary calculations -
that temperatures are not high enough to produce elements heavier than helium.
There only remains the possibility that some of the white dwarf material gets
mixed with the accreted mass and is subsequently ejected. Theory further shows
that diffusion of elements, convection, turbulent mixing or a combination of these
mechanisms, do indeed result in mixing of white dwarf and accreted material.
230 11 Interacting binary stars

which is blown away at outburst. Thus the conclusion as to the decrease of the
white-dwarf mass stands on two firm legs. It may still be possible that under
special circumstances the white dwarf will manage to retain part of the accreted
mass, so this scenario has not yet been entirely abandoned.
An alternative scenario involves merging of two carbon-oxygen white dwarfs
with a combined mass in excess of the Chandrasekhar mass, known as the double­
degenerate model. From the stellar evolution point of view, the occurrence of
white-dwarf binaries should be quite natural. The white dwarf phase, with which
stars of intermediate mass end their lives, lasts practically indefinitely. Hence
binary components of different initial masses may reach it at different times, but
will meet there eventually. Not every white dwarf binary will end up merging,
and not every merger will have a sufficiently high mass (considering that the
average white-dwarf mass is less than half A/ch), but theoretical estimates show
that the merger rate is quite high and consistent with the observed rate of Type la
supernovae. However, this promising scenario has its own problems, for it seems
that under certain circumstances, mergers may end up in collapse rather than
explosion.
Merging of two stars is the ultimate form of stellar interaction and thus an
appropriate place for ending this chapter, in which we have only touched briefly
upon the wealth of phenomena arising in close, interacting binary systems.
 12

The stellar life cycle

12.1 The interstellar medium

Although to all intents and purposes a single or binary star may be regarded
as evolving isolated in empty space, not only is it a member of a very large
system of stars - a galaxy - but it is also immersed in a medium of gas and dust,
the interstellar medium. This background material (mostly gas) amounts, in our
Galaxy, to a few percent of the galactic mass, some 109A/s, concentrated in a very
thin disc, less than 103 light-years in thickness (we recall that 1 ly~ 9.5 x 10l?m),
and ~105 light-years in diameter, near the galactic midplane. Its average density
is extremely small, about one particle per cubic centimetre, corresponding to a
mass density of 10-21 kgm-3 (10-24 gem-3); in an ordinary laboratory it would
be considered a perfect ‘vacuum’. The predominant component of galactic gas -
of which stars are formed - is hydrogen, amounting to about 70% of the mass,
either in molecular form (H2), or as neutral (atomic) gas (H I) or else as ionized gas
(HII), depending on the prevailing temperature and density. Most of the remaining
mass is made up of helium. The interstellar material is not uniformly dispersed,
but resides in clouds of gas and dust, also known as nebulae. We have already
encountered special kinds of such nebulae: planetary nebulae, supernova remnants
and nova shells. These expanding nebulae are, however, relatively short-lived and
after dissipating into the interstellar medium, their material mixes with other,
larger ones. There are relatively dense clouds, with number densities reaching up
to a few thousand particles per cubic centimetre, and there is a diffuse intercloud
medium, where densities can be much lower than one particle per cubic centimetre.
The interstellar medium is extremely rich and diverse, which makes its exploration
all the more fascinating.
When we speak of temperature in the interstellar medium, we refer to the
kinetic temperature of the gas. The radiation that fills the medium, emitted
by the vast number of stars within it, is not in equilibrium with the gas, as it
is in the stellar interiors. Nevertheless, it is this radiation that determines the

231
232 12 The stellar life cycle

gas temperature. The UV photons ionize the hydrogen atoms and the resulting
free electrons collide with the ions. Although the mean free path of particles
in the interstellar medium is about lO'-'m, comparable to the diameter of the
entire solar system, this amounts to only ~10-3 light-years, a minute fraction of
the typical cloud dimensions of tens to hundreds of light-years. Hence thermody­
namic equilibrium is indeed achieved for the gas. and temperature is a meaningful
concept.
Partly ionized gas clouds surrounding hot stars (such as massive main-
sequence stars) may attain temperatures of the order of 104K over regions of
tens of light-years. The extent of such a region is obtained by requiring ionization
balance: the number of absorbed ionizing photons must be equal to the number
of recombinations per unit volume per unit time. The H 1 zones of the interstel­
lar medium (identified by the detection ofthe famous 21-cm radio line emitted
by atomic hydrogen) have temperatures of 50-100 K. Roughly, the pressures
within the different types of clouds are comparable: it is possible that the cold
clouds, which are not gravitationally bound, are held together against their inter­
nal pressure by the hot gas component of the interstellar medium, which exerts a
counter-pressure. Hence the densities are in inverse proportion to the temperature.
Typical number densities are ~l07-108m-3 for the cold clouds and ~105m-3
for the hot gas. Besides the cold and hot clouds of neutral and ionized hydrogen,
there are giant, dense, and dust-rich molecular clouds, where temperatures can be
as low as 10 K. and number densities are in the range 1-3 x 10s m-3 and more.
Their masses may reach 106Mo and their sizes are of the order of 100 light-years.
It is in these giant gaseous clouds that stars arc born.

12.2 Star formation

The process of star formation constitutes one of the problems at the frontier of
modern theoretical astrophysics. We shall not deal with the complicated stages
that turn a fragment of an interstellar cloud into a star, but only address the
question of the basic phenomenon of fragmentation.
Interstellar gaseous clouds are often subject to perturbations that are due, for
example, to propagating shock waves originating in a nearby supernova explosion,
or to collisions with other clouds. Consider an ideal case of a low-density cloud
of uniform temperature T, in a state of hydrostatic equilibrium. If at some place
a random perturbation will produce a region of higher density, the gravitational
pull will increase in that region. The gas pressure will increase as well, but not
necessarily in the amount required to maintain the hydrostatic equilibrium. The
outcome ofthe perturbation will depend on the dynamical stability of that region.
Our purpose is to derive the condition for stability for a region of volume V
(which, for simplicity, may be assumed spherical), containing a given mass M.
 12.2 Star formation 233

Denoting the radius (characteristic length) by R, we may use the partial virial
theorem (Equation (2.24) of Section 2.4), as we did in Section 9.4, to obtain

f , GM2
p cIV = P,V + ^a—~, (12.1)

where Ps is the pressure at the region’s boundary, exerted by the surrounding

gas. and « is a constant of the order of unity (depending on the mass distribution
within the region considered). We may assume an ideal gas equation of state
(Equation (3.28)), which yields

f R. f 'R
/ PdV = —T / pdV = —TM. (12.2)
J P J P
Combining Equations (12.1) and (12.2), we obtain

R. . GM2
— TM = PsV + ^a------- . (12.3)
p R
Now. both Ps and V are positive quantities, and hence, obviously, the left-hand
side of Equation (12.3) must exceed the second term on the right-hand side, which
means
a pGM
~ 3 R.T
Equality is obtained when the entire cloud is involved. A critical radius (dimen­
sion) R) may thus be defined by

known as the Jeans radius, after Sir James H. Jeans, who was the first to investigate
instabilities of this kind (in 1902). It constitutes a lower limit for the dimension
of a stable region of temperature T. containing a given mass M, within a gaseous
cloud. Contraction below this limit will cause the perturbed region to collapse:
the gas pressure will be insufficient for balancing gravity. Conversely, we may
obtain an upper limit for the mass that can be contained in hydrostatic equilibrium
within a region of given volume, the Jeans mass bf. With pav = M/ V,

where n is the number of gas particles per nr'.

Inserting in Equation (12.5) characteristic values of T and n for galactic-
gaseous nebulae, the Jeans mass turns out to be of the order of thousands to tens
of thousands of solar masses, typical of stellar clusters rather than individual
stars. Ordinary interstellar clouds have masses below this limit and hence they are
stable. Only giant gas and dust complexes are prone to collapse. When collapse
234 12 The stellar life cycle

on such a scale is triggered, the question is how will it develop, and whether it will
eventually stop. This is one of the crucial questions of the star formation theory.
Consider a collapsing cloud: both the density and the temperature increase
and hence the value of the critical mass is expected to change. If the Jeans mass
increases (inefficient cooling), wc are faced with two possibilities: either the
increase in ,Wj is sufficient for the stability criterion to be satisfied, in which case
the collapse will halt, or M] is still smaller than the cloud’s mass, in which
case the collapse will continue. If, on the other hand, Mj decreases (efficient
cooling), the violation of the stability criterion is yet more severe: it may now
happen that regions within the cloud violate the stability criterion and start col­
lapsing, inducing fragmentation of the cloud. The fragmentation process may go
on to smaller and smaller scales, down to the stellar mass scale. Such a hierarchi­
cal model was first suggested by Hoyle in 1953. Which of the possible situations
will actually occur depends on the ratio between the timescale of collapse, which
is the dynamical timescale of the cloud (of the order of i/y/Gpm), and the cool­
ing (thermal) timescale. Since cloud densities are many orders of magnitude
lower than those prevailing in stars, these timescales are comparable and hence
an accurate evaluation of the processes involved in the collapse is required. As
cloud fragments become increasingly denser and hotter, they eventually become
opaque and cooling becomes inefficient. At some point, the Jeans mass starts
increasing. Thus, depending on local conditions, a minimum Jeans mass exists,
which defines a lower limit to fragments of clouds that are bound to contract and
form stars. A schematic illustration of fragmentation is given in Figure 12.1 and
observational evidence for the process of collapse and fragmentation is shown in
Figure 12.2.

Exercise 12.1: Estimate the minimum Jeans mass of a collapsing isothermal gas
cloud of temperature T, on the assumption that the radiation temperature is lower
than the gas temperature (since there is not sufficient time for thermodynamic
equilibrium to be achieved).

A fragment of a gas cloud bound by self-gravity, which has a mass in the stellar
mass range, may be regarded as a nucleus of a future star. The mass continues to
grow by accretion of gas from the surroundings. The gravitational energy released
as material accretes is turned into thermal energy. The increase in both density
and temperature raises the opacity of the gas. When the contracting gas becomes
opaque to its own radiation, it has reached the status of a stellar embryo, the
photosphere defining the boundary between the inside and the outside of the star
in the making. When hydrostatic equilibrium is achieved, the embryo becomes
a protostar (see Section 9.1). Eventually, the central temperature reaches the
hydrogen ignition threshold and the protostar becomes a star, assuming its place
on the main sequence of the H-R (log Te(( , log L) diagram appropriate to its mass.
 12.2 Star formation 235

Figure 12.1 Schematic illustration of the fragmentation of a gas cloud.

Figure 12.2 Observational evidence of fragmentation in a gaseous cloud: starbirth clouds

and gas pillars in the Eagle Nebula (Ml6), a star forming region 7000ly away. The tallest
pillar (left) is about 1 ly long from base to tip. Note the small globules of denser gas buried
within the pillars (photograph by J. Hester and P. Scowen, Arizona State University, taken
with NASA's Hubble Space Telescope).
236 12 The stellar life cycle

12.3 Stars, brown dwarfs and planets

The process of star formation has nothing to do with the ability of a star to
ignite hydrogen when the turbulent stages leading up to ignition are finally over.
Hence we cannot grant the protostellar cloud the prescience of having to end up
with a mass above the lower stellar mass limit of about 0.08 MQ. Indeed, the
estimated minimum Jeans mass is about an order of magnitude lower than the
lower stellar mass limit. Therefore, smaller objects should be expected to form by
the same process that creates stars, only to start cooling before they could ignite
hydrogen. Such objects have been observed, or their existence has been indirectly
inferred from its effect on a binary companion. They are called brown dwarfs, to
be distinguished from the common, bright white dwarfs, which will eventually
become extinct black dwarfs, and from the lower main-sequence stars that are
often referred to as red dwarfs, due to their reddish colour, resembling that of red
giants. In the H-R diagram brown dwarfs descend the Hayashi track, but they
turn away from the main sequence toward lower effective temperatures. In the
(log 7'c. log pc) diagram, they start by contracting and heating up, as stars do. but
their tracks bend into the degeneracy zone before crossing the hydrogen burning
threshold. Subsequently, they behave much in the same way as giant planets.
Planets, however, form in a different way: they separate out of circumstellar discs
surrounding very young stars, by aggregation of larger and larger particles and by
accretion of gas.
Hence brown dwarfs constitute a transitional class of objects, between stars
and planets: they are born like stars; they evolve like planets. In fact, they may
even have a claim to stardom, since they do briefly ignite deuterium, primordial
deuterium being present in very small amounts, of order 10"5, in the initial
composition of all stars. The evolution of the luminosity for objects in the mass
range 0.0003-0.2Mq resulting from model calculations is shown in Figure 12.3.
The early flat part of the tracks, between 106 and 10s yr, is due to deuterium
burning; this phase is very short in the more massive stars, but can last as long as
108 yr in an object of ~().()1 MQ, at the lower mass limit for deuterium burning.
After about 108 yr the stars among these objects reach a plateau luminosity upon
settling on the main sequence. For planets, on the other hand, the luminosity
decreases continuously. Brown dwarfs fall in between, with a brief period of
constant luminosity, followed by a steady decline.
Thus another distinction may be made between brown dwarfs and planets, not
according to birth, but according to whether or not they have ever burnt nuclear
fuel. Strangely enough, both definitions - although having nothing in common -
result in similar lower limits for brown dwarf masses, 0.01 ± 0.003M©. Yet a
further distinction may be made according to structure. In very low-mass stars and
brown dwarfs the internal pressure is supplied mainly by the degeneracy pressure
of electrons, similarly to white dwarfs, except that white dwarfs are much closer
to complete degeneracy and hence, in a way, simpler to model, and they are made
 12.3 Stars, brown dwarfs and planets 237

log1cAge (yr)

Figure 12.3 Evolution of the luminosity of red-dwarf stars (solid curves), brown dwarfs
(dashed curves) and planets (dash-dotted curves). Brown dwarfs are here identified as
those objects that burn deuterium. Curves are labelled according to mass, the lowest three
corresponding to the mass of Jupiter, then half of Jupiter’s mass and finally the mass of
Saturn (from A. Burrows et al. (1997), Astrophys. J. 491).

of elements heavier than hydrogen. We have seen that for objects dominated by
degeneracy pressure radii increase with decreasing mass (Section 5.4). But such
behaviour cannot go on indefinitely. We know, for example, that for terrestrial
planets, which are governed by much more complicated equations of state, radii
decrease with the mass. Therefore, a mass must exist for which the radius, as
a function of mass, reaches a maximum. The mass-radius relation for spheres
of low mass based on an accurate equation of state is shown in Figure 12.4 for
different compositions.
As it turns out, the mass corresponding to the maximal radius is very close to
Jupiter’s. Hence Jupiter’s mass, Mjup % 0.001 MQ, may be regarded as a borderline
between two classes of objects. Indeed, brown dwarf masses are often expressed
in units of A/Jup. ranging from about 80Mjup down to about IOA/Jup (or less?).
However, none of the criteria mentioned above for distinguishing brown
dwarfs from planets can be applied observationally; they arc all based on his­
tory or internal structure. In order to identify brown dwarfs we need to specify
surface characteristics, such as spectral signatures. These are difficult to deter­
mine because opacities at low temperatures are complicated by the formation of
molecules and dust grains. In fact, the interest in these small and faint objects
has been aroused by their kinship to planets, which are currently at the focus
of astronomical research, in the attempt to answer the intriguing question of
extraterrestrial life. We have yet a great deal to learn about the nature of brown
dwarfs, about giant planets, and about the formation of stars and planets, until
we shall be able to sort out and fully understand the variety of substellar objects,
238 12 The stellar life cycle

Log(M/M0)

Figure 12.4 Mass-radius relation for low-mass objects (following H. S. Zapolsky &
E. E. Salpeter, Astrophys. J. 158). Different curves correspond to different compositions, as
indicated. The locations of several planets - Earth, Jupiter, Saturn, Uranus and Neptune -
are marked by the planets’ symbols. Also marked are the locations of two white dwarfs,
Sirius B (§) and 40 Eridani B (f) (data from D. Koester (1987). Astrophys. J.. 322).

even before we address the question of the origin of life. Another reason for the
increasing interest in brow n dwarfs is their potential contribution to the galactic
mass budget in the form of ‘dark’ matter. For this contribution to be significant,
their number must be considerable. This brings us to the question of the stellar
mass distribution.

Note: Dark matter is matter that we do not see (in any wavelength), but we have
other indications to presume it is there. These come mainly from the gravitational field that
such matter would generate, just as in the case of black holes. On the galactic scale, the
evidence is provided by fast-moving stars and gas clouds at the very edge of the revolving
galactic disc, where Keplerian velocities should be much smaller, if the gravitational field
were due to visible matter alone. At such velocities these stars and clouds should have
long dispersed, unless pulled in by the gravitational field of an invisible material halo. On
larger scales, a similar phenomenon is observed in clusters of galaxies, as was pointed
out by Zwicky in the 1930s. The random motions of galaxies within a cluster tend to
disperse it, while the mutual gravitational pull would cause them to fall to the centre. Thus
balance is established, with the random velocities being related to the cluster’s mass (as
in the virial theorem that applies to a self-gravitating gas; Section 2.4). As it turns out,
the observed velocities (deduced from Doppler shifts) of cluster members exceed by far
those that correspond to the visible mass. In order to keep them confined to the cluster, a
mass exceeding their own by a factor of almost ten would be required. Hence the quest
for ‘dark’ matter.
 12.4 The initial mass function 239

12.4 The initial mass function

Continual star formation results in a steady decrease in the population of massive,

luminous stars. As these stars have vanishingly short life spans on the galactic
timescale, their relative number is at each instant correlated with the fractional
amount of gas in a galaxy. Thus even if they were created with the same probability
as low-mass stars, massive stars would have become rarer in the course of galactic
evolution. All the more so, if the probability of formation of massive stars is
small relative to that of low-mass stars, as turns out to be the case. Assuming star
formation to be independent of galactic age or location, the number of stars formed
at a given time within a given volume, with masses in the range (M. M + dM),
is solely a function of M:

dN = <t>(M)dM. (12.6)

The so-called birth function was derived by Salpeter as early as 1955 and
it has hardly changed since:

4>(M) oc M~235. (12.7)

The related initial mass function l-(M) is defined as follows: the amount of mass
locked up in stars with masses in the interval (M. M + dM), formed at a given
time within a given volume, is

MdN = $(M)dM (12.8)

and combining relations (12.6)—(12.8), we have

/ M V1'35
e(M) <x — . (12.9)
\ Mq)
The semi-empirical derivation of relation (12.7) was based on observations
of main-sequence star luminosities in the solar neighbourhood. It involved the
division of the luminosity range of main-sequence stars into intervals, counting
the total number of stars with luminosities in each interval, using the mass­
luminosity relation (1.6), and, finally, assuming that the duration of the main-
sequence phase was proportional to M/L. This assumption implies that stars
leave the main sequence as soon as they have burnt a fixed fraction of their mass,
as indicated by the stellar evolution theory. In spite of the enormous increase in
observational data since the early 1950s and the refinements of stellar evolution
theory, the conclusion remains that over a mass range spanning more than two
orders of magnitude, from ~().3A7O to ~6OA/0, the birth function is of the form
<t>( A7) oc with y 1.5 ± 0.3. An example based on main-sequence stars
of the galactic disc population around the Sun is given in Figure 12.5. There are
many theories that try to explain this empirical birth function, but so far none has
been generally accepted.
240 12 The stellar life cycle

Log (M/M0)

Figure 12.5 The initial mass function of main-sequence stars in the solar neighbourhood.
The Salpeter slope is indicated by the straight line (data from N. C. Rana (1987), Astron.
Astrophys., 184).

We note that at the low-mass end the initial mass function deviates con­
siderably from the inverse power law (12.9) and becomes almost flat and even
decreasing with mass. The difficulties involved in observing the faint low-mass
stars and brown dwarfs and obtaining complete samples make the derivation of
a birth function in this range rather uncertain. However, it is already clear that
the total mass of objects with M Z 0.3M© can account for less than 20% of the
total mass of stars with M i 0.3Mo. Thus the solution to the missing mass prob­
lem should probably be sought elsewhere. We should also mention that recent
observations indicate a conspicuous change in slope for the initial mass function
around the transition mass between brown dwarfs and planets. This strengthens
the hypothesis that these two types of objects were formed by radically different
processes.
With the aid of the initial mass function, rough estimates may be derived for the
mass exchange between stars and their environment, and for stellar distributions
within a volume of the galaxy. Consider, for example, one generation of stars
formed at a given time in some part of the galaxy. The fractional amount of mass
returned to the galactic medium by this generation of stars may be computed as
follows. Let < be the mass initially locked up in these stars, whose masses are in
the range A/min < M < Mmax. Then

ę = MdN = / $(M)dM. (12.10)

J 'ttinin v ■ttmin
Let St denote the fraction of initial mass that a star ejects in the course of its
evolution. As we have seen in Chapter 10, stars that end their lives in a supernova
explosion eject more than 80% of their mass. For the purpose of a rough estimate,
 12.4 The initial mass function 241

we may assume that stars of initial mass above A/sn ~ 10A7o return their entire
mass to the galactic medium. Stars of initial mass below A/ms 0.7A4s will
still be in the main-sequence phase, as we have seen in Section 9.2. These stars
have lost, therefore, only a negligible fraction of their initial mass. Stars in the
intermediate range A/ms < M < A7sn may be taken to have turned instanta­
neously into white dwarfs, since the time elapsed between the main-sequence
phase and the white-dwarf phase is relatively short (see Sections 9.4-9.7).
These stars have thus ejected all but the remnant white dwarf’s mass (A/wd ~
0.6A7o). Consequently,

1 ; M > A/sn
(M - A/\is < M < A/sn (12.11)
0 ; M < Mms

and the mass returned bv generation of stars to the interstellar medium is

-■ this c
/’ Mmax 'V/vvn
rj = / $(M)M(M)dM = / $(M)dM - / ——$(M)dM.
JMmin JmMs JMms M
(12.12)
The fractional mass returned is obtained by dividing Equation (12.12) by
Equation (12.10),

„ - MWD M~235dM
2 _ J AAis_____________________ u Mis_____________
(12.13)
C fu"’" M-'-^dM
J *wmin

which amounts to ~ 1 /3, for A/mjn — 0.1 Afe and Afniax = 60A/©.

Test the sensitivity of the above estimate for rj/ę to the stellar mass
Exercise 12.2:

range assumed, by repeating the calculations for all combinations of Mmjn = 0.05
and 0.2Mq, and A/max = 30 and 1 20MQ.

We may also estimate the number of white dwarfs relative to the number of
main-sequence stars in a population of stars formed at a given time, such as a
stellar cluster. All we need to know is the mass corresponding to the upper end
of the main sequence in the H-R diagram of the cluster - the mass of the turnoff
point Mlp. The number of main-sequence stars is then given by

(V.MS = <J>(M)dM. (12.14)

The number of white dwarfs is obtained by assuming, as before, that the

transition from the end of the main-sequence state to the white-dwarf state is
242 12 The stellar life cycle

Figure 12.6 Ratio of number of white dwarfs to number of main-sequence stars for a
stellar ensemble of given age, where the age is given by the mass corresponding to the
main-sequence turnoff point in the H-R diagram (see Fig. 9.5).

instantaneous;
/■Msn p.WsN
Mvd = / dN = / <P(M)dM. (12.15)
7;W,p

Hence the ratio

*V35 - a^n135
Mvd
(12.16)
Nms Mmi'n35 - M-'-35’

which amounts to only a few percent, is a function of Mlp, or the cluster’s age, as
shown in Figure 12.6.

Exercise 12.3: Assume the brightness of a stellar cluster to be mainly determined

by the summed luminosities of its main-sequence stars. Calculate by what factor
would a cluster’s brightness decrease, as the turnoff point of its main sequence
moves down from 1.3MS to 0.85/WQ.

A similar estimate of the number of supernovae (or, equivalently, neutron

stars) relative to that of main-sequence stars yields a lower ratio by more than a
factor of 10. Wc should keep in mind, however, that supernovae would be visible
only during the early evolution of the cluster, up to about the main-sequence life
span of a star of mass Msn, and if no further star formation occurs, no stellar
explosions should be seen thereafter. Stellar statistics, which take into account
both variations due to different stellar masses and variations due to different ages,
involve far more complicated calculations and represent a separate field of study -
a very important one, considering that many of the tests of the stellar evolution
theory are of a statistical nature.
 12.5 The global stellar evolution cycle 243

Figure 12.7 Sketch of the stellar evolution cycle.

12.5 The global stellar evolution cycle

On large scales, the process of stellar evolution is a cyclic process: stars are
born out of gaseous clouds within galaxies, and in the course of their lives
they return to the galactic medium a large fraction of the mass they have
temporarily trapped. This material blends with the interstellar matter and con­
tributes, in turn, to the formation of new generations of stars. This is sketched in
Figure 12.7.
The term ‘generation of stars’ is somewhat misleading, for we have seen that
stellar lifetimes differ by as much as four orders of magnitude, depending on the
initial mass. Thus a succession of a great many generations of massive stars may
coincide with only one single generation of low-mass stars. The different ways
by which stars return material to the interstellar medium are illustrated by the
images of Figure 12.8. where the shell ejected by a nova outburst (Section 11.6) is
shown in addition to the wind from a massive star (Section 9.9), another example
of a planetary nebula (Section 9.7), and the shell ejected by supernova SN1987A
(Section 10.3). We note the conspicuous similarity of these images, despite the
huge differences in length and time scales. The ejected material has been pro­
cessed. however, and its composition differs from the prevailing composition of
the galactic gas. Thus, later generations of stars have, al birth, increasingly larger
abundances of heavy elements (or metals). The survivors of the entire evolution
process are dense compact stars - white dwarfs, neutron stars, and. possibly,
black holes - as well as brown dwarfs and low-mass main-sequence stars, whose
main-sequence life spans exceed the present age of the universe. In the end, when
the entire gas reservoir will have been locked up in these small and mostly faint
stars, star formation will cease.
244 12 The stellar life cycle

Figure 12.8 Illustration of mass loss by images taken with NASA’s Hubble Space Tele­
scope: (a) nebula (Pistol) ejected by a massive star (estimated at ~IOO;W3) extending
in radius to ~4 ly (photograph by D. F. Figer, University of California at Los Angeles);
(b) mass ejected by SN1987A: the ring of gas, about 1.5 ly in diameter, was expelled
by the progenitor star some 2 x IO4yr before the supernova explosion. At its centre,
the glowing gas ejected in the explosion expands at a speed of 3000 km s-1 (pho­
tograph by P. Garnavich. Harvard-Smithsonian CFA); (c) planetary nebula (Henize
1357). the youngest known so far, extending to a radius of less than 0.1 ly (pho­
tograph by M. Bobrowsky, Orbital Science Corp.); (d) mass shells ejected by nova
T Pyxidis, forming more than 2000 gaseous blobs, which extend to a diameter of
about I ly (photograph by M. Shara, R. Williams and D. Zurek, Space Telescope Sci­
ence Institute; R. Gilmozzi, European Southern Observatory; and D. Prialnik. Tel Aviv
University).

The main evolutionary processes that take place on the galactic scale as a
result of individual stellar evolution may be summarized as follows:

1. The amount of free gas decreases. Nebulae and gas clouds become
sparse.
2. The galactic luminosity - made up of the individual stellar luminosities -
declines, as the relative number of massive stars decreases at the expense
ofthe growing proportion of compact, faint stars.
3. The composition becomes enriched in heavy elements, created in stars and
returned to the galaxy by the various processes of mass ejection.
 12.5 The global stellar evolution cycle 245

Figure 12.9 Relative contributions of different types of stars to the heavy clement content
of the interstellar medium (adapted from C. Chiosi & A. Maeder (1986), Ann. Rev. Astron.
Astrophys., 24).

Exercise 12.4: Let Y(t) be the fractional amount of gas in the Galaxy as a
function of time, satisfying the initial condition Y(0) = 1. Assume the rate of
decrease of free gas as a result of star formation to be proportional to Y2. Find
the function Y(r), if at present, t = rp, the gas constitutes 0.05 of the galactic
mass. At what time in the past (fraction of tp) was the mass of free gas half the
entire mass? At what time was it one tenth of the entire mass? At what future
time will the gas mass have decreased to half its present value?

As galactic material is continually enriched in heavy elements, the relative abun­

dance of these elements in newly formed stars increases with time elapsed from
the formation of the galaxy. Despite their relatively low birth rate, it is the massive
stars that provide the overwhelming contribution to heavy element enrichment,
first because they eject a larger fraction of their initial mass than do low-mass
stars, secondly because this fraction constitutes a much larger amount of mass,
and thirdly because the mass is returned practically instantaneously. The contri­
butions of stars of different masses to interstellar helium and metal enrichment by
different processes are illustrated in Figure 12.9. These yields have to be weighted
by the initial mass function to properly derive the galactic chemical evolution.
The rate of heavy element enrichment of the galactic medium has been far
from constant. Most of the enrichment occurred at early limes, and the enrichment
rate has markedly decreased with time. As an illustration, we note that the age of
the Sun is about one third of the age of the Galaxy, and its heavy element mass
246 12 The stellar life cycle

fraction (metallicity) Z is nearly 0.02. The metallicity of the youngest stars is about
0.04. that of the oldest, about 0.0003. Thus Z has increased a hundredfold during
the first two thirds of the galactic lifetime and only twofold during the last third.
Although the change in initial abundance is gradual, it has become cus­
tomary to divide stars into two populations, Population I (Pop I, for short) and
Population II (Pop II), according to composition and hence to age. The stars of Pop
I are young and metal rich, those of Pop 11 are old and metal poor. If we reverse the
time arrow from the present backward, into the past, we first encounter the Pop 1
stars and then those belonging to Pop II. This could be taken as the rationale for
ordering the populations. Thus old Pop I stars are those stars formed in between
Pop I and Pop II. And sometimes reference to Population III stars may be found,
meaning that we have to go further down the time arrow, passing the extreme Pop II
stars, toward the very beginning of galactic evolution. On this time arrow Z
decreases, with older populations corresponding to lower Z values.

Exercise 12.5: Using dimensional analysis as in Section 7.4, but taking into
account the effect of heavy element abundance on the opacity law and the energy
generation rate, compare Pop I and Pop II main-sequence stars in terms of
temperature, density and luminosity. Assume a Kramers opacity law of the form
k — KoZpT~1/2, and an energy generation rate of the general form q = qfZpT".

However, the initial classification of stars into distinct populations, dating

back to 1944, when it was introduced by Baade, was based neither on initial
composition nor on age, but on the location of stars in the galaxy. A typical spiral
galaxy, such as the Milky Way, consists of a spherical distribution of stars, which
includes the central bulge and the galactic halo, and a flattened disc distribution,
which is not uniform, but divides into spiral arms. The disc stars made up Pop
I, while the stars in the central region of the galaxy and in the globular clusters
forming the galactic halo constituted Pop II. Location, age and composition are
thus interconnected. The galactic disc, which contains most of the free gas and
dust, is still harbouring star formation. Hence, not surprisingly, young stars are
abundant in the disc. The halo is a relic of the original distribution of matter in
the galaxy, before most of the material collapsed to form the disc. It contains the
old high-velocity stars and globular clusters that formed before the collapse of
the disc and retained their high kinetic energy. The H-R diagrams of the different
populations are consistent with the inference of their relative ages.
But even in the youngest stars the mass fraction of heavy elements amounts
to only a few percent. On the one hand, this is too little to affect a star’s evolution
considerably, although the evolutionary paths traced by stars of different metal-
licities in the H-R diagram do differ to some extent. On the other hand, the total
absolute mass of heavy elements in a star is rather considerable; in the Sun, for
 12.5 The global stellar evolution cycle 247

example, it exceeds the mass ofthe entire solar system, including planets, moons,
comets, asteroids and other star formation debris. From where we stand this can­
not be considered negligible. In fact, except for the giant planets, which contain
a significant amount of (primordial) gas, all the other bodies in the solar system
arc made precisely of some of that small fraction of heavy elements present in
the protosolar nebula. And, as we recall that the source of these elements has
been nuclear burning, we come to the awesome conclusion that most atoms in our
bodies, the atoms in the air that we breathe, and. in short, the elements making
up every object around us, have belonged to a star at some time in the past and.
in all probability, have witnessed a gigantic stellar explosion.
And steadfast as Keats’ Eremite,
Not even stooping from its sphere,
It asks a little of us here.
It asks of us a certain height,
So when at times the mob is swayed
To carry praise or blame too far,
We may choose something like a star
To stay our eyes on and be staid.
 Appendix A

The equation of radiative transfer

Consider a small cylinder of length dt and cross-section dS at a distance r from

the centre of a star, with its axis in the direction 0 with respect to the radius
vector. The projected length of the cylinder is thus dr = dteas9 (as shown in
Figure A. 1). The radiation within the cylinder obeys energy conservation. We
define the intensity of radiation l(r,0) such that l(r,0)da) is the energy flux
(energy per unit area per unit time) moving inside a cone of directions defined by
the solid angle da> = 2rr sinddd around the direction off?. Since the radiation is
composed of photons of different frequencies and since interactions between mat­
ter and radiation depend on frequency, we consider the monochromatic intensity
Iv(r. 0) defined such that
/»0O
/(r, 0) — / lv(r,0)dv, (A.l)
Jo
and we apply energy conservation in each frequency.
The different contributions to the radiation energy during a time interval dt
are as follows:

1. The energy entering the cylinder at the bottom is

dQm = Ifr,9)d(odSdt.

2. The energy leaving the cylinder at the top is

dQ0Ut = —Iv(.r + dr, 0)da>dSdt.

In fact, along the cylinder, the angle between the axis and the radial
direction decreases and there should be a difference dO between the top and
the bottom. For simplicity, we neglect this difference, which is tantamount
to adopting the plane parallel approximation. (In the general case, the same
basic relations are reached as we shall obtain here, following the same line
of reasoning, but the mathematics is a little more complicated.)

251
252 Appendix A The equation of radiative transfer

Figure A. I Cylindrical volume element within a star; conservation of radiation energy

within it leads to the radiation transfer equation.

3. The absorbed radiation within the cylinder is

rZ<2abs = —Kvplv(r.9)da)dSd£dt.

We distinguish between true absorption and absorption caused by scatter­

ing and label the opacity coefficients accordingly; thus kv — /ca i, + /cs
4. The emitted radiation by the mass within the cylinder is

dQem = pjvdcodSdtdt

where jv is the total radiation emitted per unit mass per unit time. We
include in this term radiation emitted by the mass within the cylinder,
./ein.v- as well as radiation scattered into the cylinder. js-M. The latter is
obtained by integrating K\vlv(r. O') over all directions 0' from which pho­
tons are scattered into our cylinder, assuming that the scattering process
does not change the photon frequency. This is usually a complicated task.
In the simple isotropic case, that is. when the scattered radiation is emitted
equally into equal solid angles, / l[.doj /4.t.
Conservation of energy requires

YdQ = (),

and hence

|—lv(r + dr, 0) + Iv(r, 0) — Kvplv(r, 0)dt + pjvd£\dSda>dt — 0. (A.2)

Substituting

, „ dlv(r. 0) dlv(r,0)
Iv(r + dr, 0) — !v(r. 0) = —- ----- dr — ———dtcosO
ar dr
 Appendix A The equation of radiative transfer 253

and dividing Equation (A.2) by dtdSdcodt, we obtain the radiative transfer

equation in the form

1 d/,.(r.0)
------- ------ COS0 + KvIv(r, 0) - jv — 0. (A.3)
p dr

Note that the scattering term /s l, turns the transfer equation into a integro­
differential equation. In order to solve it, we have to evaluate y'em.v, itself a
function of
In thermodynamic equilibrium, the radiation field is given by the Planck
(blackbody) distribution

2h v- I
BV(T) = — . • (A.4)

which is isotropic, and there is perfect balance between absorption and emission
of radiation (known as Kirchhoff’s law'); we then have 7v(r) = BV(T) and jcm.v =
VZ?V(7'). In stars, however, the radiation field is not perfectly isotropic, and hence
we have to consider the different contributions to the emission of radiation. It was
Einstein who recognized that these must be of two kinds: spontaneous emission,
determined by the temperature, and induced (or stimulated) emission, which is
caused by the radiation field itself. The relationship between them and between
emission and absorption may be easily understood by considering a simple case
of two discrete energy levels 1 and 2, such that Ei — E) + hv. Let n, and ah be
the number densities of particles in the energy states E\ and E^, respectively. In
thermodynamic equilibrium a second condition is satisfied: particle densities are
related by Boltzmann’s formula

'll = = Sle-^kT^ (A 5)
"I gl gl

where the factors gi j represent the statistical weights of the energy states (essen­
tially, the number of states with different quantum numbers that correspond to
the same energy level). Transition of a particle from level 2 to level 1 involves
the emission of a photon of energy hv; similarly, the reverse transition occurs
by absorption of such a photon, as show'n schematically in Figure A.2. The rate
of spontaneous emission is proportional to the number of particles in the high
energy state >?21 the rate of induced emission, on the other hand, depends on both
112 and the radiation field BV(T). Finally, the rate of absorption is proportional
to the number of particles in the low energy state n\ and to the radiation field.
Introducing the appropriate coefficients - A21 for spontaneous emission, /Li for
induced emission and B\2 for absorption - and applying Kirchhoff’s law. we
obtain Einstein’s equation:

A21/12 + BiiniBlI) = BV(T), (A.6)

254 Appendix A The equation of radiative transfer

hv
E2-Ei=hv

absorption

Figure A.2 Schematic representation of emission and absorption in a two energy-level

system.

where we identify on the right-hand side B\tti\ = /ca v. Multiplying by (e/n7K — 1),
defining av = 2/tv3/c2 and substituting 112 from Equation (A.5), we have

g2A21(eAv/ir - l) + g2fl21au = g|B12a1,e"v/A'r (A.7)

This equation holds for any temperature, regardless of photon frequency; hence
temperature-dependent and temperature-independent terms must balance sepa­
rately. We thus obtain the Einstein relations between the three coefficients:

g2A2| = gl ^I2«r (A.8a)

g2S2i =gi#i2 (A.8b)

(the second follows from the first and A2j = Bz\av), which leave only one inde­
pendent coefficient.
Now comes the crucial point of the discussion: these relations must hold
whether or not the system is in thermodynamic equilibrium. This is because they
are connected to the microscopic state of the system - the nature of individual
emitters-absorbers - whereas thermodynamic equilibrium is a macroscopic prop­
erty. Individual particles are unaware, as it were, of the general state of the system.
Consequently, for any radiation field intensity Iv the emission is given by

Jem.v = A2iH2 + Biynily = — B|2O'VH2 + —5|2«2fv = — (QT + Iv)-

g2 g2 g2 nI
(A.9)
Substituting Equations (A.5) and (A.4) into Equation (A.9) yields

Jem.v = Ka.v(l ~ e~hv/kT)BV(T) + >Ca.l.e-',v/*7 Zv, (A.10)

and it is easy to see that in the case of thermodynamic equilibrium this relation
reduces to jemv = KavBv(T), because then /,, — BV(T).
Defining a reduced absorption opacity coefficient by

-e-',v/kTl (A.ll)
 Appendix A The equation of radiative transfer 255

and substituting expression (A. 10) into Equation (A.3), we obtain the transfer
equation in the form

'y>eos^ + <„[7„(/-.^)- Bv(,T)] + Ks.vIv(r,9)-js,v, = 0, (A.12)

p dr
which can be solved for given opacity coefficients in all radiation frequencies.
To obtain a solution, we expand Iv(r, 9) in Legendre polynomials P,,(cos9):

Iv(r, 9) = lv^r)P(} + A..i(r)Pi(cosP) + Iv2(r)P2(c^9) + • • •. (A. 13)

recalling that Po = I and P\ =cos0. Since matter and radiation are in local
thermodynamic equilibrium in stars (see Section 2.1), we know that the first
(isotropic) term in the expansion is none other than the Planck distribution BV(T\
Substituting it into the transfer equation (A.12) yields
I dBv(T) 1 dlv i
------- P\ + -COS9 P, + • • • + 1 Pl + <„4.2Pl + • • •
p dr p dr

+ Ks.\ BV(T) + /cs.vA>.i Pi + • • • — js.„ — 0.

We now use the recurrence relation of Legendre polynomials,

cos 9 Pn =

to obtain
1 dBv(T) \^dlv.„/' n n+l
-------7---- ~ , “3— I S—TT r,i-i + t—AT r"+1
p-------------------- P c‘r + 1 2n + 1

00

+ (<„ + ^s.i>) A'.H Pn + ks.vBv(T) — js v = 0. (A. 14)

n=l

Equating coefficients of the corresponding polynomials leads to the following

series of equations:

+ (A.15)
3 p dr
assuming isotropic scattering, that is, Js v independent of 9,

+n + (. + ); = 0 (A,i6a)
p dr 5 p dr
|-^- + |-^+«l,+/Cs.v)A-.2 = 0. (A.16b)
3 p dr 7 p dr
or, generally, for n > 1:
256 Appendix A The equation of radiative transfer

We may evaluate the ratio between successive terms of the expansion by

replacing dlv„_\ by and dr by R (the stellar radius) in Equation (A. 16c)
and neglecting factors of the order of unity. Thus to order of magnitude, we have

---- + Z,.,) ~ /,, ] (A. 17a)

Kp R

1
---- q!A.I + fv.3) A-.2 (A. 17b)
KpR
and, generally, for n > 1,

1
7t(A',h-I + *v.n+l) ff.ii- (A. 17c)
KpR
Since the deviation from isotropy is small in stellar interiors, there is some 8 < 1,
such that < eBv(T) for all n > 1. The question is how many terms of the
expansion should we retain. The following argument is due to Eddington. In all
Equations (A. 17c) with n >2 the left-hand side is smaller than 8Bv(T)/(KpR),
neglecting factors of the order of unity. But
1 R2
io-10.
KpR kM

for average opacities k and typical stellar densities and radii, and hence for /v,2
and all subsequent coefficients we have

/r.„>2 < 10-|0£Bv(7’).

We now repeat the argument using this result in relation (A. 17c) with n > 3 and
obtain Iv.,,>t, < s Br(T) and again, < I()“3OeZ?v(7'), and so forth.
As to j, from relation (A. 17a) it follows that it is of the order of \(Ti0Bv(T).
Clearly, the power series (A. 13) converges very rapidly.

io-10.

meaning that the deviation from isotropy is indeed very small and we may discard
all but the first two terms of the expansion. (Obviously, we cannot discard the
second term as well, for that would leave us with an isotropic radiation field
with no net flux.) This approximation is called the diffusion approximation. The
solution of the transfer equation (A. 12) is thus
1 dBv(T)
Iv(r, 6) = BV(T) + /,. ,(r)cos0 = B}.(T)-------------------------- —- cos0,
(<,.+ «-s.i-)P dr
(A.18)
where we have eliminated 7, j from Equation (A. 16a). Finally,
dBv(T) _ dBvdT
dr ~ ~dT~dr (A. 19)
 Appendix A The equation of radiative transfer 257

For the theory of stellar structure, knowing Iv(r. 0) does not suffice; we are
interested in H(r) (introduced in Section 3.7) - the total radiation flux (in all
frequencies) in the radial direction. In order to eliminate the dependence on 0,
we consider moments of the radiation intensity field /(r, 0), which relate to the
physical quantities that we have already encountered. The flux H(r) is obviously
given by

//(/•) = j I(r, 0) cosOda>.

Inserting Equation (A. 18) into definition (A. I) and noting that f cos Odea — 0, we
have
4ttJT 1 dB'<T\h
(A.20)
3p dr JQ «•*,, +ks.v dT

The radiation pressure Ąa(i (introduced in Section 3.4) is due to the fact that each
photon carries a momentum hv/c. Hence the radiation flux in the 0 direction
across a surface element dS transfers momentum of amount /(r, 0)cos(9/c in
the radial direction, incident on an area element dS cos 0 perpendicular to it. The
resulting pressure in the radial direction is therefore given by the next moment of
l(r.O)-.

^rad(f) = COS’ Odd),

leading, with Equation (A. 18) and f cos3 Oda> = 0, to

1 f00 4.t
Prad(r) = - / —Bv(T)dv = \aT4. (A.21)
<■' Jtt ->
Finally, differentiating Pia(j with respect to r,
JPrad 4ndT CxdBv(T) ,
(A.22)
dr 3c dr dT
and dividing Equation (A.20) by Equation (A.21), we obtain
.. C dPrat|
H = — —------------- , (A.23)
Kp dr
where
f00_____ L_____dB^dv
JO K-t1 — e~/n kT )+ks.v dT
(A.24)
Jo ilT 1

is called the Rosseland mean opacity, after its originator. Svein Rosseland. Sub­
stituting Prad from Equation (A.21), we finally obtain the diffusion equation for
radiation in the simple form:
4a c T3 dT
H =----------------- (A.25)
3k p dr
258 Appendix A The equation of radiative transfer

It is the same as Equation (3.67), derived from simplistic arguments, but it includes
a rigorous treatment of the interaction between matter and radiation, expressed by
k, which is the essence of the behaviour of stellar matter. We note that the harmonic
nature of the Rosseland mean gives highest weight to the lowest opacities. At the
same time, the weighting factor dBv/dT becomes small at very low and very high
frequencies; it peaks at v = AkT /h. In the Sun, for example, the corresponding
wavelength A = c/v is about 6000 A (within the visible range) at the surface,
where T % 6000 K, and about 2.4 A (in the X-ray range) at the centre, where
T 1.5 x IO7 K. The optimal radiative transfer efficiency would be attained if
the lowest opacities occurred at frequencies near ĄkT /h. This, however, is not
necessarily the case.
 Appendix B

The equation of state for

degenerate electrons

Consider the electrons in some volume of a star as constituting a gas that satisfies
the following assumptions: (a) the electrons are free, that is, interactions are
negligible both among them and between them and the ions; (b) the distribution
in space is homogeneous; (c) the distribution of velocities is isotropic; (d) the
entire system is in thermodynamic equilibrium, which enables the calculation of
all thermodynamic properties as functions of temperature T and density p for
a specified composition (see Section 2.1); (e) the atoms arc completely ionized,
so that density and composition determine the electron number density ne (see
Equations (3.23) and (3.24)). We are interested in the equation of state of the
electrons, that is, their contribution to the pressure as a function of T and ne,
as well as the electron contribution to the internal energy, both quantities being
required for solving the equations of evolution of the stellar structure.
The concept of pressure implies transfer of momentum. Internal energy of
a free gas is the kinetic energy of the particles - a direct function of momen­
tum. In fact, according to statistical mechanics, which provides the link between
macroscopic thermodynamic properties of a system and the microscopic state
of its constituent particles, any thermodynamic quantity may be derived from
the distribution of particle momenta in the three-dimensional momentum space.
By assumptions (b) and (c), the momentum space may be regarded as spherical;
therefore, an element of space is d2 p = 4jtpzdp. The distribution function f
determines the number of electrons per unit volume that have momenta in the
interval (p, p + dp), corresponding to kinetic energy values e(p) in that interval,
hence regardless of direction. We denote this number by n(p),

n(p)dp — f(e(p)-,T. \I/)4ttp2dp. (B.l)

Obviously, f may depend on temperature, which is one of the independent ther­

modynamic properties. The other parameter, which we have denoted by i//, is
meant to take account of restrictions imposed by quantum mechanics on the

259
260 Appendix B The equation of state for degenerate electrons

distribution of’ momenta, and we may expect it to depend on nc, the other inde­
pendent properly.
By definition, the electron number density is given by

«e = / n(p)dp = I f(€(p);T,^)47tp2dp. (B.2)

Jo Jo

The pressure exerted by the electron gas is obtained from the pressure integral (3.4)

Pc = | [ v(p)pn(p)dp = | [ v(p)pf(e(p):T.xl/)47rp 2dp (B.3)

Jo ' Jo

and the specific internal energy, from the energy integral (3.42),

ue — - I n(p)e(p)dp, (B.4)
P Jo

where the integral pue is the energy density. The additional relations required in
order to perform the integrals are the relativistic formulae

de(p)
U(/7) = (B.6)
dp

The distribution of electrons over the six-dimensional phase space is a direct

consequence of the Pauli exclusion principle, which states that if an element
of space 4n:p2dpdV is divided into cells of volume A3, then each cell can be
occupied by at most two electrons (with opposite spins). Particles that are subject
to this restriction are called fermions and they are governed by the Fermi-Dirac
distribution function

/(6(p);T, + (B-7)

where V/ may assume any value between —oc and -Foo. Wc note that / has a
maximum of 2//?\ reached in the limit xj/ +oc, which expresses the Pauli
exclusion principle. In this limit. f(p) becomes a step function, which means
that electrons tend to occupy the lowest available energy (momentum) states. The
parameter is thus known as the degeneracy parameter.
Substituting the distribution function (B.7) in the expressions for the electron­
number density, electron pressure and internal-energy density (B.2)-(B,4),
 Appendix B The equation of state for degenerate electrons 261

we obtain
8.t f00 p2dp
Ip /o ~'l' + I (B.8)

8% v(p)p'dp
(B.9)
3/P Jo + | ’

8tt e(p)p2dp
P«C = — (B.10)
/P Jo et(^/AT-v- + ] ’
Thus, the general procedure for obtaining the equation of state and related quan­
tities, given the temperature and the electron-number density, is as follows: first
V/ is determined using equation (B.8), and then the pressure is obtained from
(B.9) by substituting (B.6) for v(p), and the specific energy from (B. 10). by
substituting (B.5) for c(p). Any other thermodynamic quantity of interest may
be calculated with the aid of Pc and mc. In principle, this procedure appears sim­
ple and straightforward; in practice, as we shall see shortly, the calculations are
quite complicated, especially if analytical expressions are sought, so as to gain
some physical insight.
A general relation may already be derived between Pe and puc, which follows
from the equality
-^-[6(/?)p’] = ^-^//+ 3e(/?)p2, (B.ll)
dp dp
connecting the numerators in the integrands of (B.9) and (B.10). It is thus easily
verified that

8tt Z”0
= 3( Ą. + puc). (B.12)
IP Jo + |

In the non-relativistic limit, e(p) = p2 /2mc and ■~i[c(p)p2\ — ^p4/2me, so that

the left-hand side of (B.12) reduces to the form of (B.9), yielding

—=3(Pc + p«c) => p(/e = ^pe. (B.I3)

In the extreme relativistic limit e(p) = pc and -^[((.plp'] — 4p-c, so that the
left-hand side of (B.12) reduces to the form of (B.10), yielding

4pue ~2>(Pe + pue) pua = 3Pe. (B.14)

We note that these relations are independent of the value of that is, unaffected
by the degree of degeneracy.
Clearly, integrals (B.8)-(B. 10) do not have simple general expressions. Even
approximations are not entirely obvious because they may be of two distinct and
262 Appendix B The equation of state for degenerate electrons

independent kinds. These are related to the two different and independent effects
that determine the state of the electron gas. quantum-mechanical and relativistic,
each ranging from very weak to very strong.
Essentially, an approximation is obtained by expanding a function in terms of
a scaling dimensionless variable and therefore we would like to have an indepen­
dent variable for each effect. The natural free variables that define our system -
temperature and electron number density - are, unfortunately, not appropriate for
this task, for the strengths of both relativistic and quantum effects usually depend
on both variables. One such dimensionless scaling parameter is ty(T, n^), which
is clearly associated with quantum-mechanical effects. For relativistic effects we
must find a scaling parameter - that we shall denote by $ - so that £ —> 0 in the
nonrelativistic limit v/c 1. and £ -+ oo in the extreme relativistic limit v c.
Obviously, it will be a function of temperature and electron number density,
$ = %(T, nc), but its definition may differ in different i/r ranges. For example, a
natural relativistic measure is kT/mec2 for scaling energy, which is equivalent to
£(7') = JmekT/mcc for scaling momentum, but when the strength of relativistic
effects depends mainly on the electron number density (a situation that we shall
shortly encounter), this scaling is not appropriate and a different one must be
sought. We shall return to the transition between different forms of £ later on. The
parameter space spanned by [y/, £] covers all possible physical regimes.
Thus, the distribution function and the thermodynamic quantities derived
from it may be expressed in terms of § and y/ and asymptotic expansions in these
parameters may be obtained. The full expansions are by no means simpler or more
transparent than just evaluating integrals (B.8)-(B. 10) numerically. The limiting
cases and first approximations that may be derived in this fashion are, however,
instructive.

The non-relativistic non-degenerate case: | 1 and —y/ I

In the nonrelativistic limit p ,nevande % p2/2me. In the limit — y/ » 1 (infact

e-^ » 1 suffices) we have ee/k r+^' 1 for any value of e and of T, so that the
unity term may be neglected in the denominator of the Fermi-Dirac distribution
function. It is therefore natural to introduce a new variable x = p/^/2mekT for the
integrals (B.8)-(B. 10), and thus to scale relativistic effects by £ = y/mekT/mec.
As mentioned above, the first step towards obtaining Pe is using (B.8) to derive
V/. Thus, substituting the new variable at, we may cast the integral in (B.8) in the
form e~'x2dx — V^r/4, and obtain

(B.15)
2 (2nmckT)W
Note that the left-hand side of equation (B.15) is a very small number, which
means that we are in a regime of low density and relatively high temperature. An
 Appendix B The equation of state for degenerate electrons 263

extremely high temperature, however, would be incompatible with £ I. Using

the definition of £ to eliminate T in (B. 15), we obtain

_ (A/nec)3 nc
(B.16)
" 2(2^)3/2p’

which means that the limit 1// —> —oo and £ —> 0 strictly applies to very low
electron number densities.
Inserting (B.15) into the distribution function (B.7), we have

He c-e(p)/kT
,!\P- T) = (B.17)

which we immediately recognize as the Maxwell-Boltzmann distribution function

for an ideal gas. The pressure is obtained by

P' "e P-/2»lcrr47r/?2J/A

--------------------- — c
me (27rmeA:T)3/2

which, after changing variables to x and substituting f^e X'x4dx = 3s/n/8,

yields

Pe = nekT, (B.18)

and it is easily verified that the energy density satisfies pue = 3 Pe, as in the
general case (B.13). We have thus recovered the thermodynamics of an ideal
(nondegenerate) classical (nonrelativistic) gas, as given in Section 3.2.

The strongly-degenerate relativistic case: arbitrary and V/ I

For the 1 regime (here e''/ J>> 1 suffices), it is useful to define an energy e0
by the requirement

= (B.19)

and write the distribution function (B.7) as

2 1
/(«(p);r.») = ji;(!>|<Wti_ll + 1,

which stresses the high sensitivity of this function to the sign of [e(p)/eo — 1] in
the exponent. The shape of the distribution function is shown in Figure B.l.
Thus

1 1 for 6 < 6(),

(B.20)
e(!r|c(p)/«0-lJ _|_ ]
0 for e > e().

The meaning of the distribution in this limit is that all energy states up to <?o are
occupied, while all states beyond e0 are empty, a state of degeneracy. In fact, the
264 Appendix B The equation of state for degenerate electrons

Figure B. I The Fermi-Dirac distribution function with respect to e(p)/eo- Increasingly

steeper curves correspond to increasing values of as marked. Alternatively, for given
they correspond to decreasing temperatures - in proportion to the reciprocals of
these numbers. Thus the deviation of the distribution function from a step-function is
proportional to A T.

limit (B.20) describes the state of extreme or complete degeneracy, that we have
considered in Section 3.3.
We denote the momentum associated with 6y by py,

(B.21)

Parameters ey (known as the Fermi energy) and py (known as the Fermi

momentum) arc interchangeable as measures of degeneracy. As t// oc, so are
fo and py. The simple limiting form of f from (B.20) - 2/ h} for p < py and zero
beyond py - leads to:

8.T f"" 3 8,7 ,

/zL. (B.22)
T./„ =
and yields py(ne)- Temperature ceases to play a role in this extreme regime.
In order to obtain the pressure and energy density, we express v(p) and e(p) in
terms of the dimensionless variable x = p/mec. Since p is limited, the relativistic
scaling parameter is readily defined by

$ = ptt/mec, while 0 < x < £. (B.23)

 Appendix B The equation of state for degenerate electrons 265

Replacing p by xmec in the corresponding integrals, we obtain the pressure and

energy density from

f1’0 , &Tm4ec5 f" x4dx

(B.24)
3/r Jo 3/?3 Jo (1 +-V-)1 -

and

puc=— / e(p)p2dp = —-/ 1(1+x2),/2-lU-Vr. (B.25)

/;•’ Jo Jo

The integral in (B.24), which we denote by J\$), is

£ _ £ , C for 0
5 14 24
^) = |l^(2e2 -3)(r’+ l)1/2 + 3sinh '$]
fj. _ 3?: ,
4 S'' for £ —> oo

(B.26)

and the integral in (B.25), which we denote by </(§), is

3ę? _ , £
10 56 48
for § 0
P(£) = £3[(£2 + D'/2 11 - ^(?)
4
- 5t3 + ZŁ
g for $ —> oo.
(B.27)
Retaining only the first term in each expansion series, we obtain the expressions
for electron pressure and internal energy density in the nonrelativistic and extreme
relativistic limits, as derived in Section 3.3:

and pue = for -> 0 (B.28)

and puc = 3PC, for £ —> oc. (B.29)

What is the physical meaning of these limiting cases? Combining definitions

(B.19), (B.21) and (B.23), we obtain a relation among the parameters that deter­
mine the regime we are considering (T. xp and £) of the form:
•>
mecL
iA = ~kT~ (yi+^-i). (B.30)

According to this relation, the nonrelativistic limit £ —> 0 together with strong
degeneracy x!/ —>■ oc implies T —> 0. Thus, nonrelativistic complete degeneracy
strictly applies to a system at vanishing temperature. We shall return to this point
shortly. On the other hand, as £ increases, xjr tends to infinity with it, regardless
of temperature.
266 Appendix B The equation of state for degenerate electrons

The nonrelativistic degenerate case: | << I and arbitrary i//

In the nonrelativistic limit £ —> 0, it is possible to consider analytically the entire

range of the degeneracy parameter x]/. This requires, however, the replacement of
p by e = p2/2me, as the integration variable in equations (B.8)—(B. 10) that yield
ne, and puc. Accordingly, u = ,/2e//ne and dp = Jmc/2e de. In addition,
we define functions of the form
f00 x"dx
(B.31)
/'■„(’A) = *o + 1’

which are known as Fermi-Dirac integrals. With the new integration variable, we
have
47r(2/nc)3/2 f00 6l/2r/f 4rr(2Wcr/-)3/2
(B.32)
/?3 Jo + I /?3

87T(2/He)3/2 ^kT(2mckTY'2
(B.33)
3/z3 Jo e</*7’-V' + 1 3/?3

and again pwc = ^Pc. It is easily verified that

-Fyipty)
Pc = nekT (B.34)

The Fermi-Dirac integrals may be expanded in powers of <A (or e^) in the
limits i// —oo and x// oo, much as the functions of and in the
case considered above were expanded in powers of £. The derivation of these
expansions is quite complicated and we shall only give the results here:

for —* —oo
(B.35)
for xk -> oo

for xk —> —oo

/?3/2(’A) (B.36)
for xjr oo.

Retaining only the first terms in the limit xk -> —oo, we recover the equation
of state of an ideal classical gas Pc — nckT. The opposite limit x// oo (while
$ -» 0) wc have already reached from a different direction. Although we attain
it here through a different approximation, the final result is of course the same as
(B.29). as can be easily verified. But here we gain additional insight. It is only
in the lowest approximation that the relation between Pe and «e is independent
of temperature in the strong degeneracy limit. The second approximation, which
 Appendix B The equation of state for degenerate electrons 267

we shall now attempt, will give us an idea on the effect of temperature on this
relation.
Using (B.32) with the strongly degenerate form of (B.35) in the lowest approx­
imation, we have

87T(2me*7')3/2
V/3/2.
3/P

We now use (B.33) with the strongly degenerate form of (B.36) in the second
approximation to obtain

lr \2/' 5/3 F 4O.t2 '”e*2 7'2 1

(B.37)
20me J e |_ (3/tt)4/3 /?4 n4/3_

When T and ne are such that the second term in the square brackets, which may
be regarded as a temperature-dependent correction, approaches unity, the electron
degeneracy will be lifted. Wc note that this criterion is similar - up to a numerical
factor of order unity - to (B. 15), which estimated the effect of degeneracy from
the opposite limit, that is, when may a gas be considered ideal. Furthermore, it can
be easily verified that the ratio between the two definitions of § at the extremes
ofthe i// range is also, not surprisingly, of this form.
Wc may interpret the criterion for degeneracy, as expressed by (B.15) or
(B.37) in yet another way. The average momentum of an electron in an ideal gas
is of the order of Jm^kT, and hence the corresponding de Broglie wavelength of
the electron is A = h/ JiipkT. The average distance between electrons is given
by d = ne ’ . Therefore, the right-hand side of (B.I5) is of the order of (c//a)
and the correction term in (B.37). ofthe order of (r//X)4, while the ratio between
the two expressions for £ is d/k. Quantum-mechanical effects become important
as the intcrparticlc distance shrinks towards the de Broglie wavelength (d Ź a).
So long as particles are much farther apart than their wavelength (d X). the gas
may be considered ideal.
If we express the temperature in units of IOSK, denoting it 7X, and the
density in units of 109kgm-3 (106gcm-3), denoting it /?9. and assume /ze 2,
the correction term in (B.37) is 1.7 x 10-2T82/pg'3. This means that at densities
typical of dense stellar cores or white dwarfs, electrons will be degenerate even
at temperatures well in excess of IO8 K, and the pressure will barely be affected
by changes in temperature.

The extreme-relativistic degenerate case: £ I and arbitrary x!/

In the extreme relativistic limit £ —> oc, we change variables again from p to r =
pc, and substitute v = c and dp = deje. Using again the Fermi-Dirac integral
268 Appendix B The equation of state for degenerate electrons

oo

Figure B.2 Schematic representation of the [tf/, £ | parameter space and the various approx­
imation domains.

notation, we obtain

fkT\'
= Stf — F2(>A) (B.38)
\ /

8>tAT
~hc) (B.39)

and. of course. pue = 3Pe. The expansion of the Fermi-Dirac integrals in the
tA -» oc limit is

FfT’A) = _|(A? + (B.40)

/W) = j’A4 + IttV2. (B.41)

Retaining only the first terms in (B.40) and (B.4I), eliminating (A between (B.38)
and (B.40) and substituting the result in (B.39). we recover (B.29) - the expression
already obtained when arriving at this corner of the parameter space from the
perpendicular direction (increasing $ at high iA). Repeating the procedure used
for the nonrelativistic case above, we obtain the second approximation for Pc,

(B.42)

Here, however, the correction term is 6.6 x 10 l6( T^/p^)2^, and thus completely
negligible for any temperature value (at which electrons still exist - see below).
 Appendix B The equation of state for degenerate electrons 269

There remains the nondegenerate (or weakly degenerate) extreme-relativistic

case. In this regime, however, a complication arises, which causes some of the
basic assumptions that we made at the beginning of this treatment to break down.
The electrons can no longer be decoupled from the radiation field and the ion
sea in which they are embedded, and the electron number density is no longer
an independent parameter. The reason is that at the high temperatures implied by
= kT/mec- » 1. the radiation field in the presence of ions creates electron­
positron pairs, as discussed in Section 4.9. Thus the number density of electrons
changes as a function of temperature, being constrained by the requirement of
global charge neutrality (the charge density of ions plus that of positrons must
equal the electron charge density). More advanced physics is required in order
to deal with these processes. Fortunately, few stars, or more accurately, few
evolutionary phases reach this regime and survive it, for it is dynamically unstable,
as explained in Section 6.4.
We have now covered all sides of the parameter box, shown in Fig­
ure B.2. and wc have identified the overlapping corners. The middle part - the
moderately degenerate and moderately relativistic regime - requires more com­
plicated numerical procedures and has no analytically transparent results; it is
thus beyond the scope of this text.
 Appendix C

Solutions to all the exercises

Exercise 1.1: Consider a mass element A/tt containing 10 000 hydrogen atoms and let
the mass unit be the mass of a hydrogen atom. Then

Am ~ 10000 x I + 1000 x 4 + 8 x 16 + 4 x 12 + 1 x 14 + I x 20.

according to the data given in the text (since elements heavier than neon are neglected, a
small error is introduced). Now, by definition,
10000 x 1
X =-------------
= 0.7037
Am

1000 x 4
Y = = 0.2815,
Am
and similarly, Zc = 0.0034, ZN = 0.0010, Zo = 0.0090. and ZNe = 0.0014.

Exercise 1.2:

(a) Substituting p(r) in Equation (1.5) we have

m(r) = 4nr2p(r)dr — 4itpc

z,.3 ,.5 X
- 4jTpc----------- - .
\3 5R2)

(b) M = m(R) = &rpcR'/i5.

(c) By definition, p = and substituting (b) for ;W, we obtain p = 0.4pc.

Exercise 2.1:

(a) For a uniform density, p = p and

47rr’
»»(r) = —— p

270
 Appendix C Solutions to all the exercises 271

Substituting m(r) in the hydrostatic equation (2.14) and integrating from the
centre (P - Pc) to the surface ( P = 0), we have
/»(/-), 3GM2 GM2
Pc~GPJo ~ r~ 8-tK4 > 8^‘

(b) Using p(r), m(r) and M(R) from Exercise 1.2, we integrate Equation (2.14) to
obtain
l'R ■> fR F / r \21 / r r' \
= G

\5GM2 GM2
I6.7/?4 ’ XttR4'

Exercise 2.2: If we imagine the star compressed into a sphere of uniform density pc, the
new central pressure P' must exceed Pc, since by bringing the matter closer together we
increase the gravitational attraction between its parts, that is, the force to be balanced by
this pressure. The new central pressure is obtained, as in Exercise 2.1, by integrating the
hydrostatic equation (2.14), with in — 4rrr3pc/3, up to R - (3M/4npc)''3, which yields

P’ = l(4.T/3)l/3GM2/3p4/3.

In conclusion, Pc < P' leads to

Ą. < (4rr)1/30.347GAf2/3pc4/3.

Exercise 2.3:

(a) Inserting /n(r) = 4nr3p/3 and dm = 4nr2pdr into Equation (2.20) and per­
forming the integration, we obtain, after eliminating p.
3 GM2
Q =
5 R
whence a = 0.6.
(b) Using m(r) from Exercise 1.2 and dm = p(r)4jtr2dr in Equation (2.20), we
obtain
5 GM2
Q = -4.7 Gp2
7 R '
whence a = 0.71.

Exercise 2.4: The rate of change of the energy, as given by Equation (2.43). is
E = —L. Assuming hydrostatic equilibrium, we have from the virial theorem E =
(Equation (2.44)) with Q = —aGM2/R (Equation (2.27)). Hence

E = -|aGM2(-| = -L.
2 \R)
272 Appendix C Solutions to all the exercises

Setting t — 0 and R = Ro at the beginning of contraction, we obtain by integration

2Ł
aGM2 ''
R Ro
which yields

Ro/t aGM-
(t/r + I)2 2RoL
For / )?> r, —Ror/t2.

Exercise 3.1: For a degenerate electron gas to be considered perfect, the Coulomb energy
per particle, eę, must be smaller than the kinetic energy, in this case, p^/hne. where p()
is given by Equation (3.32). The average distance between electrons is nJ1 \ where ne is
the electron number density. Hence 6c K /4.t6() and the condition is
■>1/3 .-> /o x 2/3
e ne h / 3nc \
4/reo 2we \ 8,t /
Thus the electron number density must satisfy

Exercise 3.2: By definition (Equations (3.11) and (3.12)), P^.M = ftP and /’r;ili =
(I — j8)/’. and is assumed constant throughout the star. The specific energy of a (non­
relativistic) gas, whether ideal or degenerate, is given by Equation (3.44),

and the specific energy of radiation is given by Equation (3.47),

Prill R
Mrad = 3— =3(1 — /J) — .
P P
Hence
3 P P 2
Mgas + Wrad — z(2 — ft) ■ ----- < — 77r(“gas + Wrad)-
2/2 p 3(2 - ft)
Using the virial theorem in the form (2.23), we have
2 [("gas + »rad)</»' = ~ ~ V-
Q = -3
(2^ J (2.- ft)
Now, E = U + Q and, substituting the relation between Q and U, we finally obtain

which tends to zero when the radiation pressure predominates (ft —> 0) and to the well-
known relations E = Q/2 = — U. when radiation pressure is negligible (ft 1). If the
 Appendix C Solutions to all the exercises 273

change in gravitational potential energy is AQ. the change in total energy, which is the
energy radiated away (Equation (2.43)), is AE = (/J/2)AQ, while the energy that serves
to heat the star is At/ = -[(2 - /?)/2]AQ. For /J = I, the amounts are equal. As ft
decreases, the radiated energy fraction decreases.

Exercise 3.3: The hydrostatic equation (2.14) may be written in the form

dP
~d7 = -Pg<

where we have used the definition of the local gravitational acceleration, g - Gm/r.
Dividing both sides by Kp and using the definition of optical depth dr = —Kpdr, we
obtain the desired equation. Since the mass and thickness of the photosphere arc negligible
compared with the stellar mass and radius, we may assume g to be constant throughout
the photosphere, gR — G M / R-. Taking for the constant opacity its value kk at E. we may
integrate the hydrostatic equation in its new form, to obtain

GM r
KR clp = ~pr dx-
Joo K Joo

The integral on the right-hand side is unity by definition; on the left-hand side, the pressure
vanishes far away from the star. Thus,
GM
«rPr — gR => Pr — —
krR-

In Section 2.4 we obtained a lower limit for the central pressure of a star: Pc >
Therefore the ratio Pr!Pc is at most

PR ^R-
P, < KrM '

For the Sun, this means that the surface and central pressures are more than 11 orders of
magnitude apart.

Exercise 4.1: Consider a mass element A/n of helium, half of which turns into carbon and
half into oxygen, by nuclear processes that can be expressed as 3a l2C and 4a 16O.
The energy released in the first process is Q3» = 7.275 MeV (see text), while the energy
released in the second is given by adding to it the energy released by a capture on a
l2C nucleus, 7.162 MeV (see text), amounting to = 14.437 MeV. The number of l2C
nuclei produced is given by

().5A/n
«(12C) =
12/hh

and, similarly.
0.5 Am
n(l6O) =
16m 11
274 Appendix C Solutions to all the exercises

Hence the total energy released per unit mass is

„ n(l2C)e3a+n(16O)24o 2W24 + Q^/32 inHIb-i

Q =-------------------------------=------------------------ = 7.3 x 10 J kg .
A/?/ «'H

Exercise 4.2: Using the results of Exercises 1.2 and 2.1, in which the same density
distribution is assumed, we have

15/W 15G.W2
Pc =------ e and Pc =-------- r.
8.t/?3

Combining these results, and using the equation of state for an ideal gas (3.28), we obtain
the central temperature

(Ex.l)
c 2 R R~

where /x = 0.61 for a solar composition (see Section 3.3). The assumption of non­
degeneracy implies that for the electrons, the ideal gas pressure (3.27) is higher than
the degeneracy pressure (3.34),

—PcTc > K\ (Ex.2)

Re

where ~ 1.17 for a solar composition (see Section 3.3). Using Equation (Ex.l), we
express pc in terms of 7'c and M,

15 \3 T3
Pe = — rG) Jr-'

and insert the expression into inequality (Ex.2). We thus obtain an upper limit for Tc,
given the stellar mass M:

< R2 Re 3 G2 m4/3

Vi5/ RA-;

The desired lower limit for the stellar mass required for each nuclear burning process
is obtained by reversing this relation and substituting for Tc the appropriate threshold
temperatures given in Table 4.1.

Exercise 5.1: If we adopt r as the independent space variable, the Taylor expansion near
r =■ 0 for any function f(r) is

r2 + l
dr- 6
 Appendix C Solutions to all the exercises 275

and we retain only the first nonvanishing term besides fc. For the mass m(r) we have
mc — 0 (boundary condition) and from Equation (5.2) on the left
/dm \ , ,
I — I - 4rr(r2p)c = 0
\dr

(2rp + r2 — ) =0

= 4?r ^2p+4r~ + r2^-^\ = &rpc.

\ dr dr-dc
Therefore near the centre

m(j) = |7Tpe'‘\

as if the density were uniform and equal to the central value. For the pressure P(r) we
have from Equation (5.1) on the left and the result obtained for m(r)
fdP\ / Gm\ P4nGp2r\
H- = ~ p— =_ —i— = °
\ /c V f" / c s 3 /c

/d2P\ /dp Gm G dm Gm\ 4nGp2

I tt I =- t—r + f>~~i—-p~r I =—7—•
ydr1 ) \dr rL r* dr r- /c 3
Therefore near the centre

P(r) = Pc- jnGp2

cr2.

For the luminosity F(r) we have Fc = 0 (boundary condition) and from Equation (5.4)
on the left
dF\
= 4?r(r2pr/)c = 0
77 / c

-TV = 4rr 2rpq + r —q + r p — 1 = 0

ydr1 ) c \ dr dr / c

= 4n\2pq + r(...) + r2(... )]t = Snpcqc.

Therefore near the centre

F(r) = i7rpcqcr\

For the temperature T(r) we have from Equation (5.3) on the left and the result obtained
for F(r)
'dT\ 3 / Kp F\
Jc 16jT«c \ T3 r~ / c 4ac \ /3 /c
=0
d2l \ 3 Kp d +L tL(k±\ 1 KePę<F
Jr7 Ą. lÓTrac T^7r r2dr \T^ 4ac T2"
276 Appendix C Solutions to all the exercises

Therefore near the centre

1 Kc p,. r/c o
T(r) = Ą.---------- 2.
Sac T?

Note that these relations hold regardless of the functional dependences P(p. T). q(p, T)
and k(p, T).

Exercise 5.2:

(a) For n = 0, the Lane-Emden equation (5.17) becomes

£ ( 2d0\ = 2
<in v dd

Integrating, we obtain

where C is an integration constant. Dividing by £2 and integrating again, we obtain

the solution

- + n.
o = -^2-C
ę

where D is a second integration constant. Since we cannot accept solutions that are
singular at the origin, we must assume C = 0, and since 0 = 1 at the origin (by
definition), I) = 1. The solution for n = 0 is therefore

I -

Obviously, = $(0 = 0) — 76 and — —$i/3 = —72/3. Substituting

into Equation (5.20) and using Equation (5.18) to eliminate a. we obtain M =
4.t W3pd?) (and Do = 0, which shows that a n = 0 polytrope describes a config­
uration of uniform density.
(b) For n = I and a variable / defined as % = 1-0, the Lane-Emden equation (5.17)
becomes

whose general solution is

X = C sin(| - 15),

where C and <5 are constants of integration. Hence

H = C sin(| — <5)
 Appendix C Solutions to all the exercises 277

We must assume <5=0, for otherwise the solution is singular at the origin, and since
0 = 1 at the origin, C — 1. The solution for n = I is therefore
sin f
0(5) = -p

which has its first zero at || = ,t (and is monotonically decreasing in the interval
(0, tt)). Differentiating, we obtain
/ c/6> \ /cos$ sin|\ I
\ )i=„ ~ tt'

We now use Equations (5.18) and (5.20) to obtain M — ĄR'pc/n (noting that D\ =
tt2/3, consistent with the entry in Table 5.1).

Exercise 5.3: Forgiven M and Pc, we have from Equation (5.28)

Pc, 1.5 = / AV'4

Pc,3 \#l.5/

For given M, we obtain the ratio of radii /?(») from Equation (5.21) and Table 5.1

R( 1.5) / Di.., pc,3 \1/3 _ / Di s \1/3 / Bi.5 \1/4 _ /5.991 \ 1/3 / 0.206 \ | A
~ \~d7 ) \ih J ~ \ 54.81 J \ 0.157/

and therefore

K(3) > «(l.5).

Exercise 5.4: The central density is readily given by Equation (5.21): pt = 1.2 x
102 kg m-3. In order to obtain the central pressure as a function of M and R. we eliminate
pc between Equations (5.21) and (5.28):
GM-
Pe = -r^Tl(3D,I)4'3B„|.
4,t R'
The term in square brackets exceeds unity for all n and hence
GM- GM2
Pc > ------ r > ------ 7-
4tt/?4 8,t/?4
Thus inequality (2.18) is generally satisfied by polytropic models. For Capella, with
n = 3. P. = 6.1 x 10l2Nm-2.

Exercise 5.5: The critical mass is obtained from the relativistic-degenerate equation of
state (3.36). Hence at the stellar centre both Equations (5.28) and (3.36) are satisfied, both
being of the form Pc oc p4'3. Equating coefficients and isolating M, we obtain

I
M = (4^

The term in square brackets reduces to /T " v I .5/32,t.

278 Appendix C Solutions to all the exercises

Exercise 5.6: In radiative equilibrium, the radiation pressure gradient is obtained from
Equations (5.3) and (3.40):

dPra<i _ Kp F
dr c 4nr2

(In the case of convection, this relation is still correct, provided the flux F on the right­
hand side is taken to be the radiative flux, rather than the total flux, of which the bulk is
due to convection.) Substituting into the hydrostatic equation (5.1) P = Pias + Ąad, we
obtain

dP„M GM KpF GM / kF
~T~ = ~P~r + 3—5“ = ~p~~ 1 - 3—77-
dr r- r- \ MrcGm

So long as condition (5.34) is satisfied, the gas pressure decreases outward. When it
is violated, the density is bound to increase outward, if the temperature is decreasing
outward. This would lead to instability (of the Rayleigh-Taylor type).

Exercise 5.7:

(a) Equation (5.24) for a n = 3 polytrope may be written as

M2 = —-----777T- & ,

Where My is given in Tabic 5.1. Substituting K from Equation (5.45), we obtain

the quartic equation in the form

M
~M~.
m4^4.

where

4My'R2
M.
y/iiaj3G^2'

With a = 8rr5k4/(l5c3h3) and 'R — k/mH, we have

w 3/TÓM,
Mt - ------ — I8.3M0.

(b) The Chandrasekhar mass, given by Equation (5.31), may be expressed as

rr2 ,
8/15
 Appendix C Solutions to all the exercises 279

Exercise 6.1: In terms of the dimensionless variables 0 < y = P/Pc < 1 and po/Pc <
x = p/pc < 1, the equation of state reads:

, _ xr> - (po/p,:y'~r2xn

that is, a one-parameter equation of the form

(1 — a)y = xZl — axn,

where the dimensionless parameter a is defined by

« = (Po/Pc)y'-K!-

Since P is solely a function of p, the adiabatic exponent is obtained by taking the derivative

pdP xdy YiXv'-aY2Xn

y., =------- =------- =-------------------- . (Ex.3)
P dp y dx x?' — axy-

It is easy to show that ya is a monotonically decreasing function of x:

s,on\dYa/dx] = sgn|-a(yi ~ 7z2)2-vXl”l“z?“1].

therefore negative, and thus the minimum value of /a is obtained at the centre, where
x — 1. Hence the stability condition (6.23), /a > 4/3, will be satisfied everywhere, if it
is satisfied for x = 1. Substituting x = 1 in Equation (Ex.3), we obtain the condition

3/i — 4 > o'(3/2 — 4).

(a) Since a is positive, it follows that either /,, y? > 4/3 or Y\, Y2 < ^/J>-
(b) Since a < 1 and Y2 < /i, if /,, Y2 > 4/3, the stability condition is satisfied
regardless of the value of a. However, if /,, /2 < 4/3, then a must satisfy:

4 - 3/i
a > .
4-3/2
which means

Po > /4 — 3/i \ l/<Zl~Z2>

pc > \4 - 3/2/

For example, taking y, = 2/3 and y2 = 1 /3, the restriction imposed is A)/A? >
0.3.

Exercise 6.2: For an adiabatic process (8Q = 0), changes in energy are due to radial
perturbations 8r, or 8V = 47tr28r. An adiabatic change in the internal energy U is thus
obtained by combining Equations (2.2) and (2.3):

d
8U = — I P—(8V)dm.
Jo dm
280 Appendix C Solutions to all the exercises

Since P vanishes at the surface and <5V at the centre, integration by parts yields

, dP
8U — I 4nr~8r—dm.
Jo dm
A change in the gravitational potential energy is given by

Gmdm Gm
<5Q = -8 —r-didm

Adding these expressions, we have

8E = 8U +8Q = 8rdm (Ex.4)

and since 8r is arbitrary, it follows that the integrand vanishes with 8E. This means that
hydrostatic equilibrium corresponds to an extremum (stationary point) of the total energy.
Stability requires it to be a minimum.

Note: This property of hydrostatic equilibrium may be used in numerical calcu­

lations of stellar models as an efficient way for obtaining a hydrostatic configuration,
given arbitrary intial conditions, as shown by Rakavy, Shaviv and Zinamon in 1967. The
procedure, known as ‘quasi-dynamic’, is to put the velocity, rather than the acceleration,
proportional to the force in the equation of motion (2.13), as if the star were embedded
in an external viscous medium. Using r to denote the quasi-time variable, so as not to
confuse it with the real time /, we have
dr ~,dP Gm
k— = —47rr‘-----------
dr dm r2
where k is a positive constant. Multiplying both sides by dr/dr, integrating over the mass
and using (Ex.4) obtained above, we have

</£
dr
Thus the total (static) energy decreases with quasi-time and hence integrating the quasi-
dynamic equation over quasi-time, will lead to a minimum of the energy, if such a
minimum exists, and therefore to a hydrostatic configuration. If a minimum does not
exist, it will mean that there is no stable hydrostatic configuration for the given entropy
distribution of the model star.

Exercise 6.3: According to Equation (3.60). the adiabatic exponent for a partially ionized
gas is a function of T and x. Since the expression is symmetrical with respect to x = 0.5,
it has an extremum al this value of x for any given temperature, and it may be easily
shown that this extremum is a minimum (32ya/9x2 > 0 there). We may now regard the
 Appendix C Solutions to all the exercises 281

minima as a continuous function of 7’:

Za.minfT') =

It may be shown that ya.min(M) decreases monotonically with increasing /' < 0).
The critical lower limit for stability will thus be obtained by setting ya.min(^) = 4/3. This
results in a quadratic equation for the variable z — x/kT:

4z2- 122-63 = 0,

which has only one positive root, corresponding to T = 2.75 x IO4 K. Thus only below
this temperature may partially ionized hydrogen become dynamically unstable, the lower
the temperature, the larger the range of x corresponding to ya < 4/3.

Exercise 6.4: Adiabatic processes satisfy Equation (3.48):

. P dp
di< + Prf ( - | = 0 du =------
\pj P P
from which relations may be derived between any two of the thermodynamic functions
P, p and T. For gas and radiation we define adiabatic exponents T i and T2 by

(IP r dp
= li — (Ex.5)
P P

dP _ r2 dT
(Ex.6)
~p ~ r2 - 1 T"’
noting that both are equal to the ya of conditions (6.26) and (6.28) in the case of gas
without radiation. Now, for an ideal gas we have from Equations (3.28), (3.44) and (3.47)

3 7? aT4
u --T 4--------
2p p

R. . 4
p = P&s + Mad = —p l' + \ai 4
p
and from Equations (3.11) and (3.12),

P^ = PP Mad=(l-£)F.

Hence
3R 4aT4 aT4 3 P dT
PdT PdT „ P dp
du — -—dT + ------ dT _ dp = - p)-— -3(\ -
2 /z p p~ 2 p T p I p p P P
which, substituted into the condition for adiabaticity, leads to
24 — 21/3^/7' „ dp
----- Z—- — = (4 - 3p)~ (Ex.7)
2 7 p
282 Appendix C Solutions to all the exercises

For the pressure we have

dT dp dT
dP= P^— + Pgas— +4Prad —,
7 p T
leading to

dP dT dp
— =(4-30)—+ 0—. (Ex.8)
P Ip
Eliminating dT/T between Equations (Ex.7) and (Ex.8), we obtain

dP = |~2(4-3/3)2 I dp
P 24-210 p ’

Comparing this result with Equation (Ex.5), we have

_ 32 - 24/1 - 302
' ~ 24 — 21/3

Similarly, by eliminating dp/p between Equations (Ex.7) and (Ex.8) and comparing to
Equation (Ex.6), we obtain

_ 32 - 24)3 - 302
2 “ 24- 180 —302

For 0 = 1 (pure gas), F| = F2 = 5/3; for fi = 0 (pure radiation), F> = T2 = 4/3; for
0 = |, T, = 1.43, while r2 = 1.35.

Note: The adiabatic exponent T( for matter and radiation was introduced by Edding­
ton in 1918; T2, as well as a further adiabatic exponent Fj, which relates T and p, were
later introduced by Chandrasekhar.

Exercise 6.5: Let Mc be the mass of the convective core. The temperature gradient at its
boundary is given on the one hand by the adiabatic gradient (as in the core),
dT _ ya - 1 T dP _ ya - 1 T GM.p
dr ya P dr ya P r2
after substituting the pressure gradient from the hydrostatic equation, and on the other
hand by the radiative diffusion equation (5.3),
dT 3 kp F
dr 4ac T3 4nr2
Continuity of dT/dr (imposed by the continuity of the radiative flux) requires equality
of the right-hand sides of these equations:
ya — 1 T „ 3 k F
——~-C>Mc =------- -—.
ya P 4ac T3 4rr
 Appendix C Solutions to all the exercises 283

Since there are no energy sources outside the core, we may take F — L. Substituting

\aT4 Pnd
1----- = - = !-«,
P P
dividing by M. and rearranging terms, we obtain

_ ya kL
M 4(ya - 1)(1 - P) 4ttcGM

Now, if k is constant up to the surface, then 4ncGM/k is the Eddington luminosity

Z,Edd, and if P is constant, we have from (5.42) L/L^m = 1 - P- which yields the desired
expression for the core mass fraction. Note that the ratio depends indirectly on M through
the adiabatic exponent, which depends on p, where p = P(M).

Exercise 7.1: Inserting relation (7.36) into Equation (7.28) on the right, we obtain

/ "-I \ ~4 10-
P. oc M~ \M" ' j => P, oc M «+•’ .

We have P„ oc M2/1 torn = 4, (that is, P, increases with A7), whereas P. oc M 22/19 for
n — 16 (that is, P. decreases with increasing stellar mass).
Inserting relation (7.36) into Equation (7.33), we obtain

T, oc

which yields T. oc M4/1 for n = 4 and 7'. oc M4/l9 for n — 16. Note the weak dependence
of 7’. on M corresponding to stars that burn hydrogen by the CNO cycle, which means
that the main sequence of these stars may be taken to represent a line of constant central
temperature.

Exercise 7.2: The effective temperature of a star of known L and R is obtained from
Equation (1.3):

/. = 4it R2aT4n.

Using relation (7.35) for the luminosity at the lower end of the main sequence,

/ .
Aralii
/M
/ 7Kfrnm \
\3

and relation (7.36) for the radius (calibrated to the solar radius, with n = 4),

Rq \ )

we obtain
/ A/ ■ \3 / M ■ \b/1
L ------ = kx I ------- a 7\ff _ ■.
284 Appendix C Solutions to all the exercises

Substituting Teff 3 — | Lq/(4tt R2 a)]1/4 % 5800 K. and Afmjn % 0.1 ,WS, we have

/M \15/28
Tefl.min = Teff.0(-^) % 1700K.
\ Mq )

Exercise 7.3: First, we write the condition L < 4ncGM/Ks as

L 4ttcGMq M
Lq KsLQ Mq
Next, we calibrate relation (7.35) between luminosity and mass:
£. _ / M \3
\^e/
Substituting it into the previous relation, we obtain an upper limit for the mass of main-
sequence stars

M 4ncGMQ
= 180.
m~

assuming «s is the electron scattering opacity Keso (Equation (3.64)). Using the mass­
luminosity relation, we obtain the corresponding upper limit for the luminosity of main-
sequence stars: L < 5.8 x 106 Ls. The radius of a 180/Wo star may be obtained from
the calibrated relation (7.36), taking n — 16. appropriate to the upper main sequence.
The effective temperature results from L = 4rras in Exercise 7.2, which yields
7cfl = 3.7 x IO4 K.

Exercise 7.4: Some of the relations between starred quantities (Equations (7.28), (7.29)
and (7.33)) are independent of the opacity or the nuclear energy generation laws. These
are
M Gp M
(Ex.9)
ft4
From the Kramers opacity law and u = 4 we obtain two additional relations, using
Equations (7.31) and (7.32):
ac T™R4
F.=----- ------- (Ex. 10)
K() p„M

F. = qQpJtM. (Ex. 11)

Substituting relations (Ex.9), we have

M55 Mb
F* a ^ÓT and
«.7’
which, combined, yield a relation between radius and mass in the form

R. (x Ml/'3.
 Appendix C Solutions to all the exercises 285

This, in turn, enables the derivation of a mass-luminosity relation, as well as a radius­

luminosity relation. With the aid of the latter, the main-sequence slope may be derived as
in the text. Thus,

L a. A/546

log L = 4.12 log Teff + constant.

In conclusion, different opacity laws result in different main-sequence slopes (even assum­
ing the same n), 4.12 for a Kramers opacity law, as compared to 5.6 for a constant opacity
(Equation (7.39)).

Exercise 7.5: We substitute relations (Ex.9) into Equations (Ex. 10) and (Ex. 11) to obtain
M5-5m7-5 M('/i4
/-, oc-----
^0.5
— and /■. oc —R7 .

Combined, these relations result in

Eliminating R between this relation and one of the relations for F above, we obtain a
mass-luminosity relation that includes the effect of /z,

L oc M5-46m7'77.

Reversing this relation to obtain M(E, /z) and inserting the result into the relation R(M. n),
we may derive R(L. /F), which, combined with L <x R2T^, yields

log I. =4.12 log 7jr —1.18 log /z + constant.

Thus, w ith increasing Y at the expense of X, if a star maintains its luminosity, its effective
temperature will decrease and hence the star will move to the right in the H-R diagram.

Exercise 7.6:

(a) Assume an amount of mass 8m is burnt during a time interval St (and added to the
core). The nuclear energy supplied is Q 8m; this energy is radiated by the star at a
rate L and hence Q 8m = L 8t. Therefore the rate of core growth is Mc = L/Q.
Since L and Q are constants, and = 0 at t = 0, we get by integration
We(t)=-|r. (Ex. 12)

(b) The envelope loses mass at its inner boundary at the same rate as the core gains
mass due to nuclear burning. It also loses mass at its outer boundary - at the
mass loss rate of the star. Thus
E / 1 \
Mt. — —Mc + M = —■— a I. = — E I — + a 1 .
286 Appendix C Solutions to all the exercises

Integrating and using the initial condition Me = Mq at t = 0, we have

Me(t) = Mo - L (Ex. 13)

(c) The core mass attained when the envelope mass is exhausted is obtained by
setting Mc(z) = 0 and eliminating t between Equations (Ex.12) and (Ex.13),

'W«
M,. = ---------
1 + aQ
(d) For the star to become a white dwarf this core mass must satisfy Me < Mch,
which imposes an upper limit on the initial mass of the star:

Mo < MCh(l +a<2) Mo < 9 Mo.

Exercise 8.1: The equation of state for the gas is

7?
Pgas —

and the constant temperature (dT/dr = 0) implies F = 0. Thus substituting u = vs in

Equation (8.20), we have
27?.T0 <^M
prc rc ? ''

and hence
_ GMii
'c “ 27?.To ’

The isothermal sound speed is given by

t’s = <7? To/ /I.

while the escape velocity at the critical point is

b’ese = y/2GM/re.

Substituting the expression above for rc, we obtain

Vs = ^esc-

Exercise 8.2:

(a) For a polytropic pressure (implying vanishing radiation pressure), we have with
the aid of Equation (8.18),

dP \dp_ n 1 (1P 1 dv 2\
— =yKp Y~
dr dr
= yP--r
p dr
-yP
v dr
+- .
r/
 Appendix C Solutions to all the exercises 287

and substituting it in the momentum equation, we obtain

y P \ dv 2yP GM
v - Tr
P pr r~

which is the same as Equation (8.20), if is defined as ■jyP/p.

(b) With the substitution

I dP _ y d_ Y P
p dr y — dr Tr Y - 1 P )'

the momentum equation (8.15)

dv 1 r/P GM
dr p dr —=°
may be integrated to yield
GM
I1' + —f- = A = constant.
Y - 1 P
On the other hand, the energy equation (8.16) reads

5P GM\
-------------- + F = constant.
2 P r )

Combining these equations, we obtain

■ r p 5 Y
M A+- + /■' = constant.
P 2 y - 1

The term in parentheses is (3y — 5)/2(y — 1) and vanishes for y = 5/3. In this
case, since M and A are constants, it follows that F = constant, which means
that the flow does not absorb nor release heat, that is, the flow is adiabatic.

Exercise 8.3: The wind emanated by the Sun crosses any spherical surface centred on the
Sun (just as the radiation emitted by the Sun does); otherwise matter would accumulate at
some place; hence m = constant. Conservation of mass (in spherical symmetry) requires
that an amount of mass 8m crossing a spherical surface of radius r during a time interval
8t equal the density at r multiplied by the volume of this mass, 5 V — 4rtr28r. Since
8r = v8t, where v is the (radial) velocity of the wind, we have at any distance r

8m = 4xr2pv8t.

Dividing by <5/, we obtain

m — 4rtr2pv.

As the contribution of electrons to the mass (density) is negligible, we may assume the
wind density to be p » npwtH, where np is the proton number density. The measurements
at Earth (r = 1 AU) thus yield m 1.3 x IO9 kgs-1 ~2x 10~l4Afo yr-1.
288 Appendix C Solutions to all the exercises

Exercise 8.4:

(a) The mass-loss timescale may be estimated by M/M (Equation (2.55)). The
thermal timescale is given by Equation (2.59), r,h » GM2/RL. Using Equation
(8.31) for M, we obtain
_ M _ I c GM2 1 c
r"’-' “ m ~ 0 twT'h'

Generally, vesc << c and certainly 0vesc c; therefore we may conclude that
Tn—1 Th-

(b) The energy required for removing an amount of mass 8m from the surface of a star
is equal to the gravitational binding energy of this element, <5£grav = GM8m/R
and if the mass is removed during a time interval <5z, then the rate of energy
supply (<5Egrav/<5t) is

where M was substituted from Equation (8.31). As argued in (a), £grav <£ L.
(c) From estimate (2.61), rnuc eMc2/L, where e amounts to a few times 0.001.
Using the result of (a), we have
Tn-I = 1 1 GM
Tiuc 0 ttCscC 6 R
Substituting on the right-hand side GM / R = r’jsc/2, we obtain
Tn—I 1 tVsc 1

Tiuc 0 t 2e
If t'esc < 0.001c (as is mostly the case) and if 0 is not a too small fraction (as,
indeed, observations indicate), then r,n_| < rnuc.

Exercise 8.5: In Section 7.4 we have seen that, for main-sequence stars, global quantities
may be expressed as power laws ofthe stellar mass. These may be easily reverted to power
laws of the luminosity. Thus

L <x Ma' M a £,/a‘.

R <x Mu: => R ex La2/a'.

A parametrization of the mass-loss rate of the form (8.31) would result in

M ex —— oc .
GM
Using the results of Section 7.4, we have oq = 3 (relation (7.35)) and cn = (« — 1)/
(n -I- 3) (relation (7.36)). whence

M CX £<3n+5)/(3’’+9)
 Appendix C Solutions to all the exercises 289

We note that this is very close to a linear dependence, particularly for massive stars, which
burn hydrogen by means of the CNO cycle (n 16).

Exercise 9.1:

(a) Assume n helium nuclei are produced in the Sun per unit time, of which n, are
produced by the p — p 1 chain, nj by the p — p 11 chain and by the p — p
111 chain. Thus n = n\ + n2 + «3 and the branching ratios are ip/n (1 < i < 3),
respectively. The neutrino fluxes intercepted at Earth, fvi (I < i < 3)- listed in
the second column of Table 9.3 - are a fraction a = (4nd2)~l (where d = 1 AU)
of those produced per unit time in the Sun. In the production of a helium nucleus
by the p — p 1 chain, two p — p neutrinos are emitted; by the p — p II chain,
one p — p neutrino and one Be neutrino, and by the p — p III chain, one p — p
neutrino and one SB neutrino (see Section 4.3). Therefore

A.1 = a(2ni +«2 +«.O

fv.i = otn2

fV3 = a'i 3

fv.1 + fv.2 + A.3 = 2an.

Eliminating W; from these relations we obtain the branching ratios:

21 -= JvA---- Aj---- As3 _ 0 85 (p- pl)

n A.1+A.2 + A.3
H2. 2A-2
= , , *0.15 (p pH)
n A.l + A.2 + Jv.3
Il 3 _ 9r
= ------- —---- — ~2 x IO-4 (p-p III).
n A. 1 + fv.2 + fv.3
(b) The average energy carried by each neutrino type, Qvj is listed in the last column
of Table 9.3. The neutrino luminosity of the Sun is given by the total neutrino
energy flux at Earth, multiplied by 4.tc/2:

= 4tt</2(/,,i Cv.i + ,I\.,2Q>.2 + A.3CA3) = 8.9 x 1024 Js"1 = 0.023Lo.

(c) If the branching ratios of the p — p chain were not known, then the neutrino
energy lost for each helium nucleus produced would vary between a minimum
value of 2 x 0.263 MeV (corresponding to the p — p I chain) and a maximum
value of (0.263 + 7.2) MeV (corresponding to the p — p III chain). The net
energy released in the production of a helium nucleus (that would ultimately
be radiated by the Sun) would range between (2 max = 26.73 — 2 x 0.263 =
290 Appendix C Solutions to all the exercises

26.20 MeV and (2min = 26.73 — 0.263 — 7.2 = 19.27 MeV. Since the luminos­
ity of the Sun is known, the number of helium nuclei that should be produced
per unit time in order to supply it can be calculated in each case. The number of
neutrinos emitted is twice as much. Therefore
«.nin = 2-^ = 1.84 X 1038S-'
max

«,„ax = 2-^- = 2.50 x 1038 s 1

(Zinin

Exercise 9.2: First, we integrate Equation (5.2) in order to obtain the core mass M\.

(Ex.14)

Next, we integrate Equation (5.2) in order to obtain the mass outside the core:

Dividing Equation (Ex. 15) by Equation (Ex. 14) and substituting x, = pjp\ and y, =
M j Mi, we have

Now, since R\ < R. we may neglect (7?|//?)' with respect to 1 in the denominator on the
left-hand side; hence exponentiating, we obtain

_ % e((.V|-l)(2.V|+3)+5|/l5
/?!

which yields R/Ry ~ 3 x IO4 forxi = 10 and yi = 7.5. Thus, if the core radius is of the
order of a white dwarf’s, /?i ~0.01 RQ, the resulting stellarradius is ~ 300 illustrating
the possibility of having a compact core and a very extended envelope.

Exercise 9.3:

(a) In the outer layer of a white dwarf we have by Equation (9.36) P — P(T). We
may thus write the equation of hydrostatic equilibrium (9.32) as
dP dT _ GM
TtTt ~ ~P~'
Using the ideal gas equation of state (appropriate to this layer), we substitute
p = (/i/'R)(P/T) to obtain
d In P dT p GM
(Ex. 16)
d In T 77 ~ ~7~
 Appendix C Solutions to all the exercises 291

From Equation (9.36) we have

c/ In P _ 17
d\nT ~ T’
and hence, integrating Equation (Ex. 16) and using the boundary condition
T(R) = 0, we obtain the required relation (9.42).
(b) We may write this relation for Tc = Tb = T(rb) in the form
7?.. _ 4 GM R-rb
(Ex.17)
~i c ~ T7 ~r 7T-
The left-hand side represents (roughly) the ion energy per unit mass, P\/p. The
term GM j R on the right-hand side is, according to the virial theorem, the total
energy per unit mass (or P/p). Since for degenerate electrons, P % Pc )§> Pb
we must have

R - rb « rb < R.

(c) Since we have shown that £ = P — rb « P, then rb P and we may write

Equation (Ex.17) as
4 p GM „
Tc = — => Tc <x €,
17 7?. P2
Using relation (9.39) between L and Tc, we obtain (. oc L2’7, and hence
2/7
€| (Ly
IL2 = 1002/7 % 3.7.

Exercise 10.1: The equation of motion for free fall (a motion governed by the gravitational
field without any - or with negligible - opposition exerted by pressure) is, according to
Equation (2.12),
Gm
r(m. t) = -
r(m. t)2'
Multiplying both sides by r(m. t), we obtain

[p2(»t. /)]’ = Gm[\/r(m, /)]',

or, since m and t are independent variables,

rp:(m, t) - Gm/r(m. r)]' = 0.

Integrating, we have

t) - Gm/r(m. t) — —C.

where —C is an integration constant (independent of time). If the collapse starts from

rest, that is, r(m. 0) - 0 everywhere, then C = Gm/r(m, 0). We choose C = 0. implying
collapse from a very extended initial configuration. All solutions will converge with
time to that corresponding to C = 0, since the term Gm/r(m, t), which increases with
292 Appendix C Solutions to all the exercises

time, will eventually become dominant. For a uniform density, in — y-r(m, t^p, where
p = p(t). and hence
&nGp(t)
r2(m, /) = (Ex.18)
3
Therefore

= — v/8ttGp(/)/3 r(in, t).

where we have chosen the negative root, appropriate to collapse. This shows that at any
given time the velocity changes linearly with distance from the centre.

Note: The same equation of motion applies to the universe (in the Newtonian
approach) and describes its expansion - when the positive root of Equation (Ex. 18) is
chosen. The resulting linear dependence of velocity on distance - describing the relative
motion of galaxies - is known as the Hubble law, which was first discovered from
observations.

Exercise 10.2: We proceed as in Exercise 10.1 to obtain the first integral of the equation
of motion.

where r = r(m.t) and ro = r(m.O). From the condition of uniform initial density, which
we denote by po, we have
4,t 3
- —r0Po.

Substituting m in the former relation, we obtain

8?r Gw2
(Ex. 19)
3
where we have chosen the negative root to describe the collapse. In order to solve this
equation, we introduce a new variable, x(m, r ). defined by
COS2 A- = -,
f()

noting that x = 0 at t = 0. We also define a constant K = ^/SzrGpo/S. It is easy to see

that Equation (Ex. 19) becomes

x cos2 x = IK,

which may be directly integrated to yield

x 4- 5 sin 2x = Kt.
 Appendix C Solutions to all the exercises 293

Now, the solution x(/), or r/ro, is the same for all in. meaning that any part of the core will
take the same amount of time to contract to a given fraction of its former radial distance
from the centre. The density will thus remain uniform. It is noteworthy that the time of
collapse is finite: when r(m, t) = 0, x = tt/2 and t = n/2K (which is of the order of the
dynamical timescale 1 /y/Gpo). Hence the solution has a singularity, the density becoming
infinite at t — n/2K.

Exercise 10.3: Let N\(t) denote the number of 56Ni nuclei, initially /Vo = IA/e/56ffln —
2.15 x 1055, and A^Iz), the number of 56Co nuclei, initially 0. The characteristic decay
time is obtained from the half-life time by r = Z|y2/ln2, which yields T| = 8.8 days for
56Ni and T2 = 111 days for 56Co. Using the mass-excess table (Appendix D), we obtain
energy release in the amount 0, = 2.136 MeV for 56Ni ->■ 56Co and 02 = 4.564 MeV
for 56Co -> 56Fc. The rate of decay of 56Ni is given by Ń\ — leading with the
initial condition to /Vj (/) = Noe_'/ri. The rate of change of 56Co - by build-up from the
decay of 56Ni and its own decay - is given by

N2 = /V|/t, - N2/t2,

leading with the corresponding initial condition to the solution

,V2(Z) = /Vo(e“'/r' - e-'/T-)T2/(T2 - n).

The luminosity, resulting from the decay of both 5flNi and v’Co, is given by

L(t) = V|(z)£?|/ri + N2(t)Q2/r2.

Substituting the expressions for the numbers of nuclei, we obtain

and inserting parameter values.

L(t) = 2.5 x IOl()e"'/T,|l +0.184(eIL6,/r2 - 1)]L0.

Exercise 11.1: Using Equations (11.2), we may express d in terms of d\:

A7,
d = d\ d~> = d\ T di---- — d| (<V71 T ;V/■>)/ Mi.
M2
Substituting the result in the equation of motion for the first star. Equation (11.3 )-left, we
obtain an equation of the form (11.4), where (M\ + M2) in the numerator is replaced by
+ M2)2. The same procedure may be applied to the second star.
Since the stars and the centre of mass (focal point of the orbits) are always aligned,
both stars pass at the same time through pericentre (shortest distance from the centre) and
through apocentre (longest distance from the centre). In each case - as at every point of
the stars’ orbits - their respective distances from the centre relate as the inverse mass ratio
294 Appendix C Solutions to all the exercises

(by Equation (11.2)-left). Denoting by «i and «2 the respective semi-major axes and by
e\ and e2. the eccentricities, we thus have

«i(l ~<'t) _ ^2
a2(l - e2) M\

«i(l +f|) M2
<r2( 1 + e2) ;W|

Equating the left-hand sides of these equations, we obtain e\ — e2.

Exercise 11.2: Let M be the total mass of the system, M = M\ + ,W2. and a the separation.
Denoting by rq and ri2 the distances ofthe two stars from the centre of mass, respectively,
wc have by Equations (11.2) a = a\ + a2,(i\/a — M2jM and«2/« — M\/M. Let rube the
common angular velocity of the stars (which is constant for circular orbits); w = 2tt/P.
Denoting by V| and v2 the velocities of the two stars, respectively, we have = rq sin i
and v„,2 = v2 sin r. In addition.
2.t«i 2tt 2rta2
and
i't CO 1’2 co

and hence
M2 _ «!_ _ U|_ U„.|

«2 1'2 1-O.2

which provides one relation between the desired masses and observables. Another relation
is obtained from the equations of motion of the two stars,

, GM,M2 2 GM2Mt
=----- -— and = -———,
a- a-
which we may add to obtain:

2 GM
<i>~ — ——

(Kepler’s third law). Substituting a — ci\ + «2 = iq/ru + u2/rw = (u„ i + n(,2)/(<usin/),

we have

Exercise 11.3:

(a) The Roche-lobe radius of M2 is given by Equation (11.25) in the form

/ <W2 \l/3
= 0.5a
\ M| 4~ A/2 /
 Appendix C Solutions to all the exercises 295

Conservative mass transfer means that (M| + Mi) = constant, and hence taking
the logarithmic time derivative of this relation, we obtain

ft ii 1 M2 it I Mi
rL « 3 Mi a 3 Mi
since M2 = — M\. For the first term on the right-hand side we use Equation
(11.32), which reads
it M1
- =2(1 -q)-±
a M2
to obtain
4 Z5 \ Ml
— = ~ — 2« -—.
rL \3 7 W2
Shrinkage of the Roche lobe means rL < 0, which requires the term in parentheses
on the right-hand side to be negative, since M| is positive. Thus the condition on
the mass ratio is q1 > 7.
o
(b) Assuming a relation between mass and radius of the form R oc M~^", we have

/?-> 1 M2 1 M,
— =------- - =----- - > 0.
Ri n Mi n Mi
Clearly, if q is such that its Roche lobe shrinks (the condition found above), the
star will keep overflowing its Roche lobe while losing mass. If the Roche lobe
expands, on the other hand, then the lobe radius must increase at a slower rate
than the rate of growth of the stellar radius. If initially Ri ~ rL, the requirement

ń. >
n Ri
yields q > (5/3 — \./n)/2 as the condition for the donor star to continue over­
flowing its Roche lobe, which is less restrictive than q >

Exercise 11.4: Applying to the white-dwarf centre the equation of state for degenerate
matter to first order in temperature (B.37), we have

\ Pc
where
fi2 (3.t2)2/3 _ 5m}k- (7?; 11Ate)4'3
5mc (»iHMc)5/3 a 6fi4 (37T2)1/3

and 2. Substituting this expression into Equation (5.28) that gives the central pressure
in hydrostatic equilibrium with 13\ 5 = 0.206, we obtain a relation between pc, Tc and M.
of the form

fip\li + af3 — = yM2/3,

Pc
296 Appendix C Solutions to all the exercises

where y — (4.T)1 '0.206G. Taking the time derivative, we obtain

JL <W = +
2pt \ 3 pc I dt Pc dt 3 dt

Since a is a very small number, deriving from the small correction to the degenerate
equation of state due to temperature, the coefficient of the density derivative on the left­
hand side is positive for (almost) any temperature and density values. The terms on the
right-hand side have opposite signs, since both dTJdt and dM/dt are negative. Since Tc
changes rapidly with time at the beginning and very slowly thereafter (sec Section 9.8),
while the mass decreases at a constant rate, the left-hand side will change sign from
positive to negative at some point and thus the density will go through a maximum.
We note, however, that for a sufficiently high rate of mass loss, the second term on the
right-hand side will always dominate and the central density will decrease monotonically.
On the other hand, without mass loss, the central density will increase steadily with
decreasing temperature, tending asymptotically to the value (7.13), which depends solely
on the stellar mass.

Exercise I 1.5:

(a) Assume a white dwarf of mass M and radius /?(M) has an outer layer of solarcom­
position and of mass Ahi <K M (and negligible thickness). The energy required
to expel this layer is equal to the gravitational binding energy GM&m/R(M). If
Q is the energy released per unit mass of burnt hydrogen (from Section 4.3,
Q * 6 x 10l4Jkg_|), and the hydrogen mass fraction in the outer layer is
X0 % 0.7, then the amount of hydrogen mass burnt is f/MnX, satisfying

GM t\m
= f^mXQQ.
R(M)

(b) The R(M) relationship (5.29) for white dwarfs, appropriate to a nonrelativistic
equation of state, that is, for M < M(:h, may be calibrated with the aid of the
provided data:

R
0.01/?0

Combining these results, we have

GMq
0.01 R. XqQ

Note that for typical white dwarf masses this fraction is very small, despite the
strong gravitational field that must be overcome.
 Appendix C Solutions to all the exercises 297

Exercise 12.1: Consider a cloud of mass equal to the Jeans mass A/j and temperature T.
According to Equation (12.4), its radius is
a iiGMj „„
/? = -------- -. (Ex.20)
3 7?T
The rate of gravitational energy release in collapse may be estimated by the potential
gravitational energy, of the order of G Mj IR (Equation (2.27)), divided by the free-fall or
dynamical time (Equation (2.56)). Thus,

g R \ R3 J J
Since the radiation temperature is lower than the gas temperature T. the rate at which
energy is radiated at the cloud’s surface, or the cloud's luminosity L, may be taken as

L = e^R-(yT\

where e < 1. As the radiated energy is supplied by the gravitational energy released in
collapse, we have

c4.t/?2<7 7'4 = aV2G3/2Mj/2R~5/2. (Ex.21)

Substituting Equation (Ex.20) into Equation (Ex.21) yields

r4.0(K/M)9/4l 7’1'4
J L o'-'^G3/2 J el/2

Since e < 1, taking 6=1 on the right-hand side provides an approximate lower limit
Af, ^5.6 x 1O-3Ti/4A/0.

Exercise 12.2: From Equation (12.13) we have

>1 _ (M^35 - M-™) - A/wd(Mm^5 - Ms-J-35)^
C
Substituting A/m$ = O.7A/0, Afsis = lOA/0 and A/wd = O.6Af0, we obtain
2 = O.89(Afmax/A/0)035 - 1
< (Mmax/Minin)0^ - 1 ’
which yields for A/„lax = 30 A70:
- = 0.23 (A/,nin = O.O5Af0) ’j = 0.40 (Afmin = O.2OAf0),

while for Afinax = !2OAf0:

= 0.26 (Afmin = ().05Afo) 'j = 0.45 (A7min = ().2O/W0).

In conclusion, the ratio >//( is far more sensitive to A/,njn than to A7max.

Exercise 12.3: As we have seen in Section 1.4, and again in Section 7.4. the luminosity of
main-sequence stars is a function of the stellar mass in the form of a power law, L a M".
298 Appendix C Solutions to all the exercises

If the cluster’s luminosity /< is the sum of the luminosities of its main-sequence stars,
which have masses in the range /Wmin < M < Mtp, then

yjW,p /..W,p
Lc= L(M)dN = L(M)Q(M)dM oc Mv~235dM.
‘^•'^min ‘'Minin ” Mmin

The relative change in Lę from Z-c.i, say< to ^c.2, as the main-sequence turnoff point
decreases from Mlp.i — 1.3Af.s to Mtp,2 = 0.85MQ, is given by

Lc.2 = f^ Mv-23SdM = (Mtp.2/Mmin)‘-'-35 - 1 % ZM,p.2y-135

/-c.i j'"'' Mv~235dM (^p.i/M™,,)1’-135 - 1 ~~ \ ,WlpJ )

Thus the cluster’s luminosity decreases by a factor of ~2, if we adopt v = 3, and by a

factor of ~5. if v = 5.

Exercise 12.4: The function Y(r) satisfies the equation

Y = -ctY2,

where a is a constant to be determined from the given data. Integrating and using the
initial condition Y(0) = 1, we have

1
y - 1 = CUt.

Substituting Y(rp) = Yp = 0.05, we may eliminate a to obtain

Substituting Y = 0.5 yields t/tp = 0.053, meaning that when the Galaxy was ~5% of its
present age. the gas content amounted to half the galactic mass. It decreased to a tenth of
the galactic mass when the Galaxy reached about half its present age. Decreasing further,
it will reach half its present mass (that is, Y = 0.025) when the Galaxy will be about
twice its present age.

Exercise 12.5: Substituting the opacity law and the nuclear energy generation rate in
Equations (7.31) and (7.32), we obtain as in Exercise 7.4,

c ac T2R*
F* —-------------
/C()Z p.M

F. = q0Zp.T"M.

Using relations (Ex.9) and retaining the dependence on Z, we have

,w5.5 ZM"+2
F. oc---- —T and /■„ oc--------—.
Z/?05 Rn+3
 Appendix C Solutions to all the exercises 299

which, combined, yield the dependence of radius on mass and heavy clement content in
the form
^2/(n+2.5) ;^<.n—3.5)/(n+2.5)

Substituting this relation back into either of the relations for F above, we derive the
dependence of the luminosity L on Z, for a given mass:

L <x Z"("+3'5)/("+2'5).

Similarly, substitution of R(Z) in relations (Ex.9) yields

T. <x z-2/(',+2-5) and p, <x z~6/<"+2'5).

Thus, for main-sequence stars of the same mass but different Z, the higher-Z star will have
lower internal temperatures and densities and a lower luminosity. Hence, it will spend
a longer time as a main-sequence star. Our conclusion is based on highly simplifying
assumptions; nevertheless, it is generally true that Pop I stars have considerably longer
main-sequence lifetimes than Pop II stars.
 Appendix D

Physical and astronomical

constants and conversion factors

Table D. I Fundamental constants

Units

Constant Symbol Value SI cg.v

Speed of light c 2.99792458 10s m s-1 I0locms 1

Permeability JUO 4/r 10 7C”2Ns2 1
Permittivity so 1/4,7 107/c2C2N"1 m~2 1
Gravitational G 6.67259 10-"m3kg-'s-2 l(rscm'g 1 s 2
Planck h 6.6260755 10 ,4Js IO-27 ergs
Boltzmann k 1.3X0658 io-2'jk 1 10 16 erg K~'
Stefan-Boltzmann O’ 5.67051 IO-8Jm-2s-1 K~4 10 5 erg cm-2 s~' K"4
Radiation a 7.5646 IO"l6Jm“3 K 4 10 l5ergcm-3K-4
Wien 2.897756 lO-’mK 10 1 cm K
Avogadro A'a 6.0221367 IO23 mor1 1023 inol 1
Atomic mass unit "tH 1.6605402 IO-27 kg io-24g
Ideal gas 7? 8.314510 lO-’Jkg'1 K 1 KFergg 1 K“'
Electron charge e 1.60217733 10-|9C 10 20cesu
Electron mass mc 9.109.3897 IO"31 kg 10 2Sg
Proton mass nip 1.6726231 10-27 kg 10 24 g
Neutron mass m„ 1.6749286 10 27 kg io-24g
Angstrom A 1 10 1,1 m 10"8 cm

Note: a = Aa/c, mH = l/NA, R. = k/m^. Fundamental constants arc from E. R. Cohen and
B. N. Taylor. (1987), Rev. Mod. Phys. 59. p. 1121; CODATA Bulletin (1986). 63 (Nov.); Physics
Today (1995). Part 2. BG9 (Aug.).

300
 Appendix D Physical and astronomical constants and conversion factors 301

Table D.2 Astronomical constants

Units

Constant Symbol Value SI Cg5

Solar mass M 1.9891 IO30 kg 1033 g

Solar radius 6.9598 10s m IO10 cm
Solar luminosity I- 3.8515 lO^Js- 1 1033ergs_|
Year (solar) yr 3.1558 107s 107s
Light-year >y 9.463 10l5m 10l7cm
Parsec PC 3.086 1016m 10l8cm
Astronomical Unit AU 1.496 10"m IO13 cm
Earth mass 5.976 1024kg 1027g
Earth radius 6.378 10('m 10s cm

Note: Astronomical constants are from C. Caso et al., (1998),

European Physical Journal. C3. p. 1.

Table D.3 Energy conversion factors

Units erg eV 5-' cm 1 K

erg 1 1.602177331-12) 6.6260755(-27) I.9864475(-I6) 1.3806581-16)

eV 6.2415064(1 1) 1 4.1356692(-15) 1.23984244(-4) 8.617385(-5)
s_| 1.50918897(26) 2.41798836(14) 1 2.99792458(10) 2.083674(10)
cm 1 5.0341125(15) 8.0655410(3) 3.3356409521-11) 1 6.950387(1)
K 7.242924(15) 1.160445(4) 4.7992161-11) 1.438769 1

Note: Powers of 10 are given in parentheses. The units of energy are related as follows:
I J = IO7 erg; I erg = 1/eeV = 1/hs 1 = l/(/tc)cm 1 = \/k K. Energy conversion factors
are from E. R.Cohen & B.N.Taylor. Rev. Mod. Phys. 59. p. 1121 (1987); CODATA Bulletin.
63 (Nov. 1986); Physics Today, Part 2, BG9 (Aug. 1995). Values within the same column are
equivalent.
302 Appendix D Physical and astronomical constants and conversion factors

Table D.4 Mass excesses, asterisks indicating unstable isotopes.

z Element A A.M (MeV) 7 Element A AA4 (MeV)

0 n 1 8.071 13 Al 26* -12.210

1 II 1 7.289 27 -17.197
D 2 13.136 14 Si 28 -21.493
2 He 3 14.931 29 -21.895
4 2.425 30 -24.433
3 Li 6 14.086 15 P 31 -24.441
7 14.908 16 s 31* -19.045
4 Be 9 11.348 32 -26.016
5 B 10 12.051 33 -26.586
11 8.668 34 -29.932
6 C 12 0 17 Cl 35 -29.013
13 3.125 37 -31.761
14* 3.020 18 Ar 37* -30.948
7 N 13* 5.345 40 -35.039
14 2.863 19 K 39 -33.806
15 0.101 40* -33.534
8 0 15* 2.855 25 Mn 55 -57.708
16 -4.737 26 Fe 56 -60.601
17 -0.809 27 Co 56’ -56.037
18 -0.782 59 -62.224
9 F 17* 1.951 28 Ni 56’ -53.901
19 -1.487 58 -60.223
10 Ne 20 -7.042 60 -64.470
21 -5.732 31 Ga 69 -69.322
22 -8.024 71 -70.139
II Na 23 -9.530 32 Gc 70 -70.561
12 Mg 23* -5.473 71* -69.906
24 -13.933 72 -72.583
25 -13.193 73 -71.295
26 -16.215 74 -73.423

Note: Published by J. K. Tuli, National Nuclear Data Center, Brookhaven National Laboratory.
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 Index

accretion Bethe, Hans A.. 57. 59. 61. 126

disc. 220-223 Biermann, Ludwig, 100. 135
luminosity. 214. 215, 228 binary
rate. 214. 215 astrometric. 210
adiabatic exponent. 45-46. 93. 102-103. 147. contact. 218
205. 280-283 detached. 218
adiabatic gradient. 102. 145. 282 eclipsing, 211,214
adiabatic process. 44. 93. 97. 281 semidetached, 218
age spectroscopic. 211,214
stellar cluster, 121. 154 visual. 210
Sun, 30, 153. 245 black dwarf. 182, 236
universe. 33. 154. 155. 243 black hole, 200, 205. 215, 238, 243
Algol blackbody radiation. 4. 16. 42, 49. 50. 124.
binary. 211.214 215,'253
paradox. 213,214. 220 Boltzmann constant, 22, 300
aluminium, radioactive, 199 Boltzmann formula, 253
Andromeda galaxy. 189 Bondi. Hermann, 135
angular momentum. 17, 201-202. 219 boundary conditions, 29, 71,73, 135
antineutrino. 26, 66 Brahe. Tycho, 191
apparent brightness. 2. 155. 167 bremsstrahlung. 48. 53
Archimedes buoyancy principle (law), 97. 101 brown dwarf, 236-238, 240, 243
argon, radioactive, 157 Burbidge, Geoffrey R., 67
Aristotle, 191 Burbidgc, Margaret E.. 67
Astronomical Unit (AU), 3. 301
asymptotic giant branch ( AGB). 168-173. 174. carbon. 6. 26. 59. 61-6.3, 66. I 15, 125, 145. 185
177 burning. 63-65, 108, 185
Atkinson. Robert. 55 detonation. I 15. 1 16. 204-205
atomic mass unit. 26. 52. 300 carbon-oxygen core. 125, 168, 171-174, 176
atomic nucleus. 25-28. 53 54, 66-67. 68. 200 cataclysmic variable, 220, 223
Cepheid, 166-167. 189
Baade. Walter, 200, 246 period-luminosity relation. 167
Bahcall, John N„ 156. 160 Chamberlin. Thomas C., 32
baryon. 25 Chandrasekhar, Subramanyan. 71,77. 85. 135,
number. 25.26. 54 160, 162. 282
Bell. Jocelyn, 201 mass. 77-78. 94. I 12. I 15. I 16. 127. 174.
P decay. 53, 55. 60. 61.67. 198, 226 191. 193. 200. 229-230. 278

308
 Index 309

charge (atomic number). 25. 35. 38, 55. 56. deuterium. 53, 57, 59, 158, 236
67 burning, 236
Cherenkov light. 157. 159 diffusion approximation. 256
Clayton. Donald D., 57, 68 dimensional analysis, 117
cluster Doppler
globular. 164. 246 effect. 8. 130.211
Hyades, 13. 154 shift. 211,2)4, 238
M3. 13. 154 dredge-up, 153, 164, 168, 171
Pleiades, 12. 13. 154
stellar. 11. 12. 13. 121. 127-128. 154. 155. Earth, 10. 16. 156, 191. 196, 202, 238. 287
187. 233 Eddington. Sir Arthur S., 15. 32. 34. 50. 71. 82.
47 Tucanae, 12 83. 188. 256. 282
CNO cycle (bi-cycle), 59-61. 114. 119. 162. luminosity. 78-80, 122. 135, 142. 172. 183.
225-226 226. 229, 283
cobalt. 198. 204 quartic equation. 81. 82
Colgate, Stirling A.. 195 standard model. 80-83. 100. 127
collapse. 30, 94, 103. I 15. 116. 148. 185. 194. effective temperature. 4, 9. 116. 120, 121. 149,
200. 233. 291-293 163, 166, 182.212
comet. 1, 135. 191,224. 247 Eggleton, Peter. 218
composition (abundances). 7. 25, 27. 58. 60. Einstein. Albert, 253
67. 72. 185. 245 General Theory of Relativity. 200, 206
cosmic. 62. 198 mass-energy relation. 32, 51
mass fraction. 6. 7. 16. 17. 26. 28. 39 radiation equation, 253
solar. 6. 7, 39. 49 relations, 254
conservation law electron-positron pair, 52, 67
angular momentum. 17. 202. 219 energy. 23-25. 88-89, 94, 163, 194
baryon number, 27. 52, 57 gravitational potential (binding), 8, 21-23.
charge. 27. 57 25, .31,88. 124. 148. 194. 214. 215. 234
energy. 17. 25, 67. 124, 134 kinetic. 24. 195. 203. 228
lepton number. 27. 57. 195 nuclear. 18. 33. 89. 120. 173, 194, 228
mass, 17, 18. 132-133 nuclear binding. 51-53.54
momentum. 19,67. 133 radiation. 43. 88. 94. 195. 228
continuity equation. 70. 132 rest-mass. 29. 32. 51.67. 200
convection. 79.96-102. 145. 152. 164. 168. specific (internal, per unit mass). 17. 43.
170. 226 44
Coulomb thermal (internal). 17.22.31. 124. 181.
barrier. 54. 55. 62-64. 66 234
energy. 35. 41. 272 energy equation, 17-19. 23. 52. 134
field (force). 26. 53 entropy. 44. 105
interaction. 35. 181 equation of motion. 19-21. 133.209-210
Cowling. Thomas G.. 84. 86. 99 equation of state. 35-37. 90. 105-107. 117.
point-source model. 83-85 146. 180
degenerate electron gas, 40-42. 259-269
dark matter (missing mass). 238. 240 ideal gas. 37-39. 263
Darwin. Charles. 32 polytropic. 73. 76.81. I 10
Davis. Raymond, 156. 160 equilibrium
chlorine experiment. 156-157 hydrostatic. 20. 21.25. 30. 70. 73. 76. 79.
Debye temperature, 181 90-92, 103, 110. 146. 148. 151. 232
degeneracy pressure. 40-42. 114. 164. 236. nuclear (statistical). 28.65-66. 108. 198
259-269 radiative. 79-80. I 17. 152
density thermal. 19. 25. 33. 70. 83. 89. 114. 120.
average. 2.3. 30. 35. 75, 163 149. 151, 17.3,214
central. 75-76. 90. 104. 110-113. 163 thermodynamic. 15-16. 37,42. 232, 253
310 Index

escape velocity. 30. 137. 139. 143, 205 flash, 164. 165. 171
Euler. Leonhard. 133 main sequence. 166
equation. 133 Helmholtz. Hermann von. 31
evolution equations, set of. 28, 33 Henyey, Louis, 144
Herschel. Sir Frederick William. 175. 208
Fowler, William A.. 67. 77 Hertz.sprung, Ejnar, 9. 167
fragmentation. 232. 234-235 gap. 162. 163
free fall. 148. 194. 291 Hewish. Anthony, 201
Hipparchus of Nicea, 3
galactic disc. 231.238. 246 Hipparcos satellite. 3
galactic halo. 246 homologous contraction, 194
Galaxy (Milky Way). 2. 127. 167. 190. 199. homology. 119. 162
215. 231.246 horizontal branch. 166. 168
galaxy. 2. 189. 193, 231.238. 243 Houtermans, Fritz. 55
y-ray. 199. 215 Hoyle. Sir Fred. 61.62. 67. 86. 125. 194. 201.
Gamow, George. 55 234
peak. 56 Hubble. Edwin P.. 189
gas law. 292
classical (nonrelativistic). 44 hydrogen. 6. 23. 53. 83. 183. 193. 198. 231
degenerate electron. 43. 44, 73. 77. 105. 113. 21-cm line. 232
114. 178. 193. 225, 259-269 atomic, 147.231-232
degenerate neutron. 194, 200 burning. 57-61.65. 108. 114. 116, 123. 125,
ideal (nondegenerate). 22. 25. 35, 37. 38. 44. 149. 151. 165. 169. 225-229
73.81.88, 105. 112. 146. 178.263 molecular, 147, 148
perfect (nonintcracting, free). 35. 41,43, hydrostatic (equilibrium) equation. 20, 47, 70,
181. 272 99. 111. 117. 146. 179
relativistic degenerate electron. 41.44. 73.
94. 107. 259-269 Iben, Icko Jr.. 144
gas constant, 38. 300 infrared radiation. 4
germanium, 158 initial mass function, 239-242, 245
Gold. Thomas. 201. 203 instability
Goodrickc. John, 214 convective. 97. 98-100
gravitation dynamical. 94-96. 109. 166. 191.205
acceleration. 47 Jeans. 234
constant. 8, 300 Schonberg-Chandrasekhar. 161. 164
force (field). 19. 30. 112. 142. 148, 206. 209. thermal. 89-92. 109. 171
213.216. 238. 291 thermonuclear, 115, 116. 226
thin shell. 90-92. 171.225
H-R diagram. 9-14. 85. 116. 144. 150, 152. interstellar medium. 7. 231-232, 241,243,
154, 155, 162-164, 166, 169, 182-183, 245
186-187. 196. 234. 236. 246 ionization. 23. 38. 45. 49. 68. 95. 99. 148. 166.
Harm. Richard. 91 232
Hayashi. Chushiro. 145 degree, 45
forbidden zone. 145-148. 164. 169 potential. 45, 148
track. 145. 151. 164. 236 iron. 53. 67. 83. 145
heavy element (metal). 7. 38. 39. 66. 153. 171. core. 1 16. 185. 191. 193. 195. 198. 200
197. 243-246 group, 65, 108, 185, 198, 204
Heisenberg uncertainty principle. 39 photodisintegration. 68-69. 95. 108. 109.
helium. 4. 6. 23.53. I 15. 168. 184. 185. 198. 115. 116. 194
231 isothermal core. 160-162. 170. 181
burning, 61-63. 65. 108. 1 14. 125. 165. 168,
169. 171. 185 Jeans. Sir James H„ 233
core, 123. 164. 165-168. 177, 183 mass. 233, 235
 Index 311

mass, minimum, 234, 236 mass, atomic, 26, 38, 53, 59

radius. 233 excess, 52. 59. 302
Jupiter, 35, 237, 238 mean, 38, 83, 119, 161. 179-181
unit. 26.52. 300
Kanicl, Shmuel, 98 mass, space variable. 8. 16-17.20. 70, 117-118
Kelvin, Lord (William Thomson), 31 mass, stellar, 4, 33, 35. 66. 73, 75. 77. 110-113.
Kelvin-Helmholtz timescale. 31, 102. 149. 117, 122, 152, 160, 205,211
162. 234 initial, 7. II. 128, 150, 153, 166. 174, 177,
Kepler, Johannes, 191 183. 193. 197.241
laws, 191,216,294 solar, 4. 301
Kirchhoff's law. 253 mass limit
Kopal, Zdenek, 218 Chandrasekhar, 77-78. 94. 112. 115. 116.
Kovetz, Attay, 98, 105 127, 174, 191, 193, 200, 229-230, 278
Kraft, Robert, 225 Jeans, 233. 235
Kramers, Hendrick A.. 49. 83 main sequence, lower, 121,236
opacity law, 49, 99, 122, 179, 285 main sequence, upper. 80, 122, 155, 284
neutron star, 200
Lagrangian point, 217-218. 220-222. Schonberg-Chandrasekhar, 161-162. 164
225 mass loss (wind)
Landau. Lev, 200 critical point, 136
Lane-Emden equation. 74. 75. 276 equations. 131-136
Laplace, Pierre-Simon, Marquis de, 206 rate, 139-143, 173-174
Large Magellanic Cloud (LMC). 190. 192. 196, solution, isothermal, 136-139
" 197 ” mass-luminosity relation, 12, 14, 80, 82, 83.
Leavitt. Henrietta Swan. 167 85, 117, 119, 122, 239
Lee. Tsung-Dao, 90 mass-radius relation, 76. 77. 78. 200. 237.
Legendre polynomials, 255 238
lepton. 26 Maxwellian velocity distribution, 15. 37. 40.
number, 26-7 54. 263
light-year, 3. 301 McCrea. Sir William H.. 142. 161
local thermodynamic equilibrium (LTE). 16. mean free path. 16. 47. 49. 53, 155, 232
255 mean free (collision) time. 16, 61
luminosity. 2. 4. 9. 11.31.49. 51, 81.83, 108, mean molecular weight, see mass, atomic,
116. 119-121, 149, 164-166, 170, 171, mean
180-182, 189, 195. 236, 239.244 Messier, Charles, 175
accretion. 214. 215. 228 Mestel. Leon. 90. 128, 178
Eddington. 78-80, 122. 135, 142. 172, 183, Michell, John, 206
226, 229. 283 Mira variable, 174
neutrino. 156, 289 mixing-length theory. 100-102
nuclear. 19. 32, 89, 91-92. 124-125 parameter, 101
solar, 2, 156. 301 molecular cloud. 232
Lyman, Theodore. 7 Montanari, Geminiano. 214
Lyttleton, Raymond A.. 88
nebula, 16. 149, 231,244
Magellanic Clouds. 167 Crab, 191. 192. 203
magnetic dipole radiation. 203 Eagle, 235
magnetic field. 8. 201-203 Pistol, 244
main sequence. 10. 11. 80. 116-122. 127. 149, neon. 6, 65. 185
151-155, 169. 170. 183.234, 239 Neptune, 238
fitting. 155 neutrino. 26. 52, 58-60. 72, 128, 145. 164. 185,
lifetime, 120. 153-154 195. 196
turnoff point. 154, 241.242 photoneutrino. 52
zero-age, 124 solar. 155-160
312 Index

neutrino experiments P-Cygni line. 130-131

BOREXINO. 160 Pacini. Franco. 203
Davis, chlorine. 156-157 Paczyński. Bohdan. 172
GALLEX. 158 Payne-Gaposhkin, Cecilia. 7. 224
I MB. 196 pair annihilation. 52, 67
Kamiokandc, 157. 196 pair production. 67. 95. 109. 115, 116. 205. 269
MINOS. 160 parallax, 3
SAGE. 158 Parker. Eugene, 135
SNO. 158-159 parsec. 3, 301
neutrino-antineutrino pair. 52 Pauli exclusion principle, 40, 260
neutron capture. 66-67, 171 photodisintegration. 65, 95
neutron star. 200-202. 205. 215. 242. iron, 68-69.95. 108. 109. 115, I 16. 194
243 photoionization. 48. 65
Newton. Sir Isaac. 18.51. 192.206 photosphere. 4. 38. 47. 140. 145-146. 179, 234
second law. 19, 50 Pickering, Edward. 214
third law. 209 Planck constant, 15.300
nickel. 198. 204 Planck distribution function. 16, 42,48. 67,
nitrogen, 6, 59, 168, 184 253, 255
Nobel Prize. 61.77. 160, 201 planet. I. 35. 149. 191.236-238, 240, 247
nova. 90. 189. 223-229. 243 planetary nebula. 175-177, 181,231,243
T Pyxidis. 244 Helix nebula. 176
Vul 1670. 225 Hcnizc 1357 nebula. 244
nuclear burning, 65. 79. 96. 104, 108-109. 164. nucleus (PNN), 175. 177
177 plasmon, 53
nuclear energy production rate, 18, 52, 56. 71. polytrope, 73-76. 77. 103. 1 10. 145, 147, 182,
79, 99, 108 200, 278
nuclear reaction. 25-27. 51-53. 57-67. 72. 83. polytropic index. 73, 110. 147
85-86 Population 1, II. 246
branching ratio. 58. 59.64. 156.289 positron, 26. 52. 53. 68
rate. 27. 52. 53. 56. 89. 108 p - p chain. 57-59. 114, 119. 121. 156. 289
resonant, 56. 62 pre-main-sequence, 145, 148
r-process. 66-67, 68 pressure,
.V-process, 66-67. 69. 171 central. 21,76, 90. 110
nucleosynthesis, 56, 69. 108. 197-200 electron. 38-42, 180
gas. 37. 39, 81 -83. 93. 96, 99-100. 135-136
opacity coefficient, 47-49. 50. 71,80, 83. 84. ion. 37-38
96.99. 149.237 radiation. 37. 42-43. 50, 81,83. 88. 94. 97.
bound-bound, 48 99, 107. 135-136.257
bound-free. 48 pressure integral, 35-36, 40. 41
electron-scattering, 48. 80. 122 proton mass. 25,41,300
free-free. 48-49 protostar. 148. 234
Kramers law. 49, 99, 122. 179. 285 pulsar. 200-203
reduced. 254 Crab. 201,203
Rosseland mean, 257 Vela. 201
Oppenheimer. Robert J.. 200
optical depth. 47. 140. 146.212 quasi-static process, 31, 70. 124
orbital motion, 209-210
circular orbit. 210. 219 radiation
eccentricity, 210 absorption. 4. 44, 46-48. 254
period. 216 emission. 4. 44. 48. 253, 254
semi-major axis. 210 energy, see energy, radiation
oxygen, 6. 59. 62, 66. I 15, 185. 205 field. 50. 52. 53, 65, 66. 142, 253
burning. 63-65. 109. 205 flux, 46-47, 50. 257
 Index 313

intensity, 251 shell flash, 171

momentum, 15.43.50. 143,257 shock wave. 143, 196. 198. 232
pressure, see pressure, radiation silicon. 65. 185
temperature, 16, 215, 258 burning, 65-66. 108
radiation constant. 42. 300 Small Magellanic Cloud (SMC). 167
radiative diffusion. 70, 79 spectrum. 4, 10, 130-131, 153. 169. 177. 193,
radiative transfer equation. 46-50. 70. 99, 135. 211
146, 251-258 speed
radiative zero solution. 180 light. 3, 15,41. 157. 205,300
radius sound (isothermal). 136, 286
Jeans. 233 thermal, 101,221
Schwarzschild. 215 spherical symmetry, 1.8-9. 17. 20. 72. 92. 132
solar. 4. 153, 301 stability-
stellar. 4, 10. 35.47. 74, 75.77. 90, 119, 149, convective. 96-98
163, 182, 202, 205. 215 dynamical, 92-93, 102, 115. 147, 232
Rakavy. Gideon. 106. 280 secular (thermal). 88-89,91
recombination. 48. 95. 232 standard candle. 167. 204. 228
red dwarf. 155, 236. 237 star, 1-2, 236-238
red giant. 10. 11. 116. 124-125. 140. 160-165, compact. 4. 8. 30, 77. 116. 215. 243
173 fully convective. 97. 145. 151. 179
branch. 125, 164. 166. 169 intermediate-mass, 155. 163, 166-170. 186
Reimers, Dieter. 173 low-mass, 85, 100. 114. 120. 145. 151, 155,
formula. 173, 174 164, 165, 177, 186. 236, 243
Riccioli, Giambattista, 208 main-sequence. 10. II, 14. 116. 117. 119.
Roche. Edouard Albert, 217 120, 140-141, 149, 152,215.239-243
equipotential, 217, 222 massive. 82. 114-115, 120, 127, 140. 153.
lobe. 218. 220-223. 225 155. 182-186. 191. 193, 200, 205, 239.
radius. 218. 223 243-245
Rosscland, Svcin, 257 star formation, 29. 232-235, 239. 243. 246
mean opacity, 257 stars, individual
rotation. 8. 201. 219 Algol (fl Persei). 214
RR Lyrae variable, 166 Capella. 77
Ruderman, Malvin, 178 Eta Carinae, 184, 207
Russell, Henry Norris, 7, 9 Mizar. 208
Ryle. Martin. 201 P Cygni, 130
Proxima Centauri. 3. 6
Saha, Meghnad, 45 Sirius. 210
equation, 45 Stefan-Boltzmann constant, 4. 300
Salpeter. Edwin E.. 61.62. 125, 239 stellar model. 71-72. 86. 99. 127, 144, 187
birth function. 239 point-source (Cowling). 83-85
Sandage. Allan R.. 125 polytropic, 72-76
Saturn. 238 standard (Eddington). 80-83. 100. 127
Schonberg. Mario. 160. 162 Stromgren, Bengt. 83, 85
Schonberg-Chandrasekhar limit, 161-162, strong nuclear force (interaction). 26, 55. 57.
164 59, 65. 200
Schatzman, Evry. 90 structure equations. 70-71, 117, 146, 179
Schwarzschild. Karl. 96. 206 Sun, 2. 4, 8. 10,21,33,38. 121, 131. 135, 151,
convective stability criterion. 96-97 201.239, 246, 258
radius, 215 supcradiabaticity, 102, 145
Schwarzschild. Martin. 13.91. 124. 125 supergiant, 125, 170, 173
self-gravity, 1.234 supernova. 30. 116. 155. 185. 189-193. 197.
Shapley, Harlow, 7 203.242
Shaviv. Giora. 105. 280 historical, 190
314 Index

supernova (cant.) thermal (Kelvin-Helmholtz), 31, 102, 149,

light curve. 196. 199. 205. 206 162, 234
nucleosynthesis, 197-200 triplc-o' (3a) reaction, 61-63, 108, 125, 184
paur-production, 205-207 tunnelling, 55
progenitor. 126. 185. 193. 197
remnant, 191,201,231,244 Uranus, 238
Type I. 193, 204-205. 229-230 UV radiation. 196. 215
Type 11. 193-197
supernovae, individual virial theorem. 21-23, 25, 31, 88. 94. 124. 149,
IC4182, 190 161,233,238
N132D. 192 Vogel, Hermann Carl. 214
SN19691, 196 Volkoff, George. 200
SN1987A, 196-199. 243.244
SN2006gy, 206-207 weak force (interaction). 26, 57, 195
yr 1054 (Crab). 192. 201 Weizsacker, Carl-Friedrich von. 59
superwind. 173-174 Wheeler. John Archibald. 203
synchrotron radiation, 202 white dwarf. 10, 11, 30, 100, 116, 127, 128,
165, 174, 177-192, 215, 225, 236,
T Tauri star, 149 241-243
Tayler. Roger J., 100 cooling, 179
temperature mass, 116. 127-128, 178
average. 23, 149 white dwarfs, individual
central, 90, 104, I 10-113. 121. 161-162. 40 Eridani B. 238
163, 180-181 Sirius B. 238
temperature-density diagram. 104-115, White. Richard H.. 195
122-123. 127. 236 Wien constant. 4. 300
thermal equilibrium equation, 19, 71. 83-84, Wien’s law, 4
117 wind
thermal pulse. 168-173, 177 equations, see mass loss (wind), equations
thermonuclear reaction, 27, 56 solar, 16. 135. 141. 153
thermonuclear runaway. 90. 103. 115, stellar. 79. 83. 128. 130, 142, 165. 170. 173,
226 177. 183, 184, 243, 244
threshold (ignition) temperature. 56. 66, 108, Wolf-Rayet star. 183. 184
114, 115, 125. 164.234
timescale. 29 X-ray, 4. 215, 258
dynamical. 29-31. 102. 148. 198, 205, 234.
293 Zeeman effect, 8
nuclear, 32-33 Zwicky. Fritz, 189, 200, 238