Stellar Evolution

Andrey Nikitin
May 2023

Table of Contents
1 Introduction 2

2 Background 2
2.1 Hydrostatic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 HR Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Stellar Interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Stellar Evolution 13
3.1 Protostars: A Star is Born . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Main Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Red Giant Branch (RGB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Horizontal Branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.5 Asymptotic Giant Branch (AGB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Dead Stars 18
4.1 Planetary Nebulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 White Dwarfs (WD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4 Neutron stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.5 Black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.6 Binary systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Binary stars 25
5.1 Types of binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2 Binary evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.3 Recurrent Novae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.4 Mathz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Conclusion 30

7 Resources 30

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1 Introduction
Stars are like people - they have a life that comes in distinct stages. Stars are ”born” from
clouds of gas, spending their adolescent years going through many changes. Then, they spend the
majority of their lives as an adult on the main sequence before they eventually meet their end.
Furthermore, where and how a star is born determines where they will end up. Some go out with
an explosive bang, while others slowly dim before they even had a chance to shine.
Together, these transitions are known collectively as stellar evolution (this differs from bi-
ological evolution in that it describes changes in a single individual rather than a population).
Since a star goes through many stages in their life, and these stages differ depending on the mass
and metallicity a star begins with, it can be daunting to try and understand their lives. This
handout will begin by explaining some general properties of stars. With these tools in hand, it
will be easier to understand and predict how a star evolves

• Bolded terms are important

• Gray text is advanced info and can be safely skipped without impacting understanding

2 Background
2.1 Hydrostatic Equilibrium
Stars are under the constant influence of two forces: gravity pulling inwards and radiation pres-
sure pushing outwards. When these two forces are balanced, the star is said to be in hydrostatic
equilibrium.
Some general trends:

• Radiation pressure increases with temperature. Radiation pressure is an outward

force caused by the momentum of photons produced in the stellar interior being transferred
to matter within the star, causing the star to expand. Higher temperatures means more
fusion occurs and more photons are produced, resulting in greater radiation pressure.

– In addition, as temperature increases, molecules in the Sun vibrate more, increasing

the gas pressure. Therefore, higher temperatures cause both radiation pressure and gas
pressure to increase, creating an outward force that pushes the star to expand.

• Getting smaller, getting hotter: As a star contracts, gravitational potential energy is

converted to kinetic energy, increasing the temperature (average kinetic energy) of the star.

– This is a result of the Virial theorem, which states that for most systems in equilib-
rium, Ek = − 12 Ep . This is because the energy conditions in such a system are similar
to those found in a circular orbit. This trick is useful for simplifying energy scenarios
and can show up in competitions like USAAAO.

• Getting larger, getting cooler: This is the opposite of the previous idea. A star’s surface
will get cooler as it expands.

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• More massive stars evolve quicker: This is because they will burn through their fuel
supply quicker (There is a mass-luminosity relation L ∝ M 3.5 , so luminosity [or the rate
at which fuel is used] increases faster than mass [or the amount of fuel] does) and move on
to the next stage of their life. This also means more massive stars have shorter life
spans!

• Stefan-Boltzmann Law: Describes the power radiated by an object; used to describe

luminosity of a star.

P = ϵσAT 4

– P is the absolute luminosity,

– ϵ is the emissivity, and is 1 for perfect blackbodies like stars (NOTE: while stars are
often approximated as perfect blackbodies, they are not actually perfect blackbodies)
– σ = 5.67 × 10−8 mW
2 K 4 is the Stefan-Boltzmann constant.

– A is the surface area, and is 4πR2 for spheres

– T 4 is the temperature raised to the 4th power.

This means that

– Bigger stars appear brighter.

– Hotter stars appear brighter.

2.2 HR Diagram
In order to make sense of stars, we first must develop a system to classify them. The two main
properties used to classify stars stars are absolute luminosity (since we can directly measure ap-
parent magnitude and then calculate absolute magnitude, and thus luminosity, if we know the
distance to the star), and temperature (which can be found from the star’s emission spectra).
Classification schemes:

• Harvard spectral classification: Uses the strength of H I Balmer lines to classify stars
based on temperature.
Due to the Stefan-Boltzmann law, a higher temperature should result in stronger H I lines,
since the star becomes more luminous. So, stars were assigned to classes A through Z where
A had the strongest lines and believed to be hottest, while Z had the weakest and believed
to be coldest. Over time, most of the classes fell out of use and only a few remain.
However, Annie Jump Cannon realized while looking at spectra of stars that some stars
that had weak HI lines also had really strong H II lines. This means that these stars are
so hot that it ionized the H, which is why H II (singly ionized) lines were strong and H I
(neutral) lines were weak. She reorganized the classes based on temperature and came up
with a mnemonic to help remember the order:
Oh Be A Fine Girl, Kiss Me

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The dwarf star classifications L, T, Y were later added to the end, so the order of stars from
hottest to coolest is OBAFGKMLTY
A single digit is added after the letter to specify how hot it is within that class, with 0 being
the hottest and 9 being the coolest.
Thus, a B0 star is hotter than a B3 star, but a B5 star is hotter than an A2 star.
Other weird stars:

– W: Wolf-Rayet stars, these are young, very hot and large stars that lack H
– C: Carbon stars, these stars have a lot of carbon in their atmosphere and often make
up Mira variables
– S: These stars are intermediate between normal stars and C stars

• Yerkes spectral classification: Classifies based on size of star.

– Ia: luminous supergiants

– Ib: supergiants
– II: luminous giants
– III: giants
– IV: subgiants
– V: main sequence stars
– VI: subdwarfs
– VII: white dwarfs

• Morgan-Keenen System: The MK system combines the Harvard and Yerkes systems into
one name.
For example, our Sun is G2V, making it a G-type main sequence star.

• Populations: A star is formed from a cloud of interstellar gas - the remnants of a previous
star that exploded. Since a star converts light elements like H to heavier elements during
its lifetime via fusion, we would expect the stars of later generations to start off with all
of the metals (In astronomy, metal refers to any element heavier than He) formed by its
predecessors. With this idea, we can split stars up into generations, known as populations:

– Population I: These are stars like our Sun that formed recently. Therefore they are
young, typically bluer in color, and have a high metallicity (Note: a ”high” metallicity
is still only around 1.4%)
– Population II: These are stars that formed a long time ago. Therefore, they are old,
typically redder, and have a low metallicity.
– Population III: These are a hypothetical class of stars that formed soon after the Big
Bang. These stars have virtually no metal and are incredibly old and red. None have
been observed yet.

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Now we can talk about how we classify stars. Stars are plotted on the Hertzsprung-Russell
diagram where the x-axis is temperature, with hottest on the left and coolest on the right, and
the y-axis is absolute magnitude or luminosity, with brighter stars higher up and dimmer stars on
the bottom.

• Instead of temperature, the x-axis may have spectral class, color, or B-V index (which all
depend on temperature).

• It may seem weird that temperature is hottest on the left, but it is done this way so that the
diagram looks nicer and the main sequence doesn’t just abruptly end, as you will soon see.

• Stars of the same size follow a diagonal relationship. Due to the Stefan-Boltzmann Law,
hotter stars appear brighter. So, for a given size, we would expect luminosity to increase
(moving up) as temperature increases (moving left). This also means that as a star increases
in size, we would expect it to move perpendicular to these diagonals, so up and the the
right. This relationship becomes apparent when we see where red giants and white dwarfs
are located and when we track the evolution of stars on the HR diagram.

If we look at all the stars in the sky and plot them, we notice that they fit into certain groups
on the HR diagram:

• Main sequence: This is where stars will spend most of their life. It is a nearly straight
line that runs from the upper-left (blue giant stars) to lower-right corners (red dwarf stars)
of the diagram.
When we look at the sky, 90% of stars are main-sequence, which means that a star will
spend about 90% of its life on the main-sequence. Why is this true? It’s like a census! If
we take a snapshot of the population by recording everyone’s age, we’ll see that there are
some adolescents and senior citizens, but a LOT of working age adults. This is because
people spend most of their lives working. Similarly, if we take a snapshot of stars by looking
at the sky, most of them are main-sequence because they spend most of their lives on the
main-sequence.
The lower end of the main-sequence is more densely populated than the top end because
high mass stars evolve quicker and spend less time on the main sequence.

• Red giants and supergiants: These stars are clumped in the upper-right corner of the
diagram. The red giants are really large, which is why they are so bright, and since as stars
get larger, their surface gets cooler, these large stars have very cool, or red surfaces.

• White dwarfs: These stars are clumped in the lower-left corner of the diagram and are
the opposite of red giants. They are small, and therefore not very luminous. However, since
getting smaller gets hotter, these stars are also very hot, giving them their white color.

• Instability strip: This is where variable stars are located - stars that are unstable and
vary regularly in brightness. More on this in another handout.

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Figure 1: The HR diagram, showing where stars are located

Source: COSMOS

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Figure 2: The HR diagram, showing the evolutionary path of stars of different masses. In the
bottom left-corner, there are two lines showing the diagonal relationship a star’s size follows on
the HR diagram.
Source: Fundamental Astronomy by Karttunen

2.3 Fusion
To see how stars survive, we first need to understand where they get their energy from. Simply
put, nuclear fusion converts mass into energy by the relation E = mc2 . By smashing smaller
nuclei together, stars can make heavier elements (nucleosynthesis). Generally, fusion of heavier
elements occur at higher temperatures because more energy is required to smash them together
(since there are more protons so there is a higher Coloumb barrier... quantum tunneling
helps overcome this barrier). There are many different types of fusion that are classified based on
what elements are involved, but only the main 3 are important:

• Deuterium fusion

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This process occurs in protostars, slowing their contraction by increasing radiation pressure.
Not considered stars yet because not fusing H.
Occurs at lower temperature than proton-proton chain. Acts mainly as a thermostat, regu-
lating the contraction of the protostar, rather than a source of energy
Possible in planets > 13MJ

Figure 3: Proton-proton chain (pp1)

Source: Wikipedia

• Proton-proton chain: 4H → He
As the name says, we take H nuclei (protons) and smash them together one after another (a
chain of reactions) to eventually make He.
The first step releases neutrinos νe (which escape the star, carrying away energy) and a
positron e+ , which quickly combines with an electron to release 2 gamma rays γ
The second step releases a gamma ray.
Dominant process in Sun-like main sequence stars.
There are actually many reactions that could happen (pp1, pp2, pp3, pp4[Hep], PEP) but
usually pp1 is used.

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Figure 4: CNO cycle

Source: Wikipedia

• CNO Cycle: 4H → He using a catalyst!

T > 20 mil K
This is an alternative to the proton-proton chain that uses a catalyst, C N and O.
Dominant process in main sequence stars > 1.3M⊙ (some sources say 1.5M⊙ ) because CNO
cycle reaction increases more rapidly with temperature than proton-proton chain (17th power
vs 4th).
AKA: Bethe-Weizsäcker cycle

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Figure 5: Triple alpha process

Source: Wikipedia

• Triple alpha process: 3He → C

T > 108 K
A Helium flash occurs when stars begin this process, marking the transition from the red
giant branch to horizontal branch (explained later)

Any process that has ”burning” in the name just means the star keeps on fusing that element
to make heavier elements until it runs out.

• Alpha process (Helium burning): Keep adding He to make heavier elements

• Carbon burning: After He is used up, keep adding C to make heavier elements
T > (5 − 8) × 1010 K
Occurs in stars > 8M⊙

• Neon burning: Same thing after C, keep adding Ne

• Oxygen burning: Even though O is lighter than Ne, comes after because O16 is doubly
magic

• Silicon burning: Makes 56

Ni and 56
Fe
The production of elements heavier than Fe requires an input of energy, and thus does not
occur except during explosive stellar deaths.

• Lithium burning: Occurs in brown dwarfs, which are cool enough to still have Li

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The next few processes are used to make elements heavier than Fe and follow the nuclear/pro-
ton drip line (when atoms pass this limit they begin to decay).

• r-process: Rapidly adds neutrons 1 after another to make elements heavier than Fe
Occurs when stars explode.

• s-process: Adds neutrons slowly enough that β − decay can happen

Occurs in AGB stars (explained later)

• rp-process: Rapidly adds protons

Tin-antimony-tellurium cycle is the upper limit.

• p-process: Creates neutron-deficient isotopes, process currently unkown

At T > 109 K, photons contain enough energy that they can break up nuclei into lighter
elements (photodisintegration)

2.4 Stellar Interior

Stars are like onions, they have layers.

There are 3 main layers to a star, as you go deeper, the layers get hotter:

• Core: Where most of the fusion happens.

• Zones where energy is transferred to the outside.

– Conduction: only happens in very dense objects, like white dwarfs and neutron stars
(More later)
– Convection takes over radiation when the temperature gradient becomes too high or
the star is too opaque for radiation to occur. Convection also mixes the contents within
the star, causing higher abundances of heavy metals in outer layers.
– .08M⊙ < M < .26M⊙ : These stars are convective throughout.
– Low-mass (Sun-like) stars: They have a radiative core since the pp chain occurs
throughout the core and isn’t super concentrated. This creates a low temperature
gradient allowing for radiation.
They have a convective envelope since the low temperature makes the envelope
opaque, preventing radiation from transfering the energy.
– High-mass stars (> 1.5M⊙ ): They have a convective core because the CNO cycle
is temperature-sensitive and concentrated in the center where it is hottest. This makes
a steep temperature gradient, requiring convection in the inner layers.
They have a radiative envelope because no nuclear reactions are occuring in the
envelope and it is hot enough to allow for radiation.

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Figure 6: Layers in stars of different masses. C’s represent convective layers while unfilled
layers are radiative
Source: Fundamental Astronomy by Karttunen

• The atmosphere. The layers of the atmosphere from innermost to outermost are:
– Photosphere: This is the part of a star we actually see. When we talk about temper-
ature of star, we are talking about the surface temperature of the photosphere.
– Chromosphere: This is an outer, reddish layer that is only seen during solar eclipses.
– Corona: This layer extends very far but has very few particles. Since temperature is
the average kinetic energy of particles, the corona is the hottest layer since it has so few
particles to carry the energy.

Figure 7: Layers of the Sun and surface features

Source: NASA

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Now that we understand how stars work and can describe them using the HR diagram, we
have all the tools necessary to track the evolution of a star through its life.

3 Stellar Evolution
Through this section you will see that a star will stay in its current phase until a change happens.
Whether it is a change in composition or energy transfer, it will cause the star to advance to the
next stage of its life. For each section, we will first describe what happens for a Sun-like star and
then see how the process changes when mass or metallicity changes.

3.1 Protostars: A Star is Born

A star isn’t officially a star until it starts fusing hydrogen in its core. Before that, it is known
as a protostar. Protostars form when a cloud of interstellar gas (known as a stellar nebula)
contracts. As it gets smaller, it gets hotter, causing the temperature to rise until the cloud is fully
convective and becomes a protostar. At this point, we start plotting it on the HR diagram. Since
fusion has not yet begun, the surface temperature is cool. Also, the gas cloud is large, so it has a
high luminosity. Thus, our protostar is in the upper-right corner of the HR diagram.
At this point, the star begins to follow a line in the HR diagram called the Hayashi track
going basically straight down as the protostar continues to contract and get hotter. Eventually,
the center gets hot enough that its opacity decreases and radiation can begin in the core. At this
point, the protostar gets off the Hayashi track and begins going left on the HR diagram along the
Henyey track as its surface temperature increases. It continues along this path until hydrogen
fusion begins in the core and the protostar becomes a star on the main-sequence.
For higher mass stars, the change from Hayashi to Henyey happens earlier, so their protostar
will appear to follow a mostly horizontal path while lower mass protostars will apear to follow a
mostly vertical path.
A T Tauri star is an example of a protostar in this phase. It has a high abundance of Li,
indicating that they are new stars that have not reached temperatures high enough to destroy Li.
When these stars form, they develop an accretion disk around them. As a result, radiation
is forced to exit on opposite sides that the disk does not obscure, creating polar jets of stellar
wind. These jets may collide with clouds of gas and dust near a T Tauri star, creating small bright
nebulae called Herbig-Haro (HH) objects.

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Figure 8 (left): the path of a protostar on the HR diagram.

Source: Fundamental Astronomy by Karttunen
Figure 9 (right): HH47, a Herbig-Haro object from a T Tauri star. The dark cloud in the
middle is the accretion disk, obscuring the protostar from sight.
The bright clouds on either end show the polar jets creating HH objects.
Source: Wikipedia

Note: There is an interesting balance involved whenever we have matter accreting onto a stellar
object. The more mass that accretes, the more energy the central object emits (or more friction
in the accretion disk radiates more energy). The outflow of energy and stellar wind prevents more
matter from accreting, starving the newborn star of fuel. Thus, protostars must reach a balance
between accreting enough matter to produce enough energy to become a star, but not producing
so much energy that it will drive away its food source. This also applies to other stellar objects
that form accretion disks, such as black holes.

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3.2 Main Sequence

Stars spend most of their life on the main sequence, fusing hydrogen in their core. Low-mass
stars (Sun-like) do so via the proton-proton chain, while high-mass stars do so via the CNO cycle,
causing differences in their radiative and convective layers (described in 2.4 Stellar Interior).

3.3 Red Giant Branch (RGB)

As stars fuse hydrogen in their core, helium begins to accumulate in the center. Eventually, enough
helium accumulates that the hydrogen core is replaced by a helium core! However, the core is still
not hot enough for helium fusion to occur, so there is hydrogen fusion in a shell around the core.
Since a shell has more surface area than the core, fusion occurs at a faster rate than before,
causing more radiation pressure to push out on the outer layers, making the star expand. The
star grows rapidly in size, causing its luminosity to increase. As a result, the surface (the core
continues to get hotter!) will decrease in temperature, causing the star to get redder. This is what
makes a red giant.
On the HR diagram, stars on the red giant branch move up and to the right.

Figure 10: The evolutionary track of the Sun on the HR diagram. Here we can see that from
the main sequence, the Sun travels along a slight curve known as the Subgiant branch before
moving up and to the right along the Red Giant Branch.
Source: Chandra

In low-mass (Sun-like) stars, the transition from main sequence to RGB is gradual and known
as the Subgiant branch. Since high-mass stars evolve quickly, there is no noticable subgiant
branch.

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3.4 Horizontal Branch

As helium continues to accumulate in the core, eventually it gets hot enough for the triple alpha
process to occur! This rapid start to helium fusion in the core causes an explosion in the core
known as the helium flash. However, the energy from this explosion does not cause the star to
expand. Instead, it is used to make the degenerate (basically just super dense) helium core expand
and become not degenerate.
As a result, the star’s size does not change much. In fact, luminosity may actually decrease
as energy is used to expand the degenerate core. Surface temperature increases as this increase
in fusion releases more energy. Therefore, on the HR diagram, our red giant will move to the left
(and a bit down) along the Horizontal branch.
The helium flash is super important because it marks the transition from the RGB to the
Horizontal branch!
The horizontal branch is actually pretty complicated because it can take a bunch of different
paths based on the star’s mass and metallicity:

• Low-mass (Sun-like): The horizontal branch is actually split into two segments, red and
blue, which are separated by an RR Lyrae gap where the stars lie on the instability strip.
In globular clusters with low metallicity, the blue horizontal branch is prominent.
In solar metallicity stars, the horizontal branch is reduced to a short stump called the red
clump.

Figure 11: The evolutionary track of a Sun-like star on the HR diagram, showing the red
clump.

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• Intermediate-mass (2.3M⊙ ≤ M ≤ 8M⊙ ): In these stars, the central temperature is high

enough that the helium core is never degenerate. Instead, these stars will follow a blue loop,
increasing in temperature towards bluer colors and then cooling back towards the RGB.
The blue loop causes these stars to cross the instability strip where they become Cepheid
variables.

Figure 12: The evolutionary track of an intermediate-masss star on the HR diagram,

showing the blue loop.

• Massive stars: These stars begin helium burning before they even reach the RGB. This
causes the star to continue moving to the right to become a red supergiant. Stars in this
phase have massive stellar winds and are known as Luminous Blue Variables (LBVs).
The LBVs may loose too much mass to become a red supergiant and will instead turn to the
right to become a Wolf-Rayet star.

3.5 Asymptotic Giant Branch (AGB)

This stage is very similar to the RGB except instead of H, it involves He.
Eventually, carbon from the triple alpha process will accumulate in the center and the helium
core will be replaced by a carbon core! However, the core is still not hot enough for carbon burning
to occur, so there is helium fusion in a shell around the core.
Since a shell has more surface area than the core, fusion occurs at a faster rate than before,
causing the star to expand. This expansion causes surface temperatures to cool and luminosity to
increase. This evolutionary process closely follows the RGB, and is therefore called the asymptotic
giant branch.
After this stage, a lack of material to fuse means that the star will be unable to sustain itself
and will die.

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4 Dead Stars
4.1 Planetary Nebulae
Planetary Nebulae have nothing at all to do with planets! Early astronomers were dumb and
thought that they looked like where planets formed, but that’s not true (planets usually form
alongside stars in a stellar nebula). Planetary nebulae are a place of death, not birth. When a
star reaches the end of the AGB and can no longer perform fusion, gravity takes over radiation
pressure and the star begins to contract. The star continues to contract until it hits a degenerate
core that can not contract any further. When the outer layers of the star hit this ”wall”, they
bounce off and go exploding into space! To us, this appears as a pretty nebulae as the star sheds
off its outer layers.

Figure 13: Crab Nebula, the remnant of one of the earliest recorded supernovae, SN 1054.
Source: Wikipedia

4.2 Supernovae
The explosion of a star near the end of its life is known as a nova. When a supermassive star
dies, the explosion is super bright and known as a supernovae. There is an even brighter type of
explosion called a kilonova, but this occurs during a binary neutron star or black hole merger.
Supernovae are classified based on what elements are found in their spectra:

• Type I: no H

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– Type Ia: no H, no He, strong Si II

Caused by a white dwarf exceeding the Chandrasekhar limit.
Type Ia supernovae are used as ”standard candles” because they all have the same
peak brightness (-19.5). The reason for this consistency is that a white dwarf will
always have the same mass (1.44M· ) when it collapses. This makes them insanely
useful for determining distances to far-away places.
Light comes from 56 Ni decaying to 56 Co to 56 Fe
– Type Ib: no H, strong He, no Si
– Type Ic: no H, no He, no Si
Both Type Ib and Type Ic supernovae are likely caused by a massive WR star collaps-
ing.
• Type II: has H
Caused by a supermassive star collapsing.

– Type II-P: has a plateau in its light curve

– Type II-L: loses H faster than a Type II-P supernova, so has a linear decline instead
of a plateau.
– Type IIn: has narrow emission lines because the surrounding cloud is dense.
– Type IIb: starts off as a Type II supernovae but transitions into a type Ib supernovae
as H is lost.

Figure 14: Light curve of Type II SN.

Source: Wikipedia

4.3 White Dwarfs (WD)

At the end of the AGB, radiation pressure is no longer enough to counter gravity and the star
contracts. It continues to contract until the atoms within the core get so close to each other that

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the electrons begin to overlap. Due to the Pauli exclusion principle, it is not possible for two
electrons to occupy the same space. Thus, there is an electron degeneracy pressure preventing
further contraction. A white dwarf is sustained by electron degeneracy pressure.
In the explosion that forms a white dwarf, the outer layers bounce off the degenerate core, so
only a C-O core remains. In higher mass stars, C fusion occurs and the white dwarf is a O-Ne
white dwarf.
A white dwarf can exist on its own and slowly radiate away energy until its surface cools
(moving to the right on the HR diagram) to the point that the white dwarf stops emitting light
and becomes a black dwarf. However, this process takes so long that it is believed no black
dwarfs exist yet.
Alternatively, many white dwarfs are found in binary systems. Here, the white dwarf will
accrete mass from its companion star, creating short bursts of energy known as recurrent novae.
If the white dwarf accretes enough mass that it exceeds the Chandrasekhar limit (1.44M⊙ ),
then electron degeneracy pressure no longer becomes enough to resist the force of gravity. The
white dwarf will continue to contract, pushing the atoms so close together that electrons will begin
to combine into protons to form neutrons. This process releases a large amount of energy, creating
a Type Ia supernovae and leaving behind a core of neutrons.

4.4 Neutron stars

If a star is massive enough (> 8M⊙ at the end), then electron degeneracy pressure will not be
enough to counter the force of gravity. At this point, only a core of neutrons will remain after
the supernovae - a neutron star. Although most sources will say that neutron degeneracy
pressure, a concept similar to electron degeneracy pressure but for neutrons, keeps a neutron star
from further collapse, this is not true! For M > .7M⊙ , repulsive nuclear forces prevent further
collapse rather than neutron degeneracy pressure.
Generally, the more massive an object is, the more intense radiation it will produce. Neutron
stars and black holes are the main source of X-rays we detect in space! This is usually
caused by intense friction in the accretion disk (matter falling into the central body forms a disk
around it) raising the matter to super high temperatures, resulting in X-rays.
Neutron stars also have a mass limit of their own, known as the Tolman-Oppenheimer-
Volkoff (TOV) limit and is ∼ 2.5M ⊙ most sources say 3M⊙ is a safe upper limit. If the neutron
star exceeds this mass, nuclear forces can no longer counter gravity and the neutron star will
collapse into a black hole.
When a star collapses into a neutron star, the conservation of angular momentum says
that the star must spin faster like how a figure skater spins quicker when they pull their arms in.
As a result, some neutron stars spin really fast.
In addition, magnetic fields must be conserved! So, as a neutron star collapses, the magnetic
fields get stronger. This magnetic field prevents ejected material from escaping from the sides and
forces it to exit through the poles in astrophysical jets. In addition, since neutron stars spin,
where these jets point changes over time. If one of the jets happens to hit Earth, we observe it as
a bright flash of light. As the neutron star spins around, we see these flashes happen at regular
intervals. In this case, the neutron star is called a pulsar. If a pulsar spins really quickly, it is
known as a millisecond pulsar. A pulsar with an incredibly strong magnetic field is called a
magnetar.
Since the neutron star is radiating away energy, that causes its rotation to slow down over

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time (spin-down). However, the neutron star can increase its rotation speed by accreting mass
(spin-up). A sudden spin-up is known as a glitch and its cause is unknown. Likewise, a sudden
spin-down is an anti-glitch.
Although the contents of the interior of a neutron star is unkown, it is believed that the crust
is made up of a substance called nuclear pasta, which comes in gnocchi, spaghetti, lasagna,
bucatini and Swiss cheese phases. If it exists, nuclear pasta would be the strongest material in
the universe.

Figure 15: Pulsar

Source: Astronomy.com

4.5 Black holes

If a star is massive enough, then no force will be able to resist the collapse caused by gravity. The
star will continue to collapse, getting smaller and smaller, until all that mass is squished into a
single point in space known as a singularity.

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Black holes are so dense that nothing, not even light, can escape its gravitational pull. The
event horizon marks the boundary at which light can no longer escape and is 1 Schwarzschild
radius away from the center. We can calculate the value for the Schwarzschild radius by setting
the escape velocity equal to the speed of light.
r
2GM 2GM
vescape = = c, R =
R c2
The photon sphere, or the minimum radius at which photons can have circular orbits around
the black hole, is 1.5R. The Innermost Stable Circular Orbit (ISCO) for other particles is
3R.
The massive gravitational force of black holes allows for several interesting effects.
• As an object falls into a black hole, they will experience intense tidal forces. The side closer
to the black hole will be pulled much more than the side away from the black hole, causing
the object to elongate indefinitely, a process known as spaghetification.

• In addition, gravity near a black hole is so strong that it bends light around it, a process
known as gravitational lensing. This allows observers to see objects located behind the
black hole. Thus, it is impossible to actually see a black hole, since it envelops itself in space
and light. However, we do have ways of detecting when a black hole is there. Gravitational
lensing causes objects near the black hole to appear to multiply and spread out.

• In addition, due to general relativity, a black hole will bend the space around it. For rea-
sons too complicated to discuss, it effectively causes light to travel a longer path in the
same amount of space, causing its wavelength to appear to get longer, a process known as
gravitational redshift.
Thus, when weird stuff happens, we know a black hole is there!

Figure 16: Gravitational lensing

Source: FastForward

Since light can not escape a black hole, any information an object contains is lost when it
goes into a black hole (since we have no way of observing that info). This idea that a black hole
has no information is known as the no hair theorem (info is hair ig?). What happens to this
information? I have no clue!

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• The Holographic principle may offer an explanation. It is the theory that volume itself is
an illusion. Information is contained to a 2D surface and the 3D world we experience is merely
a ”holographic projection” of that information, turning it into a form we can understand. In
this sense, it may be that information that goes into a black hole is not lost. Instead, it is
trapped on the 2D surface of the black hole. As a result, the information is not in a format
we can process and so we perceive that information as being destroyed.

The only 3 properties of a black hole that we can observe are mass, spin, and charge. We use
these three properties to classify black holes:

• Schwarzschild: Defined only by mass.

• Kerr: Defined by mass and spin.

Due to its spin, a Kerr black hole has a lot of weird properties like a 2nd event horizon and
an ergosphere - a region where space-time is literally pulled by a black hole.

– In other words, NOTHING can remain still inside the ergosphere. This phenomenon is
known as frame-dragging (AKA: Lense-Thirring effect).
– Since frame-dragging causes space-time to move, if we had a Kerr black hole, we could
theoretically shoot 2 photons into the ergosphere and have them split so one falls into the
black hole while the other escapes with more momentum (gained from frame-dragging)
than the 2 photons originally had. In other words, we can FARM ENERGY from a
black hole! This is the Penrose process
– The Blandford-Znajek process explains how astrophysical jets can form from
black holes, leading to a lot of cool things like quasars!
– Kerr black holes have also been involved in a lot of legit time travel theories which are
waaay too complicated to discuss here. If you’re curious, just watch Steins;Gate!!!

• Reissner-Nordstrom: Defined by mass and charge

• Kerr-Newman: Defined by mass, charge, and spin

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Figure 17: Black Hole types

Source: iFunny - messylustofscience

Although light can not escape a black hole, they still emit radiation! This is called Hawking
radiation and causes black holes to ”evaporate”, or get smaller. Since black holes radiate, we
can assign a temperature to them. What’s even weirder is that smaller black holes emit more
Hawking radiation, so as a black hole evaporates, it gets hotter and evaporates even quicker until
it eventually reaches an infinite temperature and has evaporated out of existence.
Why does Hawking radiation exist? It is likely due to virtual particles. The simple expla-
nation is that at the event horizon, one virtual particle gets sucked in and one real particle gets
expelled. We observe that real particle as Hawking radiation.

4.6 Binary systems

Binary systems of black holes or neutron stars are really weird phenomena, but their most inter-
esting property is that as their orbits get closer together, they emit something known as Gravi-
tational waves, allowing us to detect them!
Gravitational waves are wave-like distortions in space-time, effectively creating virtual ”gravity
wells” in space time and deforming things. However, these distortions are so subtle that in order
to detect them, scientists use 4km long lasers at detectors like LIGO.

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5 Binary stars
Everything we’ve dealt with up until now has involved solitary stars. However, binary systems (two
stars orbiting around each other) complicate things. As we have seen, mass plays an important
role in the evolution of a star, but as we will see, stars in a binary system can exchange mass!
Older sources say that as much as 85% of stars exist in a binary system, but with improving
technology allowing us to observe more faint solitary stars, that estimate is likely to decrease. Still,
binaries contain plenty of unique properties that make them important to study.

5.1 Types of binaries

Binary stars are classified based on how we observe them:
• Visual binaries: These are the simplest to understand. If you see two stars orbiting each
other, then it is a visual binary! However, most stars tend to be so far away, or one of the
components is too faint, for the system to be resolved as two distinct stars.
• Astrometric binaries: These are similar to visual binaries, except only the brighter star
is observable. Then, by detecting variations in its orbit, we can calculate the mass of its
invisible companion.
• Spectroscopic binaries: These systems cannot be resolved visually. Instead, we rely on
the Doppler effect to detect the binary system! When the brighter star moves towards us
in its orbit, the light emitted from the star will be blueshifted slightly, and when the star
moves away from us, the light will be redshifted slightly. By measuring these variations in
the spectrum, we can calculate the period of the orbit and even the mass!
• Photometric binaries: We rely on changes in brightness during the orbit to detect these
systems. We can classify these binaries even further depending on the reason for the change
in brightness:
– Algol stars: In these binaries, the dips in brightness are caused by eclipses. When
the two stars are separate, both of their luminosities contribute to the total luminosity.
When the brighter star (called the primary) eclipses the fainter companion (called the
secondary), that creates a small dip in brightness since we no longer see light from the
secondary. When the secondary eclipses the primary, we still get the brightness from
the secondary, but we don’t get any brightness from the portion of the primary that
is covered up be the secondary. Since the primary is brighter than the secondary, this
creates a larger dip in brightness. We will do an example problem in Mathz to explain
how we can calculate the properties of the system from these observations.
– β Lyrae stars: These systems are so close that the stars pull each other into ellipsoid
shapes. As a result, the brightness varies continuously as the shape of the stars change
during the orbit. The graph of brightness will still have two dips, but now the curve
will be smoothed out.
– W UMa stars: In these systems, the two dips in the light curve have nearly the same
minima. These systems are usually contact binaries. Additional variations in the
light curve can be caused by many effects (shape of stars distorted by tidal effects, limb
darkening, gravity darkening, reflection of stars off each other, mass transfer).

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We also classify binaries based on how the two stars share their mass:

• The Roche lobe is the region of space around a star where the gravity of the star causes
any mass withing the Roche lobe to belong to that star. For example, if a moon falls within
the Roche lobe of its planet, tidal forces from the planet will tear the moon apart until it all
falls into the planet. There are 3 types of binaries based on how their Roche lobes interact:

• Detached: Each star lies within their Roche lobe. Therefore, no mass transfer occurs and
the two stars are ”detached” from one another.

• Semi-detached: One star exceeds its Roche lobe (like what happens when a RGB grows
too big). As a result, mass transfers from that star to its smaller companion star.

• Contact: Both stars exceed their Roche lobe. As a result, the surfaces of the two stars
contact each other and the stars act as conjoined twins.

Figure 18 (left): Types of binaries

Source: Spectral Atlas for Amateur Astronomers by Richard Walker
Figure 19 (right): Illustration of a binary system where a white dwarf accretes matter from its
companion star
Source: Chandra

5.2 Binary evolution

Most of the interesting binary systems you will see involve a white dwarf and a companion star,
usually a red dwarf. However, sometimes we see a white dwarf with a blue main-sequence
star. This is super weird because we would expect the more massive star to evolve and become a
white dwarf first. So how is there still a blue star?

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1. Two stars are in a binary system. We’ll just say one is a blue main sequence, Bob, and the
other is a red main sequence, Carl.

2. The Bob evolves first, becoming a red giant. At this stage, it exceeds its Roche lobe and
begins feeding matter to Carl.

3. Bob has been stripped of its outer layers, leaving behind a white dwarf. Meanwhile, Carl has
just increased its mass. As a result, it is a massive, and therefore blue, main sequence star.

4. Carl will live out their life and eventually die to become a stellar remnant, likely a white
dwarf. At this point, we will have two white dwarfs orbiting each other.

This general process also applies to a bunch of other scenarios (neutron star binaries, black
hole binaries, etc.).

5.3 Recurrent Novae

WDs in a binary system can release explosions of energy in two ways:

1. When matter from its companion star falls onto the accretion disk, it heats up due to
friction and radiates energy.

2. When matter from the accretion disk finally reaches the surface of the WD, it suddenly
becomes hot enough to achieve fusion! This creates a flash of energy. Since it can happen
multiple times, these are often known as recurrent novae.

5.4 Mathz
Binary systems provide a wealth of information that we would not be able to detect from a singular
body.
Kepler’s laws can be extremely useful when finding info on the components of the binary based
on their orbit:

Example 1 (USAAAO First Exam 2022)

Follow up: Now let’s say that Ek and Do are stars, with masses of 7M⊙ and 1.4M⊙ ,
respectively.
If the system is observed to have an orbital period of 8.4 yrs, what is the distance between
Ek and Do? Further, what is the distance of Ek and Do, respectively, to the center of mass
of the orbit?

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Solution: This problem requires 1 key insight - the angular velocity for both components
of a binary is the same. This is because the centripetal force is provided by gravity. Since
gravity acts between the two bodies, they must always be on opposite sides of each other in the
orbit (since centripetal force must point towards the center of orbit).
Now, let’s use this idea to solve this question. Since the centripetal (gravitational) force is the
same for both objects (by Newton’s 3rd law), then we know:

m1 ω12 r1 = m2 ω22 r
Since ω1 = ω2 (this is our insight), that means

m1 r1 = m2 r2

This means the more massive an object is, the closer it is to the center of orbit. If you are
familiar with mass points, it’s the same concept!
Plugging our values for the mass is, we see that Ek, which is 5 times as massive (7 = 1.4 × 5),
must have an orbit that is 5 times as close to the center of mass.
Now, to solve for angular momentum we use the formula

L = Iω = mr2 ω = (mr)rω

Since we solved that mr and ω are the same for both, this means that Do will have 5 times the
angular momentum of Ek. Therefore, our answer is 5+15
= 65 , which is C!
Now, for the bonus question, we need to use Kepler’s third law:

a3 ∝ M p 3

Where a is the semimajor-axis of the orbit, p is the period of the orbit, and M is the combined
mass of the system. This form is very helpful since we can always compare everything to Earth’s
orbit, which has a = 1AU, p = 1yr, and M = 1M⊙
Thus, we can just plug in the numbers the problem gave us to get

a3 = 8.4(8.4)2 = 8.43

Therefore, the distance between Ek and Do is 2a, or 16.8AU! Since we already established that
the radii of Ek’s and Do’s orbits are in a 1 : 5 ratio, that means the distances are 2.8AU for Ek,
and 14AU for Do.
Orbit dynamics questions involving binary systems will use some variation of these questions.
So, if you understand how we solved this question you should be in good shape!

Now it’s time to look at a question about a binary system’s luminosity:

Example 2 (USAAAO First Exam 2022)

17. An astronomer observes an eclipsing binary star system from Earth, and he plots the
following light curve.

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18. Assume that the smaller star in the above binary star system is brighter than the larger
star. What is the ratio of the radius of the smaller star to the radius of the larger star?

Solution: A light curve gives us 2 important pieces of info - the brightness and the time.
Since we are given the distance and we can find the period by looking at the graph, we can
just use Kepler’s 3rd law to find the total mass of the system! To find the period, just find the
time between two dips. This will be half of the period. The primary dip happens a bit before 2015
and the secondary dip happens around 2019, which gives us a time of ∼ 4.2 yrs. Thus, the total
period is 4.2 × 2 = 8.4 yrs!

14.83 = M (8.4)2
14.83
So M = 8.42
∼ 46M⊙

Now, for the next question, we need to make use of the luminosity shown in the graph. There are
three luminosities we need to consider:

1. The combined luminosity of both stars. This would be when they appear side by side to us
in their orbit. It is represented by the flat line at the top of the orbit, so ∼ 6 in this case.

2. The luminosity of the larger star. This is when the larger star eclipses and completely covers
the smaller star so that we no longer receive any light from it. Therefore, there will be a dip

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in brightness since it’s only the larger star’s luminosity rather than the combined luminosity.
This can be either the larger (if the smaller star is brighter) or smaller dip (if the larger star
is brighter) in our graph. Here, it is the larger dip (since the smaller star is brighter) and
appears to be ∼ 7.5

3. The luminosity of the smaller star plus part of the larger star. This is when the smaller star
eclipses, but does not completely cover the larger star. Again, this dip in brightness can be
either the larger (if the larger star is brighter) or smaller dip (if the smaller star is brighter)
in our graph. Here, it is the smaller dip (since the smaller star is brighter) and appears to
be ∼ 6.2

We can use the definition of magnitudes (sorta like distance modulus) to compare the lumi-
nosities in different cases. Start of by comparing the combined luminosity to when the smaller star
is eclipsed, so between 1 and 2 in this case. We see:
2
10− 5 (7.5−6) ∼ .25

This means that the larger star contributed 14 of the combined luminosity, so the smaller star
contributed 43 of the combined luminosity.
Now, let’s compare when the larger star is partially eclipsed.
2
10− 5 (6−6.2) ∼ .83

If the larger star contributes .25 of the combined luminosity, then that means 1−.83
.25
= .17
.25
= 68%
2
of the larger star was covered. Since A ∝ r , we take the sqrt of this to find the ratio of the radii.
√
.68 ∼ .82

6 Conclusion
Stars, like teenagers, live complicated lives. However, by understanding a few general concepts and
rules, we were able to make sense of it all! This reflects a broader trend in astronomy: astronomers
are able to use their ingenuity to interpret their observations. From these observations, they create
simple rules that allow us to describe THE ENTIRE UNIVERSE!
Given you’ve made it this far, it’s clear that you’re interested in astro, so here’s some advice:
seek to understand why things happen; once you understand the rules that govern reality, the rest
becomes intuitive. Good luck in your pursuit of knowledge!

7 Resources
For a better explanation of Hawking radiation, check out this video! ScienceClic takes confusing
concepts in astronomy and quantum physics and explains our weird reality in a form that is easy to
understand. If stellar remnants interest you, then I highly recommend checking out this channel:
https://www.youtube.com/watch?v=isezfMo8kWQ

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