Kevin Zhou Physics Olympiad Handouts
Modern III: Matter, Astro, and Cosmo
Chapters 35 and 36 of Blundell cover astrophysics, and chapter 15 of Krane covers cosmology. For
solid state physics, see chapter 49 of Halliday and Resnick, chapter III-14 of the Feynman lectures, or
section 5.3 of Griffiths’ Introduction to Quantum Mechanics (3rd edition). For more on magnetism,
see chapters II-34 through II-37 of the Feynman lectures. For a detailed introduction to the physics
of stars and compact objects, see chapters 10 and 16 of An Introduction to Modern Astrophysics
by Carroll and Ostlie. (Carroll and Ostlie is also a great introduction to astrophysics in general,
accessible with just Olympiad physics knowledge.) There is a total of 87 points.
1 Condensed Matter
Condensed matter is an enormous field, touching everything from solid state physics to biophysics
and atomic physics, and contains more than half of all physicists. However, you hear less about it
in the news, and in Olympiad problems, because it requires a substantial amount of background
to explain. The following problems cover some classic ideas in condensed matter, using a mix of
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modern physics and waves.
[5] Problem 1. APhO 2016, problem 3. A good question on the quantum mechanics of super-
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conductivity.
[5] Problem 2. USAPhO+ 2021, problem 3. A nice question which derives the integer quantum
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Hall effect using classical electromagnetism and Bohr quantization.
[5] Problem 3. APhO 2015, problem 1. A question on the fractional quantum Hall effect, which
is even subtler than the integer quantum Hall effect. This question is not very clearly written, and
requires a good amount of educated guessing; however, I include it to give you practice with this
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type of Olympiad question, and some exposure to a very important topic of current research.
[5] Problem 4. IZhO 2021, problem 2. A problem on the thermodynamics of plasmas, reviewing
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some material in T2.
[5] Problem 5. APhO 2021, problem 2. A very technical problem that fully explains how a
famous test of quantum mechanics was conducted. Requires material from E8, W1, W3, and X1.
2 Stars
This section contains problems involving stars and star formation. You might think this is a lot about
stars, but all of the problems below use a mix of mechanics, electromagnetism, thermodynamics,
relativity, and modern physics; they are excellent review for the entire course. For a beautiful
graphical overview of these objects, see this paper.
Example 1: PTD 44
The density of stars in the central region of the galaxy is about n = 106 pc−3 , and their
speeds are about v = 200 kms−1 . Could an advanced civilization develop in this region?
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Kevin Zhou Physics Olympiad Handouts
Solution
Impacts between solar systems will occur frequently. For concreteness, suppose catastrophic
effects will happen to an Earth-like planet if a different star passes within the equivalent of
Jupiter’s orbit, which has radius r. The typical time between such events is
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t∼ ∼ 2.4 × 106 years.
n(πr2 )v
Some argue that this is too short a time for advanced civilization to develop, so the center
of the galaxy is outside of the so-called galactic habitable zone.
Example 2: CPhO 2013.3
For stars not too much heavier than the Sun, the luminosity scale with mass as L ∝ M 3.5 .
If all of these stars release the same fraction α of their rest mass energy by nuclear burning,
then how does the lifetime of the star scale with M ?
Solution
The amount of energy available is αM . The lifetime is thus
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τ= ∝ M −2.5
L
so heavier stars live shorter lives. Incidentally, the luminosity scales as L ∝ R2 T 4 by the
Stefan–Boltzmann law. By considering the details of the interior of the star, we can find how
all of these quantities scale with mass, a principle known as stellar homology.
[3] Problem 6. 01^ USAPhO 2018, problem B3.
[5] Problem 7. 01h
IPhO 2012, problem 3. This elegant and tricky problem covers the early stages
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of star formation, and serves as a review of T1.
[5] Problem 8. GPhO 2017, problem 1. This problem covers the physics of fusion in main
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sequence stars, relying on X1 and X2.
[5] Problem 9. IPhO 2007, problem “pink”. This problem covers binary stars, with a strong
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emphasis on data analysis methods, as covered in P2.
[5] Problem 10. GPhO 2016, problem 3. This problem covers the physics of intense magnetic
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fields, using the results of E8 and X1.
[5] Problem 11. APhO 2015, problem 2. A nice but somewhat hard to read problem on the
aurora and the solar wind.
3 Compact Objects
Compact objects such as white dwarfs and neutron stars must be handled with quantum statistical
mechanics, as introduced in X1.
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Kevin Zhou Physics Olympiad Handouts
[4] Problem 12. 01^Do the following JPhO problem. This pedagogical problem reviews the physics
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of white dwarf stars, deriving the Chandrasekhar limit, using the techniques of X1.
[3] Problem 13. USAPhO 2024, problem A2. Rough estimates of the dynamics of stars and
white dwarfs.
Remark
The estimates performed in the previous problem are quite rough, basically treating the
white dwarf as being homogeneous, with uniform density and pressure. In reality, we have
∇p = −ρg just like in any situation in hydrostatic equilibrium, where the degeneracy
pressure p is determined by the local density n. It’s like the gaseous atmospheres you dealt
with in T1, but with a different equation of state.
If you additionally allow the white dwarf to have a net charge, then there is an additional
contribution from the electrostatic force, and the resulting equations are called the Thomas–
Fermi equations of structure. They can also be used to model many-electron atoms, when
you can neglect the discreteness of the electrons.
[3] Problem 14. The Bekenstein–Hawking formula states that a black hole has an entropy of
A
S=
4
where A is the area of its event horizon. The radius of an uncharged, nonrotating black hole is
R = 2M.
In these equations, ℏ, c, and G have all been set to one; you do not need to restore these factors.
(a) Compute the temperature and heat capacity of such a black hole.
(b) Two uncharged, nonrotating black holes begin very far apart from each other, then merge
into a single black hole, emitting gravitational waves in the process; assume there is no initial
angular momentum, so the final black hole is nonrotating as well. Find the maximum possible
efficiency of this process, defined as the fraction of the initial energy that is converted into
gravitational waves, for any set of initial black hole masses.
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It may also be useful to review IPhO 2007, problem “blue”, which we originally covered in P1.
[4] Problem 15. US TST 2022, problem 3. Rough estimates of gravitational wave emission.
[3] Problem 16. NBPhO 2017, problem 4. Rough estimates of gravitational wave detection.
Remark
The discovery of gravitational waves by LIGO has been one of the most important results
this decade, so it’s naturally a popular question topic. Once you finish the above problems,
you can check out a few others with a different take on the same idea. GPhO 2016, problem
2 (solutions here) does a rougher treatment of gravitational wave emission, while IPhO 2018,
problem 1 gives a more accurate treatment using more of the language of general relativity.
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Kevin Zhou Physics Olympiad Handouts
For another way to estimate gravitational wave emission, see section 9.3 of The Art of Insight,
and for some followup questions, see this paper.
Remark
In 1931, after building a sensitive short-wave radio receiver, Karl Jansky heard an unusual
noise on his receiver from a direction that moved across the sky about once a day. He
therefore initially thought it was from the Sun. However, over time he found that the
direction of the noise moved across the sky only once every 23 hours and 56 minutes.
This was a huge difference! To understand why, note that the length of a day is the time it
takes for the same side of the Earth to face the Sun again; it depends on both the Earth’s
spin and its orbital motion about the Sun. The period of the Earth’s spin alone, the so-called
sidereal period, is only 23 hours and 56 minutes. Thus, a signal with this period indicates
an origin from outside the solar system. Jansky later found that the source was the center
of the galaxy; today we know it is due to the supermassive black hole there, Sagittarius A*.
The early days of radio astronomy were full of dramatic discoveries like this. To hear about
the discovery of pulsars, see this talk.
4 Cosmology
Cosmological is a rather technical topic because a proper treatment requires general relativity, but
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one can derive special cases of some of the results using just Newtonian gravity.
[2] Problem 17. AuPhO 2014, problem 14. A quick problem on the basics of dark matter and
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galaxy measurements.
[5] Problem 18. IPhO 2017, problem 1. This long but straightforward problem covers some of
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the evidence for dark matter.
[5] Problem 19. APhO 2016, problem 2. This straightforward problem introduces the basic
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equations of cosmology, such as the Friedmann equation.
[5] Problem 20. IPhO 2017, problem 3. This somewhat more difficult problem expands on the
previous one, leading up to a discussion of cosmological inflation.
