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Question 1:. We previously made the comparison between gas pressure and radiation pressure at
the center of the Sun. Let us make a similar comparison between gas pressure and electron degeneracy
pressure. Approximate that at the center of the sun, the gas is fully ionized, and has a density of
1.5 × 105 kilograms per cubic meter. You may assume that the gas at the core of the Sun is half
hydrogen, and half helium by mass (this is just a rough estimate, based on the sun being half-way
through its main-sequence lifetime).
a. Calculate the number density of electrons at the core of the Sun
b. use this to calculate the Fermi energy; i.e. the energy of the highest occupied electron state if we
ignore temperature.
c. Compare this to 3/2kB T - the average kinetic energy per electron, and make a statement about
whether electron degeneracy pressure is important for the structure of the Sun.

Question 2
Question 2. Dust grains (small solid particles) in our solar system slowly spiral in towards the Sun
due to interaction with the radiation field. This interaction with the radiation field is known as
Poynting-Robertson drag. For simplicity, assume the grain absorbs and re-emits light as a blackbody
and is on a circular orbit with orbital radius r.
a. In the frame of the Sun, photons are traveling radially. Show that in the frame of the grain, they
arrive with an angle θ ≈ v/c, where v is the orbital velocity (and I have dropped higher-order terms
in the small quantity v/c).
b. In terms of the luminosity of the Sun L⊙ , the orbital radius r, and the physical radius R of
the grain, calculate the radiation pressure force on the grain. Include both the (dominant) radial
component, and the component that opposes the grain velocity.
c. Note that both the radiation pressure force and the gravitational force scale inversely with r2 , and
therefore their ratio is independent of orbital radius. This ratio is typically denoted β. Calculate β
as a function of grain radius and density.
d. Considering just the radial component of the radiation pressure force, solve for the orbital velocity
in terms of β (hint - do a radial force balance including gravity, radiation pressure, and centrifugal
force).
e. Since there is also a component of the force that opposes the grain motion, it is doing work on the
grain, lowering its energy. Calculate the rate at which the orbital radius changes in response to this
f. (extra credit - 5 points). In the frame of the Sun, the photons are moving radially and therefore
carry no angular momentum (assume they are not circularly polarized). How does the effect work
then - why does the grain experience a transverse component of the force?

Question 3
At the end of their lifetime, stars may lose an appreciable fraction of their mass. Let’s consider in
two idealized scenarios how this may affect the orbits of planets.
a. Consider a planet initially on a circular orbit at 1 AU around a star with one solar mass. Let’s
suppose the mass loss is very slow (i.e. it takes place over many orbits of the planet). You may ignore
any interaction between the planet and the material leaving the star, and assume that the planetary
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orbit evolves only because the mass of the star is changing. Assuming the orbit remains circular as
the star loses mass, calculate the orbital radius as a function of the mass of the star.
b. Let’s take the opposite extreme, and assume the mass-loss to be instantaneous. This is ap-
propriate for example for a supernova explosion. Let’s assume the planet was initially on a circular
orbit at 1 AU around a 5 solar-mass star. Instantaneously, the star blows up and leaves a black hole
with mass MBH . Again, ignoring any interaction between the planet and the shell of outward-moving
material, calculate the semi-major axis and eccentricity of the resulting orbit in terms of MBH . Hint
- since the mass-loss happens instantaneously, the position and velocity of the planet do not change
during the mass-loss process. Therefore, you just need to convert that position and velocity into
semi-major axis and eccentricity given the new value for the stellar mass.
c. Supernova explosions are in generally slightly asymmetric. This means that the black hole
also experiences a recoil “kick”. Suppose the planet is originally in a circular orbit with radius 1
AU around a 5 solar mass star. This star blows up, leaving behind a 2 solar mass black hole, and
receiving a kick in a random direction with a magnitude of 50 kilometers per second. What is the
probability that the planet remains bound to the star? Assume that the direction of the kick is drawn
randomly from a uniform distribution over the entire unit sphere.

Question 4
A typical meteor hits the Earth’s atmosphere at about 20 kilometers per second, however there is
a significant range of speeds depending on the orbit of the meteor. What is the slowest and fastest
possible relative velocity between a meteor and the Earth at the time the meteor is impacting the
atmosphere? You may assume that the meteor came from material that is gravitationallly bound to
the Sun. Don’t forget to include the gravitational potential of the Earth itself.