Homework 2, due Thurs Sep 29, by class time (we will go over it
before the quiz)
1) a) Find an expression for the Keplerian radius (the radius
where circular orbits have the same orbital angular
velocity as the angular velocity of the solid-body rotation
of the star), which should depend on the mass M,
the angular momentum J, and the radius R, of the star,
given that the rotational inertia is 0.07 MR^2.
b) Calculate the angular momentum of Jupiter's orbit around
the Sun, looking up its attributes. This represents about
70 percent of the total orbital angular momentum of all the
planets.
c) Use the above to calculate the ratio of the Keplerian
radius to the Sun's radius, if the Sun had all the angular
momentum that is now in the planets.
d) Draw some conclusion from your result to (c), not necessarily
a logical deduction, but just a suggestion about what these
numbers might imply.
2) Assume the Sun has an opacity (cross section per gram) of
ionized hydrogen gas (which is about 0.4 cm^2 / g). This means
we are neglecting the opacity of bound electrons, which is not a
good approximation in "cool stars" like the Sun
but it will simplify some useful concepts.
a) Estimate the time it takes a photon to diffuse (via random
walk) from the center of the Sun to the surface.
These are rough estimates, really just scaling laws with
the correct units.
(Neglect the existence of the convection zone.)
b) Note that
such a photon would start out as an X-ray photon, given the
high T in the solar core, and be constantly thermalized into
more and more photons of smaller and smaller energy as it
diffuses outward (because it is constantly being thermalized
to lower and lower T as it goes outward). What aspect of
the opacity we are assuming implies that we do not need to
worry about what energy the photon has or how many there are,
we can still get a meaningful time for the escape of these
photons?
c) Estimate the radiative energy within the Sun at any given
time. Given the luminosity of the Sun, does this estimate make
sense given your answer to (a)?
d) Now estimate the total kinetic energy within the Sun (using
the virial theorem). By about what factor
is this larger than the radiative energy
is this? Comment on the validity of textbooks that say
radiation pressure holds up the fusing core of stars like the Sun.
e) Take your above estimate of total kinetic energy inside the Sun and
divide by the Sun's luminosity. This is a time called
the "Kelvin-Helmholz time", and is the timescale for the Sun to
undergo significant change if it did not have an internal heat
source to balance what is lost to its luminosity.
Compare your result to the age of the Earth inferred from
geologic radioactive dating. Why did that comparison create a
conundrum in understanding the age of the Sun, before the discovery
of nuclear fusion?
3) Let us consider gases and radiation fields in a two-dimensional
universe placed into contact with a thermal reservoir at T.
a) Use the Boltzmann factor and the way to count states in a two-dimensional
momentum phase space (so in circular shells instead of spherical shells) to
derive the 2D Maxwell-Boltzmann distribution over nonrelativistic
kinetic energy (i.e., the distribution of E = p^2 / 2m). (Do this in a
unit volume, so the spatial volume is not an important part of the phase
space, and note the 2D volume of a state in phase space is h^2, where we
are not including the two spin states of fermions because it is not
obvious how spin would work in a truly 2D universe!)
b) Now do the same again for photons, to derive the Planck function in
a 2D universe. (The 3D Planck function is often denoted B(nu,T), so
you are looking for the 2D version of that function.)
4) In the late 17th century,
Newton and Huygens had a famous disagreement about the nature of light.
Newton thought is was comprised of particles that were pulled into
the glass as they passed across a boundary with air, while Huygens
thought light was a wave that slows down as it passes into glass.
Both pictures can correctly account for the angle of refraction.
a) Newton's picture of how light particles refract
in glass got which of the following right, and which ones wrong:
(the momentum of the refracted particles, the speed of the refracted
particles, whether or not particles of light exist).
b) Knowing what you do today, if you could be a fly whispering in Newton's
and Huygen's ear at the time of that debate, and you could only tell them
one sentence to help them understand the extent to which
they were either right or wrong, what would that sentence be?