Homework #1 ..... Due Tues, Sept 6
1) The mass-to-light (M/L) ratio in a galaxy is the
mass of the galaxy, in units of solar mass, divided by the luminosity of
the galaxy, in units of solar luminosity.
Most of the mass in M/L comes from dark matter, a conclusion which
arises from first determining what M/L should be for just the
baryonic matter (i.e., for the stars alone).
Use the Salpeter IMF, and the way luminosity scales with mass (assume all
the stars are main-sequence stars for simplicity), to find:
a) the stellar M/L for a starburst galaxy approximated as a huge cluster that just formed.
b) the stellar M/L for a constant-state galaxy whose star-formation and death rates have
been constant for the lifetime of all the stars in the galaxy.
c) which of the above is more like the Milky Way, given that the Milky Way has stellar M/L = 4 in the
general vicinity of the Sun? (By the way, the actual M/L of the Milky Way, counting
dark matter, depends on where you put the edge of the galaxy, but is in the range 20 - 100.
This is why we conclude the Milky Way is mostly dark matter.)
2) Imagine a heavily weighted coin that has a lower center of gravity when it
comes up "tails" then when it comes up "heads".
If it is flipped in a consistent and random way, let's say
it comes up tails 68 million times and heads 17 million times.
Now imagine that same coin is flipped in
the same way on a planet that is in every way the same as Earth
except that its gravity is twice that of Earth.
What distribution do you now expect after 85 million flips?
Homework #2 ..... Due Thurs, Sep 15
1) Imagine that an O star magically turns on instantly inside a cloud
of neutral hydrogen gas with some specified low density.
The cloud absorbs all the ionizing
photons from the star (assume the latter are emitted at a known and
constant rate), and there is always a sharp front between mostly ionized
and mostly neutral regions.
a) First, consider timescales much shorter than
the recombination time for an ionized proton, so prior to any establishment
of a global ionization equilibrium, and find how the radius of the ionized
region depends on time.
b) Also find, still on short times well before any equilibrium is
established, how the radius at any given time would depend on
the given (constant) flux of ionizing photons from the star.
c)
Now if you are told the recombination rate in ionized gas,
what is the radius of the
front when the global ionization equilibrium is eventually established?
d)
Given the result from above, estimate how long it takes for the front
to reach that radius. Does that make sense in terms of the recombination
rate?
2. Imagine some gas in the ISM is heated by photoionization by a
radiation field of mean intensity J that is (for simplicity)
all at one frequency f.
Assume hf > X + 3kT/2, where X is the ionization energy of the atom.
Consider that
hf-X of heat energy is deposited per ionizing event, and 3kT/2 (on
average) is removed per recombination event, and assert that the
gas remains in ionization equilibrium throughout, and at fixed T.
The dominant
rates are radiative ionization and radiative recombination.
a) If the gas is mostly neutral, find how the net heating rate will scale
with J and f, if we imagine adjusting these parameters and letting the
ionization balance readjust (while maintaining constant pressure and
electron temperature).
b) Do this again if the gas is almost completely ionized.
c) Use results from (a) and (b) to mock up a rough graph that shows
how the net heating rate depends on J, at fixed f > X/h, and
for fixed pressure and electron temperature.
Give a physical reason for the somewhat surprising behavior you see.
Homework #3 due Thurs Sep 29
1) Use the Jeans mass in a giant molecular cloud to
give a reason why such clouds can make some stars
with much higher masses than the Sun.
Also, such clouds make many stars with considerably lower masses-- give a
reason from the same gravitational instability considerations of why a
contracting region from such a cloud could fragment
into smaller Jeans masses
as it evolves.
2) In the early universe, the kinetic temperature of the gas cooled with
the temperature of the CMB, which went like the inverse of the scale
parameter. This means the temperature of the gas scaled like its density
raised to the 1/3 power.
a) find how the ambient gas pressure in the universe, prior to the formation
of density inhomogeneities like galaxies, depends on temperature. Use the
fact that there are about a billion as many CMB photons as gas particles,
and look up the radiation pressure as a function of T.
b) estimate the Jeans mass during the epoch of galaxy formation, assuming no
dark matter.
c) in a matter-dominated universe at the critical density, the age of the universe scales like the
density to the -1/2 power. Is that consistent with the fact that observations show that galaxies
did not form for a long time after isothermal contraction would have been possible (i.e., a long
time after the recombination era)?
Homework #4 Due Thurs, Oct 27
1) Imagine a cloud with a given heating per particle Q, but Q varies smoothly from place
to place in the cloud, so the temperature T does also. Q is distributed over particles by
some f(Q) = dN/dQ (so there are dN particles receiving within dQ of heating Q ergs
per second), and given is a standard radiative cooling function Lambda(T) (which when
multiplied by density gives the rate of cooling per particle in ergs per second).
a) Assuming you are also given the particle density n, which is constant everywhere, find the
distribution of particles over T, i.e., find an expression for g(T) = dN/dT that is based on
knowledge of f(Q), Lambda(T), and n.
b) More realistic would be to assume you are given a constant known pressure P. Do part (a)
again using constant P rather than constant n.
2) Imagine a cloud of given length scale R and pressure P. Assume the cloud has a spatially
contant temperature T, but you don't know what it is, so treat T and particle density n as
free parameters. The radiative cooling function
Lambda(T) depends on lines of various optical depths, but focus on Lambda_i, the
contribution from some line i. Lambda_i also depends on T, so assume you know this
function in the effectively thin case. Assume the line is highly optically thick,
and can even become effectively thick below some temperature T_thick. Assume
hydrogen is virtually entirely ionized, so the free-electron density (responsible
for collisional rates) is simply proportional to the total density n.
a) Describe how you would determine T_thick, given information about the atomic rates
in the line, neglecting any T dependence in both the collisional de-excitation rate
and the line opacity (so the destruction probability and line optical depth depend
only on n and not on T). How does T_thick vary with changes in the given R and P?
b) Describe how your answer to part (a) could be used to find a T-dependent correction
factor to include with Lambda_i. Try to do better than just a cutoff at T_thick-- can
you use a probabilistic argument to get a general reduction to Lambda_i that depends
smoothly on the escape probability and the destruction probability per photon scattering?
Homework #5 (due Tues, Nov 15)
1) Find the cross section per gram of a power-law distribution of Mie
scatterers, power -n, between a lower radius cutoff R- and an upper
radius cutoff R+. Approximate the scattering cross section
as being the geometric cross section for wavelengths less than 2 pi R,
and being effectively zero for larger wavelengths.
Assume R+ >> R-, and just look at the two regimes
lambda << R-, and R- << lambda << R+. Break into regimes in n as
necessary. Also, make some effort to provide simple short explanations
for any simple outcomes you discover.
(2) Imagine a massive evolved star whose core is mostly helium and whose envelope is mostly
hydrogen. Its luminosity is gradually rising, until it exceeds the Eddington limit for its
envelope.
a) Describe what you think will happen to this star, assuming its luminosity stays fixed.
Consider two cases-- one where the helium core is a small fraction of the mass of the star,
and another where the helium core holds the majority of the mass of the star. (Hint: what
significance is there in the fact that helium has a neutron per electron, and hydrogen does not?)
b) Find a way to estimate how long it will take for the process that you describe to play out.
(Hint: can energy considerations be your guide?)
Homework #6 (Due Tuesday Dec 6)
1)a) According to the isothermal Parker wind solution, what would happen to the mass-loss rate and speed at
1 AU of the solar wind if the T of the wind was doubled? What would happen to them if the T stayed the same
but the rate of heat deposition into the corona doubled? What would happen to them if half the surface of
the Sun was closed off by loops of strong magnetic field, but the open-field regions were unaffected?
b) Explain (briefly) why an isothermal gas-pressure driven cool-star wind does not reach a finite maximum speed, but a radiatively
driven hot-star wind does, and why a gas-pressure driven wind (like the Parker solution) does not require knowledge of
the mass-loss rate to know the wind acceleration, but a radiative line-driven wind does.
2) Explain (briefly) why cool molecular gas can still emit light in the radio regime. What kind of charges are emitting that light,
and is this the usual state of affairs in astronomy?