ISM  final exam

1)  Consider a plane-parallel slab of dilute cold neutral H gas in thermodynamic
equilibrium with a reservoir at some temperature T.  Find how the following depend 
on T, in the limit that T >> E/k, where E is the energy of the 21 cm line:
a) the ratio of radiative upward to radiative downward transitions in that line
b) the ratio of populations in the excited to ground state of that line
c) the optical depth in the line, counting stimulated emission as negative
d) Given these results, why is the 21 cm line good for inferring the column
depth of the slab, but not good for inferring the T of the slab?

2) Consider H in ionization equilibrium.  
a) Derive the population ratio of ionized to neutral H if all collisional
rates balance (remember that collisional recombination is a three-body
process) given some temperature T.  Do this in two ways-- using a rate
balance approach, and using a statistical mechanics approach (i.e., counting
states).  Leave any functions of T, and all constants, unspecified, we only
want to know where T appears, and how the answer depends on density.
b) now derive these dependences if all radiative rates balance, using a 
thermal radiation field that is simply diluted by some factor W.  You do not
need to do this more than one way.
c) derive these dependence again if all upward rates are collisional, all 
downward rates are radiative, and stimulated downward rates are negligible.
 
3) Imagine a self-gravitating isothermal ideal monatomic gas in a spherical
box that subjects the gas to an adjustable external pressure P.
a) Find how much P is required to cause gravitational collapse if the gas in
the box has a mass M that is much less than the Jeans mass at nominal
external pressure.  The gas remains isothermal at T.
b) Derive an expression for an estimate of the timescale for the collapse.
c) Describe how the internal density structure would change if P were very
gradually increased from essentially zero to the value in part (a).  Here I
am looking for a qualitative description of how the density would vary with
radius.
d) If you were given the P(t) and T(t) as a function of age t in a Big
Bang model of the universe, how would you use the above information to 
determine at what age galaxies form?

4) Imagine a star with a flux N of ionizing photons (per unit time) turns
on in a region of cold neutral H at number density n.  The average energy per
photon is E and the temperature of the post-ionized gas is held fixed at T.
a) Contrast the sound crossing time with the age of the growing HII region,
and interpret the meaning of when those times cross.
b) Recognize that the HII region will be dynamical, and wait until the
age is long and the ionization equilibrium has set in to find how the
radius of the ionized region will vary with time.  The hot gas will pile
up ISM gas like a supernova blast wave, so treat the density in three
regimes-- internal to the cavity (assume the appropriate characteristic 
density throughout), external to the cavity, and in a narrow shell piling
up at the edge of the cavity.  Find how the radius of that shell depends
on time, neglecting any thermal pressure in the external gas.
 
5) Imagine a dust population with two sizes only, one whose radius is similar 
to red wavelengths and one whose radius is similar to blue wavelengths.
Given a number ratio N between them, describe the ISM reddening that would
occur when seeing distant stars through this dust population, given the
optical depth in the blue component.  First find the optical depth ratio
between the dust components by approximating Mie scattering as we did in
class, using the better approximation that the cross section is inversely
proportional to wavelength when wavelength crosses the boundary discussed.