Summary: ASTR 5120: Problem Set 2
1. A young star of radius R # and e#ective temperature T eff is surrounded by a flat and
razorthin disk of gas and dust that extends from infinity to the equator. The disk
absorbs all intercepted radiation from the star and reemits it locally as thermal
radiation at temperature T disk (r). By making use of (a) the definition of flux in
terms of an integral of intensity over solid angle and (b) the fact that intensity is
constant along rays, derive the temperature profile in the disk (which will be of the
form T disk /T eff = f(r, R # ).
[You can assume that the star is a sphere of uniform brightness, and will need to
know the standard result that the flux from a surface of brightness B is F = #B.]
2. The interiors of large planets are (or were) molten, whereas the same is not true of
very small bodies. First of all, explain physically why this should be so?
Let us try and construct a simple model to estimate how large a body needs to
get before the interior melts. Suppose that the source of heat is radioactive decay,
which results in a release of energy at a rate —
u per unit mass [units erg s -1 g -1 ].
The heat is transported di#usively out of the interior according to the equation,
q = -##T
where q is the heat flux, and # the thermal conductivity. For a body of mass m,
radius r and specific heat capacity c, estimate (or derive exactly, if you prefer!) the