UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Physics
2008 FALL STARS Professor Martin
Midterm Exam
October 17, 2008
This exam is closed book. No calculators are allowed. Please turn off your cell phone before
the exam begins. The last page includes constants and equations which may be useful. Please
explain what you are doing with words as well as equations. It makes it much easier for us to
follow your logic and give partial credit. The exam ends promptly at 2:50pm.
1. Simple Stellar Models
Consider a star of mass Mand radius Rin hydrostatic balance. Assume the density is a
constant, ρ(r) = ρ0= 3M/4πR3. You found previously that the gravitational binding energy of
such a star is EGR =−3
5GM2/R, and you may use this result.
(a,2pts) Find the average pressure inside the star.
(b,4pts) Calculate the central pressure in the star.
(c, 2pts) Assume the required pressure is supplied by ionized hydrogen, i.e. by an ideal gas
with γ= 5/3. Find the internal energy of the star, Ein, and the total energy of the star,
Etot =Ein +EGR.
(d, 2pts) Assume the required pressure is supplied by radiation pressure, i.e. by a photon gas.
Find the internal energy of the star, Ein, and the total energy of the star, Etot =Ein +EGR.
2. Simple Stellar Atmospheres
Consider a plane-parallel atmosphere of pure, ionized hydrogen with constant gravity, g,
pointing down, so that ~g =−gˆz. Presume that the heat transport is dominated by radiation
and that Thomson scattering sets the opacity.
(a, 1pt) What is the pressure of this gas where the density is ρand the temperature T?
(b, 1pt) Write down the heat transport equation and, presuming that the outward flux, F, is
constant and that the gas is ideal, find dT/dz.
(c, 4pts) Now find dT/dP, which is the change of temperature with pressure in the atmosphere,
and show that the relation between density and temperature in the atmosphere is ρ∝T3.
(d, 2pts) At a fixed pressure in this atmosphere or, equivalently, a fixed location, how does the
temperature depend on the composition of the atmosphere? Specfically, if the atmosphere were
composed of completely ionized Helium instead of ionized Hydrogen, by what factor would the
temperature change?
(e, 2pts) We assumed pressure support from an ideal gas in this atmosphere. Let’s check whether
this assumption is likely to hold at all heights in the atmosphere. Calculate the ratio of the
radiation pressure to the gas pressure as a function of height, z. If our assumption is good at
the base of the atmosphere, is it likely to hold higher up?
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