Physics 441/541: Homework Questions Due February 22nd

1
Consider a star of volume V held together by gravity. If the gas is ideal and non-relativistic, then
a)    show that the average pressure is given by:
<P>=2/3  Ek.e./V,
where Ek.e. is the kinetic energy due to the translational motion of all the particles.

b)  multiply the equation for hydrostatic equilibrium by 4*pi*r**3 and proceed to show that <P>=-1/3  Egr/V, where Egr is the gravitational energy of the system. Does this result agree with what you know about the virial theorem?

2
Imagine that you calculate a model for a stellar interior that doesn't include radiation pressure. You derive a certain radius for the star, R1. You then realize your mistake and put in an appropriate accounting of the new, increased pressure. You then proceed to calculate a new radius, R2. Qualitatively, arguing from the virial theorem, would you expect R2 to be greater, or smaller, than R1? Is this surprising? Why?

3
As the Sun evolved towards the main sequence, it contracted under the influence of gravity while remaining close to hydrostatic equilibrium, and its internal temperature changed from about 30,000 K to about 6 x 106 K. (This stage of stellar evolution is called the Kelvin-Helmholtz stage.) FInd the total energy radiated during this contraction, assuming that the luminosity during this stage is the same as the present luminosity. Then, estimate the time taken to reach the main seuence.

4
The very crude estimates we made in class for the central conditions in the sun are a bit unappealing. To derive more realistic bounds, consider a star of mass M and radius R. Show that in hydrostatic equilibrium, the function:

P(r)+Gm(r)2/(8*pi*r4)

decreases as r increases. Hence, show that the central pressure satisfies the inequality:

Pc>1/6 (4*pi/3)1/3 G <rho>4/3 M2/3,

where <rho> is the average density.