Chapter 1.6, Problem 55P

Heat capacities are normally positive, but there is an important class of exceptions: system of particles held together by gravity, such as stars and star clusters.

1. Consider a system of just two particles, with identical masses, orbiting in circles about their center of mass. Show that the gravitational potential energy of this system is -2 times the total kinetic energy.
2. The conclusion of part (a) turns out to be true, at least on average, for any system of particles held together by mutual gravitational attraction: ${\bar{U}}_{potential}=-2{\bar{U}}_{kineti{c}_{}}.$
3. Here each $\bar{U}$ refers to the total energy (of that type) for the entire system, averaged over some sufficiently long time period. This result is known as the virial theorem. (For a proof, see Carroll and Ostlie (1996), Section 2.4.) Suppose, then, that you add some energy to such a system and then wait for the system to equilibrate. Does the average total kinetic energy increase or decrease? Explain.
4. A star can be modeled as a gas of particles that interact with each other only gravitationally. According to the equipartition theorem, the average kinetic energy of the particles in such a star should be $\frac{3}{2}kT$, where T is the average temperature. Express the total energy of star in terms of its average temperature, and calculate the heat capacity. Note the sign.
5. Use dimensional analysis to argue that a star of mass M and radius R should have a total potential energy of $-G{M}^2/R,$ times some constant of order 1.
6. Estimate the average temperature of the sun, whose mass is $2\times {10}^{30}kg$ and whose radius is $7\times {10}^{8}m$. Assume, for simplicity, that the sun is made entirely of protons and electrons.