Heat capacities are normally positive, but there is an important class of exceptions: systems of particles held together by gravity, such as stars and star clusters.
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-times the total kinetic energy.
The conclusion of part turns out to be true, at least on average, for any system of particles held together by mutual gravitational attraction:
Here each 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 (), Section .) 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.
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 , where is the average temperature. Express the total energy of a star in terms of its average temperature, and calculate the heat capacity. Note the sign.
Use dimensional analysis to argue that a star of mass and radius should have a total potential energy of , times some constant of order
Estimate the average temperature of the sun, whose mass is and whose radius is . Assume, for simplicity, that the sun is made entirely of protons and electrons.
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