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Question: Heat capacities are normally positive, but there is an importan…
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.
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a. consider a system of just two particles, with identical masses, orbiting in a circle about their Center of
mass. So that the gravitational potential energy of this system is -2 times the total kinetic energy. (Up =
-2Uk)
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b. the conclusion of part A turns out to be true, at least on average, for any system up arkles held together
by Mutual gravitational attraction:
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U(potential) = -2U(kinetic)
Here each Ui refers to the total energy for the entire system, averaged over some sufficiently long time
period. this result is known as the virial theorem. 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?
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c. a star can be modeled as a gas particle 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 (3/2)Kat,
where T is the average temperature of the star. Express the total energy U ofa star in terms of its average
temperature, and then calculate the heat capacity
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d. for a single star, its gravitational potential energy is given by -((GM^2)/R). estimate the average
temperature of the Sun, whose mass is 2 x 10^30 kg and whose radius is 7 x 10^30 m. assume for
Simplicity, that the sun is made up entirely of hydrogen atoms. Compare this value to the temperature of
the Sun at the surface (6000K) and at the core (2 x 10^7 K)
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