Question

A self-gravitating system has a negative heat capacity - explain what this statement means and give one example of when this property is important.

Answer 1

Most physical systems exhibit a positive heat capacity. However, even though it can seem paradoxical at first, there are some systems for which the heat capacity is negative. These are inhomogeneous systems that do not meet the strict definition of thermodynamic equilibrium. They include gravitating objects such as stars and galaxies, and also sometimes some nano-scale clusters of a few tens of atoms, close to a phase transition. A negative heat capacity can result in a negative temperature.

Negative heat capacity would mean that when a system loses energy, its temperature increases.

According to the virial theorem, for a self-gravitating body like a star or an interstellar gas cloud, the average potential energy Upot and the average kinetic energy Ukin are locked together in the relation U(pot)=2U(kin)

The total energy U (= Upot + Ukin) therefore obeys

U=-U(kin)

If the system loses energy, for example, by radiating energy into space, the average kinetic energy actually increases. If a temperature is defined by the average kinetic energy, then the system therefore can be said to have a negative heat capacity.

A more extreme version of this occurs with black holes. According to black-hole thermodynamics, the more mass and energy a black hole absorbs, the colder it becomes. In contrast, if it is a net emitter of energy, through Hawking radiation, it will become hotter and hotter until it boils away.