# Entropy of a star

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.15.

The heat capacity of a star was estimated using the virial theorem as

$\displaystyle C_{V}=-\frac{3}{2}Nk \ \ \ \ \ (1)$

where ${N}$ is the number of particles (typically dissociated protons and electrons) in the star. A negative heat capacity is typical of gravitationally bound systems.

We can use this to work out the entropy from the formula

 $\displaystyle S$ $\displaystyle =$ $\displaystyle \int\frac{C_{V}\left(T\right)}{T}dT\ \ \ \ \ (2)$ $\displaystyle$ $\displaystyle =$ $\displaystyle -\frac{3}{2}Nk\ln T+f\left(N,V\right) \ \ \ \ \ (3)$

where ${f}$ is some function that depends on ${N}$ and the volume ${V}$, but not on ${T}$.

The total energy of a gravitationally bound system is negative and, from the virial theorem, we have

$\displaystyle U=-K=-\frac{3}{2}NkT \ \ \ \ \ (4)$

where ${K}$ is the kinetic energy and the formula is obtained from the equipartition theorem. Thus the entropy can be written in terms of the energy as

$\displaystyle S=-\frac{3}{2}Nk\ln\left|\frac{2U}{3Nk}\right|+f\left(N,V\right) \ \ \ \ \ (5)$

We can incorporate everything inside the logarithm except for ${U}$ into the function ${f}$ (call the new function ${g}$, say), so that

$\displaystyle S=-\frac{3}{2}Nk\ln\left|U\right|+g\left(N,V\right) \ \ \ \ \ (6)$

The general shape of this curve is like this (units on the axes are arbitrary as I’m just trying to show the shape of the graph):

The graph is concave upwards, which is typical of systems with negative heat capacity as we discussed earlier. For sufficiently low (large negative) values of ${U}$, the graph would go negative, but I would guess that the temperature at that point would be higher than anything found in real stars so that would never happen. Note that as ${U\rightarrow0}$, ${S}$ effectively becomes infinite. At ${U=0}$, however, the system becomes gravitationally unbound, so the particles would presumably then be able to wander over the entire universe, giving them an infinite number of possible states.