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.

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