# Gravity vs. Degeneracy Pressure

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## Main Question or Discussion Point

Okay, I noticed that my OP got 39 views but no responses, so let me change my strategy. Here is my question:

An object made out of degenerate matter (e.g. white dwarf) will collapse if more gravitational energy is lost in getting smaller than the energy that is gained due to electron degeneracy. So, the sum of these two energies will decrease. But total energy must be conserved. So, what happens to the rest of the GPE that is lost (the part that doesn't go into doing work to overcome degeneracy pressure during contraction)? Is it just converted to heat?

My OP is below if you want more details on what I'm asking about.

My prof. was giving a sort of heuristic outline of how the Chandrasekhar mass limit arises. He started with the basic "equilibrium equation" that the energy E = Ekin + Epot should be minimized. This is in the context of a "cold" object, where most of "Ekin" is due to electron degeneracy. In other words, there is no thermal pressure support. All of the support against gravity comes from degeneracy pressure. He then worked out how each of these energies varies with the volume (or radius) of the object, and showed that in the case of non-relativistic electrons, there is an equilibrium point (the sum of the energies E is minimized for some finite radius R), and in the relativistic case, there is no global minimum, the solution is unchecked collapse. I looked into this further, and I realized that another way to think about this is in terms of pressure balance. You can equate the "gravitational pressure" to the negative of the degeneracy pressure, where

Pgrav = -∂Epot/∂V

and

Pdeg = -∂Ekin/∂V

This equation of pressures results in

∂(Epot + Ekin) / ∂V = 0

showing that pressure balance does indeed occur at a minimum in the sum of the energies.

Here is my question: in the situation where collapse occurs, gravitational pressure exceeds degeneracy pressure, meaning that the rate of change of Epot and Ekin with radius (or volume) is such that more GPE is lost in contracting than the energy that is gained due to electron degeneracy. In other words, not all of the GPE that is lost is "used up" in doing work against degeneracy pressure. The sum E is actually reduced. But total energy must be conserved right? The sum E must not be the total energy of the system. So what happens to the rest of that GPE?

An obvious answer would seem to be: "it is converted into heat." Is that the case? If so, then I have another related question...

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## Answers and Replies

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Ich
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It's converted to kinetic energy from ordered inward directed motion in the first place. If the star settles down again, it would become heat.

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Heh, inwardly directed motion. How did I miss that?