Are there widespread misconceptions about degeneracy pressure?

[virgil1612]

Thank you for the link to that document. It is exactly what I was looking for.

Not quite, one would normally assume the temperatures of He ions and electrons is equilibrated, so they both rise. The rising degeneracy just means that the kT of the electrons is way less than the average kinetic energy of each electron. That's why the kinetic energy is in the degenerate electrons, not the ideal-gas ions, a standard effect of electron degeneracy.
Just one temperature, the key is that kT only reflects the kinetic energy of the ions, it is way less than the kinetic energy of each electron. That's what you mean by a rising degeneracy parameter.

kT is no longer measuring the kinetic energy of the electrons? So because electrons are degenerate, there's another equation for calculating their kinetic energy? It was said more than once that degeneracy lowers the temperature of the electrons. While this happens, T of the ions increases because of the compression. And now you say there is only one temperature. I really don't understand. Maybe after I read that paper...

 

[Ken G]

kT is no longer measuring the kinetic energy of the electrons? So because electrons are degenerate, there's another equation for calculating their kinetic energy?

Yes, that's precisely what is different with degeneracy. It's a thermodynamic effect, relating to temperature, not a mechanical effect, relating to pressure. It only ends up connecting to the pressure indirectly, because temperature influences heat transport. There are so many places that promote the misconception that degeneracy is a type of pressure.

It was said more than once that degeneracy lowers the temperature of the electrons.

Yes, in the sense of lowering it compared to E/k I mean-- not necessarily lowering it compared to what it was before. The problem is that we often use complete degeneracy as a kind of benchmark, to get approximate results, but formally, complete degeneracy means T=0. So that benchmark isn't actually achieved, since T tends to keep rising, so complete degeneracy is just a useful signpost.

While this happens, T of the ions increases because of the compression. And now you say there is only one temperature. I really don't understand. Maybe after I read that paper...

It is certainly a subtle point. As the core loses heat and contracts, its degeneracy rises. So the kT of the electrons goes way below their E, so much so that we can approximate the situation by setting T=0. However, this won't work for the ions, we could not understand why they undergo helium fusion at all. So we keep track of the ion T, know that it is the same as the electron T, but only use it for the ions-- for the electrons, we approximate the situation with T=0 to get the overall mass-radius relationship and so on. The latter is just a benchmark-- the actual electron T matches the ion T. These are tricks of approximation. The key subtlety is that as T rises, the electron pressure is only increased by a fractional amount of order (kT/E)², which is negligible. This is why so many sources incorrectly say the pressure does not rise-- what they mean is that the electron pressure does not rise. But that's only because the heat goes into the ions, which is the whole point of what is going on there. The total pressure rises completely normally, it's mechanical not thermodynamic. I have tried this argument on half a dozen referees already, none seem able to grasp it sadly.

 

[virgil1612]

I know that you can only define a temperature in a system in equilibrium. Could it be that when electrons go degenerate you can no longer talk about thermal equilibrium between them and the ions?

 

[Ken G]

I know that you can only define a temperature in a system in equilibrium. Could it be that when electrons go degenerate you can no longer talk about thermal equilibrium between them and the ions?

It is OK for the electrons and ions to have a temperature, indeed it is crucial that they have the same temperature. This is what regulates the amount of kinetic energy in each, so is what is involved in the helium flash. For example, let us imagine the opposite limit of no thermal contact at all between electrons and ions. Then when helium fusion initiates, the heat will go into the electrons (it is largely released as gamma rays, which interact more with electrons than ions). If the ion T did not need to equilibrate with the electron T, there would be no reason for any significant fraction of that heat to end up in the ions, so there would not be a helium flash. Remarkably, what happens in the limit of T equilibration is that most of the added heat ends up in the ions, and the electrons actually lose kinetic energy. The reason the electrons don't end up receiving much heat is an issue of heat capacity-- whenever you have two substances in thermal contact (i.e., same T), and you add heat, the heat ends up partitioning in proportion to the heat capacity of the substances. Degenerate electrons have a tiny heat capacity-- you need to add very little heat to them to get a big jump in temperature, because adding heat breaks the degeneracy.