L23 Degeneracy pressure

L23 Degeneracy pressure#

Week 13, Thursday

Material covered and references#

We know turn out attention to whether the ideal gas law is always valid, or whether there are other possible sources of significant pressure. Today we will talk about the degeneracy pressure, that arises from the Fermi principle.

We first review the concept of Fermi momentum and Fermi energy – that is, if you consider a set of electrons confined to a box such that all the energy levels (we are here talking about the momemtum quantization of free electrons in a volume, not electrons bound to atoms) are occupied. The momentum and energy of the very last electron is the “Fermi momentum” and the “Fermi energy”.

Here, we are mostly interested in the Fermi momentum

In the textbooks:

  • Best reviewed in the Schroder Thermal Physics textbook

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Now, we can find an expression for the pressure that arises from these free electrons (i.e. the fact that you cannot compress them into an infinitesimal volume because of the Pauli exclusion principle). We can use the fact that a pressure can be represented by a momentum flux through a surface. From this, we found a general expression for the pressure.

Now we can split into two limiting cases, non-relativistic and relativistic. See equations in slides, and 3.65 and 3.66 in the Hansen textbook.

In the textbooks:

  • Hansen section 3.6

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We then set to find the conditions (density and temperature) where the ideal gas pressure and the degeneracy pressure are equally important. In the notebook, we shaded the regions where non-relativistic degeneracy pressure and relativistic degeneracy pressure are important on a log(T)-log(rho) graph (see fig 3.6 in Hansen)

In the textbooks:

  • Hansen section 3.5.3

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Finally, the equation of state for the degeneracy pressure (P = ….) does not depend on temperature – this means that all of the equations we derived for polytrops are valid here! Objects for which degeneracy pressure is important, and even dominant, should be cold and dense (according to our log(T)-log(rho) graph). In the notebook, you will look at the typical density of a white dwarf.

You will also use the equations for polytropes to determine what would be the radius of white dwarf stars of various masses, in the non-relativistic case, and in the relativistic case (expect that something weird happens in the relativistic case :)

In the textbooks:

  • The Hansen has the best description in section 7.2.8, in the polytrop chapter.