We can see this using two different approaches:
The Virial Theorem
In both cases we make use of scaling behavior.
The first case below is for Hydrostatic Equilibrium and the second on is for the virial theorem. Its important that you fully understand the power of examining scaling relations and their implications (this is also a good question for the midterm ...)
Note in the case of application of the virial theorem, the total internal energy, T, of a degenerate gas goes as its Volume times the energy density in the gas.
In the case of relativistic degeneracy, you should confirm that that you will get a scaling relation of the form:
That is, there is some unique mass that is required for the relativist degeneracy limit. This is called the Chandresekhar mass.
Convenient scaling relations for white dwarfs and neutron stars are given below. In general, for a fully ionized electron gas, the mean mass per free electron is 2, so the second term on the right hand side becomes 1. In a partially ionized gas, which will occur as the degenerate core cools, the radius will adjust itself to a new value.
Post Main Sequence Evolution.
As the core of the sun becomes fully Helium, its density will increase to about 15,000 g/cc at which point degenerate electrons will comprise about 50% of the total internal pressure.
The core temperature as this point is insufficient for the fusion of Helium and so the core contracts and heats, but the degeneracy condition of the core means that the heating does not produce additional pressure (expansion) of the core.
This leads to a thermal runaway, culuminating in the Helium Flash. The effects of the Helium flash are to greatly increase available phase space for the electrons and hence to remove the degeneracy from the core.
The core settles down, back to an ideal gas, with Helium to Carbon fusion as the energy source.
Throughout the period of core collapse, the heat generated by this collapse is transferred to shells around the star which begins to burn Hydrogen, and later Helium. Eventually you will get to a stage of double shell burning that will blow the outer layers of the star away, revealing a hot stellar core (which then ionizes these outer layers).
The outer layers glow through recombination emission and the result is one of the more spectacular objects in the sky, namely a Planetary Nebulae:
Good Summary of Post Main Sequence Evolution This summarizes everything said in class.