Why are more massive white dwarfs smaller?

Question

I understand that a white dwarf is supported by the Pauli Exclusion Principle and that the larger the gravitational force against them, the closer the electrons must pack.

But I have two queries: One is that for more massive white dwarf stars, there would be more particles present in the star's core, which should take up more space. Is there any way in seeing why this isn't a significant effect?

Also, we haven't studied quantum mechanics and the Pauli Exclusion Principle in detail so it seems weird to me that electrons can just pack closer. This is my understanding, the Pauli Exclusion Principle means no two electrons with the same spin can share the same quantum state. For this effect to produce a force, wouldn't it mean that the electrons were already the closest they could be, as any closer they would be in the same quantum state. How does the Pauli Exclusion Principle work to counteract gravity here?

Answer 1

The Pauli Exclusion Principle is that indistinguishable fermions cannot occupy the same quantum state. The quantum state of an electron in an ideal degenerate gas is just specified by its spin and momentum. As you pack the electrons closer together, in order to avoid occupying the same spin/momentum state, then the electrons must occupy ever higher momentum states (since only two electrons, spin up/down, can occupy each momentum state). It is this increasing momentum which leads to electron degeneracy pressure.

As for why a more massive white dwarf becomes smaller, this is just a consequence of the particular equation of state. If the pressure is proportional to density to the power of $\alpha$ with $4/3 <\alpha < 2$, then the equilibrium radius is a decreasing function of mass. Essentially, the requirement of an increased pressure gradient to support the increased weight can only be satisfied by a decrease in radius, because the central pressure does not increase enough with density. As a consequence of being supported by electron degeneracy pressure, $\alpha$ varies between about 5/3 in a low-mass, low density white dwarf to nearly 4/3 in a more massive, high density white dwarf.