White Dwarfs and the Chandrasekhar Mass

I've previously written a bit about evolution of main sequence stars once they leave the main sequence, but what I didn't mention is that basically any star lighter than a few solar masses will eventually peter off into a white dwarf star. These are 'stars' that no longer produce heat. They are entirely at equilibrium between gravitational contraction pulling in and electron degeneracy pressure pushing out. The only reason we can see them at all is that they still radiate some of the thermal energy left over from the days when they were still undergoing fusion. As they do this, they cool and dim, until eventually they are (pretty much) unobservable. These compact bodies will exist pretty much forever, unless other objects interact with them.
We know that these stars are entirely supported by degeneracy pressure, which we can quantify in terms of density and composition of the star. We can also calculate the necessary pressure in the center of the star to avoid collapse, which primarily depends on the radius and density of the star. Setting these equal allows us to derive the mass-radius relation for white dwarfs, which describes the preferred radius of a white dwarf with a given mass. It turns out that in general, for stars in which the electrons are nonrelativistic, $MR^3$ is a constant. That seems a little strange at first - normally we expect stars to increase in size when they get more mass. What it's saying is that if the mass of the white dwarf increases, its radius has to decrease in order to increase the density enough to provide the necessary supporting degeneracy pressure. If we keep adding mass, the electrons pack in tighter and tighter, which means they have higher and higher energies. Eventually our assumption that the electrons are nonrelativistic starts to break down, and we end up with a new expression for degeneracy pressure in ultrarelativistic white dwarfs. (This means the electrons are moving close to the speed of light, not that the star itself is careening off through space.) In this expression, the degeneracy pressure depends less on the density of the electrons in the star. When we try to use it to find a new and improved mass-radius relation, we find that with a given mass, a star has no preferred radius! The star is now unstable, and if we compress it a little, the extra degeneracy pressure isn't enough to combat the compression. It instead starts an unstable collapse until the protons in the nuclei start to capture electrons. As a result, the whole thing becomes a neutron star.
Luckily this doesn't happen for all white dwarfs, just the ultrarelativistic ones. And a white dwarf can only go ultrarelativistic if it has a sufficiently high mass, called the Chandrasekhar mass, of around 1.4 solar masses.
Okay...we now have a neutron star supported by neutron degeneracy pressure. The pressure equation takes the same form as for electrons, so the mass-radius relation (for nonrelativistic neutron stars, anyway) is the same - $MR^3$ is a constant, albeit a somewhat different constant than in the electron case. But once again, if the neutron star passes a certain mass (probably around 3 solar masses), the neutrons start to go ultrarelativistic, and we have an unstable collapse again, this time into a black hole.
We have some (very) fuzzy ideas about what the inside of a neutron star is like - I've heard rumors of neutrons being simultaneous superfluids and superconductors, but it's still very much an active area of research. Keep your eyes peeled!