I wanted to get a rough picture of how white dwarfs and the Chandrasekhar limit work. I wound up with an argument nearly identical to this one on Wikipedia up through the non-relativistic white dwarf. That is, I estimated the energy in a degenerate electron gas at 0K and found the regime where that energy is comparable to the gravitational potential energy of the white dwarf. This serves as an approximation of the true equilibrium condition.
However, when considering the limit where white dwarfs cannot exist, Wikipedia suggests going to the ultrarelativistic limit, where $p = E/c$ for an electron, and noting that the new equation for energy balance equation gives a unique mass, which we interpret as the limit of white dwarf mass.
My thought was that instead, as we add mass to the white dwarf, there will be a point where the kinetic energy per electron is similar to the energy needed to go from a proton and an electron to a neutron. At his point, the proton and electron combine to form a neutron. Almost all the kinetic energy of the electron disappears because the neutron is much more massive, so the point where this happens is where the electron kinetic energy is equal to $c^2$ times the mass difference between a neutron and an electron+proton pair. This mass difference is about 1.5 electron masses, so my condition has electrons being moderately relativistic ($\gamma \approx 1.5$) instead of ultrarelativistic ($\gamma = \infty$) as in Wikipedia.
Plugging in my condition, I got the same expression as Wikipedia for the Chandrasekhar limit (modulo some constant factor), but it seems like the physics is different. What's going on?
Some possible answers I haven't been able to fully evaluate yet:
- the protons can't simply capture an electron; you need to emit a neutrino as well, and this pushes the energy required up into the ultrarelativistic regime
- The energy of nuclei is more complicated than just the mass difference between protons and neutrons; we need to consider nuclear binding energy as well
- I'm ignoring the entropy loss when the electron is captured, but we can work at 0K and ignore this without sacrificing a basic understanding of white dwarfs, right?
- perhaps the mass required to get to $\gamma = 1.5$ (or some other number accounting for above effects) turns out not to be very different from the mass required for $\gamma = \infty$, so the Wikipedia calculation was right just by accident; as we add mass to the white dwarf, it actually becomes a neutron star before $\gamma \to \infty$, but the estimate for the transition mass is still basically right if we use $\gamma \to \infty$.