# Why don't electrons and protons in a white dwarf combine before the electrons become ultrarelativistic?

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 $\gamma \to \infty$.

You have independently discovered that the Chandrasekhar mass, defined as the ultrarelativistic limit for the electron degeneracy pressure, $$M_{\rm Ch} \simeq 5.8 \mu_e^{-2}\ M_{\odot},$$ where $\mu_e$ is the number of mass units per electron in the gas, is never reached in practice. This Chandrasekhar mass is appropriate for an ideal gas of non-interacting, completely degenerate electrons and is calculated for Newtonian gravity.

The real "Chandrasekhar mass", if defined as the upper limit to the mass of a stable white dwarf star, is lower and may be due to interactions or General Relativity.

The phenomenon you describe is called inverse beta decay or sometimes, neutronisation. Electrons are indeed captured by protons in nuclei once the Fermi energy of the electrons becomes high enough.

The threshold for this reaction to occur is a (total) electron Fermi energy of 1.29 MeV if the capture is on to free protons. However free protons are not present in significant quantities in the interiors of white dwarf stars. Instead, the majority of protons (and neutrons) are inside the nuclei of carbon, oxygen (and perhaps for more massive white dwarfs, magnesium and neon) ions. The threshold to cause neutronisation of protons inside these nuclei is significantly higher, because the new nucleus that is formed has lower binding energy.

The neutronisation threshold electron energy for carbon is 13.9 MeV ($\gamma = 27$) and for oxygen it is 10.9 MeV ($\gamma=21$). If the Fermi energies of the electrons are this high then the electrons can be considered highly relativistic. These Fermi energies translate directly into a threshold density for the occurrence of neutronisation. For carbon this density is $3.9\times10^{13}$ kg/m$^3$ and for oxygen it is $1.9 \times 10^{13}$ kg/m$^3$.

White dwarfs with interior densities as high as this are actually very close to the "traditional" Chandrasekhar mass at about 1.37-1.38$M_{\odot}$. It is thus possible that neutronisation is what triggers an instability in a massive white dwarf, perhaps leading to a type Ia supernova. The situation is very confusing because General Relativity also causes an instability at almost exactly the same density in a carbon white dwarf and it is also thought that pyconuclear fusion reactions between carbon nuclei may commence at these densities too. At present it is unclear what actually determines the upper mass limit for a stable white dwarf, it seems likely to be GR for carbon white dwarfs, neutronisation for oxygen white dwarfs, but it is certainly below $1.4M_{\odot}$.

A very useful paper to look at is by Rotondo et al. (2011).