White dwarf stars: limits to stability

Question

The Chandrasekhar mass limit $M_\text{Ch}$ for a cold, non-rotating white dwarf star is derived from the hydrostatic equilibrium assuming Newtonian gravity and a Lane-Emden polytrope with n=3. However, $M_\text{Ch}$ is not a realistic limit because it implies a vanishing star radius and infinite density.

Several authors have calculated more realistic stability limits, e.g. Rotondo et al. https://arxiv.org/abs/1012.0154. Among their improvements on the stability limit are: (a) the effect of general relativity using the Tolman-Oppenheimer-Volkoff (TOV) equation; (b) inverse bèta decay (=electron capture, neutronization); (c) Coulomb interactions of electrons and nuclei.

Question 1: is the TOV equation analogous to the classical hydrostatic equilibrium, but using GR instead of Newtonian gravity (and applying the TOV equation to degenerate neutrons instead of electrons when considering a neutron star instead of a white dwarf)?

Question 2: what causes the GR effects? Is it correct to say that the high density in a heavy white dwarf adds mass-energy, reducing the maximum stable mass?

Question 3: how can we explain without detailed calculations that the Coulomb interactions reduce the maximum stable mass?

Rotondo et al. calculate the maximum stable mass of white dwarfs with various compositions. One of them is a Fe-56 white dwarf, quite different from the usual He, C, O suspects when a low or intermediate mass star dies.

Question 4: is the Fe-56 white dwarf meant to be the electron degenerate core of a heavy main sequence star ($>8-10 M_\odot$) in a stage before it collapses by electron capture to become a neutron star (degenerate neutrons, not electrons)?

Answer 1

1. The TOV equation is just the differential equation expression for hydrostatic equilibrium in General Relativistic conditions. It can be applied for any equation of state.

2. The TOV equation can be written as $$\frac{dP}{dr} = -\left( \frac{Gm(r)\rho}{r^2}\right) \left( \frac{ (1 + P/\rho c^2)(1 + 4\pi r^3 P/m(r)c^2)}{1 - 2Gm(r)/rc^2}\right),$$ where $m(r)$ is the mass inside radius $r$, $P$ is the pressure and $\rho$ the density.

The first bracket on the RHS is the Newtonian version of hydrostatic equilibrium. The second bracket contains three factors: the two in the numerator become $>1$ when $P \rightarrow \rho c^2$ and the denominator becomes $<1$ when $r \rightarrow 2Gm(r)/c^2$ [the Schwarzschild radius]. This means the required pressure gradient in a dense star becomes larger than the same star in Newtonian gravitation. Ultimately though, a large pressure gradient requires a large interior pressure and/or a small radius and thus the terms on the right hand side become even more significant. Eventually increasing the pressure becomes self-defeating because the extra pressure (which is like a kinetic energy density) also increases the RHS and no stable solution can be found beyond a threshold mass at a finite density and pressure.

3. The Coulomb interactions have nothing to do with the TOV equation of state. In a white dwarf, the positive charge is concentrated in ions, whereas the negative charge is in the form of almost uniformly distributed electrons. If you calculate the electrostatic potential energy per unit volume of this configuration, it is negative. This is because the self-repelling electrons are, on average, further away from each other than they are from a positive nucleus and this makes the gas more compressible than if all the charge (negative and positive) were spread uniformly.

The Coulomb energy of a spherical, neutral "Wigner-Seitz" cell of radius $r_0$, taking account of the self potential energy of the electrons and the potential energy of the electrons with the nucleus is easily shown to be $$ E_C = -\frac{9}{40 \pi \epsilon_0}\frac{Z^2 e^2}{r_0},$$ where $Z$ is the atomic number of the nuclei. If each Wigner-Seitz cell has a volume $V= 4\pi r_0^3/3$ then the pressure is given by $$ P_C = -\frac{dE_c}{dV} = -\frac{dE_C}{dr_0}\frac{dr_0}{dV} = -\frac{9Z^2 e^2}{160\pi^2 \epsilon_0 r_0^4}$$ and thus the pressure contribution is negative.

Since the number density of electrons $n_e \propto r_0^{-3}$, then $P_C \propto -n_e^{4/3}$, which is the same dependence as ultrarelativistic degeneracy pressure, but negative. Thus at high densities, the Coulomb interactions simply reduce the degeneracy pressure at any density, thus leading to a lower threshold mass that can be supported.

4. The core of a massive star just before it collapses as a supernova resembles an "iron white dwarf". The core would be iron (and other iron-peak elements) and supported (briefly) by electron degeneracy pressure.

However, I think Rotondo et al. study this as an intellectual curiosity. A cold, degenerate equation of state is not really ideal for treating the core of a pre-supernova massive star. It was thought (back in the 90s), shortly after the release of Hipparcos parallax measurements for nearby white dwarfs, that some of them (particularly Procyon B) were too small to be explained in terms of a carbon/oxygen composition. Something with a larger number of mass units per electron was indicated - i.e. iron. These results have subsequently been explained in terms of poor radius estimates based on an incomplete understanding of white dwarf atmospheres at the time. Most white dwarfs are almost certainly carbon/oxygen, though He white dwarfs are possible through various binary evolution channels and some high mass stars may be able to leave behind a degenerate Ne/Mg core.