Why can't the degeneracy pressure self-adjust itself to resist gravitational collapse?

Question

After a star becomes a White dwarf, it resists gravitational collapse mainly due to the electron degeneracy pressure. If the mass of the white dwarf is greater than the Chandrasekhar limit, the degeneracy pressure cannot resist the collapse any longer and is doomed to become a neutron star or a black hole. Why can't the degeneracy pressure keep on self-adjusting itself to resist collapse forever?

Answer 1

The basic problem is that for a sufficiently massive star, the electrons become relativistic. The fine details of this calculation are rather complicated, but you can get a qualitative sense of the argument as follows:

For non-relativistic fermions at zero temperature, it is possible to show that the total energy of $N$ particles in a box of volume $V$ is proportional to $N^{5/3}/V^{2/3}$. This can be done via counting the density of states, and using the fact that the energy of a non-relativistic particle obeys $E \propto |\vec{p}|^{2}$. For a spherical volume of radius $R$, we have $R \propto V^{1/3}$, and the number of fermions present is proportional to the mass. This means that the total energy of the fermions is proportional to $M^{5/3}/R^2$. This energy is positive.

On the other hand, the gravitational energy of a solid sphere is negative and proportional to $M^2/R$. This means that the total energy is the sum of a negative $R^{-1}$ term and a positive $R^{-2}$ term, and such a function will have a minimum somewhere. This will be the equilibrium point. At smaller radii, the energy of the degeneracy grows faster than the binding energy decreases, pushing the radius back to larger values. At larger radii, the reverse occurs. This means that the star will be stable.

This argument doesn't hold up to arbitrarily large energies, though, because eventually the Fermi energy of the electrons exceeds the rest energy of the electron; in other words, the electrons become relativistic. This changes the relationship between energy and momentum of the electrons. For highly relativistic electrons, we have $E \propto |\vec{p}|$ instead; and going through the same calculations (neglecting the electron mass entirely), we find that the total energy of a relativistic fermion gas is proportional to $N^{4/3}/V^{1/3} \propto M^{4/3}/R$.

The gravitational binding energy, on the other hand, remains negative and proportional to $M^2/R$. This implies that the overall energy is itself proportional to $1/R$, and there is no extremum of the total energy of the system. Since the fermion energy and the binding energy always increase or decrease at exactly the same rate, there will be no stable equilibrium radius. The star will either blow itself apart or collapse in on itself, depending on whether the kinetic energy of the fermions or the gravitational binding energy wins out.

Answer 2

An alternative: As the white dwarf mass increases, the electrons become ultra-relativistic. Hydrostatic equilibrium is not possible for ultra-relativistic degeneracy pressure.

Hydrostatic equilibrium requires: $$\frac{dP}{dr} = - \rho g \ . $$ Working just with proportionalities, non-relativistic degeneracy pressure $\propto \rho^{5/3} \propto M^{5/3} R^{-5}$, where $\rho$ is density, $M$ is mass and $R$ is radius. So the LHS and RHS of the hydrostatic equilibrium equation can be written \begin{eqnarray} M^{5/3} R^{-6} & \propto & (M R^{-3})(M R^{-2}) \\ & \propto & M^2 R^{-5}\ . \end{eqnarray} For a given mass, the radius can be adjusted to find an equilibrium.

For a more massive star, that equilibrium radius goes as $R \propto M^{-1/3}$, so a more massive star has a smaller radius, higher density; the electron Fermi energy increases and the electrons become ultra-relativistic.

Ultra-relativistic electron degeneracy pressure is proportional to $\rho^{4/3} \propto M^{4/3} R^{-4}$. Inserting this into the hydrostatic equilibrium equation we see \begin{eqnarray} M^{4/3} R^{-5} & \propto & (M R^{-3})(M R^{-2}) \\ & \propto & M^2 R^{-5}\ , \end{eqnarray} and thus there is no possible adjustment in radius that can make this equation balance. It is satisfied (but unstable) for just one mass - the Chandrasekhar mass.

Edit:

Note that this simple argument is conservative. In practice, there is a lower mass beyond which a stable configuration is not possible, for at least two reasons.

1. At small radii and high densities, the electron Fermi energies become high enough to cause electron-capture reactions (or neutronisation). This removes electrons from the gas, and lowers the adiabatic index below $4/3$ and collapse (or thermonuclear detonation) ensues.

2. The above treatment uses a Newtonian treatment of hydrostatic equilibrium. This is not appropriate for very small white dwarfs. The Tolman-Oppenheimer-Volkoff General Relativistic equation of hydrostatic equilibrium should be used instead. This also features pressure on the right hand side. This means increasing pressure demands an ever-increasing pressure gradient, which leads to instability at a finite density and at a mass lower than the non-General-Relativistic Chandrasekhar mass.