Chandrasekhar limit

Remarks

At high density the electrons are ultra-relativistic, the pressure tends to $K \rho^{4/3}$, and the star tends to a polytrope of index $3$. For a polytrope of index $3$ the mass $4 \pi \alpha^3 \rho_c \, \omega_3$ does not depend on the central density $\rho_c$, since $\alpha^3$ is proportional to $\rho_c^{-1}$; this common value is $M_{\mathrm{Ch}}$. For a white dwarf of carbon and oxygen, $\mu_e = 2$ and $M_{\mathrm{Ch}}$ is about $1.4$ solar masses. Without special relativity the pressure stays proportional to $\rho^{5/3}$, the star is a polytrope of index $3/2$ with $M \propto \rho_c^{1/2}$, and there is no upper bound on the mass. Real white dwarfs become unstable a little below $M_{\mathrm{Ch}}$, because of general relativity and the capture of electrons by nuclei.

Going deeper

For a student

The limit comes from comparing two power laws. Gravity needs a central pressure that scales like $M^{2/3} \rho^{4/3}$, and a relativistic degenerate gas supplies $K \rho^{4/3}$. The power of $\rho$ is the same, so equilibrium fixes the mass and not the density. A star whose core grows past the limit must collapse further, to a neutron star or a black hole, which is the start of a core-collapse supernova. For neutron stars the analogous bound is the Tolman–Oppenheimer–Volkoff limit, which needs general relativity.