Chandrasekhar's white dwarf equation

Remarks

It expresses hydrostatic equilibrium for a spherical ball of cold, fully degenerate electron gas. Let $x$ be the Fermi momentum of the electrons in units of $m_e c$, with $m_e$ the electron mass and $c$ the speed of light. Let $x_0$ be the central value of $x$. Then $\varphi = \sqrt{1 + x^2} / \sqrt{1 + x_0^2}$ and $C = 1/(1 + x_0^2)$. So $C$ near $1$ means a low central density and $C$ near $0$ a very high one. Example: take $C = 1/2$. The central value $\varphi(0) = 1$ satisfies $\varphi^2 \ge C$, and the solution decreases from $1$ until it reaches $\sqrt{1/2}$ at a finite radius $\eta_1$. It describes a white dwarf whose central electrons have Fermi momentum $x_0 = 1$. Limit case: for $C = 0$ the equation becomes $\eta^{-2} (\eta^2 \varphi')' + \varphi^3 = 0$, the Lane–Emden equation of index $3$, which governs the mass limit. Non-example: the constant $\varphi = 1$ is not a solution, since the left side equals $(1 - C)^{3/2} > 0$. For $C$ close to $1$, writing $\varphi^2 - C = (1 - C) \theta$ and rescaling $\eta$ turns the equation, to leading order in $1 - C$, into the Lane–Emden equation of index $3/2$.