Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 317 / 3 / ii

For the nonrelativistic white dwarf, combine the hydrostatic pressure support equation and enclosed mass equation:Eliminating $m$ yieldsIntroduce the Lane-Emden variables for a stellar polytrope,The resulting Lane-Emden equation isThe white dwarf surface is at the first zero ${\xi }_1$ of $\theta$, with $R=a{\xi }_1$.

Let $\omega =-{\xi }_1^2{\theta }^{\prime }({\xi }_1)$ be the Lane-Emden surface mass constant. The Lane-Emden mass formula givesThus $R\propto {\rho }_c^{-1/6}$ while $M\propto {\rho }_c^{1/2}$ at fixed composition. Eliminating central mass density gives the polytropic mass-radius relationMore explicitly,Increasing the mass of a nonrelativistic white dwarf decreases its radius. This scaling ceases to apply when relativistic electron degeneracy pressure becomes important.