Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 14B / d / Solution

Since $R=a{\xi }_1$, the scaling in part (b) givesSubstitute this into $M=4\pi (R/{\xi }_1{)}^3{\rho }_c{Y}_n$. After collecting powers,whereThis is the polytropic mass-radius relation.

An $n=3$ polytrope models a white dwarf whose electrons are an ultrarelativistic degenerate gas. For $n=3$, the radius exponent vanishes and the mass is independent of both $R$ and ${\rho }_c$. This fixed limiting value is the Chandrasekhar mass: increasing the central density can no longer support a larger mass, so a cold nonrotating white dwarf above the limit cannot remain in hydrostatic equilibrium.