Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 317 / 3

Write the mean interior density asSincethe assumed outward decrease of $\bar{\rho }$ implies $\rho \le \bar{\rho }$. In mass coordinates, hydrostatic equilibrium isFor every interior mass $m\le m_r$, monotonicity givesIntegrating from the centre and using ${\int }_0^{m_r}{m}^{-1/3}dm=3m_r^{2/3}/2$ proves

Let $\mathcal{R}$ be the gas constant per mole. At the centre,Eliminating $T_c$ gives the Eddington quartic relation in its central form,

At the surface, set $P=0$ and $m_r=M$ in the upper pressure bound. Cubing it givesThe function $(1-\beta )/{\beta }^4$ decreases strictly as $\beta$ increases on $0<\beta <1$. Define ${\beta }^{\ast }$ by equality in the resulting bound:Then ${\beta }_c\ge {\beta }^{\ast }$, orand rearrangement gives exactly

Radiative diffusion in a star can be writtenDividing by hydrostatic equilibrium giveswhich is constant by assumption. Since both $P_{\mathrm{rad}}$ and $P$ vanish at the surface, integration gives $P_{\mathrm{rad}}=(1-\beta )P$ with constant $\beta$.

Eliminating $T$ between the gas and radiation equations of state now givesThus the star is an $n=3$ stellar polytrope. PutThe structure equations become the Lane-Emden equationThe surface is the first zero ${\xi }_1\simeq 6.89685$. Writingthe Lane-Emden mass formula givesThereforewhereThe additional radiative-equilibrium relation is $1-\beta =\nu L/(4\pi cGM)$.