Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 317 / 2

Suppose first that the gas-pressure fraction $\beta =P_g/P$ is spatially constant, with $0<\beta <1$. This is a sufficient condition for a stellar polytrope. Sincethe ratio $P_{\mathrm{rad}}/P_g=(1-\beta )/\beta$ givesThereforewithFor an ordinary positive polytropic index one also assumes $0\le s<3$.

The radiative diffusion in a star equation can be written asBecause $P_{\mathrm{rad}}=(1-\beta )P$ and $\beta$ is constant, hydrostatic equilibrium givesEquating them and using $\eta =(L_r/L)/(m_r/M)$ yields

For $P=K{\rho }^{1+1/n}$, introduce the Lane-Emden equation variablesCombining mass conservation with hydrostatic equilibrium givesRegularity and normalization imposeThe stellar surface is the first positive zero ${\xi }_1$ of $\theta$. Integrating the equation once givesand hence

When $s=0$, part i gives the polytropic index $n=3$. Stellar homology givesbecause gas pressure is the fraction $\beta$ of the pressure required by hydrostatic equilibrium. The Kramers opacity law therefore scales asWith constant $\eta$, the opacity relation from part i gives

For an $n=3$ polytrope, $M\propto (K/G{)}^{3/2}$. The explicit $K(\beta )$ of part i obeysso the Eddington quartic relation isat fixed composition. Substitution gives