Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 317 / 1

The four stellar structure equations aretogether with the perfect-gas equation of stateHere $a_{\mathrm{rad}}$ is the radiation constant, $ϵ$ is the specific stellar energy-generation rate, and $\kappa$ is the opacity.

Put $x=r/R$. Integrating the prescribed density gives the enclosed massSince $m_R=M$,Integrating hydrostatic equilibrium inward from $P(R)=0$ givesThe ideal-gas law then givesIn particular,

At $x=1/2$, defineDifferentiating the temperature profile givesSince the luminosity is already constant there, the radiative diffusion in a star equation yields

For the Kramers opacity law $\kappa ={\kappa }_0\rho T^{-7/2}$, substitution giveswhereThus $\alpha =11/2$ and $\beta =-1/2$ in the notation of the question.

For constant electron-scattering opacity ${\kappa }_{\mathrm{es}}$,whereHence $\delta =3$ and $\gamma =0$: the radius cancels, and the mass dependence is shallower than under Kramers opacity.