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

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.