Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 317 / 1 / i / Solution

Assume stellar homology: fixed dimensionless profiles, uniform fixed composition, negligible radiation pressure, ideal gas support, and radiative transport throughout the model. Mass conservation gives ${\rho }_c\propto M/R^3$. The hydrostatic pressure support equation gives $P_c\propto G{M}^2/R^4$; combining with $P\propto \rho T$ gives $T_c\propto M/R$.

Integrating specific proton–proton chain energy generation over mass givesFor radiative diffusion in a star, $T_c/R\propto {\kappa }_c{\rho }_{c}L/(R^2{T}_c^3)$, hence $L_{\mathrm{rad}}\propto R{T}_c^4/({\kappa }_c{\rho }_c)$. With the Kramers' opacity law ${\kappa }_c\propto {\rho }_c{T}_c^{-7/2}$,Equating generated and transported luminosity gives $M^{1/2}\propto R^{13/2}$, so the radiative homology with proton-proton burning and Kramers opacity relation isThe proportionality constants are fixed only within the adopted homologous family, not for arbitrary evolving stars.