Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 317 / 2 / a / Solution

Use stellar homology: dimensionless profiles and composition are fixed, so proportionality constants do not change along the model sequence. Scaling mass conservation and the stellar hydrostatic equation giveswhere the last relation uses the ideal gas equation of state with fixed mean molecular weight. The specific stellar energy-generation rate is $ϵ\propto \rho T^{\eta }$, soBy stellar radiative diffusion, $T_{\ast }/R\sim {\kappa }_{\ast }{\rho }_{\ast }L/R^2{T}_{\ast }^3$, henceEquate the generated and transported luminosity in thermal equilibrium. The homology relations for radiative stars giveThis division presumes a nonzero denominator. If it vanishes, the scaling balance instead imposes a condition on $M$ or leaves the radius scaling undetermined, depending on the numerator. For later use, the corresponding mass-luminosity relation exponent is