Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 58 / 1 / b / Solution

Evaluate radiation pressure on the gas-supported hydrostatic equilibrium solution. At the centre,The factors $R^{-4}$ cancel between $T_c^4$ and $P_c$, so at fixed mean molecular weight the ratio depends only on mass. Equivalently it is $(M/M_{\mathrm{eq}}{)}^2$, withFor $H\simeq 1.6605\times 10^{-27}\mathrm{kg}$ this is approximately $14.6{\mu }^{-2}{M}_{\odot }$, or $40.5M_{\odot }$ if $\mu =0.6$. If $H$ denotes hydrogen-atom mass instead, use the corresponding mean molecular weight convention consistently; the coefficient is about $14.4{\mu }^{-2}{M}_{\odot }$. Equality marks the breakdown of the gas-pressure approximation. It is a diagnostic based on the gas-only model, not an exact mixed-pressure stellar solution.