Past exam of the short options paper of the physics Prelims of the University of Oxford / 2022 / S02 / 2 / b / Solution

Write $m(r)$ for enclosed mass. The stellar hydrostatic equation is $dP/dr=-Gm\rho /r^2$. If radiation pressure dominates, compare it directly with the radiative diffusion result for $d{P}_r/dr$:This is the limiting Eddington luminosity scaling, assuming fixed composition and electron-scattering opacity.

If gas pressure dominates, use stellar homology: characteristic density $\rho \sim M/R_{\ast }^3$ and pressure $P\sim G{M}^2/R_{\ast }^4$. The ideal gas law then givesEstimating $|dT/dr|\sim T/R_{\ast }$ in radiative diffusion givesThus at fixed $\mu$ and constant electron-scattering opacity, $\boxed{L\propto M^3}$. These are stellar homology estimates for radiative stars, not exact relations valid for arbitrary composition and internal transport.