Constant-opacity gas-pressure stellar homology ($L\propto {\mu }^4{M}^3/\kappa$)

For homologous gas-pressure-supported stars, hydrostatic equilibrium gives $T_c\propto \mu M/R$ and ${\rho }_c\propto M/R^3$. Radiative diffusion gives $L\propto R{T}_c^4/(\kappa {\rho }_c)$, so the radius cancels and yields the displayed cubic mass scaling. Composition and opacity are held fixed. Strong radiation pressure alters this gas-pressure relation towards the Eddington luminosity scaling.