= Solution
Use characteristic stellar values in the supplied radiative <temperature> gradient: $T_c/R\sim\kappa\rho L/(R^2T_c^3)$. Hence $L\propto T_c^4R/(\kappa\rho)$. With Kramers <opacity>, $\kappa=\kappa_0\rho T_c^{-3.5}$,
$$
L\propto\kappa_0^{-1}\frac{T_c^{7.5}R}{\rho^2}
\propto\kappa_0^{-1}\frac{T_c^{7.5}R^7}{M_s^2}.
$$
Hydrostatic support by an <ideal gas> gives $T_c\propto\mu M_s/R$, or $R\propto\mu M_s/T_c$. Substitution then yields
$$
\boxed{L\propto\kappa_0^{-1}\mu^7M_s^5T_c^{1/2}.}
$$
The result assumes homologous <density> and <temperature> profiles, comparable structural constants, gas-pressure support and radiative transport throughout. It is not simultaneously an electron-scattering-opacity result; replacing Kramers <opacity> changes the exponents. Convection, degenerate cores or a different <opacity> regime limit quantitative application to real evolved <helium> <stars>.
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