Degenerate-core shell-burning mass-luminosity relation ($L\propto M_c^{32/3}$)

In a thin-shell giant model with a nonrelativistic degenerate core, $R_c\propto M_c^{-1/3}$. For a gas-pressure radiative layer with $\kappa \propto \rho T^{-3}$ and specific energy generation $ϵ\propto \rho T^{10}$, the deep profile has $P\propto T^4$, $\rho \propto T^3$ and $T\propto M_c/r$. Integrating the strongly concentrated shell power gives $L\propto C^2{M}_c^{16}{R}_c^{-13}$, while radiative transport gives $C^2\propto M_c/L$. Therefore $L^2\propto M_c^{17}{R}_c^{-13}$, yielding the displayed exponent. This is a model-specific scaling, not a universal core-mass law; it fails when envelope stripping removes the shell fuel or the confining structure.

A negligible-mass radiative envelope surrounding a core of mass $M_c$, carrying constant luminosity and having constant opacity, obeys $dP/dT=16\pi acG{M}_c{T}^3/(3{\kappa }_{0}L)$. Neglecting the outer pressure integration constant gives $\Lambda =4\pi acG{M}_c/(3{\kappa }_{0}L)$. The gas pressure fraction $\beta$ is constant because $P_{\mathrm{rad}}=a{T}^4/3$, and $\rho =\mu \beta \Lambda T^3/\mathcal{R}$. Hydrostatic balance gives $T=T_s+\mu \beta G{M}_c(1/r-1/R_s)/(4\mathcal{R})$; dropping this outer-boundary constant gives the displayed deep-envelope approximation. A positive gas-pressure fraction requires $L<4\pi cG{M}_c/{\kappa }_0$.

With specific heating $ϵ={ϵ}_0\rho T^{16}$, the constant-opacity radiative-zero giant envelope has $T\propto \mu \beta M_c/r$ and $\rho \propto \mu \beta M_c{T}^3/L$. Therefore the shell power $4\pi r^2\rho ϵ$ is proportional to $(\mu \beta M_c{)}^{24}{L}^{-2}{r}^{-20}$. Integrating from the core radius gives $L^3\propto (\mu \beta M_c{)}^{24}{R}_c^{-19}$. The steep radial dependence justifies a thin-shell estimate dominated by the base. Using the outer radiative profile throughout the emitting layer is an approximation; a resolved shell can change the numerical coefficient. Also $\beta =1-{\kappa }_{0}L/(4\pi cG{M}_c)$, so retaining radiation pressure makes the relation implicit in luminosity.