High-power shell burning in a constant-opacity giant envelope ($L\propto (\mu \beta M_c{)}^8{R}_c^{-19/3}$)

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.