= High-power shell burning in a constant-opacity giant envelope
{title2=$L\propto(\mu\beta M_c)^8R_c^{-19/3}$}
With specific heating $\epsilon=\epsilon_0\rho T^{16}$, the <constant-opacity radiative-zero giant envelope> has $T\propto\mu\beta M_c/r$ and $\rho\propto\mu\beta M_cT^3/L$. Therefore the shell power $4\pi r^2\rho\epsilon$ 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 $\beta=1-\kappa_0L/(4\pi cGM_c)$, so retaining radiation pressure makes the relation implicit in luminosity.
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