= Fixed-radius-core shell-burning luminosity relation
{title2=$L\propto M_c^{97/8}\quad(R_c\text{ fixed})$}
Use a <Kramers radiative-zero envelope around a stellar core> and specific heating $\epsilon=\epsilon_0\rho T^{67/4}$. Since $\rho\propto CT^{13/4}$ and $T\propto M_c/r$, $\epsilon\propto Cr^{-20}$ and $\epsilon(1.05R_c)/\epsilon(R_c)=1.05^{-20}\simeq0.377$. The volume heating is $\rho\epsilon\propto C^2T^{93/4}$. Integrating $4\pi r^2\rho\epsilon$ from $R_c$ outwards gives $L\propto C^2M_c^{93/4}R_c^{-81/4}$, while radiative transport gives $C^2\propto M_c/L$. Therefore $L^2\propto M_c^{97/4}R_c^{-81/4}$, or $L\propto M_c^{97/8}R_c^{-81/8}$. For a core sequence with fixed $R_c$, the <mass> exponent is $97/8$. The steep integrand confines most heating to a thin shell; using the exterior constant-luminosity profile throughout it estimates the coefficient rather than resolving the shell structure.
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