= Deep radiative boundary beneath a power-law-opacity convective envelope
{title2=$T_b\propto T_e^{-(8\beta+2)/7}$}
In a negligible-mass, non-burning stellar envelope with $P=KT^{5/2}$ and <Kramers' opacity law> $\kappa=\kappa_1\rho T^{-7/2}$, the <stellar radiative temperature gradient> is $\nabla_{\mathrm{rad}}=3\kappa_1\mu K^2L/(16\pi ac\mathcal RGM T^{7/2})$. It decreases inward as the temperature rises, so the inner edge of efficient <convection> is where it equals $2/5$. If outer matching gives $K\propto g^{1/2}T_e^{-2\beta-5/2}$, then $T_b^{7/2}\propto K^2L/M\propto T_e^{-4\beta-1}$, since $g=GM/R^2$ and $L=4\pi R^2\sigma T_e^4$. Hence $T_b\propto T_e^{-(8\beta+2)/7}$.
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