Deep radiative boundary beneath a power-law-opacity convective envelope ($T_b\propto T_e^{-(8\beta +2)/7}$)

In a negligible-mass, non-burning stellar envelope with $P=K{T}^{5/2}$ and Kramers' opacity law $\kappa ={\kappa }_1\rho T^{-7/2}$, the stellar radiative temperature gradient is ${\nabla }_{\mathrm{rad}}=3{\kappa }_1\mu K^{2}L/(16\pi ac\mathcal{R}GM{T}^{7/2})$. 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 $T_b^{7/2}\propto K^{2}L/M\propto T_e^{-4\beta -1}$, since $g=GM/R^2$ and $L=4\pi R^2\sigma T_e^4$. Hence $T_b\propto T_e^{-(8\beta +2)/7}$.