Fully convective homology with tenth-power surface opacity ($R\propto M^{5/8},L\propto M^2$)

For a monatomic ideal gas fully convective star, $P=K{\rho }^{5/3}$ and $K\propto R{M}^{1/3}$. Matching its adiabat to a photosphere with $P\sim g/\kappa$ and $\kappa \propto \rho T^{10}$ gives $T_{\mathrm{eff}}^{14}\propto M^{2}R$, and hence surface luminosity $L\propto M^{4/7}{R}^{16/7}$. Specific heating $ϵ\propto \rho T^5$ gives $L\propto M^7{R}^{-8}$. Equality yields the displayed laws and $T_{\mathrm{eff}}\propto M^{3/16}$, so the theoretical HR slope is $32/3$. Fixed composition, efficient convection and the surface matching approximation are essential.