= Fully convective homology with tenth-power surface opacity
{title2=$R\propto M^{5/8},\quad L\propto M^2$}
For a monatomic <ideal gas> <fully convective star>, $P=K\rho^{5/3}$ and $K\propto RM^{1/3}$. Matching its adiabat to a photosphere with $P\sim g/\kappa$ and $\kappa\propto\rho T^{10}$ gives $T_{\rm eff}^{14}\propto M^2R$, and hence surface luminosity $L\propto M^{4/7}R^{16/7}$. Specific heating $\epsilon\propto\rho T^5$ gives $L\propto M^7R^{-8}$. Equality yields the displayed laws and $T_{\rm eff}\propto M^{3/16}$, so the theoretical HR slope is $32/3$. Fixed composition, efficient convection and the surface matching approximation are essential.
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