Hayashi relation with eighth-power temperature opacity ($L\propto R^{98/47}{M}^{28/47}$)

Match a fully convective stellar polytrope of index $3/2$ to the grey-atmosphere convection onset with eighth-power temperature opacity. At the fixed matching optical depth, $T_b\propto T_e$ and $P_b\propto g^{2/3}{T}_e^{-16/3}$. The ideal-gas adiabat $P=K{\rho }^{5/3}$ gives $K\propto T_b^{5/3}{P}_b^{-2/3}\propto T_e^{47/9}{g}^{-4/9}$. The polytropic mass-radius relation gives $K\propto R{M}^{1/3}$; with $g\propto M/R^2$, this yields $T_e^{47}\propto R{M}^7$. The Stefan–Boltzmann law then gives the displayed luminosity. Composition and opacity normalization are held fixed between models.