Hayashi relation with thirteenth-power opacity ($L\propto M^{8/17}{R}^{38/17}$)

Match an efficient, monatomic gas, fully convective star to the grey-atmosphere convection onset with thirteenth-power opacity. Its convective relation is $P=K_T{T}^{5/2}$. At the matching temperature $T_b\propto T_e$, the stellar atmosphere gives $K_T^2\propto M{L}^{-1}{T}_e^{-13}\propto M{R}^{13/2}{L}^{-17/4}$ using the Stefan–Boltzmann law. The ideal-gas relation converts this to $P=K_{\rho }{\rho }^{5/3}$ with $K_{\rho }=(\mathcal{R}/\mu {)}^{5/3}{K}_T^{-2/3}$. The polytropic mass-radius relation at index $3/2$ gives $K_{\rho }\propto G{M}^{1/3}R$, hence $K_T^2\propto M^{-1}{R}^{-3}$. Equating the two expressions proves $L^{17/4}\propto M^2{R}^{19/2}$, or $L\propto M^{8/17}{R}^{38/17}$. The entropy coefficient is constant throughout the convective envelope, but can change between models.