= Hayashi relation with thirteenth-power opacity
{c}
{title2=$L\propto M^{8/17}R^{38/17}$}
Match an efficient, <monatomic gas>, fully convective star to the <grey-atmosphere convection onset with thirteenth-power opacity>. Its convective relation is $P=K_TT^{5/2}$. At the matching <temperature> $T_b\propto T_e$, the <stellar atmosphere> gives $K_T^2\propto ML^{-1}T_e^{-13}\propto MR^{13/2}L^{-17/4}$ 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 $K_\rho\propto GM^{1/3}R$, hence $K_T^2\propto M^{-1}R^{-3}$. Equating the two expressions proves $L^{17/4}\propto M^2R^{19/2}$, or $L\propto M^{8/17}R^{38/17}$. The entropy coefficient is constant throughout the <convective envelope>, but can change between models.
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