= Hayashi relation with eighth-power temperature opacity
{c}
{title2=$L\propto R^{98/47}M^{28/47}$}
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 $K\propto RM^{1/3}$; with $g\propto M/R^2$, this yields $T_e^{47}\propto RM^7$. The <Stefan–Boltzmann law> then gives the displayed luminosity. Composition and opacity normalization are held fixed between models.
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