= Hayashi contraction with seventeenth-power opacity
{c}
{title2=$L\propto M^{8/21}R^{46/21},\quad L\simeq\frac{9GM^2}{67Rt}$}
Match an $n=3/2$ <stellar polytrope> to the <grey-atmosphere convection onset with seventeenth-power opacity>. At each instant the convective entropy coefficient obeys $P=KT^{5/2}$, $K\propto M^{-1/2}R^{-3/2}$ at fixed composition, while the boundary gives $K\propto g^{1/2}T_e^{-21/2}$. Thus $T_e^{21/2}\propto MR^{1/2}$ and the first displayed <luminosity> law follows. The <stellar virial theorem> for a monatomic gas gives total energy $-3GM^2/(7R)$, so <Kelvin-Helmholtz contraction> has $\dot R=-7LR^2/(3GM^2)$. Integrating at fixed mass gives $R^{-67/21}-R_0^{-67/21}\propto t$. The second displayed relation holds after the initial-radius term is negligible. The entropy coefficient is spatially constant but changes during contraction.
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