 Hayashi contraction with seventeenth-power opacity

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 $P=K{T}^{5/2}$, $K\propto M^{-1/2}{R}^{-3/2}$ at fixed composition, while the boundary gives $K\propto g^{1/2}{T}_e^{-21/2}$. Thus $T_e^{21/2}\propto M{R}^{1/2}$ and the first displayed luminosity law follows. The stellar virial theorem for a monatomic gas gives total energy $-3G{M}^2/(7R)$, so Kelvin-Helmholtz contraction has $\dot{R}=-7L{R}^2/(3G{M}^2)$. 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.