= Convective contraction with power-law opacity
{title2=$T_e^8\propto\mu^4M^2R$}
For a monatomic fully convective <ideal gas> star, uniform <specific entropy> gives $P=KT^{5/2}$ with $K\propto\mu^{-5/2}M^{-1/2}R^{-3/2}$. Combining this with the <stellar surface boundary condition> $\kappa P=2g/3$ and $\kappa=\kappa_0\rho^aT^b$ gives
$$
T_e^{(3a+5)/2+b}\propto
\mu^{(3a+5)/2}M^{(a+3)/2}R^{(3a-1)/2}.
$$
All composition comparisons here hold $\kappa_0$ fixed. For $a=1,b=4$, $T_e^8\propto\mu^4M^2R$, and the <Stefan–Boltzmann law> gives $L\propto\mu^2MR^{5/2}$. The <stellar virial theorem> and <Kelvin-Helmholtz contraction> then give $R^{-7/2}-R_0^{-7/2}\propto\mu^2M^{-1}(t-t_0)$, with a positive coefficient. Once the initial-radius term is negligible, $R\propto\mu^{-4/7}M^{2/7}(t-t_0)^{-2/7}$.
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