 Convective contraction with power-law opacity

For a monatomic fully convective ideal gas star, uniform specific entropy gives $P=K{T}^{5/2}$ 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 $\kappa ={\kappa }_0{\rho }^a{T}^b$ givesAll composition comparisons here hold ${\kappa }_0$ fixed. For $a=1,b=4$, $T_e^8\propto {\mu }^4{M}^{2}R$, and the Stefan–Boltzmann law gives $L\propto {\mu }^{2}M{R}^{5/2}$. The stellar virial theorem and Kelvin-Helmholtz contraction then give $R^{-7/2}-R_0^{-7/2}\propto {\mu }^2{M}^{-1}(t-t_0)$, with a positive coefficient. Once the initial-radius term is negligible, $R\propto {\mu }^{-4/7}{M}^{2/7}(t-t_0{)}^{-2/7}$.