Contraction law with density-linear fourth-power stellar opacity ($R^{-7/2}-R_0^{-7/2}=C{\mu }^2{M}^{-1}(t-t_0)$)

For a fixed-mass, fully convective monatomic gas protostar, write $P=K{T}^{5/2}$ and $K=K_0{\mu }^{-5/2}{M}^{-1/2}{R}^{-3/2}$. Matching the photosphere to $\kappa P=2g/3$ with $\kappa ={\kappa }_0\rho T^4$ gives $T_e^8=2G\mathcal{R}{\mu }^4{M}^{2}R/(3{\kappa }_0{K}_0^2)$. The Stefan–Boltzmann law gives $L=\pi ac[2G\mathcal{R}/(3{\kappa }_0){]}^{1/2}{K}_0^{-1}{\mu }^{2}M{R}^{5/2}$. The stellar virial theorem gives total energy $-3G{M}^2/(7R)$, so $L=-3G{M}^2\dot{R}/(7R^2)$. Integrating proves $C=(49\pi ac/6)[2\mathcal{R}/(3{\kappa }_{0}G){]}^{1/2}{K}_0^{-1}$. This holds before stellar nuclear fusion, changing opacity or electron degeneracy pressure invalidates the assumed contraction law.