= Contraction law with density-linear fourth-power stellar opacity
{title2=$R^{-7/2}-R_0^{-7/2}=C\mu^2M^{-1}(t-t_0)$}
For a fixed-mass, fully convective <monatomic gas> <protostar>, write $P=KT^{5/2}$ 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 $T_e^8=2G\mathcal R\mu^4M^2R/(3\kappa_0K_0^2)$. The <Stefan–Boltzmann law> gives $L=\pi ac[2G\mathcal R/(3\kappa_0)]^{1/2}K_0^{-1}\mu^2MR^{5/2}$. The <stellar virial theorem> gives total energy $-3GM^2/(7R)$, so $L=-3GM^2\dot R/(7R^2)$. Integrating proves $C=(49\pi ac/6)[2\mathcal R/(3\kappa_0G)]^{1/2}K_0^{-1}$. This holds before <stellar nuclear fusion>, changing <opacity> or <electron degeneracy pressure> invalidates the assumed contraction law.
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