= Surface boundary condition for a power-law-opacity grey atmosphere
{title2=$P\kappa/g=4\alpha(1-2^{-\beta})/(3\beta)$}
For a <plane-parallel atmosphere> in <radiative equilibrium>, constant <surface gravity of a star> $g$ and <opacity> $\kappa=\kappa_0P^{\alpha-1}T^{4-4\beta}$, match the <stellar structure> interior where the <radiative flux> satisfies $F=\sigma T^4$. The <grey atmosphere> law then puts the surface at <optical depth> $\tau=2/3$ with $T^4=2T_0^4$. Integrating <hydrostatic equilibrium in optical depth> from $P(0)=0$ gives $P^\alpha=2\alpha g(T^{4\beta}-T_0^{4\beta})/(3\beta\kappa_0T_0^4)$. Multiplication by $\kappa_0T^{4-4\beta}/g$ proves the displayed boundary condition for $\alpha>0$, $\beta\ne0$. At $\beta=0$ the integral is logarithmic and the continuous <limit> is $P\kappa/g=4\alpha\log2/3$. The relation assumes negligible <radiation pressure> and describes the <Eddington closure approximation>, rather than exact angular transfer.
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