= Grey-atmosphere pressure with power-law opacity
{title2=$P^\alpha=\frac{2\alpha g}{3\beta\kappa_0T_0^4}(T^{4\beta}-T_0^{4\beta})$}
In a <grey atmosphere> with $T^4=T_0^4(1+3\tau/2)$ and $\kappa=\kappa_0P^{\alpha-1}T^{4-4\beta}$, <hydrostatic equilibrium in optical depth> gives $dP/d\tau=g/\kappa$. With $\alpha,\beta>0$ and $P(0)=0$, integration gives the displayed pressure-temperature relation. At the matching surface $T=T_e$, the <Eddington surface boundary condition> gives $T_e^4=2T_0^4$, hence $P\kappa/g=4\alpha(1-2^{-\beta})/(3\beta)$.
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