 Grey-atmosphere pressure with power-law opacity

In a grey atmosphere with $T^4=T_0^4(1+3\tau /2)$ and $\kappa ={\kappa }_0{P}^{\alpha -1}{T}^{4-4\beta }$, 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 $P\kappa /g=4\alpha (1-2^{-\beta })/(3\beta )$.