 Hydrostatic equilibrium in optical depth

In a plane-parallel atmosphere, take the vertical coordinate $z$ positive outward, inward-increasing optical depth $\tau$, positive opacity $\kappa$ and approximately constant downward gravitational acceleration $g$. ThenThus the hydrostatic equilibrium equation becomes a relation between pressure and optical depth, rather than geometric height. Its integrated form isIf $g$ and $\kappa$ vary little, $P(\tau )\simeq P(0)+g\tau /\kappa$. The familiar $P\simeq g\tau /\kappa$ additionally neglects the external pressure $P(0)$. For the Eddington surface boundary condition at $\tau =2/3$, this gives $P_{\mathrm{phot}}\simeq 2g/(3\kappa )$ under the same assumptions. A grey atmosphere with strongly pressure- or temperature-dependent opacity requires the integral, as in the grey-atmosphere pressure with power-law opacity calculation.