 Solution

The pressure-linear grey opacity model interprets $\kappa$ as a mass absorption coefficient. If it is instead a molecular absorption cross section, replace $\kappa$ below by that cross section divided by molecular mass. Assume a plane-parallel atmosphere, hydrostatic balance, local thermodynamic equilibrium, and thermal absorption and emission without scattering of light. With height $z$ increasing upwards,Integrating upwards from the surface givesThusThe vertical optical depth is dimensionless. The atmosphere is optically thick when ${\tau }_{\infty }\gg 1$ and optically thin when ${\tau }_{\infty }\ll 1$. A ray with zenith-angle cosine $\mu$ sees optical depth ${\tau }_{\infty }/\mu$, so even a vertically thin layer can be opaque along a sufficiently grazing ray.