= Solution
{c}
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,
$$
d\tau=\kappa\rho\,dz=-\frac{\kappa}{g}\,dp.
$$
Integrating upwards from the surface gives
$$
\tau(p)=\frac{\kappa_s}{gp_s}\int_p^{p_s}p'\,dp'
=\frac{\kappa_sp_s}{2g}\left[1-\left(\frac p{p_s}\right)^2\right].
$$
Thus
$$
\boxed{\tau_\infty=\frac{\kappa_sp_s}{2g},\qquad
\tau(p)=\tau_\infty[1-(p/p_s)^2].}
$$
The vertical <optical depth> is dimensionless. The <atmosphere> is optically thick when $\tau_\infty\gg1$ and optically thin when $\tau_\infty\ll1$. 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.
Back to article page
