= Strong-line limit
{title2=$C\gg1$}
Consider a plane-parallel <atmosphere> in <hydrostatic equilibrium>, with constant <gravitational acceleration> $g$ and a uniformly mixed absorber of <mass fraction> $q$. Let its mass <absorption coefficient> be the <Lorentz line shape> $\kappa_\nu=(S/\pi)\Delta/[(\nu-\nu_0)^2+\Delta^2]$, with pressure-independent integrated strength $S$ and half-width $\Delta(p)=\Delta_s p/p_s$. This assumes a fixed-<temperature> <pressure broadening> model over the pressures used.
For vertical propagation from the top to pressure $p$, <optical depth> is $\tau_\nu=(q/g)\int_0^p\kappa_\nu(p')dp'$. Integration gives
$$
\tau_\nu(p)=C\log\!\left[1+\frac{\Delta(p)^2}{(\nu-\nu_0)^2}\right],\qquad C=\frac{qSp_s}{2\pi g\Delta_s}.
$$
The strong-line limit is $C\gg1$. At <optical depth> one, $|\nu-\nu_0|=\Delta(p)/\sqrt{e^{1/C}-1}\simeq\Delta(p)\sqrt C$, so the escaping <frequencies> lie in the far wings. Expanding the logarithm there recovers $\tau_\nu\simeq qS\Delta_s p^2/[2\pi gp_s(\nu-\nu_0)^2]$. This identifies both the dimensionless control parameter and the physical approximation; <pressure broadening> still requires a suitable collisional regime.
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