Strong-line limit ($C\gg 1$)

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 ${\tau }_{\nu }=(q/g){\int }_0^p{\kappa }_{\nu }(p^{\prime })d{p}^{\prime }$. Integration givesThe strong-line limit is $C\gg 1$. 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 ${\tau }_{\nu }\simeq qS{\Delta }_s{p}^2/[2\pi g{p}_s(\nu -{\nu }_0{)}^2]$. This identifies both the dimensionless control parameter and the physical approximation; pressure broadening still requires a suitable collisional regime.