Lorentz line shape (${\kappa }_{\nu }=\frac{S}{\pi }\frac{\Delta }{(\nu -{\nu }_0{)}^2+{\Delta }^2}$)

A normalized collisional line model ${\kappa }_{\nu }=(S/\pi )\Delta /[(\nu -{\nu }_0{)}^2+{\Delta }^2]$, with integrated strength $S$ and half-width $\Delta$. The centre scales as $S/\Delta$ and far wings as $S\Delta /(\nu -{\nu }_0{)}^2$. In a dilute pressure-broadened gas $\Delta$ scales approximately with pressure.

A regime where optical-depth-one transmission occurs in the far wings of a strongly absorbing line. With a uniformly mixed absorber, $\Delta ={\Delta }_{s}p/p_s$ and $C=qS{p}_s/(2\pi g{\Delta }_s)\gg 1$, the radiative-wing optical depth is $\tau \simeq qS{\Delta }_s{p}^2/[2\pi g{p}_s(\nu -{\nu }_0{)}^2]$. This condition does not make pressure broadening valid in the arbitrarily rarefied upper atmosphere.