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Take $S$ as the integrated line mass absorption coefficient, and $q$ as the absorber mass fraction, assumed constant; for a pure absorbing gas set $q=1$. All spectral intervals below use the paper's wavenumber coordinate. The normalized Lorentz line shape isIf instead an absorption coefficient per length is wanted, multiply by absorber density. Neglecting temperature dependence, pressure broadening gives $\Delta (p)={\Delta }_{s}p/p_s$. In the line wings, with $d=|\nu -{\nu }_0|\gg \Delta (p)$,Define the line-opacity parameter and characteristic wavenumber widthAt an emitting level ${\tau }_{\nu }\sim 1$, the wing estimate implies $d/\Delta (p)\sim \sqrt{C}$. Thus the strong-line limit relevant to outgoing radiation isIt ensures that the opaque-to-transparent transition lies in the Lorentz wings, rather than the narrow central core, and the characteristic absorption width is much larger than ${\Delta }_s$. At a grazing ray angle the criterion is even easier to satisfy because of the larger optical depth. This is a criterion for the radiatively important wings, not a claim that collision broadening remains physical at arbitrarily small pressure.

For a vertical ray, the transmittance from $p_2$ to space is ${\mathcal{T}}_{\nu }(p_2)=e^{-{\tau }_{\nu }(p_2)}$. Direct integration of the full pressure-dependent Lorentz line shape gives, for $d>0$,The exact Lorentz-model transmittance and its strong-line limit are thereforeThe approximation is justified at the radiating level by $C\gg 1$. At zenith-angle cosine $\mu$, replace $\tau$ by $\tau /\mu$; if a hemispheric flux transmission is desired, its normalized value is $2{\int }_0^1\mu e^{-\tau /\mu }d\mu =2E_3(\tau )$. The pure Lorentz model formally gives infinite line-centre optical depth from every nonzero $p_2$ to $p=0$, because ${\kappa }_{{\nu }_0}\propto 1/p$. This is an unphysical central-core singularity, addressed in part (d), rather than a divergence of the integrated outgoing flux.