 Solution

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.