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For a vertical ray, the formal Schwarzschild equation solution isIn the wing model $\tau \propto p^2$, the contribution per interval $d\ln p$ is proportional to $2\tau e^{-\tau }$ when the source varies slowly, and is concentrated near $\tau =1$. This motivateslimited to $p_s$ in transparent spectral regions. Rays contributing to the hemispheric flux replace this by $p_s\sqrt{\mu }d/w$, an order-one angular refinement.

For $T_s>T_{\mathrm{strat}}$, put $p_t=p_s(T_{\mathrm{strat}}/T_s{)}^{1/x}$ and $d_t=w{p}_t/p_s$. With the vertical optical-depth-one estimate, the spectral outgoing longwave radiation is approximatelyThe cold flat central portion comes from the isothermal stratosphere; intermediate wings sample progressively warmer, deeper tropospheric levels; far outside the line the warm surface is seen. The transitions are smooth in the exact transfer solution. If $T_s=T_{\mathrm{strat}}$, the entire column and ground are isothermal and there is no line contrast. If $T_s<T_{\mathrm{strat}}$, the given max-profile has no cooler troposphere: the atmosphere can instead produce an emission feature above the colder surface background. Assuming a narrow isolated line permits the local Planck function background to be treated as nearly constant across its width. The original schematic below also compares the optical-depth-one estimate with angular and source-function integration for illustrative, explicitly chosen parameters.

Use a convective layer $T(z)=T_s-\Gamma z$ capped at an isothermal minimum $T_t$ at height $z_t$. Let the initial optical depth be $\tau (z)={\tau }_s{e}^{-z/H_{\tau }}$ and ${\tau }_t={\tau }_s{e}^{-z_t/H_{\tau }}$. Within the lapse-rate layer, $d{B}_{\lambda }/d\tau =B^{\prime }\Gamma H_{\tau }/\tau$; above it the derivative vanishes. The supplied Schwarzschild equation therefore givesAssume the additional carbon dioxide multiplies the relevant opacity by $a=1.1$, leaving temperatures, the lapse rate and $H_{\tau }$ initially fixed. All material surfaces then acquire $a$ times their previous optical depth. HenceFor ${\tau }_t\ll 1$ and ${\tau }_s\gg 10$, the specified approximation givesFor illustration, $\Gamma =6.5\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$ and $H_{\tau }=7\mathrm{km}$ give $-4.34\mathrm{K}$; these are illustrative, not supplied data. The inequalities put the effective emitting level $\tau \sim 1$ inside the decreasing-temperature layer, well above an opaque surface and below the thin isothermal cap. Increasing opacity raises that emitting level by $H_{\tau }\ln 1.1$, so it is colder. In a spectral window or above a temperature inversion the approximation and possibly the sign change. Opacity proportional to mixing ratio is also an assumption, not exact for every absorption mechanism.