 Solution

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.