= Solution
For a vertical ray, the formal <Schwarzschild equation> solution is
$$
I_\nu(0)=B_\nu(T_s)e^{-\tau_{\nu,s}}+\int_0^{\tau_{\nu,s}}B_\nu[T(t)]e^{-t}\,dt.
$$
In 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 motivates
$$
\boxed{p_{\rm rad}(\nu)\sim p_s\frac{|\nu-\nu_0|}{w},}
$$
limited 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 $T_s>T_{\rm strat}$, put $p_t=p_s(T_{\rm strat}/T_s)^{1/x}$ and $d_t=w p_t/p_s$. With the vertical optical-depth-one estimate, the spectral <outgoing longwave radiation> is approximately
$$
\boxed{F_\nu\sim\begin{cases}
\pi B_\nu(T_{\rm strat}),&d\lesssim d_t,\\
\pi B_\nu\!\left[T_s(d/w)^x\right],&d_t\lesssim d\lesssim w,\\
\pi B_\nu(T_s),&d\gtrsim w.
\end{cases}}
$$
The 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 $T_s=T_{\rm strat}$, the entire column and ground are isothermal and there is no line contrast. If $T_s<T_{\rm strat}$, 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.
\Image[/past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2019-line-emission.png]
{title=An isolated absorption line samples an isothermal stratosphere, a warmer troposphere and the surface}
{description=Illustrative narrow-line model with $T_s=300\,\mathrm K$, $T_{\mathrm{strat}}=200\,\mathrm K$, $x=2/7$ and line centre $667\,\mathrm{cm^{-1}}$; horizontal offsets are normalized by $w$. No measured spectrum is reproduced.}
Back to article page
