= Solution
$I_+$ and $I_-$ are the positive upward and downward hemispherically integrated infrared fluxes, in $\mathrm{W\,m^{-2}}$, rather than directional specific intensities. The source is the <blackbody> hemispheric flux $\sigma T^4$. Absorption removes a fraction proportional to the flux along a ray, and thermal emission replaces it according to Kirchhoff's law. <Local thermodynamic equilibrium> sets that emission source to the local <Planck function>.
A plane-parallel, non-scattering, <grey atmosphere>[gray atmosphere] with a <two-stream approximation>[two-stream/angular closure] reduces the angle- and frequency-dependent transfer equation to these two flux equations. The angular path factors are absorbed into the \i[effective] <optical depth>. Since the supplied coordinate increases upwards, the upward flux loses $I_+\,d\tau$ while gaining $\sigma T^4d\tau$ in moving upward; the downward ray traverses the coordinate in the opposite direction, reversing the differential signs in its equation. The equations therefore encode emission as well as absorption and are consistent with an isothermal source approaching <thermal radiation> in an optically thick medium. The source's use of “<radiance>” for these integrated quantities should not introduce an extra factor of $\pi$.
Back to article page
