Schwarzschild's equation for radiative transfer

In the study of heat transfer, Schwarzschild's equation is used to calculate radiative transfer (energy transfer via electromagnetic radiation) through a medium in local thermodynamic equilibrium that both absorbs and emits radiation. The incremental change in spectral intensity, (dIλ, [W/sr/m2/μm]) at a given wavelength as radiation travels an incremental distance (ds) through a non-scattering medium is given by: d I λ = n σ λ B λ ( T ) d s − n σ λ I λ d s = n σ λ [ B λ ( T ) − I λ ] d s {\displaystyle {\begin{aligned}dI_{\lambda }&=n\sigma _{\lambda }B_{\lambda }(T)\,ds-n\sigma _{\lambda }I_{\lambda }\,ds\\[1ex]&=n\sigma _{\lambda }\left[B_{\lambda }(T)-I_{\lambda }\right]\,ds\end{aligned}}} where n is the number density of absorbing/emitting molecules (units: molecules/volume) σλ is their absorption cross-section at wavelength λ (units: area) Bλ(T) is the Planck function for temperature T and wavelength λ (units: power/area/solid angle/wavelength - e.g.

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