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, ( $dI λ$ , [W/sr/m²/μm]) at a given wavelength as radiation travels an incremental distance ( $ds$ ) through a non-scattering medium is given by:

$dI_{\lambda }=n\sigma _{\lambda }B_{\lambda }(T)\,ds-n\sigma _{\lambda }I_{\lambda }\,ds=n\sigma _{\lambda }[B_{\lambda }(T)-I_{\lambda }]\,ds$

where

$n$ is the number density of absorbing/emitting molecules (units: molecules/volume)
$σ λ$ is their absorption cross-section at wavelength $λ$ (units: area)
$B λ (T)$ is the Planck function for temperature $T$ and wavelength $λ$ (units: power/area/solid angle/wavelength - e.g. watts/cm²/sr/cm)
$I λ$ is the spectral intensity of the radiation entering the increment $ds$ with the same units as $B λ (T)$

This equation and various equivalent expressions are known as Schwarzschild's equation. The second term describes absorption of radiation by the molecules in a short segment of the radiation's path ( $ds$ ) and the first term describes emission by those same molecules. In a non-homogeneous medium, these parameters can vary with altitude and location along the path, formally making these terms $n (s)$ , $σ λ (s)$ , $T (s)$ , and $I λ (s)$ . Additional terms are added when scattering is important. Integrating the change in spectral intensity [W/sr/m²/μm] over all relevant wavelengths gives the change in intensity [W/sr/m²]. Integrating over a hemisphere then affords the flux perpendicular to a plane ( $F$ , [W/m²]).

Schwarzschild's equation is the formula by which you may calculate the intensity of any flux of electromagnetic energy after passage through a non-scattering medium when all variables are fixed, provided we know the temperature, pressure, and composition of the medium.