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: where n is the density of absorbing/emitting molecules, σλ is their absorption cross-section at wavelength λ, Bλ(T) is the Planck function for temperature T and wavelength λ, Iλ is the spectral intensity of the radiation entering the increment ds. This equation and a variety of 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), σλ(s), T(s), and Iλ(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 contains the fundamental physics needed to understand and quantify how increasing greenhouse gases (GHGs) in the atmosphere reduce the flux of thermal infrared radiation to space. If no other fluxes change, the law of conservation of energy demands that the Earth warm (from one steady state to another) until balance is restored between inward and outward fluxes. Schwarzschild's equation alone says nothing about how much warming would be required to restore balance. When meteorologists and climate scientists refer to "radiative transfer calculations" or "radiative transfer equations" (RTE), the phenomena of emission and absorption are handled by numerical integration of Schwarzschild's equation over a path through the atmosphere.