Radiative transfer (also called radiation transport) is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by absorption, emission, and scattering processes. The equation of radiative transfer describes these interactions mathematically. Equations of radiative transfer have application in a wide variety of subjects including optics, astrophysics, atmospheric science, and remote sensing. Analytic solutions to the radiative transfer equation (RTE) exist for simple cases but for more realistic media, with complex multiple scattering effects, numerical methods are required. The present article is largely focused on the condition of radiative equilibrium.
Definitions The fundamental quantity that describes a field of radiation is called spectral radiance in radiometric terms (in other fields it is often called specific intensity). For a very small area element in the radiation field, there can be electromagnetic radiation passing in both senses in every spatial direction through it. In radiometric terms, the passage can be completely characterized by the amount of energy radiated in each of the two senses in each spatial direction, per unit time, per unit area of surface of sourcing passage, per unit solid angle of reception at a distance, per unit wavelength interval being considered (polarization will be ignored for the moment). In terms of the spectral radiance, I ν {\displaystyle I_{\nu }} , the energy flowing across an area element of area d a {\displaystyle da} located at r {\displaystyle \mathbf {r} } in time d t {\displaystyle dt} in the solid angle d Ω {\displaystyle d\Omega } about the direction n ^ {\displaystyle {\hat {\mathbf {n} }}} in the frequency interval ν {\displaystyle \nu \,} to ν + d ν {\displaystyle \nu +d\nu \,} is
d E ν = I ν ( r , n ^ , t ) cos θ d ν d a d Ω d t {\displaystyle dE_{\nu }=I_{\nu }(\mathbf {r} ,{\hat {\mathbf {n} }},t)\cos \theta \ d\nu \,da\,d\Omega \,dt}
where θ {\displaystyle \theta } is the angle that the unit direction vector n ^ {\displaystyle {\hat {\mathbf {n} }}} makes with a normal to the area element. The units of the spectral radiance are seen to be energy/time/area/solid angle/frequency. In MKS units this would be W·m−2·sr−1·Hz−1 (watts per square-metre-steradian-hertz).
The equation of radiative transfer The equation of radiative transfer simply says that as a beam of radiation travels, it loses energy to absorption, gains energy by emission processes, and redistributes energy by scattering. The differential form of the equation for radiative transfer is:
1 c ∂ I ν ∂ t + Ω ^ ⋅ ∇ I ν + ( k ν , s + k ν , a ) ρ I ν = j ν ρ + k ν , s ρ 4 π ∫ Ω I ν d Ω {\displaystyle {\frac {1}{c}}{\frac {\partial I_{\nu }}{\partial t}}+{\hat {\Omega }}\cdot \nabla I_{\nu }+\left(k_{\nu ,s}+k_{\nu ,a}\right)\rho I_{\nu }=j_{\nu }\rho +{\frac {k_{\nu ,s}\rho }{4\pi }}\int _{\Omega }I_{\nu }\,d\Omega }
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