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.
[1][2] 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).
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).
The differential form of the equation for radiative transfer is: where
term represents radiation scattered from other directions onto a surface.
Solutions to the equation of radiative transfer form an enormous body of work.
The differences however, are essentially due to the various forms for the emission and absorption coefficients.
If scattering is ignored, then a general steady state solution in terms of the emission and absorption coefficients may be written: where
: A particularly useful simplification of the equation of radiative transfer occurs under the conditions of local thermodynamic equilibrium (LTE).
It is important to note that local equilibrium may apply only to a certain subset of particles in the system.
In this situation, the absorbing/emitting medium consists of massive particles which are locally in equilibrium with each other, and therefore have a definable temperature (Zeroth Law of Thermodynamics).
The radiation field is not, however in equilibrium and is being entirely driven by the presence of the massive particles.
is the black body spectral radiance at temperature T. The solution to the equation of radiative transfer is then: Knowing the temperature profile and the density profile of the medium is sufficient to calculate a solution to the equation of radiative transfer.
The two-stream approximation assumes that the intensity is constant with angle in the upward hemisphere, with a different constant value in the downward hemisphere.
The Eddington approximation instead assumes that the intensity is a linear function of
Note that expressing angular integrals in terms of
appears in the Jacobian of integrals in spherical coordinates.
The Eddington approximation can be used to obtain the spectral radiance in a "plane-parallel" medium (one in which properties only vary in the perpendicular direction) with isotropic frequency-independent scattering.
Extracting the first few moments of the spectral radiance with respect to
yields Thus the Eddington approximation is equivalent to setting
Higher order versions of the Eddington approximation also exist, and consist of more complicated linear relations of the intensity moments.
This extra equation can be used as a closure relation for the truncated system of moments.
Note that the first two moments have simple physical meanings.
at local thermodynamic equilibrium is given by Integrating over all angles yields Premultiplying by
, and then integrating over all angles gives Substituting in the closure relation, and differentiating with respect to
allows the two above equations to be combined to form the radiative diffusion equation This equation shows how the effective optical depth in scattering-dominated systems may be significantly different from that given by the scattering opacity if the absorptive opacity is small.