Radiative transfer 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.
Contents
- Definitions
- The equation of radiative transfer
- Solutions to the equation of radiative transfer
- Local thermodynamic equilibrium
- The Eddington approximation
- References
The present article is largely focused on the condition of radiative equilibrium.
Definitions
The fundamental quantity which 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,
where
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, and redistributes energy by scattering. The differential form of the equation for radiative transfer is:
where
Solutions to the equation of radiative transfer
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 solution in terms of the emission and absorption coefficients may be written:
where
Local thermodynamic equilibrium
A particularly useful simplification of the equation of radiative transfer occurs under the conditions of local thermodynamic equilibrium (LTE). In this situation, the absorbing/emitting medium consists of massive particles which are in equilibrium with each other, and therefore have a definable temperature. The radiation field is not, however in equilibrium and is being entirely driven by the presence of the massive particles. For a medium in LTE, the emission coefficient and absorption coefficient are functions of temperature and density only, and are related by:
where
Knowing the temperature profile and the density profile of the medium is sufficient to calculate a solution to the equation of radiative transfer.
The Eddington approximation
The Eddington approximation is a special case of the two stream approximation. It 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. It assumes that the intensity is a linear function of
where
Extracting the first few moments of the spectral radiance with respect to
Thus the Eddington approximation is equivalent to setting
Note that the first two moments have simple physical meanings.
The radiative transfer through an isotropically scattering medium at local thermodynamic equilibrium is given by
Integrating over all angles yields
Premultiplying by
Substituting in the closure relation, and differentiating with respect to
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.