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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 RTE is a differential equation describing radiance (, ^,).It can be derived via conservation of energy.Briefly, the RTE states that a beam of light loses energy through divergence and extinction (including both absorption and scattering away from the beam) and gains energy from light sources in the medium and scattering directed towards the beam.
The radiation field thereby maintains the blackbody intensity appropriate for the local temperature. At equilibrium, I λ = B λ (T) and therefore dI λ = 0 even when the density of the GHG (n) increases. This has led some to falsely believe that Schwarzschild's equation predicts no radiative forcing at wavelengths where absorption is "saturated".
The radiative transfer equation is a monochromatic equation to calculate radiance in a single layer of the Earth's atmosphere. To calculate the radiance for a spectral region with a finite width (e.g., to estimate the Earth's energy budget or simulate an instrument response), one has to integrate this over a band of frequencies (or wavelengths).
This is equivalent to modeling photon transport analytically by the radiative transfer equation (RTE), which describes the motion of photons using a differential equation. However, closed-form solutions of the RTE are often not possible; for some geometries, the diffusion approximation can be used to simplify the RTE, although this, in turn ...
The intensity field can in principle be solved from the integrodifferential radiative transfer equation (RTE), but an exact solution is usually impossible and even in the case of geometrically simple systems can contain unusual special functions such as the Chandrasekhar's H-function and Chandrasekhar's X- and Y-functions. [3]
The grey atmosphere (or gray) is a useful set of approximations made for radiative transfer applications in studies of stellar atmospheres (atmospheres of stars) based on the simplified notion that the absorption coefficient of matter within a star's atmosphere is constant—that is, unchanging—for all frequencies of the star's incident radiation.
Putting this into the equation for radiative transfer we get = where s is the distance measured along the path traveled by the beam. The minus sign on the left hand side shows that the intensity decreases as the beam travels, due to the absorption of photons.
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