Conservative Scattering

A single-scattering albedo of 1 is sometimes referred to as conservative scattering, which is not a comment on the political affiliation of photons but rather signals that radiant energy is conserved; nonzero absorption implies that radiant energy is converted into other forms although, as always, total energy is conserved. As noted at the end of the previous paragraph w = 1 does not exist in nature, but the assumption that it does sometimes leads to no serious errors and simplifies solutions to the equations of transfer. Indeed, with this assumption and the assumption that g is independent of t, the difference of irradiances in Eq. (5.47) is constant, and their sum in Eq. (5.48) is a linear function of t, which yields

where B and C are constants determined by conditions at the upper (t = 0) and lower (t = t) boundaries of the medium. Although written without a subscript, here optical depth is scattering optical depth. The quantity t* = t(1 — g) is the scaled optical depth. Scaled here means scaled to isotropic scattering. Two media, one composed of isotropic scatterers (g = 0), the other of anisotropic scatterers (g = 0), are equivalent if the optical depth of the latter is the optical depth of the former divided by 1 — g. The optical thickness of the entire medium (from 2 = 0 to 2 = h)is t = i ¡3d2. (5.50)

Optical thickness and optical depth often are used interchangeably, but we try to be careful to reserve optical thickness for the optical depth of an entire medium.

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