A METHOD FOR THE INTEGRATION OF THE RADIATIVE-TRANSFER EQUATION

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Abstract
The equations for radiative transfer are integrated exactly for a band of spectral lines which do not overlap and for an Elsasser band. The result of the two-fold integration of the absorption over frequency and over the atmospheric path can be expressed in terms of the Legendre functions. Here it is assumed that the mixing ratio is constant, that the line intensity is independent of temperature, and that the Lorentz line shape is valid. Asymptotic forms of the Legendre functions are used to obtain the solutions to the following problems from these exact results. The regions of validity of the single-line and strong-line approximations are precisely stated. It is shown that the strongest line in the band absorbs more radiation than any other line in an atmospheric layer, when the overlap of the lines can be neglected. For an Elsasser band, an expression is derived for the line strength that gives the maximum absorption in an atmospheric layer for radiation emitted either by a black body at another level or by another atmospheric layer.

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