gcos6 Jp1 S(t0)
Note that for weak lines, the averaged transmission is independent of the line width. From the expression for T we can define the equivalent width of the line, W = S(To )iw. To understand the meaning of the equivalent width, imagine that absorption takes all of the energy out of the incident beam within a range of wavenumbers of width W, leaving the rest of the spectrum. undisturbed. The equivalent width W is defined such that the amount of energy thus removed is equal the amount removed by the actual absorption, which takes just a little bit of energy out of each wavenumber throughout the spectrum.
When the layer of atmosphere between p1 and p2 is optically thick at the line center, the transmission is reduced to nearly zero there. This defines the strong line limit. For strong lines, there is essentially no transmission near the line center; all the transmission occurs out on the wings of the lines. Since essentially nothing gets through near the line centers anyway, there is little loss of accuracy in replacing the line shape by it's far-tail form, n-1Sy/v/2. With this approximation to the line shape, the band-averaged transmission may be written:
1 fCm 1
where X = y7S(To)j(po,To)£s/n, and the weighted path for strong lines is
The third line in the expression for T comes from introducing the rescaled dummy variable ( = v'/y/X; the limit of integration then becomes Zm = A/(2^fX) Unless the path is enormous, (m will be very large, because the averaging interval A is invariably taken to be much larger than the typical line width (otherwise there would be little point in averaging). For Qm >> 1, the integral in the last line can be evaluated analytically, and is f Cm 1
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