for values of w < 10, the above series converges after 25 terms with an error less than 1%. For values w > 10 the absorptance can be computed, with a maximum error of 3%, from the expression
A = 2A0S(0.7523b3/2 + 0.6513b1/2 + 0.3013b-1/2 + 0.1231), (4.78)
where b = ln w, based on an asymptotic expansion for single lines given in Struve and Elvey (1934). Thus, given the total line strength of the band, k, or band strength, the absorber amount (in units corresponding to the band strength), the bandwidth parameter, Ao, and the Doppler shift, yd, we can calculate the band absorptance, A. If we now define an effective width of the band, weff, which defines the wave number range where the rotational line strengths become significant, then the mean band transmission over this spectral interval can be estimated from t = 1 — A/ueff. For example, for the most important infra-red band of O3 (the v3 band) with band centre wo = 1042 cm-1 or 9.6 ¡m, with a band strength 376 cm-2 atm-1 STP (standard temperature, 273 K, and pressure, 1 atm =1013.25 mbar), a midlatitude ozone amount of 0.345 atm cm STP, we can calculate the absoptance at 300 K using a yd = 1.075 x 10-6wo cm-1, mean line spacing, d = 0.1 cm-1, and bandwidth parameter, Ao = 30 cm-1, we find that w = 109 and so A =11 cm-1, assuming Doppler-broadened rotational lines. This is a fair assumption considering that most of the ozone is located in the stratosphere and hence at low pressures. We note that the weak bands of ozone contribute about another 6 cm-1 to the absorptance, bringing the total ozone absorptance to 16 cm-1 (see Table 5, Vardavas and Carver 1984). If we adopt an effective bandwidth = 100 cm-1, then we obtain a mean atmospheric transmission in the spectral interval 992-1092 cm-1 due to this band of 0.89.
We have seen that the line absorptance for collisionally broadened lines has two limits for its variation with absorber amount, X. For small X the variation is linear, while at large X it is a square-root law. We have also seen that for Doppler broadening, the linear variation also holds for small X, while for large X the variation is logarithmic. For collisionally broadened lines, the band absorptance has three limits for its variation with absorber amount, the linear, square root and logarithmic. The linear limit holds for small absorber amounts, as it does for line absorptance and also for bands with Doppler-broadened lines. For large absorber amounts, the band absorptance varies as the logarithm of the amount. For intermediate absorber amounts, certain conditions need to be fulfilled for the variation to be a square-root law.
A band-absorptance formulation that satisfies the above limits is
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