that transitions of the pure rotation spectrum of H2 O dominate the atmospheric opacity at the frequency, ν = 500.5 cm−1 . Fig. 8(b) shows an extreme blanket frequency, with an optical depth τmp = 51, 687, and an emission height ze = 85.3 km, just below the mesopause. As one can see in Fig. 9, the opacity at the frequency, ν = 667.4 cm−1 , is mostly due to a Q-branch vibration-rotation transition of CO2 , with rotational angular momentum j = 4 in the initial and final states. The intense cooling rate, −R̃ = ∂ Z̃/∂z, near the top of the atmosphere helps to drive down the temperature of the mesopause. Fig. 8(c) shows an extreme window frequency, ν = 971 cm−1 , where the optical depth is only τmp = 0.016. At this frequency, and in the absence of clouds, surface radiation reaches space with negligible attenuation by greenhouse gases. One of the infrared images of the Earth’s disk, provided by geosynchronous satellites[39], is in a band centered on the wavelengh λ = 1/(971 cm−1 ) = 10.3 µm, the “clean infrared window.” Fig. 8(d) shows a blanket frequency, ν = 1016.2 cm−1 , with a moderate optical depth τmp = 7.11 in the O3 band. Not surprisingly, the emission height, ze = 34 km is in the upper stratosphere, where Fig. 2 shows that the O3 concentration maximizes and heating by ultraviolet sunlight keeps the temperature higher than in the lower stratosphere. Downward spectral flux from the emission height contributes to pronounced spectral heating of the lower stratosphere. This is in contrast to the radiative cooling that characterizes most frequencies and most altitudes.
3.2
Frequency dependence of Z̃ and I˜
˜ for θ = 0. Fig. 9 shows the spectral flux, Z̃, and π times the vertical spectral intensity, I, These were calculated at the mesopause altitude zmp = 86 km, the top of the radiative atmosphere, with (44) and (34), for frequencies from 667.3 cm−1 to 667.5 cm−1 , where most of the absorption is due to CO2 molecules. The five prominent resonances of Fig. 9 are Q-branch absorption lines of the most abundant isotopologue, 16 O12 C16 O, near the mesopause, where the pressure is low enough that Doppler broadening (66) determines the line width. The rotational quantum numbers j for ∆j = 0 transitions from lower vibrational states of the bending mode and upper states with one unit of vibrational excitation, are j = 2, 4, 6, . . .. Odd values of j are forbidden because of the identical 16 O atoms with nuclear spin I = 0. The resonances are thermalinfrared analogs of the Fraunhofer dark lines of the sun [38]. The resolution of most satellite spectrometers is far too coarse to resolve these Q absorption lines. For frequencies between the Q resonances, most of the flux and intensity is from near the stratopause at an altitude of zsp = 47 km, where there is a temperature maximum T (z) = 271.2 K, as can be seen from Fig. 2. The very weak fluxes for frequencies near the minima of the narrow Q lines are from molecules near the mesopause, at an altitude of 86 km, where the temperature reaches its minimum value of 187.5 K. An example is shown in Fig. 8(b). The dashed red line shows the flux, π B̃, of a blackbody at the temperature of T = 288.7 K of the surface. This is the flux that would be observed at the top of a hypothetical atmosphere with no greenhouse gases. ˜ A spectrometer in a satellite can measure the upward spectral intensity I(θ) of (34) at 24
