After examining the calculated spectra, we selected five gases in an attempt to minimize remaining windows. These five gases' strong vibrational bands in the range of 670-1,400 cm1 are listed in Table 2. Although some of these gases have been synthesized or observed elsewhere (e.g., SF6 in ref. 4 and SF5CF3 in ref. 14), we use B3LYP outputs for all of them to work with a consistent set of statistics.
Based on various analogs such as fluorinated ethers (15), the Earth's 9.6-µm band of ozone, or the calibrations described above, we estimate bandwidths for the super greenhouse gases to have full-width half-max between 16 and 30 cm1. Rather than use Lorentzian band shapes and give ourselves the benefit of far wings that may not exist, we model all bands as triangles. We do not increase bandwidths with gas amount, so that the equivalent width of any given band has an upper-limit independent of concentration. In this way we hope to avoid crediting a saturated band with filling in nearby windows.
For a first guess at column amounts, we identified a constraining band of each gas not overlapping strong absorptions of any of the other gases. Because the band intensities are calculated at standard temperature and pressure (STP), we are not accounting for the reduction of pressure in the Martian upper troposphere and stratosphere properly; however, we note that the 9.6-µm ozone band in the Earth's atmosphere contributes an optical thickness comparable to the product of its STP band intensity and its column density divided by its bandwidth, even though most of Earth's ozone is not in the troposphere. We doubled the spectral optical thickness of terrestrial gases, because Mars's lower surface gravity (0.38 that of Earth) requires increased airmass to achieve 100 kPa at the surface. We added spectral optical thicknesses of the five selected super greenhouse gases to the spectral opacity of the doubled terrestrial gases. Without regard to the variation of the Planck function, we then stepped through the spectrum 700 to 1,400 cm1 at 1-cm1 intervals adding optical thicknesses and calculating spectral transmission. We iterated with the intent of minimizing the total number of special greenhouse molecules within the constraint that the gray optical thickness be at least 3.
The minimum column amounts resulting when bandwidth (full-width half-max) is estimated as 30 cm1 are given as the second column in Table 3. The total molecular column density of the mixture is less than 5 × 1022 m2, which would be about 240 parts per billion by volume (ppbv) of Earth's atmosphere. If the bandwidth is estimated as 16 cm1, achieving = 3 would require the addition of some NF3 and CF3NF2 to the five gases in Table 2 and a total molecular column of 1.7 × 1023 m2 (about 810 ppbv of Earth's atmosphere).
Fig. 1 shows spectral absorption through the 700-1,400 cm1 window when the gas mixture of Tables 2 and 3 is added (solid curve) to the spectrally doubled terrestrial gases. The dotted curve represents absorption by the super greenhouse gases only. The three humps just to the left of the deepest trough may be identified roughly with bands of SF4(CF3)2, SF5CF3, and SF6 seen in Table 2. The contribution of ozone accounts for the difference between the two curves in the interval between 1,000 and 1,080 cm1. Other humps are hybrid absorptions of several gases. It's worth noting that experiments on SF6 (4) show its peak absorption between 940 and 950 cm1, which would make our problem a little easier to solve.