We are now equipped to revisit the runaway greenhouse phenomenon, this time using the absorption spectrum of actual gases in place of the idealized grey gas employed in Section 4.3.3. The setup of the problem is essentially the same as in the grey gas case. We consider a condensible greenhouse gas, optionally mixed with a background gas which is transparent to infrared and noncondensing. A surface temperature Tg is specified, and the corresponding moist adiabat is computed. The temperature and the greenhouse gas concentration profiles provide the information necessary to compute the OLR, in the present instance using the homebrew exponential sums radiation model in place of the greygas OLR integral. As before, the OLR is plotted as a function of Tg for the saturated atmosphere, and the Kombayashi-Ingersoll limit is given by the asymptotic value of OLR at large surface temperature.

We'll begin with water vapor. Figure 4.37 shows the results for a pure water vapor atmosphere, computed for various values of the surfaced gravity. The overall behavior is very similar to the grey gas result shown in Fig. 4.3: the OLR attain a limiting value as temperature is increased, and the limit - defining the absorbed solar radiation above which the planet goes into a runaway state - becomes higher as the surface gravity is increased, and for precisely the same reasons as invoked in the grey gas case. The result, however, is now much easier to apply to actual planets, since with the real gas calculations we have the real numbers in hand for water vapor, and not for some mythical gas characterized by a single absorption coefficient. Several specific applications will be given shortly.

As in the grey-gas case the limiting OLR increases as surface gravity is increased. It would

280 300 320 340 360 380 400 Surface Temperature (K)

Figure 4.37: OLR vs surface temperature for a saturated pure water vapor atmosphere. The numbers on the curve indicate the planet's surface gravity. The calculation was done with the homebrew exponential sums radiation code, incorporating both the 1000 cm-1 and 2200 cm-1 continua, but neglecting temperature scaling of absorption outside the continua. 20 terms were used in the exponential sums, and wavenumbers out to 5000 cm-1 were included; the atmosphere was considered transparent to higher wavenumbers

280 300 320 340 360 380 400 Surface Temperature (K)

Figure 4.37: OLR vs surface temperature for a saturated pure water vapor atmosphere. The numbers on the curve indicate the planet's surface gravity. The calculation was done with the homebrew exponential sums radiation code, incorporating both the 1000 cm-1 and 2200 cm-1 continua, but neglecting temperature scaling of absorption outside the continua. 20 terms were used in the exponential sums, and wavenumbers out to 5000 cm-1 were included; the atmosphere was considered transparent to higher wavenumbers be useful if the real gas result could be represented in terms of an equivalent grey gas, but first one must generalize Eq. 4.39 to incorporate the increase of absorption coefficient with pressure. This is done formally in Problem ??, but the qualitative derivation runs as follows. We need to determine the pressure p1 where the optical thickness to the top of the atmosphere is unity, and then evaluate the temperature at that point, along the one-component saturated adiabat. For linear pressure broadening or the continuum the optical thickness requirement implies 1 Kop\/pog = 1, where po is the reference pressure to which the absorption is referred (generally 100mb for the data given in our survey of gaseous absorption properties). Substituting the resulting p1 into the expression for T(p) we infer an expression of the form

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