In our new model, the atmosphere continues to infinite altitude, getting thinner and thinner as we go. The lowest layer of the atmosphere is the

*troposphere*. Above the troposphere is the

*tropopause*. The tropopause is the layer that radiates heat into space. Above the tropopause, the atmosphere is so thin that it is transparent.

For now, we ignore day and night, and we ignore clouds. We ignore heat flowing through the solid mass of the planet. Short-wave radiation arriving from the sun passes straight through the atmosphere and is of uniform and constant power

*E*. The temperature and pressure at the planet surface are

_{S}*T*and

_{p}*P*respectively. A near-adiabatic convection cycle carries heat up through the troposphere to the tropopause, where it is radiated into space. The temperature and pressure of the tropopause are

_{p}*T*and

_{T}*P*respectively.

_{T}In our Extreme Greenhouse, the atmosphere was entirely opaque to all long-wave radiation. In our new Planetary Greenhouse, the atmosphere is transparent to all radiation with wavelength less than some threshold, λ

_{T}. At greater wavelengths, a trace greenhouse gas in the atmosphere absorbs radiation. The greater the concentration of this greenhouse gas, the shorter the absorption length of the atmosphere for wavelengths greater than λ

_{T}. As we saw in The Upper Gas, the shorter the absorption length, the lower the tropopause pressure. As we saw in Atmospheric Convection, the lower the tropopause pressure, the lower its temperature.

As in our Extreme Greenhouse, we assume that the planet radiates like a black body. The radiation it emits below λ

_{T}passes out into space. This radiated power is

*E*. As we described in Black Bodies, we can use Plank's Law to determine the power radiated at all wavelengths, so we can calculate the power radiated below λ

_{P}_{T}as a function of

*T*.

_{P}The atmospheric convection cycle allows us to calculate the temperature of the tropopause given only

*T*and the ratio

_{P}*P*/

_{T}*P*. The tropopause radiates power

_{P}*E*into space. Using Plank's Law and radiative symmetry, we can calculate the power radiated by the tropopause as a function of

_{T}*T*. The atmosphere emits radiation only at wavelengths greater than λ

_{T}_{T}.

For thermal equilibrium, we must have the total power radiated outward by the body and the tropopause equal to the total power arriving from the sun. Thus we must have

*E*=

_{S}*E*+

_{P}*E*. And so we arrive at a robust model of the planetary greenhouse effect when the greenhouse gas has a step-like absorption spectrum. Given only λ

_{T}_{T}and

*P*, we can plot a graph of

_{P}*E*+

_{P}*E*versus temperature. This graph crosses

_{T}*E*at the equilibrium value of

_{S}*T*. We wrote a computer program to calculate

_{P}*E*+

_{P}*E*versus

_{T}*T*for a range of tropopause pressures. You will find the program here.

_{P}In our next post, we will choose useful values for λ

_{T}and

*E*, and we will see how doubling the concentration of our greenhouse gas affects the temperature of the planet.

_{S}
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