Our objective in this chapter is to treat the computation of a planet's energy loss by infrared emission in sufficient detail that the energy loss can be quantitatively linked to the actual concentration of specific greenhouse gases in the atmosphere. Unlike the simple model of the greenhouse effect described in the preceding chapter, the infrared radiation in a real atmosphere does not all come from a single level; rather, a bit of emission is contributed from each level (each having its own temperature), and a bit of this is absorbed at each intervening level of the atmosphere. The radiation comes out in all directions, and the rate of emission and absorption is strongly dependent on frequency. Dealing with all these complexities may seem daunting, but in fact it can all be boiled down to a conceptually simple set of equations which suffice for a vast range of problems in planetary climate.

It was shown in Chapter 3 that there is almost invariably an order of magnitude separation in wavelengths between the shortwave spectrum at which a planet receives stellar radiation and the longwave (generally infrared) spectrum at which energy is radiated to space. This is true throughout the Solar system, for cold bodies like Titan and hot bodies like Venus, as well as for bodies like Earth that are habitable for creatures like ourselves. The separation calls for distinct sets of approximations in dealing with the two kinds of radiation. Infrared is both absorbed and emitted by an atmosphere, at typical planetary temperatures. However, the long infrared wavelengths are not appreciably scattered by molecules or water clouds, so scattering can be neglected in many circumstances. One of the particular challenges of infrared radiative transfer is the intricate dependence of absorption and emission on wavelength. The character of this dependence is linked to the quantum transitions in molecules whose energy corresponds to infrared photons; it requires an infrared-specific description.

In contrast, planets do not emit significant amounts of radiation in the shortwave spectrum, though shortwave scattering by molecules and clouds is invariably significant; absorption of shortwave radiation arises from quite different molecular processes than those involved in infrared absorption, and its wavelength-dependence has a correspondingly different character. Moreover, solar radiation generally reaches the planet in the form of a nearly parallel beam, whereas infrared from thermal emission by the planet and it's atmosphere is more nearly isotropic. The approximations pertinent to shortwave radiation will be taken up in Chapter 5, where we will also consider the effects of scattering on thermal infrared.

We'll begin with a general formulation of the equations of plane-parallel radiative transfer without scattering, in Section 4.2. Though we will be able to derive certain general properties of the solutions of these equations, the equations are not very useful in themselves because of the problem of wavelength dependence. To gain further insight, a detailed examination of an idealized model with wavelength-independent infrared emissivity will be presented in Section 4.3. A characterization of the wavelength dependence of the absorption of real gases, and methods for dealing with that dependence, will be given in Sections 4.4 and 4.5.

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