Exact results in modeling planetary atmospheres-II. Semi-gray atmospheres

We solve the radiative transfer equation for a semi-gray planetary atmosphere in radiative equilibrium, in an attempt to define an entirely analytical non-gray model atmosphere of finite optical thickness. The salient feature of the model is that the incident solar radiation is partitioned between two adjacent spectral domains-the “visible” and the “infrared”-in each of which the atmosphere's (effective) opacity is assumed to be independent of frequency (the semi-gray assumption). We envisage a plane-parallel atmosphere illuminated by a beam of parallel radiation and bounded below by a partially reflecting and emitting ground. The former emits infrared radiation, induced by the absorption of radiation both visible and infrared, deriving from the external irradiation as well as from the emission of the planet's surface layer. For an atmosphere with given single-scattering albedos and optical thicknesses in both the visible and infrared domains, we compute the temperature at every depth of the atmosphere, as well as the ground's temperature.