Radiative Transfer in Planetary Atmospheres

The apparent brightness temperature of a deep atmosphere is related to the physical parameters of the atmosphere, such as pressure, temperature, and composition, through the equation of radiative transfer. To a good approximation at radio wavelengths, the equation of radiative transfer for a ray making an angle cos with the vertical in a lossy medium is 

Measurements with single-dish antennas generally have insufficient angular resolution to determine the brightness distribution across the planetary disk. The mean disk brightness temperature To can be calculated by integration of Eq. (13) over all angles of incidence. This yields 

This equation is usually used to calculate the average disk brightness temperature for a model atmosphere for comparison with observations. Unlike the blackbody radiation model, this model predicts a wavelength dependence of 

FIGURE 3 Theoretical effective temperatures of the planets for two models. The higher temperature model is for the case where the planet is not rotating [Eq. (12)]. The lower temperature model is for the case where the planet is rapidly rotating [Eq. (11)]. 

The absorption may either vary slowly with frequency or exhibit an abrupt change over a narrow frequency range. The two kinds of absorption are referred to as nonreso-nant and resonant absorption, respectively. The standard classical theory of nonresonant molecular absorption is due to Debye. Resonant absorption is produced by the discrete transitions from one energy level to another in a molecule that cause the molecule to absorb or emit at particular frequencies. The study of resonant absorption lines in planetary atmospheres is referred to as planetary spectroscopy. 

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Meador, W.E., and W.R. Weaver, Two-stream approximations to radiative transfer in planetary atmospheres A unified description of existing methods and a new improvement. J Atmos Sci 37, 630, 1980. 

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