Exact and Approximate Solutions for Multiple Scattering by Cloudy and Hazy Planetary Atmospheres

Abstract

Solutions are obtained for the problem of multiple scattering by a plane parallel atmosphere with anisotropic phase functions typical of cloud and haze particles. The resulting albedos, angular distributions of intensities, and planetary magnitudes are compared to solutions obtained with approximate analytic phase functions and, in the case of the cloud phase function, to the solution obtained with the forward diffraction peak omitted from the phase function. It is shown that the cloud phase function with the truncated peak yields results practically identical to those obtained with the complete cloud phase function, not only for albedos and magnitudes, but also for the angular distribution; the approximation introduces errors of several per cent in the angular distribution for direct backscattering (the region of the glory), for emergent angles near grazing regardless of the incident angle, and, of course, a larger error occurs for total scattering angles near 0°. However, the errors are unimportant for many applications, and hence a large reduction in computer time is possible. This is particularly useful, for example, in making practical the computations needed for interpreting the phase curve, limb darkening and spectral reflectivity of Venus. It is shown that the Henyey-Greenstein phase function, based on the asymmetry factor ?cos??, yields spherical and plane albedos and planetary magnitudes (for optically thick atmospheres) close to those obtained with the cloud and haze phase functions. The Kagiwada-Kalaba phase function, based on the ratio of forward to backward scattering, gives significantly less satisfactory results for the same quantities. Neither of the two analytic phase functions can accurately duplicate the true angular distribution of scattering by thin clouds; however, the results are better with thick layers, especially for hazes. The results indicate that the Henyey-Greenstein phase function may be useful for problems such as line formation in planetary atmospheres.