Effects of Rayleigh-scattering polarization on reflected intensity: a fast and accurate approximation method for atmospheres with aerosols

Solar radiation reflected by the atmospheres of Neptune and Uranus is dominated by Rayleigh scattering at visible wavelengths, and thus subject to the effects of polarization. Ignoring these effects can lead to errors in reflected intensity of more than 9% in a clear atmosphere. But solving the full vector equation of transfer is computationally very costly, forcing approximations with limitations that are not well understood and not generally applicable to spatially resolved observations and complex atmospheric structures. Using accurate vector radiation transfer calculations, it is here shown that differences between vector and scalar results near zero phase angle have systematic dependencies on optical depth, single scattering albedo, and angle, that provide a basis for accurate approximation of the reflected intensities. With little computational cost, it is possible to calculate corrected spatially resolved scalar intensities that closely match vector intensities, with individual errors rarely exceeding 1%, and mean and RMS errors generally within a few tenths of 1%. The correction method accounts for the attenuating effects of clouds and molecular absorption.