Radiative transfer: exact Rayleigh scattering series and a formula for daylight

Daylight, or sky light, is sunlight Rayleigh scattered by the atmosphere onto the ground. This random scattering propagation through clear air is governed by 'radiative transfer'. Beyond the single-scattering approximation, the famous virtuoso analysis of Chandrasekhar formulated the problem and offered exact, but rather involved, and ultimately numerical, algorithms for its solution. However, there is no real difficulty in writing down directly the exact Rayleigh scattering series in integrals. Its practical utility is limited to fairly small thicknesses T of atmosphere ( compared with the mean free path), but the Earth has just such. Here even the next order beyond single scattering ( error order T-2) supplies a formula for the brightness and partial polarization of daylight across the sky, which captures the essential topology of the polarization pattern, and also remains uniformly valid in the small thickness limit, for all elevations of the Sun and viewing angles. The status of the mathematical polarization direction pattern invented by Berry, Dennis and Lee as the simplest fit to the required topology is clarified.