Radiative transfer : exact Rayleigh scattering series and a daylight formula

Daylight, or sky light, is sunlight Rayleigh scattered by the atmosphere onto the ground. This random scattering propagation through clear air is well described by "radiative transfer." Beyond the single scattering approximation, algorithms that are rather involved and ultimately numerical have been required to solve the equations. However, avoiding the use of "Stokes parameters," which are not helpful, there is no real difficulty in writing down directly the exact Rayleigh scattering series in integrals: the initial 3n-dimensional one for an n-fold scattering reducing to an n-dimensional one. The derivation of the series here, by iteration, considerably simplifies my recent probabilistic one. The next order beyond single scattering supplies a formula for the brightness and partial polarization of daylight across the sky, which captures the essential topology of the polarization direction pattern.