Simplified radiative transfer for combined Rayleigh and isotropic scattering

We consider the vector equation of radiative transfer describing the Stokes parameters of light, with a scattering law corresponding to a combination of Rayleigh and isotropic scattering. We show that this vector description can be reduced to a renormalized scalar equation of transfer for the intensity I in the asymptotic limit of either near-thermodynamic equilibrium or that corresponding to a source-free, weakly absorbing system. A simple quadrature result is also obtained for the state of polarization of the light. We apply this analysis to the classic diffuse reflection problem, and numerical results indicate an improvement in accuracy over the usual scalar equations of transfer predictions for this scattering law. As part of our solution methodology for the diffuse reflection problem, we introduce a new analytic technique based upon the source function in the equation of transfer.