Boundary conditions for polarized radiative transfer with incident radiation

Abstract

Context. Polarized radiative transfer in the presence of scattering in spectral lines and/or in continua may be cast in a reduced form for six reduced components of the radiation field. In this formalism, the six components of the reduced source function are angle-independent quantities. It thus drastically reduces the storage requirement of numerical codes and it is very well suited to solving polarized non-local thermodynamic equilibrium radiative transfer problems in 3D media. Aims. This approach encounters a fundamental problem when the medium is illuminated by a polarized incident radiation, because there is a priori no way of relating the known (and measurable) Stokes parameters of the incident radiation to boundary conditions for the reduced equations. The origin of this problem is that there is no unique way of deriving the radiation-reduced components from its Stokes parameters (only the inverse operation is clearly defined). The method proposed here aims at enabling to work with arbitrary incident radiation field (polarized or unpolarized). Methods. In previous studies, an ad-hoc treatment of the boundary conditions, applied to cases where the incident radiation is unpolarized, has been used. In this paper, we show that it is possible to account for the incident radiation in a rigorous way without any assumption on its properties by expressing the radiation field as the sum of a directly transmitted radiation and of a diffuse radiation. This approach was first used by Chandrasekhar to solve the problem of diffuse reflection by planetary atmospheres illuminated by their host star. Results. The diffuse radiation field obeys a transfer equation with no incident radiation that may be solved in the reduced form. The first scattering of the incident radiation introduces primary creation terms in the six components of the reduced source function. Once the reduced polarized transfer problem is solved for the diffuse radiation field, its Stokes parameters can be computed. The full radiation field is then obtained by adding the directly transmitted radiation field computed in the Stokes formalism. Conclusions. In the case of an unpolarized incident radiation, the diffuse field approach allows us to validate the previously introduced ad-hoc expressions. The diffuse field approach however leads to more accurate computation of the source terms in the case where the incident radiation is anisotropic. It is the only possible approach when the incident radiation field is polarized. We perform numerical computations of test cases, showing that the emergent line-polarization may be significantly affected by the polarization of the incident radiation. © 2013 ESO.