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Boundary conditions for NLTE polarized radiative transfer with incident radiation

Polarized NLTE radiative transfer in the presence of scattering in spectral lines and/or in continua may be cast in a so-called 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 reduces drastically the storage requirement of numerical codes. 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). In previous works an ad-hoc treatment of the boundary conditions, applying to cases where the incident radiation is unpolarized, has been used. In this note 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 so-called diffuse radiation. 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. We perform numerical computations of such cases, showing that the emergent line-polarization may be significantly affected by the polarization of the incident radiation.