## Boundary conditions for polarized radiative transfer with incident radiation

^{1}
UMR 7293 Lagrange, Université de Nice Sophia Antipolis, CNRS, Observatoire
de la Côte d’Azur,
Campus Valrose,
06108
Nice,
France

e-mail:
marianne.faurobert@unice.fr

^{2}
Astronomical observatory Belgrade, Volgina 7,
11060
Belgrade,
Serbia

^{3}
Department of Astronomy, Faculty of Mathematics, University of
Belgrade, Studentski Trg
16, 11000
Belgrade,
Serbia

Received:
17
July
2013

Accepted:
18
September
2013

*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.

Key words: polarization / radiative transfer / scattering / methods: analytical

*© ESO, 2013*