Polarized line formation in Multi-dimensional media-I: Decomposition of Stokes parameters in arbitrary geometries

Abstract

The solution of the polarized line radiative transfer (RT) equation in muti-dimensional geome- tries has been rarely addressed and only under the approximation that the changes of frequencies at each scattering are uncorrelated (complete frequency redistribution). With the increase in the resolution power of telescopes, being able to handle radiative transfer in multi-dimensional structures becomes absolutely necessary. In the present paper, our first aim is to formulate the polarized RT equation for resonance scattering in multi-dimensional media, using the elegant technique of irreducible spherical tensors T K Q (i, ). Our second aim is to develop a numerical method of solution based on the polarized approximate lambda iteration (PALI) approach. We consider both complete frequency redistri- bution (CRD) as well as partial frequency redistribution (PRD) in the line scattering. In a multi-D geometry the radiation field is non-axisymmetrical even in the absence of a symmetry breaking mechanism such as an oriented magnetic field. We generalize here to the 3D case, the decomposition technique developed for the Hanle effect in a 1D medium which allows one to represent the Stokes parameters I,Q,U by a set of 6 cylindrically symmetrical functions. The scattering phase matrix is expressed in terms of T K Q (i, ), (i = 0, 1, 2,K = 0, 1, 2,−K Q +K), with , being the direction of the outgoing ray. Starting from the definition of the source vector, we show that it can be represented in terms of 6 components SK Q independent of . The formal solution of the multi-dimensional transfer equation shows that the Stokes parameters can also be expanded in terms of the T K Q (i, ). Because of the 3D-geometry, the expansion coefficients IK Q remain -dependent. We show that each IK Q satisfies a simple transfer equation with a source term SK Q and that this transfer equation provides an efficient approach for handling the polarized transfer in multi-D geometries. A PALI method for 3D, associated to a core-wing separation method for treating PRD, is developed. It is tested by comparison with 1D solutions and several benchmark solutions in the 3D case are given.