Astron. Astrophys. 332, 610-628 (1998)

Astron. Astrophys. 332, 610-628 (1998)

1. Introduction

In Paper I of this series (Faurobert-Scholl et al. 1997), we have introduced an iterative numerical method of the Approximate Lambda Iteration (ALI) type to solve polarized radiative transfer equations describing the linear resonance polarization of spectral lines formed with complete frequency redistribution. The method described in Paper I can be applied to spectral lines formed in non-magnetic regions or in the presence of a weak isotropic turbulent magnetic field. It is tailored for radiation fields with an axial symmetry. The gain in memory space and computing time with respect to standard methods which do not make use of an Approximate Lambda Operator is quite significant.

Here we show that the method of Paper I can be generalized to handle the Hanle effect which describes the action of a weak magnetic field on resonance polarization. It applies when the magnetic sublevels of transition are sufficiently close in frequency that the natural linewidths of the sublevels overlap significantly. Permanent phase coherences between the Zeeman sublevels are partially destroyed leading to changes in the degree of linear polarization and in the orientation of the plane of polarization of the scattered radiation. The diagnostic potential of the Hanle effect with optically thick lines is already quite impressive. It was first used for solar prominences (Landi Degl'Innocenti et al. 1987; Bommier et al. 1989), then for the upper solar atmosphere (see the review in Faurobert-Scholl 1996). Other references can be found in Stenflo (1994). More recently the Hanle effect has been considered for the detection of weak magnetic fields in stellar envelopes (Ignace et al. 1997). Efforts to increase the efficiency of numerical methods able to treat this effect promise to be rewarding. The presence of an oriented magnetic field breaks the axial symmetry of the problem. The required generalization of the method of Paper I is achieved by means of an azimuthal Fourier expansion of the radiation field. As in Paper I we restrict ourselves to the approximation of complete frequency redistribution in the line.

In Sect. 2we describe the polarized line transfer equations, and an azimuthal Fourier expansion method which can handle any depth-dependent magnetic field. It is more general than the decomposition used in Faurobert-Scholl (1991), henceforth denoted by FS91, which was restricted to a magnetic field with constant azimuthal angle. In Sect. 3we present a reduced radiative transfer problem for a six-dimensional vector radiation field. It is the starting point for the PALI-H operator perturbation method tested in Sect. 4on a bench-marking problem with a uniform magnetic field. In Sect. 5we study the effects of changing the direction and strength of a uniform vector magnetic field. A case of a vector magnetic field with a depth dependent azimuthal angle is also considered. We also discuss the basic symmetries of the problem. In Sect. 6we apply the PALI-H method to construct various polarization diagrams. We study their dependence on the direction of the line of sight and on the vector magnetic field strength and orientation. We remark on a simple approximation to the full Hanle problem and show that it can be used for initial rough estimations of the magnetic field parameters in an inversion code for polarimetric observations. Some concluding remarks are presented in Sect. 7.

Online publication: March 23, 1998
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