DOI: 10.1051/0004-6361:20042540

## The Hanle effect in a random medium

**H. Frisch**

Laboratoire Cassiopée (CNRS, UMR 6202), Observatoire de la Côte d'Azur, BP 4229, 06304 Nice Cedex 4, France

e-mail: frisch@obs-nice.fr

(Received 15 December 2004 / Accepted 21 September 2005 )

** Abstract **

This paper considers the Hanle effect produced by a
turbulent magnetic field. To overcome the simplified microturbulent
treatment whereby the Hanle phase matrix is locally averaged over some
magnetic field distribution, we consider a turbulent magnetic field
with a finite correlation length. We assume that the magnetic field
along each individual photon path can be represented by a
Kubo-Anderson process (KAP) and study the stationary solution as time
goes to infinity. A KAP is a discontinuous Markov process. The random
magnetic field is characterized by a correlation length and a
distribution function of the magnetic field vector; both can be chosen
arbitrarily. The microturbulent limit is recovered when the
correlation length goes to zero. A non-stochastic integral equation of
the Wiener-Hopf type is obtained for a mean conditional source
vector. This integral equation yields explicit expressions for the
mean Stokes parameters, provided one makes physically realistic
approximations, namely neglect the effect of the magnetic field on
Stokes *I*, keep only the contributions from *I* and *Q* in the
source terms for Stokes *Q* and Stokes *U* and solve the
integral equation for *Q* with a two-scattering approximation. The
final expressions involve mean values and
correlation functions of some of the elements of the Hanle phase
matrix and show the dependence on the correlation length of the random
magnetic field.

The combined effects of a turbulent velocity field and a turbulent magnetic field with finite correlation lengths is also studied. The velocity field is represented by a KAP with the same correlation length as the magnetic field. Some of the velocity field effects are treated with an effective medium approximation as in Frisch & Frisch (1976, MNRAS, 175, 157). Explicit expressions are obtained for the mean Stokes parameters. They can account for correlations between velocity field and magnetic field fluctuations.

**Key words:**line: formation

**--**polarization

**--**magnetic fields

**--**turbulence

**--**radiative transfer

**©**

*ESO 2006*