The Hanle effect in a random medium

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.