A&A 442, 11-28 (2005)

DOI: 10.1051/0004-6361:20042493

## Stochastic polarized line formation

##### I. Zeeman propagation matrix in a random magnetic field

**H. Frisch**

^{1}, M. Sampoorna^{1, 2, 3}and K. N. Nagendra^{1, 2}^{1}Laboratoire Cassiopée (CNRS, UMR 6202), Observatoire de la Côte d'Azur, BP 4229, 06304 Nice Cedex 4, France

^{2}Indian Institute of Astrophysics, Koramangala Layout, Bangalore 560 034, India

^{3}JAP, Dept. of Physics, Indian Institute of Science, Bangalore 560 012, India

(Received 7 December 2004 /Accepted 4 July 2005 )

** Abstract **

This paper considers the effect of a random magnetic field
on Zeeman line transfer, assuming that the scales of fluctuations of
the random field are much smaller than photon mean free paths
associated to the line formation (micro-turbulent limit). The mean
absorption and anomalous dispersion coefficients are calculated for
random fields with a given mean value, isotropic or anisotropic
Gaussian distributions azimuthally invariant about the direction of
the mean field. Following Domke & Pavlov (1979, Ap&SS, 66, 47), the averaging
process is carried out in a reference frame defined by the direction
of the mean field. The main steps are described in detail. They
involve the writing of the Zeeman matrix in the polarization matrix
representation of the radiation field and a rotation of the line of
sight reference frame. Three types of fluctuations are considered :
fluctuations along the direction of the mean field, fluctuations
perpendicular to the mean field, and isotropic fluctuations. In each
case, the averaging method is described in detail and fairly explicit
expressions for the mean coefficients are established, most of which
were given in Dolginov & Pavlov (1972, Soviet Ast., 16, 450) or Domke & Pavlov (1979, Ap&SS, 66, 47).
They include the effect of a microturbulent velocity field with zero
mean and a Gaussian distribution.

A detailed numerical investigation of the mean coefficients illustrates the two effects of magnetic field fluctuations: broadening of the -components by fluctuations of the magnetic field intensity, leaving the -components unchanged, and averaging over the angular dependence of the and components. For longitudinal fluctuations only the first effect is at play. For isotropic and perpendicular fluctuations, angular averaging can modify the frequency profiles of the mean coefficients quite drastically with the appearance of an unpolarized central component in the diagonal absorption coefficient, even when the mean field is in direction of the line of sight. A detailed comparison of the effects of the three types of fluctuation coefficients is performed. In general the magnetic field fluctuations induce a broadening of the absorption and anomalous dispersion coefficients together with a decrease of their values. Two different regimes can be distinguished depending on whether the broadening is larger or smaller than the Zeeman shift by the mean magnetic field.

For isotropic fluctuations, the mean coefficients can be expressed in
terms of generalized Voigt and Faraday-Voigt functions *H*^{(n)} and
*F*^{(n)} introduced by Dolginov & Pavlov (1972, Soviet Ast., 16, 450). These functions
are related to the derivatives of the Voigt and Faraday-Voigt
functions. A recursion relation is given in an Appendix for their
calculation. A detailed analysis is carried out of the dependence of
the mean coefficients on the intensity and direction of the mean
magnetic field, on its root mean square fluctuations and on the
Landé factor and damping parameter of the line.

**Key words:**line: formation

**--**polarization

**--**magnetic fields

**--**turbulence

**--**radiative transfer

**©**

*ESO 2005*