A&A 485, 275-287 (2008)

DOI: 10.1051/0004-6361:20079160

## Zeeman line formation in solar magnetic fields

##### Studies with empirical probability distribution functions

**M. Sampoorna**

^{1, 2}, K. N. Nagendra^{1, 2}, H. Frisch^{2}, and J. O. Stenflo^{3}^{1}Indian Institute of Astrophysics, Koramangala, Bangalore 560 034, India

e-mail: sampoorna@iiap.res.in

^{2}Laboratoire Cassiopée, CNRS, Université de Nice, Observatoire de la Côte d'Azur, BP 4229, 06304 Nice Cedex 4, France

^{3}Institute of Astronomy, ETH Zürich, 8092 Zürich, Switzerland

Received 28 November 2007 / Accepted 11 April 2008

** Abstract ***Context. *Numerical simulations of magneto-convection and analysis of solar
magnetograms provide probability distribution functions (PDFs)
for the magnetic field strength.*Aims. *In the paper, we explore the effects of these PDFs on
Zeeman line formation.*Methods. *We calculate the mean Stokes parameters for a Milne-Eddington
atmosphere in the limit of optically thin (micro-turbulent) and
thick (macro-turbulent) magnetic structures and also the dispersion around
the mean profiles in the optically thick limit. Several types of
PDFs are considered: (a) Voigt function and stretched exponential
type PDFs for fields with fixed direction but fluctuating
strength; (b) a cylindrically symmetrical power law for the angular
distribution of magnetic fields with given field strength; (c) composite PDFs accounting for randomness in both strength and
direction obtained by combining a Voigt function or a stretched
exponential with an angular power law. For optically thin
structures, explicit expressions are given for the mean values of the
Zeeman absorption matrix elements. We also describe how the averaging
technique for a normal Zeeman triplet may be generalized to the more
common case of anomalous Zeeman splitting patterns.*Results. *We show that, for magnetic field rms fluctuations of the order of
6 G, consistent with observational data, Stokes *I* is essentially
independent of the shapes of the PDFs but Stokes *Q*, *U*, and *V*
and also the dispersion around the mean values are quite
sensitive to the tail behavior of the PDF. We confirm a
previous result that Stokes *V* is less sensitive to the scale of the
magnetic structures than Stokes *Q* and *U*. The composite PDF
proposed for the fluctuations of the magnetic field vector has an
angular distribution peaked about the vertical direction for strong
fields, and is isotropically distributed for weak fields;
it can be used to mimic solar surface random fields.

**Key words:**line: formation

**--**polarization

**--**magnetic fields

**--**turbulence

**--**radiative transfer

**--**Sun: atmosphere

**©**

*ESO 2008*