Zeeman line formation in solar magnetic fields. Studies with empirical probability distribution functions

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