The Hanle effect in a random magnetic field

Abstract

Context. The Hanle effect is used to determine weak turbulent magnetic fields in the solar atmosphere, usually assuming that the angular distribution is isotropic, the magnetic field strength constant, and that micro-turbulence holds, i.e. that the magnetic field correlation length is much less than a photon mean free path. Aims. To examine the sensitivity of turbulent magnetic field measurements to these assumptions, we study the dependence of Hanle effect on the magnetic field correlation length, its angular, and strength distributions. Methods. We introduce a fairly general random magnetic field model characterized by a correlation length and a magnetic field vector distribution. Micro-turbulence is recovered when the correlation length goes to zero and macro-turbulence when it goes to infinity. Radiative transfer equations are established for the calculation of the mean Stokes parameters and they are solved numerically by a polarized approximate lambda iteration method. Results. We show that optically thin spectral lines and optically very thick ones are insensitive to the correlation length of the magnetic field, while spectral lines with intermediate optical depths (around 10-100) show some sensitivity to this parameter. The result is interpreted in terms of the mean number of scattering events needed to create the surface polarization. It is shown that the single-scattering approximation holds good for thin and thick lines but may fail for lines with intermediate thickness. The dependence of the polarization on the magnetic field vector probability density function (PDF) is examined in the micro-turbulent limit. A few PDFs with different angular and strength distributions, but equal mean value of the magnetic field, are considered. It is found that the polarization is in general quite sensitive to the shape of the magnetic field strength PDF and somewhat to the angular distribution. Conclusions. The mean field derived from Hanle effect analysis of polarimetric data strongly depends on the choice of the field strength distribution used in the analysis. It is shown that micro-turbulence is in general a safe approximation.

Spectral line polarization with angle-dependent partial frequency redistribution

“…Two are of the accelerated lambda iteration (ALI) type; they are generalizations of the frequency-by-frequency (FBF) and core-wing methods originally formulated for scalar PRD transfer in Paletou & Auer (1995). The other is a scattering expansion (Neumann series) method developed for the Hanle effect with complete frequency redistribution in Frisch et al (2009).…”

“…5, we show the convergence history for the ratio Q/I. We have not performed a systematic investigation of the variation in the speed of convergence with the slab thickness as was done for CRD in Frisch et al (2009), but the calculations that we have performed suggest that the behavior observed with CRD will also hold for angle-dependent and angle-averaged PRD, namely, a very good approximation to the emergent polarization is provided by the single scattered solution for optically thin slabs (T 10) and optically very thick ones (T ≥ 2 × 10 6 ), but is not reliable for intermediate optical thicknesses. As shown here, the problem can be cured by considering higher order terms in the scattering expansion.…”

“…It is based on a Neumann series expansion of the components of the source vector contributing to the polarization. This Neumann series amounts to an expansion in the mean number of scattering events (see Frisch et al 2009). Its first term yields the so-called single scattering solution.…”

Nonlinear force-free field extrapolation of the coronal magnetic field using the data obtained by the Hinode satellite

“…[5] Up to now, while many efforts have been devoted to the direct measurements of magnetic fields in the solar chromosphere and corona [e.g., Judge, 1998;Solanki et al, 2003;Lagg et al, 2004;Lin et al, 2004;Kramar et al, 2006;Trujillo Bueno and Asensio Ramos, 2007;Liu and Lin, 2008;Cargill, 2009;Frisch et al, 2009], the extrapolation based on NLFFF model is still a commonly used and necessary approach to understand the 3-D configuration of the coronal magnetic field. That is, the field can be reconstructed by solving the force-free and divergence-free differential equations (2) and (3) or equivalent equations (4) and (5), in which the observed photospheric vector magnetogram is taken as the bottom boundary condition.…”

Spectral line polarization with angle-dependent partial frequency redistribution

“…In the following section we present an iterative method called the SEM to solve Eq. (11) efficiently (see Frisch et al 2009;Sampoorna et al 2011). …”