Some aspects of polarized line formation in magneto-turbulent media

Abstract

Observations and numerical simulations of magneto-convection show a highly variable solar magnetic field. Using a statistical approach, we analyze the effects of random magnetic fields on Stokes profiles of spectral lines. We consider the micro and macro-turbulent regimes, which provide bounds for more general random fields with finite scales of variations. The mean Stokes parameters are obtained in the micro-turbulent regime, by first averaging the Zeeman propagation matrix $\widehat{\mathbf{\Phi }}$ over the probability distribution function P(B) of the magnetic field and then solving the concerned radiative transfer equation. In the macro-turbulent regime, the mean solution is obtained by averaging the emergent solution over P(B). It is assumed that B has a Gaussian distribution defined by its mean field B₀, angular distribution and dispersion. Fluctuations parallel and perpendicular to B₀ are considered. Spectral lines are parameterized by their strength β, which is varied over the range 1–10⁴. A detailed comparison of micro and macro-turbulent limit with mean field solution shows that differences are important for β ⩾ 10. When β increases, the saturation behavior of micro-turbulent profiles are significantly different from that of mean field profiles. The Stokes profiles shapes are explained in terms of the non-linear β-dependence of the Unno–Rachkovsky solution using approximate expressions for the mean absorption coefficients. These expressions when inserted in the Unno–Rachkovsky solution can predict Stokes profiles that match with the numerical result to a good approximation.

Introduction

Quantitative analysis of spectro-polarimetric data entered an active phase with the analytical solution of Unno (1956). This solution considers only the absorption/emission of polarized radiation in a magnetized medium. The extension by Rachkovsky, 1962a, Rachkovsky, 1962b includes magneto-optical effects due to differential shifts between orthogonal polarization states, which appear during the propagation through the medium. Magneto-optical effects are important when Zeeman shifts are of the order of Doppler widths and affect Stokes parameters mainly around the line center. The analytical solution of the Stokes vector transfer equation known as Unno–Rachkovsky solution (hereafter referred to as UR solution) implies a uniform magnetic field and approximations regarding the atmospheric model known as Milne–Eddington approximations – namely that the line strength is independent of the depth in the atmosphere, and that the line source function varies linearly with optical depth. The Milne–Eddington approximation has provided insight into the physical processes taking place in line formation. Its specific analytical character is its most powerful feature. A new area in the analysis of polarization spectra was opened with numerical solutions of the polarized radiative transfer equation for realistic atmospheres involving depth-dependent physical quantities. It started with the work by Beckers, 1969, Wittmann, 1974, Landi Degl’Innocenti, 1976. See Rees (1987) for a historical review.

However the UR solutions continue to be used in astrophysics, in particular in inversion codes aimed at the automatic reconstruction of magnetic fields and atmospheric parameters from large sets of polarimetric data (e.g. UNNO-FIT technique – Landi Degl’Innocenti and Landolfi, 2004, p. 634, and references cited therein; Bellot Rubio, 2006 and references cited therein). It can, as recently shown, provide a systematic approach to evaluate the sensitivity of Stokes profiles to atmospheric and magnetic field parameters (Orozco Suárez and del Toro Iniesta, 2007). Let us also mention that a widely used atlas of theoretical Stokes profiles was constructed with the help of UR solution (Arena and Landi Degl’Innocenti, 1982). An excellent description of UR solution, extensions and practical applications, are presented in del Toro Iniesta, 2003, Landi Degl’Innocenti and Landolfi, 2004.

In this work, we present a systematic study of the UR solution for random magnetic fields. The UR solution, can be employed for random fields in two limiting regimes: (i) the regime of micro-turbulence in which the characteristic scale of variation of the random magnetic field is much smaller than a typical photon mean free path and (ii) the regime of macro-turbulence where one has the opposite situation. The micro-turbulent approach, suggested in Stenflo (1971), has been employed in, e.g. Stenflo and Lindegren, 1977, Sánchez Almeida et al., 1996. Multi-components models are special versions of the macro-turbulent limit (see, e.g. Stenflo, 1994 and references cited therein). Here, we assume that the magnetic field fluctuations are described by a probability distribution function P(B). In the micro-turbulent regime, the coefficients of the polarized transfer equation, in particular the Zeeman propagation matrix (also called absorption matrix), can be locally averaged over P(B). Dolginov and Pavlov, 1972, Domke and Pavlov, 1979 were the first to examine Zeeman line transfer for micro-turbulent magnetic fields and proposed explicit expressions for the mean values of the coefficients of the Zeeman propagation matrix. An up-to-date presentation of their results and some extensions can be found in Frisch et al. (2005, hereafter referred to as Paper I). In the macro-turbulent regime, the magnetic field is uniform over the region where the spectral line is formed but takes random values distributed according to P(B). The averaging over P(B) is performed on the emergent UR solution itself.

The micro and macro-turbulent limits cannot describe situations where the mean free path of photons is of the same order as the characteristic scale of variation of the magnetic field. This more general situation requires the solution of polarized radiative transfer equations with stochastic coefficients (Frisch et al., 2006a, hereafter referred to as Paper II; see also Frisch et al., 2007, Carroll and Kopf, 2007). The corresponding mean Stokes parameters always lie between the micro and macro-turbulent limits. The latter have thus a significant interest for assessing the effects of random magnetic fields.

In this paper we examine the micro and macro-turbulent limits for isotropic and anisotropic Gaussian magnetic field distributions. The velocity field is assumed to be micro-turbulent, and uncorrelated to the magnetic field. The results are compared to the UR solution corresponding to the mean magnetic field, henceforth referred to as the mean field solution. The comparison is carried out for lines with different line strength β = κ_l/κ_c (κ_l the frequency averaged line absorption coefficient, κ_c the continuum absorption coefficient). In a Milne–Eddington atmosphere β is a constant. The UR solution varies linearly with β when β is small or around unity, but non-linearly when β becomes large. We investigate in detail this non-linear behavior for turbulent magnetic fields.

In Section 2, we consider the micro-turbulent limit and in Section 3 the macro-turbulent limit for longitudinal and transverse propagation. In these sections and all the following ones, the results are shown for a residual Stokes vector r = (r_I, r_Q, r_U, r_V)^T, independent of the slope of the source function. In Section 4, we compare micro and macro-turbulence effects for an arbitrary orientation of the mean field and in Section 5 we discuss mean Stokes profiles calculated with isotropic and anisotropic distributions. In Section 6, concluding remarks are given. An Appendix is devoted to describe the basic equations.