Polarized Line Formation In Turbulent And Scattering Media

Abstract

This thesis is devoted to improve our knowledge on the theory of polarized line formation in a magneto-turbulent medium, and in a scattering dominated magnetized medium, where partial redistribution (PRD) effects become important. Thus the thesis consists of two parts. In the first part we carry out a detailed investigation on the effect of random magnetic fields on Zeeman line radiative transfer. In the second part we develop the theory of polarized line formation in the presence of arbitrary magnetic fields and with PRD. We present numerical methods of solution of the relevant transfer equation in both part-I and II. In Chapter I we give a general introduction, that describes the basic physical concepts required in both parts of the thesis. Chapters 2-6 deal with the part-I, namely stochastic polarized Zeeman line formation. Chapters 7-10 deal with part –II, namely the theory and numerics of polarized line formation in scattering media. Chapter II is devoted to the future outlook on the problems described in part-I and II of the thesis. Appendices are devoted to additional mathematical details. Part-I of the Thesis: Stochastic polarized line formation in magneto-turbulent media Magneto-convection on the Sun has a size spectrum that spans several orders of magnitudes and hence develops turbulent elements or eddies the sizes of which are much smaller than the spatial resolution of current spectro-polarimeters (about 0.2 arcsec or 150km at the photospheric level). We were thus strongly motivated to consider the Zeeman effect in a medium where the magnetic field is random with characteristic scales of variation comparable to the radiative transfer characteristic scales. In Chapter 2, we consider the micro-turbulent limit and study the mean zeeman absorption matrix in detail. The micro-turbulent limit refers to the case when the scales of fluctuations of the random field are much smaller than the photon mean free paths associated to the line formation. The ‘mean’ absorption and anomalous dispersion coefficients are calculated for random fields with a non-Zero mean value - isotropic or anisotropic Gaussian distributions that are azimuthally invariant about the direction of the mean field. The averaging method is described in detail, and fairly explicit expressions for the mean coefficients are established. A detailed numerical investigation of the mean coefficients illustrates two simple effects of the magnetic field fluctuations: (i) broadening of the components by fluctuations of the field strength, leaving the π-components unchanged, and (ii) averaging over the angular dependence of the π and components. Angular averaging can modify the frequency profiles of the mean coefficients quite drastically, namely, the appearance of an unpolarized central component in the diagonal absorption coefficient, even when the mean field is in the direction of the line-of-sight. For isotropic fluctuations, the mean coefficients can be expressed in terms of generalized Voigt and Faraday-Voigt functions, which are related to the derivatives of the Voigt and Faraday-Voigt functions. In chapter 3, we study these functions in detail. Simple recurrence relations are established and used for the calculation of the functions themselves and of their partial derivatives. Asymptotic expansions are also derived. In Chapter 4, we consider the Zeeman effect from a magnetic field which has a finite correlation length(meso-turbulence) that can be varied from zero to infinity and thus made comparable to the photon mean free-path. The random vector magnetic field B is modeled by a Kubo-Anderson process – a piecewise constant Markov process characterized by a correlation length and a probability distribution function(PDF) for the random values of the magnetic field. The micro- and macro-turbulent limits are recovered when the correlation length goes to zero or infinity respectively. Mean values and rms fluctuations around the mean values are calculated numerically for a random magnetic field with isotropic Gaussian fluctuations. The effects of a finite correlation length are discussed in detail. The rms fluctuations of the Stokes parameters are shown to be very sensitive to the correlation length of the magnetic field. It is suggested to use them as a diagnostic tools to determine the scale of unresolved features in the solar atmosphere. In Chapter 5, using statistical approach, we analyze the effects of random magnetic fields on Stokes line profiles. 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 absorption matrix Φ over the PDF 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). In this chapter, we consider the same Gaussian PDFs that are used to construct (Φ) in chapter 2. Numerical simulations of magneto-convection and analysis of solar magnetograms provide the empirical PDF for the magnetic field line-of-sight component on the solar atmosphere. In Chapter 6, we explore the effects of different kinds of PDFs on Zeeman line formation. We again consider the limits of micro and macro-turbulence. The types of PDFs considered are: (a) Voigt function and stretched exponential type PDFs for fields with fixed direction but fluctuating strength. (b) 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. The composite PDF proposed has an angular distribution peaked about the vertical direction for strong fields and is nearly isotropically distributed for weak fields, which could mimic solar surface random fields. 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. Part-II of the Thesis: Polarized line formation in scattering media-Theory and numerical methods Many of the strongest and most conspicuous lines in the Second Solar Spectrum are strong lines that are formed rather high, often in the chromosphere above the temperature minimum. From the standard, unpolarized and non-magnetic line-formation theory such lines are known to be formed under the conditions that are very far from local thermodynamic equilibrium. They are characterized by broad damping wings surrounding the line core. Doppler shifts in combination with collisions cause photons that are absorbed at a given frequency to be redistributed in frequency across the line profile in a complex way during the scattering process. Two idealized, limiting cases to describe this redistribution are “frequency coherence” and “complete redistribution” (CRD), but the general theory that properly combines these two limiting cases goes under the name “partial frequency redistribution” (PRD). Resonance lines which are usually strong can be properly modeled only when PRD is taken into account. To use these strong lines for magnetic field diagnostics we need a line scattering theory of PRD in the presence of magnetic fields of arbitrary strength. In the second part of the thesis we develop such a theory and derive the polarized PRD matrices. These matrices are then used in the polarized line transfer equation to compute the emergent Stokes parameters. Polarized scattering in spectral lines is governed by a 4 x 4 matrix that describes how the Stokes vector is scattered in all directions and redistributed in frequency within the line. In Chapter 7, using a classical approach we develop the theory for this redistribution matrix in the presence of magnetic fields of arbitrary strength and direction, and for a J = 0 → 1 → 0 transition. This case of arbitrary magnetic fields is called the Hanle-Zeeman regime, since it covers both the partially overlapping weak and strong-field regimes, in which the Hanle and Zeeman effects respectively dominate the scattering polarization. In this general regime the angle-frequency correlations that describe the so-called PRD are intimately coupled to the polarization properties. We also show how the classical theory can be extended to treat atomic and molecular scattering transitions for any combinations of J quantum numbers. In chapter 8 , we show explicitly that for a J = 0 → 1 → 0 scattering transition there exists an equivalence between the Hanle-Zeeman redistribution matrix that is derived through quantum electrodynamics(Bommier 1997b) and the one derived in Chapter 7 starting from the classical, time-dependent oscillator theory of Bommier & Stenflo (1999). This equivalence holds for all strengths and directions of the magnetic field. Several aspects of the Hanle-Zeeman redistribution matrix are illustrated, and explicit algebraic expressions are given, which are of practical use for the polarized line transfer computations. In chapter 9, we solve the polarized radiative transfer equation numerically, taking into account both the Zeeman absorption matrix and the Hanle-Zeeman redistribution matrix. We compute the line profiles for arbitrary field strengths, and scattering dominated line transitions. We use a perturbation method (see eg. Nagendra et al. 2002) to solve the Hanle-Zeeman line transfer problem. The limiting cases of weak field Hanle scattering and strong field Zeeman true absorption are retrieved. The ilntermediate regime, where both Zeeman absorption and scattering effects are important, is studied in some detail. Numerical method used to solve the Hanle-Zeeman line transfer problem in Chapter 9 is computationally expensive. Hence it is necessary to develop fast iterative methods like PALI (Polarized Approximate Lambda Iteration). As a first step in this direction we develop such a method in Chapter 10 to solve the transfer problem with weak field Hanle scattering. We use a ‘redistribution matrix’ with coupling between frequency redistribution and polarization and no domain decomposition. Such a matrix is constructed by angle-averaging the frequency dependent terms in the exact weak field Hanle redistribution matrix for a two-level atom with unpolarized ground level (that can be obtained by taking the weak field limit of the Hanle-Zeeman redistribution matrix). In the past, the PALI technique has been applied to redistribution matrices in which frequency redistribution is ‘decoupled’ from scattering polarization, the decoupling being achieved by an adequate decomposition of the frequency space into several domains. In this chapter, we examine the consequences of frequency space decomposition, and the resulting decoupling between the frequency redistribution and polarization, on the solution of the polarized transfer equation for the Stokes parameters.