K. N. Nagendra

Polarized Line Formation with J-state Interference in the Presence of Magnetic Fields. I. Partial Frequency Redistribution in the Collisionless Regime

Quantum interference phenomena play a fundamental role in astrophysical spectra that are formed by coherent scattering processes. Here we derive a partial frequency redistribution (PRD) matrix that includes J-state interference in the presence of magnetic fields of arbitrary strength. The paper focuses on PRD in the collisionless regime, which in the traditional PRD terminology is referred to as Hummers type-II scattering. By limiting the treatment to the linear Zeeman regime, for which the Zeeman splitting is much smaller than the fine-structure splitting, it is possible to formulate analytical expressions for the PRD matrices. In the special case of non-magnetic scattering we recover the redistribution matrix derived from an independent quantum electrodynamic formulation based on the metalevel theory.

Generalization of the Last Scattering Approximation for the Second Solar Spectrum Modeling: The Ca I 4227 Å Line as a Case Study

To model the second solar spectrum (the linearly polarized spectrum of the Sun that is due to coherent scattering processes), one needs to solve the polarized radiative transfer (RT) equation. For strong resonance lines, partial frequency redistribution (PRD) effects must be accounted for, which make the problem computationally demanding. The “last scattering approximation” (LSA) is a concept that has been introduced to make this highly complex problem more tractable. An earlier application of a simple LSA version could successfully model the wings of the strong Cai 4227 A resonance line in Stokes Q/I (fractional linear polarization), but completely failed to reproduce the observed Q/I peak in the line core. Since the magnetic field signatures from the Hanle effect only occur in the line core, we need to generalize the existing LSA approach if it is to be useful for the diagnostics of chromospheric and turbulent magnetic fields. In this paper, we explore three different approximation levels for LSA and compare each of them with the benchmark represented by the solution of the full polarized RT, including PRD effects. The simplest approximation level is LSA-1, which uses the observed center-to-limb variation of the intensity profile to obtain the anisotropy of the radiation field at the surface, without solving any transfer equation. In contrast, the next two approximation levels use the solution of the unpolarized transfer equation to derive the anisotropy of the incident radiation field and use it as an input. In the case of LSA-2, the anisotropy at level τλ = μ, the atmospheric level from which an observed photon is most likely to originate, is used. LSA-3, on the other hand, makes use of the full depth dependence of the radiation anisotropy. The Q/I formula for LSA-3 is obtained by keeping the first term in a series expansion of the Q-source function in powers of the mean number of scattering events. Computationally, LSA-1 is 21 times faster than LSA-2, which is 5 times faster than the more general LSA-3, which itself is 8 times faster than the polarized RT approach. A comparison of the calculated Q/I spectra with the RT benchmark shows excellent agreement for LSA-3, including good modeling of the Q/I core region with its PRD effects. In contrast, both LSA-1 and LSA-2 fail to model the core region. The RT and LSA-3 approaches are then applied to model the recently observed Q/I profile of the Cai 4227 A line in quiet regions of the Sun. Apart from a global scale factor both give a very good fit to the Q/I spectra for all the wavelengths, including the core peak and blend line depolarizations. We conclude that LSA-3 is an excellent substitute for the full polarized RT and can be used to interpret the second solar spectrum, including the Hanle effect with PRD. It also allows the techniques developed for unpolarized three-dimensional RT to be applied to the modeling of the second solar spectrum.

Hanle-Zeeman Redistribution Matrix. II. Comparison of Classical and Quantum Electrodynamic Approaches

The Hanle-Zeeman redistribution matrix accounts for the intricately coupled correlations in frequency, angle, and polarizationbetweentheincomingandoutgoingradiationandembodiesthephysicsof thescatteringprocess.Weshow explicitly for a J ¼ 0 ! 1 ! 0 scattering transition the equivalence between the Hanle-Zeeman redistribution matrix that is derived through quantum electrodynamics and the one derived through classical, time-dependent oscillator theory. This equivalence holds for all strengths and directions of the magnetic field. Several aspects of the Hanle-Zeeman redistribution matrix are illustrated, andexplicit algebraic expressions are given, which are of practical use for the polarized line transfer computations. While the efficiency of the Hanle effect is usually confined to the line core, we show how elastic collisions can produce a ‘‘wing Hanle effect’’ as well under favorable conditions in the solar atmosphere. Subject headingg atomic processes — line: formation — polarization — radiative transfer

The Hanle effect in a random magnetic field Dependence of the polarization on statistical properties of the magnetic field

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.

An Operator Perturbation Method for Polarized Line Transfer

In this paper we present an operator perturbation method to solve the Hanle effect problem in one-dimensional (1D) media with partial frequency redistribution (PRD) and assuming a two-level atom without ground-level polarization. This is a generalization of our PALI-H method (Nagendra et al., 1998) to the case of PRD. We make use of the Fourier azimuthal decomposition of the radiation field to treat the non axi-symmetry caused by the presence of an arbitrarily oriented magnetic field. We work with the approximation that frequency redistribution is decoupled from angular redistribution.

Hanle-Zeeman Redistribution Matrix. I. Classical Theory Expressions in the Laboratory Frame

Polarized scattering in spectral lines is governed by a 4; 4 matrix that describes how the Stokes vector is scattered and redistributed in frequency and direction. Here we develop the theory for this redistribution matrix in the presence of magnetic fields of arbitrary strength and direction. This general magnetic field case is called the Hanle- Zeeman regime, since it covers both of the partially overlapping weak- and strong- field regimes in which the Hanle and Zeeman effects dominate the scattering polarization. In this general regime, the angle-frequency correlations that describe the so-called partial frequency redistribution (PRD) are intimately coupled to the polarization properties. We develop the theory for the PRD redistribution matrix in this general case and explore its detailed mathematical properties and symmetries for the case of a J = 0 -> 1 -> 0 scattering transition, which can be treated in terms of time-dependent classical oscillator theory. It is shown how the redistribution matrix can be expressed as a linear superposition of coherent and noncoherent parts, each of which contain the magnetic redistribution functions that resemble the well- known Hummer- type functions. We also show how the classical theory can be extended to treat atomic and molecular scattering transitions for any combinations of quantum numbers.

POLARIZED LINE FORMATION IN MULTI-DIMENSIONAL MEDIA. I. DECOMPOSITION OF STOKES PARAMETERS IN ARBITRARY GEOMETRIES

The solution of the polarized line radiative transfer (RT) equation in multi-dimensional geometries has been rarely addressed and only under the approximation that the changes of frequencies at each scattering are uncorrelated (complete frequency redistribution). With the increase in the resolution power of telescopes, being able to handle RT in multi-dimensional structures becomes absolutely necessary. In the present paper, our first aim is to formulate the polarized RT equation for resonance scattering in multi-dimensional media, using the elegant technique of irreducible spherical tensors . Our second aim is to develop a numerical method of a solution based on the polarized approximate lambda iteration (PALI) approach. We consider both complete frequency redistribution and partial frequency redistribution (PRD) in the line scattering. In a multi-dimensional geometry, the radiation field is non-axisymmetrical even in the absence of a symmetry breaking mechanism such as an oriented magnetic field. We generalize here to the three-dimensional (3D) case, the decomposition technique developed for the Hanle effect in a one-dimensional (1D) medium which allows one to represent the Stokes parameters I, Q, U by a set of six cylindrically symmetrical functions. The scattering phase matrix is expressed in terms of , with Ω being the direction of the outgoing ray. Starting from the definition of the source vector, we show that it can be represented in terms of six components SK Q independent of Ω. The formal solution of the multi-dimensional transfer equation shows that the Stokes parameters can also be expanded in terms of . Because of the 3D geometry, the expansion coefficients IK Q remain Ω-dependent. We show that each IK Q satisfies a simple transfer equation with a source term SK Q and that this transfer equation provides an efficient approach for handling the polarized transfer in multi-dimensional geometries. A PALI method for 3D, associated with a core-wing separation method for treating PRD, is developed. It is tested by comparison with 1D solutions, and several benchmark solutions in the 3D case are given.

POLARIZED LINE FORMATION IN MULTI-DIMENSIONAL MEDIA. III. HANLE EFFECT WITH PARTIAL FREQUENCY REDISTRIBUTION

In two previous papers, we solved the polarized radiative transfer (RT) equation in multi-dimensional (multi-D) geometries with partial frequency redistribution as the scattering mechanism. We assumed Rayleigh scattering as the only source of linear polarization (Q/I, U/I) in both these papers. In this paper, we extend these previous works to include the effect of weak oriented magnetic fields (Hanle effect) on line scattering. We generalize the technique of Stokes vector decomposition in terms of the irreducible spherical tensors , developed by Anusha & Nagendra, to the case of RT with Hanle effect. A fast iterative method of solution (based on the Stabilized Preconditioned Bi-Conjugate-Gradient technique), developed by Anusha et al., is now generalized to the case of RT in magnetized three-dimensional media. We use the efficient short-characteristics formal solution method for multi-D media, generalized appropriately to the present context. The main results of this paper are the following:? (1) a comparison of emergent (I, Q/I, U/I) profiles formed in one-dimensional (1D) media, with the corresponding emergent, spatially averaged profiles formed in multi-D media, shows that in the spatially resolved structures, the assumption of 1D may lead to large errors in linear polarization, especially in the line wings. (2) The multi-D RT in semi-infinite non-magnetic media causes a strong spatial variation of the emergent (Q/I, U/I) profiles, which is more pronounced in the line wings. (3) The presence of a weak magnetic field modifies the spatial variation of the emergent (Q/I, U/I) profiles in the line core, by producing significant changes in their magnitudes.

Observations of the forward scattering Hanle effect in the Ca I 4227 Å line

Chromospheric magnetic fields are notoriously difficult to measure. The chromospheric lines are broad, while the fields are producing a minuscule Zeeman-effect polarization. A promising diagnostic alternative is provided by the forward-scattering Hanle effect, which can be recorded in chromospheric lines such as the He i 10830 A and the Ca i 4227 A lines. We present a set of spectropolarimetric observations of the full Stokes vector obtained near the center of the solar disk in the Ca i 4227 A line with the ZIMPOL polarimeter at the IRSOL observatory. We detect a number of interesting forward-scattering Hanle effect signatures, which we model successfully using polarized radiative transfer. Here we focus on the observational aspects, while a separate companion paper deals with the theoretical modeling.

J-state interference signatures in the second solar spectrum - Modeling the Cr i triplet at 5204–5208 Å

The scattering polarization in the solar spectrum is traditionally modeled with each spectral line treated separately, but this is generally inadequate for multiplets where J-state interference plays a significant role. Through simultaneous observations of all the 3 lines of a Cr i triplet, combined with realistic radiative transfer modeling of the data, we show that it is necessary to include J-state interference consistently when modeling lines with partially interacting fine structure components. Polarized line formation theory that includes J-state interference effects together with partial frequency redistribution for a two-term atom is used to model the observations. Collisional frequency redistribution is also accounted for. We show that the resonance polarization in the Cr i triplet is strongly affected by the partial frequency redistribution effects in the line core and near wing peaks. The Cr i triplet is quite sensitive to the temperature structure of the photospheric layers. Our complete frequency redistribution calculations in semi-empirical models of the solar atmosphere cannot reproduce the observed near wing polarization or the cross-over of the Stokes Q/I line polarization about the continuum polarization level that is due to the J-state interference. When however partial frequency redistribution is included, a good fit to these features can be achieved. Further, to obtain a good fit to the far wings, a small temperature enhancement of the FALF model in the photospheric layers is necessary.