L. S. Anusha

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.

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.

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.

ORIGIN OF SPATIAL VARIATIONS OF SCATTERING POLARIZATION IN THE WINGS OF THE Ca I 4227 A line

Polarization that is produced by coherent scattering can be modified by magnetic fields via the Hanle effect. This has opened a window to explorations of solar magnetism in parameter domains not accessible to the Zeeman effect. According to standard theory the Hanle effect should only be operating in the Doppler core of spectral lines but not in the wings. In contrast, our observations of the scattering polarization in the Ca i 4227 u line reveals the existence of spatial variations of the scattering polarization throughout the far line wings. This raises the question whether the observed spatial variations in wing polarization have a magnetic or non-magnetic origin. A magnetic origin may be possible if elastic collisions are able to cause sufficient frequency redistribution to make the Hanle effect effective in the wings without causing excessive collisional depolarization, as suggested by recent theories for partial frequency redistribution (PRD) with coherent scattering in magnetic fields. To model the wing polarization we bypass the problem of solving the full polarized radiative-transfer equations and instead apply an extended version of the technique based on the “last scattering approximation” (LSA). It assumes that the polarization of the emergent radiation is determined by the anisotropy of the incident radiation field at the last scattering event. We determine this anisotropy from the observed limb darkening as a function of wavelength throughout the spectral line. The empirical anisotropy profile is used together with the single-scattering redistribution matrix, which contains all the PRD, collisional, and magnetic-field effects. The model further contains a continuum opacity parameter, which increasingly dilutes the polarized line photons as we move away

Polarized Line Formation in Multi-dimensional Media. II. A Fast Method to Solve Problems with Partial Frequency Redistribution

In the previous paper of this series (Anusha & Nagendra 2010), we presented a formulation of the polarized radiative transfer equation for resonance scattering with partial frequency redistribution (PRD) in multi-dimensional media for a two-level atom model with unpolarized ground level, using the irreducible spherical tensors T K Q (i,) for polarimetry. We also presented a polarized approximate lambda iteration (PALI) method to solve this equation using the Jacobi iteration scheme. The formal solution used was based on a simple finite volume technique. In this paper, we develop a faster and more efficient method which uses the projection t echniques applied to the radiative transfer equation (the Stabilized Preconditioned Bi-Conjugate Gradient method). We now use a more accurate formal solver, namely the well known 2D (two dimensional) short characteristics method. Using the numerical method developed in Anusha & Nagendra (2010), we can consider only simpler cases of finite 2D slabs due to computational limitations. Using the method developed in this paper we could compute PRD solutions in 2D media, in the more difficult context of semi-infinite 2D slabs as well. We present several solutions which may serve as benchmarks in future studies in this area. Subject headings: line: formation – radiative transfer – polarization – scattering– Sun: atmosphere

Preconditioned Bi-Conjugate Gradient Method for Radiative Transfer in Spherical Media

A robust numerical method called the Preconditioned Bi-Conjugate Gradient (Pre-BiCG) method is proposed for the solution of the radiative transfer equation in spherical geometry. A variant of this method called Stabilized Preconditioned Bi-Conjugate Gradient (Pre-BiCG-STAB) is also presented. These are iterative methods based on the construction of a set of bi-orthogonal vectors. The application of the Pre-BiCG method in some benchmark tests shows that the method is quite versatile, and can handle difficult problems that may arise in astrophysical radiative transfer theory.

An efficient decomposition technique to solve angle-dependent Hanle scattering problems

Abstract Hanle scattering is an important diagnostic tool to study weak solar magnetic fields. Partial frequency redistribution (PRD) is necessary to interpret the linear polarization observed in strong resonance lines. Usually angle-averaged PRD functions are used to analyze linear polarization. However, it is established that angle-dependent PRD functions are often necessary to interpret polarization profiles formed in the presence of weak magnetic fields. Our aim is to present an efficient decomposition technique, and the numerical method to solve the concerned angle-dependent line transfer problem. Together with the standard Stokes decomposition technique, we employ Fourier expansion over the outgoing azimuth angle to express in a more convenient form, the angle-dependent PRD function for the Hanle effect. It allows the use of angle-dependent frequency domains of Bommier to solve the Hanle transfer problem. Such an approach is self-consistent and accurate compared to a recent approach where angle-averaged frequency domains were used to solve the same problem. We show that it is necessary to incorporate angle-dependent frequency domains instead of angle-averaged frequency domains to solve the Hanle transfer problem accurately, especially for the Stokes U parameter. The importance of using angle-dependent domains has been highlighted by taking the example of Hanle effect in the case of line transfer with vertical magnetic fields in a slab atmosphere. We have also studied the case of polarized line formation when micro-turbulent magnetic fields are present. The difference between angle-averaged and angle-dependent solutions is enhanced by the presence of micro-turbulent fields.

Modeling the center-to-limb variation of the Ca i 4227 Å line using FCHHT models

The Ca i 4227 Å is a chromospheric line exhibiting the largest degree of linear polarization near the limb, in the visible spectrum of the Sun. Modeling the observations of the center-to-limb variations (CLV) of different lines in the Second Solar Spectrum helps to sample the height dependence of the magnetic field, as the observations made at different lines of sight sample different heights in the solar atmosphere. Supriya et al. (2014) attempted to simultaneously model the CLV of the (I , Q/I) spectra of the Ca i 4227 Å line using the standard 1-D FAL model atmospheres. They found that the standard FAL model atmospheres and also any appropriate combination of them, fail to simultaneously fit the observed Stokes (I , Q/I) profiles at all the limb distances (μ) satisfying at the same time all the observational constraints. This failure of 1-D modeling approach can probably be overcome by using multi-dimensional modeling which is computationally expensive. To eliminate an even wider choice of 1-D models, we attempt here to simultaneously model the CLV of the (I , Q/I) spectra using the FCHHT solar model atmospheres which are updated and recent versions of the FAL models. The details of our modeling efforts and the results are presented.