Author name code: anusha ADS astronomy entries on 2022-09-14 =author:"Anusha, L.S." ------------------------------------------------------------------------ Title: Chromospheric extension of the MURaM code Authors: Przybylski, D.; Cameron, R.; Solanki, S. K.; Rempel, M.; Leenaarts, J.; Anusha, L. S.; Witzke, V.; Shapiro, A. I. Bibcode: 2022A&A...664A..91P Altcode: 2022arXiv220403126P Context. Detailed numerical models of the chromosphere and corona are required to understand the heating of the solar atmosphere. An accurate treatment of the solar chromosphere is complicated by the effects arising from non-local thermodynamic equilibrium (NLTE) radiative transfer. A small number of strong, highly scattering lines dominate the cooling and heating in the chromosphere. Additionally, the recombination times of ionised hydrogen are longer than the dynamical timescales, requiring a non-equilibrium (NE) treatment of hydrogen ionisation.
Aims: We describe a set of necessary additions to the MURaM code that allow it to handle some of the important NLTE effects. We investigate the impact on solar chromosphere models caused by NLTE and NE effects in radiation magnetohydrodynamic simulations of the solar atmosphere.
Methods: The MURaM code was extended to include the physical process required for an accurate simulation of the solar chromosphere, as implemented in the Bifrost code. This includes a time-dependent treatment of hydrogen ionisation, a scattering multi-group radiation transfer scheme, and approximations for NLTE radiative cooling.
Results: The inclusion of NE and NLTE physics has a large impact on the structure of the chromosphere; the NE treatment of hydrogen ionisation leads to a higher ionisation fraction and enhanced populations in the first excited state throughout cold inter-shock regions of the chromosphere. Additionally, this prevents hydrogen ionisation from buffering energy fluctuations, leading to hotter shocks and cooler inter-shock regions. The hydrogen populations in the ground and first excited state are enhanced by 102-103 in the upper chromosphere and by up to 109 near the transition region.
Conclusions: Including the necessary NLTE physics leads to significant differences in chromospheric structure and dynamics. The thermodynamics and hydrogen populations calculated using the extended version of the MURaM code are consistent with previous non-equilibrium simulations. The electron number and temperature calculated using the non-equilibrium treatment of the chromosphere are required to accurately synthesise chromospheric spectral lines.

Movies associated to Fig. 2 are only available at https://www.aanda.org Title: MPS-ATLAS: A fast all-in-one code for synthesising stellar spectra Authors: Witzke, V.; Shapiro, A. I.; Cernetic, M.; Tagirov, R. V.; Kostogryz, N. M.; Anusha, L. S.; Unruh, Y. C.; Solanki, S. K.; Kurucz, R. L. Bibcode: 2021A&A...653A..65W Altcode: 2021arXiv210513611W Context. Stellar spectral synthesis is essential for various applications, ranging from determining stellar parameters to comprehensive stellar variability calculations. New observational resources as well as advanced stellar atmosphere modelling, taking three dimensional effects from radiative magnetohydrodynamics calculations into account, require a more efficient radiative transfer.
Aims: For accurate, fast and flexible calculations of opacity distribution functions (ODFs), stellar atmospheres, and stellar spectra, we developed an efficient code building on the well-established ATLAS9 code. The new code also paves the way for easy and fast access to different elemental compositions in stellar calculations.
Methods: For the generation of ODF tables, we further developed the well-established DFSYNTHE code by implementing additional functionality and a speed-up by employing a parallel computation scheme. In addition, the line lists used can be changed from Kurucz's recent lists. In particular, we implemented the VALD3 line list.
Results: A new code, the Merged Parallelised Simplified ATLAS, is presented. It combines the efficient generation of ODF, atmosphere modelling, and spectral synthesis in local thermodynamic equilibrium, therefore being an all-in-one code. This all-in-one code provides more numerical functionality and is substantially faster compared to other available codes. The fully portable MPS-ATLAS code is validated against previous ATLAS9 calculations, the PHOENIX code calculations, and high-quality observations. Title: Radiative Transfer with Opacity Distribution Functions: Application to Narrowband Filters Authors: Anusha, L. S.; Shapiro, A. I.; Witzke, V.; Cernetic, M.; Solanki, S. K.; Gizon, L. Bibcode: 2021ApJS..255....3A Altcode: 2021arXiv210413661A Modeling of stellar radiative intensities in various spectral passbands plays an important role in stellar physics. At the same time, direct calculation of the high-resolution spectrum and then integration of it over the given spectral passband is computationally demanding due to the vast number of atomic and molecular lines. This is particularly so when employing three-dimensional (3D) models of stellar atmospheres. To accelerate the calculations, one can employ approximate methods, e.g., the use of opacity distribution functions (ODFs). Generally, ODFs provide a good approximation of traditional spectral synthesis, i.e., computation of intensities through filters with strictly rectangular transmission functions. However, their performance strongly deteriorates when the filter transmission noticeably changes within its passband, which is the case for almost all filters routinely used in stellar physics. In this context, the aims of this paper are (a) to generalize the ODFs method for calculating intensities through filters with arbitrary transmission functions, and (b) to study the performance of the standard and generalized ODFs methods for calculating intensities emergent from 3D models of stellar atmospheres. For this purpose we use the newly developed MPS-ATLAS radiative transfer code to compute intensities emergent from 3D cubes simulated with the radiative magnetohydrodynamics code MURaM. The calculations are performed in the 1.5D regime, i.e., along many parallel rays passing through the simulated cube. We demonstrate that the generalized ODFs method allows accurate and fast syntheses of spectral intensities and their center-to-limb variations. Title: Nonequilibrium Equation of State in Stellar Atmospheres Authors: Anusha, L. S.; van Noort, M.; Cameron, R. H. Bibcode: 2021ApJ...911...71A Altcode: 2021arXiv210413650A In the stellar chromospheres, radiative energy transport is dominated by only the strongest spectral lines. For these lines, the approximation of local thermodynamic equilibrium (LTE) is known to be very inaccurate, and a state of equilibrium cannot be assumed in general. To calculate the radiative energy transport under these conditions, the population evolution equation must be evaluated explicitly, including all time-dependent terms. We develop a numerical method to solve the evolution equation for the atomic-level populations in a time-implicit way, keeping all time-dependent terms to first order. We show that the linear approximation of the time dependence of the populations can handle very large time steps without losing accuracy. We reproduce the benchmark solutions from earlier, well-established works in terms of non-LTE kinetic equilibrium solutions and typical ionization/recombination timescales in the solar chromosphere. Title: Power spectrum of turbulent convection in the solar photosphere Authors: Yelles Chaouche, L.; Cameron, R. H.; Solanki, S. K.; Riethmüller, T. L.; Anusha, L. S.; Witzke, V.; Shapiro, A. I.; Barthol, P.; Gandorfer, A.; Gizon, L.; Hirzberger, J.; van Noort, M.; Blanco Rodríguez, J.; Del Toro Iniesta, J. C.; Orozco Suárez, D.; Schmidt, W.; Martínez Pillet, V.; Knölker, M. Bibcode: 2020A&A...644A..44Y Altcode: 2020arXiv201009037Y The solar photosphere provides us with a laboratory for understanding turbulence in a layer where the fundamental processes of transport vary rapidly and a strongly superadiabatic region lies very closely to a subadiabatic layer. Our tools for probing the turbulence are high-resolution spectropolarimetric observations such as have recently been obtained with the two balloon-borne SUNRISE missions, and numerical simulations. Our aim is to study photospheric turbulence with the help of Fourier power spectra that we compute from observations and simulations. We also attempt to explain some properties of the photospheric overshooting flow with the help of its governing equations and simulations. We find that quiet-Sun observations and smeared simulations are consistent with each other and exhibit a power-law behavior in the subgranular range of their Doppler velocity power spectra with a power-law index of ≈ - 2. The unsmeared simulations exhibit a power law that extends over the full range between the integral and Taylor scales with a power-law index of ≈ - 2.25. The smearing, reminiscent of observational conditions, considerably reduces the extent of the power-law-like portion of the power spectra. This suggests that the limited spatial resolution in some observations might eventually result in larger uncertainties in the estimation of the power-law indices. The simulated vertical velocity power spectra as a function of height show a rapid change in the power-law index (at the subgranular range) from roughly the optical depth unity layer, that is, the solar surface, to 300 km above it. We propose that the cause of the steepening of the power-law index is the transition from a super- to a subadiabatic region, in which the dominant source of motions is overshooting convection. A scale-dependent transport of the vertical momentum occurs. At smaller scales, the vertical momentum is more efficiently transported sideways than at larger scales. This results in less vertical velocity power transported upward at small scales than at larger scales and produces a progressively steeper vertical velocity power law below 180 km. Above this height, the gravity work progressively gains importance at all relevant scales, making the atmosphere progressively more hydrostatic and resulting in a gradually less steep power law. Radiative heating and cooling of the plasma is shown to play a dominant role in the plasma energetics in this region, which is important in terms of nonadiabatic damping of the convective motions. Title: Fast Iterative Techniques for Polarized Radiative Transfer in Spherically Symmetric Moving Media Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Anusha, L. S.; Sankarasubramanian, K. Bibcode: 2020ApJ...903....6M Altcode: For a more precise modeling of polarized spectral lines formed in extended and expanding stellar atmospheres, the solution of the radiative transfer equation for the Stokes vectors must be obtained in a spherical geometry rather than in a planar geometry. In this paper, we present the modern iterative techniques based on operator perturbation to solve the spherically symmetric polarized radiative transfer equation with velocity fields. We consider scattering on a two-level atom and account for partial frequency redistribution. An accurate numerical solution to such problems requires the use of spatial grids with higher resolution. Consequently, Jacobi-based methods lead to slower convergence rate. The convergence rate can be improved by a factor of 2 or more when fast iterative schemes based on Gauss-Seidel (GS) and successive overrelaxation (SOR) methods are used over the Jacobi-based method. Here we present the Jacobi, GS, and SOR iterative techniques for solving the abovementioned problem, and discuss their convergence behavior. Title: The Dimmest State of the Sun Authors: Yeo, K. L.; Solanki, S. K.; Krivova, N. A.; Rempel, M.; Anusha, L. S.; Shapiro, A. I.; Tagirov, R. V.; Witzke, V. Bibcode: 2020GeoRL..4790243Y Altcode: 2021arXiv210209487Y How the solar electromagnetic energy entering the Earth's atmosphere varied since preindustrial times is an important consideration in the climate change debate. Detrimental to this debate, estimates of the change in total solar irradiance (TSI) since the Maunder minimum, an extended period of weak solar activity preceding the industrial revolution, differ markedly, ranging from a drop of 0.75 W m-2 to a rise of 6.3 W m-2. Consequently, the exact contribution by solar forcing to the rise in global temperatures over the past centuries remains inconclusive. Adopting a novel approach based on state-of-the-art solar imagery and numerical simulations, we establish the TSI level of the Sun when it is in its least-active state to be 2.0 ± 0.7 W m-2 below the 2019 level. This means TSI could not have risen since the Maunder minimum by more than this amount, thus restricting the possible role of solar forcing in global warming. Title: Importance of Angle-dependent Partial Frequency Redistribution in Hyperfine Structure Transitions Under the Incomplete Paschen-Back Effect Regime Authors: Nagendra, K. N.; Sowmya, K.; Sampoorna, M.; Stenflo, J. O.; Anusha, L. S. Bibcode: 2020ApJ...898...49N Altcode: 2020arXiv200704044N Angle-frequency coupling in scattering of polarized light on atoms is represented by the angle-dependent (AD) partial frequency redistribution (PRD) matrices. There are several lines in the linearly polarized solar spectrum, for which PRD combined with quantum interference between hyperfine structure states play a significant role. Here we present the solution of the polarized line transfer equation including the AD-PRD matrix for scattering on a two-level atom with hyperfine structure splitting and an unpolarized lower level. We account for the effects of arbitrary magnetic fields (including the incomplete Paschen-Back effect regime) and elastic collisions. For exploratory purposes we consider a self-emitting isothermal planar atmosphere and use atomic parameters that represent an isolated Na I D2 line. For this case we show that the AD-PRD effects are significant for field strengths below about 30 G, but that the computationally much less demanding approximation of angle-averaged PRD may be used for stronger fields. Title: Resonance Line Polarization in Spherically Symmetric Moving Media: a Parametric Study Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Anusha, L. S.; Sankarasubramanian, K. Bibcode: 2020arXiv200106201M Altcode: In the present paper we consider the problem of resonance line polarization formed in the spherically symmetric expanding atmospheres. For the solution of the concerned polarized transfer equation we use the comoving frame formulation, and apply the Accelerated Lambda Iteration (ALI) method. We restrict ourselves to the non-relativistic regime of velocities wherein mainly Doppler shift effects are significant. For our studies, we consider the scattering on a two-level atom, including the effects of partial frequency redistribution (PFR). We present the dependence of linearly polarized profiles on different atmospheric and atomic parameters. Title: Polarized Line Formation in Arbitrary Strength Magnetic Fields: The Case of a Two-level Atom with Hyperfine Structure Splitting Authors: Sampoorna, M.; Nagendra, K. N.; Sowmya, K.; Stenflo, J. O.; Anusha, L. S. Bibcode: 2019ApJ...883..188S Altcode: 2019arXiv191010913S Quantum interference effects, together with partial frequency redistribution (PFR) in line scattering, produce subtle signatures in the so-called Second Solar Spectrum (the linearly polarized spectrum of the Sun). These signatures are modified in the presence of arbitrary strength magnetic fields via the Hanle, Zeeman, and Paschen-Back effects. In the present paper we solve the problem of polarized line formation in a magnetized atmosphere taking into account scattering in a two-level atom with hyperfine structure splitting together with PFR. To this end we incorporate the collisionless PFR matrix derived in Sowmya et al. in the polarized transfer equation. We apply the scattering expansion method to solve this transfer equation. We study the combined effects of PFR and the Paschen-Back effect on polarized line profiles formed in an isothermal one-dimensional planar atmosphere. For this purpose, we consider the cases of D2 lines of Li I and Na I. Title: Resonance Line Polarization in Spherically Symmetric Moving Media: a Parametric Study. Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Anusha, L. S.; Sankarasubramanian, K. Bibcode: 2019spw..confE..14M Altcode: No abstract at ADS Title: Polarized Line Transfer in the Incomplete Paschen-Back Effect Regime with Angle-dependent Partial Frequency Redistribution. Authors: Nagendra, K. N.; Sowmya, K.; Sampoorna, M.; Stenflo, J. O.; Anusha, L. S. Bibcode: 2019spw..confE..13N Altcode: No abstract at ADS Title: Polarized Line Formation in Spherically Symmetric Expanding Atmospheres Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Anusha, L. S.; Sankarasubramanian, K. Bibcode: 2019ASPC..519...27M Altcode: We consider the problem of polarized line formation in the spherically symmetric expanding atmospheres. The velocity fields in line forming regions produce Doppler shift, aberration of photons and also gives rise to advection. These in turn can modify the amplitudes and shapes of the emergent Stokes profiles. However, here we consider only non-relativistic regime, wherein mainly Doppler shift effects are significant. Thus only Doppler shift terms are considered in the polarized transfer equation. For the solution of the concerned polarized transfer equation we use the comoving frame formulation, and apply the Accelerated Lambda Iteration (ALI) method. We present the results by considering the scattering on a two-level atom, including the effects of partial frequency redistribution (PFR). The polarized line profiles are shown for few velocity laws, representative of expanding spherical atmospheres. It is shown that the degree of polarization in the lines depends sensitively on the extendedness R of the spherical atmosphere. We also present a comparison of polarized profiles computed under complete frequency redistribution (CFR) and PFR in the case of static as well as expanding atmospheres. Title: Polarized Line Formation with Incomplete Paschen-Back Effect and Partial Frequency Redistribution Authors: Sampoorna, M.; Nagendra, K. N.; Sowmya, K.; Stenflo, J. O.; Anusha, L. S. Bibcode: 2019ASPC..519..113S Altcode: Quantum interference between the hyperfine structure states is known to depolarize the cores of some of the lines in the linearly polarized spectrum of the Sun (the Second Solar Spectrum). The presence of external magnetic fields in the line forming regions modify these signatures through the Hanle, Zeeman, and incomplete/ complete Paschen-Back effects (PBE), depending on the strength of the magnetic field. In an earlier paper, Sowmya et al. (2014) derived the relevant collisionless partial frequency redistribution (PFR) matrix for scattering on a two-level atom with hyperfine structure splitting (HFS) and in the presence of arbitrary strength magnetic fields (including the PBE regime). In the present paper we solve the problem of polarized line transfer in a magnetized atmosphere, including this PFR matrix. For this purpose, we apply a scattering expansion method which is based on orders of scattering approach. We present the results on the combined effects of PBE and PFR on the polarized line profiles using the atomic parameters relevant to the Na I D2 line. Title: Solution of Multi-dimensional Polarized Radiative Transfer Equation with PFR Authors: Anusha, L. S. Bibcode: 2019ASPC..519..105A Altcode: Solar observations and numerical simulations have proved the existence of enormous inhomogeneous structures in the solar atmosphere. Solving polarized radiative transfer (RT) equation is a powerful means of understanding the effects of these inhomogeneities on the spectrum emerging from the Sun. To take spatial inhomogeneities into account, a solution of the transfer equation in multi-dimensional (multi-D) geometries is necessary. Here we present a summary of the developments we carried out in a series of papers, to solve multidimensional polarized RT equation with partial frequency redistribution (PFR) and the application of these methods to analyze the observations. Title: Polarized Line Formation in Spherically Symmetric Atmospheres with Velocity Fields Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Anusha, L. S.; Sankarasubramanian, K. Bibcode: 2019ApJ...879...48M Altcode: The plane-parallel approximation of the stellar atmospheres cannot be applied to model the formation of optically thick lines formed in extended atmospheres. To a good approximation these atmospheres can be represented by a spherically symmetric medium. Such extended stellar atmospheres are dynamic, in general, due to the systematic motions present in their layers. Macroscopic velocity fields in the spectral line forming regions produce Doppler shift, aberration of photons, and also give rise to advection. All of these effects can modify the amplitudes and shapes of the emergent Stokes profiles. In the present paper we consider the problem of polarized line formation in spherically symmetric media, in the presence of velocity fields. Solving the radiative transfer problem in the observer’s frame is a straightforward approach to handle the presence of velocity fields. This method, however, becomes computationally prohibitive when large velocity fields are considered, particularly in the case of the line formation with partial frequency redistribution (PFR). In this paper we present a polarized comoving frame method to solve the problem at hand. We consider nonrelativistic radial velocity fields, thereby accounting only for Doppler shift effects and neglecting advection and aberration of photons. We study the effects of extendedness, velocity fields, and PFR on the polarized line profiles formed in highly extended atmospheres. Title: Estimation of the Magnetic Flux Emergence Rate in the Quiet Sun from Sunrise Data Authors: Smitha, H. N.; Anusha, L. S.; Solanki, S. K.; Riethmüller, T. L. Bibcode: 2017ApJS..229...17S Altcode: 2016arXiv161106432S Small-scale internetwork (IN) features are thought to be the major source of fresh magnetic flux in the quiet Sun. During its first science flight in 2009, the balloon-borne observatory Sunrise captured images of the magnetic fields in the quiet Sun at a high spatial resolution. Using these data we measure the rate at which the IN features bring magnetic flux to the solar surface. In a previous paper it was found that the lowest magnetic flux in small-scale features detected using the Sunrise observations is 9 × 1014 Mx. This is nearly an order of magnitude smaller than the smallest fluxes of features detected in observations from the Hinode satellite. In this paper, we compute the flux emergence rate (FER) by accounting for such small fluxes, which was not possible before Sunrise. By tracking the features with fluxes in the range {10}15{--}{10}18 Mx, we measure an FER of 1100 {Mx} {{cm}}-2 {{day}}-1. The smaller features with fluxes ≤slant {10}16 Mx are found to be the dominant contributors to the solar magnetic flux. The FER found here is an order of magnitude higher than the rate from Hinode, obtained with a similar feature tracking technique. A wider comparison with the literature shows, however, that the exact technique of determining the rate of the appearance of new flux can lead to results that differ by up to two orders of magnitude, even when applied to similar data. The causes of this discrepancy are discussed and first qualitative explanations proposed. Title: Statistical evolution of quiet-Sun small-scale magnetic features using Sunrise observations Authors: Anusha, L. S.; Solanki, S. K.; Hirzberger, J.; Feller, A. Bibcode: 2017A&A...598A..47A Altcode: 2016arXiv160808499A The evolution of small magnetic features in quiet regions of the Sun provides a unique window for probing solar magneto-convection. Here we analyze small-scale magnetic features in the quiet Sun, using the high resolution, seeing-free observations from the Sunrise balloon borne solar observatory. Our aim is to understand the contribution of different physical processes, such as splitting, merging, emergence and cancellation of magnetic fields to the rearrangement, addition and removal of magnetic flux in the photosphere. We have employed a statistical approach for the analysis and the evolution studies are carried out using a feature-tracking technique. In this paper we provide a detailed description of the feature-tracking algorithm that we have newly developed and we present the results of a statistical study of several physical quantities. The results on the fractions of the flux in the emergence, appearance, splitting, merging, disappearance and cancellation qualitatively agrees with other recent studies. To summarize, the total flux gained in unipolar appearance is an order of magnitude larger than the total flux gained in emergence. On the other hand, the bipolar cancellation contributes nearly an equal amount to the loss of magnetic flux as unipolar disappearance. The total flux lost in cancellation is nearly six to eight times larger than the total flux gained in emergence. One big difference between our study and previous similar studies is that, thanks to the higher spatial resolution of Sunrise, we can track features with fluxes as low as 9 × 1014 Mx. This flux is nearly an order of magnitude lower than the smallest fluxes of the features tracked in the highest resolution previous studies based on Hinode data. The area and flux of the magnetic features follow power-law type distribution, while the lifetimes show either power-law or exponential type distribution depending on the exact definitions used to define various birth and death events. We have also statistically determined the evolution of the flux within the features in the course of their lifetime, finding that this evolution depends very strongly on the birth and death process that the features undergo. Title: Flux emergence rate in the quiet Sun from Sunrise data Authors: Smitha, H. N.; Anusha, L. S.; Solanki, S. K.; Riethmüller, T. L. Bibcode: 2017psio.confE.106S Altcode: No abstract at ADS Title: Scattering Polarization of the Spectral Lines in the Solar Chromosphere Authors: Anusha, L. S.; Nanjundarao, N. K. Bibcode: 2015AGUFMSH43B2452A Altcode: Linear polarization produced by scattering of light and its magnetic field modification (namely, Hanle effect) in spectral lines can be used as diagnostic tools to understand the structuring of the solar atmosphere and the physical processes that produce and modify the linear, scattering poalrization signals. Modeling observations of strong spectral lines such as Ca II K at 3933 A using model solar atmosphers can give information on the less understood solar chromosphere, where this line is formed. Modeling the observations of the scattering poalrization in spectral lines requires the solution of polarized radiative transfer equation. Although solution is easier in one-dimenion, to be more realistic we must go for multi-dimensional geometries. In particular Ca II K line is appropriate for chromospheric diagnostics as it contains information on the less understood topics such as (1) scattering mechansim called partial frequency redistribution, (2) effect of magnetic field on line polarization in the chromosphere through Hanle effect, and (3) spatial structuring of the chromosphere that cause modifications in line polarization. Our aim here is to address these three topics through modeling the observations of Ca II K line using different model solar atmospheres. Title: Modeling the center-to-limb variation of the Ca i 4227 Å line using FCHHT models Authors: Supriya, H. D.; Smitha, H. N.; Nagendra, K. N.; Stenflo, J. O.; Bianda, M.; Ravindra, B.; Ramelli, R.; Anusha, L. S. Bibcode: 2015IAUS..305..381S Altcode: 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. Title: Study of resonance scattering polarization in O i 130 nm lines Authors: Anusha, L. S.; Nagendra, K. N.; Uitenbroek, Han Bibcode: 2015IAUS..305..234A Altcode: Here we address the importance of frequency cross-redistribution on the scattering polarization of the O i line at 130.2 nm. We compute the polarized profiles of this line with ordinary partial frequency redistribution and cross-redistribution using a two-dimensional radiative transfer code. Title: Evolution of Small Scale Magnetic Structures from Sunrise Data Authors: Anusha, L. S.; Feller, A.; Hirzberger, J.; Solanki, S. K. Bibcode: 2014ASPC..489...83A Altcode: We present the results of an analysis of small scale magnetic features in the quiet Sun, observed with the Sunrise balloon borne telescope. Our aim is to understand the contribution of different physical processes that drive the evolution of magnetic features in quiet regions of the photosphere. To this end, we study the rearrangement, addition, and removal of magnetic flux through splitting, merging, cancellation, and emergence of magnetic fields. Title: Multi-Dimensional Polarized Radiative Transfer: Methods and Solar Applications Authors: Anusha, L. S.; Nagendra, K. N. Bibcode: 2014ASPC..489..225A Altcode: Observations of the Sun using modern telescopes as well as numerical simulations of the Sun reveal existence of enormous inhomogeneous structures in the solar atmosphere. The polarized spectrum of the Sun (Second Solar Spectrum), produced due to anisotropic scattering helps to infer the temperature structure, magnetic fields, and other physical properties of the solar atmosphere more accurately. Analysis of the Second Solar Spectrum requires solution of the polarized radiative transfer equation. To take spatial inhomogeneities into account, a solution of the transfer equation in multi-dimensional geometries is necessary. In this paper we present a specialized review of recent developments in the methods to solve multi-dimensional polarized radiative transfer equation and an application of these methods to analyze the observations. Title: Effect of Cross-redistribution on the Resonance Scattering Polarization of O I Line at 1302 Å Authors: Anusha, L. S.; Nagendra, K. N.; Uitenbroek, H. Bibcode: 2014ApJ...794...17A Altcode: 2014arXiv1407.8456A Oxygen is the most abundant element on the Sun after hydrogen and helium. The intensity spectrum of resonance lines of neutral oxygen, namely, O I (1302, 1305, and 1306 Å), has been studied in the literature for chromospheric diagnostics. In this paper, we study the resonance scattering polarization in the O I line at 1302 Å using two-dimensional (2D) radiative transfer in a composite atmosphere constructed using a 2D magneto-hydrodynamical snapshot in the photosphere and columns of the one-dimensional FALC atmosphere in the chromosphere. The methods developed by us recently in a series of papers to solve multi-dimensional polarized radiative transfer have been incorporated in our new code POLY2D, which we use for our analysis. We find that multi-dimensional radiative transfer including XRD effects is important in reproducing the amplitude and shape of scattering polarization signals of the O I line at 1302 Å. Title: Center-to-limb Observations and Modeling of the Ca I 4227 Å Line Authors: Supriya, H. D.; Smitha, H. N.; Nagendra, K. N.; Stenflo, J. O.; Bianda, M.; Ramelli, R.; Ravindra, B.; Anusha, L. S. Bibcode: 2014ApJ...793...42S Altcode: 2014arXiv1407.5461S The observed center-to-limb variation (CLV) of the scattering polarization in different lines of the Second Solar Spectrum can be used to constrain the height variation of various atmospheric parameters, in particular the magnetic fields, via the Hanle effect. Here we attempt to model the nonmagnetic CLV observations of the Q/I profiles of the Ca I 4227 Å line recorded with the Zurich Imaging Polarimeter-3 at IRSOL. For modeling, we use the polarized radiative transfer with partial frequency redistribution with a number of realistic one-dimensional (1D) model atmospheres. We find that all the standard Fontenla-Avrett-Loeser (FAL) model atmospheres, which we used, fail to simultaneously fit the observed (I, Q/I) at all the limb distances (μ). However, an attempt is made to find a single model which can provide a fit to at least the CLV of the observed Q/I instead of a simultaneous fit to the (I, Q/I) at all μ. To this end we construct a new 1D model by combining two of the standard models after modifying their temperature structures in the appropriate height ranges. This new combined model closely reproduces the observed Q/I at all μ but fails to reproduce the observed rest intensity at different μ. Hence we find that no single 1D model atmosphere succeeds in providing a good representation of the real Sun. This failure of 1D models does not, however, cause an impediment to the magnetic field diagnostic potential of the Ca I 4227 Å line. To demonstrate this we deduce the field strength at various μ positions without invoking the use of radiative transfer. Title: Multi-dimensional Radiative Transfer to Analyze Hanle Effect in Ca II K Line at 3933 Å Authors: Anusha, L. S.; Nagendra, K. N. Bibcode: 2013ApJ...767..108A Altcode: 2013arXiv1308.3437A Radiative transfer (RT) studies of the linearly polarized spectrum of the Sun (the second solar spectrum) have generally focused on line formation, with an aim to understand the vertical structure of the solar atmosphere using one-dimensional (1D) model atmospheres. Modeling spatial structuring in the observations of the linearly polarized line profiles requires the solution of multi-dimensional (multi-D) polarized RT equation and a model solar atmosphere obtained by magnetohydrodynamical (MHD) simulations of the solar atmosphere. Our aim in this paper is to analyze the chromospheric resonance line Ca II K at 3933 Å using multi-D polarized RT with the Hanle effect and partial frequency redistribution (PRD) in line scattering. We use an atmosphere that is constructed by a two-dimensional snapshot of the three-dimensional MHD simulations of the solar photosphere, combined with columns of a 1D atmosphere in the chromosphere. This paper represents the first application of polarized multi-D RT to explore the chromospheric lines using multi-D MHD atmospheres, with PRD as the line scattering mechanism. We find that the horizontal inhomogeneities caused by MHD in the lower layers of the atmosphere are responsible for strong spatial inhomogeneities in the wings of the linear polarization profiles, while the use of horizontally homogeneous chromosphere (FALC) produces spatially homogeneous linear polarization in the line core. The introduction of different magnetic field configurations modifies the line core polarization through the Hanle effect and can cause spatial inhomogeneities in the line core. A comparison of our theoretical profiles with the observations of this line shows that the MHD structuring in the photosphere is sufficient to reproduce the line wings and in the line core, but only line center polarization can be reproduced using the Hanle effect. For a simultaneous modeling of the line wings and the line core (including the line center), MHD atmospheres with inhomogeneities in the chromosphere are required. Title: An efficient decomposition technique to solve angle-dependent Hanle scattering problems Authors: Supriya, H. D.; Sampoorna, M.; Nagendra, K. N.; Ravindra, B.; Anusha, L. S. Bibcode: 2013JQSRT.119...67S Altcode: 2013arXiv1304.5321S 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. Title: Forward-scattering Hanle effect in the solar Ca I 4227 Å line Authors: Frisch, H.; Anusha, L. S.; Bianda, M.; Holzreuter, R.; Nagendra, K. N.; Ramelli, R.; Sampoorna, M.; Smitha, H. N.; Stenflo, J. O. Bibcode: 2012EAS....55...59F Altcode: High sensitivity spectropolarimetric observations of the four Stokes parameters of the solar Ca I 4227 Å line have been performed in October 2010 at IRSOL with the ZIMPOL polarimeter, near the disk center, outside an active region (Bianda et al. 2011). They were analyzed in Anusha et al. 2011 with a combination of detailed radiative transfer modelling of the Hanle effect for the linear polarization and weak field Zeeman approximation for the circular polarization. This approach made possible a unique determination of the magnetic field vector at various positions along the slit of the spectrograph. A summary of the observations and of their analysis is presented here. Title: J-state interference signatures in the second solar spectrum. Modeling the Cr i triplet at 5204-5208 Å Authors: Smitha, H. N.; Nagendra, K. N.; Stenflo, J. O.; Bianda, M.; Sampoorna, M.; Ramelli, R.; Anusha, L. S. Bibcode: 2012A&A...541A..24S Altcode: 2012arXiv1203.4934S 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. Title: Polarized Line Formation in Multi-dimensional Media. V. Effects of Angle-dependent Partial Frequency Redistribution Authors: Anusha, L. S.; Nagendra, K. N. Bibcode: 2012ApJ...746...84A Altcode: 2013arXiv1308.3443A The solution of polarized radiative transfer equation with angle-dependent (AD) partial frequency redistribution (PRD) is a challenging problem. Modeling the observed, linearly polarized strong resonance lines in the solar spectrum often requires the solution of the AD line transfer problems in one-dimensional or multi-dimensional (multi-D) geometries. The purpose of this paper is to develop an understanding of the relative importance of the AD PRD effects and the multi-D transfer effects and particularly their combined influence on the line polarization. This would help in a quantitative analysis of the second solar spectrum (the linearly polarized spectrum of the Sun). We consider both non-magnetic and magnetic media. In this paper we reduce the Stokes vector transfer equation to a simpler form using a Fourier decomposition technique for multi-D media. A fast numerical method is also devised to solve the concerned multi-D transfer problem. The numerical results are presented for a two-dimensional medium with a moderate optical thickness (effectively thin) and are computed for a collisionless frequency redistribution. We show that the AD PRD effects are significant and cannot be ignored in a quantitative fine analysis of the line polarization. These effects are accentuated by the finite dimensionality of the medium (multi-D transfer). The presence of magnetic fields (Hanle effect) modifies the impact of these two effects to a considerable extent. Title: Polarized Line Formation in Multi-dimensional Media. IV. A Fourier Decomposition Technique to Formulate the Transfer Equation with Angle-dependent Partial Frequency Redistribution Authors: Anusha, L. S.; Nagendra, K. N. Bibcode: 2011ApJ...739...40A Altcode: 2013arXiv1308.3447A To explain the linear polarization observed in spatially resolved structures in the solar atmosphere, the solution of polarized radiative transfer (RT) equation in multi-dimensional (multi-D) geometries is essential. For strong resonance lines, partial frequency redistribution (PRD) effects also become important. In a series of papers, we have been investigating the nature of Stokes profiles formed in multi-D media including PRD in line scattering. For numerical simplicity, so far we have restricted our attention to the particular case of PRD functions which are averaged over all the incident and scattered directions. In this paper, we formulate the polarized RT equation in multi-D media that takes into account the Hanle effect with angle-dependent PRD functions. We generalize here to the multi-D case the method for Fourier series expansion of angle-dependent PRD functions originally developed for RT in one-dimensional geometry. We show that the Stokes source vector S = (SI , SQ , SU ) T and the Stokes vector I = (I, Q, U) T can be expanded in terms of infinite sets of components \tilde{S}^(k), \tilde{I}^(k), respectively, k in [0, +∞). We show that the components \tilde{S}^(k) become independent of the azimuthal angle (phiv) of the scattered ray, whereas the components \tilde{I}^(k) remain dependent on phiv due to the nature of RT in multi-D geometry. We also establish that \tilde{S}^(k) and \tilde{I}^(k) satisfy a simple transfer equation, which can be solved by any iterative method such as an approximate Lambda iteration or a Bi-Conjugate Gradient-type projection method provided we truncate the Fourier series to have a finite number of terms. Title: Polarized Line Formation in Multi-dimensional Media. III. Hanle Effect with Partial Frequency Redistribution Authors: Anusha, L. S.; Nagendra, K. N. Bibcode: 2011ApJ...738..116A Altcode: 2011arXiv1105.4052A 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 {T}^K_Q, 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. Title: Analysis of the Forward-scattering Hanle Effect in the Ca I 4227 Å Line Authors: Anusha, L. S.; Nagendra, K. N.; Bianda, M.; Stenflo, J. O.; Holzreuter, R.; Sampoorna, M.; Frisch, H.; Ramelli, R.; Smitha, H. N. Bibcode: 2011ApJ...737...95A Altcode: Coherent scattering of limb-darkened radiation is responsible for the generation of the linearly polarized spectrum of the Sun (the Second Solar Spectrum). This Second Solar Spectrum is usually observed near the limb of the Sun, where the polarization amplitudes are largest. At the center of the solar disk the linear polarization is zero for an axially symmetric atmosphere. Any mechanism that breaks the axial symmetry (like the presence of an oriented magnetic field, or resolved inhomogeneities in the atmosphere) can generate a non-zero linear polarization. In the present paper we study the linear polarization near the disk center in a weakly magnetized region, where the axisymmetry is broken. We present polarimetric (I, Q/I, U/I, and V/I) observations of the Ca I 4227 Å line recorded around μ = cos θ = 0.9 (where θ is the heliocentric angle) and a modeling of these observations. The high sensitivity of the instrument (ZIMPOL-3) makes it possible to measure the weak polarimetric signals with great accuracy. The modeling of these high-quality observations requires the solution of the polarized radiative transfer equation in the presence of a magnetic field. For this we use standard one-dimensional model atmospheres. We show that the linear polarization is mainly produced by the Hanle effect (rather than by the transverse Zeeman effect), while the circular polarization is due to the longitudinal Zeeman effect. A unique determination of the full \bm {B} vector may be achieved when both effects are accounted for. The field strengths required for the simultaneous fitting of Q/I, U/I, and V/I are in the range 10-50 G. The shapes and signs of the Q/I and U/I profiles are highly sensitive to the orientation of the magnetic field. Title: Observations of the forward scattering Hanle effect in the Ca I 4227 Å line Authors: Bianda, M.; Ramelli, R.; Anusha, L. S.; Stenflo, J. O.; Nagendra, K. N.; Holzreuter, R.; Sampoorna, M.; Frisch, H.; Smitha, H. N. Bibcode: 2011A&A...530L..13B Altcode: 2011arXiv1105.2157B 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 10 830 Å and the Ca i 4227 Å 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 Å 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. Title: Linear Polarization of the Solar Ca I 4227 Å Line: Modeling with Radiative Transfer and Last Scattering Approximation Authors: Anusha, L. S.; Stenflo, J. O.; Frisch, H.; Bianda, M.; Holzreuter, R.; Nagendra, K. N.; Sampoorna, M.; Ramelli, R. Bibcode: 2011ASPC..437...57A Altcode: To model the Ca I 4227 Å line polarization, radiative transfer effects with partial frequency redistribution (PRD) must be taken into account. The numerical solution of the relevant polarized radiative transfer (RT) equations is computationally very demanding. The “last scattering approximation” (LSA) is a concept allowing faster methods to be devised. It is based on the remark that a single scattering of the radiation field is sufficient for creating most of the polarization. Its key ingredient is the anisotropy of the radiation field. If the anisotropy is extracted from the observed center to limb variation of the intensity profile, only the wings of the Q/I spectrum can be modeled (Sampoorna et al. 2009). We show here that the core region may be modeled as well if one takes into account the depth variation of the anisotropy which is obtained from an unpolarized multilevel RT (Anusha et al. 2010). After a validation of the LSA approach by comparison with a polarized RT calculation, we apply both approaches to model recent observations of the Ca I 4227 Å line polarization taken on the quiet Sun. Apart from a global scaling factor, both approaches give a very good fit to the Q/I spectrum for all the wavelengths. As the LSA is 8 times faster than the RT approach, we can recommend it as an efficient method to analyze other strong resonance lines in the second solar spectrum. Title: Observations of the Solar Ca I 4227 Å Line Authors: Bianda, M.; Ramelli, R.; Stenflo, J. O.; Anusha, L. S.; Nagendra, K. N.; Sampoorna, M.; Holzreuter, R.; Frisch, H. Bibcode: 2011ASPC..437...67B Altcode: Our aim is to understand some interesting polarization features observed in the solar Ca I 4277 Å line. Here we only discuss the observational aspects. Observations have also been made in other chromospheric lines within a few hours of those in the Ca I 4227 Å line, in the same region near the north solar limb, to illustrate the potential of simultaneous observations in different lines. Title: Polarized Line Formation in Multi-dimensional Media. II. A Fast Method to Solve Problems with Partial Frequency Redistribution Authors: Anusha, L. S.; Nagendra, K. N.; Paletou, F. Bibcode: 2011ApJ...726...96A Altcode: In the previous paper of this series, 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,Ω) for polarimetry. We also presented a polarized approximate lambda iteration 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 techniques 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 two-dimensional (2D) short characteristics method. Using the numerical method developed in Paper I, 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 also. We present several solutions which may serve as benchmarks in future studies in this area. Title: Polarized Line Formation in Multi-dimensional Media. I. Decomposition of Stokes Parameters in Arbitrary Geometries Authors: Anusha, L. S.; Nagendra, K. N. Bibcode: 2011ApJ...726....6A Altcode: 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 {T}^K_Q(i, Ω). 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 {T}^K_Q(i, Ω) (i=0,1,2, K=0,1,2, -K ≤ Q ≤ +K), 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 {T}^K_Q(i, Ω). 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. Title: Generalization of the Last Scattering Approximation for the Second Solar Spectrum Modeling: The Ca I 4227 Å Line as a Case Study Authors: Anusha, L. S.; Nagendra, K. N.; Stenflo, J. O.; Bianda, M.; Sampoorna, M.; Frisch, H.; Holzreuter, R.; Ramelli, R. Bibcode: 2010ApJ...718..988A Altcode: 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 Ca I 4227 Å 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 Ca I 4227 Å 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. Title: Recent Developments in Polarized Line Formation in Magnetic Fields Authors: Nagendra, K. N.; Sampoorna, M.; Anusha, L. S. Bibcode: 2010ASSP...17..139N Altcode: 2010rast.conf..139N; 2010rasp.book..139N The nature of solar surface magnetism has been an open problem in solar physics. In this paper we address three frontline problems of spectropolarimetry of the Sun. We first review the theoretical formulation and numerical solutions of Zeeman absorption and then the Hanle scattering phenomena in 'turbulent magnetic fields'. We show that the mean emergent Stokes profiles cannot be obtained by simply averaging the scattering and absorption opacities, respectively, over a given distribution of the random field (except when the micro-turbulence prevails). A new formulation of the transfer equation is necessary to study the astrophysically interesting meso-turbulence case. Such formulations of the stochastic polarized radiative transfer problems for absorbing and scattering media are developed only in recent years. We review them and show some results computed by our new formulations.Until recent years the solution of the polarized line radiative transfer equation in LTE (Zeeman absorption in strong fields), and its NLTE counterpart (Hanle scattering in weak fields), were treated as two disparate problems. The reason for this artificial division was more due to the theoretical and numerical difficulties encountered in the solution of the combined Hanle-Zeeman radiative transfer equation. A very general form of the transfer equation was formulated only a decade ago, for the case of complete frequency redistribution. A more difficult case of partial frequency redistribution is explored by us recently. We review these developments through a study of the Hanle-Zeeman effect in arbitrary strength magnetic fields. Title: The Hanle Effect as Diagnostic Tool for Turbulent Magnetic Fields Authors: Anusha, L. S.; Sampoorna, M.; Frisch, H.; Nagendra, K. N. Bibcode: 2010ASSP...19..390A Altcode: 2010mcia.conf..390A The Hanle effect is calculated for a random magnetic field characterized by a finite correlation length and a probability density function of the magnetic field vector. It is shown that linear polarization is essentially independent of the magnetic field correlation length, but strongly depends on the distribution of the field strength. Title: Preconditioned Bi-conjugate Gradient Method for Radiative Transfer in Spherical Media Authors: Anusha, L. S.; Nagendra, K. N.; Paletou, F.; Léger, L. Bibcode: 2009ApJ...704..661A Altcode: 2009arXiv0906.2926A 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. Title: The Hanle effect in a random magnetic field. Dependence of the polarization on statistical properties of the magnetic field Authors: Frisch, H.; Anusha, L. S.; Sampoorna, M.; Nagendra, K. N. Bibcode: 2009A&A...501..335F Altcode: 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. Title: Origin of Spatial Variations of Scattering Polarization in the Wings of the Ca I 4227 Å Line Authors: Sampoorna, M.; Stenflo, J. O.; Nagendra, K. N.; Bianda, M.; Ramelli, R.; Anusha, L. S. Bibcode: 2009ApJ...699.1650S Altcode: 2009arXiv0906.1184S 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 Å line reveal 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 nonmagnetic 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." 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 from the line center, and a continuum polarization parameter that represents the observed polarization level far from the line. This model is highly successful in reproducing the observed Stokes Q/I polarization (linear polarization parallel to the nearest solar limb), including the location of the wing polarization maxima and the minima around the Doppler core, but it fails to reproduce the observed spatial variations of the wing polarization in terms of magnetic field effects with frequency redistribution. This null result points in the direction of a nonmagnetic origin in terms of local inhomogeneities (varying collisional depolarization, radiation-field anisotropies, and deviations from a plane-parallel atmospheric stratification). Title: Polarization : Proving ground for methods in radiative transfer. Authors: Nagendra, K. N.; Anusha, L. S.; Sampoorna, M. Bibcode: 2009MmSAI..80..678N Altcode: Polarization of solar lines arises due to illumination of radiating atom by anisotropic (limb darkened/brightened) radiation. Modelling the polarized spectra of the Sun and stars requires solution of the line radiative transfer problem in which the relevant polarizing physical mechanisms are incorporated. The purpose of this paper is to describe in what different ways the polarization state of the radiation `complicates' the numerical methods originally designed for scalar radiative transfer. We present several interesting situations involving the solution of polarized line transfer to prove our point. They are (i) Comparison of the polarized approximate lambda iteration (PALI) methods with new approaches like Bi-conjugate gradient method that is faster, (ii) Polarized Hanle scattering line radiative transfer in random magnetic fields, (iii) Difficulties encountered in incorporating polarized partial frequency redistribution (PRD) matrices in line radiative transfer codes, (iv) Technical difficulties encountered in handling polarized specific intensity vector, some components of which are sign changing, (v) Proving that scattering polarization is indeed a boundary layer phenomenon. We provide credible benchmarks in each of the above studies. We show that any new numerical methods can be tested in the best possible way, when it is extended to include polarization state of the radiation field in line scattering. Title: Projection methods for line radiative transfer in spherical media. Authors: Anusha, L. S.; Nagendra, K. N. Bibcode: 2009MmSAI..80..631A Altcode: An efficient numerical method called the Preconditioned Bi-Conjugate Gradient (Pre-BiCG) method is presented for the solution of radiative transfer equation in spherical geometry. A variant of this method called Stabilized Preconditioned Bi-Conjugate Gradient (Pre-BiCG-STAB) is also presented. These methods are based on projections on the subspaces of the n dimensional Euclidean space mathbb {R}n called Krylov subspaces. The methods are shown to be faster in terms of convergence rate compared to the contemporary iterative methods such as Jacobi, Gauss-Seidel and Successive Over Relaxation (SOR).