Author name code: sampoorna ADS astronomy entries on 2022-09-14 author:"Sampoorna, M." ------------------------------------------------------------------------ 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: 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: Solution of Polarized Radiative Transfer Equation with Cross-redistribution. Authors: Nagendra, K. N.; Sampoorna, M. Bibcode: 2019spw..confE...1N Altcode: No abstract at ADS Title: Effects of Angle-Dependent Partial Frequency Redistribution on Polarized Line Profiles Authors: Sampoorna, M.; Nagendra, K. N.; Frisch, H.; Stenflo, J. O. Bibcode: 2019ASPC..519..109S Altcode: Scattering of the solar limb-darkened radiation field on atoms and molecules produces linearly polarized spectrum of the Sun (Second Solar Spectrum). Partial frequency redistribution (PFR) plays a fundamental role in shaping the wings of linearly polarized profiles of strong resonance lines. Here we present the effects of the angle-dependent (AD) PFR on resonance polarization both in the presence and absence of magnetic fields. We consider scattering on a two-level atom with unpolarized lower level, and a one-dimensional isothermal atmosphere. 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: 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: Comoving Frame Method for Polarized PRD Line Transfer with Velocity Fields Authors: Sampoorna, M.; Nagendra, K. N. Bibcode: 2019ASPC..526..153S Altcode: The solution of the transfer equation in moving atmospheres is a classical problem. While the low-velocity regime can be handled in a simpler manner using the observer's frame method, the regime of high-velocity requires the comoving frame technique to be applied. We show that even in the low-velocity regime (like that prevailing in the solar atmosphere) we require the comoving frame method when linear polarization together with partial frequency redistribution (PRD) in line scattering is considered. This situation arises because of the numerical difficulties that we encounter in the observer's frame method, namely, a heavy demand on the angle, frequency, and depth grids when we consider polarized line formation with PRD and velocity fields. These difficulties can be overcome by applying the comoving frame technique. Here we present the details of this technique and its applications to solar-like conditions. Title: Polarized Scattering Matrix for Magnetic Dipole Transitions Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Sankarasubramanian, K. Bibcode: 2019ASPC..526..207M Altcode: Forbidden emission lines, produced by magnetic dipole (M1) transitions, are difficult to observe in the laboratory, but naturally arise in the highly ionized atoms present in the solar corona. The polarization of these lines is the result of anisotropic excitation processes. The polarization measurement of forbidden emission lines is the most direct method of determining the magnetic field direction in the solar corona. Here we consider the general case of M1 transitions in the presence of magnetic fields of arbitrary strength. In particular, we derive the scattering matrix for the M1 transitions using the classical magnetic dipole oscillator model of Casini & Lin (2002) and applying the scattering matrix approach of Stenflo (1998). The derived scattering matrix covers, in a continuous way, saturated Hanle, intermediate Hanle-Zeeman, and Zeeman regimes. Title: Partial Frequency Redistribution Theory with Paschen-Back Effect: Application to Li I 6708 Å Lines Authors: Sowmya, K.; Nagendra, K. N.; Sampoorna, M.; Stenflo, J. O. Bibcode: 2019ASPC..526...43S Altcode: The diagnostically important Li I D lines in the Second Solar Spectrum are governed by the quantum interference processes that take place among the magnetic substates belonging to different fine (J) and hyperfine (F) structure states. This interference gets modified in the presence of a magnetic field. The signatures of this interference in polarization contain information on the nature of the vector magnetic field in the solar atmosphere. With this motivation, we apply the polarized redistribution matrix including Paschen-Back effect, derived based on the Kramers-Heisenberg scattering matrix approach, to model the polarization profiles of the Li lines observed in the quiet Sun. We make use of the last scattering approximation which is based on the concept that the polarization of the emergent radiation is generated in the last scattering event, before the radiation escapes from the atmosphere. We present a comparison of the quiet Sun observations of the linear polarization profiles of Li I 6708 Å line system with the theoretical profiles computed using our simple modeling approach. We also present theoretical Stokes profiles in the Paschen-Back regime of magnetic fields and compare them with the single scattered profiles. Title: Effects of Lower-Level Polarization and Partial Frequency Redistribution on Stokes Profiles Authors: Supriya, H. D.; Nagendra, K. N.; Sampoorna, M.; Stenflo, J. O.; Ravindra, B. Bibcode: 2019ASPC..526...61S Altcode: The theory of polarized radiative transfer including the effects of partial frequency redistribution (PRD) for a two-level and two-term atom has been formulated in the scattering matrix approach. However there exist several enigmatic features in the Second Solar Spectrum which cannot be explained on the basis of said approach. The reason for this lies in the approximations made in this approach. One such approximation is the assumption that the lower level of the atom involved in the scattering process is unpolarized. There are alternative approaches based on the density matrix formalism to relax this assumption. It has been shown that the inclusion of the polarization of all the atomic levels involved in the scattering process is important. In our recent studies, the collisionless redistribution matrix including the effects of both PRD and lower-level polarization (LLP) was derived starting from the Kramers-Heisenberg scattering formulation. We proposed a simple numerical technique namely, the correction method, to solve the problem of polarized radiative transfer with PRD and LLP. Here we apply this technique to different atomic systems and discuss the effects of PRD and LLP on the emergent Stokes profiles. Title: Coronal magnetic field measurements using forbidden emission lines Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Sankarasubramanian, K. Bibcode: 2018IAUS..340...61M Altcode: The polarization measurement of coronal forbidden emission lines is the most promising method of determining the direction of magnetic fields in the corona. A classical theory for the forbidden lines was presented in Megha et al. (2017) for the case of arbitrary strength magnetic fields. Here we apply that theoretical formalism to study the effect of density distributions, magnetic field configurations, and velocity fields on the Stokes profiles formed in corona. For illustrations we use the atomic parameters of the [Fe xiii] 10747 Å coronal forbidden line. Title: Polarized Line Formation in Arbitrary Strength Magnetic Fields Angle-averaged and Angle-dependent Partial Frequency Redistribution Authors: Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O. Bibcode: 2017ApJ...844...97S Altcode: Magnetic fields in the solar atmosphere leave their fingerprints in the polarized spectrum of the Sun via the Hanle and Zeeman effects. While the Hanle and Zeeman effects dominate, respectively, in the weak and strong field regimes, both these effects jointly operate in the intermediate field strength regime. Therefore, it is necessary to solve the polarized line transfer equation, including the combined influence of Hanle and Zeeman effects. Furthermore, it is required to take into account the effects of partial frequency redistribution (PRD) in scattering when dealing with strong chromospheric lines with broad damping wings. In this paper, we present a numerical method to solve the problem of polarized PRD line formation in magnetic fields of arbitrary strength and orientation. This numerical method is based on the concept of operator perturbation. For our studies, we consider a two-level atom model without hyperfine structure and lower-level polarization. We compare the PRD idealization of angle-averaged Hanle-Zeeman redistribution matrices with the full treatment of angle-dependent PRD, to indicate when the idealized treatment is inadequate and what kind of polarization effects are specific to angle-dependent PRD. Because the angle-dependent treatment is presently computationally prohibitive when applied to realistic model atmospheres, we present the computed emergent Stokes profiles for a range of magnetic fields, with the assumption of an isothermal one-dimensional medium. Title: Hanle-Zeeman Scattering Matrix for Magnetic Dipole Transitions Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Sankarasubramanian, K. Bibcode: 2017ApJ...841..129M Altcode: The polarization of the light that is scattered by the coronal ions is influenced by the anisotropic illumination from the photosphere and the magnetic field structuring in the solar corona. The properties of the coronal magnetic fields can be well studied by understanding the polarization properties of coronal forbidden emission lines that arise from magnetic dipole (M1) transitions in the highly ionized atoms that are present in the corona. We present the classical scattering theory of the forbidden lines for a more general case of arbitrary-strength magnetic fields. We derive the scattering matrix for M1 transitions using the classical magnetic dipole model of Casini & Lin and applying the scattering matrix approach of Stenflo. We consider a two-level atom model and neglect collisional effects. The scattering matrix so derived is used to study the Stokes profiles formed in coronal conditions in those regions where the radiative excitations dominate collisional excitations. To this end, we take into account the integration over a cone of an unpolarized radiation from the solar disk incident on the scattering atoms. Furthermore, we also integrate along the line of sight to calculate the emerging polarized line profiles. We consider radial and dipole magnetic field configurations and spherically symmetric density distributions. For our studies we adopt the atomic parameters corresponding to the [Fe xiii] 10747 Å coronal forbidden line. We also discuss the nature of the scattering matrix for M1 transitions and compare it with that for the electric dipole (E1) transitions. Title: Importance of Cross-redistribution in Scattering Polarization of Spectral Lines: The Cases of 3P-3S Triplets of Mg I and Ca I Authors: Sampoorna, M.; Nagendra, K. N. Bibcode: 2017ApJ...838...95S Altcode: Scattering on a multi-level atomic system has dominant contributions from resonance and Raman scattering. While initial and final levels are the same for resonance scattering, they are different for Raman scattering. The frequency redistribution for resonance scattering is described by the usual partial frequency redistribution functions of Hummer, while that for Raman scattering is described by cross-redistribution (XRD) function. In the present paper, we investigate the importance of XRD on linear polarization profiles of 3P-3S triplets of Mg I and Ca I formed in an isothermal one-dimensional atmosphere. We show that XRD produces significant effects on the linear polarization profiles when the wavelength separations between the line components of the multiplet are small, like in the cases of Mg I b and Ca I triplets. Title: Polarized Line Formation in Non-monotonic Velocity Fields Authors: Sampoorna, M.; Nagendra, K. N. Bibcode: 2016ApJ...833...32S Altcode: For a correct interpretation of the observed spectro-polarimetric data from astrophysical objects such as the Sun, it is necessary to solve the polarized line transfer problems taking into account a realistic temperature structure, the dynamical state of the atmosphere, a realistic scattering mechanism (namely, the partial frequency redistribution—PRD), and the magnetic fields. In a recent paper, we studied the effects of monotonic vertical velocity fields on linearly polarized line profiles formed in isothermal atmospheres with and without magnetic fields. However, in general the velocity fields that prevail in dynamical atmospheres of astrophysical objects are non-monotonic. Stellar atmospheres with shocks, multi-component supernova atmospheres, and various kinds of wave motions in solar and stellar atmospheres are examples of non-monotonic velocity fields. Here we present studies on the effect of non-relativistic non-monotonic vertical velocity fields on the linearly polarized line profiles formed in semi-empirical atmospheres. We consider a two-level atom model and PRD scattering mechanism. We solve the polarized transfer equation in the comoving frame (CMF) of the fluid using a polarized accelerated lambda iteration method that has been appropriately modified for the problem at hand. We present numerical tests to validate the CMF method and also discuss the accuracy and numerical instabilities associated with it. Title: Polarized Line Formation with Lower-level Polarization and Partial Frequency Redistribution Authors: Supriya, H. D.; Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O.; Ravindra, B. Bibcode: 2016ApJ...828...84S Altcode: In the well-established theories of polarized line formation with partial frequency redistribution (PRD) for a two-level and two-term atom, it is generally assumed that the lower level of the scattering transition is unpolarized. However, the existence of unexplained spectral features in some lines of the Second Solar Spectrum points toward a need to relax this assumption. There exists a density matrix theory that accounts for the polarization of all the atomic levels, but it is based on the flat-spectrum approximation (corresponding to complete frequency redistribution). In the present paper we propose a numerical algorithm to solve the problem of polarized line formation in magnetized media, which includes both the effects of PRD and the lower level polarization (LLP) for a two-level atom. First we derive a collisionless redistribution matrix that includes the combined effects of the PRD and the LLP. We then solve the relevant transfer equation using a two-stage approach. For illustration purposes, we consider two case studies in the non-magnetic regime, namely, the J a = 1, J b = 0 and J a = J b = 1, where J a and J b represent the total angular momentum quantum numbers of the lower and upper states, respectively. Our studies show that the effects of LLP are significant only in the line core. This leads us to propose a simplified numerical approach to solve the concerned radiative transfer problem. Title: Polarized Scattering of Light for Arbitrary Magnetic Fields with Level-crossings from the Combination of Hyperfine and Fine Structure Splittings Authors: Sowmya, K.; Nagendra, K. N.; Sampoorna, M.; Stenflo, J. O. Bibcode: 2015ApJ...814..127S Altcode: 2015arXiv151207736S Interference between magnetic substates of the hyperfine structure states belonging to different fine structure states of the same term influences the polarization for some of the diagnostically important lines of the Sun's spectrum, like the sodium and lithium doublets. The polarization signatures of this combined interference contain information on the properties of the solar magnetic fields. Motivated by this, in the present paper, we study the problem of polarized scattering on a two-term atom with hyperfine structure by accounting for the partial redistribution in the photon frequencies arising due to the Doppler motions of the atoms. We consider the scattering atoms to be under the influence of a magnetic field of arbitrary strength and develop a formalism based on the Kramers-Heisenberg approach to calculate the scattering cross section for this process. We explore the rich polarization effects that arise from various level-crossings in the Paschen-Back regime in a single scattering case using the lithium atomic system as a concrete example that is relevant to the Sun. Title: Paschen-Back effect involving atomic fine and hyperfine structure states Authors: Sowmya, K.; Nagendra, K. N.; Sampoorna, M.; Stenflo, J. O. Bibcode: 2015IAUS..305..154S Altcode: The linear polarization in spectral lines produced by coherent scattering is significantly modified by the quantum interference between the atomic states in the presence of a magnetic field. When magnetic fields produce a splitting which is of the order of or greater than the fine or hyperfine structure splittings, we enter the Paschen-Back effect (PBE) regime, in which the magnetic field dependence of the Zeeman splittings and transition amplitudes becomes non-linear. In general, PBE occurs for sufficiently strong fields when the fine structure states are involved and for weak fields in the case of hyperfine structure states. In this work, we apply the recently developed theory of PBE in the atomic fine and hyperfine structure states including the effects of partial frequency redistribution to the case of Li i 6708 Å doublet. We explore the signatures of PBE in a single scattering event and their applicability to the solar magnetic field diagnostics. Title: A revisit to model the Cr i triplet at 5204-5208 Å and the Ba ii D2 line at 4554 Å in the Second Solar Spectrum Authors: Smitha, H. N.; Nagendra, K. N.; Stenflo, J. O.; Bianda, M.; Sampoorna, M.; Ramelli, R. Bibcode: 2015IAUS..305..372S Altcode: In our previous attempt to model the Stokes profiles of the Cr i triplet at 5204-5208 Å and the Ba ii D2 at 4554 Å, we found it necessary to slightly modify the standard FAL model atmospheres to fit the observed polarization profiles. In the case of Cr i triplet, this modification was done to reduce the theoretical continuum polarization, and in the case of Ba ii D2, it was needed to reproduce the central peak in Q/I. In this work, we revisit both these cases using different standard model atmospheres whose temperature structures closely resemble those of the modified FAL models, and explore the possibility of synthesizing the line profiles without the need for small modifications of the model atmosphere. Title: Polarized Line Formation in Moving Atmospheres with Partial Frequency Redistribution and a Weak Magnetic Field Authors: Sampoorna, M.; Nagendra, K. N. Bibcode: 2015ApJ...812...28S Altcode: The dynamical state of the solar and stellar atmospheres depends on the macroscopic velocity fields prevailing within them. The presence of such velocity fields in the line formation regions strongly affects the polarized radiation field emerging from these atmospheres. Thus it becomes necessary to solve the radiative transfer equation for polarized lines in moving atmospheres. Solutions based on the “observer’s frame method” are computationally expensive to obtain, especially when partial frequency redistribution (PRD) in line scattering and large-amplitude velocity fields are taken into account. In this paper we present an efficient alternative method of solution, namely, the comoving frame technique, to solve the polarized PRD line formation problems in the presence of velocity fields. We consider one-dimensional planar isothermal atmospheres with vertical velocity fields. We present a study of the effect of velocity fields on the emergent linear polarization profiles formed in optically thick moving atmospheres. We show that the comoving frame method is far superior when compared to the observer’s frame method in terms of the computational speed and memory requirements. Title: Partial Redistribution Effects on Polarized Lines Formed in Moving Media in the Presence of a Weak Magnetic Field Authors: Sampoorna, M.; Nagendra, K. N. Bibcode: 2015IAUS..305..387S Altcode: Macroscopic velocity fields in stellar atmospheres significantly affect the shapes of the emergent Stokes profiles. The inextricable coupling between the angle and frequency variables becomes more complex in a moving medium when compared to a static medium. In this paper we consider both complete frequency redistribution (CRD) and partial frequency redistribution (PRD) in the line scattering of a two-level atom in the presence of an external weak magnetic field. For simplicity we consider empirical velocity laws to represent motion of the atmospheric layers. We present emergent Stokes profiles computed with CRD, angle-averaged PRD, and angle-dependent PRD. We show that angle-dependent PRD effects are important both in non-magnetic and magnetized scattering when vertical velocity gradients are present in the atmosphere. The results are presented for simple atmospheric models. They are expected to be of relevance to polarized line formation in slowly expanding chromospheric layers. Title: Electron Scattering Redistribution Effect on Atomic Line Polarization Authors: Supriya, H. D.; Nagendra, K. N.; Ravindra, B.; Sampoorna, M. Bibcode: 2014ASPC..489..117S Altcode: The scattering of line photons on free electrons modifies the polarization of the atomic spectral lines. Hence it is important to treat scattering by electrons as a redistribution process and to study in detail its effect on line formation. The numerically difficult problem of evaluation and the use of angle-dependent atomic and electron scattering redistribution functions in the line transfer equation is considered. Two numerical methods, namely approximate lambda iteration and scattering expansion method, are used to solve the relevant polarized transfer problem. A study of the polarized line formation in a standard two-level atom picture including an exact treatment of electron scattering redistribution shows the importance of the latter in the analysis of polarized line profiles emitted by solar and stellar atmospheres. The effect of electron scattering turns out to be extremely important in the interpretation of very far wing line polarization of solar and stellar spectral lines. Title: Polarized Line Formation with Angle-Dependent Partial Frequency Redistribution Authors: Sampoorna, M. Bibcode: 2014ASPC..489..197S Altcode: The linear polarization of spectral lines seen in the solar limb observations is created by the scattering of the anisotropic radiation field by atoms and molecules. The partial frequency redistribution (PRD) effects in line scattering are necessary ingredients for interpreting the linear polarization observed in strong resonance lines. This polarization is sensitive to the form of the PRD function used in the polarized line transfer equation. The use of angle-averaged PRD function is quite common in the literature on polarized transfer, as it greatly reduces the computing efforts. In this paper we present our recent work on the importance of the angle-dependent PRD in polarized line transfer. First we present a brief historical background on polarized transfer with angle-dependent PRD. To simplify the numerical work needed to handle angle-dependent PRD function a Stokes vector decomposition technique was developed by Frisch (2009, 2010). After briefly recalling this technique, we present two numerical methods developed to solve the polarized line transfer with angle-dependent PRD. These are (1) polarized accelerated lambda iteration method and (2) the scattering expansion method. Through illustrative examples, we show that while angle-dependent effects are somewhat less important for scattering polarization in the absence of magnetic fields, they play an important role in the presence of a weak magnetic field. Title: Solar Polarization 7 Authors: Nagendra, K. N.; Stenflo, J. O.; Qu, Z. Q.; Sampoorna, M. Bibcode: 2014ASPC..489.....N Altcode: No abstract at ADS Title: The Role of Quantum Interference and Partial Redistribution in the Solar Ba <font size=2>II D2 4554 Å Line Authors: Smitha, H. N.; Nagendra, K. N.; Stenflo, J. O.; Sampoorna, M. Bibcode: 2014ASPC..489..213S Altcode: 2014arXiv1409.0465S The Ba <font size=2>II D2 line at 4554 Å is a good example, where the F-state interference effects due to the odd isotopes produce polarization profiles, which are very different from those of the even isotopes that do not exhibit F-state interference. It is therefore necessary to account for the contributions from the different isotopes to understand the observed linear polarization profiles of this line. In this paper we present radiative transfer modeling with partial frequency redistribution, which is shown to be essential to model this line. This is because complete frequency redistribution cannot reproduce the observed wing polarization. We present the observed and computed Q/I profiles at different limb distances. The theoretical profiles strongly depend on limb distance (μ) and the model atmosphere which fits the limb observations fails at other μ positions. Title: Intrinsically Polarized Blend Lines Authors: Sowmya, K.; Nagendra, K. N.; Sampoorna, M. Bibcode: 2014ASPC..489..125S Altcode: The Second Solar Spectrum formed by coherent scattering processes in the Sun, is highly structured. It is characterized by numerous blend lines, both intrinsically polarizing and depolarizing, superposed on the background continuum. These blend lines play an important role in the interpretation of the Second Solar Spectrum. Since blend lines affect the shapes of the neighboring spectral lines they have to be treated in a sophisticated manner in order to efficiently model a given spectral line of interest. The depolarizing blend lines - mostly considered to be formed under LTE conditions - depolarize the background continuum and thereby affect the absolute scale of the polarization measurement. An understanding of the influence of the blend lines leads to a proper determination of the zero-point of the polarization scale. With this motivation we extend a previously developed framework to include many blend lines formed under NLTE conditions, in the radiative transfer equation. The results are shown for the particular case of two blend lines situated on either side of the main spectral line. Title: Polarized Light Scattering with the Paschen-Back Effect, Level-crossing of Fine Structure States, and Partial Frequency Redistribution Authors: Sowmya, K.; Nagendra, K. N.; Sampoorna, M.; Stenflo, J. O. Bibcode: 2014ApJ...793...71S Altcode: 2014arXiv1407.5457S The quantum interference between the fine structure states of an atom modifies the shapes of the emergent Stokes profiles in the second solar spectrum. This phenomenon has been studied in great detail both in the presence and absence of magnetic fields. By assuming a flat-spectrum for the incident radiation, the signatures of this effect have been explored for arbitrary field strengths. Even though the theory which takes into account the frequency dependence of the incident radiation is well developed, it is restricted to the regime in which the magnetic splitting is much smaller than the fine structure splitting. In the present paper, we carry out a generalization of our scattering matrix formalism including the effects of partial frequency redistribution for arbitrary magnetic fields. We test the formalism using available benchmarks for special cases. In particular, we apply it to the Li I 6708 Å D1 and D2 line system, for which observable effects from the Paschen-Back regime are expected in the Sun's spectrum. Title: Polarized Scattering with Paschen-Back Effect, Hyperfine Structure, and Partial Frequency Redistribution in Magnetized Stellar Atmospheres Authors: Sowmya, K.; Nagendra, K. N.; Stenflo, J. O.; Sampoorna, M. Bibcode: 2014ApJ...786..150S Altcode: 2015arXiv151207731S F-state interference significantly modifies the polarization produced by scattering processes in the solar atmosphere. Its signature in the emergent Stokes spectrum in the absence of magnetic fields is depolarization in the line core. In the present paper, we derive the partial frequency redistribution (PRD) matrix that includes interference between the upper hyperfine structure states of a two-level atom in the presence of magnetic fields of arbitrary strengths. The theory is applied to the Na I D2 line that is produced by the transition between the lower J = 1/2 and upper J = 3/2 states which split into F states because of the coupling with the nuclear spin Is = 3/2. The properties of the PRD matrix for the single-scattering case is explored, in particular, the effects of the magnetic field in the Paschen-Back regime and their usefulness as a tool for the diagnostics of solar magnetic fields. Title: Line-interlocking Effects on Polarization in Spectral Lines by Rayleigh and Raman Scattering Authors: Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O. Bibcode: 2013ApJ...770...92S Altcode: The polarized spectrum of the Sun and stars is formed from the scattering of anisotropic radiation on atoms. Interpretation of this spectrum requires the solution of polarized line transfer in multilevel atomic systems. While sophisticated quantum theories of polarized line formation in multilevel atomic systems exist, they are limited by the approximation of complete frequency redistribution in scattering. The partial frequency redistribution (PRD) in line scattering is a necessary component in modeling the polarized spectra of strong lines. The polarized PRD line scattering theories developed so far confine themselves to a two-level or a two-term atom model. In this paper, we present a heuristic approach to the problem of polarized line formation in multilevel atoms taking into account the effects of PRD and a weak magnetic field. Starting from the unpolarized PRD multilevel atom approach of Hubeny et al., we incorporate the polarization state of the radiation field. However, the lower level polarization is neglected. Two iterative methods of solving the polarized PRD line transfer in multilevel atoms are also presented. Taking the example of a five-level Ca II atom model, we present illustrative results for an isothermal one-dimensional model atmosphere. Title: Modeling the Quantum Interference Signatures of the Ba II D2 4554 Å Line in the Second Solar Spectrum Authors: Smitha, H. N.; Nagendra, K. N.; Stenflo, J. O.; Sampoorna, M. Bibcode: 2013ApJ...768..163S Altcode: 2013arXiv1303.7304S Quantum interference effects play a vital role in shaping the linear polarization profiles of solar spectral lines. The Ba II D2 line at 4554 Å is a prominent example, where the F-state interference effects due to the odd isotopes produce polarization profiles, which are very different from those of the even isotopes that have no F-state interference. It is therefore necessary to account for the contributions from the different isotopes to understand the observed linear polarization profiles of this line. Here we do radiative transfer modeling with partial frequency redistribution (PRD) of such observations while accounting for the interference effects and isotope composition. The Ba II D2 polarization profile is found to be strongly governed by the PRD mechanism. We show how a full PRD treatment succeeds in reproducing the observations, while complete frequency redistribution alone fails to produce polarization profiles that have any resemblance to the observed ones. However, we also find that the line center polarization is sensitive to the temperature structure of the model atmosphere. To obtain a good fit to the line center peak of the observed Stokes Q/I profile, a small modification of the FALX model atmosphere is needed, by lowering the temperature in the line-forming layers. Because of the pronounced temperature sensitivity of the Ba II D2 line it may not be a suitable tool for Hanle magnetic-field diagnostics of the solar chromosphere, because there is currently no straightforward way to separate the temperature and magnetic-field effects from each other. 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: Quantum interference with angle-dependent partial frequency redistribution: solution of the polarized line transfer in the non-magnetic case Authors: Supriya, H. D.; Smitha, H. N.; Nagendra, K. N.; Ravindra, B.; Sampoorna, M. Bibcode: 2013MNRAS.429..275S Altcode: Angle-dependent partial frequency redistribution (PRD) matrices represent the physical redistribution in the process of light scattering on atoms. For the purpose of numerical simplicity, it is a common practice in astrophysical literature to use the angle-averaged versions of these matrices, in the line transfer computations. The aim of this paper is to study the combined effects of angle-dependent PRD and the quantum interference phenomena arising either between the fine structure (J) states of a two-term atom or between the hyperfine structure (F) states of a two-level atom. We restrict our attention to the case of non-magnetic and collisionless line scattering on atoms. A rapid method of solution based on Neumann series expansion is developed to solve the angle-dependent PRD problem including quantum interference in an atomic system. We discuss the differences that occur in the Stokes profiles when angle-dependent PRD mechanism is taken into account. Title: Polarized line formation with J-state interference in the presence of magnetic fields: A Heuristic treatment of collisional frequency redistribution Authors: Smitha, H. N.; Nagendra, K. N.; Sampoorna, M.; Stenflo, J. O. Bibcode: 2013JQSRT.115...46S Altcode: 2012arXiv1209.0243S An expression for the partial frequency redistribution (PRD) matrix for line scattering in a two-term atom, which includes the J-state interference between its fine structure line components is derived. The influence of collisions (both elastic and inelastic) and an external magnetic field on the scattering process is taken into account. The lower term is assumed to be unpolarized and infinitely sharp. The linear Zeeman regime in which the Zeeman splitting is much smaller than the fine structure splitting is considered. The inelastic collision rates between the different levels are included in our treatment. We account for the depolarization caused by the collisions coupling the fine structure states of the upper term, but neglect the polarization transfer between the fine structure states. When the fine structure splitting goes to zero, we recover the redistribution matrix that represents the scattering on a two-level atom (which exhibits only m-state interference—namely the Hanle effect). The way in which the multipolar index of the scattering atom enters into the expression for the redistribution matrix through the collisional branching ratios is discussed. The properties of the redistribution matrix are explored for a single scattering process for a L=0→1→0 scattering transition with S=1/2 (a hypothetical doublet centered at 5000 Å and 5001 Å). Further, a method for solving the Hanle radiative transfer equation for a two-term atom in the presence of collisions, PRD, and J-state interference is developed. The Stokes profiles emerging from an isothermal constant property medium are computed. Title: Polarized Line Transfer with F-state Interference in a Non-magnetic Medium: Partial Frequency Redistribution Effects in the Collisionless Regime Authors: Smitha, H. N.; Sowmya, K.; Nagendra, K. N.; Sampoorna, M.; Stenflo, J. O. Bibcode: 2012ApJ...758..112S Altcode: 2012arXiv1208.6369S Quantum interference phenomena manifest themselves in several ways in the polarized solar spectrum formed due to coherent scattering processes. One such effect arises due to interference between the fine structure (J) states giving rise to multiplets. Another effect is that which arises due to interference between the hyperfine structure (F) states. We extend the redistribution matrix derived for the J-state interference to the case of F-state interference. We then incorporate it into the polarized radiative transfer equation and solve it for isothermal constant property slab atmospheres. The relevant transfer equation is solved using a polarized approximate lambda iteration (PALI) technique based on operator perturbation. An alternative method derived from the Neumann series expansion is also proposed and is found to be relatively more efficient than the PALI method. The effects of partial frequency redistribution and the F-state interference on the shapes of the linearly polarized Stokes profiles are discussed. The emergent Stokes profiles are computed for hypothetical line transitions arising due to hyperfine structure splitting of the upper J = 3/2 and lower J = 1/2 levels of a two-level atom model with nuclear spin Is = 3/2. We confine our attention to the non-magnetic scattering in the collisionless regime. Title: Polarized Partial Frequency Redistribution in Subordinate Lines. II. Solution of the Transfer Equation with Rayleigh Scattering Authors: Nagendra, K. N.; Sampoorna, M. Bibcode: 2012ApJ...757...33N Altcode: It is quite common in line formation theory to treat scattering in subordinate lines under the assumption of complete frequency redistribution (CRD). The partial frequency redistribution (PRD) in subordinate lines cannot always be approximated by CRD, especially when the polarization state of the line radiation is taken into account. Here we investigate the PRD effects in subordinate lines including scattering polarization. The line formation is described by a polarized non-LTE line transfer equation based on a two-level atom model. We use the recently derived subordinate line redistribution matrix. We devise polarized approximate lambda iteration methods to solve the concerned transfer problem. The linear polarization profiles of subordinate lines formed in non-magnetic (Rayleigh) scattering atmospheres are discussed. We consider one-dimensional isothermal planar model atmospheres. We show that in the polarized line transfer calculations of subordinate lines, PRD plays as important of a role as it does in the case of resonance lines. We also study the effect of collisions on linear polarization profiles of subordinate lines. Title: The effect of electron scattering redistribution on atomic line polarization Authors: Supriya, H. D.; Nagendra, K. N.; Sampoorna, M.; Ravindra, B. Bibcode: 2012MNRAS.425..527S Altcode: 2012MNRAS.tmp.3423S The polarization of spectral lines is generated by the scattering of angularly anisotropic incident radiation field on the atoms in the stellar atmosphere. This atomic scattering polarization is modified by frequency non-coherent scattering of line photons on free electrons. With modern spectropolarimeters of high sensitivity, it is possible to detect such changes in the spectral line polarization caused by scattering on electrons. We present new and efficient numerical techniques to solve the problem of line radiative transfer with atomic and electron scattering frequency redistribution in planar media. The evaluation and use of angle-dependent partial frequency redistribution functions (both atomic and electron scattering type) in the transfer equation require a lot of computing effort. In this paper, we apply a decomposition technique to handle this numerically difficult problem. This recently developed technique is applied for the first time to the electron scattering partial redistribution. This decomposition technique allows us to devise fast iterative methods of solving the polarized line transfer equation. An approximate lambda iteration (ALI) method and a method based on Neumann series expansion of the polarized source vector are proposed. We show that these numerical methods can be used to obtain a solution of the problem, when both atomic and electron scattering partial frequency redistribution are considered together. This is in contrast with the classical numerical methods which require a great amount of computing time. We show the importance of electron scattering redistribution in the far wing line polarization, which has practical implications in the analysis of polarized stellar or solar spectra, where non-coherent electron scattering controls the line wing transfer. Title: Blend lines in the polarized spectrum of the Sun Authors: Sowmya, K.; Nagendra, K. N.; Sampoorna, M. Bibcode: 2012MNRAS.423.2949S Altcode: 2012MNRAS.tmp.3077S; 2015arXiv151207728S Blend lines form an integral part of the theoretical analysis and modelling of the polarized spectrum of the Sun. Their interaction with other spectral lines needs to be explored and understood before we can properly use the main spectral lines to diagnose the Sun. They are known to cause a decrease in the polarization in the wings of the main line on which they superpose, or in the polarization of the continuum, when they are assumed to be formed either under the local thermodynamic equilibrium (LTE) conditions or when their intrinsic polarizability factor is zero. In this paper, we describe the theoretical framework to include the blend lines formed under non-LTE conditions, in the radiative transfer equation, and the numerical techniques to solve it. The properties of a blend line having an intrinsic polarization of its own and its interaction with the main line are discussed. The results of our analysis show that the influence of the blend lines on the main spectral lines, though small in the present context, is important and needs to be considered when interpreting the polarized spectral lines in the second solar spectrum. 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 Partial Frequency Redistribution in Subordinate Lines. I. Resonance Scattering with Collisions Authors: Sampoorna, M. Bibcode: 2012ApJ...745..189S Altcode: Using a previously established theory, we derive a suitable form of the laboratory frame redistribution matrix for the resonance scattering in subordinate lines, allowing for the radiative as well as collisional broadening of both atomic levels involved. The lower level, though broadened, is assumed to be unpolarized. The elastic collisions both in the upper and lower levels are taken into account. We show that, in situations, when elastic collisions in the lower level can be neglected, the redistribution matrix for subordinate lines takes a form that is analogous to the corresponding case of resonance lines. Further, in the case of no-lower-level interactions (i.e., infinitely sharp lower level), we recover the redistribution matrix for resonance lines. We express the redistribution matrix for subordinate lines in terms of the irreducible spherical tensors for polarimetry. For practical applications in one-dimensional polarized radiative transfer problem, we derive the azimuth averaged subordinate line redistribution matrix. Title: Radiative transfer with J-state interference in a two-term atom. Partial frequency redistribution in the non-magnetic case Authors: Smitha, H. N.; Nagendra, K. N.; Sampoorna, M.; Stenflo, J. O. Bibcode: 2011A&A...535A..35S Altcode: Context. Quantum interference phenomena play a fundamental role in the formation of linear polarization that arises from scattering processes in multiplets of the solar spectrum. In particular, the J-state interference between different line components of a multiplet (arising from transitions in a two-term atom) produces significant effects in the linearly polarized spectra.
Aims: We aim to solve the polarized radiative transfer equation for a two-term atom with the unpolarized lower term in isothermal slabs, including the effect of the interference between the upper J-states and partial frequency redistribution (PRD). We consider only the case of non-magnetic scattering.
Methods: The PRD matrix for the J-state interference derived in previous works is incorporated into the polarized transfer equation. The standard form of the two-level atom transfer equation is extended to a two-term atom. The transfer problem is then solved using a traditional polarized approximate lambda iteration method.
Results: We show how the PRD and the J-state interference together affect the shapes of the (I,Q/I) profiles. We present the benchmark solutions for isothermal, constant-property slabs of a given optical thickness. We consider a hypothetical doublet produced by an L = 0 → 1 → 0 scattering transition with spin S = 1/2. We present the results in the form of Stokes (I,Q/I) profiles for different values of (i) the line separation, (ii) optical thickness, (iii) thermalization parameter, and (iv) the continuum opacity. Title: Spectral line polarization with angle-dependent partial frequency redistribution. IV. Scattering expansion method for the Hanle effect Authors: Nagendra, K. N.; Sampoorna, M. Bibcode: 2011A&A...535A..88N Altcode: Context. The partial frequency redistribution (PRD) effects in line scattering are necessary ingredients for interpreting the linear polarization observed in strong resonance lines. It is a common practice to use angle-averaged PRD functions for simplicity (obtained by averaging over all scattering angles). It has been established that the use of angle-dependent PRD functions instead of angle-averaged functions is essential for weak fields.
Aims: Here we present an efficient iterative method to solve the polarized line radiative transfer equation in weak magnetic fields using angle-dependent PRD functions.
Methods: Based on the theory of Stokes vector decomposition for the Hanle effect combined with the Fourier azimuthal expansion of the angle-dependent PRD function, we try to formulate an efficient numerical method of solving the concerned transfer problem in one-dimensional media. This iterative method (referred to as the scattering expansion method, SEM) is based on a series expansion of the polarized source vector in mean number of scatterings (Neumann series expansion). We apply the SEM approach to handle both the exact and various approximate forms of the Hanle scattering redistribution matrix.
Results: The SEM is shown to be an efficient method to solve angle-dependent PRD problems involving the Hanle effect. We show that compared to the earlier methods such as the perturbation methods, the SEM is stable and faster. We find that angle-dependent PRD significantly affects the Stokes U parameter. 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: Spectral line polarization with angle-dependent partial frequency redistribution. III. Single scattering approximation for the Hanle effect Authors: Sampoorna, M. Bibcode: 2011A&A...532A..52S Altcode: Context. The solar limb observations in spectral lines display evidence of linear polarization, caused by non-magnetic resonance scattering process. This polarization is modified by weak magnetic fields - the process of the Hanle effect. These two processes serve as diagnostic tools for weak solar magnetic field determination. In modeling the polarimetric observations the partial frequency redistribution (PRD) effects in line scattering have to be accounted for. For simplicity, it is common practice to use PRD functions averaged over all scattering angles. For weak fields, it has been established that the use of angle-dependent PRD functions instead of angle-averaged functions is essential.
Aims: We introduce a single scattering approximation to the problem of polarized line radiative transfer in weak magnetic fields with an angle-dependent PRD. This helps us to rapidly compute an approximate solution to the difficult and numerically expensive problem of polarized line formation with angle-dependent PRD.
Methods: We start from the recently developed Stokes vector decomposition technique combined with the Fourier azimuthal expansion for angle-dependent PRD with the Hanle effect. In this decomposition technique, the polarized radiation field (I, Q, U) is decomposed into an infinite set of cylindrically symmetric Fourier coefficients tilde I(k)K_Q, where K = 0,2, with - K ≤ Q ≤ + K, and k is the order of the Fourier coefficients (k takes values from - ∞ to + ∞). In the single scattering approximation, the effect of the magnetic field on the Stokes I is neglected, so that it can be computed using the standard non-local thermodynamic equilibrium (non-LTE) scalar line transfer equation. In the case of angle-dependent PRD, we further assume that the Stokes I is cylindrically symmetric and given by its dominant term tilde I(0)0_0. Keeping only the contribution from tilde I(0)0_0 in the source terms for the K = 2 components (which give rise to Stokes Q and U), the value of k is limited to 0, ± 1, ± 2. As a result, the dimensionality of the problem is reduced from infinity to 25 for the K = 2 Fourier coefficients.
Results: We show that the single scattered solution provides a reasonable approximation to the emergent polarization computed using the polarized line transfer equation including angle-dependent PRD and the Hanle effect. While the full problem is computationally expensive, the single scattering approximation provides a faster method of solution. The presence of elastic collisions particularly enhances the domain of applicability of this approximation.

Appendices A and B are available in electronic form at http://www.aanda.org 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: Polarized Line Formation with J-state Interference in the Presence of Magnetic Fields. I. Partial Frequency Redistribution in the Collisionless Regime Authors: Smitha, H. N.; Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O. Bibcode: 2011ApJ...733....4S Altcode: 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 Hummer's 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. 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: m-state Interference with Partial Frequency Redistribution for Polarized Line Formation in Arbitrary Magnetic Fields Authors: Sampoorna, M. Bibcode: 2011ApJ...731..114S Altcode: The present paper concerns the derivation of polarized partial frequency redistribution (PRD) matrices for scattering on a two-level atom in arbitrary magnetic fields. We generalize the classical theory of PRD that is applicable to a J = 0 → 1 → 0 scattering transition, to other types of atomic transitions with arbitrary quantum numbers. We take into account quantum interference between magnetic substates of a given upper J-state. The generalization proceeds in a phenomenological way, based on the direct analogy between the Kramers-Heisenberg scattering amplitude in quantum mechanics and the Jones scattering matrix in classical physics. The redistribution matrices derived from such a generalization of classical PRD theory are identical to those obtained from a summed perturbative quantum electrodynamic treatment of the atom-radiation interaction. Our semi-classical approach has the advantage that it is non-perturbative, more intuitive, and lends itself more easily to further generalization (like the inclusion of J-state interference in the PRD theory). Title: Spectral line polarization with angle-dependent partial frequency redistribution. II. Accelerated lambda iteration and scattering expansion methods for the Rayleigh scattering Authors: Sampoorna, M.; Nagendra, K. N.; Frisch, H. Bibcode: 2011A&A...527A..89S Altcode: Context. The linear polarization of strong resonance lines observed in the solar spectrum is created by the scattering of the photospheric radiation field. This polarization is sensitive to the form of the partial frequency redistribution (PRD) function used in the line radiative transfer equation. Observations have been analyzed until now with angle-averaged PRD functions. With an increase in the polarimetric sensitivity and resolving power of the present-day telescopes, it will become possible to detect finer effects caused by the angle dependence of the PRD functions.
Aims: We devise new efficient numerical methods to solve the polarized line transfer equation with angle-dependent PRD, in plane-parallel cylindrically symmetrical media. We try to bring out the essential differences between the polarized spectra formed under angle-averaged and the more realistic case of angle-dependent PRD functions.
Methods: We use a recently developed Stokes vector decomposition technique to formulate three different iterative methods tailored for angle-dependent PRD functions. Two of them are of the accelerated lambda iteration type, one is based on the core-wing approach, and the other one on the frequency by frequency approach suitably generalized to handle angle-dependent PRD. The third one is based on a series expansion in the mean number of scattering events (Neumann series expansion).
Results: We show that all these methods work well on this difficult problem of polarized line formation with angle-dependent PRD. We present several benchmark solutions with isothermal atmospheres to show the performance of the three numerical methods and to analyze the role of the angle-dependent PRD effects. For weak lines, we find no significant effects when the angle-dependence of the PRD functions is taken into account. For strong lines, we find a significant decrease in the polarization, the largest effect occurring in the near wing maxima. Title: On the Sensitivity of Partial Redistribution Scattering Polarization Profiles to Various Atmospheric Parameters Authors: Sampoorna, M.; Trujillo Bueno, J.; Landi Degl'Innocenti, E. Bibcode: 2010ApJ...722.1269S Altcode: 2010arXiv1008.2585S This paper presents a detailed study of the scattering polarization profiles formed under partial frequency redistribution (PRD) in two thermal models of the solar atmosphere. Particular attention is given to understanding the influence of several atmospheric parameters on the emergent fractional linear polarization profiles. The shapes of these Q/I profiles are interpreted in terms of the anisotropy of the radiation field, which in turn depends on the source function gradient that sets the angular variation of the specific intensity. We define a suitable frequency integrated anisotropy factor for PRD that can be directly related to the emergent linear polarization. We show that complete frequency redistribution is a good approximation to model weak resonance lines. We also show that the emergent linear polarization profiles can be very sensitive to the thermal structure of the solar atmosphere and, in particular, to spatial variations of the damping parameter. 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: Gauss-Seidel and Successive Overrelaxation Methods for Radiative Transfer with Partial Frequency Redistribution Authors: Sampoorna, M.; Trujillo Bueno, J. Bibcode: 2010ApJ...712.1331S Altcode: 2010arXiv1002.4179S The linearly polarized solar limb spectrum that is produced by scattering processes contains a wealth of information on the physical conditions and magnetic fields of the solar outer atmosphere, but the modeling of many of its strongest spectral lines requires solving an involved non-local thermodynamic equilibrium radiative transfer problem accounting for partial redistribution (PRD) effects. Fast radiative transfer methods for the numerical solution of PRD problems are also needed for a proper treatment of hydrogen lines when aiming at realistic time-dependent magnetohydrodynamic simulations of the solar chromosphere. Here we show how the two-level atom PRD problem with and without polarization can be solved accurately and efficiently via the application of highly convergent iterative schemes based on the Gauss-Seidel and successive overrelaxation (SOR) radiative transfer methods that had been previously developed for the complete redistribution case. Of particular interest is the Symmetric SOR method, which allows us to reach the fully converged solution with an order of magnitude of improvement in the total computational time with respect to the Jacobi-based local accelerated lambda iteration method. Title: Probability Density Functions to Represent Magnetic Fields at the Solar Surface Authors: Sampoorna, M. Bibcode: 2010ASSP...19..141S Altcode: 2009arXiv0903.1911S; 2010mcia.conf..141S Numerical simulations of magneto-convection and analysis of solar magnetogram data provide empirical probability density functions (PDFs) for the line-of-sight component of the magnetic field. In this paper, we theoretically explore effects of several types of PDFs on polarized Zeeman line formation. We also propose composite PDFs to account for randomness in both field strength and orientation. Such PDFs can possibly mimic random fields at the solar surface. 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: 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: Theory of Polarized Scattering in the Mixed Hanle-Zeeman Regime Authors: Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O. Bibcode: 2009ASPC..405...93S Altcode: We present the theory of polarized partial frequency redistribution (PRD) in the presence of arbitrary magnetic fields. Starting from the classical theory of \citet{sampoorna_bom99} we derive the laboratory frame PRD matrices, which we call `Hanle-Zeeman redistribution matrices', since they cover the partially overlapping weak and strong field regimes. We show that for the simplest case of a triplet (J=0 → 1 → 0) transition, the classical and quantum \citep{sampoorna_bom97} treatments give identical results. Title: Phase Matrices for Higher Multipoles of Scattering in External Magnetic Fields Authors: Oo, Y. Y.; San, P. P.; Sampoorna, M.; Nagendra, K. N.; Ramachandran, G. Bibcode: 2009ASPC..405..119O Altcode: Scattering phase matrices for forbidden lines are derived, in the presence of an external magnetic field, using the quantum electrodynamical approach. The particular case of 0 → 2 → 0 M2 transitions are considered and Stokes profiles are shown in the strong field (Zeeman) and weak field (Hanle) limits, covering also the regime of intermediate field strengths. Title: Numerical Methods in Polarized Line Formation Theory Authors: Nagendra, K. N.; Sampoorna, M. Bibcode: 2009ASPC..405..261N Altcode: We review some numerical methods and provide benchmark solutions for the polarized line formation theory with partial redistribution (PRD) in the presence of magnetic fields. The transfer equation remains non-axisymmetric when written in the `Stokes vector basis'. It is relatively easier to develop numerical methods to solve the transfer equation for axisymmetric radiation fields. Therefore for non-axisymmetric problems it would be necessary to expand the azimuthal dependence of the scattering redistribution matrices in a Fourier series. The transfer equation in this so called `reduced form' becomes axisymmetric in the Fourier domain in which it is solved, and the reduced intensity is then transformed into the Stokes vector basis in real space. The advantage is that the reduced problem lends itself to be solved by appropriately organized PALI (Polarized Approximate Lambda Iteration) methods. We first dwell upon a frequency by frequency method (PALI7) that uses non-domain based PRD for the Hanle scattering problem, and then compare it with a core-wing method (PALI6) that uses a domain based PRD. The PALI methods use operator perturbation and involve construction of a suitable procedure to evaluate an `iterated source vector correction'. Another important component of PALI methods is the `Formal Solver' (for example Feautrier, short characteristic, DELOPAR etc.). The PALI methods are extremely fast on a computer and require very small memory. Finally, we present a simple perturbation method to solve the Hanle-Zeeman line formation problem in arbitrary strength magnetic fields. Title: Modeling the second solar spectrum . An empirical approach Authors: Sampoorna, M. Bibcode: 2009MmSAI..80..696S Altcode: We present an empirical approach to model the wing polarization of strong resonance lines. This procedure based on `last scattering approximation' (LSA) was developed by \citet{ste80,ste82}, for coherent scattering in the laboratory frame. We generalize his empirical approach to handle partial frequency redistribution (PRD). We illustrate this approach by applying it to the Ca I 4227 Å line. The LSA approach is successful in reproducing the observed Stokes Q/I polarization, including the location of the wing polarization maxima and the minima around the Doppler core, but fails to reproduce the observed spatial variations of the far wing polarization in terms of magnetic field and PRD effects. This null result points in the direction of a non-magnetic origin, which may include local deviations from a plane-parallel stratification with an inhomogeneous solar atmosphere. 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: Zeeman line formation in solar magnetic fields. Studies with empirical probability distribution functions Authors: Sampoorna, M.; Nagendra, K. N.; Frisch, H.; Stenflo, J. O. Bibcode: 2008A&A...485..275S Altcode: Context: Numerical simulations of magneto-convection and analysis of solar magnetograms provide probability distribution functions (PDFs) for the magnetic field strength.
Aims: In the paper, we explore the effects of these PDFs on Zeeman line formation.
Methods: We calculate the mean Stokes parameters for a Milne-Eddington atmosphere in the limit of optically thin (micro-turbulent) and thick (macro-turbulent) magnetic structures and also the dispersion around the mean profiles in the optically thick limit. Several types of PDFs are considered: (a) Voigt function and stretched exponential type PDFs for fields with fixed direction but fluctuating strength; (b) a cylindrically symmetrical power law for the angular distribution of magnetic fields with given field strength; (c) composite PDFs accounting for randomness in both strength and direction obtained by combining a Voigt function or a stretched exponential with an angular power law. For optically thin structures, explicit expressions are given for the mean values of the Zeeman absorption matrix elements. We also describe how the averaging technique for a normal Zeeman triplet may be generalized to the more common case of anomalous Zeeman splitting patterns.
Results: We show that, for magnetic field rms fluctuations of the order of 6 G, consistent with observational data, Stokes I is essentially independent of the shapes of the PDFs but Stokes Q, U, and V and also the dispersion around the mean values are quite sensitive to the tail behavior of the PDF. We confirm a previous result that Stokes V is less sensitive to the scale of the magnetic structures than Stokes Q and U. The composite PDF proposed for the fluctuations of the magnetic field vector has an angular distribution peaked about the vertical direction for strong fields, and is isotropically distributed for weak fields; it can be used to mimic solar surface random fields. Title: Some aspects of polarized line formation in magneto-turbulent media Authors: Sampoorna, M.; Frisch, H.; Nagendra, K. N. Bibcode: 2008NewA...13..233S Altcode: Observations and numerical simulations of magneto-convection show a highly variable solar magnetic field. Using a statistical approach, we analyze the effects of random magnetic fields on Stokes profiles of spectral lines. We consider the micro and macro-turbulent regimes, which provide bounds for more general random fields with finite scales of variations. The mean Stokes parameters are obtained in the micro-turbulent regime, by first averaging the Zeeman propagation matrix Φ^ over the probability distribution function P( B) of the magnetic field and then solving the concerned radiative transfer equation. In the macro-turbulent regime, the mean solution is obtained by averaging the emergent solution over P( B). It is assumed that B has a Gaussian distribution defined by its mean field B0, angular distribution and dispersion. Fluctuations parallel and perpendicular to B0 are considered. Spectral lines are parameterized by their strength β, which is varied over the range 1-10 4. A detailed comparison of micro and macro-turbulent limit with mean field solution shows that differences are important for β ⩾ 10. When β increases, the saturation behavior of micro-turbulent profiles are significantly different from that of mean field profiles. The Stokes profiles shapes are explained in terms of the non-linear β-dependence of the Unno-Rachkovsky solution using approximate expressions for the mean absorption coefficients. These expressions when inserted in the Unno-Rachkovsky solution can predict Stokes profiles that match with the numerical result to a good approximation. Title: Hanle-Zeeman Redistribution Matrix. III. Solution of the Polarized Line Formation Problem Authors: Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O. Bibcode: 2008ApJ...679..889S Altcode: The polarized radiative transfer equation is solved numerically, taking into account both the Zeeman absorption matrix and the Hanle-Zeeman redistribution matrix, to obtain line profiles for arbitrary magnetic field strengths, partial frequency redistribution, and scattering-dominated line transitions. The limiting cases of weak-field Hanle scattering and strong-field Zeeman true absorption are retrieved. The intermediate regime, where both Zeeman absorption and scattering effects are important, is studied in some detail. The numerical method is applied to various test cases to illustrate aspects of partial frequency redistribution on line scattering in magnetic fields of arbitrary strength and direction. Title: Polarization phase matrices for radiation scattering on atoms in external magnetic fields: The case of forbidden transitions in astrophysics Authors: Oo, Yee Yee; Phyu San, Phyu; Sampoorna, M.; Nagendra, K. N.; Ramachandran, G. Bibcode: 2008arXiv0805.3860O Altcode: Using a quantum electrodynamical approach, we derive the scattering phase matrices for polarized radiation involving forbidden line transitions and in the presence of an external magnetic field. The case of (J=0->2->0) scattering is considered as an example. The non-magnetic Rayleigh scattering phase matrix is also presented. The Stokes profiles in a single scattering event are computed for the strong field (Zeeman) and weak field (Hanle) limits, covering also the regime of intermediate field strengths (Hanle- Zeeman). Title: Hanle-Zeeman Redistribution Matrix. II. Comparison of Classical and Quantum Electrodynamic Approaches Authors: Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O. Bibcode: 2007ApJ...670.1485S Altcode: The Hanle-Zeeman redistribution matrix accounts for the intricately coupled correlations in frequency, angle, and polarization between the incoming and outgoing radiation and embodies the physics of the scattering process. We show explicitly for a J=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, and explicit 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. Title: Scattering polarization in the presence of magnetic and electric fields Authors: Oo, Yee Yee; Sampoorna, M.; Nagendra, K. N.; Ananthamurthy, Sharath; Ramachandran, G. Bibcode: 2007JQSRT.108..161O Altcode: 2007astro.ph..2312O The polarization of radiation by scattering on an atom embedded in combined external quadrupole electric and uniform magnetic fields is studied theoretically. Limiting cases of scattering under Zeeman effect, and Hanle effect in weak magnetic fields are discussed. The theory is general enough to handle scattering in intermediate magnetic fields (Hanle Zeeman effect) and for arbitrary orientation of magnetic field. The quadrupolar electric field produces asymmetric line shifts, and causes interesting level-crossing phenomena either in the absence of an ambient magnetic field, or in its presence. It is shown that the quadrupolar electric field produces an additional depolarization in the Q/I profiles and rotation of the plane of polarization in the U/I profile over and above that arising from magnetic field itself. This characteristic may have a diagnostic potential to detect steady-state and time-varying electric fields that surround radiating atoms in solar atmospheric layers. Title: Hanle-Zeeman Redistribution Matrix. I. Classical Theory Expressions in the Laboratory Frame Authors: Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O. Bibcode: 2007ApJ...663..625S Altcode: 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. Title: Turbulent magnetic field averages for the Zeeman effect . Authors: Frisch, H.; Sampoorna, M.; Nagendra, K. N. Bibcode: 2007MmSAI..78..142F Altcode: Stokes parameters measured in the Solar atmosphere are in general time or space averages over a magnetic field probability distribution function. Here we show how to write the Zeeman propagation matrix in a reference frame defined with respect to the direction of a mean magnetic field and how to average over a random magnetic field distribution. We concentrate on the case of a normal Zeeman triplet but indicate how to treat general Zeeman patterns. Numerical results are presented for Gaussian distributions having cylindrical symmetry about a mean field. Different models of probability distribution functions (PDF), are compared. Title: Polarized Spectral Line Formation in Turbulent Magnetic Fields: The Zeeman and Hanle Effects Authors: Frisch, H.; Sampoorna, M.; Nagendra, K. N. Bibcode: 2006ASPC..358..126F Altcode: We present a short summary of work carried out on the effects of random magnetic fields with finite correlation length on spectral line polarization. The magnetic field is modeled by a step-wise Markovian random process defined by a probability distribution and a correlation length. Micro- and macro-turbulent limits are recovered when this length goes to zero and infinity, respectively. For the Zeeman effect, explicit expressions have been obtained for the mean emergent Stokes parameters and for their r.m.s. fluctuations. Examples illustrate the dependence of the mean Zeeman propagation matrix on the magnetic field distribution, and the dependence of mean Stokes parameters and their r.m.s. fluctuations on the correlation length of the magnetic field. For the Hanle effect, explicit expressions have also been obtained for the mean Stokes parameters. We outline the approach and give an explicit expression for the mean value of Stokes Q. Title: Stochastic polarized line formation. II. Zeeman line transfer in a random magnetic field Authors: Frisch, H.; Sampoorna, M.; Nagendra, K. N. Bibcode: 2006A&A...453.1095F Altcode: Context: .The Zeeman effect produced by a turbulent magnetic field or a random distribution of flux tubes is usually treated in the microturbulent or macroturbulent limits where the Zeeman propagation matrix or the Stokes parameters, respectively, are averaged over the probability distribution function of the magnetic field when computing polarized line profiles.
Aims: .To overcome these simplifying assumptions we consider the Zeeman effect from a random magnetic field which has a finite correlation length that can be varied from zero to infinity and thus made comparable to the photon mean free-path.
Methods: .The vector magnetic field is modeled by a Kubo-Anderson process, a piecewise constant Markov process characterized by a correlation length and a probability distribution function for the random values of the magnetic field. The micro and macro turbulent limits are recovered when the correlation goes to zero or infinity.
Results: .An integral equation is constructed for the mean propagation operator and explicit expressions are obtained for the mean values and second-order moments of the Stokes parameters at the surface of a Milne-Eddington type atmosphere. The expression given by Landi Degl'Innocenti (1994) for the mean Stokes parameters is recovered. Mean values and rms fluctuations around the mean values are calculated numerically for a random magnetic field with isotropic Gaussian fluctuations. The effects of a finite correlation length are discussed in detail. Various extensions of the Milne-Eddington and magnetic field model are considered and the corresponding integral equations for the mean propagation operator are given.
Conclusions: .The rms fluctuations of the Stokes parameters are shown to be very sensitive to the correlation length of the magnetic field. It is suggested to use them as a diagnostic tool to determine the scale of unresolved features in the solar atmosphere. Title: Stochastic polarized line formation. I. Zeeman propagation matrix in a random magnetic field Authors: Frisch, H.; Sampoorna, M.; Nagendra, K. N. Bibcode: 2005A&A...442...11F Altcode: This paper considers the effect of a random magnetic field on Zeeman line transfer, assuming that the scales of fluctuations of the random field are much smaller than photon mean free paths associated to the line formation (micro-turbulent limit). The mean absorption and anomalous dispersion coefficients are calculated for random fields with a given mean value, isotropic or anisotropic Gaussian distributions azimuthally invariant about the direction of the mean field. Following Domke & Pavlov (1979, Ap&SS, 66, 47), the averaging process is carried out in a reference frame defined by the direction of the mean field. The main steps are described in detail. They involve the writing of the Zeeman matrix in the polarization matrix representation of the radiation field and a rotation of the line of sight reference frame. Three types of fluctuations are considered : fluctuations along the direction of the mean field, fluctuations perpendicular to the mean field, and isotropic fluctuations. In each case, the averaging method is described in detail and fairly explicit expressions for the mean coefficients are established, most of which were given in Dolginov & Pavlov (1972, Soviet Ast., 16, 450) or Domke & Pavlov (1979, Ap&SS, 66, 47). They include the effect of a microturbulent velocity field with zero mean and a Gaussian distribution. A detailed numerical investigation of the mean coefficients illustrates the two effects of magnetic field fluctuations: broadening of the σ-components by fluctuations of the magnetic field intensity, leaving the π-components unchanged, and averaging over the angular dependence of the π and σ components. For longitudinal fluctuations only the first effect is at play. For isotropic and perpendicular fluctuations, angular averaging can modify the frequency profiles of the mean coefficients quite drastically with the appearance of an unpolarized central component in the diagonal absorption coefficient, even when the mean field is in direction of the line of sight. A detailed comparison of the effects of the three types of fluctuation coefficients is performed. In general the magnetic field fluctuations induce a broadening of the absorption and anomalous dispersion coefficients together with a decrease of their values. Two different regimes can be distinguished depending on whether the broadening is larger or smaller than the Zeeman shift by the mean magnetic field. For isotropic fluctuations, the mean coefficients can be expressed in terms of generalized Voigt and Faraday-Voigt functions H(n) and F(n) introduced by Dolginov & Pavlov (1972, Soviet Ast., 16, 450). These functions are related to the derivatives of the Voigt and Faraday-Voigt functions. A recursion relation is given in an Appendix for their calculation. A detailed analysis is carried out of the dependence of the mean coefficients on the intensity and direction of the mean magnetic field, on its root mean square fluctuations and on the Landé factor and damping parameter of the line.