Partial Frequency Redistribution Research Articles

Full non-LTE spectral line formation

In the present paper we consider the full nonlocal thermodynamic equilibrium (non-LTE) radiation transfer problem. This formalism allows us to account for deviation from equilibrium distribution of both the radiation field and the massive particles. In the present study, two-level atoms with broadened upper level represent the massive particles. In the absence of velocity-changing collisions, we demonstrate the analytic equivalence of the full non-LTE source function with the corresponding standard non-LTE partial frequency redistribution (PFR) model. We present an iterative method based on operator splitting techniques that can be used to numerically solve the problem at hand. We benchmark it against the standard non-LTE transfer problem for a two-level atom with PFR. We illustrate the deviation of the velocity distribution function of excited atoms from the equilibrium distribution. We also discuss the dependence of the emission profile and the velocity distribution function on elastic collisions and velocity-changing collisions.

Exact Solutions for Radiative Transfer with Partial Frequency Redistribution

The construction of exact solutions for radiative transfer in a plane-parallel medium has been addressed by Hemsch and Ferziger in 1972 for a partial frequency redistribution model of the formation of spectral lines consisting in a linear combination of frequency coherent and fully incoherent scattering. The method of solution is based on an eigenfunction expansion of the radiation field, leading to two singular integral equations with Cauchy or Cauchy-type kernels, that have to be solved one after the other. We reconsider this problem, using as starting point the integral formulation of the radiative transfer equation, where the terms involving the coupling between the two scattering mechanisms are clearly displayed, as well as the primary source of photons. With an inverse Laplace transform, we recover the singular integral equations previously established and, with Hilbert transforms as in the previous work, recast them as boundary value problems in the complex plane. Their solutions are presented in detail for an infinite and a semi-infinite medium. The coupling terms are carefully analyzed and consistency with either the coherent or the incoherent limit is systematically checked. We recover the important result of the previous work that an exact solution exists for an infinite medium, whereas for a semi-infinite medium, which requires the introduction of half-space auxiliary functions, the solution is given by a Fredholm integral equation to be solved numerically. The solutions of the singular integral equations are used to construct explicit expressions providing the radiation field for an arbitrary primary source and for the Green’s function. An explicit expression is given for the radiation field emerging from a semi-infinite medium.

Impact of the numerical conversion to optical depth on the transfer of polarized radiation

Making the conversion from the geometrical spatial scale to the optical depth spatial scale is useful in obtaining numerical solutions for the radiative transfer equation. This is because it allows for the use of exponential integrators, while enforcing numerical stability. Such a conversion involves the integration of the total opacity of the medium along the considered ray path. This is usually approximated by applying a piecewise quadrature in each spatial cell of the discretized medium. However, a rigorous analysis of this numerical step is lacking. This work is aimed at clearly assessing the performance of different optical depth conversion schemes with respect to the solution of the radiative transfer problem for polarized radiation, out of the local thermodynamic equilibrium. We analyzed different optical depth conversion schemes and their combinations with common formal solvers, both in terms of the rate of convergence as a function of the number of spatial points and the accuracy of the emergent Stokes profiles. The analysis was performed in a 1D semi-empirical model of the solar atmosphere, both in the absence and in the presence of a magnetic field. We solved the transfer problem of polarized radiation in different settings: the continuum, the photospheric Sr i AA modeled under the assumption of complete frequency redistribution, and the chromospheric Ca i AA taking the partial frequency redistribution effects into account during the modeling. High-order conversion schemes generally outperform low-order methods when a sufficiently high number of spatial grid points is considered. In the synthesis of the emergent Stokes profiles, the convergence rate, as a function of the number of spatial points, is impacted by both the conversion scheme and formal solver. The use of low-order conversion schemes significantly reduces the accuracy of high-order formal solvers. In practical applications, the use of low-order optical depth conversion schemes introduces large numerical errors in the formal solution. To fully exploit high-order formal solvers and obtain accurate synthetic emergent Stokes profiles, it is necessary to use high-order optical depth conversion schemes.

Modeling the scattering polarization in the solar Ca I 4227Å line with angle-dependent PRD effects and bulk velocities

Context. Modeling the scattering polarization signals of strong chromospheric lines requires solving the radiative transfer problem for polarized radiation, out of local thermodynamic equilibrium, taking partial frequency redistribution (PRD) effects into account. This problem is extremely challenging from a computational standpoint and, so far, most studies have been carried out by either modeling PRD effects under the angle-average approximation or by considering academic models of the solar atmosphere. Thanks to a new solution strategy, applicable to atomic systems that allow for a linearization of the problem, accurate solutions can now be routinely obtained in realistic 1D models, taking angle-dependent (AD) PRD effects into account. Aims. This work is aimed at assessing the suitability and performance of this new approach to handling dynamic scenarios. At the same time, it aims to explore the joint impact of magnetic fields and bulk velocities on the scattering polarization profiles of strong resonance lines, accounting for AD PRD effects and considering more realistic atmospheric models than in previous investigations. Methods. By using a two-level atomic model for neutral calcium, we synthesized the intensity and polarization profiles of the Ca I 4227 Å line. Our calculations were performed in 1D atmospheric models, both semi-empirical and extracted from 3D magnetohydrodynamic simulations, including vertical bulk velocities and magnetic fields of arbitrary strength and orientation, both constant and varying with height. Results. We obtained accurate solutions after only a few iterations across all considered scenarios. Even when formulating the problem in the observer’s reference frame, the frequency and angular grids required for accurate results were easily manageable. The calculated profiles showed the expected signatures of bulk velocities: wavelength shifts, enhancement of the line-core polarization amplitude, and prominent asymmetries in the wing signals. The results obtained in atmospheric models with complex thermal, dynamic, and magnetic structures unveiled the broad diversity of features in the emergent radiation that can be expected from realistic scenarios. Conclusions. The presented results assess the suitability of the proposed solution strategy and its parallel implementation, thus supporting its generalization to the 3D case. Our applications in increasingly realistic atmospheric models showed the difficulty related to precisely establishing the individual weight of bulk velocities and magnetic fields in the shape of the emergent profiles. This highlights the need to account for both these physical ingredients to perform reliable inversions of observed scattering polarization profiles.

The Polarization of the Solar Ba ii D1 Line with Partial Frequency Redistribution and Its Magnetic Sensitivity

We investigate the main physical mechanisms that shape the intensity and polarization of the Ba ii D1 line at 4934 Å via radiative transfer numerical experiments. We focus especially on the scattering linear polarization arising from the spectral structure of the anisotropic radiation in the wavelength interval spanned by the line’s hyperfine structure (HFS) components in the odd isotopes of barium. After verifying that the presence of the low-energy metastable levels only impacts the amplitude, but not the shape, of the D1 linear polarization, we relied on a two-term atomic model that neglects such metastable levels but includes HFS. The D1 fractional linear polarization shows a very small variation with the choice of atmospheric model, enhancing its suitability for solar magnetic field diagnostics. Tangled magnetic fields with strengths of tens of gauss reduce the linear polarization, and saturation is reached at roughly 300 G. Deterministic inclined magnetic fields produce a U/I profile and, if they have a significant longitudinal component, a V/I profile, whose modeling requires accounting for HFS and the Paschen–Back effect. Because of the overlap between HFS components, the magnetograph formula cannot be applied to infer the longitudinal magnetic field. Accurately modeling the D1 intensity and polarization requires an atomic system that includes the metastable levels and the HFS, the detailed spectral structure of the radiation field, the incomplete Paschen–Back regime for magnetic fields, and an accurate treatment of collisions.

Assessment of the CRD approximation for the observer’s frame RIII redistribution matrix

Context. Approximated forms of the RII and RIII redistribution matrices are frequently applied to simplify the numerical solution of the radiative transfer problem for polarized radiation, taking partial frequency redistribution (PRD) effects into account. A widely used approximation for RIII is to consider its expression under the assumption of complete frequency redistribution (CRD) in the observer’s frame (RIII−CRD). The adequacy of this approximation for modeling the intensity profiles has been firmly established. By contrast, its suitability for modeling scattering polarization signals has only been analyzed in a few studies, considering simplified settings. Aims. In this work, we aim to quantitatively assess the impact and the range of validity of the RIII−CRD approximation in the modeling of scattering polarization. Methods. We first present an analytic comparison between RIII and RIII−CRD. We then compare the results of radiative transfer calculations, out of local thermodynamic equilibrium, performed with RIII and RIII−CRD in realistic one-dimensional atmospheric models. We focus on the chromospheric Ca I line at 4227 Å and on the photospheric Sr I line at 4607 Å. Results. The RIII−CRD approximation provides accurate results for the Ca I 4227 Å line. Only when velocities are included can some appreciable discrepancies be found, especially for lines of sight close to the disk center. The approximation performs well also for the Sr I 4607 Å line, especially in the absence of magnetic fields or when a micro-turbulent field is included. However, some appreciable errors appear when deterministic magnetic fields or bulk velocities are considered. Conclusions. Our results show that the RIII−CRD approximation is suited for the PRD modeling of the scattering polarization signals of strong chromospheric lines, both in the core and wings. With a few minor exceptions, this approximation is also suitable for photospheric lines, although PRD effects generally play a minor role in their modeling.

The Impact of Angle-dependent Partial Frequency Redistribution on the Scattering Polarization of the Solar Na i D Lines

The long-standing paradox of the linear polarization signal of the Na i D1 line was recently resolved by accounting for the atom’s hyperfine structure and the detailed spectral structure of the incident radiation field. That modeling relied on the simplifying angle-averaged (AA) approximation for partial frequency redistribution (PRD) in scattering, which potentially neglects important angle–frequency couplings. This work aims at evaluating the suitability of a PRD-AA modeling for the D1 and D2 lines through comparisons with general angle-dependent (AD) PRD calculations in both the absence and presence of magnetic fields. We solved the radiative transfer problem for polarized radiation in a 1D semiempirical atmospheric model with microturbulent and isotropic magnetic fields, accounting for PRD effects and comparing PRD-AA and PRD-AD modelings. The D1 and D2 lines are modeled separately as a two-level atomic system with hyperfine structure. The numerical results confirm that a spectrally structured radiation field induces linear polarization in the D1 line. However, the PRD-AA approximation greatly impacts the Q/I shape, producing an antisymmetric pattern instead of the more symmetric PRD-AD one while presenting a similar sensitivity to magnetic fields between 10 and 200 G. Under the PRD-AA approximation, the Q/I profile of the D2 line presents an artificial dip in its core, which is not found for the PRD-AD case. We conclude that accounting for PRD-AD effects is essential to suitably model the scattering polarization of the Na i D lines. These results bring us closer to exploiting the full diagnostic potential of these lines for the elusive chromospheric magnetic fields.

Polarized resonance line transfer in a spherically symmetric medium with angle-dependent partial frequency redistribution

ABSTRACT In a stellar atmosphere, the resonance line polarization arises from scattering of limb-darkened radiation field by atoms. This spectral line polarization gets affected particularly in the wings, when the line photons suffer scattering on electrons in thermal motion. Scattering of line photons by atoms and electrons are, respectively, described by the atomic and Thomson electron scattering redistribution functions, which in general depend on both the frequencies and directions of incident and scattered photons. In this paper, we consider the polarized spectral line formation in spherically symmetric extended and expanding media accounting for the angle-dependent partial frequency redistribution (AD-PRD) in scattering on both atoms and electrons. We solve this computationally demanding polarized transfer problem using an accelerated lambda iteration method and a method based on orders of scattering approach. In the case of expanding spherical medium, the concerned transfer problem is solved in the comoving frame. Because of the computational limitations, we consider optically thin isothermal spherically symmetric media of different extensions for the static case as well as when the velocity fields are present. For the considered model, we show that the AD-PRD effects on the linear polarization profiles are significant and have to be accounted for.

Chromospheric thermodynamic conditions from inversions of complex Mg II h & k profiles observed in flares

The flare activity of the Sun has been studied for decades, using both space- and ground-based telescopes. The former have mainly focused on the corona, while the latter have mostly been used to investigate the conditions in the chromosphere and photosphere. The Interface Region Imaging Spectrograph (IRIS) instrument has served as a gateway between these two cases, given its capability to observe quasi-simultaneously the corona, the transition region, and the chromosphere using different spectral lines in the near- and far-ultraviolet ranges. IRIS thus provides unique diagnostics to investigate the thermodynamics of flares in the solar atmosphere. In particular, the Mg II h&k and the Mg II UV triplet lines provide key information about the thermodynamics of low to upper chromosphere, while the C II 1334 & 1335 Å lines cover the upper-chromosphere and low transition region. The Mg II h&k and the Mg II UV triplet lines show a peculiar, pointy shape before and during the flare activity. The physical interpretation, i.e., the physical conditions in the chromosphere, that can explain these profiles has remained elusive. In this paper, we show the results of a non-LTE inversion of such peculiar profiles. To better constrain the atmospheric conditions, the Mg II h&k and the Mg II UV triplet lines are simultaneously inverted with the C II 1334 & 1335 Å lines. This combined inversion leads to more accurate derived thermodynamic parameters, especially the temperature and the turbulent motions (micro-turbulence velocity). We use an iterative process that looks for the best fit between the observed profile and a synthetic profile obtained by considering non-local thermodynamic equilibrium and partial frequency redistribution of the radiation due to scattered photons. This method is computationally rather expensive (≈6 CPU-hour/profile). Therefore, we use the k-means clustering technique to identify representative profiles and associated representative model atmospheres. By inverting the representative profiles with the most advanced inversion code (STiC), in addition to recover the main physical parameters, we are able to conclude that these unique, pointy profiles are associated with a large gradient in the line-of-sight velocity along the optical depth in the high chromosphere.

Hanle Effect with Angle-dependent Partial Frequency Redistribution in Three-dimensional Media

There has been a constant improvement in the observational measurement of linear polarization in chromospheric spectral lines in the last three decades. However, modeling polarized profiles of these lines still remains incomplete, due to the lack of inclusion of fundamental physics in modeling efforts. To model the observed solar spectrum and its linear polarization, we need a solution to the polarized radiative transfer equation. The polarization in strong resonance lines originates from the scattering mechanism known as the partial frequency redistribution (PFR) of an anisotropic radiation field. The understanding of the linear polarization in spatially resolved structures needs radiative transfer solutions in multidimensional geometries. In this paper, we explore the effects of angle-dependent PFR on scattering polarization profiles formed in three-dimensional (3D) media. We find that the 3D geometry combined with angle-dependent PFR produces more scattering polarization than an angle-averaged one.

Full non-LTE spectral line formation

In this article we discuss a method for obtaining a numerical solution to the so-called full nonlocal thermodynamic equilibrium (non-LTE) radiation transfer problem. More specifically, the usual numerical iterative methods for non-LTE radiation transfer are coupled with that formalism and new numerical additions are explained in detail. We benchmarked the whole process with the standard non-LTE transfer problem for a two-level atom with Hummer’s RI − A partial frequency redistribution function. Finally, we present new quantities, such as the spatial distribution of the velocity distribution function of excited atoms, which can only be accessed by adopting a more general frame for non-LTE radiation transfer, such as the one we propose here.

Scalable matrix-free solver for 3D transfer of polarized radiation in stellar atmospheres

We present an efficient and massively parallel solution strategy for the transfer problem of polarized radiation, for a 3D stationary medium out of local thermodynamic equilibrium. Scattering processes are included accounting for partial frequency redistribution effects. Such a setting is one of the most challenging ones in radiative transfer modeling. The problem is formulated for a two-level atomic model, which allows linearization. The discrete ordinate method alongside an exponential integrator are used for discretization. Efficient solution is obtained with a Krylov method equipped with a tailored physics-based preconditioner. A matrix-free approach results in a lightweight implementation, suited for tackling large problems. Near-optimal strong and weak scalability are obtained with two complementary decompositions of the computational domain. The presented approach made it possible to perform simulations for the Ca i line at 4227Å with more than 109 degrees of freedom in less than half an hour on massively parallel machines, always converging in a few iterations for the proposed tests.

The Circular Polarization of the Mn 1 Resonance Lines around 280 nm for Exploring Chromospheric Magnetism

We study the circular polarization of the Mn i resonance lines at 279.56, 279.91, and 280.19 nm (hereafter, UV multiplet) by means of radiative transfer modeling. In 2019, the CLASP2 mission obtained unprecedented spectropolarimetric data in a region of the solar ultraviolet including the Mg ii h and k resonance lines and two lines of a subordinate triplet, as well as two Mn i resonance lines. The first analysis of such data, in particular those corresponding to a plage region, allowed the inference of the longitudinal magnetic field from the photosphere to the upper chromosphere just below the transition region. This was achieved by applying the weak field approximation to the circular polarization profiles of the Mg ii and Mn i lines. While the applicability of this approximation to the Mg ii lines was already demonstrated in previous works, this is not the case for the Mn i UV multiplet. These lines are observed as absorptions between the Mg ii h and k lines, a region whose intensity is shaped by their partial frequency redistribution effects. Moreover, the only Mn i stable isotope has nuclear spin I = 5/2 and thus hyperfine structure must be, a priori, taken into account. Here we study the generation and transfer of the intensity and circular polarization of the Mn i resonance lines accounting for these physical ingredients. We analyze their sensitivity to the magnetic field by means of their response function, and we demonstrate the applicability of the weak field approximation to determine the longitudinal component of the magnetic field.

The polarization signals of the solar K I D lines and their magnetic sensitivity

Aims. This work aims to identify the relevant physical processes in shaping the intensity and polarization patterns of the solar K I D lines through spectral syntheses, placing particular emphasis on the D2 line. Methods. The theoretical Stokes profiles were obtained by numerically solving the radiative transfer problem for polarized radiation considering one-dimensional semi-empirical models of the solar atmosphere. The calculations account for scattering polarization, partial frequency redistribution (PRD) effects, hyperfine structure (HFS), J- and F-state interference, multiple isotopes, and magnetic fields of arbitrary strength and orientation. Results. The intensity and circular polarization profiles of both D lines can be suitably modeled while neglecting both J-state interference and HFS. The magnetograph formula can be applied to both lines, without including HFS, to estimate weak longitudinal magnetic fields in the lower chromosphere. By contrast, modeling the scattering polarization signal of the D lines requires the inclusion of HFS. The amplitude of the D2 scattering polarization signal is strongly depolarized by HFS, but it remains measurable. A small yet appreciable error is incurred in the scattering polarization profile if PRD effects are not taken into account. Collisions during scattering processes have a clear depolarizing effect, although a quantitative analysis is left for a forthcoming publication. Finally, the D2 scattering polarization signal is particularly sensitive to magnetic fields with strengths around 10 G and it strongly depends on their orientation. Despite this, its center-to-limb variation relative to the amplitude at the limb is largely insensitive to the field strength and orientation. Conclusions. These findings highlight the value of the K I D2 line polarization for diagnostics of the solar magnetism, and show that the linear and circular polarization signals of this line are primarily sensitive to magnetic fields in the lower chromosphere and upper photosphere, respectively.

Combined Analysis of Hydrogen and Oxygen 102.6 nm Emission at Mars

AbstractWater is lost from the Mars upper atmosphere to space as hydrogen and oxygen, both of which can be observed in scattered ultraviolet sunlight at 102.6 nm. We present Emirates Mars Mission Emirates Mars Ultraviolet Spectrometer (EMM/EMUS) insertion orbit observations of this airglow, resolving the independent altitude contributions of H and O for the first time. We present the first airglow modeling of the complete H and O 102.6 nm system and the first 3D azimuthally symmetric modeling of the O emission, retrieving temperatures and densities typical of northern spring. Our model reproduces the emission well above 200 km, but does not incorporate partial frequency redistribution needed to reproduce the observed O brightness at lower altitudes and on the disk. These results support future EMM/EMUS science orbit retrievals of H loss and the use of 102.6 nm observations to constrain planetary atmospheres across the solar system.

The transfer of polarized radiation in resonance lines with partial frequency redistribution, J-state interference, and arbitrary magnetic fields

Aims. We present the theoretical framework and numerical methods we have implemented to solve the problem of the generation and transfer of polarized radiation in spectral lines without assuming local thermodynamical equilibrium, while accounting for scattering polarization, partial frequency redistribution (due to both the Doppler effect and elastic collisions), J-state interference, and hyperfine structure. The resulting radiative transfer code allows one to model the impact of magnetic fields of an arbitrary strength and orientation through the Hanle, incomplete Paschen-Back, and magneto-optical effects. We also evaluate the suitability of a series of approximations for modeling the scattering polarization in the wings of strong resonance lines at a much lower computational cost, which is particularly valuable for the numerically intensive case of three-dimensional radiative transfer. Methods. We examine the suitability of the considered approximations by using our radiative transfer code to model the Stokes profiles of the Mg II h & k lines and of the H I Lyman-α line in magnetized one-dimensional models of the solar atmosphere. Results. Neglecting Doppler redistribution in the scattering processes that are unperturbed by elastic collisions (i.e., treating them as coherent in the observer’s frame) produces a negligible error in the scattering polarization wings of the Mg II resonance lines and a minor one in the Lyman-α wings, although it is unsuitable to model the cores of these lines. For both lines, the scattering processes that are perturbed by elastic collisions only give a significant contribution to the intensity component of the emissivity. Neglecting collisional as well as Doppler redistribution (so that all scattering processes are coherent) represents a rough but suitable approximation for the wings of the Mg II resonance lines, but a very poor one for the Lyman-α wings. The magnetic sensitivity in the scattering polarization wings of the considered lines can be modeled by accounting for the magnetic field in only the ηI and ρV coefficients of the Stokes-vector transfer equation (i.e., using the zero-field expression for the emissivity).

Numerical solutions to linear transfer problems of polarized radiation

Context. The polarization signals produced by the scattering of anistropic radiation in strong resonance lines encode important information about the elusive magnetic fields in the outer layers of the solar atmosphere. An accurate modeling of these signals is a very challenging problem from the computational point of view, in particular when partial frequency redistribution (PRD) effects in scattering processes are accounted for with a general angle-dependent treatment. Aims. We aim at solving the radiative transfer problem for polarized radiation in nonlocal thermodynamic equilibrium conditions, taking angle-dependent PRD effects into account. The problem is formulated for a two-level atomic model in the presence of arbitrary magnetic and bulk velocity fields. The polarization produced by scattering processes and the Zeeman effect is considered. Methods. The proposed solution strategy is based on an algebraic formulation of the problem and relies on a convenient physical assumption, which allows its linearization. We applied a nested matrix-free GMRES iterative method. Effective preconditioning is obtained in a multifidelity framework by considering the light-weight description of scattering processes in the limit of complete frequency redistribution (CRD). Results. Numerical experiments for a one-dimensional (1D) atmospheric model show near optimal strong and weak scaling of the proposed CRD-preconditioned GMRES method, which converges in few iterations, independently of the discretization parameters. A suitable parallelization strategy and high-performance computing tools lead to competitive run times, providing accurate solutions in a few minutes. Conclusions. The proposed solution strategy allows the fast systematic modeling of the scattering polarization signals of strong resonance lines, taking angle-dependent PRD effects into account together with the impact of arbitrary magnetic and bulk velocity fields. Almost optimal strong and weak scaling results suggest that this strategy is applicable to realistic 3D models. Moreover, the proposed strategy is general, and applications to more complex atomic models are possible.

TIC: A Stokes Inversion Code for Scattering Polarization with Partial Frequency Redistribution and Arbitrary Magnetic Fields

We present the Tenerife Inversion Code (TIC), which has been developed to infer the magnetic and plasma properties of the solar chromosphere and transition region via full Stokes inversion of polarized spectral lines. The code is based on the HanleRT forward engine, which takes into account many of the physical mechanisms that are critical for a proper modeling of the Stokes profiles of spectral lines originating in the tenuous and highly dynamic plasmas of the chromosphere and transition region: the scattering polarization produced by quantum level imbalance and interference (atomic polarization), the effects of frequency coherence in polarized resonance scattering (partial redistribution), and the impact of arbitrary magnetic fields on the atomic polarization and the radiation field. We present first results of atmospheric and magnetic inversions, and discuss future developments for the project.

Modeling the scattering polarization of the solar Ca I 4227 Å line with angle-dependent partial frequency redistribution

Context. The correct modeling of the scattering polarization signals observed in several strong resonance lines requires taking partial frequency redistribution (PRD) phenomena into account. Modeling scattering polarization with PRD effects is very computationally demanding and the simplifying angle-averaged (AA) approximation is therefore commonly applied. Aims. This work aims to assess the impact and the range of validity of the AA approximation with respect to the general angle-dependent (AD) treatment of PRD effects in the modeling of scattering polarization in strong resonance lines, with a focus on the solar Ca I 4227 Å line. Methods. Spectral line polarization was modeled by solving the radiative transfer problem for polarized radiation, under nonlocal thermodynamic equilibrium conditions, taking PRD effects into account in static one-dimensional semi-empirical atmospheric models presenting arbitrary magnetic fields. The problem was solved through a two-step approach. In step 1, the problem was solved for the intensity only, considering a multilevel atom. In step 2, the problem was solved including polarization, considering a two-level atom with an unpolarized and infinitely sharp lower level, and fixing the lower level population calculated at step 1. Results. The results for the Ca I 4227 Å line show a good agreement between the AA and AD calculations for the Q/I and U/I wings’ signals. However, AA calculations reveal an artificial trough in the line-core peak of the linear polarization profiles, whereas AD calculations show a sharper peak in agreement with the observations. Conclusions. An AD treatment of PRD effects is essential to correctly model the line-core peak of the scattering polarization signal of the Ca I 4227 Å line. By contrast, in the considered static case, the AA approximation seems to be suitable to model the wing scattering polarization lobes and their magnetic sensitivity through magneto-optical effects.

Fast Iterative Techniques for Polarized Radiative Transfer in Spherically Symmetric Moving Media

Abstract 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.