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We apply the HanleRT Tenerife Inversion Code to the spectro-polarimetric observations obtained by the Chromospheric LAyer SpectroPolarimeter. This suborbital space experiment measured the variation with wavelength of the four Stokes parameters in the near-ultraviolet spectral region of the Mg II h & k lines over a solar disk area containing part of an active region plage and the edge of a sunspot penumbra. We infer the stratification of the temperature, the electron density, the line of-sight velocity, the micro-turbulent velocity, and the longitudinal component of the magnetic field from the observed intensity and circular polarization profiles. The inferred model atmosphere shows larger temperature and electron density in the plage and the superpenumbra regions than in the quiet regions. The shape of the plage region in terms of its brightness is similar to the pattern of the inferred longitudinal component of the magnetic field in the chromosphere, as well as to that of the overlying moss observed by AIA in the 171 A band, which suggests a similar magnetic origin for the heating in both the plage and the moss region. Moreover, this heating is particularly significant in the regions with larger inferred magnetic flux. In contrast, in the superpenumbra, the regions with larger electron density and temperature are usually found in between these regions with larger magnetic flux, suggesting that the details of the heating mechanism in the chromosphere of the superpenumbra may be different to those in the plage, but with the magnetic field still playing a key role., Comment: Accepted for publication in the Astrophysical Journal

Measurable linear scattering polarization signals have been predicted and detected at the solar disk center in the core of chromospheric lines. These forward-scattering polarization signals, which are of high interest for magnetic field diagnostics, have always been modeled either under the assumption of complete frequency redistribution (CRD), or taking partial frequency redistribution (PRD) effects into account under the angle-averaged (AA) approximation. This work aims at assessing the suitability of the CRD and PRD-AA approximations for modeling the forward-scattering polarization signals produced by the presence of an inclined magnetic field, the so-called forward-scattering Hanle effect, in the chromospheric CaI 4227 A line. Radiative transfer calculations are performed in semi-empirical 1D solar atmospheres, out of local thermodynamic equilibrium (LTE). A two-step solution strategy is applied: the non-LTE RT problem is first solved considering a multilevel atom and neglecting polarization phenomena. The same problem is then solved including polarization, considering a two-level atom and keeping fixed the lower-level population calculated at the previous step. The emergent linear polarization signals calculated under the CRD and PRD-AA approximations are analyzed and compared to those obtained by modeling PRD effects in their general angle-dependent (AD) formulation. With respect to the PRD-AD case, the CRD and PRD-AA calculations significantly underestimate the amplitude of the line-center polarization signals produced by the forward-scattering Hanle effect. The results of this work suggest that a PRD-AD modeling is required in order to develop reliable diagnostic techniques exploiting the forward-scattering polarization signals observed in the CaI 4227 A line. These results need to be confirmed by full 3D calculations including non-magnetic symmetry-breaking effects.

Aims: Main goal of this work is to study the potential of He II Ly-alpha wavelength-integrated scattering polarisation for probing the magnetism of the solar upper chromosphere. Methods: Radiative transfer calculations are performed in semi-empirical 1D solar atmospheres considering a two-term atomic model and accounting for the Hanle, Zeeman, and magneto-optical effects. The problem is suitably linearised and discretised, and the resulting numerical system is solved with a matrix-free iterative method. The results obtained modelling scattering processes with three different descriptions, namely in the limit of complete frequency redistribution (CRD), and accounting for partial frequency redistribution (PRD) effects under the angle-averaged (AA) approximation and in the general angle-dependent (AD) formulation, are compared. Results: In the line-core, the synthetic Stokes profiles resulting from CRD, PRD-AA, and PRD-AD calculations show a very good agreement. On the other hand, relevant differences are observed in Q/I outside the line-core region. Besides, the precise structure of the atmospheric model does not noticeably affect the line-core profiles, but it strongly impacts the Q/I signals outside the line-core. As most of the He II Ly-alpha photons originate in the core region, it turns out that wavelength-integrated linear polarisation signals are almost insensitive to both the scattering description and the atmospheric model. Appreciable wavelength-integrated U/I signals, showing observable sensitivity to horizontal magnetic fields in the range 0-1000 G, are also found, particularly near the limb. It turns out that, while the integration time required to detect magnetic fields in the quiet chromosphere with this line is too long for sounding rocket missions, magnetic fields corresponding to typical plage areas would produce detectable signals, especially near the limb.

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 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. In this work, we aim at quantitatively assessing 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 1D atmospheric models. We focus on the chromospheric Ca i line at 4227 A and on the photospheric Sr i line at 4607 A.

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, both in the absence and presence of magnetic fields. We solved the radiative transfer problem for polarized radiation in a one-dimensional semi-empirical atmospheric model with microturbulent and isotropic magnetic fields, accounting for PRD effects, comparing PRD-AA and PRD-AD modelings. The D1 and D2 lines are modeled separately as 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.

The intensity and the linear scattering polarization profiles of the hydrogen Ly-alpha line encode valuable information on the thermodynamic and magnetic structure of the upper layers of the solar chromosphere. The Chromospheric Lyman-Alpha Spectro-Polarimeter (CLASP) sounding rocket experiment provided unprecedented spectropolarimetric data of this line, as well as two-dimensional broadband images in intensity and linear polarization. We theoretically investigate the potential of the Ly-alpha broadband polarimetric signals for probing the solar chromosphere and its magnetic fields. We analyze the synthetic Stokes profiles obtained from a series of radiative transfer (RT) calculations out of local thermodynamic equilibrium, considering semi-empirical one-dimensional models of the solar atmosphere. The wavelength-integrated linear polarization signal is found to be dominated by the contribution from the wings when considering a Gaussian weighting function with a FWHM that corresponds to the CLASP slit-jaw broadband filter. These broadband linear polarization signals are strongly sensitive to magnetic fields of strengths on the order of 50 G, via the action of magneto-optical (MO) effects, and are expected to encode information on the middle-upper chromosphere. The two-dimensional broadband intensity and linear polarization images observed by CLASP can be suitably mimicked using synthetic wavelength-integrated signals obtained considering atmospheric models and magnetic fields that are representative of solar regions with different levels of activity, provided that the impact of MO effects is taken into account. Despite the limitations of a one-dimensional RT modeling, this work illustrates the diagnostic potential of filter-polarimetric Ly-alpha signals for probing the solar chromosphere and its magnetism., Comment: 16 pages (12 main text + 4 appendix). 7 figures in the main text (2 panels in Fig.1 , 2 panels in Fig.2, 4 panels in Fig.3, 9 panels in Fig.4, 4 panels in Fig.5, 2 panels in Fig.6, single panel in Fig. 7). 3 figures in appendices (single panel in Fig.8, single panel in Fig.9, 6 panels in Fig.10). Accepted for publication in The Astrophysical Journal on 03/03/2023

Radiative transfer investigations of the solar Mg II h and k resonance lines around 280~nm showed that, while their circular polarization (Stokes V) signals arise from the Zeeman effect, the linear polarization profiles (Stokes Q and U) are dominated by the scattering of anisotropic radiation and the Hanle and magneto-optical (MO) effects. Using the unprecedented observations of the Mg II and Mn I resonance lines obtained by the Chromospheric LAyer Spectro-Polarimeter (CLASP2), here we investigate how the linear polarization signals at different wavelengths (i.e., at the center, and at the near and far wings of the k line) vary with the longitudinal component of the magnetic field ($B_{L}$) at their approximate height of formation. The $B_{L}$ is estimated from the V signals in the aforementioned spectral lines. Particular attention is given to the following quantities that are expected to be influenced by the presence of magnetic fields through the Hanle and MO effects: the sign of the U signals, the total linear polarization amplitude ($LP$) and its direction ($\chi$) with respect to a reference direction. We find that at the center and near wings of the $k$ line, the behavior of these quantities is significantly different in the observed quiet and plage regions, and that both $LP$ and $\chi$ seem to depend on $B_{L}$. These observational results are indicative of the operation of the Hanle effect, Comment: 26 pages, 18 figures, accepted for publication in the Astrophysical Journal

The CLASP2 (Chromospheric LAyer SpectroPolarimeter 2) sounding rocket mission was launched on 2019 April 11. CLASP2 measured the four Stokes parameters of the Mg II h & k spectral region around 2800 Angstroms along a 200 arcsecond slit at three locations on the solar disk, achieving the first spatially and spectrally resolved observations of the solar polarization in this near ultraviolet region. The focus of the work presented here is the center-to-limb variation of the linear polarization across these resonance lines, which is produced by the scattering of anisotropic radiation in the solar atmosphere. The linear polarization signals of the Mg II h & k lines are sensitive to the magnetic field from the low to the upper chromosphere through the Hanle and magneto-optical effects. We compare the observations to theoretical predictions from radiative transfer calculations in unmagnetized semi-empirical models, arguing that magnetic fields and horizontal inhomogeneities are needed to explain the observed polarization signals and spatial variations. This comparison is an important step in both validating and refining our understanding of the physical origin of these polarization signatures, and also in paving the way toward future space telescopes for probing the magnetic fields of the solar upper atmosphere via ultraviolet spectropolarimetry., Comment: 14 pages, 5 figures, to be published in the Astrophysical Journal (ApJ), initially submitted May 2022, revised submission July 2022

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 out of local thermodynamical equilibrium, while accounting for scattering polarization, partial frequency redistribution, J-state interference, and hyperfine structure. The resulting radiative transfer code allows modeling the impact of magnetic fields of 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 that would be particularly useful for the numerically intensive case of three-dimensional radiative transfer. We examine the suitability of the considered approximation using our radiative transfer code to model the Stokes profiles of the Mg II h & k lines and of the H I Lyman alpha line in magnetized one-dimensional models of the solar atmosphere. Neglecting Doppler redistribution in the scattering processes that are unperturbed by elastic collisions produces a negligible error in the scattering polarization wings of the Mg II resonance lines and a minor one in the Lyman alpha wings, although it is unsuitable to model the cores of these lines. For both lines, the scattering processes that are perturbed by elastic collisions give a significant contribution only to the intensity component of the emissivity. Neglecting collisional as well as Doppler redistribution represents a rough but suitable approximation for the wings of the Mg II resonance lines, but a very poor one for the Lyman alpha wings. The magnetic sensitivity in the scattering polarization wings of the considered lines can be modeled by accounting for the magnetic field only in the eta I and rho V coefficients of the Stokes-vector transfer equation., Comment: Accepted for publication in the journal Astronomy & Astrophysics. Main text has 14 pages and 7 figures. The appendices (divided into Appendix A, B, and C) have 13 pages and 5 figures (contained in Appendix B)

Observations of scattering polarization and the Hanle effect in various spectral lines are increasingly used to complement traditional solar magnetic field determination techniques. One of the strongest scattering polarization signals in the photosphere is measured in the Sr I line at 4607.3 {\AA} when observed close to the solar limb. Here, we present the first observational evidence of Hanle rotation in the linearly polarized spectrum of this at several limb distances. We observed with the Zurich IMaging POLarimeter, ZIMPOL, at the IRSOL observatory, with exceptionally good seeing conditions, allowing for long integration times. We combined the fast modulating polarimeter with a slow modulator installed in front of the telescope. This combination allows the measurement of spectropolarimetric data being highly precise and unprecedentedly accurate. Fixing the reference direction for positive Stokes $Q$ parallel to the limb, we detect singly-peaked $U/I$ signals well above the noise level. We can exclude instrumental origin for such $U/I$ signals. These signatures are exclusively found in the Sr I line, but not in the adjoining Fe I line, therefore eliminating the Zeeman effect as the mechanism responsible for their appearance. However, we find a clear spatial correlation between the circular polarization produced by the Zeeman effect and the $U/I$ amplitudes. This suggests that the detected $U/I$ signals are the signatures of Hanle rotation caused by a spatially resolved magnetic field. A novel measurement technique allows for determining the absolute level of polarization with unprecedented precision. Using this technique, high-precision spectropolarimetric observations reveal for the first time unambiguous $U/I$ signals due to Hanle rotation in the Sr I line., Comment: 7 pages, 4 figures, accepted for publication in A&A

When observed in quiet regions close to the solar limb, many strong resonance lines show conspicuous linear polarization signals, produced by scattering processes, with extended wing lobes. Recent studies indicate that, contrary to what was previously believed, the wing lobes are sensitive to the presence of relatively weak longitudinal magnetic fields through magneto-optical (MO) effects. We theoretically investigate the sensitivity of the scattering polarization wings of the Ca I 4227 {\AA} line to the MO effects, and we explore its diagnostic potential for inferring information on the longitudinal component of the photospheric magnetic field. We calculate the intensity and polarization profiles of the Ca I 4227 {\AA} line by numerically solving the problem of the generation and transfer of polarized radiation under non-local thermodynamic equilibrium conditions in one-dimensional semi-empirical models of the solar atmosphere, taking into account the joint action of the Hanle, Zeeman, and MO effects. We consider volume-filling magnetic fields as well as magnetic fields occupying a fraction of the resolution element. In contrast to the circular polarization signals produced by the Zeeman effect, we find that the linear polarization angle in the scattering polarization wings of Ca I 4227 presents a clear sensitivity, through MO effects, not only to the flux of the photospheric magnetic field, but also to the fraction of the resolution element that the magnetic field occupies. We identify the linear polarization angle in the wings of strong resonance lines as a valuable observable for diagnosing unresolved magnetic fields. Used in combination with observables that encode information on the magnetic flux and other properties of the observed atmospheric region, it can provide constraints on the filling factor of the magnetic field., Comment: 9 pages, 9 figures

Context. The correct modeling of the scattering polarization signals observed in several strong resonance lines requires taking partial frequency redistribution (PRD) phenomena into account. Aims. This work aims at assessing the impact and the range of validity of the angle-averaged 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 focus on the solar Ca i 4227 {\AA} line. Methods. Spectral line polarization is 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 is solved through a two-step approach. In step 1, the problem is solved for intensity only, considering a multi-level atom. In step 2, the problem is 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 {\AA} 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 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 {\AA} 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.

Context. Several numerical problems require the interpolation of discrete data that present various types of discontinuities. The radiative transfer is a typical example of such a problem. This calls for high-order well-behaved techniques to interpolate both smooth and discontinuous data. Aims. The final aim is to propose new techniques suitable for applications in the context of numerical radiative transfer. Methods. We have proposed and tested two different techniques. Essentially non-oscillatory (ENO) techniques generate several candidate interpolations based on different substencils. The smoothest candidate interpolation is determined from a measure for the local smoothness, thereby enabling the essential non-oscillatory property. Weighted ENO (WENO) techniques use a convex combination of all candidate substencils to obtain high-order accuracy in smooth regions while keeping the essentially non-oscillatory property. In particular, we have outlined and tested a novel well-performing fourth-order WENO interpolation technique for both uniform and nonuniform grids. Results. Numerical tests prove that the fourth-order WENO interpolation guarantees fourth-order accuracy in smooth regions of the interpolated functions. In the presence of discontinuities, the fourth-order WENO interpolation enables the non-oscillatory property, avoiding oscillations. Unlike B\'ezier and monotonic high-order Hermite interpolations, it does not degenerate to a linear interpolation near smooth extrema of the interpolated function. Conclusions. The novel fourth-order WENO interpolation guarantees high accuracy in smooth regions, while effectively handling discontinuities. This interpolation technique might be particularly suitable for several problems, including a number of radiative transfer applications such as multidimensional problems, multigrid methods, and formal solutions.

Context. Numerical solutions to transfer problems of polarized radiation in solar and stellar atmospheres commonly rely on stationary iterative methods, which often perform poorly when applied to large problems. In recent times, stationary iterative methods have been replaced by state-of-the-art preconditioned Krylov iterative methods for many applications. However, a general description and a convergence analysis of Krylov methods in the polarized radiative transfer context are still lacking. Aims. We describe the practical application of preconditioned Krylov methods to linear transfer problems of polarized radiation, possibly in a matrix-free context. The main aim is to clarify the advantages and drawbacks of various Krylov accelerators with respect to stationary iterative methods. Methods. We report the convergence rate and the run time of various Krylov-accelerated techniques combined with different formal solvers when applied to a 1D benchmark transfer problem of polarized radiation. In particular, we analyze the GMRES, BICGSTAB, and CGS Krylov methods, preconditioned with Jacobi, or (S)SOR. Results. Krylov methods accelerate the convergence, reduce the run time, and improve the robustness of standard stationary iterative methods. Jacobi-preconditioned Krylov methods outperform SOR-preconditioned stationary iterations in all respects. In particular, the Jacobi-GMRES method offers the best overall performance for the problem setting in use. Conclusions. Krylov methods can be more challenging to implement than stationary iterative methods. However, an algebraic formulation of the radiative transfer problem allows one to apply and study Krylov acceleration strategies with little effort. Furthermore, many available numerical libraries implement matrix-free Krylov routines, enabling an almost effortless transition to Krylov methods.

Context. The numerical modeling of the generation and transfer of polarized radiation is a key task in solar and stellar physics research and has led to a relevant class of discrete problems that can be reframed as linear systems. In order to solve such problems, it is common to rely on efficient stationary iterative methods. However, the convergence properties of these methods are problem-dependent, and a rigorous investigation of their convergence conditions, when applied to transfer problems of polarized radiation, is still lacking. Aims. After summarizing the most widely employed iterative methods used in the numerical transfer of polarized radiation, this article aims to clarify how the convergence of these methods depends on different design elements, such as the choice of the formal solver, the discretization of the problem, or the use of damping factors. The main goal is to highlight advantages and disadvantages of the different iterative methods in terms of stability and rate of convergence. Methods. We first introduce an algebraic formulation of the radiative transfer problem. This formulation allows us to explicitly assemble the iteration matrices arising from different stationary iterative methods, compute their spectral radii and derive their convergence rates, and test the impact of different discretization settings, problem parameters, and damping factors. Conclusions. The general methodology used in this article, based on a fully algebraic formulation of linear transfer problems of polarized radiation, provides useful estimates of the convergence rates of various iterative schemes. Additionally, it can lead to novel solution approaches as well as analyses for a wider range of settings, including the unpolarized case.

Twenty-five years ago, enigmatic linear polarization signals were discovered in the core of the sodium D1 line. The only explanation that could be found implied that the solar chromosphere is practically unmagnetized, in contradiction with other evidences. This opened a paradox that has challenged physicists for many years. Here we present its solution, demonstrating that these polarization signals can be properly explained in the presence of magnetic fields in the gauss range. This result opens a novel diagnostic window for exploring the elusive magnetism of the solar chromosphere., Comment: Main article has a length of 5 pages and contains 2 figures. Supplemental Material has a length of 6 pages and contains 3 figures. Note that the main article and the Supplemental Material have separate reference lists. Article published in open access by Physical Review Letters on 18/08/2021; Link: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.127.081101

Routine ultraviolet imaging of the Sun's upper atmosphere shows the spectacular manifestation of solar activity; yet we remain blind to its main driver, the magnetic field. Here we report unprecedented spectropolarimetric observations of an active region plage and its surrounding enhanced network, showing circular polarization in ultraviolet (Mg II $h$ & $k$ and Mn I) and visible (Fe I) lines. We infer the longitudinal magnetic field from the photosphere to the very upper chromosphere. At the top of the plage chromosphere the field strengths reach more than 300 gauss, strongly correlated with the Mg II $k$ line core intensity and the electron pressure. This unique mapping shows how the magnetic field couples the different atmospheric layers and reveals the magnetic origin of the heating in the plage chromosphere., Comment: 50 pages, 11 figures, 1 table, published in Science Advances

The magnetic activity of the Sun directly impacts the Earth and human life. Likewise, other stars will have an impact on the habitability of planets orbiting these host stars. The lack of information on the magnetic field in the higher atmospheric layers hampers our progress in understanding solar magnetic activity. Overcoming this limitation would allow us to address four paramount long-standing questions: (1) How does the magnetic field couple the different layers of the atmosphere, and how does it transport energy? (2) How does the magnetic field structure, drive and interact with the plasma in the chromosphere and upper atmosphere? (3) How does the magnetic field destabilise the outer solar atmosphere and thus affect the interplanetary environment? (4) How do magnetic processes accelerate particles to high energies? New ground-breaking observations are needed to address these science questions. We suggest a suite of three instruments that far exceed current capabilities in terms of spatial resolution, light-gathering power, and polarimetric performance: (a) A large-aperture UV-to-IR telescope of the 1-3 m class aimed mainly to measure the magnetic field in the chromosphere by combining high spatial resolution and high sensitivity. (b) An extreme-UV-to-IR coronagraph that is designed to measure the large-scale magnetic field in the corona with an aperture of about 40 cm. (c) An extreme-UV imaging polarimeter based on a 30 cm telescope that combines high throughput in the extreme UV with polarimetry to connect the magnetic measurements of the other two instruments. This mission to measure the magnetic field will unlock the driver of the dynamics in the outer solar atmosphere and thereby greatly advance our understanding of the Sun and the heliosphere., Comment: Submitted to Experimental Astronomy (on 28. Jul. 2020). Based on a proposal submitted in response to a call for white papers in the Voyage 2050 long-term plan in the ESA science programme. 36 pages, 10 figures

Modeling spectral line profiles taking frequency redistribution effects into account is a notoriously challenging problem from the computational point of view, especially when polarization phenomena (atomic polarization and polarized radiation) are taken into account. Frequency redistribution effects are conveniently described through the redistribution function formalism, and the angle-averaged approximation is often introduced to simplify the problem. Even in this case, the evaluation of the emission coefficient for polarized radiation remains computationally costly, especially when magnetic fields are present or complex atomic models are considered. We aim to develop an efficient algorithm to numerically evaluate the angle-averaged redistribution function for polarized radiation. Our proposed approach is based on a low-rank approximation via trivariate polynomials whose univariate components are represented in the Chebyshev basis. The resulting algorithm is significantly faster than standard quadrature-based schemes for any target accuracy in the range between 10^-6 and 10^-2., Comment: 12 pages, 10 figures, two appendices

Several strong resonance lines, such as H I Ly-$\alpha$, Mg II k, Ca II K, Ca I 4227 \AA\, which are characterized by deep and broad absorption profiles in the solar intensity spectrum, show conspicuous linear scattering polarization signals when observed in quiet regions close to the solar limb. Such signals show a characteristic triplet-peak structure, with a sharp peak in the line core and extended wing lobes. The line core peak is sensitive to the presence of magnetic fields through the Hanle effect, which however is known not to operate in the line wings. Recent theoretical studies indicate that, contrary to what was previously believed, the wing linear polarization signals are also sensitive to the magnetic field through magneto-optical effects (MO). We search for observational indications of this recently discovered physical mechanism in the scattering polarization wings of the Ca I 4227 \AA\ line. We performed a series of spectropolarimetric observations of this line using the Zurich IMaging POLarimeter (ZIMPOL) camera at the Gregory-Coud\'e telescope of IRSOL (Switzerland) and at the GREGOR telescope in Tenerife (Spain). Spatial variations of the total linear polarization degree and of the linear polarization angle are clearly appreciable in the wings of the observed line. We provide a detailed discussion of our observational results, showing that the detected variations always take place in regions where longitudinal magnetic fields are present, thus supporting the theoretical prediction that they are produced by MO effects., Comment: Accepted by A&A

The European Solar Telescope (EST) is a research infrastructure for solar physics. It is planned to be an on-axis solar telescope with an aperture of 4 m and equipped with an innovative suite of spectro-polarimetric and imaging post-focus instrumentation. The EST project was initiated and is driven by EAST, the European Association for Solar Telescopes. EAST was founded in 2006 as an association of 14 European countries. Today, as of December 2019, EAST consists of 26 European research institutes from 18 European countries. The Preliminary Design Phase of EST was accomplished between 2008 and 2011. During this phase, in 2010, the first version of the EST Science Requirement Document (SRD) was published. After EST became a project on the ESFRI roadmap 2016, the preparatory phase started. The goal of the preparatory phase is to accomplish a final design for the telescope and the legal governance structure of EST. A major milestone on this path is to revisit and update the Science Requirement Document (SRD). The EST Science Advisory Group (SAG) has been constituted by EAST and the Board of the PRE-EST EU project in November 2017 and has been charged with the task of providing with a final statement on the science requirements for EST. Based on the conceptual design, the SRD update takes into account recent technical and scientific developments, to ensure that EST provides significant advancement beyond the current state-of-the-art. The present update of the EST SRD has been developed and discussed during a series of EST SAG meetings. The SRD develops the top-level science objectives of EST into individual science cases. Identifying critical science requirements is one of its main goals. Those requirements will define the capabilities of EST and the post-focus instrument suite. The technical requirements for the final design of EST will be derived from the SRD., Comment: 2nd edition, December 2019, 138 pages

The Sr~{\sc i} 4607~\AA\ spectral line shows one of the strongest scattering polarization signals in the visible solar spectrum. The amplitude of this polarization signal is expected to vary at granular spatial scales, due to the combined action of the Hanle effect and the local anisotropy of the radiation field. Observing these variations would be of great interest because it would provide precious information on the small-scale activity of the solar photosphere. At present, few detections of such spatial variations have been reported. This is due to the difficulty of these measurements, which require combining high spatial ($\sim$ 0.1"), spectral ($\leq$ 20 m\AA), and temporal resolution (< 1 min) with increased polarimetric sensitivity ($\sim$ 10$^-$$^4$). Aims. We aim to detect spatial variations at granular scales of the scattering polarization peak of the Sr~{\sc i} 4607~\AA\ line at different limb distances, and to study the correlation with the continuum intensity. Methods.Using the Zurich IMaging POLarimeter (ZIMPOL) system mounted at the GREGOR telescope and spectrograph in Tenerife, Spain, we carried out spectro-polarimetric measurements to obtain the four Stokes parameters in the Sr~{\sc i} line at different limb distances, from $\mu=0.2$ to $\mu=0.8$, on the solar disk. Results.Spatial variations of the scattering polarization signal in the Sr~{\sc i} 4607~\AA\ line, with a spatial resolution of about 0.66", are clearly observed at every $\mu$. The spatial scale of these variations is comparable to the granular size. A statistical analysis reveals that the linear scattering polarization amplitude in this Sr~{\sc i} spectral line is positively correlated with the intensity in the continuum, corresponding to the granules, at every $\mu$., Comment: 8 pages, 6 figures, accepted for publication in A&A

Sr I 4607\AA spectral line shows one of the strongest scattering polarization signals in the visible solar spectrum. The amplitudes of these signals are expected to vary at granular spatial scales. This variation can be due to changes in the magnetic field intensity and orientation (Hanle effect) as well as due to spatial and temporal variations in the plasma properties. Measuring the spatial variation of such polarization signal would allow us to study the properties of the magnetic fields at subgranular region. But, the observations are challenging since both high spatial resolution and high spectropolarimetric sensitivity are required at the same time. To the aim of measuring these spatial variations at granular scale, we carried out a spectro-polarimetric measurement with the Zurich IMaging POLarimeter (ZIMPOL), at the GREGOR solar telescope at different limb distances on solar disk. Our results show a spatial variation of scattering linear polarization signals in Sr I 4607\AA line at the granular scale at every $\mu$, starting from 0.2 to 0.8. The correlation between the polarization signal amplitude and the continuum intensity imply statistically that the scattering polarization is higher at the granular regions than in the intergranular lanes., Comment: 4 pages, 3 figures, Proceeding of Third Meeting of the Italian Solar and Heliospheric Community, OCTOBER 28-31, 2018 - TURIN

The linear polarization produced by scattering processes in the hydrogen Lyman alpha line of the solar disk radiation is a key observable for probing the chromosphere-corona transition region (TR) and the underlying chromospheric plasma. While the line-center signal encodes information on the magnetic field and three-dimensional structure of the TR, the sizable scattering polarization signals that the joint action of partial frequency redistribution (PRD) and J-state interference produce in the Lyman alpha wings have generally been thought to be sensitive only to the thermal structure of the solar atmosphere. Here we show that the wings of the Q/I and U/I scattering polarization profiles of this line are actually sensitive to the presence of chromospheric magnetic fields, with strengths similar to those that produce the Hanle effect in the line core (i.e., between 5 and 100 gauss, approximately). In spite of the fact that the Zeeman splitting induced by such weak fields is very small compared to the total width of the line, the magneto-optical effects that couple the transfer equations for Stokes Q and U are actually able to produce sizable changes in the Q/I and U/I wings. The theoretical results presented here further expand the diagnostic content of the unprecedented spectropolarimetric observations provided by the Chromospheric Lyman-Alpha Spectropolarimeter (CLASP)., Comment: Accepted for publication in the Astrophysical Journal. 13 pages, 7 figures, 3 appendices

Context. The spectrum of the hydrogen atom was explained by Bohr more than one century ago. We revisit here some of the aspects of the underlying quantum structure, with a modern formalism, focusing on the limit of the Balmer series. Aims. We investigate the behaviour of the absorption coefficient of the isolated hydrogen atom in the neighbourhood of the Balmer limit. Methods. We analytically computed the total cross-section arising from bound-bound and bound-free transitions in the isolated hydrogen atom at the Balmer limit, and established a simplified semi-analytical model for the surroundings of that limit. We worked within the framework of the formalism of Landi Degl'Innocenti & Landolfi (2004, Astrophys. Space Sci. Lib., 307), which permits an almost straight-forward generalization of our results to other atoms and molecules, and which is perfectly suitable for including polarization phenomena in the problem. Results. We analytically show that there is no discontinuity at the Balmer limit, even though the concept of a "Balmer jump" is still meaningful. Furthermore, we give a possible definition of the location of the Balmer jump, and we check that this location is dependent on the broadening mechanisms. At the Balmer limit, we compute the cross-section in a fully analytical way. Conclusions. The Balmer jump is produced by a rapid drop of the total Balmer cross-section, yet this variation is smooth and continuous when both bound-bound and bound-free processes are taken into account, and its shape and location is dependent on the broadening mechanisms.

The Chromospheric Lyman-Alpha Spectro-Polarimeter (CLASP) is a suborbital rocket experiment that on 3rd September 2015 measured the linear polarization produced by scattering processes in the hydrogen Ly-$\alpha$ line of the solar disk radiation, whose line-center photons stem from the chromosphere-corona transition region (TR). These unprecedented spectropolarimetric observations revealed an interesting surprise, namely that there is practically no center-to-limb variation (CLV) in the $Q/I$ line-center signals. Using an analytical model, we first show that the geometrical complexity of the corrugated surface that delineates the TR has a crucial impact on the CLV of the $Q/I$ and $U/I$ line-center signals. Secondly, we introduce a statistical description of the solar atmosphere based on a three-dimensional (3D) model derived from a state-of-the-art radiation magneto-hydrodynamic simulation. Each realization of the statistical ensemble is a 3D model characterized by a given degree of magnetization and corrugation of the TR, and for each such realization we solve the full 3D radiative transfer problem taking into account the impact of the CLASP instrument degradation on the calculated polarization signals. Finally, we apply the statistical inference method presented in a previous paper to show that the TR of the 3D model that produces the best agreement with the CLASP observations has a relatively weak magnetic field and a relatively high degree of corrugation. We emphasize that a suitable way to validate or refute numerical models of the upper solar chromosphere is by confronting calculations and observations of the scattering polarization in ultraviolet lines sensitive to the Hanle effect., Comment: Accepted for publication in The Astrophysical Journal Letters

The numerical computation of reliable and accurate Stokes profiles is of great relevance in solar physics. In the synthesis process, many actors play a relevant role: among them the formal solver, the discrete atmospheric model, and the spectral line. This paper tests the performances of different numerical schemes in the synthesis of polarized spectra for different spectral lines and atmospheric models. The hierarchy between formal solvers is enforced, stressing the peculiarities of high-order and low-order formal solvers. The density of grid points necessary for reaching a given accuracy requirement is quantitatively described for specific situations.

On 3rd September 2015, the Chromospheric Lyman-Alpha SpectroPolarimeter (CLASP) successfully measured the linear polarization produced by scattering processes in the hydrogen Lyman-$\alpha$ line of the solar disk radiation, revealing conspicuous spatial variations in the $Q/I$ and $U/I$ signals. Via the Hanle effect the line-center $Q/I$ and $U/I$ amplitudes encode information on the magnetic field of the chromosphere-corona transition region (TR), but they are also sensitive to the three-dimensional structure of this corrugated interface region. With the help of a simple line formation model, here we propose a statistical inference method for interpreting the Lyman-$\alpha$ line-center polarization observed by CLASP., Comment: Accepted for publication in The Astrophysical Journal

Context. The scattering polarization signal observed in the photospheric Sr i 4607 {\AA} line is expected to vary at granular spatial scales. This variation can be due to changes in the magnetic field intensity and orientation (Hanle effect), but also to spatial and temporal variations in the plasma properties. Measuring the spatial variation of such polarization signal would allow us to study the properties of the magnetic fields at subgranular scales, but observations are challenging since both high spatial resolution and high spectropolarimetric sensitivity are required. Aims. We aim to provide observational evidence of the polarization peak spatial variations, and to analyze the correlation they might have with granulation. Methods. Observations conjugating high spatial resolution and high spectropolarimetric precision were performed with the Zurich IMaging POLarimeter, ZIMPOL, at the GREGOR solar telescope, taking advantage of the adaptive optics system and the newly installed image derotator. Results. Spatial variations of the scattering polarization in the Sr i 4607 {\AA} line are clearly observed. The spatial scale of these variations is comparable with the granular size. Small correlations between the polarization signal amplitude and the continuum intensity indicate that the polarization is higher at the center of granules than in the intergranular lanes., Comment: 5 pages, 4 figures

The linear polarization pattern produced by scattering processes in the Ca I resonance line at 4227 angstroms is a valuable observable for probing the solar atmosphere. Via the Hanle effect, the very significant Q/I and U/I line-center signals are sensitive to the presence of magnetic fields in the lower chromosphere with strengths between 5 and 125 G, approximately. On the other hand, partial frequency redistribution (PRD) produces sizable signals in the wings of the Q/I profile, which have always been thought to be insensitive to the presence of magnetic fields. Interestingly, novel observations of this line revealed a surprising behavior: fully unexpected signals in the wings of the U/I profile and spatial variability in the wings of both Q/I and U/I. Here we show that the magneto-optical (MO) terms of the Stokes-vector transfer equation produce sizable signals in the wings of U/I and a clear sensitivity of the Q/I and U/I wings to the presence of photospheric magnetic fields with strengths similar to those that produce the Hanle effect in the line core. This radiative transfer investigation on the joint action of scattering processes and the Hanle and Zeeman effects should facilitate the development of more reliable techniques for exploring the magnetism of stellar atmospheres. To this end, we can now exploit the circular polarization produced by the Zeeman effect, the magnetic sensitivity caused by the above-mentioned MO effects in the Q/I and U/I wings, and the Hanle effect in the line core., Comment: Accepted for publication in the Astrophysical Journal

When integrating the radiative transfer equation for polarized light, the necessity of high-order numerical methods is well known. In fact, well-performing high-order formal solvers enable higher accuracy and the use of coarser spatial grids. Aiming to provide a clear comparison between formal solvers, this work presents different high-order numerical schemes and applies the systematic analysis proposed by Janett et al. (2017), emphasizing their advantages and drawbacks in terms of order of accuracy, stability, and computational cost., Comment: 10 pages, 4 figures