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The solar atmosphere is diagnosed by solving the polarized radiative transfer problem for plasmas in Non-Local Thermodynamical Equilibrium (NLTE). A key challenge in multidimensional NLTE diagnosis is to integrate the radiative transfer equation (RTE) efficiently, but current methods are local, limited to constant propagation matrices. The formalism presented in this paper lays the foundations to achieve an efficient non-local integration of the RTE based on the Magnus expansion. Our approach starts formulating the problem in terms of rotations represented in the Lorentz-Poincare group (Stokes formalism) and motivating the use of the Magnus expansion. Then, we combine a highly detailed algebraic characterization of the propagation matrix with Magnus to reformulate the homogenous solution to the RTE. Thus, we obtain a compact analytical evolution operator supporting arbitrary variations of the propagation matrix to first and second order in the Magnus expansion and showing the way to higher orders. Finally, we reformulate the inhomogeneous part to make it solvable with the Magnus expansion too, which gives rise to an interesting new object: the inhomogeneous evolution operator. This provides the first consistent and general formal solution of the RTE that furthermore is non-local, efficient, and accurate by design, with other peculiarity: it separates the integration part from the algebraic formal solution. Our framework is verified analytically and with a computational implementation that leads to a new family of numerical radiative transfer methods and suggests several applications, e.g. accelerating NLTE calculations with non-local radiative transfer. With cosmetic changes, our results apply to other universal physical problems sharing the Lorentz-Poincare algebra of the RTE and special relativity., Comment: 15+3 pages (including Appendix), 6 figures, submitted to A&A

Aims: From a coherent interpretation of the linear polarisation detected in the spectral lines of the Mira star $\chi$ Cyg, we derive information about the dynamics of the stellar photosphere, including pulsation. Methods: From spectropolarimetric observations of $\chi$ Cyg, we perform careful analysis of the polarisation signals observed in atomic and molecular lines, both in absorption and emission, using the radiative transfer in the polarisation context, through two mechanisms: intrinsic polarisation and continuum depolarisation. We also explain the observed line doubling phenomenon in terms of an expanding shell in spherical geometry, which allows us to pinpoint the coordinates over the stellar disk with enhanced polarisation. Results: We find that the polarised spectrum of $\chi$ Cyg is dominated by intrinsic polarisation, with a negligible continuum depolarisation. The observed polarised signals can only be explained by assuming that this polarisation is locally enhanced by velocity fields. During the pulsation, radial velocities are not homogeneous over the disk. We map these regions of enhanced velocities. Conclusions: We have set an algorithm to distinguish in any stellar spectra of linear polarisation the origin of this polarisation and the way to increase signal by coherently adding many lines with an appropriated weight. Applied to the Mira star $\chi$ Cyg, we reached the unexpected result that during the pulsation, velocities are radial but not homogeneous over the disk. The reason for these local velocity enhancements are probably related to the interplay between the atmospheric pulsation dynamics and the underlying stellar convection., Comment: Accepted for publication in A&A

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

Due to the quick evolution of the solar chromosphere, its magnetic field cannot be inferred reliably without accounting for the temporal variations of its polarized light. This has been broadly overlooked in the modelling and interpretation of the polarization due to technical problems (e.g., lack of temporal resolution or of time-dependent MHD solar models) and/or because many polarization measurements can apparently be explained without dynamics. Here, we show that the temporal evolution is critic for explaining the spectral-line scattering polarization because of its sensitivity to rapidly-varying physical quantities and the possibility of signal cancellations and attenuation during extended time integration. For studying the combined effect of time-varying magnetic fields and kinematics, we solved the 1.5D NLTE problem of the second kind in time-dependent 3D R-MHD solar models and we synthesized the Hanle and Zeeman polarization in forward scattering for the chromospheric {\lambda}4227 line. We find that the quiet-sun polarization amplitudes depend on the periodicity and spectral coherence of the signal enhacements produced by kinematics, but that substantially larger linear polarization signals should exist all over the solar disk for short integration times. The spectral morphology of the polarization is discussed as a combination of Hanle, Zeeman, dynamic effects and partial redistribution. We give physical references for observations by degrading and characterizing our slit time-series in different spatio-temporal resolutions. The implications of our results for the interpretation of the second solar spectrum and for the investigation of the solar atmospheric heatings are discussed., Comment: 16 pages, 14 figures, accepted for publication in ApJ (sent in Nov 2016)

Exposing the polarization signatures of the solar chromosphere requires studying its temporal variations, which is rarely done when modelling and interpreting scattering and Hanle signals. The present contribution sketches the scientific problem of solar polarization diagnosis from the point of view of the temporal dimension, remarking some key aspects for solving it. Our time-dependent calculations expose the need of considering dynamics explicitly when modelling and observing scattering polarization in order to achieve effective diagnosis techniques as well as a deeper knowledge of the second solar spectrum., Comment: 5 pages, 4 figures, Solar Workshop Polarization 8

Is it physically feasible to perform chromospheric diagnosis using spatial maps of scattering polarization at the solar disk center? To investigate it we synthesized polarization maps (in 8542 Angstroms) resulting from MHD solar models and NLTE radiative transfer calculations that consider Hanle effect and vertical macroscopic motions. After explaining the physical con- text of forward scattering and presenting our results, we arrive at the definition of Hanle polarity inversion lines. We show how such features can give support for a clearer chromospheric diagnosis in which the magnetic and dynamic effects in the scattering polarization could be disentangled., Comment: 7 pages, 2 figures, published in proceedings of IAU symposium 305

The quantum theory of polarized light allows one to model scattering in the solar atmosphere for inferring its properties. This powerful approach has revealed two key long-standing problems in solar physics: the puzzling dilemmas between theory and observations in several anomalously polarized spectral lines and the need for inferring the ubiquitous weak chromospheric magnetic fields, which requires discriminating the Hanle effect in dynamic optically thick plasmas. However, the ever-present dynamics, i.e., the temporal evolution of heatings and macroscopic motions, has been widely disregarded when modeling and interpreting the scattering polarization. This has hindered a consistent theoretical solution to the puzzle while falsifying the Hanle diagnosis. Here, we show that the dynamical evolution is a keystone for solving both problems because its systematic impact allows an explanation of the observations from 'anomalous' instantaneous polarization signals. Evolution accounted for, we reproduce amplitudes and (spectral and spatial) shapes of the Ca I 4227 A polarization at solar disk center, identifying a restrictive arrangement of magnetic fields, kinematics, heatings, and spatio-temporal resolution. We find that the joint action of dynamics, Hanle effect, and low temporal resolutions mimics Zeeman linear polarization profiles, the true weak-field Zeeman signals being negligible. Our results allow reinterpretation of many polarization signals of the solar spectra and support time-dependent scattering polarization as a powerful tool for deciphering the spatio-temporal distribution of chromospheric heatings and fields. This approach may be a key aid in developing the Hanle diagnosis for the solar atmosphere., Comment: 5 pages, 4 figures, published in ApJL

This paper shows the first synthetic tomography of the quiet solar chromosphere formed by spatial maps of scattering polarization. It has been calculated for the CaII 8498, 8542 and 3934 A lines by solving the NLTE (non-local thermodynamical equilibrium) RT (radiative transfer) problem of the second kind in a 3D atmosphere model obtained from realistic MHD (magneto-hydrodynamical) simulations. Maps of circular polarization were calculated neglecting atomic polarization. Our investigation focuses on the linear polarization signals induced by kinematics, radiation field anisotropy and Hanle effect in forward-scattering geometry. Thus, instead of considering slit profiles at the limb as normally done in the study of the second solar spectrum, we synthetize and analyze spatial maps of polarization at disk center. It allows us to understand the spatial signatures of dynamics and magnetic field in the linear polarization for discriminating them observationally. Our results suggest new ideas for chromospheric diagnosis that will be developed throughout a serie of papers. In particular, Hanle Polarity Inversion Lines and dynamic Hanle diagrams are two concepts introduced in the present work. We find that chromospheric dynamics and magnetic field topology create spatial fingerprints in the polarization maps that trace the dynamic situation of the plasma and the magnetic field. Based on such spatial features we reconstruct the magnetic field intensity in the middle chromosphere along grooves of null linear polarization. We finally address the problems of diagnosing Hanle saturation and kinematic amplification of scattering signals using Hanle diagrams., Comment: 16 pages, 16 figures, Accepted for publication in ApJ

Velocity gradients in a stellar atmospheric plasma have an impact on the anisotropy of the radiation field that illuminates each point within the medium, and this may in principle influence the scattering line polarization that results from the induced atomic level polarization. Here we analyze the emergent linear polarization profiles of the Ca II infrared triplet after solving the radiative transfer problem of scattering polarization in time-dependent hydrodynamical models of the solar chromosphere, taking into account the impact of the plasma macroscopic velocity on the atomic level polarization. We discuss the influence that the velocity and temperature shocks in the considered chromospheric models have on the temporal evolution of the scattering polarization signals of the Ca II infrared lines, as well as on the temporally averaged profiles. Our results indicate that the increase of the linear polarization amplitudes caused by macroscopic velocity gradients may be significant in realistic situations. We also study the effect of the integration time, the microturbulent velocity and the photospheric dynamical conditions, and discuss the feasibility of observing with large-aperture telescopes the temporal variation of the scattering polarization profiles. Finally, we explore the possibility of using the differential Hanle effect in the IR triplet of Ca II with the intention of avoiding the characterization of the zero-field polarization to infer magnetic fields in dynamic situations., Comment: 12 pages, 9 figures. Accepted for publication in ApJ. New figure added, typos corrected

Magnetic field topology, thermal structure and plasma motions are the three main factors affecting the polarization signals used to understand our star. In this theoretical investigation, we focus on the effect that gradients in the macroscopic vertical velocity field have on the non-magnetic scattering polarization signals, establishing the basis for general cases. We demonstrate that the solar plasma velocity gradients have a significant effect on the linear polarization produced by scattering in chromospheric spectral lines. In particular, we show the impact of velocity gradients on the anisotropy of the radiation field and on the ensuing fractional alignment of the CaII levels, and how they can lead to an enhancement of the zero-field linear polarization signals. This investigation remarks the importance of knowing the dynamical state of the solar atmosphere in order to correctly interpret spectropolarimetric measurements, which is important, among other things, for establishing a suitable zero field reference case to infer magnetic fields via the Hanle effect., Comment: 14 pages, 10 figures, 3 appendixes, accepted for publication in ApJ