Numerical methods in polarized radiative transfer

This paper looks at three aspects of numerical methods for solving polarized radiative transfer problems associated with spectral line formation in the presence of a magnetic field. First we prove Murphy's law for Stokes evolution operators which is the basis of the efficient algorithm used in the SPSR software package to compute the Stokes line depression contribution functions. Then we use a two-stream model to explain the efficacy of the field-free method in which the non-LTE line source function in a uniform magnetic field is approximated by the source function neglecting the magnetic field. Finally we introduce a totally new and computationally efficient approach to solving non-LTE problems based on a method of sparsely representing integral operators using wavelets. As an illustration, the wavelet method is used to solve the source function integral equation for a two-level atomic model in a finite atmosphere with coherent scattering, ignoring polarization.     

The polarization-free approximation

The development of effective iterative methods capable of accurately solving NLTE Stokes transfer problems is of considerable importance for the investigation of solar and stellar magnetic fields. After briefly indicating the iterative approach which is being presently pursued for the exact solution of such problems, the particular regime where polarization signals can only be due to the Zeeman effect is considered in some detail. By means of NLTE Stokes transfer calculations for a two-level atomic model it is first shown that the currently-used field-free approximation (Rees, 1969) cannot be safely applied in the presence of magnetic field gradients. Such gradients lead to changes in the shape and width of the line profiles and they can produce non-negligible effects on the atomic level populations and line source functions. A new approximate method is then proposed, which does not require the actual solution of the Stokes vector transfer equation and is practically as fast as the field-free one. This polarization-free approximation provides a fairly good account of the effects of homogeneous and inhomogeneous magnetic fields on the statistical equilibrium and is very easy to implement in any existing non-magnetic, multi-level transfer code.On leave from the Dipartimento di Astronomia e Scienza dello Spazio, Università di Firenze, Largo E. Fermi 5, I-50125 Firenze, Italia     

Speckle spectro‐polarimetry of magnetic structures

Two‐dimensional spectrograms were obtained with the Vacuum Tower Telescope, Tenerife, in order to study small‐scale structures and faculae on the Sun. Using the speckle reconstruction method, we obtain high‐resolution images and wavelength scans. Magnetic fields can be studied from Stokes V profiles, and velocity maps are gained by the Doppler shift of the center of gravity of Stokes I. Here some results about small‐scale structures and their magnetic fields are shown.     

The line-ratio method applied to vectormagnetographic measurements

Since solar magnetic fields are inhomogeneous, the averaging of Stokes parameter I within the entrance slit of the magnetograph is different from averaging Stokes Q₀ and V, because the former contains also light from non-magnetic, while the latter only contain light from magnetic regions. If the magnetographic calibration functions are calculated for homogeneous magnetic fields, errors arise, when they are used to reduce measurements of inhomogeneous fields. Therefore, we propose to use the line-ratio method to transform magnetographic measurements into the parameters of the magnetic vector field. The Q ratios and the V ratios of two carefully selected lines are free from errors of this kind. This is also the case for the Q ratios in line core and line wings in single-line magnetographs. An iterative method is presented to calculate the magnetic field parameters using the corresponding new calibration functions. An important advantage is, that the influence of scattered light in sunspots is also eliminated in a good approximation and the filling factor in plages can be estimated. This method is now used to determine magnetic vector fields in plages and sunspots of active regions with a new double-vector magnetograph.     

Comoving frame calculations of spectral lines formed in rapidly expanding media with the partial frequency redistribution function for zero natural line width

Comoving frame calculations have been used to compute the spectral lines formed in rapidly expanding spherical media. We have employed the angle-averaged partial frequency redistribution functionR _I with a two-level atom model in non-LTE atom approximation. A linear velocity law increasing with radius has been employed with maximum velocity at Τ=0 being set equal to 30 mean thermal units. It is found that one obtains almost symmetric emission line profiles at large velocities similar to those found in quasars.     

Filaments and large-scale magnetic fields

We have used Stanford magnetic field maps to construct distributions of longitudinal magnetic field gradients in the neighbourhood of polarity inversion lines. The distributions were constructed with proper account of the type of the polarity inversion lines and of the existence or absence of dark filaments above them. It is shown that for polarity inversion lines that pass inside active regions or on their boundary, grad B_II distributions for portions of the lines with persisting filament are shifted toward lower values of gradient as compared with grad B_II distributions for portions of the lines without filaments. The influence of the spatial resolution of the magnetograms upon polarity inversion line characteristics is discussed.     

Non-LTE resonance line polarization with partial redistribution: The solar Ca II K line

The non-LTE equations for the transfer of polarized radiation are solved for a two-level model of the Ca ii K ion in the atmospheric models of Gingerich et al. (1971) and Vernazza et al. (1981). The line formation model treats partial redistribution effects, including collisions. The effects of magnetic fields and of quantum mechanical interference are neglected. The response of the intensity, percentage polarization and Q Stokes parameter to changes in the calcium abundance, van der Waals damping parameter and atmospheric models is discussed.     

Non-LTE polarized radiative transfer in intermediate magnetic fields: Numerical problems and results

This paper presents some numerical results relative to a solution, based on the density matrix formalism, of the non-LTE, polarized radiative transfer problem for a two-level atom. The results concern the atomic upper level population and alignment, and the emergent radiation Stokes profiles, for a plane-parallel, static, isothermal atmosphere embedded in a magnetic field of intermediate strength, such that the Zeeman splitting has to be taken into account in the line profile. Zeeman coherences are neglected, whereas magneto-optical effects are taken into account, resulting in a full 4×4 absorption matrix. Induced emission is neglected and complete frequency redistribution, in the rest and laboratory frames, is assumed. Pure Doppler absorption profile (gaussian shape) has also been assumed. The presentation of the results is preceded by a brief discussion of their accuracy and of the numerical difficulties that were met in the solution of the problem.On leave from the Dipartimento di Astronomia e Scienza dello Spazio, Università di Firenze, Largo E. Fermi 5, I-50125 Firenze, Italia     

Center to limb variation of penumbral Stokes V profiles

We investigated the horizontal and the vertical component of the Evershed flow (EF). To this end, we computed average Stokes V profiles for various velocity classes in penumbrae at different heliocentric angles. Our results show that for blueshifted profiles an additional lobe with the same polarity as the spot is present in the blue side of the average Stokes V profile. The amplitude of the additional lobe grows with increasing blueshift and with increasing heliocentric angle. For small redshifts, the profiles show an additional lobe with the opposite polarity as the spot on the red side of the average Stokes V profile. Even at disk center, the original polarity of the average Stokes V profile is reversed for strong redshifts. The transition between the different types of Stokes V profiles is continuous and indicates that not only the vertical, but also the horizontal EF is a magnetized stream of plasma in a magnetic background field (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Diamagnetic pumping near the base of a stellar convection zone

The property of inhomogeneous turbulence in conducting fluids to expel large‐scale magnetic fields in the direction of decreasing turbulence intensity is shown as important for the magnetic field dynamics near the base of a stellar convection zone. The downward diamagnetic pumping confines a fossil internal magnetic field in the radiative core so that the field geometry is appropriate for formation of the solar tachocline. For the stars of solar age, the diamagnetic confinement is efficient only if the ratio of turbulent magnetic diffusivity η_T of the convection zone to the (microscopic or turbulent) diffusivity η_in of the radiative interior is η_T/η_in ≥ 10⁵. Confinement in younger stars requires larger η_T/η_in. The observation of persistent magnetic structures on young solar‐type stars can thus provide evidence for the nonexistence of tachoclines in stellar interiors and on the level of turbulence in radiative cores. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Solar Optical Telescope for the Hinode Mission: An Overview

The Solar Optical Telescope (SOT) aboard the Hinode satellite (formerly called Solar-B) consists of the Optical Telescope Assembly (OTA) and the Focal Plane Package (FPP). The OTA is a 50-cm diffraction-limited Gregorian telescope, and the FPP includes the narrowband filtergraph (NFI) and the broadband filtergraph (BFI), plus the Stokes Spectro-Polarimeter (SP). The SOT provides unprecedented high-resolution photometric and vector magnetic images of the photosphere and chromosphere with a very stable point spread function and is equipped with an image-stabilization system with performance better than 0.01 arcsec rms. Together with the other two instruments on Hinode (the X-Ray Telescope (XRT) and the EUV Imaging Spectrometer (EIS)), the SOT is poised to address many fundamental questions about solar magnetohydrodynamics. This paper provides an overview; the details of the instrument are presented in a series of companion papers. M. Otsubo is a former NAOJ staff scientist.     

A new method of calculating multi-level non-LTE line formation

This paper introduces a new method of solving the equation of multi-level non-LTE radiative transfer subject to constraints. This method is based on the combination of the advantages of the complete linearization method by Auer and Mihalas (1969) and the simple separated-iteration technique (Mihalas, 1978). First, linearize the equation of radiative transfer and constraints, respectively, then solve the linearized equation of the radiative transfer and linearized constraints, separately. It overcomes the disadvantages of requiring the simultaneous solution of the corresponding equations by the complete linearization method and the poor convergence of the simple separated-iteration technique. Therefore, it not only can deal with complex models, but also has a high speed of convergence in the calculation of multi-level NLTE line formation.     

The formation of asymmetric Stokes V profiles in the presence of a magnetopause

A magnetopause that separates two regimes of different flow, additional to the separation of a magnetic field from a field-free plasma, gives rise to the formation of asymmetric Stokes profiles. Using a simple two-layer model atmosphere, where one layer comprises a magnetic field, the other being field-free, it is shown by analytical derivation that a wide variety of Stokes V profiles can be produced, having amplitude asymmetries a in the range –a. These include two-humped V profiles, which have two lobes of equal sign. For the most simple models, the asymmetry depends on the ratio of continuum intensity to the Planck radiation intensity of the magnetic layer at the wavelength of the spectral line under consideration, and on the line depth. Two-humped profiles (|a|>1) require the temperature of the magnetic layer to surpass the temperature of the line-core forming region, implying a temperature inversion, so that the V profile is partially in emission. The confrontation of this formation scenario with properties of observed one-lobe profiles of quiet-Sun network regions is inconclusive due to insufficient spatial resolution and lack of a sufficient sample of simultaneously recorded Stokes spectral lines of varying line depths. It seems, however, to be in good agreement with the observed frequent occurrence of abnormal V profiles of the very strong Nai D ₂ and D ₂ spectral line. A possible observational verification for the present formation scenario of abnormal Stokes V profiles and a novel method of Stokes inversion are discussed.     

Magnetic and velocity fields of active regions

We have observed about 15 active regions on the Sun, with the Advanced Stokes Polarimeter and Dick Dunn Telescope at NSO/SP to map the Stokes parameters in the photospheric Fe 6302.5 Å and chromospheric Mg I 5173 Å lines, during 1999‐2002. The observations are corrected for dark current, gain, instrumental polarization and cross‐talk using ASP pipeline. The wavelength calibration is carried out using the O₂ telluric line 6302 Å which is also present in the observations. The photospheric and chromospheric longitudinal magnetograms are made from the Stokes V profiles, which were intercalibrated with the Kitt Peak magnetograms. The plasma motions are inferred from the line bisector measurements at different positions of the spectral line. In this paper we present the height dependence of Doppler velocity scatter plots of a sunspot in the photospheric Fe I 6302 Å line.     

Inversion of Stokes vector profiles in terms of a 3-component model

Various spectropolarimetric observations show peculiar Stokes profiles that reveal the coexistence of at least two magnetic components in the same resolution element. An example is given by observations of the full Stokes vector in a complex active region performed with the ZIMPOL I Stokes polarimeter. In order to deduce the physical parameters of the observed regions from such measured profiles, we have extended an existing inversion code, so that it can now fit the data with models composed of up to three different atmospheric components. Two of these components are magnetic and may possess different field strengths, field geometries, temperature stratifications, and velocity fields. The third component describes the field free atmosphere surrounding the magnetic features.The so extended inversion code has then been applied to the ZIMPOL I data. In this paper we present and discuss sample fits. The code is able to reproduce the observed complex Stokes profiles with good accuracy and provides physical parameters for both of the coexisting magnetic atmospheres. Inversion tests with a 2-component model (with one magnetic and one non-magnetic component) applied to the same profiles do not reproduce the measurements sufficiently well.     

Magnetic field, helicity and the 2000 July 14 flare in solar active region 9077

In this paper we analyse the non-potential magnetic field and the relationship with current (helicity) in the active region NOAA 9077 in 2000 July, using photospheric vector magnetograms obtained at different solar observatories and also coronal extreme-ultraviolet 171-Å images from the TRACE satellite.
We note that the shear and squeeze of magnetic field are two important indices for some flare-producing regions and can be confirmed by a sequence of photospheric vector magnetograms and EUV 171-Å features in the solar active region NOAA 9077. Evidence on the release of magnetic field near the photospheric magnetic neutral line is provided by the change of magnetic shear, electric current and current helicity in the lower solar atmosphere. It is found that the 'Bastille Day' 3B/5.7X flare on 2000 July 14 was triggered by the interaction of the different magnetic loop systems, which is relevant to the ejection of helical magnetic field from the lower solar atmosphere. The eruption of the large-scale coronal magnetic field occurs later than the decay of the highly sheared photospheric magnetic field and also current in the active region.     

Projection Effects on Physical Parameters Obtained from Solar Vector Magnetograms

1 INTRODUCTION Magnetic field plays an important role in solar activity. The stressing and subsequent partialrelaxation of magnetic fields in the active regions are generally accepted to be the energy sourceof solar flares. To quantitatively study the extent of stressed magnetic field as distinct from itspotential field, Hagyard et al. (1984) defined a magnetic shear angle膖he azimuth differencebetween the observed transverse magnetic field vector and the computed potential field vectorth…     

Structure of a simple sunspot from the inversion of IR spectral data

Analysis of spectral data of two neighboring infrared lines, Fe I 15648.5 Å (g = 3) and FeI 15652.9 Å (g_eff = 1.53) are carried out for a simple sunspot when it was near the solar disk center (μ = 0.92), to understand the basic structure of sunspot magnetic field. Inversions of Stokes profiles are carried out to derive different atmospheric parameters both as a function of location within the sunspot and height in the atmosphere. As a result of the inversion we have obtained maps of magnetic field strength, temperature, line‐of‐sight velocity, field inclination and azimuth for different optical depth layers between log(τ₅) = 0 and log(τ₅) = –2.0. In this paper we present few results from our inversion for a layer averaged between log(τ5) from 0.0 to –0.5.     

Second‐order error propagation in the Mueller matrix of a spectropolarimeter

Accurate measurements of Stokes IQUV in spectral lines is required for precise reconstruction of stellar magnetic field geometries with Zeeman‐Dopper imaging. Spectral Zeeman features are intrinsically weak and subjected to a number of instrumental uncertainties. The aim of this work is to study the details of the instrumental uncertainties in the Stokes IQUV measurements in spectral lines and ways of their reduction. We make a practical comparison of the polarimetric performances of two high‐resolution échelle spectropolarimeters, namely SOFIN at the NOT, and HARPS at ESO. We show the residual spectra for both instruments to characterize the cross‐talk between the observed Stokes parameters. We employ a self‐calibrating least‐squares fit to eliminate some of the polarization uncertainties to derive the full Stokes vector from stellar spectra (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The determination of the solar vector magnetic field at the surface from Stokes profiles

A simple method is proposed to infer the vector magnetic field at the surface of the Sun from Stokes profiles. This is based on the assumption that the variations of thermodynamical and magnetic field parameters with depth near the surface are so small that the displacements of the wavelengths at which the Stokes profiles reach their extrema can be ignored. And hence the polarized radiative transfer equations are greatly reduced to a set of non-linear equations with vector magnetic field parameters (, , ) which can be solved by a numerical iteration method. By fitting the synthetic profiles, it is shown that this proposed method can produce information on the vector field at the surface. It is also used to revise the observed profiles and it is found that the observed sunspot has the magnetic field structure of the fan model with the lines of magnetic field twisted.