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     

Micro-structured magnetic atmospheres

Asymmetrical Stokes profiles are produced if the photospheric magnetic and velocity fields are structured on scales smaller than the mean-free-path of the photons. Here we put forward a compact analytical expression for the radiative transfer equation in this case. Explicitly, micro-variations of the magnetic field strength and the velocity are considered. The existence of micro-structures might have serious implications on the techniques currently used to measure solar magnetic fields. For example, we show the failure of the relationship employed to calibrate magnetographs.On leave from the Dipartimento di Astronomia e Scienza dello Spazio, Università di Firenze, Largo E. Fermi 5, I-50125 Firenze, Italy     

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.     

Net circular polarization in magnetic spectral lines produced by velocity gradients: Some analytical results

The net circular polarization in a spectral line due to the combined effect of magnetic fields and velocity gradients is analyzed for a few schematic situations. In some particular cases, its dependence on the magnetic field, velocity field and line parameters can be expressed analytically.On leave from Dipartimento di Astronomia e Scienza dello Spazio, Università di Firenze, Largo E. Fermi 5, I-50125 Firenze, Italia     

The polarization-free approximation applied to multi-level non-LTE radiative transfer

The polarization-free (POF) approximation (Trujillo Bueno and Landi Degl'Innocenti, 1996) is capable of accounting for the approximate influence of the magnetic field on the statistical equilibrium, without actually solving the full Stokes vector radiative transfer equation. The method introduces the Zeeman splitting or broadening of the line absorption profile I in the scalar radiative transfer equation, but the coupling between Stokes I and the other Stokes parameters is neglected. The expected influence of the magnetic field is largest for strongly-split strong lines and the effect is greatly enhanced by gradients in the magnetic field strength. Formally the interaction with the other Stokes parameters may not be neglected for strongly-split strong lines, but it turns out that the error in Stokes I obtained through the POF approximation to a large extent cancels the neglect of interaction with the other Stokes parameters, so that the resulting line source functions and line opacities are more accurate than those obtained with the field-free approach. Although its merits have so far only been tested for a two-level atom, we apply the POF approximation to multi-level non-LTE radiative transfer problems on the premise that there is no essential difference between these two cases. Final verification of its validity in multi-level cases still awaits the completion of a non-LTE Stokes vector transfer code.For two realistic multi-level cases (CaII and MgI in the solar atmosphere) it is demonstrated that the POF method leads to small changes, with respect to the field-free method, in the line source functions and emergent Stokes vector profiles (much smaller than for a two-level atom). Real atoms are dominated by strong ultraviolet lines (only weakly split) and continua, and most lines with large magnetic splitting (in the red and the infrared) are at higher excitation energies, i.e. they are relatively weak and unable to produce significant changes in the statistical equilibrium. We find that it is generally unpredictable by how much the POF results will differ from the field-free results, so that it is nearly always necessary to confirm predictions by actual computations.The POF approximation provides more reliable results than the field-free approximation without significantly complicating the radiative transfer problem, i.e. without solving any extra equations and without excessive computational resource requirements, so that it is to be preferred over the field-free approximation.     

The first and second order moments of the polarization profiles of hydrogen lines

The main properties of the first- and second-order moments of polarized hydrogen lines, forming in the presence of stationary electric and magnetic fields, are reviewed. The analytical results presented here apply directly to the case of optically-thin emission lines in the LTE regime. Some applications of such results to electric- and magnetic-field diagnostics in (solar) plasmas are then briefly considered.On leave from the Dipartimento di Astronomia e Scienza dello Spazio, Università di Firenze, Largo E. Fermi 5, I-50125 Firenze, Italy     

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.     

Calibration of Vector Magnetograms with the Chromospheric Mg b2 Line

Stokes polarization profiles of the Mg?b₂ 5172.68 Å spectral line on two simple sunspots are obtained with the Multi-Channel Solar Telescope (MCST) at the Huairou Solar Observing Station (HSOS). This is done by means of scanning this line over the wavelength interval from 200 mÅ redward of the line center to 200 mÅ blueward, in steps of 10 mÅ. A generalized analytic solution to the transfer equation for polarized radiation is presented. With a nonlinear least-square fitting technique, the linear calibration coefficients for the low-chromospheric longitudinal magnetic field is obtained in the weak-field case. We also discuss the problems in calibrating the transverse field with this line. It is shown that the weak-field approximation is not applicable to the chromospheric Mg?b₂ line for the transverse component of the magnetic field.     

The determination of vector magnetic fields in prominences from the observations of the Stokes profiles in the D3 line of helium

A method is presented to measure the magnetic field vector in prominences by means of the polarimetric observations in the D₃ line of He obtained with the High Altitude Observatory Stokes polarimeter. The characteristics of the observed Stokes profiles are discussed. The theory of the Hanle effect is reformulated in the representation of the irreducible tensors of the density matrix, and is generalized to derive the circular polarization profiles across the spectral line in terms of the intensity and direction of the prominence magnetic field. The circular polarization profile so deduced can be employed to obtain useful information which adds to that carried by the linear polarization observations. A non-linear least-squares algorithm is proposed to derive the measurement of the magnetic field from the observations, and a consistency check is suggested to test the adequacy of the theoretical model to describe the physics of the He I atomic excitation in prominences.On leave from: Astrophysical Observatory of Arcetri, Largo E. Fermi, 5, 50125 Firenze, Italy.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

The Imaging Vector Magnetograph at Haleakalā IV: Stokes Polarization Spectra in the Sodium D<Subscript>1</Subscript> 589.6 nm Spectral Line

The Imaging Vector Magnetograph (IVM) at the Mees Solar Observatory, Haleakalā, Maui, Hawai’i, obtained many years of vector magnetic-field data in the photospheric Fe i 630.25 nm line. In the latter period of its operation, the IVM was modified to allow routine observations in the chromospheric Na i D₁ line, as well as the Fe i line. We describe the sodium observational data in detail, including the data-reduction steps that differ from those employed for the Fe i 630.25 nm line, to obtain calibrated Stokes polarization spectra. We have performed a systematic comparison between the observational data and synthetic NLTE Na i D₁ Stokes spectra derived for a variety of solar-appropriate atmospheric and magnetic configurations. While the Na i D₁ Stokes polarization signals from the solar atmosphere are expected to be weak, they should generally be within the IVM capability. A comparison between synthetic spectra and observational data indicates that this is indeed the case.     

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.     

Monochromatic images in stokes parameters and the structure of magnetic fields in sunspots

Taking into account magneto-optical effects, we have obtained numerical solutions of the transfer equations for the Stokes parameters, calculated the linearly polarized intensity (U) and constructed its monochromatic images of unipolar sunspots. By comparison with the observational material of the vector magnetograph of the Marshall Space Flight Center, Huntsville (Alabama), we have found that the model of radial magnetic fields may give rise to U monochromatic images close to those observed. The same conclusion has been obtained previously by Landi Degl'Innocenti (1979), although his analysis was performed with the Milne-Eddington approximation instead of a detailed sunspot model. Moreover, we have shown that the model of spiral magnetic fields leads to results in contrast with observations.     

The NLTE Formation of Iron Lines Used in Solar Polarimetry

We study in some detail one-dimensional NLTE effects in solar Fei lines. The lines selected are frequently used in solar polarimetry, and also in studies of line asymmetries and for abundance determinations. Our model atom for Fei–Feii–Feiii is realistic: it takes account of multiplet structure and it includes over 200 bound–bound and bound–free transitions in detail. We use very efficient iterative methods for the self-consistent solution of the kinetic and radiative transfer equations (Auer, Fabiani Bendicho, and Trujillo Bueno, 1994). We have applied these fast methods of solution because they are suitable for the investigation of 2D and 3D NLTE transfer effects with multilevel atoms, which constitutes the next step of our ongoing research project on the iron line formation problem.     

Fe I line pairs with different magnetic sensitivity

We present a list of Fe I line pairs with different magnetic sensitivity which are suitable for measurement of solar magnetic fields using the Stokes V amplitude ratio. The list contains the spectral solar line data in the wavelength range λλ = 303–996 nm given by Gurtovenko and Kostyk.     

The derivation of vector magnetic fields from stokes profiles: Integral versus least squares fitting techniques

We compare the results of two distinct methods for deriving photospheric vector magnetic fields from the Zeeman effect as observed in the Fe I line at 6302.5 A at high spectral resolution (45 mA.). One method ignores magneto-optical effects but allows for an absolute determination of B from the integral properties of the Stokes profiles, under the assumption of weak field strength. The other method is an iterative least-squares fitting technique developed by Lites and Skumanich which fits the observed Stokes profiles to the profiles predicted by the Unno-Rachkovsky solution to the radiative transfer equation. We find empirically that for sunspot fields above 1500 gauss the two methods agree in derived azimuthal and inclination angles to within ± 20 degs., Furthermore, for such fields, the estimate of the line of sight field and the transverse field derived using the two methods agree to within ± 500 gauss. In weak field strength regions the integral method can be used with little error and computational load in the estimation of the line of sight field, the transverse field and the inclination angle but the disagreement in derived azimuthal angle is considerable ( ± 90 degs.).     

The determination of vector magnetic fields from stokes profiles

The application of Unno's (1956) solution of the transfer equation for polarized radiation to the determination of thevector magnetic field is investigated. An analysis procedure utilizing non-linear least squares techniques is developed that allows one to automate the reduction of measured spectral profiles of the Stokes parameters to determine the field angles, strength as well as other parameters. The method is applied to synthetic spectra generated using a model solar atmosphere and yields results of remarkably high accuracy. The influence of additional factors upon determination of the vector field are also considered. These factors include effects of asymmetric profiles, magneto-optical effects, magnetic field gradients, unresolved field elements, scattered light, and instrumental noise.The National Center for Atmospheric Research is sponsored by the National Science Foundation.Operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.     

Imaging Spectropolarimetry of Ti I 2231 nm in a Sunspot

Spectro-polarimetric observations at 2231 nm were made of NOAA 10008 near the west solar limb on 29 June 2002 using the National Solar Observatory McMath–Pierce Telescope at Kitt Peak and the California State University Northridge – National Solar Observatory infrared camera. Scans of spectra in both Stokes I and Stokes V were collected; the intensity spectra were processed to remove strong telluric absorption lines, and the Stokes V umbral spectra were corrected for instrumental polarization. The sunspot temperature is computed using the continuum contrast and umbral temperatures down to about 3700 K are observed. A strong Tii line at 2231.0 nm is used to probe the magnetic and velocity fields in the spot umbra and penumbra. Measurements of the Tii equivalent width versus plasma temperature in the sunspot agree with model predictions. Zeeman splitting measurements of the Stokes I and Stokes V profiles show magnetic fields up to 3300 G in the umbra, and a dependence of the magnetic field on the plasma temperature similar to that which was seen using Fei 1565 nm observations of the same spot two days earlier. The umbral Doppler velocity measurements are averaged in 16 azimuthal bins, and no radial flows are revealed to a limit of ±200 m s^–1. A Stokes V magnetogram shows a reversal of the line-of-sight magnetic component between the limb and disk center sides of the penumbra. Because the Tii line is weak in the penumbra, individual spectra are averaged in azimuthal bins over the entire penumbral radial extent. The averaged Stokes V spectra show a magnetic reversal as a function of sunspot azimuthal angle. The mean penumbral magnetic field as measured with the Stokes V Zeeman component splitting is 1400 G. Several weak spectral lines are observed in the sunspot and the variation of the equivalent width versus temperature for four lines is examined. If these lines are from molecules, it is possible that lines at 2230.67, 2230.77, and 2231.70 nm originate from OH, while the line at 2232.21 nm may originate from CN.     

NLTE formation of the resonance Ba II line λ 455.4 nm in the solar atmosphere

We consider the NLTE formation of the resonance Ba II line λ 455.4 nm in the solar spectrum for three one-dimensional and one three-dimensional hydrodynamic models of the quiet solar atmosphere. The sensitivity of the line to atomic parameters, microturbulent and macroturbulent velocities, as well as to oscillator strength and barium abundance uncertainties was examined. The wings of the barium line are shown to be most sensitive to the van der Waals broadening constant. Another important parameter is the barium abundance. Our NLTE estimate of the solar barium abundance (A _Ba = 2.16) derived with allowance made for the nonuniform solar atmosphere structure is in good agreement with earlier results. The influence of granular convective motions on the line profile shape was studied, and the profiles formed in granules and in intergranular lanes are shown to be asymmetric and differently shaped. We demonstrate that the theoretical profiles match well the observed ones when the NLTE effects and the granular structure are taken into account.     

Vertical gradients of the magnetic fields in active regions

In this paper, we describe the results of an investigation of magnetic field structures in two active regions. The photospheric magnetic fields were measured simultaneously in all three components with the Crimean vector magnetograph in the Fe i 5250 line. In our analysis, we compare the observed magnetic field with the potential field. The potential field vector was calculated according to the potential-field theory, and the H _z component was taken as a boundary condition. From these data vertical gradients are calculated from the condition div H = 0. Averaged gradients of both fields increase with the H _z field intensity and within the error limits they do not differ from one another for field strengths up to 1200 G. For larger H _z the potential field gradients become higher than those of the observed field. In large spots, observed field gradients are about two times less than those of the potential field. It is shown that this difference is connected with the observed field twisting.