Radiative transfer for polarized light: Equivalence between Stokes parameters and coherency matrix formalisms

Two formal solutions of the radiative transfer equation for polarized light have been proposed. One uses the Stokes parameters to describe the polarization, while the other uses the coherency matrix. It is shown in the present work that they are equivalent. Both can be used to compute response and contribution functions for the Stokes parameters and both require the solution of systems of differential equations with similar numbers of independent variables. New equations to solve the radiative transfer problem using the Stokes parameters formalism are presented. In addition, a computer code which synthesizes the Stokes profiles by means of these equations is described.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

A generalized theory for line formation in a homogeneous magnetic field

The formation of spectral lines in a homogeneous magnetic field has been studied. A new method for solving the transfer equations for polarized light has been found. Using this method, the Stokes parameters may be derived without any special assumptions regarding the model atmosphere. With the line formed by pure absorption, the expressions for the Stokes parameters may easily be adapted to numerical calculations. In order to illustrate the method, the line profile for the Zeeman triplet 5250 has been calculated using a photospheric model atmosphere.     

太阳大气中矢量磁场信息的推导

一种能从观测到的斯托克斯轮廓中提取太阳表面矢量磁场信息的方法在本文中提出，它利用斯托克斯轮廓非线心的极值点处相应参量对波长的导数为零这一数学事实，假设表面附近磁场矢量及热力学参量的变化梯度足够小以致所考虑的极值点的波长位置不随深度改变或此变化呆忽则使得偏振辐射围方程组得到极大的简化；再应用数值方法从此简化方程组中解出辐射表面附近的磁场矢量参数。通过拟合理论轮廓表明该法确实可以得到表面近似的磁场矢量     

A study of the magnetically sensitive line FeI λ5324.191

For the spectral line FeI λ5324.191, we solve numerically the Unno equations to get the profiles of the Stokes parameters, and found that it has a large magnetic intensification and is not excessively temperature-sensitive. We conclude that it is proper to use this line as the working line of video-magnetic graphs.     

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.     

Mangeto-optical effects in sunspots and their vector magnetic field

The system of transfer equations of the four Stokes parameters I, Q, U, V under the action of the magneto-optical effect (i.e. the Unno-Beckers equations) are numerically solved in this paper for the magneto-sensitive lines FeI λλ 6302.499 and 5324.191 using an appropriate sunspot model. The errors in the expressions for the coefficients _r and _W in Beckers' paper [2] have been corrected for. From the results of calculations, features of the profiles of the Stokes parameters dependent on the magnetic vector have been isolated. Our computations also show that the magneto-optical effect should be taken into consideration in the measurement of the vector magnetic fields.

In the fourth section of this paper we have established a simple and convenient method for obtaining-information on the magnetic vector (including the field strength B, its inclination to the line of sight γ and its azimuth χ) from the profiles of the Stokes parameters. It consists of three steps: (1) The value of B is determined from the distance of the highest point in the V-profile from the central line. (2) γ is then found from V_max, i.e maximum value of V. (3) Lastly, the angle χ is found from Q₀, i.e. the value of Q at line centre.     

Observation of Hα line impact polarization in solar flares

We present the results of studying the impact linear polarization of 32 solar flares of X-ray classes C, M, and X (two flares) observed with the Large Solar Vacuum Telescope. It has turned out that there is evidence for impact polarization only in 13 of them. The newly obtained data have confirmed that the linear Stokes parameters are predominantly 2–7%, while the spatial sizes of flaring points with nonzero Stokes parameters are small (1″-2″). Two features of the manifestation of impact polarization in flares revealed by these studies are of greatest interest: (1) at the two foot points of a single flare loop or an arcade of loops, both the Hα intensity profiles and the Stokes profiles differ in behavior; (2) based on the Hα line, we have found for the first time that the sign of the Stokes parameters changes not only across the flare ribbon but also with depth of the chromosphere.     

Alfven waves and magneto-sensitive lines

In this paper the equations of transfer of Stokes parameters are solved under the hypothesis of the propogation of Alfven waves. The effects of Alfven waves on the magneto-sensitive line FeIλ6302. 499 in the spectra of sunspots are calculated. Besides, some methods have been proposed for the verification of theoretical computations with observational material and for estimating the energy flux of Alfven waves.     

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.     

A Simplified Kinetic Element Formulation for the Rotation of a Perturbed Mass-Asymmetric Rigid Body

Euler's equations, describing the rotation of an arbitrarily torqued mass asymmetric rigid body, are scaled using linear transformations that lead to a simplified set of first order ordinary differential equations without the explicit appearance of the principal moments of inertia. These scaled differential equations provide trivial access to an analytical solution and two constants of integration for the case of torque-free motion. Two additional representations for the third constant of integration are chosen to complete two new kinetic element sets that describe an osculating solution using the variation of parameters. The elements' physical representations are amplitudes and either angular displacement or initial time constant in the torque-free solution. These new kinetic elements lead to a considerably simplified variation of parameters solution to Euler's equations. The resulting variational equations are quite compact. To investigate error propagation behaviour of these new variational formulations in computer simulations, they are compared to the unmodified equations without kinematic coupling but under the influence of simulated gravity-gradient torques.     

The polarization of continuum radiation in sunspots

Expressions are derived for the Stokes parameters of light scattered by a layer of free electrons and hydrogen atoms in a sunspot. A physically reasonable sunspot model was found so that the direction of the calculated linear polarization agrees reasonably with observations. The magnitude of the calculated values of the linear polarization agrees generally with values observed in the continuum at 5830 Å. Circular polarization in the continuum also accompanies electron scattering in spot regions; however for commonly accepted values of the longitudinal magnetic field, the predicted circular polarization is much smaller than observed.     

On the error analyses of polarization measurements of the white-light coronagraph aboard ASO-S

The Advanced Space-based Solar Observatory(ASO-S) mission aims to explore the two most spectacular eruptions on the Sun: solar flares and coronal mass ejections(CMEs), and their magnetism.For the study of CMEs, the payload Lyman-alpha Solar Telescope(LST) has been proposed. It includes a traditional white-light coronagraph and a Lyman-alpha coronagraph which opens a new window to CME observations. Polarization measurements taken by white-light coronagraphs are crucial for deriving fundamental physical parameters of CMEs. To make such measurements, there are two options for a Stokes polarimeter which have been applied by existing white-light coronagraphs for space missions. One uses a single or triple linear polarizer, the other involves both a half-wave plate and a linear polarizer. We find that the former option is subject to less uncertainty in the derived Stokes vector propagating from detector noise.The latter option involves two plates which are prone to internal reflections and may have a reduced transmission factor. Therefore, the former option is adopted as our Stokes polarimeter scheme for LST. Based on the parameters of the intended linear polarizer(s) colorPol provided by CODIXX and the half-wave plate 2-APW-L2-012 C by Altechna, it is further shown that the imperfect maximum transmittance of the polarizer significantly increases the variance amplification of Stokes vector by at least about 50% when compared with the ideal case. The relative errors of Stokes vector caused by the imperfection of colorPol polarizer and the uncertainty due to the polarizer assembly in the telescope are estimated to be about 5%. Among the considered parameters, we find that the dominant error comes from the uncertainty in the maximum transmittance of the polarizer.     

自转对恒星物理量的影响

自转本身是恒星的基本物理量之一，同时影响恒星的其它许多物物量，包括对恒星半径、形状和体积的影响；对光度、包指数、谱线轮廓、等值宽度、偏振度等观测量的影响；对引力加速度、元素丰度的影响；对恒星内部结构，如中心压力、温度、密度的影响。这些将会影响到恒星在Ｈ—Ｒ图上的位置，演化路径以及年龄和寿命等。因此，无论是理论模型还是实测研究，对自转快的恒星应该进行自转影响的修正。     

Radiation transfer through a model sunspot

We present a cylindrically symmetric model for a sunspot atmosphere using the similarity principle of Schlüter and Temesvary for the magnetic field configuration. The equations of magnetostatic equilibrium are used, augmented by a radial Evershed flow. The LTE radiative transfer equations for the Stokes vector were solved under a variety of conditions for a ray emerging from a typical penumbral point. The contribution from isolated lines to the broadband circular polarization in sunspot penumbrae is evaluated using a more realistic model sunspot atmosphere than has hitherto been considered. Results indicate that the inclusion of a velocity field along B is unable to give a net circular polarization of sufficient magnitude, although the variation with the angle between the line-of-sight and B is in qualitative agreement with observations. The corresponding results for the net linear polarization are satisfactory.     

恒星线性非绝热非径向脉动问题的退耦化处理方法

本文发展了一种解恒星线性非绝热非径向脉动问题的退耦化方法。这个方法把非绝热非径向脉动问题的六阶线性微分方程,分解为由一个代数方程联系起来的一个四阶线性微分方程和另一个二阶线性微分方程进行数值求解。这样的一个退耦处理,有利于克服以前在数值解这类问题时常常遇到的收敛域小和收敛速度慢等困难,并且为数值解方程时所采用的Henyey方法提供了一个自然和方便的初始猜测解。     

Magneto-optical effect and structure of sunspot magnetic field

On the basis of a numerical soluton of the equations of transfer of Stokes parameters including the magneto-optical effect, we calculated monochromatic images of unipolar sunspots in the linear polarization signal and compared them with the observed data obtained at Marshall Space Flight Center, U.S.A. The results show that our radial fan-shaped model of sunspot magnetic field give images that are similar to the observed ones while a spiral model gives rather different pictures. We can therefore regard the former model as being the more realistic.     

The vector approach to the problem of physical libration of the Moon. II. The nonlinear problem

By the new vector method in a nonlinear setting, a physical libration of the Moon is studied. Using the decomposition method on small parameters we derive the closed system of nine differential equations with terms of the first and second order of smallness. The conclusion is drawn that in the nonlinear case a connection between the librations in a longitude and latitude, though feeble, nevertheless exists; therefore, the physical libration already is impossible to subdivide into independent from each other forms of oscillations, as usually can be done. In the linear approach, ten characteristic frequencies and two special invariants of the problem are found. It is proved that, taking into account nonlinear terms, the invariants are periodic functions of time. Therefore, the stationary solution with zero frequency, formally supposing in the linear theory a resonance, in the nonlinear approach gains only small (proportional to e) periodic oscillations. Near to zero frequency of a resonance there is no, and solution of the nonlinear equations of physical libration is stable. The given nonlinear solution slightly modifies the previously unknown conical precession of the Moon’s spin axis. The character of nonlinear solutions near the basic forcing frequency Ω₁, where in the linear approach there are beats, is carefully studied. The average method on fast variables is obtained by the linear system of differential equations with almost periodic coefficients, which describe the evolution of these coefficients in a nonlinear problem. From this follows that the nonlinear components only slightly modify the specified beats; the interior period T ≈ 16.53 days appears 411 times less than the exterior one T ≈ 18.61 Julian years. In particular, with such a period the angle between ecliptic plane and Moon orbit plane also varies. Resonances, on which other researches earlier insisted, are not discovered. As a whole, the nonlinear analysis essentially improves and supplements a linear picture of the physical libration.     

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.     

On the observation of linear polarization of solar microwave bursts

It is known that mode coupling may occur in quasi-transverse magnetic field regions of the solar corona, which produces linear polarization at microwave frequencies. A microwave polarimeter measuring all 4 Stokes parameters at 8.918 GHz simultaneously at three different highfrequency bandwidths (40 kHz, 400 kHz and 5 MHz) has been developed in order to observe the linear component and its Faraday rotation. The respective minimum detectable changes of the Stokes parameters I, Q, U and V are 9, 3 and 1 solar flux unit at an integration time of 1 s. For burst intensities greater than 300 solar flux units, the minimum detectable degree of linear and circular polarization is 1 %–3 %, depending on the bandwidth. Observations of 68 bursts showed that most of the bursts were circularly polarized. No linear polarization could be found within the limits of accuracy of our polarimeter. Two possible explanations for this result are discussed. The possibility of mode coupling however cannot be excluded from these first observations.