Scattering physics

The theory of polarized scattering in a stellar atmosphere is formulated, first within the framework of classical physics, then in terms of quantum mechanics. The expression for the redistribution matrix that describes partial redistribution in polarization and frequency is derived for the general case when the magnetic field is of arbitrary strength. The special cases of weak fields (the Hanle limit) and zero fields (non-magnetic scattering) are discussed. Observational examples of spectral signatures in linear polarization are presented, which show effects of hyperfine structure, interference between fine structure components, and molecular scattering.     

The transfer of polarized radiation in spectral lines: Solar-type stellar atmospheres

The polarization structure in several spectral lines in solar type stars is computed using the method described by McKenna (1981, 1984a). The frequency redistribution function used for these calculations is a linear combination ofR _II andR _III. The line profiles and polarization structures have been computed for several weak solar resonance lines includingKi 7664 Å, Sri 4607 Å, Baii 4554 Å, for various polar angles along the stellar disk. Both the line profiles and polarization structures as well as the center to limb behavior of the line center polarization agree well with observations.The somewhat stronger resonance line Cai 4227 Å shows a different polarization structure when compared to the weaker solar resonance lines. It is found that for strong resonance lines the proper redistribution function to be used is a linear combination ofR _III andR _v (see McKenna, 1981, 1984b; Heinzel, 1981). The major reason for this is that for strong resonance lines both the upper and lower levels are broadened by collisions. This violates the assumptions upon which the redistribution functionsR _II andR _III are based.     

Line formation in magnetic fields

Hyder (1968) has suggested that longitudinal magnetograph measurements of weak magnetic fields in prominences have underestimated field strengths as the result of zero-field levelcrossing interference (the Hanle effect). Hyder (1968) also suggested that resonance polarization effects have sometimes led to errors in measurements of the transverse component of magnetic fields. Stenflo (1969) has pointed out some errors in Hyder's paper, while contending that the Hanle effect is implicitly included in current theories of line formation in the presence of Zeeman splitting.In the present Note these questions are re-examined using the results of a density matrix treatment of absorption, emission, and scattering processes. The basic conclusions are as follows: (1) Longitudinal magnetograph measurements using optically thin lines are not influenced by the Hanle effect. (2) Although present theories of line formation in magnetic fields do not include the Hanle effect, this omission is generally unimportant for lines formed in the photosphere and lower chromosphere due to rapid collisional depolarization of atomic levels. (3) For the same reason, other resonance polarization effects are probably too small to cause significant errors in magnetograph measurements of all but the very weakest magnetic fields, when photospheric and lower chromospheric lines are used. (4) By contrast, the general phenomenon of atomic level polarization is quite important in most prominences. As emphasized by Hyder, extreme care must be used in selecting lines for magnetograph studies of solar magnetic fields.     

Polarization of resonance lines formed in extenned spherical atmospheres

The problem of polarization of the resonance lines formed in extended spherical atmospheres is studied in detail. In this paper, the atmosphere is assumed to be at rest. The basic problem of resonance line polarization in spherical atmospheres as compared to the conventionally used plane-parallel atmospheres, is studied in Nagendra (1988). Our main interest in this paper is to understand the behaviour of polarized radiation fields in extended model spherical atmospheres so that some constraints can be placed on the model parameters in the modelling work conected with observations of polarization across resonance lines. A comparison of polarized lines formed under three kinds of line-scattering mechanisms is also made. They are CS=coherent scatteirng, CRD=complete redistribution, and PRD=partial frequency redistribution which, in the increasing order of generality, provide a good approximation in the two-level atom approach, to the resonance line polarization. The dependence of polarization on the opacity laws, extendedness and on optical depth is studied in detail. The distribution of line intensity and polarization across the visible disk of an extended model stellar atmosphere is studied, in view of the possible disk-resolved observations in future, of the extended atmospheres of the stars.     

Predictions on the application of the Hanle effect to map the surface magnetic field of Jupiter

In this paper we evaluate the possibility of detecting, for the first time, the surface magnetic field of Jupiter (∼1 bar level) by observing the change of linear polarization induced by the Hanle effect on the H Lyman-alpha (Lyα) emission line of the planet. We find that, indeed, the Hanle effect, which results from the interaction between a local magnetic field and the atomic polarization induced by absorption of anisotropic radiation, is sensitive to relatively weak values of the strength of the magnetic fields expected on planets. First, we show that for the Lyα emission backscattered by atomic H in the presence of a magnetic field, the Hanle effect is polarizing. This new result is in total contrast to the depolarizing effect predicted and observed for emission lines scattered at right angles in solar prominences. Additionally, to estimate the polarization rate for the case of Jupiter, we have considered three magnetic field models: a dipole field for reference, an O₄ based model [Connerney, J.E.P., 1981. The magnetic field of Jupiter—A generalized inverse approach. J. Geophys. Res. 86, 7679-7693], and finally, an O₆ based model [Khurana, K.K., 1997. Euler potential models of Jupiter's magnetospheric field. J. Geophys. Res. 102, 11295-11306]. In all models, we show that for the jovian backscattered Lyα line, the Hanle effect does enhance the Lyα linear polarization; the polarization rate may exceed 2% at specific regions of the jovian disc, making detection possible either remotely or from an orbiter around Jupiter. In general, depending on the instrumental sensitivity and the observing strategy used, we show that accurate mapping of the linear polarization rate at the planetary surface (thermosphere) or off-disc (corona) may provide a rather accurate estimate of the jovian total magnetic field strength on large area scales.     

Parametric generation of acoustic-gravity waves by Alfvén waves in the solar atmosphere

Based on a plane-parallel isothermal model solar atmosphere permeated by a uniform magnetic field directed against the action of gravity, we investigate the parametric generation of acoustic-gravity disturbances by Alfvén waves propagating along the corresponding field lines. We established that for a weak linear coupling of Alfvén waves, the nonlinear interaction of Alfvén waves propagating in opposite directions (rather than in the same direction) is the predominant generation mechanism of acoustic-gravity disturbances at the difference frequency. In this case, no acoustic flow (wind) was found to emerge at a zero difference frequency in the acoustic-gravity field.     

Line formation in microturbulent magnetic fields

The formation of Zeeman lines in Gaussian microturbulent magnetic fields is considered assuming LTE. General formulae are derived for the local mean values of the transfer matrix elements. The cases of one-dimensional (longitudinal), isotropic, and two-dimensional (transversal) magnetic microturbulence are studied in some detail. Asymptotic formulae are given for small mean as well as for small microturbulent magnetic fields. Characteristic effects of magnetic microturbulence on the transfer coefficients are: (i) the broadening of the frequency contours, although only for the case of longitudinal Zeeman effect and longitudinal magnetic microturbulence this effect can be described analogous to Doppler broadening, (ii) the appearance of a pseudo-Zeeman structure for nonlongitudinal magnetic microturbulence, (iii) the reduction of maximal values of circular polarization, and (iv) the appearance of characteristic linear polarization effects due to the anisotropy of the magnetic microturbulence.Line contours and polarization of Zeeman triplets are computed for Milne-Edddington atmospheres. It is shown that magnetic intensification due to microturbulent magnetic fields may be much more efficient than that due to regular fields. The gravity center of a Zeeman line observed in circularly polarized light remains a reasonable measure of the line of sight component of the mean magnetic field for a line strength47-1. For saturated lines, the gravity center distance depends significantly on the magnetic microturbulence and its anisotropy. The influence of magnetic microturbulence on the ratio of longitudinal field magnetographic signals shows that unique conclusions about the magnetic microstructure can be drawn from the line ratio measurement only in combination with further spectroscopic data or physical reasoning.     

High-precision magnetic field measurements of Ap and Bp stars

In this paper we describe a new approach for measuring the mean longitudinal magnetic field and net linear polarization of Ap and Bp stars. As was demonstrated by Wade et al., least-squares deconvolution (LSD; Donati et al.) provides a powerful technique for detecting weak Stokes V , Q and U Zeeman signatures in stellar spectral lines. These signatures have the potential to apply strong new constraints to models of stellar magnetic field structure. Here we point out two important uses of LSD Stokes profiles. First, they can provide very precise determinations of the mean longitudinal magnetic field. In particular, this method allows one frequently to obtain 1 σ error bars better than 50 G, and smaller than 20 G in some cases. This method is applicable to both broad- and sharp-lined stars, with both weak and strong magnetic fields, and effectively redefines the quality standard of longitudinal field determinations. Secondly, LSD profiles can in some cases provide a measure of the net linear polarization, a quantity analogous to the broad-band linear polarization recently used to derive detailed magnetic field models for a few stars (e.g. Leroy et al.). In this paper we report new high-precision measurements of the longitudinal fields of 14 magnetic Ap/Bp stars, as well as net linear polarization measurements for four of these stars, derived from LSD profiles.     

Polarimetry in the Mg ii h and k lines

The Ultraviolet Spectrometer and Polarimeter (UVSP) on the SMM satellite has been used to record the linear polarization profile across the Mgii h and k lines, including its center-to-limb variation. Linear polarization with an orientation of the electric vector parallel to the solar limb is detected in the line wings on the short wavelength side of the k line and on the long wavelength side of the h line, in agreement with theoretical predictions of Auer et al. (1980). The predicted negative polarization (electric vector perpendicular to the limb) between the h and k lines is however not confirmed by the observations. Instead values close to zero are indicated there, although the statistical significance of the results is marginal.We have examined possible explanations of such a discrepancy between theory and observations. After having rejected other alternatives (e.g., opacity effects, different continuum polarization, or deviations from a plane-parallel stratification), it is suggested that the solution may be found in a treatment of partial redistribution of the polarized radiation with the quantum-mechanical interference between the two scattering transitions being included as an integral part of the redistribution problem.     

Stokes II — A new polarimeter for solar observations

A Stokes polarimeter has been built at the High Altitude Observatory to obtain line profiles in both linear and circular polarization in solar spectral lines. These measurements are interpreted using the theory of radiative transfer in the presence of a magnetic field to obtain vector magnetic fields on the solar disk and using the theory of resonance scattering and the Hanle effect to obtain vector magnetic fields in prominences. The polarimeter operates on the Sacramento Peak Observatory 40 cm coronagraph. It is an extensively modified and improved version of an earlier instrument.Polarization modulation is achieved by two KD^*P Pockels cells at the coronagraph prime focus and demodulation is by a microprocessor. The instrument control and data handling is done by a minicomputer. Silicon photodiode 128 element line array detectors have replaced the two photomultipliers used on the earlier instrument. This gives a speed increase of a factor of 50.A polarization scrambler provides a chop to a reference beam of unpolarized light by time scrambling the polarization of the solar beam. This device improves sensitivity to polarizations less than 0.01%. The polarization measurements are photon noise limited in most cases. This noise is 0.1% for a typical three second observation which is about one gauss on the longitudinal field and 10 gauss on the transverse field.The National Center for Atmospheric Research is sponsored by The National Science Foundation.     

Magnetoactive lines in the medium with the velocity gradient

The magnetic splitting peculiarities of the absorption lines in the sunspot spectrum are considered. The most common and typical of them is breaking of all Stokes parameter symmetry in regard to the line center. The possible reason of this effect is the macroscopic gas motion with inhomogeneous velocity. Computed contours are given for the line Fe i λ5250 Å with various combinations of magnetic and velocity fields. Magneto-optical effects within the line which are connected with the magnetic and velocity field inhomogeneity are discussed. The observation results are discussed for longitudinal magnetic field zero lines. These observations were carried out for the sunspot and photosphere in two spectral lines Fe i λλ 5250 and 5233 Å. In the sunspot the regular displacement of one zero line with respect to the other zero line takes place whereas in the photosphere the displacements are random. The possible reason of the regular displacement is the change of the magnetic field direction in the different optical layers wherein corresponding spectral lines are formed effectively. The connection between the zero line displacement of a longitudinal magnetic field and the crossover effect is discussed. The computation results are given which agree with observations and illustrate the above-mentioned relationship. The influence of the Stokes parameter asymmetry on the measured magnetic field signals is considered.     

Polarized line formation by resonance scattering: Lorentz profile

Multiple resonance scattering of spectral line radiation is examined in atmospheres with uniformly distributed sources of unpolarized radiation. It is assumed that the profile of the absorption coefficient is lorentzian and that scattering involves complete frequency redistribution. The polarization characteristics of the emerging radiation are determined by iterative solution of a nonlinear Ambartsumyan-Chandrasekhar matrix integral equation. In particular, it is found that for pure scattering the maximum polarization at the limb of the disk is 5.421%. The polarization characteristics of the emerging radiation are compared for three different absorption profiles: Lorentz, Doppler, and rectangular (monochromatic radiation). __________ Translated from Astrofizika, Vol. 50, No. 2, pp. 199–217 (May 2007).     

Magnetic fields of active galactic nuclei and quasars with regions of polarized broad Hα lines

Estimates of magnetic fields for a number of active galactic nuclei are presented. These estimates are based on the observed polarization degrees and position angles of broad Hα lines and in the nearby continuum and on asymptotic analytical formulas for the Stokes parameters of the radiation emerging from a magnetized accretion disk (the Milne problem in a magnetized atmosphere). The characteristic observed feature of the wavelength dependence of the polarization degree inside the line—a minimum at the center and a fast increase of the position angle from one wing to another—can be explained by the superposition of resonance emission from two or more clouds located in the right (Keplerian velocity directed away from the observer) and left (Keplerian velocity directed toward the observer) parts of the orbit in the rotating magnetized accretion disk. The main component in our mechanism is the azimuthal magnetic field in the disk. The presence of a magnetic field perpendicular to the disk plane (which is usually weaker than the azimuthal field) results in the asymmetry of the distribution of the polarization degree and position angle inside the line. The inferred magnetic field strengths at the galactocentric distances where broad lines are emitted can be used to estimate the magnetic fields in the region of the centermost stable orbit and at the horizon of the central black hole, using the power-law dependence of the magnetic field strength corresponding to the standard model of the accretion disk.     

Generation mechanism of Pc3 magnetic pulsations at very low latitudes

Polarization properties of Pc3 magnetic pulsations at very low latitudes cannot be explained by existing theories which are based on the field line resonance model, because magnetic field lines at ¦Φ¦ < 22° are almost entirely in the ionosphere. In order to interpret Pc3 polarization characteristics observed at very low latitudes (¦Φ¦ < 20°), I would like to propose a possible, new qualitative model in which two superimposed ionospheric eddy currents, oscillating with slight differences in frequency in the Pc3 range and in azimuthal wave number, move azimuthally at very low latitudes. The equatorial ionospheric Pedersen eddy currents are believed to be predominantly caused by inductive electric fields of compressional Pc3 source waves which may possibly arrive in the equatorial ionosphere from the outer magnetosphere.     

An interpretation of the broad-band circular polarization of sunspots

The broad-band circular polarization of sunspots is discussed on the basis of the observations made in the Okayama Astrophysical Observatory. The observation with the spectrograph proves that it is the integrated polarization of spectral lines in the observed spectral range. A velocity gradient in the line-of-sight can produce this integrated polarization due to the differential saturation between Zeeman components of magnetically sensitive lines. The observed degree of polarization and its spatial distribution in sunspots is explained when we introduce a differentially twisted magnetic field in addition to the velocity gradient. The differential twist has the azimuth rotation of the magnetic field along the line-of-sight and generates the circular polarization from the linear polarization due to the magneto-optical effect. The required azimuth rotation is reasonable and amounts at most to 30°. The required velocity gradient is compatible with the line asymmetry and its spatial distribution observed in sunspots. The observed polarity rule leads to the conclusion that the sunspot magnetic field has the differential twist with the right-handed azimuth rotation relative to the direction of the main magnetic field, without regard to the magnetic polarity and to the solar cycle. The twist itself is left-handed under the photosphere, when the sunspot is assumed to be a unwinding emerging magnetic field.     

Some issues in diagnostics of solar chromospheric magnetic fields with the Mg b2 line

Up to now, exact measurements of chromospheric magnetic fields have not been as successful as those done in the photosphere. We are currently engaging in diagnostics of chromospheric magnetic fields with the Mg b2 line by employing the Multi-Channel Solar Telescope at Huairou Solar Observing Station. Therefore, how to improve accuracy in the measurement is the main issue of our present study. To this end, we first study linear calibration coefficients for longitudinal and transverse components of chromospheric fields, which vary with wavelength, in the case of a weak field assumption. Then the polarization crosstalk introduced by instruments is analyzed in detail with two numerical simulation methods. Comparisons of the po- larization signals between cases with and without correction are presented. The result indicates that polarization accuracy is greatly improved after crosstalk correction.     

Some issues in diagnostics of solar chromospheric magnetic fields with the Mg b_2 line

Up to now,exact measurements of chromospheric magnetic fields have not been as successful as those done in the photosphere.We are currently engaging in diagnostics of chromospheric magnetic fields with the Mg b2line by employing the Multi-Channel Solar Telescope at Huairou Solar Observing Station.Therefore,how to improve accuracy in the measurement is the main issue of our present study.To this end,we first study linear calibration coefficients for longitudinal and transverse components of chromospheric fields,which vary with wavelength,in the case of a weak field assumption.Then the polarization crosstalk introduced by instruments is analyzed in detail with two numerical simulation methods.Comparisons of the polarization signals between cases with and without correction are presented.The result indicates that polarization accuracy is greatly improved after crosstalk correction.     

Polarization of emerging resonance radiation: model profiles

The limiting polarization of a resonance line is examined for standard radiative transfer of polarized radiation in a semi-infinite scattering atmosphere with complete frequency redistribution. Two families of profiles of the line absorption coefficient, which are generalizations of Lorentz and Doppler profiles, are examined. It is shown that for both families this parameter approaches the Sobolev-Chandrasekhar limit when the fraction of absorption within the frequency interval (expressed in appropriate units) from −1 to 1 relative to the total absorption in the line approaches unity.     

A diagnostic method for probing the possible twist of magnetic field lines in sunspots

In order to study the three-dimensional structure of sunspot magnetic fields it is necessary to determine whether the field lines are twisted, i.e., if the azimuthal angle of transverse field changes with depth. For this purpose we propose the following method. At a fixed point in a spot, and in a certain wavelength interval of a magnetic-sensitive spectral line, one may measure the two Stokes parameters Q and U and then calculate the azimuthal angle of the polarization plane. If the wavelength interval of observation is moved successively from the line center to a wing, one may draw the azimuth diagram by the method proposed by Makita (1986) and refined by us (Ye Shi-hui and Jin Jie-hai, 1987). According to our theoretical calculations, described in this paper, if sunspot field lines are sufficiently strongly twisted, the curve on this diagram contains loop structures. If the twist is rather weak, the curve is approximately semi-circular. From the direction in which the curve winds (clockwise or counterclockwise) one may infer whether the magnetic field is twisting in one direction or in the opposite. In the case of no twist at all, the curve is comparatively simple and similar to a parabola.When the sensitivity of observational data is high enough, our method can also be applied to regions of weak magnetic fields outside sunspots.This work has been supported by the National Natural Science Foundation of China under grant No. 9187006-01.     

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.