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).     

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.     

Line formation in spherical media with partial frequency redistribution

The effects of partial redistribution of frequency on the formation of spectral lines in a static and spherically symmetric media have been investigated. The partial redistribution functionsR _I andR _II (Hummer, 1962) have been employed to calculate the lines for a two-level atom in non-LTE in a spherically symmetric medium with homogenous physical characteristics whose ratiosB/A (of outer to inner radii) are equal to 2 and 10. These results are compared with those formed in a plane-parallel medium withB/A=1. Two types of atmosphere are treated: (1) a pure scattering medium with =0 and =0, and (2) an atmosphere with a constant source of emission =10^–4 and =0, where is the probability per scatter that a photon will be destroyed by collisional de-excitation and is the ratioK _c/K _l of opacity due to continuous absorption per unit interval of frequency to that in the line. Lines formed in complete redistribution also have been calculated for the sake of comparison, and the total optical depth in all cases has been taken to be 10³ at the line centre.Vast differences have been found between the lines formed by complete and partial redistribution functions (which, for the sake of simplicity, we shall hereafter refer to as CRD and PRD, respectively). In the case of a purely scattering medium, a small amount of emission is observed in the wings for all cases of scattering functions in the spherical medium as a result of the combined effects of curvature and physical scattering. In the scattering medium, more photons are scattered into the cores of the lines by PRD than in the case of CRD. The lines formed in the medium with internal sources show emission in all cases with small absorption in the cores, except those lines formed by the angle-dependent PRD functions which again depend on the geometrical extension of the medium.     

Scattering of radiation by free electrons. The angle-averaged redistribution function

Elementary events of photon scattering by a chaotically moving monoenergetic electron gas are examined to obtain the redistribution function (RF). It is pointed out that in many problems and, in particular, the interaction of background radiation with the intergalactic plasmas of rich galactic clusters, it is sufficient to have the angle-averaged RF. Expressions are obtained for the averaged RF and its asymptotes which describe the redistribution function for low frequency radiation. The question of determining the frequencydependent absorption coefficient is discussed. __________ Translated from Astrofizika, Vol. 49, No. 3, pp. 475–485 (August 2006).     

A Monte Carlo study of polarization structures in the Thomson-scattered line radiation

Thomson scattering is often invoked to explain broad wing features that are seen in various objects including active galactic nuclei and symbiotic stars. Despite the wavelength-independent scattering cross-section of Thomson scattering, the line flux may exhibit wavelength-dependent linear degree of polarization, because various parts of emission wings are contributed by photons with different scattering numbers. Specifically, more scattered and hence more weakly polarized photons tend to fill the farther wing parts from the line centre, while the neighbourhood of the line centre is dominated by less-scattered photons with higher degree of polarization. Using a Monte Carlo technique, we investigate the polarization structure of Thomson-scattered line radiation. A detailed analysis of polarization structure formation is conducted by investigating the dependence of the polarization and profile width on the scattering number for various finite electron scattering slabs. Significantly varying degree of polarization is obtained when the scattering medium has Thomson optical depth  τ_Th≥ 1  . We present our high-resolution spectrum of the symbiotic star V1016 Cyg obtained with the Bohyunsan Optical Echelle Spectrograph (BOES) in order to fit the broad profile around Hα by electron scattering wings adopting an oblate spheroidal geometry with Thomson optical depth  τ_Th= 0.5  and electron temperature   T _e= 6.2 × 10⁴  K  . Local maxima in the linear degree of polarization of Thomson-scattered line radiation are expected to appear in the spectral regions characterized by the average scattering number ≃1.     

Partial frequency redistribution of polarized radiation

Resonance polarization, which is created by the scattering of an anisotropic radiation field in regions of zero or weak magnetic fields, is strongly dependent on the frequency redistribution taking place during the scatterings. Here we discuss the frequency redistribution matrix relevant to resonance lines, concentrating on linear polarization. First we analyze in detail the redistribution matrix in a zero magnetic field given by the theory of Omont, Smith and Cooper (1972), revisited by Domke and Hubeny (1988). We explain that the linear polarization maxima which may appear in the wings of the Stokes Q profiles of strong resonances lines such as the Ca I 4227 Å line are coherent frequency redistribution effects. Various approximate forms of the frequency redistribution matrix are also examined. For resonance polarization in a weak magnetic field, we suggest a new expression for the redistribution matrix which can be used at all line frequencies, and is consistent with the condition that the Hanle effect acts only in the line core.     

Elemental abundance analyses with DAO spectrograms — XVIII. The double-lined spectroscopic binary 46 Draconis

Equipartition magnetic fields can dramatically affect the polarization of radiation emerging from accretion disc atmospheres in active galactic nuclei. We extend our previous work on this subject by exploring the interaction between Faraday rotation and absorption opacity in local, plane-parallel atmospheres with parameters appropriate for accretion discs. Faraday rotation in pure scattering atmospheres acts to depolarize the radiation field by rotating the polarization planes of photons after last scattering. Absorption opacity in an unmagnetized atmosphere can increase or decrease the polarization compared to the pure scattering case, depending on the thermal source function gradient. Combining both Faraday rotation and absorption opacity, we find the following results. If absorption opacity is much larger than scattering opacity throughout the atmosphere, then Faraday rotation generally has only a small effect on the emerging polarization because of the small electron column density along a photon mean free path. However, if the absorption opacity is not too large and it acts alone to increase the polarization, then the effects of Faraday rotation can be enhanced over those in a pure scattering atmosphere. Finally, while Faraday rotation often depolarizes the radiation field, it can in some cases increase the polarization when the thermal source function does not rise too steeply with optical depth. We confirm the correctness of the analytic calculation by Silant'ev of the high magnetic field limit of the pure scattering atmosphere, which we incorrectly disputed in our previous paper.     

Determination of the single-scattering albedo of a dense rayleigh-scattering atmosphere with true absorption

A procedure is developed to determine the single-scattering albedo from polarization measurements of the angle-dependent intensity at two locations within, or on the boundaries of, a homogeneous finite atmosphere which scatters radiation according to Rayleigh's law with true absorption. The density of the atmosphere need not be known.     

Stochastic transfer of polarized radiation in finite cloudy atmospheric media with reflective boundaries

The problem of monoenergetic radiative transfer in a finite planar stochastic atmospheric medium with polarized (vector) Rayleigh scattering is proposed. The solution is presented for an arbitrary absorption and scattering cross sections. The extinction function of the medium is assumed to be a continuous random function of position, with fluctuations about the mean taken as Gaussian distributed. The joint probability distribution function of these Gaussian random variables is used to calculate the ensemble-averaged quantities, such as reflectivity and transmissivity, for an arbitrary correlation function. A modified Gaussian probability distribution function is also used to average the solution in order to exclude the probable negative values of the optical variable. Pomraning-Eddington approximation is used, at first, to obtain the deterministic analytical solution for both the total intensity and the difference function used to describe the polarized radiation. The problem is treated with specular reflecting boundaries and angular-dependent externally incident flux upon the medium from one side and with no flux from the other side. For the sake of comparison, two different forms of the weight function, which introduced to force the boundary conditions to be fulfilled, are used. Numerical results of the average reflectivity and average transmissivity are obtained for both Gaussian and modified Gaussian probability density functions at the different degrees of polarization.     

The continuous spectra of supernovae in the period immediately after eruption

We consider the problem of the influence of electron scattering on the continuous spectrum of an envelope. For the radiation flux out of the envelope we assume an expression (1) obtained by us previously [3]. Computations using formula (1) are carried out for two models of the envelope: gray andpurely hydrogen. As a result we find the values of the following quantities, which characterize the continuous spectrum: the color indices U-B, B-V, V-R, and V-I; the Balmer jump D, and the bolometric correction BC. The values of these quantities are given in tables as functions of the surface temperature To of the envelope and the parameter /, where is the coefficient of electron scattering and is the mean absorption coefficient. We draw conclusions on the role of electron scattering for each model of the envelope.Translated fromAstrofizika, Vol. 37, No. 3, 1994.     

The continuous absorption coefficient of a stellar atmosphere

It is pointed out that, when calculating the continuous absorption coefficient in a stellar atmosphere, it is advantageous to use the coefficient per particle of the most abundant element instead of the usual coefficient per gram of matter. The sources of continuous opacity considered are 1) absorption by H^-, HI, H₂^-, H ₂⁺, HeI, HeII, CI, CII, CIII, NI, NII, OI, OII, NaI, MgI, MgII, AlI, AlII, SiI, SiII, ClI, KI, CaII; and 2) Rayleigh scattering by HI, HeI, CI, NI, OI, H₂, and 3) Thomson scattering of free electrons. The calculations are illustrated by the results for a solar-type photosphere.     

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.     

Atmospheres and spectra of strongly magnetized neutron stars

We construct atmosphere models for strongly magnetized neutron stars with surface fields     and effective temperatures     . The atmospheres directly determine the characteristics of thermal emission from isolated neutron stars, including radio pulsars, soft gamma-ray repeaters, and anomalous X-ray pulsars. In our models, the atmosphere is composed of pure hydrogen or helium and is assumed to be fully ionized. The radiative opacities include free–free absorption and scattering by both electrons and ions computed for the two photon polarization modes in the magnetized electron–ion plasma. Since the radiation emerges from deep layers in the atmosphere with     , plasma effects can significantly modify the photon opacities by changing the properties of the polarization modes. In the case where the magnetic field and the surface normal are parallel, we solve the full, angle-dependent, coupled radiative transfer equations for both polarization modes. We also construct atmosphere models for general field orientations based on the diffusion approximation of the transport equations and compare the results with models based on full radiative transport. In general, the emergent thermal radiation exhibits significant deviation from blackbody, with harder spectra at high energies. The spectra also show a broad feature     around the ion cyclotron resonance     , where Z and A are the atomic charge and atomic mass of the ion, respectively; this feature is particularly pronounced when     . Detection of the resonance feature would provide a direct measurement of the surface magnetic fields on magnetars.     

Influence of relativistic effects and vacuum polarization on the transfer of gyroresonance radiation and the stability of the atmospheres of compact stars

We consider the transfer of radiation and calculate the force of its pressure in the electron gyroresonance line in the atmospheres of magnetic degenerate stars. We specify the atmospheric parameters for which an outflow of plasma is possible under radiation pressure in the cyclotron line. We show that the permittivity tensor of a mildly relativistic plasma in a strong magnetic field found by applying relativistic corrections to the cyclotron resonance condition and by taking into account the vacuum polarization and recoil effects during photon scattering should be used to obtain proper results. We have determined the real and imaginary parts of the refractive indices and the polarization coefficients for normal electromagnetic waves when scattering dominates over absorption. Relativistic effects, which change greatly the dispersion and resonant absorption of waves propagating almost perpendicular to the magnetic field, and vacuum polarization have been found to change qualitatively the gyroresonance radiation spectrum and pressure for a wide range of parameters of stellar magnetospheres.     

Depolarization of multiple scattered light in atmospheres due to anisotropy of small grains and molecules

Freely oriented small anisotropic grains and molecules depolarize radiation both in single scattering and in the process of multiple scattering. Especially large depolarization occurs for resonant scattering corresponding to the electron transitions between the energy levels with very different quantum numbers. The existence of light absorption also changes essentially the angular distribution and polarization of radiation, outgoing from an atmosphere. In the present paper we consider these effects in detail both for continuum radiation and for resonant lines. Because the term describing the depolarization deals with isotropic radiation, we consider the axially symmetric part of radiation. We derived the formulas for observed intensity and polarization using the invariance-principles both for continuum and resonant scattering. We confine ourselves to two problems—the diffuse reflection of the light beam from semi-infinite atmosphere, and the Milne problem.     

Polarization of intrinsic radiation of tidally distorted stars with electron-scattering atmospheres

Linear polarization of radiation emitted by tidally distorted stars as a function of the binary system phase is computed, taking into account true absorption and the scattering of light on free and bound electrons within hot stellar atmospheres. Computations are made both for the linear distribution of true sources across the atmospheres and for radiative-stable model atmospheres presented by Kurcuzet al. (1974) and Kurucz (1979). Polarization variability was investigated as a function of wavelength . In a number of cases, polarization variability was found to be at an observable level. The most marked variability was expected in the ultraviolet range adjacent to the boundaries of the spectral series for H and He. Near the Lyman limit of approximately =912 Å for stars with an effective temperatureT _eff35 000 K and near the ionization boundary for HeII 226 Å for stars withT _eff>35 000 K, the amplitude of polarization variability is greater than in the case of pure electron atmospheres, sometimes reaching the level of 0.5–1%. For fairly long waves where the limb-darkening coefficient falls below a certain critical valueu _cr0.5, the plane of polarization is found to be turned by 90° as compared to the case of a pure electron atmosphere. For limb-darkening coefficients far from the value ofu _cr; the form of the polarization phase curves, as well as dependence on the parameters of a binary system, remain approximately the same as those in the case of pure electron scattering.     

Radiation transfer through atmospheric aerosol media with anisotropic scattering

Radiation transfer in atmospheric aerosol media with general boundary conditions has been studied for anisotropic scattering. The considered aerosol medium assumed to have specular and diffused reflecting boundary surfaces and in the presence of internal source. The radiation transfer scattering parameters as single scattering albedo, asymmetry factor, scattering, absorption, extinction efficiencies and anisotropic scattering coefficient have been calculated using the Mie theory. The problem with general boundary conditions is solved in terms of the solution of source-free problem with simply boundary conditions. Pomraning-Eddington approximation is used to solve the source-free problem. For the sake of comparison, a weight function is introduced and used in two special forms. The calculated partial heat fluxes with the two methods are compared and showed good agreement. Some of our results are found in a good agreement with published data.     

Radiation transfer in a cylinder. III. Spectrum of basic integral equation

We present a method for computing the spectrum of the integral equation for radiation transfer in a cylinder. This method, as in the previous articles in this series, is based on a Hankel transformation applied to the equation. Calculating the spectrum then reduces to solving the equation for the auxiliary function for each eigenvalue separately. The corresponding eigenfunction is then found by an additional integration. We find asymptotic expressions for the eigenvalues and the eigenfunctions for a cylinder with a large optical radius when there is scattering in a spectral line, with complete redistribution over frequency when the absorption coefficient obeys a power law. We also derive equations determining the quantities entering into these expressions. For the simplest kernel of the equation all quantities can be expressed in terms of Bessel functions and roots of a transcendental equation.Translated from Astrofizika, Vol. 38, No. 1, pp. 75–88, January–March, 1995.     

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.     

Solution of the Milne Problem for a Magnetized Atmosphere

The Milne problem is solved numerically for a magnetized semi-infinite electron atmosphere in the case where the magnetic field is directed along the normal to the medium. The calculated angular distribution, degree of linear polarization, and positional angle of inclination of the plane of polarization of the emerging radiation are given in tables for a number of values of the Faraday rotation parameter and for degrees of intrinsic optical absorption, q=0, 0.2, and 0.4. It is assumed that the magnetic field B10⁶ G, so that scattering in the optical range is purely Thomson scattering.