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.     

Magellanic Cloud WC/WO Wolf–Rayet stars – II. Colliding winds in binaries

A search for evidence of colliding winds is undertaken among the four certain Magellanic Cloud WC/WO spectroscopic binaries found in the companion Paper I, as well as among two Galactic WC/WO binaries of very similar subtype. Two methods of analysis, which allow the determination of orbital inclination and parameters relating to the shock cone from spectroscopic studies of colliding winds, are attempted. In the first method, Lührs' spectroscopic model is fitted to the moderately strong C  iii 5696-Å excess line emission arising in the shock cone for the stars Br22 and WR 9. The four other systems show only very weak C  iii 5696-Å emission. Lührs' model follows well the mean displacement of the line in velocity space, but is unable to reproduce details in the line profile and fails to give a reliable estimate of the orbital inclination. In the second method, an alternative attempt is also made to fit the variation of more global quantities, full width at half-maximum and radial velocity of the excess emission, with phase. This method also gives satisfactory results in a qualitative way, but shows numerical degeneracy with orbital inclination. Colliding wind effects on the very strong C  iv 5808-Å Wolf–Rayet emission line, present in all six binaries, are also found to behave qualitatively as expected. After allowing for line enhancement in colliding wind binaries, it now appears that all Magellanic Cloud WC/WO stars occupy a very narrow range in spectral subclass: WC4/WO3.     

Ionospheric corrections via PIM and real-time data

We describe a method for removing ionospheric effects from single-frequency radio data a posteriori. This method is based on a theoretical climatological model developed by the USAF, which returns along the line of sight to the source. Together with a model of , ionospheric delay and Faraday rotation values ensue. If contemporaneous ionospheric data – GPS TEC observations or ionosonde profiles – exist, they can be incorporated to update the modeled n_e.     

Semi-empirical solar line blanketing

Statistical data for line blanketing in the solar atmosphere have been extracted from the Revised Rowland Table. The data are corrected for apparent observational incompleteness due to line crowding and blending and are placed in a convenient form for calculations. An attempt is made to interpret the data in terms of schematic representations of the energy levels and distribution of gf-values for the important absorbers. The analysis leads to predictions in good agreement with the forms of the statistical data and substantiates the assumption of observational incompleteness in the original data. The analysis suggests that the solar data may provide useful guidance in the blanketing of stellar atmospheres sufficiently close to solar type.A blanketing method is presented which makes use of a uniformly-smeared-line (single picket) procedure and which assumes LTE and a weak-line curve of growth. The blanketing coefficient is obtained in a simple manner from the corrected statistical data and is both depth and wavelength dependent. An alternative method utilizing a four-picket scheme and average absorption coefficient profiles is suggested. Comparison of solar models computed by the two methods shows good agreement and gives support for the smeared-line procedure. Comparison of the solar model with a non-LTE blanketed model of Athay also shows reasonable agreement.Publications of the Goethe Link Observatory, Indiana University, No. 110.     

On the Molecular Line Absorption in the Li I Doublet 6708 Å Region for Evolved Star Atmospheres

Molecular line lists from latest molecular data bases have been analysed for the absorption in the Li I doublet line 6708 Å region. Their contribution to the line opacity has been estimated by the synthetic spectra method with Kurucz late model atmospheres for various temperatures and log g values.     

Formation of fourier phase shifts in the solar Nii 6768 Å line

A formalism is developed to understand better how Doppler shifts of spectrum lines as inferred from phase shifts in the Fourier transforms of line profiles are related to the underlying velocity structures which they are intended to measure. With a standard model atmosphere and a simplified, quasi-LTE treatment of line formation, the formalism is applied to the Nii 6768 Å line which has been selected for use with a network of imaging interferometers under development by the Global Oscillations Network Group (GONG) for research in helioseismology. Fourier phase shifts are found to be a remarkably linear measure of velocity even in the presence of gradients and unresolved lateral variations in the assumed velocity field. An assumed outward increase in amplitude of a model oscillatory velocity is noticeably reflected in the center-to-limb behavior of the simulated velocity measure, and a sample model of solar granulation is found to have a strong influence on the formation of the Fourier phase.     

Magnetic configurations of the fast CP rotators HD3360, HD4778, HD5737, HD112413, and HD215441

The method of “magnetic charges” is used to model magnetic fields of five rapidly rotating stars (P < 25^d) and to find the basic parameters of their magnetic fields. A table containing all the modeling results obtained using the adopted method as well as the parameters obtained for the same stars by other authors is presented. Significant discrepances are found in a number of cases, which can apparently be explained by insufficient accuracy of the estimated inclinations of the stars to the line of sight.     

Effects of Strömgren spheres on the line formation inζ Aur systems

Aur binaries are particularly well suited for the study of the winds of late type supergiants. The analysis technique is based on theoretical modelling of flux profiles which are formed by resonance scattering in the expanding envelope. However, the hot B star ionizes the wind and can produce a considerable H II region around the B star. In order to investigate the influence of the Strömgren sphere on the line formation we have calculated theoretical P Cyg line profiles with an H II region. The line source function is solved with the Sobolev method followed by a correct formal integration (Hempe, 1982). The ionization of metals within the H II region is described by an ionization parameter which mimics the typical ionization fraction of the relevant ions. The resulting line profiles are dependent on the geometric shape of the H II region, the velocity distribution, and the opacity parameter. The reduced absorber density within the H II region produces weakened line profiles and the specific geometry of the H II region influences the emission and the absorption component in different ways. The careful analysis of model computations suggests that the region of the line formation is not only restricted to the vicinity of the B star, hence the outer integration boundary has to be taken to a sufficiently large radius in the calculation of line profiles.     

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.     

A test of the plausibility of eclipsing binary asteroids

We present an analytic method for producing the lightcurve of a system of spherical bodies in circular orbit about their common center of mass and then use it to model a lightcurve of 171 Ophelia. We find that a model in which the components have a diameter ratio of 1:2.96, have equal albedos, and circle one another in a synchronous orbit (period = 13.146 hr) inclined by 15° to our line of sight provides a reasonable fit to the observed lightcurve. An expression is given for the density in terms of the model parameters. If this lightcurve of Ophelia is produced by mutual eclipses of spherical components the mean density of the system can not exceed 1.7 g cm^?3. In view of these results we conclude that it is possible for Ophelia to be a binary.     

PHYSICAL PARAMETERS OF A FLARE ON 22 DECEMBER 1999 DERIVED FROM THE Hα LINE WITH A MODIFIED CLOUD METHOD

The flare of 22 December 1999 was observed in the H line using the imaging spectrograph in the solar tower of Nanjing University. We present a new technique different from the classical cloud model to fit the H line profile which avoids using the background profile. We obtain the four parameters of the flare chromosphere: the source function, the optical thickness at line center, the line-of-sight velocity and the Doppler width. The observed asymmetry profiles have been reproduced well by the theoretical ones based on our model. A discussion is made about the reliability of the results we have obtained using the present method.     

Electrons in the solar corona

We discuss a model for the formation of the chromospheric Ca ii K line which does not make the usual assumption of complete redistribution. Using a physically reasonable scattering model, we find significant departures due to the frequency dependence of the line source function, particularly in the relative intensity and centre-to-limb behaviour of the K₁ parts of the line and in the asymmetry produced by differential velocity fields. We conclude that the frequency dependence of the K line source function must be considered in quantitative models for the formation of the K line.     

Empirical NLTE analyses of solar spectral lines

The center-to-limb behaviour of the Ba ii 4554 resonance line is analyzed together with data from the extreme limb, flash intensities and profiles of other Ba ii lines. An empirical NLTE method is employed in which the observed profiles are compared with synthesized profiles based on a standard one-dimensional model atmosphere, with the line source function, the barium abundance, the collisional damping and the atmospheric turbulence as free parameters.The line profiles from the extreme limb furnish considerable constraints on the formation of Ba ii 4554. Its wings reverse into emission well inside the solar limb, a phenomenon which cannot be explained by any frequency-independent line source function. Accounting for effects of partially coherent scattering in the line source function is a necessary and adequate step to reproduce the observations both over the disk and near the limb. The form of the empirically derived frequency-dependent line source function is discussed.Results are given for various parameters (gf-values, solar barium abundance and isotope ratios, collisional damping, microturbulence and macroturbulence).The 4554 profile of disk center shows a depression in its blue wing resembling asymetries found in various stellar spectral lines.     

On the DC ratio in the atmosphere of Uranus

A method for deriving mixing ratios in the outer planets, mostly independent of scattering processes, is applied to Uranus. It is shown that scattering processes play a major role in the line formation in the atmospheres of Uranus and Neptune; consequently, abundance ratios derived from a reflecting-layer model can be questionable. Using our method, we derive for Uranus $\text{D}\text{C}\text{< 6 × 10}{}^{\mathrm{?3}}$, which is significantly smaller than our result on Jupiter. The simplest explanation implies a C/H enrichment by at least a factor of 6 relative to the solar value.     

A two-zone model for the broad iron line emission in MCG–6-30-15

We reanalyse the ASCA and BeppoSAX data of MCG–6-30-15, using a double-zone model for the iron line profile. In this model, the X-ray source is located around ≈10 Schwarzschild radii and the regions interior and exterior to the X-ray source produce the line emission. We find that this model fits the data with a similar reduced χ ² to the standard single-zone model. Thus we show that the presence of a broad iron line feature does not necessarily require that the X-ray source be located close to the last stable orbit or in the disc rotation axis.
Within the framework of this model, the best-fitting inclination angle of the source     for the intermediate-intensity ASCA data set is compatible with that determined by earlier modelling of optical lines. The observed variability of the line profile with intensity can be explained as variations of the X-ray source size. That several active galactic nuclei with broad lines have the peak centroid near 6.4 keV can be explained under certain conditions.
We also show that the simultaneous broad-band observations of this source by BeppoSAX rule out the Comptonization model which was an alternative to the standard inner-disc one. We thereby strengthen the case that line broadening occurs as a result of the strong gravitational influence of a black hole.     

The application of the quickest descent method to the analysis of the Hα line in loops

The quickest descent method and the multi-layer model are proposed to analyse symmetrical and asymmetrical line profiles of loops in this paper. By fitting observed lines, the physical parameters such as Doppler width, Δλ _D , the optical thickness at line centre,τ _λ0, and the line displacement, Δλ ₀, can be directly obtained, Using this method, some symmetrical and asymmetrical line profiles of both the loop prominence on February 18, 1984 and the loop on August 17, 1989 are fitted. It is found that the method mentioned in this paper is characterized by rapid convergence, high precision, and less scatter.     

The density and magnetic field of the dense cores of the molecular cloud L 1630

The density and magnetic field strength of the dense cores in the Orion B molecular cloud are derived from the observed radius and FWHM line width based on the model of a uniformly magnetic sphere. We obtain the average magnetic field strength of 110μG and the average density of 8 × 10⁴/cm³ for the 39 cores, which agree closely with the observations. The method for deriving the density and magnetic field strength is applicable to the cores with R>0.2pc.     

On the moving subfeatures in the Be star ζ Oph

We analyse a series of line profile observations of the He  i 6678 line in ζ Oph. A period analysis on these data using the mode and moments of the line profile confirms the two previously known periods. We describe a new method of mode identification for pulsating stars in which the calculated profiles are directly fitted to observed profiles. The method yields the full set of pulsational parameters including the spherical harmonic degree, ℓ, and azimuthal number, m . Application of the method to these data confirms the mode identifications previously suggested for the two periodicities. We find that the derived pulsational parameters are physically realistic and conclude that non-radial pulsation is the most likely explanation for the travelling subfeatures. However, a unique mode identification is still not possible – several non-sectorial modes fit the data as well as the usually adopted sectorial identifications. The predicted photometric amplitudes are in good accord with upper limits derived from photometric observations. We conclude that ζ Oph is a star in the β Cep instability strip in which two modes of high degree (probably ℓ=4 and ℓ=8) are excited. We present an interpretation of these findings in which the cause of the low-order line profile and light variations in periodic Be stars is corotating photospheric clouds, while the travelling subfeatures are incidental to the Be phenomenon and are a result of non-radial pulsation.