Sobolev Approximation Research Articles

Non-Sobolev modelling of radiation-pressure-driven flows in active galactic nuclei

We present a new general scheme for calculating the structure and dynamics of radiation-pressure-driven photoionized flows. The new method goes one step beyond the Sobolev approximation. It involves a numerical solution of the radiative transfer in absorption lines, including the effects of differential expansion and line interactions such as line locking and blanketing. We also present a new scheme for calculating the radiation pressure due to trapped line photons in finite, differentially expanding flows. We compare our results for the radiation pressure force with those obtained using the Sobolev approximation and show the limitations of the latter. In particular, we demonstrate that the Sobolev method gives a poor approximation near discontinuity surfaces and its neglect of line blanketing can lead to erroneous results in high-velocity flows. We combine the newly calculated radiation pressure force with self-consistent photoionization and thermal calculations to study the dynamics and spectral features of broad absorption-line flows and highly ionized gas flows in active galactic nuclei (AGN). A comparison with Sobolev-type calculations shows that the latter overestimates the terminal velocity of the flow and, conversely, underestimates its opacity. We also show that line locking on broad emission lines can have a significant effect on the dynamics and spectral features of AGN flows.

Non-spherical evolution of the line-driven wind instability

In this paper, we study the structure and stability of line-driven winds using numerical hydrodynamic simulations. We calculate the radiation force from an explicit non-local solution of the radiation transfer equation, rather than a Sobolev approximation, without restricting the flow to one-dimensional symmetry. We find that the solutions that result have complex and highly variable structures, including dense condensations, which we compare with observed variable absorption features in the spectra of early-type stars.

The Statistical Sobolev‐Rosseland Mean and the Effects of Frequency Redistribution on Wolf‐Rayet Wind Driving

The optically thick character of Wolf-Rayet winds implies that stellar continuum photons are multiply scattered, as a result of both free electron opacity and overlapping wind-broadened spectral lines. This allows the wind to accumulate a substantial excess in momentum flux relative to the driving radiation field, as is observationally required. Nevertheless, sustaining such a high degree of multiple scattering requires not only a large optical depth spatially but also substantial spectral blanketing. The latter is difficult to maintain when redistribution during scattering allows radiative flux to shift preferentially into spectral regions with fewer lines, since then the channels carrying much of the flux are also the least well blanketed. This paper parameterizes the potential severity of this effect in simple terms, using a generalization of the Rosseland mean treated in the Sobolev approximation. We show that our approach provides an informative starting point for characterizing and conceptualizing nongray effects in optically thick supersonic flows.

The Zeeman effect in the Sobolev approximation: applications to spherical stellar winds

Modern spectropolarimeters are capable of detecting subkilogauss field strengths using the Zeeman effect in line profiles from the static photosphere, but supersonic Doppler broadening makes it more difficult to detect the Zeeman effect in the wind lines of hot stars. Nevertheless, the recent advances in observational capability motivate an assessment of the potential for detecting the magnetic fields threading such winds. We incorporate the weak-field longitudinal Zeeman effect in the Sobolev approximation to yield integral expressions for the flux of circularly polarized emission. To illustrate the results, two specific wind flows are considered: (i) spherical constant expansion with v(r ) = v∞ and (ii) homologous expansion with v(r ) ∝ r . Axial and split monopole magnetic fields are used to schematically illustrate the polarized profiles. For constant expansion, optically thin lines yield the well-known ‘flat-topped’ total intensity emission profiles and an antisymmetric circularly polarized profile. For homologous expansion, we include occultation and wind absorption to provide a more realistic observational comparison. Occultation severely reduces the circularly polarized flux in the redshifted component, and in the blueshifted component, the polarization is reduced by partially offsetting emission and absorption contributions. We find that for a surface field of approximately 100 G, the largest polarizations result for thin but strong recombination emission lines. Peak polarizations are approximately 0.05 per cent, which presents a substantial although not inconceivable sensitivity challenge for modern instrumentation.

Analytical solutions for saturated P cygni type profiles II: General case

As in the first part of the present study (Sapar et al., 2002) we use the β-law for velocity of stellar wind and the Sobolev approximation for radiative transfer. Here we have succeeded to derive general and relatively simple analytical formulae in elementary functions for saturated P Cygni type line profiles if parameter 2β is arbitrary positive integer (in the first part we studied the cases 2β ≤ 4). The four terms obtained describe contributions to the line profile due to isotropic and anisotropic parts of optical thickness in the source function of the light-scattering layer followed by isotropic and anisotropic parts of multiple scattering. The limits of acceptability of the Sobolev approximation for β-law have been discussed and specified.

Line-driven winds in the presence of strong gravitational fields

We propose a general physical mechanism which could contribute to the formation of fast line-driven outflows at the vicinity of strong gravitational field sources. We argue that the gradient of the gravitational potential plays the same role as the velocity gradient plays in the Sobolev approximation. Both the Doppler effect and gravitational redshifting are taken into account in the Sobolev approximation. The radiation force becomes a function of the local velocity gradient and the gradient of the gravitational potential. The derived equation of motion has a critical point that is different from that of Castor, Abbott and Klein (CAK). A solution, which is continuous through the singular point, is obtained numerically. A comparison with CAK theory is presented. It is shown that the developed theory predicts terminal velocities which are greater than those obtained from the CAK theory.

Influence of opacity on gain coefficients in static, and fast moving neon-like krypton plasmas

X-ray laser gains and the level populations in collisionally pumped neon (Ne)-like krypton (Kr XXVII) ions have been investigated considering 27 levels of the 2p6, 2p5 3s, 2p5 3p, and 2p5 3d configurations in fast moving cylindrical plasma. Effects of opacity of the 3d 1P1→2p6 1S0, 3d 3D1→2p6 1S0, 3d 3P1→2p6 1S0, 3s 3P1→2p6 1S0, and 3s 1P1→2p6 1S0 transitions on the gain coefficients are considered using escape probability factors in both the static, and dynamic plasmas under the Sobolev approximation, that includes effect of the large velocity gradient. Results indicate that the gain values are significantly affected by plasma opacity, and the velocity gradient allows us to obtain a broader gain region in dense plasmas.

Effect of Opacity on Neon-like Argon, and Krypton Line Intensity Ratios in Static, and Cylindrically Imploding High-Density Plasmas

In connection with X-ray lasers, theoretical line intensity ratios of neon (Ne)-like argon (ArIX) and krypton (KrXXVII) ions in high energy density plasmas have been investigated considering 27 levels of the 2s22p6, 2s22p53s, 2s22p53p, and 2s22p53d configurations. It was found that some line intensity ratios are sensitive to the electron density, the electron temperature, and also opacity in the interest regime of ArIX and KrXXVII X-ray lasers. Our attention is devoted to studying the opacity effects of the important radiative transitions for the lasing condition, namely; those of the 3d 1P1–2p61S0, 3d 3D1–2p61S0, 3d 3P1–2p61S0, 3s 3P1–2p61S0, and 3s 1P1–2p61S0 transitions on the line intensity ratios in both static and cylindrically imploding plasmas under the Sobolev approximation that includes the effect of the large velocity gradient.

Profiles of the resonance doublets formed in bipolar winds in symbiotic stars

We compute the profiles of resonance doublet lines (S1/2‐P1/2,3/2) formed in bipolar winds with velocity greater than the doublet separation in symbiotic stars. Particular attention has been paid to the doublet line ratio, where an essential role is played by the conversion of the short-wavelength component arising from the S1/2‐P3/2 transition into the long-wavelength component for the transition S1/2‐P1/2. We adopted a Monte Carlo technique and the Sobolev approximation. Our bipolar winds take the form of a cone and are characterized by the terminal wind velocity, the mass-loss rate and the opening angle of the cone. When an observer is in the polar direction and the Sobolev optical depth τSob � 1, we mainly obtain profiles with inverted flux line ratios, where the short-wavelength component is weaker than the long-wavelength component. When an observer is in the equatorial direction, we find that the profiles are characterized by two broad components, where the long-wavelength component is the broader and stronger of the two. We conclude that the profiles obtained in our model provide a qualitative understanding of the broad profiles and inverted intensity ratios of the doublets in symbiotic

Resonance Line Profile Calculations Based on Hydrodynamical Models of Cataclysmic Variable Winds

We describe a method of calculating synthetic line profiles using a generalized version of the Sobolev approximation. We apply this method to calculate line profiles predicted by the models of two-dimensional line-driven winds from luminous disks due to Proga, Stone, & Drew. We describe the main properties of the model line profiles and compare them with recent Hubble Space Telescope observations of the cataclysmic variable IX Vel. The model wind consists of a dense, slow outflow that is bounded on the polar side by a high-velocity stream. We find that these two wind components produce distinct spectral features. The fast stream produces profiles that show features consistent with observations. These include the appearance of the classical P Cygni shape for a range of inclinations, the location of the maximum depth of the absorption component at velocities less than the terminal velocity, and the transition from net absorption to net emission with increasing inclination. However, the model profiles have too little absorption or emission equivalent width compared to observed profiles. This quantitative difference between our models and observations is not a surprise because the line-driven wind models predict a mass-loss rate, mostly due to the fast stream, that is lower than the rate required by the observations. We note that the model profiles exhibit a double-humped structure near the line center that is not echoed in observations. We identify this structure with a nonnegligible redshifted absorption that is formed in the slow component of the wind where the rotational velocity dominates over expansion velocity. We conclude that the next generation of disk wind models, developed for application to cataclysmic variables, needs to yield stronger wind driving out to larger disk radii than do the present models.

A Complete Analytic Inversion of Supernova Lines in the Sobolev Approximation

We show that the shape of P-Cygni line profiles of photospheric phase supernova can be analytically inverted to extract both the optical depth and source function of the line -- i.e. all the physical content of the model for the case when the Sobolev approximation is valid. Under various simplifying assumptions, we derive formulae that give S(r) and tau(r) in terms of derivatives of the line flux with respect to wavelength. The transition region between the minimum and maximum of the line profile turns out to give especially interesting information on the optical depth near the photosphere. The formulae give insights into the relationship between line shape and physical quantities that may be useful in interpreting observed spectra and detailed numerical calculations.

OH inversions in star-forming regions and ALI radiation transfer

I introduce a new model of OH maser pumping in star-forming regions, based on exact radiation transfer models of the interface between an expanding ucHII region and undisturbed molecular material. The model incorporates the most recent set of rate-coefficients for collisions of OH with molecular hydrogen, and an improved model of continuum emission from dust. FIR line overlap is incorporated into the model in a manner which avoids the inconsistencies associated with the Sobolev approximation.

Analytical solutions for saturated P cygni type profiles

Analytical formulae for single P Cygni type saturated resonance line profiles in stellar winds have been derived. The limbdarkening and presence of underlying intrinsic atmospheric profile have been ignored. The Sobolev approximation for radiative transfer has been used and the general velocity law has been specified by widely used β parameter. The analytical formulae for the saturated resonance line profiles can be found for cases when 2β is an integer. The formulae for 2β = 1,2, 3 and 4 have been found by us. Also the formulae for calculating the line profiles in the cases of external and internal sharp truncation (cutoff) of the scattering shell have been given. Some characteristic line profiles have been presented. It has been shown that the turbulence-generated isotropic dominant backscattering of radiation in stellar winds generates wide dark plateaux in the blue wings of spectral lines, and the slopes of plateaux are shaped by turbulence.

Escape factors for laser-plasmas

A method of parameterizing escape factors (transmission factors and net radiative brackets) for conditions typical of laser-produced plasmas is introduced. The assumptions of planar geometry, exponentially decreasing emissivity and absorption coefficient with distance with a step rise at a particular point, and spatially constant Doppler broadened line profiles have been made. The effect of velocity gradients in spectrally shifting the absorption and emission line profiles relative to each other is taken into account assuming linear velocity gradients with distance. A parameter R representing the ratio of the spatial scale-length of the absorption coefficient to the Doppler decoupling length is introduced. Fitting formulae for transmission factors and net radiative brackets are given which are valid for all R and all optical depths. In the limit of small R (large Doppler decoupling length), the escape factors asymptotically approach formulae developed originally by, for example, Holstein assuming negligible plasma velocities. For large R (small Doppler decoupling length), the escape factors have been shown to asymptotically approach the Sobolev approximation. The parameterized net radiative bracket has been used in the hydrodynamic and atomic physics code ‘EHYBRID’ for the calculation of the effect of radiation trapping on population densities in laser-produced plasmas. The output of the modified EHYBRID code has then been post processed using the parameterised transmission factors to simulate 123 Ne-like and 399 F-like germanium resonance line intensities emitted in typical X-ray laser experiments. We obtain an agreement between the simulated and experimental spectra.

Line Forces in Keplerian Circumstellar Disks and Precession of Nearly Circular Orbits

We examine the effects of optically thick line forces on orbiting circumstellar disks, such as occur around Be stars. For radially streaming radiation (e.g., as from a point source), line forces are effective only if there is a strong radial velocity gradient, as occurs, for example, in a line-driven stellar wind. However, we emphasize here that, within an orbiting disk, the radial shear of the azimuthal velocity leads to strong line-of-sight velocity gradients along nonradial directions. As such, we show that, in the proximity of a stellar surface extending over a substantial cone angle, the nonradial components of stellar radiation can impart a significant line force to such a disk, even in the case of purely circular orbits with no radial velocity. Given the highly supersonic nature of orbital velocity variations, we use the Sobolev approximation for the line transfer, extending to the disk case the standard CAK formalism developed for line-driven winds. We delineate the parameter regimes for which radiative forces might alter disk properties; but even when radiative forces are small, we analytically quantify higher-order effects in the linear limit, including the precession of weakly elliptical orbits. We find that optically thick line forces, both radial and azimuthal, can have observable implications for the dynamics of disks around Be stars, including the generation of either prograde or retrograde precession in slightly eccentric orbits. However, our analysis here suggests a net retrograde effect, in apparent contradiction with observed long-term variations of violet/red line profile asymmetries from Be stars, which are generally thought to result from prograde propagation of a one-arm, disk-oscillation mode. We also conclude that radiative forces may alter the dynamical properties at the surface of the disk where disk winds originate, and in the outer regions far from the star, and may even make low-density disks vulnerable to being blown off completely.

SwSt 1: an O-rich planetary nebula around a C-rich central star

ABSTRA C T The hydrogen-deficient [WCL] type central star HD 167362 and its planetary nebula (PN) SwSt 1 are investigated. The central star has a carbon-rich emission-line spectrum, and yet the nebula exhibits a 10-mm emission feature from warm silicate dust, perhaps indicating a recent origin for the carbon-rich stellar spectrum. Its stellar and nebular properties might therefore provide further understanding as to the origin of the [WCL] central star class. The central star optical and UV spectra are modelled with state-of-the art non-LTE codes for expanding atmospheres, from which the stellar parameters are determined. Using the Sobolev approximation code ISA-Wind, we find T effa 40000 K, logO _ M=M( yr 21 Ua26:72, La 8900 L( (for a distance of 2.0 kpc), and v1 . 900 km s 21 . The abundance mass fractions for helium, carbon and oxygen are determined to be 37, 51 and 12 per cent, respectively. From this we derive C=O a 4:3 (by mass), confirming that the star suffered efficient third dredgeup. The nitrogen abundance is close to zero, while an upper limit of ,10 per cent by mass is established for H. The model uses a composite beta velocity law which allows us to reproduce the optical line profiles. The overall shape of the dereddened spectrum agrees with the V-scaledâmV a 11:48 mag, EOB 2 VUa0:46 maga model atmosphere, showing the nebularderived reddening to be consistent with the reddening indicated by the stellar analysis. We confirm our model results by using the comoving frame code CMFGEN, although a few differences remain.

Line‐blanketed Spherically Extended Model Atmospheres of Hot Luminous Stars with and without Winds

We have found that our spherical, hydrostatic, nonlocal thermodynamic equilibrium (non-LTE) metal line-blanketed model atmospheres are able to reproduce the spectral energy distributions of the bright B giants b and e Canis Majoris, including the extreme ultraviolet, where previous models have failed (J. P. Aufdenberg, P. H. Hauschildt, & E. Baron 1999, MNRAS, 302, 599; J. P. Aufdenberg, P. H. Hauschildt, S. N. Shore, & E. Baron 1998, ApJ, 498, 837). The brightest stellar source of EUV radiation longward of 400 A, e CMa is an extremely important star because it produces more hydrogen-ionizing flux than all nearby stars combined. Because of their proximity to the Sun, these two stars provide the rare opportunity to directly test predictions of theoretical stellar continua below the Lyman limit. These stars have precisely measured angular diameters, trigonometric parallaxes, and bolometric fluxes, and we have shown that the best-fitting models fall on the Hertzsprung-Russell diagram within the small error boxes provided by these observations. Within these constraints, we were able to compare our synthetic spectra to the observed spectral energy distributions of these B stars in absolute units, a rarity in stellar atmosphere analyses. Observations show a large EUV flux excess compared with plane-parallel model atmospheres (J. P. Cassinelli et al. 1995, ApJ, 438, 932; J. P. Cassinelli et al. 1996, ApJ, 460, 949). This led to the suggestion that various scenarios such as X-ray heating and the presence of a stellar wind were responsible for the mismatch between theory and observation. We discovered that a combination of metal line blanketing and spherical geometry present in our model atmospheres produced significantly more EUV flux than (otherwise similar) plane-parallel atmospheres without the need for these ad hoc mechanisms. These findings are in agreement with studies of B star H ii regions which also show evidence for excess EUV flux from B stars relative to plane-parallel model predictions (R. J. Reynolds 1985, AJ, 90, 92; R. J. Reynolds & S. L. Tufte 1997, ApJ, 439, L17). Our models for b and e CMa represent the only spherical, non-LTE line-blanketed models of B stars in the literature. Surprisingly, we have found that the combination of spherical geometry, line blanketing, and surface gravities as high as will produce significantly different model temlog g p 3.5 perature structures and synthetic EUV spectra relative to otherwise similar plane-parallel models over a range in effective temperature of 18,000–28,000 K. It is the combination of the spherical geometry and the line blanketing, not the non-LTE aspects, that is the most important factor creating this effect. We believe an important factor which shifts the surface cooling/ backwarming to greater depths in the spherical models is the reduced optical depth seen by photons in a spherical geometry relative to that in a semi-infinite plane-parallel geometry, for a given physical depth. As a result, in the spherical models, backwarming begins at a greater depth than in the plane-parallel models. These deeper layers are intrinsically warmer, so that the temperature of the EUV continuum-forming layers are at a higher temperature than the EUV continuum-forming layers in the plane-parallel models. We encourage other researchers to reproduce our findings with an independent spherical lineblanketed model atmosphere code. Our models for hot, luminous stars and their winds are thus far the only ones that include metal line blanketing and combine the inner hydrostatic layers with the outer dynamic layers of the atmosphere in a single structure. This is a significant step, particularly regarding the inclusion of full non-LTE metal line blanketing in both the computation of the model atmosphere plus wind and the synthetic spectrum. Model approaches to date have been limited by the use of a spherical gray temperature structure (not fully converged), the lack of full line blanketing, and the Sobolev approximation, which cannot treat overlapping lines or multiple scattering. In our models, the temperature structure is computed from the condition of energy conservation in the comoving frame, full line blanketing is included, and the Sobolev approximation is not used. Moving beyond these models, we have begun to develop a theoretical framework that will self-consistently obtain the velocity structure and the mass-loss rate from radiation-driven wind theory given the effective temperature, radius, and gravity of the star. We have developed a module for the PHOENIX

Sobolev's Approximation

A review presented at a seminar devoted to V. V. Sobolev's 85th birthday. The purpose of the review is to trace the development of the fundamental idea of the clearing of a medium due to the Doppler effect, on which Sobolev's method for moving envelopes of stars is based.

Scattering of cyclotron radiation in a moving plasma

We consider the scattering of cyclotron radiation in a plasma moving along a homogeneous magnetic field. The equation of radiation transfer in a co-moving frame is derived and two limiting cases are pointed out. In the first case of a “small” velocity gradient, the total Doppler frequency shift due to variations in the plasma velocity over the flow is much smaller than the width of the line. The second, opposite, case of a “large” velocity gradient is analogous to the Sobolev approximation in the theory of moving stellar envelopes. The solution of the transfer equation for a wind-type flow illuminated by radiation of a given intensity is obtained in the latter case, when the influence of the plasma motion on cyclotron scattering is most important. It is shown that cyclotron scattering in a moving plasma differs from the known (and qualitatively similar) problems of resonance scattering in moving stellar envelopes and cyclotron scattering in a motionless plasma permeated by an inhomogeneous magnetic field. In particular, a symmetric absorption band with residual intensity proportional to the velocity gradient appears in the spectrum of the outgoing radiation, while in these two other problems, the depth of the corresponding spectral features cannot exceed half the continuum level. Detailed qualitative analysis reveals that this difference is due to the particular form of the frequency redistribution for cyclotron scattering.

The optical spectrum of the planetary nebula NGC 6543

With the Hamilton echelle spectrograph at the Lick Observatory, emission-rich spectral lines of the planetary nebula NGC 6543 were secured in the wavelength range from 3550 to 10 100 Å. We chose two bright regions, ∼8 arcsec east and ∼13 arcsec north of the central star, the physical conditions and chemical abundances of which may differ as a result of the different physical characteristics involving the mass ejection of different epochs. By combining Hamilton echelle observations with archive UV data secured with the International Ultraviolet Explorer (IUE), we obtain improved diagnostics and chemical compositions for the two observed regions. The diagnostic diagram gives the average value of Te=8000∼8300 K, and the electron number density near Ne∼5000 cm−3 for most ions, while some low-excitation lines indicate much higher temperatures, i.e. Te∼10 000 K. With the construction of a photoionization model, we try to fit the observed spectra in a self-consistent way: thus, for most elements, we employ the same chemical abundances in the nebular shell; and we adopt an improved Sobolev approximation model atmosphere for the hydrogen-deficient Wolf-Rayet type central star. Within the observational errors, the chemical abundances do not seem to show any positional variation except for helium. The chemical abundances of NGC 6543 appear to be the same as in average planetary nebulae. The progenitor star may have been an object of one solar mass, most of the heavier elements of which were less plentiful than in the Sun.