Light curve analysis for eclipsing systems with exoplanets. The system HD 209458

We have analyzed precision light curves for HD 209458, a binary with an exoplanet. The parameters obtained at different epochs and different wavelengths are in good mutual agreement when confidence regions are used to calculate the uncertainty intervals. We demonstrate the effectiveness and reliability of our new method for estimating the uncertainty intervals. Reliable estimates are provided for the linear and quadratic limb-darkening coefficients of the star and their confidence intervals (uncertainties). We find that the wavelength dependence for the limb-darkening coefficients at λ = 3201?9708 Å differs significantly from the corresponding theoretical relation based on thin model stellar atmospheres.     

Masses of the visual components and black holes in X-ray novae: Effects of proximity of the components

It is shown that the approximation of the complex, tidally distorted shape of a star as a circular disc with local line profiles and a linear limb-darkening law, which is usually applied when deriving equatorial stellar rotation velocities from line profiles, leads to overestimation of the equatorial velocity V _rot sin i and underestimation of the component mass ratio q = M _x /M _v . A formula enabling correction of the effect of these simplifying assumptions on the shape of a star is used to re-determine the mass ratios q and the masses of the black holes M _x and visual components M _v in low-mass X-ray binary systems containing black holes. Taking into account the tidal–rotational distortion of the stellar shape can significantly increase the mass ratios q = M _x /M _v , reducing M _v , while M _x changes only slightly. The resulting distribution of M _v attains its maximum near M _v ? 0.35M _⊙, in disagreement with the results of population synthesis computations realizing standard models for Galactic X-ray novae with black holes. Possible ways to overcome this inconsistency are discussed. The derived distribution of M _x also differs strongly from the mass distribution for massive stars in the Galaxy.     

Nonlinear radial pulsations of Wolf-Rayet stars

The evolution of Population I stars with initial masses 60 M _⊙ ≤ M _ZAMS ≤ 120 M _⊙ is computed up to the Wolf-Rayet stage, when the central helium abundance decreases to Y _c ≈ 0.05. Several models from evolutionary sequences in the core helium-burning stage were used as initial conditions when solving the equations of radiative hydrodynamics for self-exciting stellar radial pulsations. The low-density envelope surrounding the compact core during the core helium burning is unstable against radial oscillations in a wide range of effective temperatures extending to T _eff ～ 10⁵ K. The e-folding time of the amplitude growth is comparable to the dynamical time scale of the star, and, when the instability ceases growing, the radial displacement of the outer layers is comparable to the stellar radius. Evolutionary changes of the stellar radius and luminosity are accompanied by a decrease in the amplitude of radial pulsations, but, at the effective temperature T _eff ≈ 10⁵ K, the stellar oscillations are still nonlinear, with a maximum expansion velocity of the outer layers of about one-third the local escape velocity. The period of the radial oscillations decreases from 9 hr to 4 min as stellar mass decreases from M = 28 M _⊙ to M = 6 M _⊙ in the course of evolution. The nonlinear oscillations lead to a substantial increase of the radii of the Lagrangian mass zones compared to their equilibrium radii throughout the instability region. The instability of Wolf-Rayet stars against radial oscillations is due to the action of the κ mechanism in the iron-group ionization zone, which has a temperature of T ～ 2 × 10⁵ K.     

Timescales for mechanisms for the dynamical evolution of open star clusters

We have obtained the stellar velocity dispersion in three mutually perpendicular directions in the halos and cores of clusters as a function of time for several non-stationary open-cluster models. During the dynamical evolution of the open-cluster models, the velocity dispersions undergo oscillations that do not decay during 5–10 violent-relaxation timescales, τ _vr . We estimated the time for synchronization of the rotation of the open-cluster models and their motion around the center of the Galaxy, t _s , which, depending on the model parameters, is t _s ? (5–27)τ _vr . Synchronization mechanisms for the models are discussed. The disruption of the systems in the force field of the Galaxy is strongly affected by tidal friction. We have also estimated the time for the formation of a spherical stellar-velocity distribution in the cluster models, t _σ ? (6 ? 25)τ _vr . The impact of instability in the stellar motions in a cluster on the formation of a spherical velocity distribution in the open-cluster models is discussed. We have noted a tendency for a weakening of the dependence of the coarse phase density of the cluster on small initial perturbations of the stellar phase coordinates in the model cluster cores for times about five times longer than the violent-relaxation time.     

Evolution of close stellar binaries with black holes under the action of gravitational radiation and magnetic and induced stellar winds of the donor

We show that semi-detached close binary systems with massive (4–25M_⊙) black holes are formed in the evolution of massive stellar binaries in which the initial mass of the primary exceeds ～25M_⊙. The mass exchange in such systems is maintained by the nuclear evolution of the donor and by its magnetic and induced stellar winds. The donor in such systems can be a main-sequence star, subgiant, non-degenerate helium star, or white dwarf. The evolution of corresponding systems with black-hole masses of 10M_⊙ is investigated.     

Estimation of the accuracy of methods for determining component masses for low-mass X-ray binary systems

Modern modeling of the population of low-mass X-ray binary systems containing black holes applying standard assumptions leads to a lack of agreement between the modeled and observed mass distributions for the optical components, with the observed masses being lower. This makes the task of estimating the systematic errors in the derived component masses due to imperfect models relevant. To estimate the influence of systematic errors in the derived masses of stars in X-ray binary systems, we considered two approximations for the tidally deformed star in a Roche model. Approximating the star as a sphere with a volume equal to that of the Roche lobe leads to slight overestimation of the equatorial rotational velocity V _rot sin i, and hence to slight underestimation of the mass ratio q = M _x /M _v . Approximating the star as a flat, circular disk with constant local line profiles and a linear limb-darkening law (a classical rotational broadeningmodel) is an appreciably cruder approach, and leads to overestimation of V _rot sin i by about 20%. In the case of high values of q = M _x /M _v , this approximation leads to substantial underestimation of the mass ratio q, which can reach several tens of percent. The mass of the optical star is overestimated by a factor of 1.5 in this case, while the mass of the black hole is changed only slightly. Since most estimates of component mass ratios for X-ray binary systems are carried out using a classical rotational broadening model for the lines, this leads to the need for appreciable corrections to (reductions of) previously published masses for the optical stars, which enhances the contradiction with the standard evolutionary scenario for low-mass X-ray binaries containing black holes.     

Magnetic field models for HD 116458 and HD 126515

We have modeled the magnetic fields of the slowly rotating stars HD 116458 and HD 126515 using the “magnetic charge” technique. HD 116458 has a small angle between its rotation axis and dipole axis (β = 12°), whereas this angle is large for HD 126515 (β = 86°). Both stars can be described with a decentered-dipole model, with the respective displacements being r = 0.07 and r = 0.24 in units of the stellar radius. The decentered-dipole model is able to satisfactorily explain the phase relations for the effective field, B_e(P), and the mean surface field, B_s(P), for both stars, along with the fact that the B_e(P) phase relation for HD 126515 is anharmonic. We discuss the role of systematic measurement errors possibly resulting from instrumental or methodical effects in one or both of the phase relations. The displacement of the dipole probably reflects real asymmetry of the stellar field structure, and is not due to measurement errors. Using both phase relations, B_e(P) and B_s(P), in the modeling considerably reduces the influence of the nonuniform distribution of chemical elements on the stellar surface.     

Hydrodynamical modeling of mass transfer in the close binary system <Emphasis Type="Italic">β</Emphasis> Lyr

We present results of two-dimensional hydrodynamical simulations of mass transfer in the close binary system β Lyr for various radii of the accreting star and coefficients describing the interaction of the gaseous flow and the main component (primary). We take the stellar wind of the donor star into account and consider various assumptions about the radiative cooling of the gaseous flow. Our calculations show that the initial radius of the flow corresponding to our adopted mass-transfer rate through the inner Lagrange point (L₁) of (1–4) × 10^?5M_⊙/yr is large: 0.22–0.29 (in units of the orbital separation). In all the models, the secondary loses mass through both the inner and outer (L₁ and L₂) Lagrange points, which makes the mass transfer in the system nonconservative. Calculations for various values of the primary radius show a strong dependence on the coefficient fv that models the flow-primary interaction. When the radius of the primary is 0.5, there is a strong interaction between the gas flow from L1 and the flow reflected from the primary surface. For other values of the primary radius (0.1 and 0.2), the flow does not interact directly with the primary. The flow passes close to the primary and forms an accretion disk whose size is comparable to that of the Roche lobe and a dense circum-binary envelope surrounding both the disk and the binary components. The density in the disk varies from 10¹² to 10¹⁴ cm^?3, and is 10¹⁰–10¹² cm^?3 in the circum-binary envelope. The temperature in the accretion disk ranges from 30000 to 120000 K, while that in the circum-binary envelope is 4000–18000 K. When radiative cooling is taken into account explicitly, the calculations reveal the presence of a spiral shock in the accretion disk. The stellar wind blowing from the secondary strongly interacts with the accretion disk, circum-binary envelope, and flow from L₂. When radiative cooling is taken into account explicitly, this wind disrupts the accretion disk.     

The formation and evolution of the most massive stars

We consider the formation of massive stars under the assumption that a young star accretes material from the protostellar cloud through its accretion disk while losing gas in the polar directions via its stellar wind. The mass of the star reaches its maximum when the intensity of the gradually strengthening stellar wind of the young star becomes equal to the accretion rate. We show that the maximum mass of the forming stars increases with the temperature of gas in the protostellar cloud T ₀, since the rate at which the protostellar matter is accreted increases with T ₀. Numerical modeling indicates that the maximum mass of the forming stars increases to ～900 M _⊙ for T ₀ ～ 300 K. Such high temperatures of the protostellar gas can be reached either in dense star-formation regions or in the vicinity of bright active galactic nuclei. It is also shown that, the lower the abundance of heavy elements in the initial stellar material Z, the larger the maximum mass of the star, since the mass-loss rate due to the stellar wind decreases with decreasing Z. This suggests that supermassive stars with masses up to 10⁶ M _⊙ could be formed at early stages in the evolution of the Universe, in young galaxies that are almost devoid of heavy elements. Under the current conditions, for T ₀ = (30–100) K, the maximum mass of a star can reach ～100M _⊙, as is confirmed by observations. Another opportunity for the most massive stars to increase their masses emerges in connection with the formation and early stages of evolution of the most massive close binary systems: the most massive stars can be produced either by coalescence of the binary components or via mass transfer in such systems.     

The mass of the black hole in the X-ray binary M33 X-7 and the evolutionary status of M33 X-7 and IC 10 X-1

We have analyzed the observed radial-velocity curve for the X-ray binary M33 X-7 in a Roche model. We have analyzed the dependence between the component masses and the degree of filling of the optical star’s Roche lobe to obtain the ratio of the masses of the optical star and compact object. For the most probable mass of the optical star, m _v = 70 M⊙, the mass of the compact object is m _x = 15.55 ± 3.20 M⊙. It has been shown that black holes with masses of mx = 15 M⊙ and even higher can form in binaries. We present characteristic evolutionary tracks for binary systems passing through an evolutionary stage with properties similar to M33 X-7-type objects. According to population-synthesis analyses, such binaries should be present in galaxies with masses of at least 10¹¹ M⊙. The present number of such systems in M33 should be of the order of unity. We have also studied the evolutionary status of the X-ray binary IC 10 X-1 with a Wolf-Rayet component, which may contain a massive black hole. The final stages of the evolution of the M33 X-7 and IC 10 X-1 systems should be accompanied by the radiation of gravitational waves.     

Atmospheric chemical abundances and the evolutionary status of the spectroscopic and speckle-interferometric binary 9 Cyg

We have performed speckle interferometry with the 6-m telescope of the Special Astrophysical Observatory and spectroscopy (at 3700–9200 Å) with the 2-m telescope at Peak Terskol of the spectroscopic and interferometric binary 9 Cyg, which is a composite-spectrum star with an orbital period of 4.3 yrs. The atmosphere of the system’s primary component is analyzed in detail. The luminosities of both components estimated to be L ₁ = 103.8 L _⊙, L ₂ = 55.2 L _⊙, where L _⊙ is the solar luminosity, and their effective temperatures to be T _e (1) = 5300 K and T _e (2) = 9400 K. The abundances of C, N, O, Fe, and other elements in the primary’s atmosphere have been derived. The chemical composition shows signatures of mixing of material from its atmosphere and the region of nuclear reactions. The evolutionary status of 9 Cyg has been determined. The binary’s age is about 400 million years; the brighter star is already in the transition to becoming a red giant, while the secondary is still in the hydrogen-burning stage near the zero-age main sequence. We suggest an evolutionary model for the binary’s orbit that explains the high eccentricity, e = 0.79.     

The impact of reflection effects on the parameters of the old pre-cataclysmic variables MS Peg and LM Com

We have determined the main parameters of the old precataclysmic variable stars MS Peg and LM Com. The radial velocities of the components, reflection effects in the spectra, and light curves of the systems are studied based on model stellar atmospheres subject to external irradiation. Forty-seven moderate-resolution spectra for MS Peg and 57 for LM Com obtained with the 6-m telescope of the Special Astrophysical Observatory are used to derive the refined orbital periods of 0.1736660 days and 0.2586873 days, respectively; the orbital eccentricities do not exceed e=0.04. The mass (M _w =0.49e_⊙) and radius (e _w =0.015R_⊙) of the MS Peg primary calculated using the gravitational redshift correspond to those for a cooling carbon white dwarf with a thin hydrogen envelope. The parameters of the red dwarf (M _r =0.19M_⊙, T_eff=3560 K, R _r =0.18R_⊙) are close to those derived from evolutionary tracks for main-sequence M stars with solar chemical composition. The radius (R _r =0.22R_⊙) and temperature (T_eff=3650 K) of the LM Com secondary exceed theoretical estimates for main-sequence stars with masses of M _r =0.17M_⊙. The luminosity excess of the red dwarf in LM Com can be explained by a prolonged (T>5×10⁶ yrs) relaxation of the M star to its normal state after the binary leaves the common-envelope stage. For both systems, theoretical U, B, V, and R light curves and spectra calculated using the adopted sets of parameters are generally consistent with the observations. This confirms the radiative origin of the hot spots, the unimportance of horizontal radiative transport, and the absence of large-scale velocity fields with high values (V_trans>50 km/s) at the surfaces of the secondaries. Most of the emission lines in the spectra of these objects are formed under conditions close to thermalization, enabling modeling of their pro files in an LTE approximation. A strong λ3905 Å emission line has been identified as the 3s²3p4s 1P⁰-3s²3p² 1S SiI λ3905.52 Å line formed in the atmosphere of the hot spot. The observed intensity can be explained by non-LTE “superionization” of SiI atoms by soft UV radiation from the white dwarf. We suggest a technique for identifying binaries whose cool components are subject to UV irradiation based on observations of λ3905 Å emission in their spectra.     

The nature of the unique precataclysmic variable V664 Cas with two-peaked Balmer lines in its spectrum

We analyze photometric and spectroscopic observations of the close binary system V664 Cas. All the characteristics of its radiation are consistent with the star being a cataclysmic variable with powerful reflection effects. The orbital period is refined (P=0.5816475 d) and the ephemerides of the system determined. The U, B, V, R light curves of V664 Cas display sinusoidal variations with similar amplitudes near Δm=1.1^m. This suggests that a hot spot on the surface of the secondary always dominates the optical radiation of the system. The spectra contain emission lines, two-peaked hydrogen lines, and narrow lines of helium and heavy elements in high ionization states, whose intensities vary synchronously with the brightness. The HeII λ4686 Å line has broad absorption wings that form in the atmosphere of the O subdwarf. The mass function, f(m)=0.007M_⊙, is the lowest among all precataclysmic variables: the mass of the secondary exceeds the mass of the primary by more than a factor of 1.6. A full set of fundamental parameters for V664 Cas is determined based on modeling of the spectra and light curves, taking into account reflection effects in the system. Most of the emission lines are formed under conditions of appreciable deviations from local thermodynamic equilibrium. The possibility of carrying out correct modeling of the Balmer-line profiles assuming the stellar radiation is absorbed in a planetary nebula is demonstrated.     

Dependence of the absorption-line profiles and radial-velocity curve of the optical star in an X-ray binary on the orbital inclination and component-mass ratio

Theoretical absorption-line profiles and radial-velocity curves for tidally deformed optical stars in X-ray binary systems are calculated assuming LTE. The variations in the profile shapes and radial-velocity curve of the optical star are analyzed as a function of the orbital inclination of the X-ray binary system. The dependence of the shape of the radial-velocity curve on the orbital inclination i increases with decreasing component-mass ratio q = m _x /m _v . The integrated line profiles and radial-velocity curves of the optical star are calculated for the Cyg X-1 binary, which are then used to estimate the orbital inclination and mass of the relativistic object: i < 43° andm _x = 8.2–12.8 M_⊙. These estimates are in good agreement with earlier results of fitting the radial-velocity curve of Cyg X-1 using a simpler model (i < 45°, m _x = 9.0–13.2 M_⊙).     

Evolutionary connection between low-mass detached binaries,contact W UMa systems,and blue stragglers

We analyze the distribution of close binary stars in the orbital semimajor axis—primary mass plane. The reduced spatial density of stars with semimajor axes below 10R_⊙ is confirmed. We identify the area in this plane occupied by precursors of W UMa stars, assuming that the driving force causing the components to approach each other is their magnetic stellar wind. This scenario enables us to estimate the rate of formation (0.02/year) and lifetime (10⁸ yr) of W UMa stars. We derive a theoretical estimate of the ratio of the number of blue stragglers, N _BS , and of horizontal-branch stars, N _HB , in globular clusters based on the hypothesis that all blue stragglers are the result of component mergers in W UMa contact binaries. This ratio is N _BS /N _HB =0.4, close to the observed value for 62 Galactic globular clusters. We discuss possible reasons for the considerable dispersion of the observed estimates of this ratio for different clusters in our Galaxy.     

An estimate of the probability of capture of a binary star by a supermassive black hole

A simple model for the dynamics of stars located in a sphere with a radius of one-tenth of the central parsec, designed to enable estimation of the probability of capture in the close vicinity (r < 10^?3 pc) of a supermassive black hole (SMBH) is presented. In the case of binary stars, such a capture with a high probability results in the formation of a hyper-velocity star. The population of stars in a sphere of radius <0.1 pc is calculated based on data for the Galactic rotation curve. To simulate the distortion of initially circular orbits of stars, these are subjected to a series of random shock encounters (“kicks”), whose net effect is to “push” these binary systems into the region of potential formation of hyper-velocity stars. The mean crossing time of the border of the close vicinity of the SMBH (r < 10^?3 pc) by the stellar orbit can be used to estimate the probability that a binary system is captured, followed by the possible ejection of a hyper-velocity star.     

Masses of stellar black holes and testing theories of gravitation

The paper analyzes the mass distribution of stellar black holes derived from the light and radial-velocity curves of optical stars in close binary systems using dynamical methods. The systematic errors inherent in this approach are discussed. These are associated primarily with uncertainties in models for the contribution from gaseous structures to the optical brightness of the systems under consideration. The mass distribution is nearly flat in the range 4–15M _⊙. This is compared with the mass distribution for black holes in massive close binaries, which can be manifest as ultrabright X-ray sources (L _x >10³⁹ erg/s) observed in other galaxies. If the X-ray luminosities of these objects correspond to the Eddington limit, the black-hole mass distribution should be described by a power law, which is incompatible with the flat shape derived dynamically from observations of close binaries in our Galaxy. One possible explanation of this discrepancy is the rapid evaporation of stellar-mass black holes predicted in recent multi-dimensional models of gravity. This hypothesis can be verified by refining the stellar black-hole mass spectrum or finding isolated or binary black holes with masses below ～3M _⊙.     

Redetermination of the helium abundances in HII regions in Blue Compact Dwarf Galaxies

The He/H abundances in HII regions in Blue Compact Dwarf Galaxies are redetermined using the new recombination coefficients of Benjamin et al. The electron number density n _e in the He⁺ region, optical depth τ₃₈₈₉ in the HeI λ3889 Å line, and coefficient of underlying stellar absorption a_HeI are determined using the self-consistent method of Olive and Skillman. The primordial helium abundance and its enrichment are found using the helium abundances obtained in this paper and heavy-element abundances from our recent paper I: Y _p = 0.244 ± 0.004 and dY/dZ = 8.8 ± 4.6.     

Apsidal motion and physical parameters of the eclipsing binary system AR Cas

Using the four-channel automatic photoelectric photometer of the Sternberg Astronomical Institute’s Tien Shan Mountain Observatory, we have acquired accurate (σ_obs≈0.004^m) W BV R brightness measurements for the eclipsing binary AR Cas during selected phases before eclipse ingress and after egress, as well as at the center of minima. A joint analysis of these measurements with other published data has enabled us to derive for the first time a self-consistent set of physical and geometrical parameters for the star and the evolutionary age of its components, t=(60±3)×10⁶ years. We have found the period of the apsidal motion (U_obs=1100±160 years, \(\dot \omega _{obs} = 0^\circ .327 \pm 0^\circ .049\) years^?1) and the apsidal parameter of the primary, logk _2,1 ^obs =?2.41±0.08, with the apsidal parameter being in good agreement with current models of stellar evolution. There is an ultraviolet excess in the primary’s radiation, Δ(U?B)=?0.12^m and Δ(B?V)=?0.06^m, possibly due to a metal deficiency in the star’s atmosphere.