M. K. Abubekerov

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, mv = 70 M⊙, the mass of the compact object is mx = 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 1011M⊙. 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.

Masses of X-ray pulsars in binary systems with OB supergiants

We describe the results of a statistical approach to analyzing the combined radial-velocity curves of X-ray binaries with OB supergiants in a Roche model, both with and without allowance for the anisotropy of the stellar wind. We present new mass estimates for the X-ray pulsars in the close binary systems Cen X-3, LMC X-4, SMC X-1, 4U 1538-52, and Vela X-1.

Estimate of the Black-Hole Mass and Orbital Inclination from the Radial-Velocity Curve of the X-Ray Binary Cyg X-1

The results of a statistical approach to interpreting a master radial-velocity curve for the X-ray binary Cyg X-1 are presented. The dependence of the mass of the X-ray component mx on the mass of the optical component mv is obtained in a Roche model. A method for estimating the orbital inclination from the radial-velocity curve is described. In contrast to the situation for a pointlike optical star, both the amplitude and shape of the radial-velocity curve changes as a function of the orbital inclination i in the case of a tidally deformed star with a complex temperature distribution over its surface. Thus, high-accuracy radial-velocity curves can be used to impose constraints on the mass and orbital inclination of the black hole: i<45°. Using the information on the optical light curve, radius, and luminosity of the optical star, we estimate 31°

The mass of the compact object in the X-ray binary her X-1/HZ her

We have obtained the first estimates of the masses of the components of the Her X-1/HZ Her X-ray binary system taking into account non-LTE effects in the formation of the Hγ absorption line: mx = 1.8 M⊙ and mv = 2.5 M⊙. These mass estimates were made in a Roche model based on the observed radial-velocity curve of the optical star, HZ Her. The masses for the X-ray pulsar and optical star obtained for an LTE model lie are mx = 0.85 ± 0.15 M⊙ and mv = 1.87 ± 0.13 M⊙. These mass estimates for the components of Her X-1/HZ Her derived from the radial-velocity curve should be considered tentative. Further mass estimates from high-precision observations of the orbital variability of the absorption profiles in a non-LTE model for the atmosphere of the optical component should be made.

The mass of the compact object in the low-mass X-ray binary 2S 0921-630

We interpret the observed radial-velocity curve of the optical star in the low-mass X-ray binary 2S 0921-630 using a Roche model, taking into account the X-ray heating of the optical star and screening of X-rays coming from the relativistic object by the accretion disk. Consequences of possible anisotropy of the X-ray radiation are considered. We obtain relations between the masses of the optical and compact (X-ray) components, mv and mx, for orbital inclinations i = 60°, 75°, and 90°. Including X-ray heating enabled us to reduce the compact object’s mass by ∼0.5–1 M⊙, compared to the case with no heating. Based on the K0III spectral type of the optical component (with a probable mass of mv ≃ 2.9 M⊙), we concluded that mx ≃ 2.45−2.55 M⊙ (for i = 75°−90°). If the K0III star has lost a substantial part of its mass as a result of mass exchange, as in the V404 Cyg and GRS 1905+105 systems, and its mass is mv ≃ 0.65−0.75 M⊙, the compact object’s mass is close to the standard mass of a neutron star, mx ≃ 1.4 M⊙ (for i = 75°−90°). Thus, it is probable that the X-ray source in the 2S 0921-630 binary is an accreting neutron star.

How abundant is the population of binary radio pulsars with black holes

Using ”Scenario Machine” we have carried out population synthesis of radio pulsar with black hole binaries (BH+Psr) in context of the most wide assumptions about star mass loss during evolution, binary stars mass ratio distribution, kick velocity and envelope mass lost during collapse. Our purpose is to display that under any suppositional parameters of evolution scenario BH+Psr population have to be abundant in Galaxy. It is shown that in the all models including models evolved by Heger et al. (2002), Woosley et al. (2002), Heger et al. (2003) expected number of the black holes paired with radio pulsars is sufficient enough to discover such systems within the next few years.

A universal approach to the calculation of the transit light curves

We have developed a universal approach to compute accurately the brightness of eclipsing binary systems during the transit of a planet in front of the stellar disc. This approach is uniform for all values of the system parameters and applicable to most limb-darkening laws used in astrophysics. In the cases of the linear and quadratic limb-darkening laws, we obtained analytical expressions for the light curve and its derivatives in terms of elementary functions, elliptic integrals and a piecewise-defined function of one variable. In the cases of the logarithmic and square-root laws of limb darkening, the flux and its derivatives were expressed in terms of integrals which can be efficiently computed using the Gaussian quadrature formula, taking into account singularities of the integrand. Recently several authors have developed algorithms for the calculation of transit light curves (see e.g. Mandel & Agol 2002; Pal 2008; 2012). However, the problem of the calculation of the light curves is still relevant, because the existing algorithms are not applicable to all values of the system parameters for some limb-darkening laws. Besides, they do not allow sufficiently accurate calculations of the light curves for some limb-darkening laws. In addition, calculations of derivatives of the light curve as a function of system parameters are important, because they can be used to solve the inverse problem of interpretation of the light curve. The paper Mandel & Agol (2002) contains an analytical expression of the light curve by elliptic integrals, for the cases of the linear and quadratic limb-darkening laws. In doing so, 13 variants of relations between the parameters are considered. For other limbdarkening laws (law of square root and its power) only an approximate method of light-curve calculation at the radius of the planet more than 10 times smaller than the radius of the star is being used. In this case, the accuracy is 2 per cent of the depth of the eclipse. In the paper Pal (2012), directly, there is only an expression of the light curve in the linear and quadratic limb-darkening laws, and the derivatives of the light curve are calculated by difference methods which is less favourable in terms of the time and accuracy of the computation. (This work contains no direct analytical expressions for the derivatives.) In addition, none of the above works considered the logarithmic law of darkening, which is most preferred for early-type stars (Klinglesmith & Sobieski 1970; Van Hamme 1993). The approach presented in this paper allows us to calculate a light curve and with almost machine accuracy for any values of the parameters (including near singularities). Binary system parameters are the radii of the components and the distance between the centres of the components in the projection on the picture plane. In general, the algorithm is uniform for all values of the system parameters, which significantly facilitates its implementation. We obtained analytical expressions for the transit light curve of the eclipsing binary system and for its derivatives in the cases of the linear and quadratic limb-darkening laws. These quantities are expressed in terms of a piecewise-defined function of one variable and incomplete elliptic integrals, which can be computed with methods proposed by Carlson (1995). In the cases of the logarithmic limb-darkening law and the square-root limb-darkening law, the light function is expressed through integrals that can be efficiently computed using the Gaussian quadrature formula. In this respect, the computation time of the light curve is not much more than the computation time by analytical expressions.

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.

Estimation of parameter errors in inverse problems. Determining limb-darkening coefficients in classical eclipsing systems

The estimation of parameters and their errors is considered using the observed light curve of the eclipsing binary system YZ Cas as an example. The error intervals are calculated using the differential-correction and confidence-region methods. The error intervals and reliability of the methods are investigated, and the reliability of limb-darkening coefficients derived from the observed light curve analyzed. A new method for calculating parameter errors is proposed.

Analysis of light curves of eclipsing systems with exoplanets: HD 189733

High-accuracy multicolor light curves of the binary system HD 189733, which contains an exoplanet, are analyzed. We have determined the radii of the star and the planet in the binary system, as well as the orbital inclination. The limb-darkening coefficients of the stellar disk were obtained in 10 filters in the wavelength interval λλ = 5500−10 500 Å. The uncertainties of the fitted parameters were estimated using the differential-correction and confidence-area methods. The wavelength dependence of the limb-darkening coefficients is compared to the corresponding theoretical function for a model thin stellar atmosphere. We confirm the wavelength dependence of the exoplanet’s radius found by Pont et al. (at the 1σ level). The exoplanet radius increases with decreasing wavelength, which seems to argue for the presence of an atmosphere around the planet.