We revisit the modal analysis of small perturbations in Keplerian ideal gas flows with a constant vertical magnetic field leading to magneto-rotational instability (MRI) using the nonlocal approach. In the general case, MRI modes are described by a Schro center dot dinger-like differential equation with some effective potential, including `repulsive' (1/r(2)) and `attractive' (-1/r(3)) terms, and are quantized. In shallow potentials, there are no stationary `energy levels.' In thin Keplerian accretion discs, the perturbation wavelengths lambda = 27t/k(z) are smaller than the disc semi-thickness h only in `deep' potential wells. We find that there is a critical magnetic field for the MRI to develop. The instability arises for magnetic fields below this critical value. In thin accretion discs, at low background Alfven velocity c(A) << (c(A))(cr), the MRI instability increment omega is suppressed compared to the value obtained in the local perturbation analysis, omega approximate to -root 3ic(A)k(z). We also investigate for the first time the case of a radially variable background magnetic field.

The flute instability at the inner edge of a thin diamagnetic accretion disk is analyzed. The magnetic field configuration model from Aly (1980) is used. We have analyzed a modified dispersion relation for the flute instability that takes into account the Keplerian disk rotation. We have derived the inner radius of the accretion disk within our analysis of the flute instability. We show that the inner radius does not differ from the Alfvén radius for spherical accretion to within a dimensionless coefficient, with the proportionality coefficient depending only on the turbulence alpha parameter and the relative disk thickness ( h/r ).

We revisit the modal analysis of small perturbations in Keplerian ideal gas flows leading to magneto-rotational instability (MRI) using the non-local approach. We consider the case of constant vertical background magnetic field, as well as the case of radially dependent background Alfv\'en velocity. In the case of constant Alfv\'en velocity, MRI modes are described by a Schr\"odinger-like differential equation with some effective potential including 'repulsive' ($1/r^2$) and 'attractive' ($-1/r^3$) terms. Taking into account the radial dependence of the background Alfv\'en speed leads to a qualitative change in the shape of the effective potential. It is shown that there are no stationary energy levels corresponding to unstable modes $\omega^2<0$ in ``shallow'' potentials. In thin accretion disks, the wavelength of the disturbance $\lambda=2\pi/k_z$ is smaller than the half-thickness $h$ of the disk only in ``deep'' potentials. The limiting value of the background Alfv\'en speed $(c_A)_\mathrm{cr}$, above which the magnetorotational instability does not occur, is found. In thin accretion disks with low background Alfv\'en speed $c_A\ll (c_A)_\mathrm{cr}$, the increment of the magnetorotational instability $\omega\approx -\sqrt{3}\mathrm{i}c_Ak_z$ is suppressed compared to the value obtained in the local perturbation analysis.

The work is devoted to synthetic light curves modeling for the observed binary stellar systems and an interpretation of their characteristic features including pre-eclipse humps, asymmetry in the vicinity of the eclipse and variability of light curves from revolution to revolution. The mathematical model of the accretion disk including the radiation cooling of the plasma, gravitational forces and incomplete plasma ionization has been studied. The results of the plasma flows calculation allow one to use the radiated energy for synthetic light curves construction. In a number of simulations the formation of an inclined accretion disk is noted. The inclination may be caused by the instability of the circumstellar plasma flowing around the disk at the initial stages of its formation. The disk obtained in the calculations retrogradely precesses with a period of about 40 orbital periods of the system. Due to this precession in different orbital periods the jet enters the disk in different places. This may explain the significant variability of the light curves of the binary star system as well as the presence of brightness humps at the eclipse.

In this work are presented the results of modelling of 35 d superorbital changes of B and V lightcurves and X-ray flux of HZ Her/Her X-1. The model implemented in the new code written in C programming language, with module for parameter optimisation written in Python. The model includes a tilted precessing and warped accretion disc around a freely precessing neutron star. The disc is warped near its inner edge due to interaction with the rotating neutron star magnetosphere. The magnetic torque depends on the precessional phase of the neutron star. The X-ray emission flux from the neutron star also depends on the free precession phase which modulates the X-ray illumination of the optical star atmosphere and the intensity of gas streams. We demonstrate that this model is able to well reproduce both optical observations of HZ Her and the behaviour of the 35-day X-ray cycle.

The physical conditions of convection appearance in laminar accretion flows with microscopic transport coefficients are examined. Hot sparse ionised flow with periods below an hour found to be optically thin and have convective layer. Cold sparse molecular flow with period about a year found to be optically thin too and are fully convective. Ranges of temperature, density and period of optically thick laminar accretion flow are shown.

Nikolay Shakura∗1,2, Dmitry Kolesnikov1, Konstantin Postnov1,2, Igor Volkov1,3, Ilfan Bikmaev2, Tatiana Irsmambetova1, Rüdiger Staubert4, Joern Wilms5, Eldar Irtuganov2, Pavel Shurygin2, Polina Golysheva1, Sergey Shugarov1,6, Igor Nikolenko3, Eugene Trunkovsky1, Gabriele Schönherr8, Axel Schwope8, Dmitry Klochkov4 1Sternberg Astronomical Institute, Moscow State University, 119234, Moscow, Russia 2Kazan Federal University, Kazan, Russia 3Institute of Astronomy RAS, Moscow, Russia 4Institute for Astronomy and Astrophysics, Tubingen, Germany, 5Astronomical Institute of the University of Erlangen-Nuremberg, Bamberg, Germany, 6Astronomical Institute of the Slovak Academy of Scienses, Tatranska Lomnica, Slovakia 7Crimean Astrophysical Observatory, Nauchny, Russia 8Leibniz Institute for Astrophysics, Potsdam, Germany E-mail: nikolai.shakura@gmail.com, kolesnikovkda@gmail.com

2070 unique, homogeneous photometric and polarization observations of the microquasar in a binary system with a black hole V404 Cyg/GS2023+338 obtained in 2015 with the MASTER global network of robotic telescopes (16 robotic telescopes located at eight points on the Earth in Russia, Spain, South Africa, and Argentina) are presented. MASTER was the first telescope network to obtain optical observations of the microquasar after its gamma-ray outburst in 2015. Observations were carried out from 18:34:09 UT on June 15, 2015 until December 2015 in four polarizations and in the four standard BV RI filters. The paper presents the results of these observations and a comparative analysis of optical and X-ray data. The observations confirm the previously discovered super-long delays of the optical radiation relative to the X-ray outbursts. Possible mechanisms causing the delay in the optical variations relative to the X-ray variations are discussed. Variability of the optical polarization discovered earlier is confirmed another similar episode reported.

We present several thousands homogeneous photometric (WBVRI and polarization) observations of the microquasar black hole binary V404 Cyg, obtained by the MASTER Global Robotic Network (Lipunov et al. 2010). MASTER were the first telescopes to optically monitor the microquasar after its gamma-ray onset from 18 h 34 m 09 s UT on June 15, 2015 (Barthelmy et al. 2015a, Lipunov et al. 2015a, 2016a), until December 31, 2015. We report the results of the analysis of these observations and compare them with the observed X-ray and gamma-ray activity. We confirm the optical emission is correlated with the hard X-ray radiation on timescales of less than 1 min to several hundreds of seconds sec and some anomalously long timescales of up to 25-30 min (Rodriguez et al., 2015). We find no correlation between the delays and the orbital phase. We confirm polarization variability, published earlier (Lipunov et al., 2016a), and present new polarization variability episode.

The observed evolution of the broad-band spectral energy distribution (SED) in NS X-ray Nova Aql X-1 during the rise phase of a bright FRED-type outburst in 2013 can be understood in the framework of thermal emission from unstationary accretion disc with temperature radial distribution transforming from a single-temperature blackbody emitting ring into the multi-colour irradiated accretion disc. SED evolution during the hard to soft X-ray state transition looks curious, as it can not be repro- duced by the standard disc irradiation model with a single irradiation parameter for NUV, Optical and NIR spectral bands. NIR (NUV) band is correlated with soft (hard) X-ray flux changes during the state transition interval, respectively. In our interpreta- tion, at the moment of X-ray state transition UV-emitting parts of the accretion disc are screened from direct X-ray illumination from the central source and are heated primary by hard X-rays (E > 10 keV), scattered in the hot corona or wind possibly formed above the optically-thick outer accretion flow; the outer edge of multi-colour disc, which emits in Optical-NIR, can be heated primary by direct X-ray illumination. We point out that future simultaneous multi-wavelength observations of X-ray Nova systems during the fast X-ray state transition interval are of great importance, as it can serve as 'X-ray tomograph' to study physical conditions in outer regions of accretion flow. This can provide an effective tool to directly test the energy-dependent X-ray heating efficiency, vertical structure and accretion flow geometry in transient LMXBs.

This small methodological chapter is devoted to considering the motion of particles along spherical geodesical trajectories around rotating black holes. The study of this motion is necessary for understanding the inner structure of the disc tilted to the equatorial plane of the rotating black hole. Moreover, this chapter uses a special approach to find out how the values that are measured in a local Lorentz frame of observers falling freely in an axially symmetric gravitational field are related to each other. This approach allows us to understand better the basic principles of measuring physical values in general relativity. These basic principles, which are systematically presented in the next chapter, are required for a more comprehensive understanding the structure of relativistic tilted accretion discs.

In this chapter, conditions of the Velikhov-Chandrasekhar magneto-rotational instability (MRI) in ideal and non-ideal plasmas are examined. A linear WKB analysis of hydromagnetic axially symmetric flows shows that in the Rayleigh-unstable hydrodynamic case where the angular momentum decreases with radius, the MRI branch becomes stable, and the magnetic field suppresses the Rayleigh instability at small wavelengths. We investigate the limiting transition from hydromagnetic flows to hydrodynamic flows. The Rayleigh mode smoothly transits to the hydrodynamic case, while the Velikhov-Chandrasekhar MRI mode completely disappears without the magnetic field. The effects of viscosity and magnetic diffusivity in the plasma on the MRI conditions in thin accretion discs are studied. We find the limits on the mean free-path of ions allowing MRI to operate in such discs.

Accretion discs are powerful energy factories in our Universe. They effectively transform the potential energy of gravitational interaction to emission, thereby unraveling the physics of distant objects. This is possible due to the presence of viscosity, driven by turbulent motions in accretion discs. In this chapter, we describe the equations for disc accretion in the framework of the standard model. We outline basic elements of the theory of turbulent viscosity and the emergence of the a-parameter. We further describe the radial and vertical structure of thin stationary accretion discs, and present analytical solutions to the basic equation of the evolution of a viscous accretion disc for both an infinite disc and for a disc in a binary system. Finally, we present a numerical method to solve the equations of disc evolution and vertical structure simultaneously.

This graduate-level book highlights selected topics of standard and modern theory of accretion onto black holes and magnetized neutron stars. The principle author and volume editor, Nikolay Shakura, is one of the founders of the disc accretion theory.

As early as in 1937, Zwicky wrote about gravitational lenses acting as 'space telescopes', allowing the observation of faint and distant objects, the fluxes from which may be considerably enhanced due to the lensing. It is clear today that gravitational lensing may be helpful in performing another important task, one of the main purposes of telescopic observations, namely, increasing spatial resolution. The images of strongly lensed QSOs are affected by microlensing effects in the halo of the lensing galaxy. In contrast to the classical strong lensing, these effects are sensitive to the size and form of an object. The purpose of this chapter is to give a general introduction to quasar microlensing and to illustrate the capabilities of the method, with a review of the latest results in this field, concentrating especially on the results obtained in our three recent papers.

A theory of quasi-spherical subsonic accretion onto slowly rotating magnetized neutron stars is presented. In this regime, the accreted matter settles with subsonic velocities onto the rotating magnetosphere forming an extended quasi-spherical shell. The accretion rate in the shell is determined by the ability of the plasma to enter the magnetosphere due to the Rayleigh-Taylor instability with account for cooling. This accretion regime may be established for moderate X-ray luminosities, corresponding to accretion rates. (M) over dot < (M) over dot dagger similar or equal to 4 x 10(16) g s(-1). For higher accretion rates a free-fall gap appears, due to strong Compton cooling of the flow above the magnetosphere, and accretion becomes highly non-stationary. Observations of spin-up and spin-down in equilibrium wind-fed X-ray pulsars with known orbital periods (like GX 301-2 and Vela X-1) enable the determination of the basic dimensionless model parameters and estimation of the neutron star magnetic field. In equilibrium pulsars with independently measured magnetic fields, the model enables the stellar wind velocity to be independently estimated. For nonequilibrium pulsars, there exists a maximum spin-down rate of the accreting neutron star. The model can also explain bright flares in Supergiant Fast X-ray Transients if stellar winds of the O-supergiant companions are magnetized.

Approximate gravitational potentials are often used to describe analytically the motion of particles near black holes (BHs), as well as to study the structure of an accretion disk. Such 'pseudo-Newtonian' potentials are used with the flat-metric equations. Here we consider the motion of a free particle near a non-rotating BH in the context of an exact `logarithmic' gravitational potential. We show how the logarithmic potential gives an exact solution for a mechanical problem and present the relativistic Bernoulli equation for the fluid in the Schwarzschild metric.

This brief review is dedicated to academician Yakov Borisovich Zeldovich, the science of his epoch and the creation of modern accretion theory.