Studies of accretion flows around rotating black holes – II. Standing shocks in the pseudo-Kerr geometry

Standing, propagating or oscillating shock waves are common in accretion and winds around compact objects. We study the topology of all possible solutions using the pseudo-Kerr geometry. We present the parameter space spanned by the specific energy and angular momentum and compare it with that obtained from the full general relativity to show that the potential can work satisfactorily in fluid dynamics also, provided the polytropic index is suitably modified. We then divide the parameter space depending on the nature of the solution topology. We specifically study the nature of the standing Rankine–Hugoniot shocks. We also show that as the Kerr parameter is increased, the shock location generally moves closer to the black hole. In future, these solutions can be used as guidelines to test numerical simulations around compact objects.     

Radiatively driven winds from effective boundary layer around black holes

Matter accreting onto black holes suffers a standing or oscillating shock wave in much of the parameter space. The post-shock region is hot, puffed up and reprocesses soft photons from a Keplerian disc to produce the characteristic hard tail of the spectrum of accretion discs. The post-shock torus is also the base of the bipolar jets. We study the interaction of these jets with the hard photons emitted from the disc. We show that radiative force can accelerate outflows but the drag can limit the terminal speed. We introduce an equilibrium speed υ_eq as a function of distance, above which the flow will experience radiative deceleration.     

Behaviour of dissipative accretion flows around black holes

We investigate the behaviour of dissipative accreting matter close to a black hole, as this provides important observational features of galactic and extragalactic black hole candidates. We find a complete set of global solutions in the presence of viscosity and synchrotron cooling. We show that advective accretion flow can have a standing shock wave and the dynamics of the shock is controlled by the dissipation parameters (both viscosity and cooling). We study the effective region of the parameter space for standing as well as oscillating shock. We find that the shock front always moves towards the black hole as the dissipation parameters are increased. However, viscosity and cooling have opposite effects in deciding the solution topologies. We obtain two critical cooling parameters that separate the nature of the accretion solution.     

Mini-discs around spinning black holes

Accretion on to black holes in wind-fed binaries and in collapsars forms small rotating discs with peculiar properties. Such 'mini-discs' accrete on the free-fall time without the help of viscosity and nevertheless can have a high radiative efficiency. The inviscid mini-disc model was previously constructed for a non-rotating black hole. We extend the model to the case of a spinning black hole, calculate the structure and radiative efficiency of the disc and find their dependence on the black hole spin. If the angular momenta of the disc and the black hole are anti-aligned, a hydrodynamic analogue of Penrose process takes place.     

Steady shocks around black holes produced by sub-Keplerian flows with negative energy

We discuss a special case of formation of axisymmetric shocks in the accretion flow of ideal gas on to a Schwarzschild black hole: when the total energy of the flow is negative. The result of our analysis enlarges the parameter space for which these steady shocks are exhibited in the accretion of gas rotating around relativistic stellar objects. Since Keplerian discs have negative total energy, we guess that, in this energy range, the production of the shock phenomenon might be easier than in the case of positive energy. So our outcome reinforces the view that sub-Keplerian flows of matter may significantly affect the physics of the high energy radiation emission from black hole candidates. We give a simple procedure to obtain analytically the position of the shocks. The comparison of the analytical results with the data of one-dimensional (1D) and two-dimensional (2D) axisymmetric numerical simulations confirms that the shocks form and are stable.     

Smoothed particle hydrodynamic simulations of viscous accretion discs around black holes

Viscous Keplerian discs become sub-Keplerian close to a black hole since they pass through sonic points before entering into it. We study the time evolution of polytropic viscous accretion discs (both in one- and two-dimensional flows) using smoothed particle hydrodynamics. We discover that for a large region of the parameter space spanned by energy, angular momentum and polytropic index, when the flow viscosity parameter is less than a critical value, standing shock waves are formed. If the viscosity is very high then the shock wave disappears. In the intermediate viscosity, the disc oscillates very significantly in the viscous time-scale. Our simulations indicate that these centrifugally supported high density regions close to a black hole play an active role in the flow dynamics, and consequently, the radiation dynamics.     

Model dependence of transonic properties of accretion flows around black holes

We analytically study how the behaviour of accretion flows changes when the flow model is varied. We study the transonic properties of the conical flow, a flow of constant height and a flow in vertical equilibrium, and show that all these models are basically identical, provided that the polytropic constant is suitably changed from one model to another. We show that this behaviour is extendible even when standing shocks are produced in the flow. The parameter space where shocks are produced remains roughly identical in all these models when the same transformation among the polytropic indices is used. We present applications of these findings.     

Emission-line profiles from self-gravitating toroids around black holes

We have computed line profiles from self-gravitating toroids around black holes. The specific angular momentum of the toroids is assumed to be constant in space. The images of the toroids show peculiar features in the far sides of the black holes. Concerning the line profiles, the red wing extends to the very low frequency region because the location of the inner edge is rather near the event horizon of the black hole and consequently the velocity of the inner edges of the toroids can be greater than that of the Kepler discs.     

Spectral variability in transonic discs around black holes

Transonic discs with accretion rates relevant to intrinsically bright Galactic X-ray sources ( L ≈10³⁸–10³⁹ erg s⁻¹) exhibit a time-dependent cyclic behaviour due to the onset of a thermal instability driven by radiation pressure. In this paper we calculate radiation spectra emitted from thermally unstable discs to provide detailed theoretical predictions for observationally relevant quantities. The emergent spectrum has been obtained by solving self-consistently the vertical structure and radiative transfer in the disc atmosphere. We focus on four particular stages of the disc evolution, the maximal evacuation stage and three intermediate stages during the replenishment phase. The disc is found to undergo rather dramatic spectral changes during the evolution, emitting mainly in the 1–10 keV band during outburst and in the 0.1–1 keV band off-outburst. Local spectra, although different in shape from a blackbody at the disc effective temperature, may be characterized in terms of a hardening factor f . We have found that f is more or less constant, both in radius and in time, with a typical value ∼ 1.65.     

Dynamical properties of a centrifugally driven outflow from a disc near a black hole

The inclination of the field lines at the surface of the disc plays a crucial role on the nature of the magnetically driven flows. For the non-relativistic case, a centrifugally driven outflow of matter from the disc is possible, if the poloidal component of the magnetic field makes an angle of less than a critical 60° with the disc surface (Blandford and Payne 1982). We investigate the dynamical properties of the magnetically driven flows from the disc near a non-rotating black hole and find that the critical angle becomes larger than 60° when the flows start from the region near the black hole.     

Gravitational instability of discs with dissipative coronae around supermassive black holes

We study the dynamical structure of a self-gravitating disc with coronae around a supermassive black hole. Assuming that the magnetorotational instability responsible for generating the turbulent stresses inside the disc is also the source for a magnetically dominated corona, a fraction of the power released when the disc matter accretes is transported to and dissipated in the corona. This has a major effect on the structure of the disc and its gravitational (in)stability according to our analytical and self-consistent solutions. We determine the radius where the disc crosses the inner radius of gravitational instability and forms the first stars. Not only the location of this radius which may extend to very large distances from the central black hole, but also the mass of the first stars highly depends on the input parameters, notably the viscosity coefficient, the mass of the central object and the accretion rate. For accretion discs around quasi-stellar objects (QSOs) and the Galactic Centre, we determine the self-gravitating radius and the mass of the first clumps. Comparing the cases with a corona and without a corona for typical discs around QSOs or the Galactic Centre, when the viscosity coefficient is around 0.3, we show that the self-gravitating radius decreases by a factor of approximately 2, but the mass of the fragments increases with more or less the same factor. The existence of a corona implies a more gravitationally unstable disc according to our results. The effect of a corona on the instability of the disc is more effective when the viscosity coefficient increases.     

The runaway instability of self-gravitating tori with non-constant specific angular momentum around black holes

We discuss the runaway instability of axisymmetric tori with non-constant specific angular momentum around black holes, taking into account self-gravity of the tori. The distribution of specific angular momentum of the tori is assumed to be a positive power law with respect to the distance from the rotational axis. By employing the pseudo-Newtonian potential for the gravity of the spherical black hole, we have found that self-gravity of the tori causes a runaway instability if the amount of the mass which is transferred from the torus to the black hole exceeds a critical value, i.e. 3 per cent of the mass of the torus. This has been shown by two different approaches: (1) by using equilibrium models and (2) by dynamical simulations. In particular, dynamical simulations using an SPH code have been carried out for both self-gravitating and non-self-gravitating tori. For non-self-gravitating models, all tori are runaway stable. Therefore we come to the conclusion that self-gravity of the tori has a stronger destabilizing effect than the stabilizing effect of the positive power-law distribution of the angular momentum.     

Dissipative accretion flows around a rotating black hole

We study the dynamical structure of a cooling dominated rotating accretion flow around a spinning black hole. We show that non-linear phenomena such as shock waves can be studied in terms of only three flow parameters, namely the specific energy     , the specific angular momentum (λ) and the accretion rate     of the flow. We present all possible accretion solutions. We find that a significant region of the parameter space in the     plane allows global accretion shock solutions. The effective area of the parameter space for which the Rankine–Hugoniot shocks are possible is maximum when the flow is dissipation-free. It decreases with the increase of cooling effects and finally disappears when the cooling is high enough. We show that shock forms further away when the black hole is rotating compared to the solution around a Schwarzschild black hole with identical flow parameters at a large distance. However, in a normalized sense, the flow parameters for which the shocks form around the rotating black holes are produced shocks closer to the black hole. The location of the shock is also dictated by the cooling efficiency in that higher the accretion rate     , the closer is the shock location. We believe that some of the high-frequency quasi-periodic oscillations may be due to the flows with higher accretion rate around the rotating black holes.     

Warp and eccentricity propagation in discs around black holes

We consider the inward propagation of warping and eccentric disturbances in discs around black holes under a wide variety of conditions. In our calculations, we use secular theories of warped and eccentric discs and assume the deformations to be stationary and propagating in a disc model similar to regions (a) and (b) of Shakura & Sunyaev discs. We find that the propagation of deformations to the innermost regions of the disc is facilitated for low viscous damping and high accretion rate. We relate our results to the possible excitation of trapped inertial modes, and to the observations of high-frequency quasi-periodic oscillations (QPOs) in black hole systems in the very high spectral state.     

Variation of the gas and radiation content in the sub-Keplerian accretion disk around black holes and its impact to the solutions

We investigate the variation of the gas and the radiation pressure in accretion disks during the infall of matter to the black hole and its effect to the flow. While the flow far away from the black hole might be non-relativistic, in the vicinity of the black hole it is expected to be relativistic behaving more like radiation. Therefore, the ratio of gas pressure to total pressure (β) and the underlying polytropic index (γ) should not be constant throughout the flow. We obtain that accretion flows exhibit significant variation of β and then γ, which affects solutions described in the standard literature based on constant β. Certain solutions for a particular set of initial parameters with a constant β do not exist when the variation of β is incorporated appropriately. We model the viscous sub-Keplerian accretion disk with a nonzero component of advection and pressure gradient around black holes by preserving the conservations of mass, momentum, energy, supplemented by the evolution of β. By solving the set of five coupled differential equations, we obtain the thermo-hydrodynamical properties of the flow. We show that during infall, β of the flow could vary up to ∼300%, while γ up to ∼20%. This might have a significant impact to the disk solutions in explaining observed data, e.g. super-luminal jets from disks, luminosity, and then extracting fundamental properties from them. Hence any conclusion based on constant γ and β should be taken with caution and corrected.