The role of flow geometry in influencing the stability criteria for low angular momentum axisymmetric black hole accretion

Using mathematical formalism borrowed from dynamical systems theory, a complete analytical investigation of the critical behaviour of stationary flows in low angular momentum axisymmetric black hole accretion, provides significant insight about the nature of the phase trajectories corresponding to transonic accretion in the steady state, without taking recourse to any explicit numerical method commonly reported in the literature on multi-transonic black hole accretion discs and related astrophysical phenomena. Investigation of an accretion process around a non-rotating black hole, forming different geometrical configurations of the flow structure under the influence of various pseudo-Schwarzschild potentials, reveals that the general profile of the parameter space divisions describing multi-critical accretion, is roughly equivalent for various flow geometries. However, a mere variation of the polytropic index of the flow cannot map a critical solution from one flow geometry to another, since the numerical domain of the parameter space responsible for producing multi-critical accretion does not undergo a continuous transformation in multi-dimensional parameter space. The stationary configuration used to demonstrate the aforementioned findings is shown to be stable under time-dependent linearised perturbations for all kinds of flow geometries, driven by any pseudo-Schwarzschild potential, and using a standard equation of state. Finally, the structure of the acoustic metric corresponding to the propagation of the linear perturbation is discussed for various flow geometries used.     

Model dependence of the multi-transonic behaviour,stability properties and the corresponding acoustic geometry for accretion onto rotating black holes

Stationary, multi-transonic, integral solutions of hydrodynamic axisymmetric accretion onto a rotating black hole have been compared for different geometrical configurations of the associated accretion disc structures described using the polytropic as well as the isothermal equations of state. Such analysis is performed for accretion under the influence of generalised post Newtonian pseudo Kerr black hole potential. The variations of the stationary shock characteristics with black hole spin have been studied in details for all the disc models and are compared for the flow characterised by the two aforementioned equations of state. Using a novel linear perturbation technique it has been demonstrated that the aforementioned stationary solutions are stable, at least upto an astrophysically relevant time scale. It has been demonstrated that the emergence of the horizon related gravity like phenomena (the analogue gravity effects) is a natural consequence of such stability analysis, and the corresponding acoustic geometry embedded within the transonic accretion can be constructed for the propagation of the linear acoustic perturbation of the mass accretion rate. The analytical expression for the associated sonic surface gravity κ has been obtained self consistently. The variations of κ with the black hole spin parameter for all different geometric configurations of matter and for various thermodynamic equations of state have been demonstrated.     

Black hole spin dependence of general relativistic multi-transonic accretion close to the horizon

We introduce a novel formalism to investigate the role of the spin angular momentum of astrophysical black holes in influencing the behavior of low angular momentum general relativistic accretion. We propose a metric independent analysis of axisymmetric general relativistic flow, and consequently formulate the space and time dependent equations describing the general relativistic hydrodynamic accretion flow in the Kerr metric. The associated stationary critical solutions for such flow equations are provided and the stability of the stationary transonic configuration is examined using an elegant linear perturbation technique. We examine the properties of infalling material for both prograde and retrograde accretion as a function of the Kerr parameter at extremely close proximity to the event horizon. Our formalism can be used to identify a new spectral signature of black hole spin, and has the potential of performing the black hole shadow imaging corresponding to the low angular momentum accretion flow.     

Standing shocks in magnetized dissipative accretion flow around black holes

We explore the global structure of the accretion flow around a Schwarzschild black hole where the accretion disc is threaded by toroidal magnetic fields. The accretion flow is optically thin and advection dominated. The synchrotron radiation is considered to be the active cooling mechanism in the flow. With this, we obtain the global transonic accretion solutions and show that centrifugal barrier in the rotating magnetized accretion flow causes a discontinuous transition of the flow variables in the form of shock waves. The shock properties and the dynamics of the post-shock corona are affected by the flow parameters such as viscosity, cooling rate and strength of the magnetic fields. The shock properties are investigated against these flow parameters. We further show that for a given set of boundary parameters at the outer edge of the disc, accretion flow around a black hole admits shock when the flow parameters are tuned for a considerable range.     

Study of accretion disk dynamics in presence of cooling

We examine the behaviour of accretion flow around a rotating black hole in presence of cooling. We obtain global flow solutions for various accretion parameters that govern the accreting flow. We show that standing isothermal shock wave may develop in such an advective accretion flow in presence of cooling. This shocked solution has observational consequences as it successfully provides the possible explanations of energy spectra as well as generation of outflows/jets of various galactic and extra-galactic black hole candidates. We study the properties of isothermal shock wave and find that it strongly depends on the cooling efficiency. We identify the region in the parameter space spanned by the specific energy and specific angular momentum of the flow for standing isothermal shock as a function of cooling efficiencies and find that parameter space gradually shrinks with the increase of cooling rates. Our results imply that accretion flow ceases to contain isothermal shocks when cooling is beyond its critical value.     

Effective sound speed in relativistic accretion discs around Schwarzschild black holes

For low angular momentum axially symmetric accretion flow maintained in hydrostatic equilibrium along the vertical direction, the value of the Mach number at the critical points deviates from unity, resulting in the non-isomorphism of the critical and the sonic points. This introduces several undesirable complexities while analytically dealing with the stationary integral accretion solutions and the corresponding phase portraits. We propose that the introduction of an effective dynamical sound speed may resolve the issue in an elegant way. We linear perturb the full spacetime-dependent general relativistic Euler and the continuity equations governing the structure and the dynamics of accretion disc in vertical equilibrium around Schwarzschild black holes and identify the sonic metric embedded within the stationary background flow. Such metric describes the propagation of the linear acoustic perturbation inside the accretion flow. We construct the wave equation corresponding to that acoustic perturbation and find the speed of propagation of such perturbation. We finally show that the ordinary thermodynamic sound speed should be substituted by the speed of propagation of the linear acoustic wave which has been obtained through the dynamical perturbation. Such substitution will make the value of Mach number at the critical point to be equal to unity. Use of the aforementioned effective sound speed will lead to a modified stationary disc structure where the critical and the sonic points will be identical.     

Critical properties of spherically symmetric accretion in a fractal medium

Spherically symmetric transonic accretion of a fractal medium has been studied in both the stationary and the dynamic regimes. The stationary transonic solution is greatly sensitive to infinitesimal deviations in the outer boundary condition, but the flow becomes transonic and stable when its evolution is followed through time. The evolution towards transonicity is more pronounced for a fractal medium than it is for a continuum, and in the former case the static sonic condition is met on relatively larger length scales. The dynamic approach also shows that there is a remarkable closeness between an equation  of motion for a perturbation in the flow, and the metric of an analogue acoustic black hole. The stationary inflow solutions of a fractal medium are as much stable under the influence of linearized perturbations as they are for the fluid continuum.     

黑洞吸积的双模式特征

黑洞吸积必定是跨声速的。对于静态、绝热吸积流,比能量E、比角动量L和质量吸积率M都是空间的常量。跨声速解的非奇异条件,F(E,L,M)=0,使独立参数减为只有两个。对于一对给定的E和L的符合条件E_c>E>E_(Barr)的值(这里E_c是一临界值,E_(Barr)是引力和离心力的联合势垒),上述非奇异条件给出两个不同的吸积率值,对应着两个不同的吸积流声速点位置。然而,物理上合理的整体解却是唯一的,它总是使两个吸积率值中之较小者得到实现。 对于一个不转动的黑洞,吸积以两种模式之一进行。一是类球吸积或称Bondi吸积,角动量的影响和相对论效应均微不足道;另一是盘吸积,这两个因素起决定性作用。两种模式之间的转换是基于声速点位置的间断性跳跃,而这种跳跃是由吸积流参数(例如角动量)的连续变化所引发。Bondi吸积可称为高态而盘吸积为低态,因为前者总对应着较高的吸积率。 随时间变化的吸积流很可能在这两种模式之间来回振荡,呈现出周期性或准周期性或无规则行为。这可以用来解释天鹅座X-1和若干活动星系核的光变现象,从而为黑洞的存在提供有力的观测依据。     

On some transonic aspects of general relativistic spherical accretion on to Schwarzschild black holes

The equations governing general relativistic, spherically symmetric, hydrodynamic accretion of polytropic fluid on to black holes are solved in the Schwarzschild metric to investigate some of the transonic properties of the flow. Only stationary solutions are discussed. For such accretion, it has been shown that real physical sonic points may form even for flow with   γ <4/3  or   γ >5/3  . The behaviour of some flow variables in the close vicinity of the event horizon is studied as a function of specific energy and the polytropic index of the flow.     

Global solutions of adiabatic accretion flows with isothermal shocks in Kerr black hole geometry

This paper presents global solutions of adiabatic accretion flows with isothermal shocks in Kerr black hole geometry. It is known that in the previously studied cases, where the flow including the shock is either entirely adiabatic or entirely isothermal, there can be no more than one stable shock solution, and the solution can only be of α –x type. However, the solution topology in the present case shows remarkable new characteristics: for the same flow parameters there can be two stable shock solutions satisfying physical boundary conditions, and the solution can be of three types, namely α– x , x –α and α–α type. In addition, shocks in the present case occur for a parameter region different from that for Rankine–Hugoniot shocks. These results greatly increase the possibilities of shock formation in astrophysical flows. It is also significant that the effects of frame-dragging of a rapid Kerr black hole on the shock formation are discovered. Finally, a brief comparison is made between shocked inviscid flows and two types of shock-free viscous flows, namely those of Shakura & Sunyaev and Narayan & Yi, and some comments are made about the fact that numerous authors who have studied transonic global solutions of accretion flows have found no shocks.     

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.     

Critical properties of spherically symmetric black hole accretion in Schwarzschild geometry

The stationary, spherically symmetric, polytropic and inviscid accretion flow in the Schwarzschild metric has been set-up as an autonomous first-order dynamical system, and it has been studied completely analytically. Of the three possible critical points in the flow, the one that is physically realistic behaves like the saddle point of the standard Bondi accretion problem. One of the two remaining critical points exhibits the strange mathematical behaviour of being either a saddle point or a centre-type point, depending on the values of the flow parameters. The third critical point is always unphysical and behaves like a centre-type point. The treatment has been extended to pseudo-Schwarzschild flows for comparison with the general relativistic analysis.     

Quasi-viscous accretion flow – I. Equilibrium conditions and asymptotic behaviour

In a novel approach to studying viscous accretion flows, viscosity has been introduced as a perturbative effect, involving a first-order correction in the α-viscosity parameter. This method reduces the problem of solving a second-order non-linear differential equation (Navier–Stokes equation) to that of an effective first-order equation. Viscosity breaks down the invariance of the equilibrium conditions for stationary inflow and outflow solutions, and distinguishes accretion from wind. Under a dynamical systems classification, the only feasible critical points of this 'quasi-viscous' flow are saddle points and spirals. On large spatial scales of the disc, where a linearized and radially propagating time-dependent perturbation is known to cause a secular instability, the velocity evolution equation of the quasi-viscous flow has been transformed to bear a formal closeness with Schrödinger's equation with a repulsive potential. Compatible with the transport of angular momentum to the outer regions of the disc, a viscosity-limited length-scale has been defined for the full spatial extent over which the accretion process would be viable.     

Instability of spherical accretion — I. Shock-free Bondi accretion

We examine the spatial stability of spherical adiabatic Bondi accretion on to a point gravitating mass against external perturbations. Both transonic critical and subsonic subcritical accretion are shown to be stable against purely radial acoustic, vortex or entropy perturbations. In the case of non-radial perturbations the amplitude of the perturbations grows without limit with smaller radii. Instability manifests itself only if the size of the accreting body is much less than the Bondi radius so that the inflow is highly supersonic or highly subsonic at the surface of the accretor in the case of critical or subcritical accretion respectively. These asymptotics hold and consequently the instability may develop for adiabatic index of accreting gas γ < 5/3. We suggest that this instability may lead to an essential thermalization of accreting flow thus, particularly, solving the problem of otherwise inefficient energy release in spherical accretion on to a black hole.     

Asymptotic Orbits at the Triangular Equilibria in the Photogravitational Restricted Three-Body Problem

We study numerically the asymptotic homoclinic and heteroclinic orbits associated with the triangular equilibrium points L ₄ and L ₅, in the gravitational and the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these critical points, are also presented. Hundreds of asymptotic orbits for equal mass of the primaries and for various values of the radiation pressure are computed and the most interesting of them are illustrated. In the Copenhagen case, which the problem is symmetric with respect to the x- and y-axis, we found and present non-symmetric heteroclinic asymptotic orbits. So pairs of heteroclinic connections (from L ₄ to L ₅ and vice versa) form non-symmetric heteroclinic cycles. The termination orbits (a combination of two asymptotic orbits) of all the simple families of symmetric periodic orbits, in the Copenhagen case, are illustrated.     

Time variability of viscosity parameter in differentially rotating discs

We propose a mechanism to produce fluctuations in the viscosity parameter (α) in differentially rotating discs. We carried out a nonlinear analysis of a general accretion flow, where any perturbation on the background α was treated as a passive/slave variable in the sense of dynamical system theory. We demonstrate a complete physical picture of growth, saturation and final degradation of the perturbation as a result of the nonlinear nature of coupled system of equations. The strong dependence of this fluctuation on the radial location in the accretion disc and the base angular momentum distribution is demonstrated. The growth of fluctuations is shown to have a time scale comparable to the radial drift time and hence the physical significance is discussed. The fluctuation is found to be a power law in time in the growing phase and we briefly discuss its statistical significance.     

Spectral properties of the accretion discs around rotating black holes

We study the radiation properties of an accretion disc around a rotating black hole. We solve the hydrodynamic equations and calculate the transonic solutions of accretion disc in the presence of shocks. Then we use these solutions to generate the radiation spectrum in the presence of radiative heating and cooling processes. We present the effect of spin parameter of the black hole on the emitted radiation spectrum. In addition, attention has also been paid to the variation in energy spectral index with Kerr parameter and accretion rate. We find that spectral index becomes harder as the spin parameter changes from negative (accretion disc is counter-rotating with respect to the black hole spin) to a positive value. Finally, we compute and compare the spectral characteristics due to a free-fall flow and a transonic flow. We notice significant differences in high energy contributions from these two solutions.     

Study of magnetized accretion flow with cooling processes

We have studied shock in magnetized accretion flow/funnel flow in case of neutron star with bremsstrahlung cooling and cyclotron cooling. All accretion solutions terminate with a shock close to the neutron star surface, but at some regions of the parameter space, it also harbours a second shock away from the star surface. We have found that cyclotron cooling is necessary for correct accretion solutions which match the surface boundary conditions.     

Linearized perturbation on stationary inflow solutions in an inviscid and thin accretion disc

The influence of a linearized perturbation on stationary inflow solutions in an inviscid and thin accretion disc has been studied here, and it has been argued that a perturbative technique would indicate that all possible classes of inflow solutions would be stable. The choice of the driving potential, Newtonian or pseudo-Newtonian, would not particularly affect the arguments which establish the stability of solutions. It has then been surmised that in the matter of the selection of a particular solution, adoption of a non-perturbative technique, based on a more physical criterion, as in the case of the selection of the transonic solution in spherically symmetric accretion, would give a more conclusive indication concerning the choice of a particular branch of the flow.