Studies of accretion flows around rotating black holes – III. Shock oscillations and an estimation of the spin parameter from QPO frequencies

In the present communication of our series of papers dealing with the accretion flows in the pseudo-Kerr geometry, we discuss the effects of viscosity on the accretion flow around a rotating black hole. We find the solution topologies and give special attention to the solutions containing shocks. We draw the parameter space where standing shocks are possible and where the shocks could be oscillating and could produce quasi-periodic oscillations (QPOs) of X-rays observed from black hole candidates. In this model, the extreme locations of the shocks give the upper limits of the QPO frequencies  (ν_QPO)  which could be observed. We show that both the viscosity of the flow and the spin of the black hole a increase the QPO frequency while, as expected, the black hole mass reduces the QPO frequencies. Our major conclusion is that the highest observed frequency gives a strict lower limit of the spin. For instance, a black hole exhibiting  ν_QPO∼ 400  and  700 Hz  must have the spin parameters of   a > 0.25  and  >0.75  , respectively, provided viscosity of the flow is small. We discuss the implications of our results in the light of observations of QPOs from black hole candidates.     

Limit-Cycle Behaviour of Thermally Unstable Accretion Flows on to Black Holes

Non-linear time-dependent calculations are being carried out in order to study the evolution of vertically integrated models of non-self-gravitating, transonic accretion discs around black holes. In this paper we present results from a new calculation for a high-α model similar to one studied previously by Honma, Matsumoto &38; Kato who found evidence for limit-cycle behaviour connected with thermal instability. Our results are in substantial agreement with theirs but, in our calculation, the disc material does not always remain completely optically thick and we include a suitable treatment for this. We followed the evolution for several cycles and determined the period of the cycle as being about 780 s. Advective cooling is dominant in the region just behind the outward-moving peak of surface density. The behaviour of this model is significantly different from what we saw earlier for low-α models (which we discussed in a previous paper) and we contrast and compare the two situations.     

Hydrodynamic simulations of propagating warps and bending waves in accretion discs

We present the results of a study of propagating warp or bending waves in accretion discs. Three-dimensional hydrodynamic simulations were performed using smoothed particle hydrodynamics (SPH), and the results are compared with calculations based on the linear theory of warped discs.
We examine the response of a gaseous disc to an initially imposed warping disturbance under a variety of physical conditions. We consider primarily the physical regime in which the dimensionless viscosity parameter α < H r , where H r is the disc aspect ratio, so that bending waves are expected to propagate. We also performed calculations for disc models in which α > H r , where the warps are expected to evolve diffusively. Small-amplitude (linear) perturbations are studied in both Keplerian and slightly non-Keplerian discs, and we find that the results of the SPH calculations can be reasonably well fitted by those of the linear theory. The main results of these calculations are: (i) the warp in Keplerian discs when α < H r propagates with little dispersion, and damps at a rate expected from estimates of the code viscosity; (ii) warps evolve diffusively when α > H r ; (iii) the slightly non-Keplerian discs lead to a substantially more dispersive behaviour of the warps, which damp at a similar rate to the Keplerian case, when α < H r .
Initially imposed higher amplitude, non-linear warping disturbances were studied in Keplerian discs. The results indicate that non-linear warps can lead to the formation of shocks, and that the evolution of the warp becomes less wave-like and more diffusive in character.
This work is relevant to the study of the warped accretion discs that may occur around Kerr black holes or in misaligned binary systems, and is mainly concerned with discs in which α < H r . The results indicate that SPH can model the hydrodynamics of warped discs, even when using rather modest numbers of particles.     

Models for jet power in elliptical galaxies: a case for rapidly spinning black holes

The power of jets from black holes is expected to depend on both the spin of the black hole and the structure of the accretion disc in the region of the last stable orbit. We investigate these dependencies using two different physical models for the jet power: the classical Blandford–Znajek (BZ) model and a hybrid model developed by Meier. In the BZ case, the jets are powered by magnetic fields directly threading the spinning black hole while in the hybrid model, the jet energy is extracted from both the accretion disc as well as the black hole via magnetic fields anchored to the accretion flow inside and outside the hole's ergosphere. The hybrid model takes advantage of the strengths of both the Blandford–Payne and BZ mechanisms, while avoiding the more controversial features of the latter. We develop these models more fully to account for general relativistic effects and to focus on advection-dominated accretion flows (ADAFs) for which the jet power is expected to be a significant fraction of the accreted rest mass energy.
We apply the models to elliptical galaxies, in order to see if these models can explain the observed correlation between the Bondi accretion rates and the total jet powers. For typical values of the disc viscosity parameter  α∼ 0.04 –0.3  and mass accretion rates consistent with ADAF model expectations, we find that the observed correlation requires   j ≳ 0.9  ; that is, it implies that the black holes are rapidly spinning. Our results suggest that the central black holes in the cores of clusters of galaxies must be rapidly rotating in order to drive jets powerful enough to heat the intracluster medium and quench cooling flows.     

Parallel tracks in infrared versus X-ray emission in black hole X-ray transient outbursts: a hysteresis effect?

We report the discovery of a new hysteresis effect in black hole X-ray binary state transitions, that of the near-infrared (NIR) flux (which most likely originates in the jets) versus X-ray flux. We find, looking at existing data sets, that the IR emission of black hole X-ray transients appears to be weaker in the low/hard state rise of an outburst than the low/hard state decline of an outburst at a given X-ray luminosity. We discuss how this effect may be caused by a shift in the radiative efficiency of the inflowing or outflowing matter, or variations in the disc viscosity or the spectrum/power of the jet. In addition we show that there is a correlation (in slope but not in normalization) between IR and X-ray luminosities on the rise and decline, for all three low-mass black hole X-ray binaries with well-sampled IR and X-ray coverage:   L _NIR∝ L ^0.5–0.7_X  . In the high/soft state this slope is much shallower;   L _NIR∝ L ^0.1–0.2_X  , and we find that the NIR emission in this state is most likely dominated by the viscously heated (as opposed to X-ray heated) accretion disc in all three sources.     

Self-consistent computation of γ-ray spectra due to proton–proton interactions in black hole systems

In the inner regions of an accretion disc around a black hole, relativistic protons can interact with ambient matter to produce electrons, positrons and γ-rays. The resultant steady-state electron and positron particle distributions are self-consistently computed taking into account Coulomb and Compton cooling,  e⁻ e⁺  pair production (due to γ–γ annihilation) and pair annihilation. While earlier works used the diffusion approximation to obtain the particle distributions, here we solve a more general integro-differential equation that correctly takes into account the large change in particle energy that occurs when the leptons Compton scatter off hard X-rays. Thus this formalism can also be applied to the hard state of black hole systems, where the dominant ambient photons are hard X-rays. The corresponding photon energy spectrum is calculated and compared with broad-band data of black hole binaries in different spectral states. The results indicate that the γ-ray spectra  ( E > 0.8 MeV)  of both the soft and hard spectral states and the entire hard X-ray/γ-ray spectrum of the ultrasoft state could be due to p–p interactions. These results are consistent with the hypothesis that there always exists in these systems a γ-ray spectral component due to p–p interactions that can contribute between 0.5 and 10 per cent of the total bolometric luminosity. The model predicts that GLAST would be able to detect black hole binaries and provide evidence for the presence of non-thermal protons, which in turn would give insight into the energy dissipation process and jet formation in these systems.     

Quasi-stars: accreting black holes inside massive envelopes

We study the structure and evolution of 'quasi-stars', accreting black holes embedded within massive hydrostatic gaseous envelopes. These configurations may model the early growth of supermassive black hole seeds. The accretion rate on to the black hole adjusts so that the luminosity carried by the convective envelope equals the Eddington limit for the total mass,   M _*+ M _BH≈ M _*  . This greatly exceeds the Eddington limit for the black hole mass alone, leading to rapid growth of the black hole. We use analytic models and numerical stellar structure calculations to study the structure and evolution of quasi-stars. We show that the photospheric temperature of the envelope scales as   T _ph∝ M ^−2/5_BH M ^7/20_*  , and decreases with time while the black hole mass increases. Once   T _ph < 10⁴ K  , the photospheric opacity drops precipitously and T _ph hits a limiting value, analogous to the Hayashi track for red giants and protostars, below which no hydrostatic solution for the convective envelope exists. For metal-free (Population III) opacities, this limiting temperature is approximately 4000 K. After a quasi-star reaches this limiting temperature, it is rapidly dispersed by radiation pressure. We find that black hole seeds with masses between 10³ and  10⁴ M_⊙  could form via this mechanism in less than a few Myr.     

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.     

Constraining the number of compact remnants near Sgr A*

Due to dynamical friction stellar mass black holes and neutron stars are expected to form high-density cusps in the inner parsec of our Galaxy. These compact remnants, expected to number around 20 000, may be accreting cold dense gas present there, and give rise to potentially observable X-ray emission. Here we build a simple but detailed time-dependent model of such emission. The possibility that these accretion flows are radiatively inefficient is taken into account and brings in some uncertainty in the conclusions. Despite this uncertainty, we find that at least several X-ray sources of this nature should be detectable with Chandra at any one time. Turning this issue around, we also ask a question of what current observational constraints might be telling us about the total number of compact remnants. In our 'best guess' model, a cusp of ∼40 000 remnants overpredicts the number of discrete sources and the total X-ray luminosity of the inner parsec, and is hence ruled out. In the most radiatively inefficient scenario that we consider, the radiative efficiency is set to be as small as  ɛ= 10⁻⁵  . In this rather unlikely scenario, a cusp of ∼40 000 black holes would be allowed by the data, but several individual sources should still be visible. Future observations of the distribution and orbits of the cold ionized gas in the inner parsec of our Galaxy will put tighter constraints on the cusp of compact remnants.     

The runaway instability of thick discs around black holes I. The constant angular momentum case

We present results from a numerical study of the runaway instability of thick discs around black holes. This instability is an important issue for most models of cosmic gamma-ray bursts, where the central engine responsible for the initial energy release is such a system consisting of a thick disc surrounding a black hole. We have carried out a comprehensive number of time-dependent simulations aimed at exploring the appearance of the instability. Our study has been performed using a fully relativistic hydrodynamics code. The general relativistic hydrodynamic equations are formulated as a hyperbolic flux-conservative system and solved using a suitable Godunov-type scheme. We build a series of constant angular momentum discs around a Schwarzschild black hole. Furthermore, the self-gravity of the disc is neglected and the evolution of the central black hole is assumed to be that of a sequence of exact Schwarzschild black holes of varying mass. The black hole mass increase is thus determined by the mass accretion rate across the event horizon. In agreement with previous studies based on stationary models, we find that by allowing the mass of the black hole to grow the disc becomes unstable. Our hydrodynamical simulations show that for all disc-to-hole mass ratios considered (between 1 and 0.05), the runaway instability appears very fast on a dynamical time-scale of a few orbital periods, typically a few 10 ms and never exceeding 1 s for our particular choice of the mass of the black hole (2.5 M) and a large range of mass fluxes  ( m 10^-3 M s^-1)  . The implications of our results in the context of gamma-ray bursts are briefly discussed.     

Spectral variability of ultraluminous X-ray sources

We study spectral variability of 11 ultraluminous X-ray sources (ULX) using archived XMM–Newton and Chandra observations. We use three models to describe the observed spectra: a power law, a multicolour disc (MCD) and a combination of these two models. We find that seven ULXs show a correlation between the luminosity L _X and the photon index Γ. Furthermore, four out of these seven ULXs also show spectral pivoting in the observed energy band. We also find that two ULXs show an   L _X–Γ  anticorrelation. The spectra of four ULXs in the sample can be adequately fitted with a MCD model. We compare these sources to known black hole binaries (BHB) and find that they follow similar paths in their luminosity–temperature diagrams. Finally, we show that the 'soft excess' reported for many of these ULXs at ∼0.2 keV seems to roughly follow a trend   L _soft∝ T ^−3.5  when modelled with a power law plus a 'cool' MCD model. This is contrary to the   L ∝ T ⁴  relation that is expected from theory and what is seen for many accreting BHBs. The observed trend could instead arise from disc emission beamed by an outflowing wind around a  ∼10 M_⊙  black hole.     

High-frequency X-ray variability as a mass estimator of stellar and supermassive black holes

There is increasing evidence that supermassive black holes in active galactic nuclei (AGN) are scaled-up versions of Galactic black holes. We show that the amplitude of high-frequency X-ray variability in the hard spectral state is inversely proportional to the black hole mass over eight orders of magnitude. We have analysed all available hard-state data from RXTE of seven Galactic black holes. Their power density spectra change dramatically from observation to observation, except for the high-frequency (≳10 Hz) tail, which seems to have a universal shape, roughly represented by a power law of index −2. The amplitude of the tail,   C _M   (extrapolated to 1 Hz), remains approximately constant for a given source, regardless of the luminosity, unlike the break or quasi-periodic oscillation frequencies, which are usually strongly correlated with luminosity. Comparison with a moderate-luminosity sample of AGN shows that the amplitude of the tail is a simple function of black hole mass,   C _M = C / M   , where   C ≈ 1.25 M_⊙ Hz⁻¹  . This makes   C _M   a robust estimator of the black hole mass which is easy to apply to low- to moderate-luminosity supermassive black holes. The high-frequency tail with its universal shape is an invariant feature of a black hole and, possibly, an imprint of the last stable orbit.     

Hot accretion discs with thermal Comptonization and advection in luminous black hole sources

We solve for the structure of a hot accretion disc with unsaturated thermal Comptonization of soft photons and with advection, generalizing the classical model of Shapiro et al. The upper limit on the accretion rate due to advection constrains the luminosity to ≲ 0.15 y^3/5 α^7/5 of the Eddington limit, where y and α are the Compton and viscosity parameters, respectively. The characteristic electron temperature and Thomson optical depth of the inner flow at accretion rates within an order of magnitude of that upper limit are ∼ 10⁹ K and ∼ 1, respectively. The resulting spectra are then in close agreement with the X-ray and soft γ-ray spectra from black hole binaries in the hard state and Seyferts. At low accretion rates, bremsstrahlung becomes the dominant radiative process.     

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.     

4U 1957+11: a persistent low-mass X-ray binary and black hole candidate in the high state?

We report on several pointed Rossi X-ray Timing Explorer observations of the enigmatic low-mass X-ray binary (LMXB) 4U 1957+11 at different X-ray luminosities. The luminosity of the source varied by more than a factor of 4 on time-scales of months to years. The spectrum of the source tends to become harder when its luminosity increases. Only very weak  (1–2  per cent rms amplitude,  0.001–10 Hz  ,  2–60 keV)  rapid X-ray variability was observed during the observations. A comparison of the spectral and temporal behaviour of 4U 1957+11 with other X-ray binary systems, in particular LMC X-3, indicates that 4U 1957+11 is likely to be a persistent LMXB harbouring a black hole and it is persistently in the black hole high state. If confirmed, it would be the only such system known.     

The quiescent light curve and the evolutionary state of GRO J1655–40

We present ellipsoidal light-curve fits to the quiescent B , V , R and I light curves of GRO J1655–40 (Nova Scorpii 1994). The fits are based on a simple model consisting of a Roche-lobe-filling secondary and an accretion disc around the black hole primary. Unlike previous studies, no assumptions are made concerning the interstellar extinction or the distance to the source; instead these are determined self-consistently from the observed light curves. In order to obtain tighter limits on the model parameters, we used the distance determination from the kinematics of the radio jet as an additional constraint. We obtain a value for the extinction that is lower than was assumed previously; this leads to lower masses for both the black hole and the secondary star of  5.4±0.3  and  1.45±0.35 M_⊙  , respectively. The errors in the determination of the model parameters are dominated by systematic errors, in particular arising from uncertainties in the modelling of the disc structure and uncertainties in the atmosphere model for the chemically anomalous secondary in the system. A lower mass of the secondary naturally explains the transient nature of the system if it is in either a late case A or early case B mass-transfer phase.     

Shock oscillation model for quasi-periodic oscillations in stellar mass and supermassive black holes

We numerically examine centrifugally supported shock waves in 2D rotating accretion flows around a stellar mass  (10 M_⊙)  and a supermassive  (10⁶ M_⊙)  black holes over a wide range of input accretion rates of     . The resultant 2D shocks are unstable with time and the luminosities show quasi-periodic oscillations (QPOs) with modulations of a factor of 2–3 and with periods of a tenth of a second to several hours, depending on the black hole masses. The shock oscillation model may explain the intermediate frequency QPOs with 1–10 Hz observed in the stellar mass black hole candidates and also suggest the existence of QPOs with the period of hours in active galactic nuclei. When the accretion rate     is low, the luminosity increases in proportion to the accretion rate. However, when     greatly exceeds the Eddington critical rate     , the luminosity is insensitive to the accretion rate and is kept constantly around  ∼3 L _E  . On the other hand, the mass-outflow rate     increases in proportion to     and it amounts to about a few per cent of the input mass-flow rate.