Models for KHz QPOs in black hole candidates

One of the strongest motivations for studying X-ray binaries has been the hope that these systems could be used as probes of fundamental physics. Stellar-mass black holes represent a laboratory for studying strong-field general relativity, while neutron stars allow us to test equations of state for nuclear matter. After the launch of the Rossi X-ray Timing Explorer (RXTE) satellite in 1996, astronomers have discovered high frequency quasi periodic oscillations (HFQPOs) in bright low mass X-ray binaries containing neutron stars or black holes, which are now held forth as one of the most promising diagnostics, with the potential to measure, for example, the masses and radii of neutron stars or the masses and spins of black holes. A great deal more work has been done explaining the HFQPO phenomena in neutron star systems than those in black hole systems, and this is largely motivated by the much larger number of observed systems and the richer data sets of these systems.

There has been a long debate whether Quasi Periodic Oscillations (QPOs) in neutron star systems and black hole systems have a common origin or not. For example, a correlation between the frequencies of the HFQPOs and the break frequency for the broad-band noise component of the Fourier power spectrum that fits both neutron stars and black holes (Psaltis, Belloni & van der Klis 1999) has been suggested to provide evidence that the same mechanism must be taking place in all these systems.

The Psaltis-Belloni-van der Klis-Mauche correlation suggests a common physical mechanism for both neutron stars and black holes

More recently, though, additional phenomenology has emerged which indicates some fundamental differences between the neutron star and black hole systems, and may suggest that different models apply to these sources after all. The kilohertz QPOs are typically seen in multiples. In the neutron star systems, the separation in the HFQPO frequencies is, in general, nearly constant, with the frequency separation shrinking as the frequencies increase. High frequency QPOs have been detected in 7 X-ray binaries containing a BH candidates. Four of them exhibit pairs of QPOs with the puzzling feature of appearing in the fixed ratio 2:3, or 1:2:3

On the contrary, kHz QPOs in neutron star sources do not follow the 2:3 relation

Low mass X-ray binaries containing a black hole are usually transient sources that remain in quiescence for periods of months and occasionally show outbursts, produced by sudden changes in the mass accretion rate. Unlike kHz QPOs in neutron stars, those in black hole candidates are much less variable and do not show clear correlations with QPOs at lower frequencies.

Several models have been proposed over the years to explain the phenomenology observed, and most of these models have originated in the context of disk oscillations.

THE RESONANCE MODEL

According to this model, QPOs, both in neutron stars and BH systems, are produced by orbital resonances in a geometrically thin accretion disc. In the linear approximation, small deviations from circular equatorial orbits obey the two equations

If non linear terms are taken into account, the equation to solve is:

Since near black holes Wr < wtheta, the strongest resonance happens for n=3, and according to this idea v_high = v_theta, v_low = v_r. The resonance occurs at the precise radial position where the radial and the latitudinal epicyclic frequencies are in 2:3 ratio. It has been demonstrated that the vertical mode corresponds to a periodic displacement in which the whole torus moves as a rigid body up and down the equatorial plane, i.e., each fluid element has a vertical velocity that periodically changes in time, but does not depend on the position. The frequency of the mode is equal to the vertical epicyclic frequency that a fictitious free particle would have if orbiting the circle of maximum pressure in the torus equilibrium position.

The resonance model is rather sensitive to observational constraints. For instance, in the case of GRO 1655-40, XTE 1550-564, H 1743-322, the data excludes the 3:2 "Keplerian" resonance as a possible explanation of twin peak QPOs. All other resonances are consistent with the existing data, but it is plausible that future observations may narrow down the choice of a resonance.

THE DISCOSEISMIC MODEL

According to this model, QPOs are produced by g-modes, namely scillation modes that arise in accretion discs and have gravitational and centrifugal forces as restoring forces. Such modes become trapped in the inner regions of a Keplerian disc around a Kerr black hole. The size of the region where the modes are trapped depends on both the mass and the spin of the accreting black hole. Additional frequencies of oscillation should be expected from p (pressure) modes and c (corrugation) modes. Given pairs of HFQPOs and a proper identification of the frequencies with the particular modes, one can measure both the black hole mass and the spin to relatively high accuracy. This model is physically well motivated but does not predict harmonic relations among modes. Therefore, it could be more interesting in explaining QPOs rather than kHz QPOs in black hole candidates.

A NEW MODEL : HFQPOs AS P-MODES IN THICK ACCRETION DISCS

In contrast to a Keplerian accretion disc, which is in principle infinitely extended, a non-Keplerian disc can easily be constructed to have a finite size, the extent being determined uniquely by the distribution of the specific angular momentum and by the pressure gradients. Because pressure gradients play such an important role, non-Keplerian discs tend to be geometrically thick and look like tori rather than thin discs. Depending then on the pressure gradients and angular momentum distribution, the orbiting fluid is confined to a finite-size region which can behave as a cavity in which global oscillation modes could be trapped. When a disc initially in equilibrium is perturbed in some way, restoring forces appear to compensate for the perturbation. A first restoring force is the centrifugal force, which is responsible for inertial oscillations of the orbital motion of the disc and hence for epicyclic oscillations. A second restoring force is the gravitational field in the direction vertical to the orbital plane and which will produce a harmonic oscillation across the equatorial plane if a portion of the disc is perturbed in the vertical direction.

A third restoring force is provided by pressure gradients, and the oscillations produced in this way are closely related to the sound waves propagating in a compressible fluid. In a geometrically thick disc, the vertical and horizontal oscillations are in general coupled, and more than a single restoring force can intervene for the same mode. Numerical simulations show that, when perturbing the equilibrium model of a thick disc around a Schwarzschild or a Kerr black hole, global modes of oscillations are excited, which manifest in a periodic accretion of matter through the cusp of the disc.

The central density of the disc is another tracer quantity used to monitor the periodic oscillation induced by the perturbation.

Click on image to see animation of the rest mast density

The numerical spectrum shows a sequence of modes, given by a fundamental model f, the overtones o1 and o2, plus their non-linear couplings. Remarkably, the ratio o1/f is very close to 3:2

In order to understand the physical origin of these modes it is necessary to perform a (linear) perturbative analysis of the axisymmetric modes of oscillation of relativistic tori in a Schwarzschild space-time. One of the important results of the perturbative investigation, performed in the Cowling approximation, is that the eigenfunctions and eigenfrequencies found are those corresponding to the p modes of the torus. Furthermore, the fundamental frequencies converge to the epicyclic frequency ?r at the position of maximum density in the torus r max as the size of the torus is progressively reduced to zero. Finally and most importantly, the eigenfrequencies computed both for the fundamental mode of oscillation and for the first few overtones are found to be in a harmonic sequence 2:3:4:. . . to within 5-10 per cent, the exact value depending on the specific model for the torus.

Proposed explanation: HFQPOs in X-ray binaries containing a BHC can be interpreted in terms of p-mode oscillations of a small-size torus orbiting near the BH. In the limit of vanishing torus size, the value of the fundamental p-mode frequency tends to the radial epyciclic frequency at the position of the maximum density.

Requirements of the model:
1) Deviation from a Keplerian motion; there is plenty of physical phenomena that can do this near the black hole
2) Stability of the disc to non axisymmetric perturbations, such as the Papaloizou-Pringle instability. This can be achieved by a certain amount of accretion.

Advantages of the model:
1) Being global oscillations, the harmonicity is present at all radii, removing the difficulty of small radial extensions.
2) By construction the frequencies scale like 1/M, as reported by osbervations.
3) The observed frequency jitter is explained naturally in terms of small variations in the size of the torus; these variations would preserve harmonicity.

Essential bibliography:

Abramowicz M., Kluzniak, W., A&A, 374, L19, 2001

Belloni T., Mendez M., Homan J., A&A, 437, 209, 2005

Kato Y., PASJ, 56, 931, 2004

Remillard R. et al., ApJ, 580, 1030, 2002

Rezzolla L., Yoshida S., Maccarone T., Zanotti O., MNRAS, 344, L37, 2003

Torok G., Abramowicz A., Kluzniak W., Stuchlik Z., A&A, 436, 1, 2005

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