A new simple model for high-frequency quasi-periodic oscillations in black hole candidates

L. Rezzolla 1 2 S'i. Yoshida 1 2 T. J. Maccarone 0 2 O. Zanotti 2 0 Astronomical Institute 'Anton Pannekoek', University of Amsterdam , Kruislaan 403, 1098 SJ, Amsterdam, the Netherlands 1 INFN, Sezione di Trieste, Via Valerio, 2 34127 Trieste, Italy 2 SISSA, International School for Advanced Studies , Via Beirut, 2 34014 Trieste, Italy A B S T R A C T Observations of X-ray emissions from binary systems have long since been considered important tools to test general relativity in strong-field regimes. The high-frequency quasi-periodic oscillations (HFQPOs) observed in binaries containing a black hole candidate, in particular, have been proposed as a means to measure more directly the properties of the black hole, such as its mass and spin. Numerous models have been suggested to explain the HFQPOs and the rich phenomenology accompanying them. Many of these models rest on a number of assumptions and are at times in conflict with the most recent observations. We here propose a new, simple model in which the HFQPOs result from basic p-mode oscillations of a small accretion torus orbiting close to the black hole. We show that within this model the key properties of the HFQPOs can be explained simply, given a single reasonable assumption. We also discuss observational tests that can refute the model. 1 I N T R O D U C T I O N 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. The high-frequency quasi-periodic oscillations (HFQPOs) seen from these sources have been held forth as one of the most promising diagnostics, with the potential to measure, for example, the masses and radii of neutron stars (e.g. Kluzniak, Michelson & Wagoner 1990; Miller, Lamb & Psaltis 1998) or the masses and spins of black holes (e.g. Wagoner, Silberlgleit & Ortega-Rodriguez 2001; Abramowicz & Kluzniak 2001). 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 motivated in large part by the much larger number of observed systems and the richer data sets of these systems. 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, hereafter PBK) has been suggested to provide evidence that the same mechanism must be taking place in all these systems. The extension of this correlation to include white dwarf systems (Mauche 2002) bolstered these claims. If the PBK and Mauche (2002) correlations are indeed universal and the interpretation that a single mechanism is at work in all cases is correct, then the mechanism cannot depend on the presence of a stellar surface, of a strong magnetic field, or of a strong (i.e. relativistic) gravitational field. Instead, a model taking advantage of the properties of a non-uniform rotating fluid would be a strong candidate for explaining the HFQPOs (see e.g. Osherovich & Titarchuk 1999; Titarchuk 2003). 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 (see e.g. van der Klis et al. 1997; Mendez et al. 1997). In the black hole systems, on the other hand, the kilohertz QPO frequencies seem to drift by much smaller amounts (Strohmayer 2001a,b) and to be found in ratios of small integers (i.e. 1:2, 2:3 or 1:2:3: Abramowicz & Kluzniak 2001; Remillard et al. 2002b; Homan et al. 2003b). There are also some claims for harmonic structure in XTE J1650 500 (Homan et al. 2003a), with peaks seen at 110, 140, 210 and 270 Hz, but the identification of a harmonic structure is not as clear here as the frequencies are not all identified simultaneously and, in fact, seem to drift. In addition to this, recent observations of the probable black hole transient XTE J1550 564 indicate that its HFQPOs do not always fit on the PBK correlation (Remillard et al. 2002b). These differences suggest that the PBK correlation may not apply to all of the HFQPOs seen; models requiring a stellar surface or general relativistic effects, or both (e.g. Stella & Vietri 1999), need not be rejected. As we wish to explain the integer ratios of the frequencies of HFQPOs from the dynamically confirmed black hole candidates, we will concentrate here on models that do not require a stellar surface. A model applied primarily to the HFQPOs from black hole candidates is the discoseismic model which asserts that g modes should become trapped in the potential well of a Keplerian disc in a Kerr potential (e.g. Nowak et al. 1997). 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. The predictions of the model are well summarized by Kato (2001). 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 (Wagoner et al. 2001, and references therein). The discoveries of three systems where the HFQPOs show a harmonic structure with relatively strong peaks seen in integer ratios 1:2, 2:3 or 1:2:3 seem to cast some doubt upon this model. However, the discoseismic model remains viable for the intermediate-frequency QPOs in GRS 1915+105, seen at 67 Hz (Morgan, Remillard & Greiner 1997) and 40 Hz (Strohmayer 2001b). We emphasize that such harmonic structure has been seen only from systems thought to contain a black hole as the compact accretor. Being the first to point out that the frequencies of QPOs in some black hole and neutron star sources were in a ratio of small integers, Abramowicz & Kluzniak (2001), Abramowicz et al. (2002) and Abramowicz et al. (2003a) have proposed the resonance model, in which a harmonic relationship in the HFQPO frequencies can be produced as a result of orbital resonances. In particular, the model suggests that an initial perturbation is amplified at a radius where the radial epicyclic frequency for point-like masses is in resonance with the latitudinal epicyclic frequency, with the two frequencies being in (small) integer ratios. (In a Schwarzschild spacetime the latitudinal epicyclic frequency and the orbital one coincide.) These annuli tend to be close to the black hole event horizon for the observed (...truncated)