Power spectra observed in LMXBs, with either a neutron star or a black hole, are characterized by an overall power law shape with superimposed several Lorentzian peaks at different frequencies. Twin peak high frequency QPOs have central frequencies typical of the Keplerian motion for matter orbiting the compact object within 10 rg. It is still unclear why the change of source’s state is accompanied by the change of frequency, coherence, and amplitude of the peaks. Moreover, the central frequencies of this couple of peaks are characterised by the ∼ 3:2 ratio. Low frequency QPOs are observed as well, whose features are thought to be strictly linked with twin high frequency QPOs. First models of these phenomena were proposed by Miller et al. and Stella et al..
We consider clumps of material orbiting a Schwarzschild black hole, that are deformed by tidal interaction.The numerical code we use, allows to study the effects of the strong gravitational field of a black hole on small objects. The appearance of a spherical blob during its tidal evolution, as calculated by numerical simulation, is shown in Figure 1. The blob of size of ∼ 0.01 rg (e.g.∼ 150 m for a 10 M⊙ black hole) is orbiting a Schwarzschild black hole on an orbit with eccentricity of e ∼ 0.05 and periastron of rp ∼ 6.1 rg. The observer is located 20◦ above the orbital plane, similar to the inclination of the source XTE J1550-564 constrained by Orosz et al.. Both, the blob and its second-order image due to gravitational lensing, are shown. The blob is squeezed and stretched by tidal forces into a ring-like shape along the orbit. The images are coloured according to the blue/redshift. A blob orbiting the black hole on a quasi-circular orbit makes radial oscillations along its orbit. From the estimates that the energy release during resonant oscillations can be as high as ∼ 0.1 mc2. Hence, a ∼ 150 m blob of matter orbiting a 10 M⊙ black hole in the inner part of the accretion disk can radiate energy of ∼ 1034 erg.
Figure 1: The evolution of tidal deformations of a low-mass satellite orbiting a Schwarzschild black hole as seen by an observer
20◦ above the orbital plane. Due to strong gravitational lensing, two images of the satellite can be seen. The colors correspond
to the redshift.
Figure 2: Simulated light curve produced by a clump of matter
orbiting a Schwarzschild black hole.
Figure 2 shows the light curve produced by the previous blob. In this region of space-time, the Keplerian frequency (νk) is almost three times the radial one (νr). During tidal evolution, the light curve shows an overall increase in luminosity which is a consequence of increasing emitting area of the source. The luminosity is normalised to its initial value L0. The time is expressed in rg/c units. For a 10 M⊙ black hole, the light curve lasts for ∼ 1 s. The inset in the figure is an enlargement of a small part of the light curve: each peak takes place on a time scale which is almost the expected Keplerian period at this radius. The peaks are due to relativistic effects, such as gravitational lensing, Doppler boosting and blue/redshift.
Figure 3: Fit of the simulated power spectrum
(thin line) and the high frequency part of the
observed one of the X-ray binary XTE J1550-564
Figure 3 shows the fit of the power spectrum of the simulated light curve and the high frequency part of the one observed in LMXB XTE J1550-564. Both twin peak QPOs are reproduced by the model. The lower peak corresponds to νk and the upper one to νk+νr. The simulation shows that, at this radial coordinate, the peaks are in a (νk+νr)/νk = 1.26 ratio. Our model reproduces the observed power law without any further assumption. From this fit, we deduce a Schwarzschild black hole mass of ∼ 11 M⊙, in agreement with Orosz et al..