The Harmonic Structure of High-Frequency Quasi-periodic Oscillations in Accreting Black Holes

Observations from the Rossi X-Ray Timing Explorer have shown the existence of high-frequency quasi-periodic oscillations (HFQPOs) in the X-ray flux from accreting black hole binary systems. In at least two systems, these HFQPOs come in pairs with a 2 : 3 frequency commensurability. We employ a simple "hot spot" model to explain the position and amplitude of the HFQPO peaks. Using the exact geodesic equations for the Kerr metric, we calculate the trajectories of massive test particles, which are treated as isotropic, monochromatic emitters in their rest frames. Photons are traced from the accretion disk to a distant observer to produce time- and frequency-dependent images of the orbiting hot spot and background disk. The power spectrum of the X-ray light curve consists of multiple peaks at integral combinations of the black hole coordinate frequencies. In particular, if the radial frequency is one-third of the azimuthal frequency (as is the case near the innermost stable circular orbit), beat frequencies appear in the power spectrum at two-thirds and four-thirds of the fundamental azimuthal orbital frequency, in agreement with observations. In addition, we model the effects of shearing the hot spot in the disk, producing an arc of emission that also follows a geodesic orbit, as well as the effects of nonplanar orbits that experience Lens-Thirring precession around the black hole axis. By varying the arc length, we are able to explain the relative amplitudes of the QPOs at either 2ν or 3ν in observations from XTE J1550-564 and GRO J1655-40. In the context of this model, the observed power spectra allow us to infer values for the black hole mass and angular momentum and also constrain the parameters of the model, such as the hot spot size and luminosity.