Wave–Warp Resonant Interactions in Relativistic Disks and kHz QPOs

Abstract

Non-linear resonant interactions between disk oscillations and relativistic disks are examined under the assumption that the disks are deformed by a warp. There are three types of resonances: horizontal and vertical resonances of $g$-mode oscillations, and horizontal resonances of $p$-mode oscillations. In the limit of long-wavelength oscillations, the resonances occur at $3 \,r_{\mathrm{g}}, 3.62 \,r_{\mathrm{g}}, 4 \,r_{\mathrm{g}}$, and $6.46 \,r_{\mathrm{g}}$, depending on the types of resonances and the wave modes. The frequencies of the resonant oscillations at these points also depend on the modes of oscillations. There are sets of resonant oscillations whose frequency ratio is close to $2:3$. The frequencies of the resonant oscillations can change as a wave packet propagates in the radial direction.