Resonant Excitation of Disk Oscillations by Warps: A Model of kHz QPOs

The amplification of disk oscillations resulting from non-linear resonant couplings between the oscillations and a warp is examined. The disks are geometrically thin and general relativistic with non-rotating central objects. By using a Lagrangian formulation, a general stability criterion is derived. The criterion is applied to horizontal and vertical resonances of g-mode oscillations and to horizontal resonances of p-mode oscillations. The results of analyses show that g-mode oscillations (including p-mode oscillations of $n=0$) are amplified by horizontal resonances with the warp. Other modes of oscillations with other resonances are all damped by resonances. The amplified g-mode oscillations are located around the radius $4r_{\rm g}$, the radius of the epicyclic frequency being maximum, i.e., $\kappa_{\rm max}$. The frequencies of amplified oscillations are harmonics of $\kappa_{\rm max}$, and well explain the 2 : 3 pairs of observed QPOs for reasonable masses (and spins) of the central objects