A geometric origin for quasi-periodic oscillations in black hole X-ray binaries

We expand the relativistic precession model to include nonequatorial and eccentric trajectories and apply it to quasi-periodic oscillations (QPOs) in black hole X-ray binaries (BHXRBs) and associate their frequencies with the fundamental frequencies of the general case of nonequatorial (with Carter's constant, $Q\neq 0$) and eccentric ($e\neq 0$) particle trajectories, around a Kerr black hole. We study cases with either two or three simultaneous QPOs and extract the parameters \{$e$, $r_p$, $a$, $Q$\}, where $r_p$ is the periastron distance of the orbit, and $a$ is the spin of the black hole. We find that the orbits with $\left[Q=0-4\right]$ should have $e\lesssim 0.5$ and $r_p \sim 2-20$ for the observed range of QPO frequencies, where $a \in [0,1]$, and that the spherical trajectories \{$e=0$, $Q \neq0$\} with $Q=2-4$ should have $r_s \sim 3-20$. We find nonequatorial eccentric solutions for both M82 X-1 and GROJ 1655-40. We see that these trajectories, when taken together, span a torus region and give rise to a strong QPO signal. For two simultaneous QPO cases, we found equatorial eccentric orbit solutions for XTEJ 1550-564, 4U 1630-47, and GRS 1915+105, and spherical orbit solutions for BHXRBs M82 X-1 and XTEJ 1550-564. We also show that the eccentric orbit solution fits the Psaltis-Belloni-Klis correlation observed in BHXRB GROJ 1655-40. Our analysis of the fluid flow in the relativistic disk edge suggests that instabilities cause QPOs to originate in the torus region. We also present some useful formulae for trajectories and frequencies of spherical and equatorial eccentric orbits.