| Abstract : | The study of bound particle trajectories around a rotating black hole is crucial to understanding many astrophysical processes. We present new closed-form analytic solutions, which are straightforward and numerically fast, for the non-equatorial eccentric bound particle trajectories, $\{ \phi \left( r, \theta \right)$, $\ t \left( r, \theta \right),\ r \left( \theta \right) \}$, and their fundamental frequencies, in the Kerr spacetime by using the transformation $1/r=\mu \left(1+ e \cos \chi \right)$. The trajectories are expressed in the eccentricity, inverse-latus rectum, spin, and Carter's constant ($e$, $\mu$, $a$, $Q$) parameter space. We derive the necessary bound orbit conditions for ($e$, $\mu$, $a$, $Q$) and specialized formulae for equatorial, spherical, and non-equatorial separatrix orbits. We apply these results to develop a generalized relativistic precession model (GRPM) for Quasi-periodic oscillations (QPOs) in black hole X-ray binaries (BHXRB) to associate the fundamental frequencies of the general eccentric trajectories with the QPOs. We show that the trajectories within the range of parameter errors, giving frequencies in the range of the width of the QPO, span a torus region that should give rise to a strong QPO signal. Our analysis of the fluid flow in the relativistic disk edge suggests that instabilities cause QPOs to originate in this region, which follows geodesic in the torus region. We apply the GRPM to X-ray QPOs seen in Seyferts and deduce orbital parameters: the radius of the emission region, $a$, and $Q$. We attribute $\gamma$-ray and optical band QPOs seen in Blazars to plasma motion in the corona or jets of these AGN. Based on the Hamiltonian formulation with a power-law distribution in the energy of the plasma, we show that the resulting PSD has a break corresponding to the energy at ISCO.
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