Astrophysically relevant bound trajectories around a Kerr black hole

Abstract

We derive alternate and new closed-form analytic solutions for the non-equatorial eccentric bound trajectories, $\{ ϕ\left( r, θ\right)$, $\ t \left( r, θ\right),\ r \left( θ\right) \}$, around a Kerr black hole by using the transformation $1/r=μ\left(1+ e \cos χ\right)$. The application of the solutions is straightforward and numerically fast. We obtain and implement translation relations between energy and angular momentum of the particle, ($E$, $L$), and eccentricity and inverse-latus rectum, ($e$, $μ$), for a given spin, $a$, and Carter's constant, $Q$, to write the trajectory completely in the ($e$, $μ$, $a$, $Q$) parameter space. The bound orbit conditions are obtained and implemented to select the allowed combination of parameters ($e$, $μ$, $a$, $Q$). We also derive specialized formulae for spherical and separatrix orbits. A study of the non-equatorial analog of the previously studied equatorial separatrix orbits is carried out where a homoclinic orbit asymptotes to an energetically bound spherical orbit. Such orbits simultaneously represent an eccentric orbit and an unstable spherical orbit, both of which share the same $E$ and $L$ values. We present exact expressions for $e$ and $μ$ as functions of the radius of the corresponding unstable spherical orbit, $r_s$, $a$, and $Q$, and their trajectories, for ($Q\neq0$) separatrix orbits; they are shown to reduce to the equatorial case. These formulae have applications to study the gravitational waveforms from EMRIs besides relativistic precession and phase space explorations. We obtain closed-form expressions of the fundamental frequencies of non-equatorial eccentric trajectories that are equivalent to the previously obtained quadrature forms and also numerically match with the equivalent formulae previously derived. We sketch several orbits and discuss their astrophysical applications.