Astrophysically relevant bound trajectories around a Kerr black hole

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Abstract
We derive alternate and new closed-form analytic solutions for the non-equatorial eccentric bound trajectories, , around a Kerr black hole by using the transformation . The application of the solutions is straightforward and numerically fast. We obtain and implement translation relations between the energy and angular momentum of the particle, ( E , L ), and eccentricity and inverse-latus rectum, ( e , ), for a given spin, a , and Carter’s constant, Q , to write the trajectory completely in the ( e , , a , Q ) parameter space. The bound orbit conditions are obtained and implemented to select the allowed combination of parameters ( e , , a , Q ). We also derive specialized formulae for equatorial, 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 as functions of the radius of the corresponding unstable spherical orbit, r s , a , and Q , and their trajectories, for ( ) separatrix orbits; they are shown to reduce to the equatorial case. These formulae have applications to study the gravitational waveforms from extreme-mass ratio inspirals (EMRIs) using adiabatic progression of a sequence of Kerr geodesics, 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 non-equatorial eccentric, separatrix, zoom-whirl, and spherical orbits, and discuss their astrophysical applications.

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