2019/01/21 by Prerna Rana, A. Mangalam · 1 citation
Physics and Astronomy · #Angular momentum #Astrophysical Phenomena and Observations #Astrophysics #Astrophysics and Cosmic Phenomena #Bifurcation #Circular orbit #Classical mechanics #Eccentricity (behavior) #Homoclinic orbit #Mathematical physics #Orbit (dynamics) #Orbital eccentricity #Phase space #Physics #Precession #Pulsars and Gravitational Waves Research #Quantum mechanics #Rotating black hole #Stars #astro-ph.HE #gr-qc
paper · pdf · doi:10.1088/1361-6382/ab004c
published as Class. Quantum Grav. 36, 045009 (2019) · Typos corrected in this version; reference to the final published article given below; 49 pages, 12 figures, 36 sub-figures; includes 7 appendices referred to in the journal article
openalex publication_date 2019/01/21 · arxiv created 2019/02/11 · arxiv updated 2019/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.