Spherical orbits around a Kerr black hole

TL;DR

This paper provides a comprehensive analysis of spherical orbits around Kerr black holes, including parameter space, stability, and explicit solutions, extending previous research on these unique orbits.

Contribution

It systematically studies and extends the understanding of spherical orbits around Kerr black holes, including stability analysis and analytic solutions.

Findings

Delineation of stable and unstable orbit boundaries

Analytic solutions of geodesic equations provided

Illustrative examples of orbits included

Abstract

A special class of orbits known to exist around a Kerr black hole are spherical orbits -- orbits with constant coordinate radii that are not necessarily confined to the equatorial plane. Spherical time-like orbits were first studied by Wilkins almost 50 years ago. In the present paper, we perform a systematic and thorough study of these orbits, encompassing and extending previous works on them. We first present simplified forms for the parameters of these orbits. The parameter space of these orbits is then analysed in detail; in particular, we delineate the boundaries between stable and unstable orbits, bound and unbound orbits, and prograde and retrograde orbits. Finally, we provide analytic solutions of the geodesic equations, and illustrate a few representative examples of these orbits.