Spherical timelike orbits around Kerr black holes

TL;DR

This paper analyzes the properties of spherical timelike orbits around Kerr black holes, revealing how their radii depend on black hole spin and inclination, and introduces an approximate analytic solution method.

Contribution

It derives a polynomial equation for the orbit radii, identifies a critical inclination angle affecting orbit behavior, and applies the Lagrange-Bürmann method for approximate solutions.

Findings

Retrograde orbit radii have nonmonotonic dependence on black hole spin above a critical inclination.

Capture cross-section of retrograde orbits decreases with increasing black hole spin.

An explicit equation for the critical inclination angle is provided.

Abstract

We study the order ten polynomial equation satisfied by the radius of the spherical timelike orbits for a massive particle with a generic energy around a Kerr black hole. Even though the radii of the prograde and retrograde orbits at the equatorial or polar plane for particles with zero or unit energy have a monotonic dependence on the rotation parameter of the black hole, we show that there is a critical inclination angle above which the retrograde orbits have a nonmonotonic dependency on the rotation of the black hole. Thus the capture cross-section of these retrograde orbits decrease with increasing black hole spin. Hence their efficiency to reduce the black hole's spin is decreased. We also provide an equation for the critical inclination angle that shows exactly at which point the nonmonotonicity starts. In addition, we employ the Lagrange-B\"{u}rmann method to find approximate analytic solutions from the known exact solutions.