A Study of Elliptical Last Stable Orbits About a Massive Kerr Black Hole

TL;DR

This paper extends the understanding of the last stable orbit (LSO) to elliptical orbits around Kerr black holes, providing new analytical and numerical tools to calculate the LSO radius and latus rectum, aiding gravitational wave research.

Contribution

It introduces a novel analytical expression for the LSO radius for circular orbits and a numerical method for elliptical orbits around Kerr black holes, expanding previous models.

Findings

Analytical formulae for LSO radius and latus rectum show excellent agreement with existing data.

New methods enable accurate calculation of elliptical LSO parameters.

Results improve understanding of gravitational wave emission from inspiraling objects.

Abstract

The last stable orbit (LSO) of a compact object (CO) is an important boundary condition when performing numerical analysis of orbit evolution. Although the LSO is already well understood for the case where a test-particle is in an elliptical orbit around a Schwarzschild black hole (SBH) and for the case of a circular orbit about a Kerr black hole (KBH) of normalised spin, S (|J|/M^2, where J is the spin angular momentum of the KBH); it is worthwhile to extend our knowledge to include elliptical orbits about a KBH. This extension helps to lay the foundation for a better understanding of gravitational wave (GW) emission. The mathematical developments described in this work sprang from the use of an effective potential (V) derived from the Kerr metric, which encapsulates the Lense-Thirring precession. That allowed us to develop a new form of analytical expression to calculate the LSO Radius for circular orbits (R_LSO) of arbitrary KBH spin. We were then able to construct a numerical method to calculate the latus rectum (l_LSO) for an elliptical LSO. Abstract Formulae for E^2 (square of normalised orbital energy) and L^2 (square of normalised orbital angular momentum) in terms of eccentricity, e, and latus rectum, l, were previously developed by others for elliptical orbits around an SBH and then extended to the KBH case; we used these results to generalise our analytical l_LSO equations to elliptical orbits. LSO data calculated from our analytical equations and numerical procedures, and those previously published, are then compared and found to be in excellent agreement.