A study of elliptical last stable orbits about a massive Kerr black hole

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 normalized spin, (|J|/M2, 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 derived from the Kerr metric, which encapsulates the Lense–Thirring precession. This allowed us to develop a new form of analytical expression to calculate the LSO Radius for circular orbits (RLSO) of arbitrary KBH spin. We were then able to construct a numerical method to calculate the latus rectum for an elliptical LSO. Formulae for (square of normalized orbital energy) and (square of normalized orbital angular momentum) in terms of eccentricity, ε, and latus rectum, , were previously developed by others for elliptical orbits around an SBH and then extended to the KBH case; we used these results to generalize our analytical 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.