The Carter constant for inclined orbits about a massive Kerr black hole: I. Circular orbits

Abstract

In an extreme binary black hole system, an orbit will increase its angle of inclination (ι) as it evolves in Kerr spacetime. We focus our attention on the behaviour of the Carter constant (Q) for near-polar orbits, and develop an analysis that is independent of and complements radiation-reaction models. For a Schwarzschild black hole, the polar orbits represent the abutment between the prograde and retrograde orbits at which Q is at its maximum value for given values of the latus rectum () and the eccentricity (e). The introduction of spin () to the massive black hole causes this boundary, or abutment, to be moved towards greater orbital inclination; thus, it no longer cleanly separates prograde and retrograde orbits. To characterize the abutment of a Kerr black hole (KBH), we first investigated the last stable orbit (LSO) of a test-particle about a KBH, and then extended this work to general orbits. To develop a better understanding of the evolution of Q we developed analytical formulae for Q in terms of , e and to describe elliptical orbits at the abutment, polar orbits and LSOs. By knowing the analytical form of at the abutment, we were able to test a 2PN flux equation for Q. We also used these formulae to numerically calculate the of hypothetical circular orbits that evolve along the abutment. From these values we have determined that . By taking the limit of this equation for , and comparing it with the published result for the weak-field radiation reaction, we found the upper limit on for the full range of up to the LSO. Although we know the value of at the abutment, we find that the second and higher derivatives of Q with respect to exert an influence on . Thus the abutment becomes an important analytical and numerical laboratory for studying the evolution of Q and ι in Kerr spacetime and for testing current and future radiation-back-reaction models for near-polar retrograde orbits.