The Carter Constant for Inclined Orbits About a Massive Kerr Black Hole: I. circular orbits

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 latus rectum (l̃) and eccentricity (e). The introduction of spin (S ̃ = |J|/M2) 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 characterise 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 un-derstanding of the evolution of Q we developed analytical formulae for Q in terms of l̃, e, and S ̃ to describe elliptical orbits at the abutment, polar orbits, and last stable orbits (LSO). By knowing the analytical form of ∂Q/∂l ̃ at the abutment, we were able to test a 2PN flux equation for Q. We also used these formulae to numerically calculate the ∂ι/∂l̃ of hypothetical circular orbits that evolve along the abutment. From these values we have determined that ∂ι/∂l ̃ =