The Carter Constant for Inclined Orbits About a Massive Kerr Black Hole: near-circular, near-polar orbits

Abstract

In an extreme mass-ratio binary black hole system, a non-equatorial orbit will list (i.e. increase its angle of inclination, {\iota}) as it evolves in Kerr spacetime. The abutment, a set of evolving, near-polar, retrograde orbits, for which the instantaneous Carter constant (Q) is at its maximum value (Q_{X}) for given values of latus rectum (l) and eccentricity (e), has been introduced as a laboratory in which the consistency of dQ/dt with corresponding evolution equations for dl/dt and de/dt might be tested independently of a specific radiation back-reaction model. To demonstrate the use of the abutment as such a laboratory, a derivation of dQ/dt, based only on published formulae for dl/dt and de/dt, was performed for elliptical orbits on the abutment. The resulting expression for dQ/dt matched the published result to the second order in e. We believe the abutment is a potentially useful tool for improving the accuracy of evolution equations to higher orders of e and l^{1}.