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 latus rectum (l) and eccentricity (e). The introduction of spin (S = |J|/M 2 ) 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 understanding of the evolution of Q we developed analytical formulae for Q in terms ofl, e, andS 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/∂l of hypothetical circular orbits that evolve along the abutment. From these values we have determined that ∂ι/∂l = − 122.7S − 36S 3 l −11/2 − 63/2S + 35/4S 3 l −9/2 − 15/2Sl −7/2 − 9/2Sl −5/2 . By taking the limit of this equation forl → ∞, and comparing it with the published result for the weak-field radiation-reaction, we found the upper limit on ∂ι/∂l for the full range ofl up to the LSO. Although we know the value of ∂Q/∂l at the abutment, we find that the second and higher derivatives of Q with respect tol exert an influence on ∂ι/∂l. Thus the abutment becomes an important analytical and numerical laboratory for studying the evolution 1 arXiv:1007.4189v4 [gr-qc] 19 Jan 2011of Q and ι in Kerr spacetime and for testing current and future radiation back-reaction models for near-polar retrograde orbits.

The Carter constant for inclined orbits about a massive Kerr black hole: near-circular, near-polar orbits

“…In Section 2 we shall analytically derive the formula for ι for elliptical orbits on the abutment, and thus confirm the result for ∂ι/∂˜ min [42], which was derived numerically for circular orbits. In addition, we shall analytically derive ∂ι/∂ for elliptical orbits that evolve on the abutment.…”

“…It was found that X ± = − X 2 ± in the vicinity of the abutment (see Section 3.5 in Komorowski et al [42]); therefore,…”

“…The abutment, which lies outside the event horizon, corresponds to a set of orbits for which Z 9 = 0 [42], i.e.…”

“…Therefore one must view the abutment as a set of contiguous points rather than a path to be followed by an evolving orbit; and it is at these points that the derivatives, ∂Q X /∂˜ and ∂Q X /∂ , fix the corresponding slopes of Q path . But as reported in [42], the second-order effect 2 must be included when working with ι at the abutment.…”

“…A set of essential analytical formulae for the orbital constants of motion has been derived in [42]: the Carter constant at the abutment (Q X ), the orbital energy (Ẽ), and the quantity, X =L −SẼ, as well as an analytical formula for ι in terms of these constants of motion. .…”