An Analytical and Numerical Treatment of the Carter Constant for Inclined Elliptical Orbits about a Massive Kerr Black Hole

In an extreme binary black hole system, an orbit will increase its angle of inclination (i) as it evolves in Kerr spacetime. We focus our attention on the behaviour of the Carter constant (Q) for near-polar orbits. The value of Q for bound orbits is non-negative; and an increase in Q corresponds to an increase in i. For a Schwarzschild black hole, the polar orbit represents the boundary between the prograde and retrograde orbits at which Q is at its maximum value. The introduction of spin (S = |J|/M2) to the massive black hole causes this boundary, or Abutment, to be moved towards the retrograde orbits. We consider this characteristic to be important for understanding the evolution of Q for near-polar orbits. We have developed analytical formulae for Q in a polar orbit and at the last stable orbit (LSO) for given values of latus rectum (l) and eccentricity (e). The Abutment is an important analytical and numerical laboratory that allows us to make a detailed investigation of the evolution of Q for a test particle near its LSO.