We derive the secular evolution of the orbital elements of a stellar-massobject orbiting a spinning massive black hole. We use the post-Newtonianapproximation in harmonic coordinates, with test-body equations of motion forthe conservative dynamics that are valid through 3PN order, includingspin-orbit, quadrupole and (spin)$^2$ effects, and with radiation-reactioncontributions linear in the mass of the body that are valid through 4.5PNorder, including the 4PN damping effects of spin-orbit coupling. The evolutionequations for the osculating orbit elements are iterated to high PN ordersusing a two-timescale approach and averaging over orbital timescales. We derivea criterion for terminating the orbit when its Carter constant drops below acritical value, whereupon the body plunges across the event horizon at the nextclosest approach. The results are valid for arbitrary eccentricities andarbitrary inclinations. We then analyze numerically the orbits of objectsinjected into high-eccentricity orbits via interactions within a surroundingstar cluster, obtaining the number of orbits and the elapsed time betweeninjection and plunge, and the residual orbital eccentricity at plunge as afunction of inclination. We derive an analytic approximation for the time toplunge in terms of initial orbital variables. We show that, if the black holeis spinning rapidly, the flux of gravitational radiation during the final orbitbefore plunge may be suppressed by as much as three orders of magnitude if theorbit is retrograde on the equatorial plane compared to its progradecounterpart.
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