This paper derives analytic formulas for how three orbital parameters — energy, angular momentum, and the Carter constant (a measure of motion out of the equatorial plane) — slowly change when a small object orbits a spinning black hole. The calculations apply to general bound orbits, including eccentric and inclined paths around a Kerr black hole (the solution that describes a rotating black hole). The results are given at leading order in the small mass ratio and pushed to the 6th post-Newtonian (6PN) order in a velocity expansion and to the 16th order in orbital eccentricity.

What the authors did and how they did it: they used black-hole perturbation theory and the Teukolsky formalism, together with flux-balance arguments, to obtain formulas for secular (long-term) changes of the constants that characterize a geodesic orbit in Kerr spacetime. They work in the adiabatic approximation, which treats the inspiral as a slow drift through geodesic orbits due to gravitational-wave emission. The paper keeps the dependence on orbital inclination exact and allows any black-hole spin. To make the analytic expressions practical, the team validated them against high-precision numerical Teukolsky results and studied how well the post-Newtonian and eccentricity expansions converge.

Why this matters: future space-based gravitational-wave detectors such as the Laser Interferometer Space Antenna (LISA), and proposed missions like Taiji and TianQin, aim to detect extreme mass ratio inspirals (EMRIs). EMRIs can spend many thousands to millions of orbital cycles in band, so waveform models must be both fast and extremely accurate in phase. Analytic high-order post-Newtonian inputs are useful because fully numerical self-force waveforms across the whole parameter space are computationally expensive. The formulas here extend prior analytic work (previously up to about 5PN and eccentricity terms to e^10) and are intended as building blocks for fast, analytic adiabatic inspiral and waveform models.