Action-angle coordinates for black-hole geodesics I: Spherically symmetric and Schwarzschild

Abstract

Action-angle coordinates are a tool commonly used in celestial mechanics to systematically parametrize and store general solutions of the equations of motion of astrophysical bodies. I perturbatively construct action-angle coordinates for bound test particle motion in static, spherically symmetric space-times using a post-circular expansion. Then I specialise the expressions to the motion in the gravitational fields of Schwarzschild black holes and give explicit formulas for the Hamiltonian to the 10th power in the radial action (20th power in eccentricity), and the transformation to angle coordinates up to the 8th harmonic with respect to a relativistic orbital anomaly. The results provide a closed-form perturbative solution for the orbital motion parametrized by coordinate time that will find applications in the modelling of compact binary inspirals and other fields of astrophysics.