Orbital evolution of a test particle around a black hole: indirect determination of the self-force in the post-Newtonian approximation

Comparing the corrections to Kepler's law with orbital evolution under a self-force, we extract the finite, already regularized part of the latter in a specific gauge. We apply this method to a quasi-circular orbit around a Schwarzschild black hole of an extreme mass ratio binary, and determine the first- and second-order conservative gravitational self-force in a post-Newtonian expansion. We use these results in the construction of the gravitational waveform, and revisit the question of the relative contribution of the self-force and spin-orbit coupling.