Constants of motion and fundamental frequencies for elliptic orbits at fourth post-Newtonian order

Abstract

In the case of nonspinning compact binary systems on quasi-elliptic orbits, I obtain the conservative map between the constants of motion (energy and angular momentum) and the fundamental (radial and azimuthal) frequencies at the fourth post-Newtonian order, including both instantaneous and tail contributions. This map is expressed in terms of an enhancement function of the eccentricity, which is appropriately resummed to ensure accuracy for any eccentricity; in particular, I recover known results for circular orbits. In order to obtain this map, the local dynamics are expressed using an action-angle formulation. The tail term is treated as a perturbation, which is first localized in time, then Delaunay-averaged. Both operations require a contact transformation of the phase-space variables, which I explicitly control. Using the first law of binary black hole mechanics, I then obtain the orbit-averaged redshift invariant for eccentric orbits at fourth post-Newtonian order; when properly accounting for the tail contributions, it perfectly agrees with analytical self-force at postgeodesic order [arXiv:2203.13832]. Finally, I use these results to re-express the fluxes of energy and angular momentum obtained at third post-Newtonian order in [arXiv:0711.0302] and [arXiv:0908.3854] in terms of fundamental frequencies.