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Third post-Newtonian angular momentum flux and the secular evolution of orbital elements for inspiralling compact binaries in quasi-elliptical orbits

The angular momentum flux from an inspiralling binary system of compact objects moving in quasi-elliptical orbits is computed at the third post-Newtonian (3PN) order using the multipolar post-Minkowskian wave generation formalism. The 3PN angular momentum flux involves the instantaneous, tail, and tail-of-tails contributions as for the 3PN energy flux, and in addition a contribution due to non-linear memory. We average the angular momentum flux over the binary's orbit using the 3PN quasi-Keplerian representation of elliptical orbits. The averaged angular momentum flux provides the final input needed for gravitational wave phasing of binaries moving in quasi-elliptical orbits. We obtain the evolution of orbital elements under 3PN gravitational radiation reaction in the quasi-elliptic case. For small eccentricities, we give simpler limiting expressions relevant for phasing up to order $e^2$. This work is important for the construction of templates for quasi-eccentric binaries, and for the comparison of post-Newtonian results with the numerical relativity simulations of the plunge and merger of eccentric binaries.