Spin effects in the phasing of gravitational waves from binaries on eccentric orbits

We compute here the spin-orbit and spin-spin couplings needed for an accurate computation of the phasing of gravitational waves emitted by comparable-mass binaries on eccentric orbits at the second post-Newtonian (PN) order. We use a quasi-Keplerian parametrization of the orbit free of divergencies in the zero-eccentricity limit. We find that spin-spin couplings induce a residual eccentricity for coalescing binaries at 2PN, of the order of 10(-4)-10(-3) for supermassive black hole binaries in the LISA band. Spinorbit precession also induces a nontrivial pattern in the evolution of the eccentricity, which could help to reduce the errors on the determination of the eccentricity and spins in a gravitational wave measurement.