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We compute here the spin-orbit and spin-spin couplings needed for an accuratecomputation of the phasing of gravitational waves emitted by comparable-massbinaries on eccentric orbits at the second post-Newtonian (PN) order. We use aquasi-Keplerian parametrization of the orbit free of divergencies in the zeroeccentricity limit. We find that spin-spin couplings induce a residualeccentricity for coalescing binaries at 2PN, of the order of$10^{-4}$-$10^{-3}$ for supermassive black hole binaries in the LISA band.Spin-orbit precession also induces a non-trivial pattern in the evolution ofthe eccentricity, which could help to reduce the errors on the determination ofthe eccentricity and spins in a gravitational wave measurement.

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