Self-acceleration of Hardening Binaries

A Keplerian binary immersed in a bath of lighter particles hardens by ejecting them through gravitational slingshots. This process drives, for example, the evolution of supermassive black hole binaries following galaxy mergers, and has long been described with just two parameters: the hardening rate and the eccentricity growth rate. Here we show that the secular dynamics is substantially richer. Combining symmetry arguments with extensive three-body scattering experiments, we demonstrate that the medium exerts a net force on the binary's center of mass (CoM), induces apsidal precession, and rotates the orbital plane when the CoM velocity has an out-of-plane component. Remarkably, these deterministic effects persist even in a perfectly uniform and isotropic medium, as the binary's own asymmetry provides the propulsion. The interplay of self-acceleration, precession, and dynamical friction drives the CoM along an outward spiral. For supermassive black hole binaries, this displacement dominates over Brownian motion and approaches the radius of influence, suggesting they may be significantly offset from their host galaxies' centers. The displacement also enlarges the stellar loss cone, with direct implications for the final-parsec problem. We further show that the previously reported circularization of small-mass-ratio binaries is a numerical artifact of truncating long-lived encounters: all binaries undergo eccentricity growth. Our results enrich the standard picture of binary hardening and have implications in a variety of astrophysical contexts, including gravitational-wave source populations.