Interaction of massive black hole binaries with their stellar environment. II. Loss cone depletion and binary orbital decay

We study the long-term evolution of massive black hole binaries (MBHBs) at the centers of galaxies using detailed scattering experiments to solve the full three-body problem. Ambient stars drawn from an isotropic Maxwellian distribution unbound to the binary are ejected by the gravitational slingshot. We construct a minimal, hybrid model for the depletion of the loss cone and the orbital decay of the binary and show that secondary slingshots - stars returning on small-impact parameter orbits to have a second superelastic scattering with the MBHB - may considerably help the shrinking of the pair in the case of large binary mass ratios. In the absence of loss cone refilling by two-body relaxation or other processes, the mass ejected before the stalling of a MBHB is half the binary reduced mass. About 50% of the ejected stars are expelled in a "burst" lasting ∼ 104 yr M61/4, where M6 is the binary mass in units of 106 M ⊙. The loss cone is completely emptied in a few bulge crossing timescales, ∼ 107 yr M61/4. Even in the absence of two-body relaxation or gas dynamical processes, unequal mass and/or eccentric binaries with M6 ≳ 0.1 can shrink to the gravitational wave emission regime in less than a Hubble time and are therefore "safe" targets for the planned Laser Interferometer Space Antenna.