The dynamical evolution of massive black hole binaries - I. Hardening in a fixed stellar background

The stellar ejection rate and the rates of change of the binary semimajor axis and eccentricity are derived from scattering experiments for the restricted three-body problem. They are used to study the evolution of binaries in simple models for galactic nuclei, starting soon after the black holes become bound and continuing until the evolution is dominated by the emission of gravitational radiation, or until the ejected mass is too large for the galaxy to be considered fixed. The eccentricity growth is found to be unimportant unless the binary forms with a large eccentricity. The scattering results are compared with predictions from Chandrasekhar's dynamical-friction formula and with previous work on the capture and scattering of comets by planetary systems. They suggest that a binary with masses $m_1\geq m_2$ should not be considered hard until its orbital velocity exceeds the background velocity dispersion by a factor that scales as $(1+m_1/m_2)^{1/2}$.