Merging of galaxies with central black holes II. Evolution of the black hole binary and the structure of the core

We investigated the evolution of the black hole binary formed by the merging of two galaxies each containing a central massive black hole. Our main goal here is to determine if the black hole binary can merge through the hardening by dynamical friction and the gravitational wave radiation. We performed $N$-body simulations of merging of two galaxies with wide range of total number of particles to investigate the effect of the number of particles on the evolution of the black hole binary. We found that the evolution timescale was independent of the number of particles in the galaxy $N$ until the separation reaches a critical value. After the separation became smaller than this critical value, the evolution timescale was longer for larger number of particles. Qualitatively, this behavior is understood naturally the result the ``loss-cone'' effect. However, the dependence of the timescale on $N$ is noticeably weaker than the theoretical prediction. In addition, the critical separation is smaller than the theoretical prediction. The timescale of evolution through gravitational radiation at this critical separation is longer than the Hubble time. We discuss the reason of these discrepancy and the implication of the present result on the structure of the ellipticals and QSO activities.