Orbital Evolution of a Massive Black Hole Pair by Dynamical Friction

We investigate the evolution of a massive black hole pair under the action of dynamical friction (DF) by a uniform background of light stars with isotropic velocity distribution. In our scenario, the primary black hole $M_1$ sits, at rest, in the center of the spherical star distribution (with mass $M_c$) and the secondary less massive companion $M_2$ moves along bound orbits determined by the background gravitational field; it loses energy $E$ and angular momentum $J$ by DF. We investigate mostly analytically the secular evolution of the orbital parameters when the motion of $M_2$ is determined either by the mean field generated by the uniform stellar distribution or by the gravitational field of $M_1$. We find that $J$ and $E$ are lost so as to cause the increase of the eccentricity $e$ during the orbital decay of $M_2$. When $M_2$ enters the region where the gravitational field of $M_1$ dominates, the evolution depends on the ratio between the black hole velocity $v$ and the stellar dispersion velocity $\sigma.$ We explore both cases $v<\sigma$ and $v>\sigma$. Only for $v>\sigma$ the rise of $e$ would be severe but this transition occurs close to the cusp radius below which our description becomes invalid. Energy losses by gravitational wave emission become comparable to those by dynamical friction at a critical distance that depends sensitively on the ratio $M_1/M_c$. The braking index $n$ is calculated in this transition region.