Orbital evolution of a massive black hole pair by dynamical friction

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Abstract
We investigate the evolution of a massive black hole pair under the action of dynamical friction by a uniform background of light stars with isotropic velocity distribution. In our scenario, the primary black hole M1 sits, at rest, in the center of the spherical star distribution, and the secondary less massive companion M2 moves along bound orbits determined by the background gravitational field. M1 loses energy and angular momentum by dynamical friction, on a timescale longer than the orbital period. The uniform star core has total mass M(c) and radius r(c), and the following inequality M(c) > M1 > M2 holds. In this paper, we investigate mostly analytically the secular evolution of the orbital parameters and find that angular momentum (J) and energy (E) are lost so as to cause the increase of the eccentricity e with time, during the orbital decay of M2. In the region of the core where the motion of M2 is determined by the mean field of the uniform stellar distribution, E and J are lost exponentially on a timescale approximately tau(DF) determined by the properties of the ambient stars. The rise of e establishes instead on a longer time approximately tau(DF)(r(c)/r(A)2, increasing as the apocenter distance r(A) decreases. With the progressive decay of the orbit, M2 enter the region r < r(B) approximately (M1/M(c))1/3r(c) where the gravitational field of the primary black hole dominates, and the star background is uniform (to first approximation). Inside r(B), a key parameter of the calculation is the ratio between the black hole velocity v and the stellar dispersion velocity sigma. (1) If upsilon < sigma, energy and angular momentum are lost exponentially on a timescale tau(DF). The growth of e occurs instead on a time tau(e) longer than tau(DF) by a factor approximately (sigma/upsilon)2. Therefore, e rises weakly during orbital decay. The time tau(e) is also found to be a function of e and increases as e --> 1. (2) In the opposite limit, i.e., when upsilon > sigma, the evolution of E and J is close to a power law and establishes on a timescale approximately (upsilon/sigma)3tau(DF). The eccentricity grows on a time tau(e) comparable to this scale. Along an evolutionary path, e increases significantly: This rise leads the pericenter distance to diminish exponentially, in this limit. The time tau(e) is a function of e and decreases as e --> 1. This limit (upsilon > sigma) is attained close to the cusp radius r(cusp) approximately (M1/M(c))r(c), i.e., the distance below which the stellar distribution is affected by the gravitational field of M1. Below r(cusp) our description is invalid, and we terminate our analysis. Energy losses by gravitational wave emission becomes comparable to those by dynamical friction at a critical distance that depends on the ratio M1/M(c): consistency with the model assumptions implies M1 much less than M(c). The braking index n is calculated in this transition region: a measurable deviation from the value of 11/3 corresponding to pure gravitational wave losses provides ideally an indirect way for probing the ambient medium.
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