5 Multiple Black Hole Systems

If binary decay stalls, an uncoalesced binary may be present in a nucleus when a third SBH, or a second binary, is deposited there following a subsequent merger. The multiple SBH system that forms will engage in its own gravitational slingshot interactions, eventually ejecting one or more of the SBHs from the nucleus and possibly from the galaxy and transferring energy to the stellar fluid.

If the infalling SBH is less massive than either of the components of the pre-existing binary, $m < (m ,m ) 3 1 2$ , the ultimate outcome is likely to be ejection of the smaller SBH and recoil of the binary, with the binary eventually returning to the galaxy center. The lighter SBH is ejected with a velocity roughly $1/3$ the relative orbital velocity of the binary [191 , 92 ], and the binary recoils with a speed that is lower by $m3/(m1 + m2)$ . Each close interaction of the smaller SBH with the binary increases the latter’s binding energy by $<E/E > ~~ 0.4m3/(m1 + m2)$ [89 ]. If $m3 > m1$ or $m3 > m2$ , there will most often be an exchange interaction, with the lightest SBH ejected and the two most massive SBHs forming a binary; further interactions then proceed as in the case $m3 < (m1, m2)$ .

During the three-body interactions, both the semi-major axis and eccentricity of the dominant binary change stochastically. Since the rate of gravity wave emission is a strong function of both parameters ( $-4 2 -7/2 E oc a (1 - e )$ ), the timescale for coalescence can be enormously shortened. This may be the most promising way to coalesce SBH binaries in the low-density nuclei of massive galaxies, where stalling of the dominant binary is likely.

This process has been extensively modelled using the PN2.5 approximation to represent gravitational wave losses [166 ] and assuming a fixed potential for the galaxy [214 , 147 , 215 ]. In these studies, there was no attempt to follow the pre-merger evolution of the galaxies or the interaction of the binary SBHs with stars. In two short non-technical contributions (submissions for the IEEE Gordon Bell prizes in 2001 and 2002), J. Makino and collaborators mention two $N$ -body simulations of triple SBH systems at the centers of galaxies using the GRAPE-6, and (apparently) a modified version of NBODY1. Relativistic energy losses were neglected and the SBH particles all had the same mass. Plots of the time evolution of the orbital parameters of the dominant binary show strong and chaotic eccentricity evolution, with values as high as 0.997 reached for short periods. Such a binary would lose energy by gravity wave emission very rapidly, by a factor $~ 108$ at the time of peak $e$ compared with a circular-orbit binary with the same semi-major axis.

In a wide, hierarchical triple, $m3 « (m1, m2)$ , the eccentricity of the dominant binary oscillates through a maximum value of $V~ ------------- ~ 1- 5cos2 i/3$ , $V~ ---- |cos i| < 3/5$ , with $i$ the mutual inclination angle [107 ]. One study [21 ] estimates that the coalescence time of the dominant binary in hierarchical triples can be reduced by factors of $~$ 10 via the Kozai mechanism.

If the binary SBH is hard when the third SBH falls in, the ejected SBH can gain enough velocity to escape the galaxy. If the three masses are comparable, even the binary can be kicked up to escape velocity. One study [217 ] estimates (based on a very simplified model of the interactions) that the recoil velocity of the smallest SBH is larger than galactic escape velocities in 99% of encounters and that the binary escapes in 8% of encounters. Thus a significant fraction of nuclei could be left with no SBH, with an offset SBH, or with a SBH whose mass is lower than expected based on the $M$ - $s$ or $M$ - $Lbulge$ relations.

There is a need for simulations of multiple-SBH systems that include both gravitational loss terms, accurate (regularized) interactions between the SBH particles, and the interactions of SBH particles with stars.