We have simulated the evolution of a galactic nucleus containing triple massive black holes. As we stated in the introduction, currently available numerical simulations suggest that when two galaxies merge, their central black holes will be left as a binary system. Thus, when the remnant of a galaxy merger again merges with a third galaxy, the final merged object is likely to host three massive black holes.
The evolution of a triple black hole system is entirely different from that of a binary system, since it is very unlikely that three black holes form a dynamically stable system. There are basically two possibilities for the final outcome. One is that one or more black holes will be ejected from the galaxy through three-body interaction of black holes (gravitational slingshot, [SVA74]). The other possibility is that during the three-body interaction two black holes come close enough that they merge through gravitational wave radiation [HR92,ME94]. Thus, even after black holes have kicked out all nearby stars, they can still evolve in a complex way. On the other hand, a bound pair of two black holes is always dynamically stable. Thus, once a binary has kicked out all nearby stars, the evolution would be completely halted.
As we have described in the introduction, several numerical studies on the evolution of the massive black hole binary have been published, but to our knowledge there is no self-consistent N-body simulation of the black hole triples embedded in the galactic core, though the potential importance was first pointed out a number of years ago [SVA74] and studied in a number of papers both theoretically and numerically as a pure three-body problem in the fixed background potential (see [Val99] for a recent example).
The primary reason for this non-existence of the direct numerical work is the same difficulty of simulation as we described in the introduction for the case of the black hole binary. Though the evolution of the black holes might be different, the numerical difficulties are all the same. Since the difficulties are the same, the same solution, the combination of GRAPE hardware and individual timestep algorithm, works rather well for this problem.
In our simulation, the galaxy is modeled with 786,432 equal-mass stars. The black holes are modeled as three point-mass particles with a mass of 1% of the total mass of the system. The relativistic effects were neglected in our simulation. The gravitational force from black holes, both on field stars and on black holes, are calculated on the host computer to ensure the best possible accuracy. The force from field particles is calculated on GRAPE-6.
We performed a simulation for 7.875 dynamical time units, for which the number of individual steps was . The whole simulation, including on-the-fly analysis of the orbital elements of black holes and file operations, took 29.88 hours. The total number of floating point operations is , since one particle-particle interaction amounts to 57 floating point operations. The resulting average computing speed is 1.349 Tflops.
The calculation reported here is the first large simulation of black hole triples in the core of the galaxy, and as such, the result turned out to be rather surprising. Figure 12 shows the evolution of the orbital elements for the most strongly bounded pair of binaries. The criterion of the selection of the pair is that the binding energy is the maximum. The binding energy of two black holes i and j is defined as
where M is the total mass , is the reduced mass defined as , and and are the relative distance and relative velocity of the pair. We regard this most strongly bounded pair as binary and plotted the time evolution of its semi-major axis and eccentricity. The evolution of the semi-major axis is not much different from those in the simulations of black hole binaries. However, the behavior of the eccentricity is quite different. In simulations of black hole binaries, it was found that the eccentricity remains almost constant during the evolution, and typically eccentricity is not very large. Thus, black holes do not come close to each other, and the effect of gravitational radiation remains small. In figure 12, however, we can see that the eccentricity e comes very close to 1 rather frequently (note that we plotted 1-e). The effect of the gravitational radiation is proportional to . Thus, if , the effect of gravitational radiation is times stronger than in the case of a circular binary. Thus, it is quite likely that this bound pair emits strong gravitational radiation resulting in the merging of two black holes.
Figure 12: The time evolution of the semi-major axis (left panel) and eccentricity (right panel) of the most strongly bounded pair of black holes.
Since the variation of the eccentricity is highly stochastic, we need more simulations from different initial conditions, and it is desirable to incorporate the effect of the gravitational radiation directly into the simulation to see if it actually works. However, our present result clearly demonstrates the importance of the triple interaction.
As a short benchmark run, we tried the same simulation with 1,400,000 stars for 0.25 time units. For this run, the total number of timesteps is , and calculation took 2.17 hours. The average calculation speed was 1.64 Tflops.