In this section, we briefly describe our resent work on the massive central black holes in the centers of galaxies (Makino and Ebisuzaki , Makino , Nakano and Makino [1999a,1999b]). The problem is what happens when two galaxies, each with a central black hole, merge. The merging of two galaxies is a rather common event. According to the standard scenario of the structure formation in the universe (hierarchical gravitational clustering), galaxies are constructed ``bottom up'' from smaller objects through merging. On the other hand, almost all galaxies except for some dwarfs seem to have central black holes. Thus, if two such galaxies merge, the merger remnant would have two black holes. These black holes would settle in the center of the merger remnant, and would eventually form a binary. Two questions arise: (a) What is the structure of the central region of the merger galaxy with two black holes? and (b) What will happen to the binary itself? Will two black holes eventually merge?
Makino and Ebisuzaki () studied the first problem, assuming that black holes would eventually merge. They performed the simulation of hierarchical (repeated) merging similar to those by Farouki et al. (), but with central black holes. What Farouki et al. found is that the core radius remains almost unchanged after merging, even though the half-mass radius almost doubles after each merging event. This result is consistent with the theoretical prediction based on the conservation of the central phase space density, but in complete contradiction with the observations of luminous elliptical galaxies. Ground-based observations in 1980s demonstrated that there is strong correlation between the effective radius and the ``core'' radius of luminous ellipticals (Lauer ). HST observations (Gebhaldt et al. ) have shown that the ``cores'' are actually all cusps with the profile , where is the volume luminosity density.
Figure 7 show the result of simulations with GRAPE-4. The central region of the merger with central black hole looks like the shallow cusps observers found in luminous ellipticals, and the radius of this cusp region is a constant fraction of the half-mass radius. Thus, numerical simulations of merging of galaxies with black holes reproduced the observed characteristic of the central regions of luminous ellipticals rather well.
Figure 7: The density profiles of the mergers with central black holes. Profiles are scaled so that the half-mass radii are the same for all remnants. The panel in the left side shows the results of runs with central black holes, and that in the right without black holes. Reproduced from Makino and Ebisuzaki (1997).
Nakano and Makino ([1999a,1999b]) investigated why the cusp is formed. First, they tried to understand the formation mechanism better by means of simplified experiments, in which just one black hole was let to sink to the center of a galaxy. Except for the case that the black hole is originally in the center, the cusp with is universally formed, and the mass within the cusp region is comparable to that of the mass of the central BH.
In the second paper (Nakano and Makino 1999b), they came up with a simple explanation for the formation mechanism. The black hole sink to the center through the dynamical friction, in the dynamical timescale. When the black hole settled at the center of the galaxies, almost no star was strongly bound to the black hole. This is because the black hole sinks to the center relatively slowly. The sinking timescale is not so slow that the adiabatic approximation can be applied to the binding energy of the stars, but sufficiently slow that the binding energy of most of the stars does not change very much. Thus, the distribution function has a sharp cutoff at the energy close to the central potential depth of the initial galaxy model.
If there is a central massive object and has a sharp cutoff, we can show that the central region is a cusp with the slope as follows.
The density is obtained by integrating the distribution function in the velocity space as follows:
where is the potential at distance r from the center, where the black hole lies. Note that we assumed that the velocity distribution is isotropic.
We assume that if . For any r with , equation (2) can be rewritten as
which can be expanded as
Therefore, we have
For the fate of the black hole binary, we performed simulations with very wide range of the number of particles (2,048 to 262,144) to investigate the dependence of the growth timescale of the binding energy of the binary on the number of particles in the galaxy. What we found was intriguing. The timescale seemed to be proportional to , while, theoretically, the timescale must be proportional to N, since the timescale should be limited by the timescale of filling the loss cone through two-body relaxation. We have not yet understood why our numerical experiments did not agree with the theoretical prediction. Simulations with larger N, which will be possible with GRAPE-6, will give us important clues for the understanding of this problem.