Ariel Chitan
The evolution of triple black hole systems was studied with emphasis placed on the effect of mass and distance of the black holes. Initial conditions followed from Valtonen et al. (1995) where black holes were placed at the vertices of Pythagorean triangles. The lifetimes, number of binary encounters and number of mergers were analysed as descriptors of the systems. Simulations were conducted on FORTRAN with the use of ARCcode by Prof. Seppo Mikkola with relativistic corrections up to the 2.5th pN order. It was found that as the mass of the systems increased, so too did the merger rate. Naturally, the lifetimes of the systems decreased with this mass increase. Smaller systems demonstrated more complicated orbits with greater numbers of binary encounters. For the distance units used (0.01 – 10000 parsecs), simulations suggest that the Newtonian systems are more affected by the change in distance unit whereas the relativistic systems are affected more by the mass of the systems and merging dominates.
The study of triple black hole systems is moving from purely theoretical to observational as triple mergers of galaxies have been found. NGC6240, in Figure 3, shows a galactic merger which is thought to contain three supermassive black holes. The research presented here studies the influence of mass and distance on the evolution of such triple systems of black holes with the use of numerical simulations. The three main parameters we focus on are: 1. The number of mergers that occur; 2. the number of binary encounters/ interaction among the three bodies; and 3. the lifetime of the systems (time until the first merge, or in cases where mergers do not occur, the time until binary pair formation and mass ejection).
Following ideas from Burrau (1913) and Valtonen et al. (1995), we re-analyse Burrau’s problem of three bodies where the mass unit, m, and distance unit, d, reflect that of a 3,4,5 Pythagorean triangle.
How does such a system move and evolve gravitationally?
This is answered with the use of Prof. Seppo Mikkola’s FORTRAN code, ARCcode6. Initial conditions are placed into the code and the position co-ordinates (w.r.t. to the center of mass of the system) are calculated (using 2.5th order Post Newtonian equations). We study how Burrau’s three body problem evolves with time i.e. the initial placement of bodies (black holes) are as Fig. 4. We then extend this study to fifteen more Pythagorean triangles. For each triangle we run 13 simulations with mass unit, m, ranging from 100 to 1012 solar masses. The effect of distance unit ,d, was also investigated where d was varied from 0.01 to 10000 parsecs.
To study triple systems of black holes then, we apply statistics to many simulations. For Figures 5. and 6., we combine data from all triangular configurations. Mass is in solar masses i.e. the mass of our Sun. From Fig. 5., a bar chart shows that as mass is increased the number of merges of black holes increase. This is in accord with theory: the larger the mass, the greater the effect of gravity and subsequently, the greater the probability of black holes coming closer and closer and eventually merging.
The opposite occurs with binary encounters. As mass is increased and merging of all bodies dominate the systems, there is less interaction and movement among the bodies as shown by the bar chart of Fig.6. We find a strong positive correlation between the number of merges and mass (0.9868), and a strong negative correlation between binary encounters and mass (-0.9494). The lifetimes of the systems decay exponentially (after curve fitting data, we obtain the equation to describe the lifetime of the systems under study 5.782×1010e−1.372x) as in Fig.8.
From Fig. 9., as the distance unit is increased for the smaller mass systems, the number of merges is reduced in the (9,40,41) startup configuration. This is until a mass unit of 106 solar masses, where even at a distance unit of 10000 parsecs, merging of all three black holes dominate the outcome.