Dr. Joanna Anosova*



   We construct and study numerically the dynamics of models of galaxies assuming that the centers of models contain very massive double black holes, surrounded by relatively low mass particles such as star clusters, gas, and dust complexes. Our work shows that the dynamical evolution of that models produce many structures similar to those observed in the nuclei of galaxies, including dumb-bell bars, rings and various types of flows, jets and spirals. The basic dynamical processes of formation of different elements of structures are formulated. It is shown that significant changes of structures often occur after the gravitational slingshot effect in the centers of systems. We found combinations of initial conditions and model parameters that produce in some time structures similar to the structures of the peculiar and interacting galaxies. In this paper, we show our results for the peculiar galaxies Arp 1, 5, 22, 29, 77  from the Atlas of Peculiar Galaxies by Halton Arp, 1966. Our figures show comparison of the model and observed structures of these galaxies as well as the past and future dynamical evolution of the models.

       Keywords: the three body problem, computer simulations, massive central

binary black holes, dynamical evolution, galaxy structure


* -



   On 14 Sept, 2015 (see Abbott et al., 2016) The LIGO reported the first direct detection of gravitational waves and the first direct observation of a binary black hole. These observations demonstrate the real existence of binary black hole systems predicted by Einstein in his General theory of relativity a century earlier. Therefore, the study of the dynamics of galaxy systems with the central massive binary black holes is especially interesting and actual now.

    It has long been believed that most of celestial objects are dominated by single  centers of attraction in them. It was more  or less natural because the solar system had been the unique model of celestial objects for a long time and even in modern times the sun and stars were the typical celestial objects.

   Recently however a lot of violent and/or complicated phenomena are discovered to take place in every part of the Universe in every scale. Many of these phenomena are caused by multiple centers of attraction: planetary rings and shepherding satellites, unusually large eccentricities of millisecond pulsars, accretion discs of various scales, peculiar structures of single galaxies, irregular and interacting galaxies, etc.

   The gravitational N body problem was applied to celestial mechanics soon after the discovery of Newton's universal law. However, in the course of three centuries the analytical studies have not provided an effective solution to the main problem. Today, numerical methods may be used for obtaining detailed solutions.

     Most dynamical studies of such complicated galaxy systems by computer simulations have utilized hydrodynamics, gravitational potential theory, stellar orbit theory, as well as the gravitational N body problem with large N (see Combes,1993; Sellwood,1987and 1993; Binney and Tremaine,1987; Yoshiaki and Keiichi,1996; Baba, 2015; Athanassoulla, 2018  as well as reviews by Kormendy, Combes, Athanassoula,Brooks, Graham, 2016, in book “Galactic Bulges”,ed. Laurikainen et al. and references therein). All these dynamical studies have assumed collision-less systems. But we are still far from a complete understanding of the dynamical structure and the kinematics of galaxies and their features.

  Computer simulations of the gravitational N body problem were first performed by von Hoerner,1960 who studied the evolution of small system (N = 4 - 25). Further advances of the direct method as well as increased computer power have yielded significant results for systems with N = 500 components (Aarseth,1974; Wielen, 1974).

    It should be noted that the study of large N body systems required considerable amount of computer time (roughly proportional to N in 3), hence, the number of different initial conditions which can be examined is relatively limited. On the other hand, small N systems offer good prospects of systematical investigations, particularly in the case of N = 3. The shortening of computer process-time permits statistical methods to be used for studying the behavior of triple systems. This is achieved by selection a representative sample of initial conditions which then reveal general features of the evolution (Agekian and Anosova,1967; Anosova,1986).

   A wide range of triple systems occur among stars and galaxies, and their kinematics and dynamics have a considerable cosmogonic interest. Moreover, studies of dynamical evolution of N body systems with N = 10 - 500 have shown that three body interactions play a crucial role in the central regions of open and globular clusters, as well as galaxy clusters. Consequently, the study of the three body problem is relevant to the subjects of celestial mechanics, stellar dynamics and astrophysics alike. 

    Many authors have studied numerically the dynamical evolution of different galaxy systems, using computer simulations in the restrict three body problem where one body is a mass-less particle: 

    -  Toomre and Toomre,1972 simulated the formation of bridges and tails due to the parabolic passage of a disturber by a galaxy surrounded by zero-mass particles;

     - Lin and Saslaw,1977 considered the flyby of a single star and a binary with associated disks of mass-less particles to explain the formation of double radio sources; 

     - Murai and Fujimoto,1980  and Lin and Lynden-Bell,1982 considered the dynamical models of the Magellanic Stream in the triple galaxy system consisting of our Galaxy and the two Magellanic Clouds;

     - Valtonen,1984a,b;1985 and Valtonen and Byrd,1979; 1986 used a similar method   (compilation of many initial conditions for the restrict three body problem) in order to study the process of dynamical formation of structures of galaxy clusters with massive central binary galaxies; 

   - Valtonen,1988 studied triple black hole systems formed in mergers of galaxies; 

   - Basu et al.,1993 modeled extragalactic jets by mergers of binary black holes surrounded by accretion disks.

    In all restricted three or four body investigations above, the motions of main bodies are either parabolic, hyperbolic, or highly elliptic ones.


   We study numerically the dynamical evolution of a galaxy model consisting of the central super-massive binary black hole with circular orbit and extended shells with numerous low mass particles inside and around the binary orbit. We carry out computer simulations in the framework of the general three body problem, considering about a million initial conditions. 

     This model can be compared with previous ones described above (Toomre and Toomre, 1972, and others; see also books by Valtonen & Karttunen, 2006 and Valtonen, Anosova et al., 2016 and references therein). In those models authors, considering the restrict three body problem, studied a single approach of the two initially independent massive bodies with parabolic, hyperbolic or highly elliptic relative motions and the mass-less particles around; in  this case the mass-less particles can have only one approach with each massive body.   

     In contrast, in our model (Anosova et al., 1994, 1995, 2000, 2006, 2017 and present) we consider a long-living central massive circular binary system; the low mass particles may interact with this binary nucleus many times during the dynamical evolution of the entire system. In such model the gravitational slingshot effect may occur many times, especially under total collapse of the system.   

     We study the dynamical evolution of our models integrating numerically the regularized equations of motions of bodies in the three body problem (see Anosova,1986), accumulating results for numerous initial conditions and compiling them. For these calculations we use the code ’TRIPLE’ due to Aarseth and Zare,1974. For our models we carry out calculations during the 40 periods rotations of the binary. 

      Each model with a massive binary black hole in the center consists of 19 shells inside and outside of the binary orbit. All shells are centered at the center of inertia of the system. Every shell has = 1000 small mass particles (such as star clusters, gas, and dust complexes) distributed there uniformly randomly. Initially, these particles have only radial velocities with respect to the center of the binary for realization there of strong interactions of the bodies. 

   This model contains the large number of particles with the low masses Mp, which do not interact to each other. For approximation of self-consistent model, we need to consider a sufficiently small value of Mp. We carried out the trial calculations with the values Mp: 0.000001, 0.0001, 0.001, 0.01, assuming that the mass of the heaviest component of the binary is M1=1. For the four values of Mp we obtained almost the same results, but the CPU time increases crucially with decreasing Mp. For basic calculations we use the value  Mp = 0.01. 

      We study the dynamical evolution of such models with numerous initial conditions and different parameters, considering the dynamics of all spherical shells together and  separately. Our method permits us to study the individual trajectories of particles, their close double and triple approaches, and inspect the time-depending structures in the models. Multiple runs of these models allow us to classify the numerous strong triple interactions of bodies resulting in various structures.   

      It is shown that first at some point in time collapse of the system takes place, when many low-mass particles have close approaches with heavy binary components and strong interactions with them. Frequently, the gravitational slingshot effect occurs. Further along in time, some part of the particles, which initially were outside the binary orbit, escapes from the system. Other particles or are captured by binary components and compile the central dumb-bell bars, or form the different types of jets, flows, open and close spirals and rings. The obtained structures are often similar to the observed structures of galaxies - spiral, elliptical, irregular, and different kinds of interacting galaxies. Symmetrical structures appear in models with the equal masses of bodies of the central binary. Globally, the systems are expanding significantly; they can expand by 300 times after P=40 period of binary rotation.



   The detail investigation of low mass particles motions shows that we often observe many near simultaneous approaches of them to the center of mass of the binary system. The resulting formation of different kinds of the structure in the model depends on the initial positions of low mass particles and their motions during close approaches with the binary:

1)  mostly particles, which initially were inside of the binary orbit, are captured by the binary components and generate a stable central dumb-bell bar;

2)  for the particles, which initially were outside of the binary orbit, the results of close approaches with the binary system depend on their motions relative to the line of apses of the binary orbit at this time:

-  in the case of almost orthogonal motions (the gravitational slingshot effect), these particles escape from the system and before an escape, form open expanding spirals;

-  in the alternative case these particles are captured by the binary black hole and compile a dumb-bell bar;

-  in the intermediate case particles create different kinds of jets, flows, open and close spirals and rings;

-  these effects are more pronounced in the plane of the binary orbit;

-  the symmetrical structure is formed only in the case of the equal masses of the binary components.

Figure 1. Basic dynamical processes in models

   Figure 1 demonstrates schematically the basic dynamical mechanisms of formation of different elements of the structure. This scheme shows the trajectories of low mass particles on the plane [X, Y ] of the binary orbit during strong interactions with this binary.

   We use the following denotes:

-    two small triangles on the horizontal coordinate X axis show locations of the binary components with the equal masses. In this case, these positions are symmetrical about the vertical coordinate Y axis;

-     filled circles indicate the trajectories of escaping particles; they are moving almost along the Y axis and we observe the gravitational slingshot effect;

-     open squares present the motions of particles that return to the center of inertia of a system after a failed ejection;

-    ticks, crosses and stars show trajectories of particles that form different types of the open and close spirals as well as the large and small rings inside the system;

-    snowflakes show the process of formation of dumb-bell bars;

-    part of the big circle in the negative quadrant (x < 0, y < 0) shows schematically the initial positions of the small particles relative to the center of mass of the binary at the time when strong interactions of bodies begin;

-     filled squares on this circle indicate roughly the regions of the positions of small particles at this time with different final motions.


   We compared obtained model structures with observed structures of many galaxies and found combinations of the initial conditions and model parameters that produce in some time similar structures in the peculiar and interacting galaxies. In this paper we demonstrate our results for peculiar galaxies Arp 1, 5, 22, 29, 77. It is interesting that different kind observations (ALMA and others - see Onishi, Iguchi, et al., 2015 and references thererein) of the galaxy Arp 77 (NGC 1097) confirm that this spiral interacting galaxy indeed has massive central binary black hole and gas and dust spirals around.

    Figures 2 - 6 show a comparison of model and observed structures of galaxies found and the past and future evolution of models. These figures consist of three parts: a) the picture of the observed galaxy; b) the best model structure; c) the 10 snapshots of the dynamical evolution of this model over the time span. The third part of each figure presents the dynamical evolution of models during the time Pf , where the unit of the time t is the period P = 1 of rotation of the binary; their semi-axis a = 1 is the unit of the distance;   t0 = 0.0 is the initial time of the evolution, dt is the time span when the model structure is fixed; the gravitational constant G = 1. For our system of unites   [P = 1, a = 1, G = 1] the masses M1&M2 of the binary components must meet the condition M1 + M2 = 2π2

   The basic parameters of models are the following ones:

-    the value M 2 = M2 from the interval [0.1, 1.0] is the ratio the mass of the binary components; the fixed value M 3 = Mp = 0.01 is the ratio of mass of small particles to the mass M1 of the heaviest component of the binary;

-   the initial radius R  of shells centered at the center of mass of  the binary; we consider R from the interval [0.25, 50.00]; the initial distribution of low mass particles on these shells (inside and outside of the central massive binary orbit) is uniformly randomly;

 - the value V is the ratio of initial radial velocities of small particles to their escape velocity Vesc; this ratio V is selected from the interval [0.0, 1.0).

      5. SUMMARY

   Obtained structures of dynamical models in some time are similar to observed structures of the peculiar galaxies Arp 1, 5, 22, 29 and 77. According to various observations, the bright galaxy Arp 77  in this list indeed has the central super-massive binary black hole with spirals.

   Figures 2 - 6 present a comparison of model and observed structures of these galaxies. We defined the parameters and initial conditions for such models and showed their past and future dynamical evolution.


Aarseth, S., 1974, Astron. Astrophys., 35, 237

Aarseth, S., & Zare, K., 1974, Celestial Mechanics, 10, 185

Abbott, B.P., et al., 2016, PhysRevLett, 116, 061102

Agekian T., Anosova J., 1967,  Astron. Zhurnal, 44, 1261

Anosova, J., 1986, Ap&SS, 124, 217

Anosova, J., & Anandarao, B., 1994, Ap&SS, 220, 83

Anosova, J., & Tanikawa, K., 1995, Ap&SS, 234, 191

Anosova, J., & Benedict, F., 2000, ASP Conference, 209, 261

Anosova, J., 2006, Annal.Univ.Turku, Finland, 358, 96

Anosova, J., 2017, AAS Meet. #230, 316.02

Arp, H., 1966, Atlas of peculiar galaxies, California Inst. Technology

Athanassoula, L., 2018.(eprint) arXiv:1801.07720 Proceed.iau.334. Manifold-

Baba, J., 2015.MNRAS, 454.3.2954

Basu, D., Valtonen, M., Valtonen, H., Mikkola, S., 1993, Astron, Astroph. 272, 417

Binney, J., Tremaine, S., 1987, Galactic Dynamics, Princeton, Princ. Univ. Press

Combes, F., 1993, In: "Physics of Galaxies: Nature or  Nurture". Ed. Balkowski. C., 137

Kormendy J., Combes F., Athanassoula L.,Brooks A., Graham A., et al.,2016, In book “Galactic Bulges”, ed. Laurikainen E. et al., Astrophysics and Space Science Library, 418

Lin, D., Saslaw, W., 1977, ApJ, 217, 958

Lin, D., Lynden-Bell D., 1982, MNRAS, 198, 707

Murai, T., Fujimoto, M., 1980. Publ. Soc. Japan, 32, 581

Onishi, K., & Iguchi, S., et al., 2015, ApJ, 806, 39

Sellwood, J., 1987, ARA&A, 25, 151

Sellwood, J., 1993, Rep. Prog. Phys., 56, 173

Toomre, A., Toomre, J., 1972, ApJ, 178, 623

Valtonen, M., 1988, Vistas Astron, 32, 23

Valtonen, M. & Anosova, J., et al., 2016, The Three-Body Problem

      from Pythagoras to Hawking, Springer International Publ

Valtonen, M., Byrd, G., 1979, ApJ,230, 655

Valtonen, M., Byrd, G., 1986, ApJ,303, 523

Valtonen, M. et al., 1984a, A&A, 130, 373

Valtonen, M. et al., 1984b, ApSS, 107, 209

Valtonen, M. et al., 1985, A&A, 143, 182

Valtonen, M., & Karttunen, H., 2006, The Three-Body Problem,

      Cambridge University Press

Von Hoerner, S., 1960, Z. Astrophys., 50, 184

Wielen, R., 1974, Astron. Rechen-Inst. Heidelberg, 77, 326

Yushiaki, T., Keiichi, W., 1996, ApJ, 469, 581