Three Body Dynamics in Dense Gravitational Systems

Open Access
Moody, Kenneth
Graduate Program:
Astronomy and Astrophysics
Doctor of Philosophy
Document Type:
Date of Defense:
August 20, 2009
Committee Members:
  • Steinn Sigurdsson, Dissertation Advisor
  • Steinn Sigurdsson, Committee Chair
  • Donald P Schneider, Committee Member
  • Eric D Feigelson, Committee Member
  • Mercedes Richards, Committee Member
  • James Kasting, Committee Member
  • computer simulations
  • dynamics
  • compact objects
  • extrasolar planets
Three body dynamics are of particular interest in clusters where the density of stars provides many opportunities for interactions. Globular clusters, which have had densities of tens to hundreds of thousands of stars per cubic parsec for billions of years, are the ideal laboratory for studying dynamics in systems which at best have solutions in only the mathematical sense of the word. Modelling these systems in a realistic way which includes all stars individually represented, with their evolution and inclusion into a comparable number of binaries as is seen in observed clusters, has driven computer hardware and software for decades (Heggie & Hut 2003). In this thesis, I have used several techniques to answer the following questions: How many black hole binaries will a cluster produce, and will they have the required properties to be seen by our gravitational wave detectors? How often does the crowded environment of star forming cluster allow the exchange of a planet between stars? To answer these questions, I have studied three scenarios: the interaction of black holes in clusters, the effect of the Kozai mechanism on pulsars in clusters, and the effect of an exchanged planetary body on a planetary system. I have examined the interactions of a system of black holes in a globular cluster in which the black holes have different masses with a more realistic distribution. This is an advance over previous studies which assumed that all black holes have the same mass, and as such when interacting tended to eject all but one or two from the cluster. The previous paradigm for black holes was that all black holes were 10 solar masses. In my thesis, black hole masses are derived from population synthesis models and span a range of a few up to 50 or 80 M$_{odot}$ depending on metallicity. My new calculations have reduced the efficiency of three-body interactions in ejecting the binary due to their non-equal masses. I also use timescales derived from earlier simulations of clusters (Sigurdsson 1995) to determine the end state of individual binaries interacting with single black holes. While N-body simulations of black hole systems such as in O'Leary et al. (2006) are less model dependent, my method can easily adapt to advances in the understanding of the processes that make black holes and rapidly produce results on rates of binary black hole mergers for gravitational wave observations and the possibilities of intermediate mass black hole seeds. Numerous black hole binaries are produced by clusters, they are hardened in the potential of the cluster, and the most massive black holes survive the interactions. Interactions with the other black holes preferentially produce binaries with higher eccentricities. I found that as many as one in seven binaries will coalesce within a Hubble time, and with the strength of signal that their higher mass gives they would rival galactic black hole binaries as a background source. Compare this to the more pessimistic forecast in Kulkarni et al. (1993) that they would not be a significant background source. I also found that the binaries are ejected from the cluster with, for the most part, a velocity just above the escape speed of the cluster which is a few tens of km/sec. These gravitational wave sources are thus constrained in their host galaxies as the galactic escape velocity is some hundreds of km/sec which only a very few binaries achieve in special cases (i.e. originally forming as a tight binary, their first three-body interaction liberates a large amount of kinetic energy). It is therefore fitting to perhaps take a census of galaxies and their clusters within the radius the binaries would be visible to LIGO to estimate the how many sources could be seen, especially considering the first extra-galactic black hole in a globular cluster being recently discovered (Maccarone et al. 07). I studied the effect of the Kozai mechanism on two pulsars, one in the globular cluster M4, and the other J1903+0327. The M4 pulsar pulsar was found to have an unusually large orbital eccentricity, given that it is in a binary with a period of nearly 200 days. This unusual behavior led to the conclusion that a planet-like third body of much less than a solar mass was orbiting the binary. Dynamical exchanges can deposit the planet in a highly inclined orbit, which can lead to eccentricity pumping by the Kozai effect. The Kozai effect requires a minimum inclination of the two orbits of about 40 degrees. I used my own code to integrate the secular evolution equations with a broad set of initial conditions to determine the first detailed properties of the third body; namely that the mass of the planet is about that of Jupiter. The second pulsar J1903+0327 consists of a 2.15ms pulsar and a near solar mass companion in an $e=0.44$ orbit. A preliminary study of this pulsar showed that the high eccentricity can be reproduced by my models, and there are three candidate clusters from which this pulsar could have originated. My third project was a study of the effect of a planet at 50 AU on the inner solar system. The origin of this planet is assumed to be from an exchange with another solar system in the early stages of the sun's life while it was still in the dense star forming region where it was born. Similar studies have been done with the exchange of stars among binaries by Malmberg et al. (2007b). The exchange once again allows the Kozai effect to bring about drastic change in the inner system. A planet is chosen as the outer object as, unlike a stellar companion, it would remain unseen by current radial velocity and direct observation methods, although it could be detected by upcoming astrometric missions. My study uses an outer body from the size of a super Earth to a brown dwarf, in various inclinations, and exerting its influence on an inner object modelled on the Earth or Jupiter. The 50 AU size of the outer orbit corresponds with the sharp drop off in Kuiper Belt objects. This result represents the first step in a much larger project to fully explore the parameter space. I found that the size of the outer orbit drastically affects the eccentricity obtained by the inner object due to the beating of the Kozai and general relativistic precessions. I also found that four-body calculation are needed for a full understanding of how the change in the outer native object's eccentricity is propagated to the inner native object, native planets being those which are formed along with their host star. Simulations of young dense star forming clusters should illustrate how planetary sized objects are exchanged between stars. I explored the dynamics of exchanges between objects and the workings of the Kozai mechanism in my first two projects. These tools prepared me for work on a crucial issue in planet formation, that of how a peculiar subset of observed planets were formed. I have shown that exchanges and the Kozai mechanism can work together to produce the observed eccentricities of exoplanets. This is a new approach to the study of the dynamics of planet formation.