Chaos in the one-dimensional gravitational three-body problem

Journal Article · Thu Apr 01 00:00:00 EST 1993 · Chaos (Woodbury, N.Y.); (United States)

DOI:https://doi.org/10.1063/1.165984· OSTI ID:6464013

Hietarinta, J ^[1]; Mikkola, S ^[2]

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1. Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States) Department of Physics, University of Turku, 20500 Turku, (Finland)
2. Turku University Observatory, 21500 Piikkioe, (Finland) and Department of Physics, University of Turku, 20500 Turku (Finland)

We have investigated the appearance of chaos in the one-dimensional Newtonian gravitational three-body system (three masses on a line with [minus]1/[ital r] pairwise potential). In the center of mass coordinates this system has two degrees of freedom and can be conveniently studied using Poincare sections. We have concentrated in particular on how the behavior changes when the relative masses of the three bodies change. We consider only the physically more interesting case of negative total energy. For two mass choices we have calculated 18 000 full orbits (with initial states on a 100[times]180 lattice on the Poincare section) and obtained dwell time distributions. For 105 mass choices we have calculated Poincare maps for 10[times]18 starting points. Our results show that the Poincare section (and hence the phase space) divides into three well defined regions with orbits of different characteristics: (1) There is a region of fast scattering, with a minimum of pairwise collisions. This region consists of scallops' bordering the [ital E]=0 line, within a scallop the orbits vary smoothly. The number of the scallops increases as the mass of the central particle decreases. (2) In the chaotic scattering region the interaction times are longer, and both the interaction time and the final state depend sensitively on the starting point on the Poincare section. For both (1) and (2) the initial and final states consist of a binary + single particle. (3) The third region consists of quasiperiodic orbits where the three masses are bound together forever. At the center of the quasiperiodic region there is a periodic orbit discovered (numerically) by Schubart in 1956. The stability of the Schubart orbit turns out to correlate strongly with the global behavior.

• DOE Contract Number: W-7405-ENG-36
• OSTI ID: 6464013
• Journal Information: Chaos (Woodbury, N.Y.); (United States), Vol. 3:2; ISSN 1054-1500
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
GRAVITATION
ONE-DIMENSIONAL CALCULATIONS
THREE-BODY PROBLEM
HAMILTONIAN FUNCTION
STABILITY
TRAJECTORIES
FUNCTIONS
MANY-BODY PROBLEM
661310* - Relativity & Gravitation- (1992-)
661100 - Classical & Quantum Mechanics- (1992-)
661300 - Other Aspects of Physical Science- (1992-)