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Numerical experiments of the planar equal-mass three-body problem: The effects of rotation

### Abstract

The planar three-body problem with angular momentum is numerically and systematically studied as a generalization of the free-fall problem (i.e., the three-body problem with zero initial velocities). The initial conditions in the configuration space exhaust all possible forms of a triangle, whereas the initial conditions in the momentum space are chosen so that position vectors and momentum vectors are orthogonal. Numerical results are organized according to the value of virial ratio defined as the ratio of the total kinetic energy to the total potential energy. Final motions are mapped in the initial value space. Several interesting features are found. Among others, binary collision curves seem to spiral into the Lagrange point, and for large , binary collision curves connect the Lagrange point and the Euler point. The existence of a lunar periodic orbit and a periodic orbit of petal-type is suggested. The number of escape orbits as a function of the escape time is analyzed for different . The behavior of this number for different time and shows most remarkably the effects of rotation of triple systems. The number of escape orbits increases exponentially for , the number seems to saturate for , and most of the escape orbits escape quickly for . The area of chaotic motion is small for .

© 2007 American Institute of Physics

Received 27 October 2006
Accepted 06 June 2007
Published online 17 August 2007

Lead Paragraph:
Systematic numerical investigations of the planar general three-body problem started 40 years ago in 1968^{1–3} as the free-fall three-body problem, i.e., with zero initial velocities. After that, no systematic studies of the problem with angular momentum were carried out except one work by Anosova *et al.* ^{4} We start in the present work with a systematic study of the planar problem with angular momentum and extend the free-fall problem. Our strategy is to restrict ourselves to unique initial conditions in the momentum space. Fixing the virial coefficient defined as the ratio of the total kinetic energy to the total potential energy, we give maximal angular momentum to triple systems as a form of pure rotation.

Acknowledgments:
The software we use for numerical integration is the chain method provided by Mikkola and Aarseth.^{21,22} The basic idea of the method is that both of the shortest two relative vectors for the three bodies are regularized by the KS regularizations. See Ref. 23 for detailed descriptions of various integration methods of the -body problem. The code with the chain method supplied by S. Aarseth and S. Mikkola is extremely efficient in the sense of integration time. We thank Dr. S. Aarseth and Dr. S. Mikkola.

Article outline:

I. INTRODUCTION AND MOTIVATION
II. THE SHAPE SPACE
A. Basic equations of motion for the planar three-body problem
B. Free-fall
C. Shape space
III. THE INITIAL CONDITIONS
A. Precise formulation of the initial conditions for the equal-mass case
B. Energy and angular momentum in the initial condition space
IV. TOOLS FOR THE NUMERICAL STUDY
A. Numerical procedure
B. Interpretation of the structure in the initial condition space
1. Evolution of the set of orbits
2. Previous researches on the relation of escape bands and collision curves
V. THE GLOBAL PROPERTY OF FINAL MOTIONS
A. The free-fall problem
B. The cases for , , and
1. The cases for and
2. The hierarchical periodic orbit in the map for
3. Periodic orbits of the petal-type in the case of
C. The case for
D. The case for
VI. ESCAPE RATE
VII. SUMMARY AND DISCUSSION

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2007-08-17

2016-09-24

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