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Resonance, Chaos and Stability in the General Three-Body Problem

Published online by Cambridge University Press:  01 September 2007

R. A. Mardling
R. A. Mardling*
Affiliation:
School of Mathematical Sciences, Monash University, Victoria, 3800, Australia email: [email protected]
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Abstract

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Three-body stability is fundamental to astrophysical processes on all length and mass scales from planetary systems to clusters of galaxies, so it is vital we have a deep and thorough understanding of this centuries-old problem. Here we summarize an analytical method for determining the stability of arbitrary three-body hierarchies which makes use of the chaos theory concept of resonance overlap. For the first time the dependence on all orbital elements and masses can be given explicitly via simple analytical expressions which contain no empirical parameters. For clarity and brevity, analysis in this paper is restricted to coplanar systems including a description of a practical algorithm for use in N-body and other applications. A Fortran routine for arbitrarily inclined systems is available from the author, and animations of stable and unstable systems are available at www.maths.monash.edu.au/~ro/Capri.

Keywords

gravitationinstabilitiesmethods: n-body simulationsplanetary systemsglobular clusters: generalbinaries (including multiple): close
Type
Contributed Papers
Information
Proceedings of the International Astronomical Union , Volume 3 , Symposium S246: Dynamical Evolution of Dense Stellar Systems , September 2007 , pp. 199 - 208
Copyright
Copyright © International Astronomical Union 2008

References

Brouwer, D. & Clements, G. M. 1961, Methods of Celestial Mechanics, New York and London: Academic PressGoogle Scholar
Chirikov, B. V. 1979, Physics Reports, 52, 263CrossRefGoogle Scholar
Heggie, D. C. 1975, MNRAS, 173, 729CrossRefGoogle Scholar
Jackson, J. D. 1975, New York: Wiley, 1975, 2nd ed.Google Scholar
Kolmogorov, A. N. 1954, Dokl. Akad. Nauk. SSSR| 98, 527Google Scholar
Mardling, R. A. 2001, in Evolution of Binary and Multiple Star Systems, eds Podsiadlowski, et al. ASP Conference Series, 229, 101Google Scholar
Mardling, R. A. 2007, MNRAS, in pressGoogle Scholar
Murray, C. D. & Dermott, S. F. 2000, Solar System Dynamics, Cambridge, UK: Cambridge University PressCrossRefGoogle Scholar
Roy, A. & Haddow, M. 2003, Cel. Mec. Dyn. Ast., 87, 411CrossRefGoogle Scholar
Walker, G. H. & Ford, J. 1969, Physical Review, 188, 416CrossRefGoogle Scholar
Wisdom, J. 1980, AJ, 85, 1122CrossRefGoogle Scholar