Resonance, chaos and stability in the general three-body problem

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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

Original languageEnglish
Pages (from-to)199-208
Number of pages10
JournalProceedings of the International Astronomical Union
Issue numberS246
Publication statusPublished - 1 Jan 2007


  • Binaries (including multiple): close
  • Globular clusters: general
  • Gravitation
  • Instabilities
  • Methods: n-body simulations
  • Planetary systems

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