Orbital Distribution Arbitrarily Close to the Homothetic Equilateral Triple Collision in the Free‐Fall Three‐Body Problem with Equal Masses

  • Hiroaki Umehara
  • Kiyotaka Tanikawa


The existence of escape and nonescape orbits arbitrarily close to the homothetic equilateral triple‐collision orbit is considered analytically in the three‐body problem with zero initial velocities and equal masses. It is proved that escape orbits in the initial condition space are distributed around three kinds of isosceles orbits. It is also proved that nonescape orbits are distributed in between the escape orbits where different particles escape. In order to show this, it is proved that the homothetic‐equilateral orbit is isolated from other triple‐collision orbits as far as the collision at the first triple encounter is concerned. Moreover, the escape criterion is formulated in the planar‐isosceles problem and translated into the words of regularizing variables. The result obtained by us explains the orbital structure numerically.

three‐body problem triple collision binary collision escape 


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© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Hiroaki Umehara
    • 1
  • Kiyotaka Tanikawa
    • 2
  1. 1.Department of Astronomical Science, School of Mathematical and Physical ScienceThe Graduate University for Advanced StudiesMitaka, TokyoJapan
  2. 2.National Astronomical ObservatoryMitaka, TokyoJapan

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