## Abstract

We study the stability of motion in the 3-body Sitnikov problem, with the two equal mass primaries (m _{1} = m _{2} = 0.5) rotating in the x, y plane and vary the mass of the third particle, 0 m _{3} < 10^{-3}, placed initially on the z-axis. We begin by finding for the restricted problem (with m _{3} = 0) an apparently infinite sequence of stability intervals on the z-axis, whose width grows and tends to a fixed non-zero value, as we move away from z = 0. We then estimate the extent of "islands" of bounded motion in x, y, z space about these intervals and show that it also increases as |z| grows. Turning to the so-called extended Sitnikov problem, where the third particle moves only along the z-axis, we find that, as m _{3} increases, the domain of allowed motion grows significantly and chaotic regions in phase space appear through a series of saddle-node bifurcations. Finally, we concentrate on the general 3-body problem and demonstrate that, for very small masses, m _{3} ≈ 10 ^{-6}, the "islands" of bounded motion about the z-axis stability intervals are larger than the ones for m _{3} = 0. Furthermore, as m _{3} increases, it is the regions of bounded motion closest to z = 0 that disappear first, while the ones further away "disperse" at larger m _{3} values, thus providing further evidence of an increasing stability of the motion away from the plane of the two primaries, as observed in the m _{3} = 0 case.

Original language | English |
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Pages (from-to) | 129-148 |

Number of pages | 20 |

Journal | Celestial Mechanics and Dynamical Astronomy |

Volume | 99 |

Issue number | 2 |

DOIs | |

Publication status | Published - Oct 1 2007 |

## Keywords

- "Islands" of bounded motion
- Extended and general Sitnikov problem
- Restricted 3-body Sitnikov problem
- Stability intervals

## ASJC Scopus subject areas

- Modelling and Simulation
- Mathematical Physics
- Astronomy and Astrophysics
- Space and Planetary Science
- Computational Mathematics
- Applied Mathematics