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**Three-dimensional, periodic, ’halo’ orbits.**
*(English)*
Zbl 0544.70013

The paper presents a wide numerical study of families of three- dimensional, periodic, ”halo” orbits nearby the collinear equilibrium points in the restricted three-body problem. Using an integration procedure of Kutta-Merson with some modifications to get the periodic orbits and determine the stability, the author has calculated about 1000 orbits for a number of mass ratios. The comprehensive analysis for orbits of all three collinear equilibrium points, provided by the author, shows that ”halo” orbits exist near these points at a wide range of mass ratios. Near the point \(L_ 1\) and \(L_ 2\) they are comparable in size for a given mass ratio, and increase in size as this ratio increases, also in the \(L_ 3\) case. Moreover, most orbits decrease in period as they approach the nearest mass. It appears that stable orbits exist approximately halfway between each equilibrium point and its nearest mass. The conclusions are very important taking into account a satellite or space station placed in ”halo” orbits.

Reviewer: A.Marciniak

### MSC:

70F07 | Three-body problems |

70-08 | Computational methods for problems pertaining to mechanics of particles and systems |

65L07 | Numerical investigation of stability of solutions to ordinary differential equations |

### Keywords:

three-dimensional, periodic, ”halo” orbits nearby the collinear equilibrium points; restricted three-body problem; integration procedure of Kutta-Merson; stability; range of mass ratios; satellite; space station### Citations:

Zbl 0544.70012
Full Text:
DOI

### References:

[1] | Breakwell, J. V. and Brown, J. V.: 1979, ?The ?Halo? Family of 3-Dimensional Periodic Orbits in the Earth-Moon Restricted 3-Body Problem?,Celest. Mech. 20, 389. · Zbl 0415.70020 |

[2] | Farquhar, R. W.: 1968, ?The Control and Use of Libration-Point Satellites?, Ph. D. Dissertation, Dept. of Aeronautics and Astronautics, Stanford University, Stanford, Calif. U.S.A. |

[3] | Farquhar, R. W.: 1970, ?The Control and Use of Libration-Point Satellites?, NASA TR R-346. |

[4] | Farquhar, R. W.: 1972, ?A Halo-Orbit Lunar Station?,Astronautics and Aeronautics, pp. 59?63. |

[5] | Farquhar, R. W. and Kamel, A. A.: 1973, ?Quasi-Periodic Orbits About the Translunar Libration Point?,Celest. Mech. 7, 458. · Zbl 0258.70011 |

[6] | Farquhar, R. W., Muhonen, D. P., and Richardson, D. L.: 1977, ?Mission Design for a Halo Orbiter of the Earth?,J. Spacecraft Rockets 14, 170. |

[7] | Richardson, D. L.: 1979, ?Halo-Orbit Formulation for the ISEE-3 Mission?,AAS Paper, 79?127. |

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