Celestial Mechanics and Dynamical Astronomy

, Volume 115, Issue 3, pp 233–259 | Cite as

Instabilities in the Sun–Jupiter–Asteroid three body problem

Original Article

Abstract

We consider dynamics of a Sun–Jupiter–Asteroid system, and, under some simplifying assumptions, show the existence of instabilities in the motions of an asteroid. In particular, we show that an asteroid whose initial orbit is far from the orbit of Mars can be gradually perturbed into one that crosses Mars’ orbit. Properly formulated, the motion of the asteroid can be described as a Hamiltonian system with two degrees of freedom, with the dynamics restricted to a “large” open region of the phase space reduced to an exact area preserving map. Instabilities arise in regions where the map has no invariant curves. The method of MacKay and Percival is used to explicitly rule out the existence of these curves, and results of Mather abstractly guarantee the existence of diffusing orbits. We emphasize that finding such diffusing orbits numerically is quite difficult, and is outside the scope of this paper.

Keywords

Hamiltonian systems Restricted problems Aubry-Mather theory Mars crossing orbits 

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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Department of MathematicsPenn State UniversityUniversity Park, State CollegeUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

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