Original Article

Celestial Mechanics and Dynamical Astronomy

, Volume 115, Issue 3, pp 233-259

First online:

Instabilities in the Sun–Jupiter–Asteroid three body problem

  • John C. UrschelAffiliated withDepartment of Mathematics, Penn State University Email author 
  • , Joseph R. GalanteAffiliated withDepartment of Mathematics, University of Maryland

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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.


Hamiltonian systems Restricted problems Aubry-Mather theory Mars crossing orbits