One to One Resonance at High Inclination

Authors: Brasser, R.1; Heggie, D.2; Mikkola, S.3

Source: Celestial Mechanics and Dynamical Astronomy, Volume 88, Number 2, February 2004 , pp. 123-152(30)

Publisher: Springer

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We report results from long term numerical integrations and analytical studies of particular orbits in the circular restricted three-body problem. These are mostly high-inclination trajectories in 1 : 1 resonance starting at or near the triangular Lagrangian L5 point. In some intervals of inclination these orbits have short Lyapunov times, from 100 to a few hundred periods, yet the osculating semi-major axis shows only relatively small fluctuations and there are no escapes from the 1 : 1 resonance. The eccentricity of these chaotic orbits varies in an erratic manner, some of the orbits becoming temporarily almost rectilinear. Similarly the inclination experiences large variations due to the conservation of the Jacobi constant. We studied such orbits for up to 109 periods in two cases and for 106 periods in all others, for inclinations varying from 0° to 180°. Thus our integrations extend from thousands to 10 million Lyapunov times without escapes of the massless particle. Since there are no zero-velocity curves restricting the motion this opens questions concerning the reason for the persistence of the 1 : 1 resonant motion. In the theory sections we consider the mechanism responsible for the observed behavior. We derive the averaged 1 : 1 resonance disturbing function, to second order in eccentricity, to calculate a critical inclination found in the numerical experiment, and examine motion close to this inclination. The cause of the chaos observed in the numerical experiments appears to be the emergence of saddle points in the averaged disturbing potential. We determine the location of several such saddle points in the (, ) plane, with  being the mean longitude difference and  the argument of pericentre. Some of the saddle points are illustrated with the aid of contour plots of the disturbing function. Motion close to these saddles is sensitive to initial conditions, thus causing chaos.

Keywords: 1:1 resonance; Chaos; Co-orbital motion; high inclination; restricted three-body problem

Document Type: Research Article


Affiliations: 1: Tuorla Observatory, University of Turku, 21500, Piikkiö, Finland, Email: 2: School of Mathematics, King’s Buildings, University of Edinburgh, Edinburgh, EH9 3JZ, UK, 3: Tuorla Observatory, University of Turku, 21500, Piikkiö, Finland,

Publication date: February 1, 2004

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