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Astron. Astrophys. 319, 290-304 (1997)

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Resonance and chaos

I. First-order interior resonances

O.C. Winter 1, 2 and C.D. Murray 2

1 Grupo de Dinâmica Orbital e Planetologia, Campus Guaratinguetá, UNESP, CP 205, CEP 12500-000, Guaratinguetá, São Paulo, Brazil
2 Astronomy Unit, Queen Mary and Westfield College, University of London, Mile End Road, London E1 4NS, UK

Received 15 May 1996 / Accepted 22 July 1996


Analytical models for studying the dynamical behaviour of objects near interior, mean motion resonances are reviewed in the context of the planar, circular, restricted three-body problem. The predicted widths of the resonances are compared with the results of numerical integrations using Poincare surfaces of section with a mass ratio of [FORMULA] (similar to the Jupiter-Sun case). It is shown that for very low eccentricities the phase space between the 2:1 and 3:2 resonances is predominantly regular, contrary to simple theoretical predictions based on overlapping resonance. A numerical study of the 'evolution' of the stable equilibrium point of the 3:2 resonance as a function of the Jacobi constant shows how apocentric libration at the 2:1 resonance arises; there is evidence of a similar mechanism being responsible for the centre of the 4:3 resonance evolving towards 3:2 apocentric libration. This effect is due to perturbations from other resonances and demonstrates that resonances cannot be considered in isolation. On theoretical grounds the maximum libration width of first-order resonances should increase as the orbit of the perturbing secondary is approached. However, in reality the width decreases due to the chaotic effect of nearby resonances.

Key words: chaos – celestial mechanics – minor planets


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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998