Resonances and close approaches. I. The Titan-Hyperion case
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The orbits of Titan and Hyperion represent an interesting case of orbital resonance of order one (ratio of periods 3/4), which can be studied within a reasonable accuracy by means of the planar restricted three-body problem. The behaviour of this resonance has been investigated by numerical integrations, of which we show the results in terms of the Poincaré mapping in the plane of the coordinates η = ✓[(2L − 2G)] cos (\(\tilde \omega \) H − t)and ξ = −,✓[(2L − 2G)] sin (\(\tilde \omega \) H −t)keeping a constant value of the Jacobi integral throughout all integrations. We find the numerical ‘invariant curves’ corresponding to low and high eccentricity resonance locking (which seem stable, at least during the limited time span of our experiments) and show that the observed libration of Hyperion's pericenter about the conjunction lies inside the stable high eccentricity region. If initial conditions are chosen outside the stable zones, we have no more stable librations, but a chaotic behaviour causing successive close approaches to Titan.
We discuss these results both from the point of view of the mathematical theory of invariant curves, and with the aim of understanding the origin of the resonance locking in this case. The tidal evolution theory cannot be rigorously tested by such experiments (because of the dissipative terms which change the Jacobi constant); however, we note that the time scale of chaotic evolution is by many orders of magnitude smaller than the tidal dissipation time scale, so that the chaotic regions of the phase space cannot be crossed by a slow and ‘smooth’ evolution. Therefore, our results seem to favour the hypothesis that Hyperion was formed via accumulation of the planetesimals originally inside a stable island of libration, while Titan was depleting by collisions or ejections the zones where the bodies could not escape the chaotic behaviour.
KeywordsChaotic Behaviour Close Approach Chaotic Region Jacobi Constant Invariant Curf
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