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Astron. Astrophys. 320, 672-680 (1997)

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1. Introduction

The 2/1 asteroidal mean-motion resonance with Jupiter coincides with the Hecuba gap. Until now, no complete explanation of the origin of this gap exists. Apart from cosmogonic conjectures, the most promising is the hypothesis that asteroids initially present in the resonant region were transferred to the high-eccentricity or Jupiter approaching orbits and were consequently ejected from the resonance.

Giffen (1973) discovered the chaotic motion in low eccentricities of the 2/1 resonance. This chaotic region was found to be confined to the low eccentricities in the three-body planar model, but numerical integrations with the four major planets (Wisdom 1987) showed escapes to high eccentricities and suggested a possible way of how the low-eccentricity chaotic region was emptied.

Lemaître & Henrard (1990) explained the existence of the chaotic zone in low eccentricities by the overlap of secondary resonances involving the circulation of longitude of perihelion [FORMULA] and the libration of critical angle

[EQUATION]

([FORMULA] and [FORMULA] are mean longitudes of a resonant asteroid and Jupiter). Morbidelli & Moons (1993) studied the effect of secular perturbations of Jupiter's orbit. They found the chaotic motion generated by the secular resonances [FORMULA] and [FORMULA] near separatrices of the 2/1 resonance and in high eccentricities. In the case of [FORMULA], they showed that, although placed near the libration centers, it does not provide any mechanism for transition to the high eccentricities.

Ferraz-Mello (1994) calculated a set of the Poincaré diagrams of the restricted, planar and averaged three-body problem clearly showing the confinement of the low-eccentricity chaotic region by regular trajectories. Moreover, he studied the spatial four-body model with Saturn. His computation of the maximum Lyapunov exponent (MLE) for a representative sample of initial conditions showed that the whole 2/1 asteroidal resonance is dominated by chaos. Typical Lyapunov times (inverse of MLE) were found between [FORMULA] and [FORMULA] years. This result raised a question whether the slow chaotic diffusion present in the model with Saturn led to significant transitions in the phase space during the solar system existence.

A recent paper of Henrard et al. (1995) explained Wisdom's integration. They localized a bridge between the secondary and secular resonances at inclinations [FORMULA] deg allowing a random walk from the low to high eccentricities. But even this detailed work did not answer completely the question of the Hecuba gap origin. They concluded: 'Orbits starting with small amplitude of libration, small inclination and eccentricities between 0.25 and 0.45 do not seem to have many possibilities to evolve. There are no resonances there except very high order resonances and the evolution through Arnold diffusion should be very, very slow.' A study of the chaotic diffusion in the moderate-eccentricity region is a main objective of this article.

A suitable tool for such a task is the frequency map analysis (FMA) introduced by Laskar (1990). This technique is based on a numerical calculation of frequencies, which do not depend on time in a regular system but are time-dependent in a chaotic system. The chaotic diffusion is then measured through time evolution of the determined frequencies. The most detailed overview of FMA was given in Laskar (1993).

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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