Abstract
A mapping model is constructed to describe asteroid motion near the 3 : 1 mean motion resonance with Jupiter, in the plane. The topology of the phase space of this mapping coincides with that of the real system, which is considered to be the elliptic restricted three body problem with the Sun and Jupiter as primaries. This model is valid for all values of the eccentricity. This is achieved by the introduction of a correcting term to the averaged Hamiltonian which is valid for small values of the ecentricity.
We start with a two dimensional mapping which represents the circular restricted three body problem. This provides the basic framework for the complete model, but cannot explain the generation of a gap in the distribution of the asteroids at this resonance. The next approximation is a four dimensional mapping, corresponding to the elliptic restricted problem. It is found that chaotic regions exist near the 3 : 1 resonance, due to the interaction between the two degrees of freedom, for initial conditions close to a critical curve of the circular model. As a consequence of the chaotic motion, the eccentricity of the asteroid jumps to high values and close encounters with Mars and even Earth may occur, thus generating a gap. It is found that the generation of chaos depends also on the phase (i.e. the angles σ andv) and as a consequence, there exist islands of ordered motion inside the sea of chaotic motion near the 3 : 1 resonance. Thus, the model of the elliptic restricted three body problem cannot explain completely the generation of a gap, although the density in the distribution of the asteroids will be much less than far from the resonance. Finally, we take into account the effect of the gravitational attraction of Saturn on Jupiter's orbit, and in particular the variation of the eccentricity and the argument of perihelion. This generates a ‘mixing’ of the phases and as a consequence the whole phase space near the 3 : 1 resonance becomes chaotic. This chaotic zone is in good agreement with the observations.
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Hadjidemetriou, J.D. Asteroid motion near the 3 : 1 resonance. Celestial Mech Dyn Astr 56, 563–599 (1993). https://doi.org/10.1007/BF00696186
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DOI: https://doi.org/10.1007/BF00696186