Asteroid motion near the 3 : 1 resonance

Symplectic mappings

“…As soon however as ej φ 0, the parameter Ν varies slowly and as a consequence a slow drift appears from one Ν = constant plane to the next. This is the mechanism by which large scale chaos appears (Hadjidemetriou 1993(Hadjidemetriou ,1995. This mechanism is also given by Wisdom (1985) and Henrard (1992), using the averaged model.…”

“…Note that this map is always symplectic by its construction. It can be proved (Hadjidemetriou, 1991(Hadjidemetriou, ,1993) that this can be achieved by adding a perturbation term F\ to Fo, which is the perturbation term of Η, given by (4), multiplied by the period 2π,…”

“…The method discussed in section 5.2 has been applied by Hadjidemetriou (1991Hadjidemetriou ( ,1993 for the study of asteroid motion at the 3:1 resonance. The starting point is also the averaged Hamiltonian of the elliptic restricted three body problem, which is of the form…”

“…The elements of Jupiter's orbit vary considerably, as shown by Laskar (1990) and Nobili et al(1989). If this variation is introduced in the mapping (22), by varying ej and u>j, we find (Hadjidemetriou 1993(Hadjidemetriou , 1995 that the chaotic behaviour appears for all values of the angles σ and i/, including those cases where the motion was ordered for a fixed orbit of Jupiter, while no appreciable effect appears outside the resonance zone.…”

“…However both models have their limitations, as we shall explain: -The averaged model on which these two mapping models are based is valid only for values of the eccentricity smaller than 0.3. This means that the high eccentricity resonances that exist in the real problem (the elliptic restricted three body problem) at eccentricities e = 0.8 are missing (Hadjidemetriou 1992 (Hadjidemetriou 1993). The corrected mapping behaves in a much different way and the eccentricity can jump to very high values of the eccentricity, 0.9 or even larger.…”