The use of symplectic maps in the study of resonant asteroid motion
John D. Hadjidemetriou
Department of Physics - University of Thessaloniki, Greece
The factors that affect the evolution of
an asteroid inside the 2/1 and the 3/2 resonances are analyzed, by
making use of a symplectic mapping model, which represents the
Poincaré map of the real system (the elliptic restricted three
body problem) on a surface of section. The effect of Saturn on the
orbit of Jupiter is also taken into account. The mapping model is
based on the averaged Hamiltonian at the corresponding resonance,
and a semianalytic method is applied to introduce in the averaged
Hamiltonian the missing high eccentricity resonances. To do this,
we made a complete analysis of the topology of the phase space of
the elliptic restricted three body problem, for three dimensional
motion, at the above resonances, by computing all the families of
resonant periodic orbits (fixed points of the Poincaré map).
These fixed points should correspond to the fixed points of the
averaged Hamiltonian and were used as a guide to correct it.
The evolution of an asteroid inside the 2/1 or the 3/2 resonance
is studied by the symplectic mapping model. This model allows us
to understand the dynamics of the factors that affect the
evolution of the asteroid, and in particular the role played by
the secondary and secular resonances. A comparative study is made
between the 2/1 and 3/2 resonances, and the results are compared
with similar results obtained by other investigators, using
different methods.