A Non-Planar Circular Model for the 4/7 Resonance |
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| Authors: | M Šidlichovský |
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| Institution: | (1) Astronomical Institute of the Academy of Sciences of the Czech Republic, Boční Il 1401, 141 31 Praha 4, Czech Republic |
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| Abstract: | An adiabatic approximation for the non-planar, circular, restricted 3BP is presented for the external resonance 4/7. It can
be used as a model for resonant Kuiper belt objects. The Hamiltonian is truncated at the fourth order in eccentricities and
inclinations. After averaging, we have a system of two degrees of freedom with two frequencies. Numerical calculations show
that the ratio of these frequencies is ~102. Having introduced suitable canonical variables, we used the adiabatic approach introduced by Wisdom in a different context.
We left slow variables frozen and after solving the pendulum problem for fast variables, we used the averaged effect of fast
variables on slow variables. In this way we obtained the guiding trajectories for slow variables as contour lines of adiabatic
invariant. We discuss the existence of a chaotic region which is formed by trajectories crossing a critical curve which corresponds
to the separatrix of fast pendulum motion, where the assumption of sharp division between fast and slow frequencies is not
correct and the adiabatic theory fails. The model works well for e ~ 0.1 and can be used for finding the chaotic regions, but for e~ 0.17 it becomes unsatisfactory due to truncation and bad convergence of the Laplace expansion. Qualitatively it can, however,
help us to understand how the protective mechanism works as the interplay of mean motion and Kozai–Lidov resonance. |
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| Keywords: | resonances Edgeworth-Kuiper belt solar system dynamics |
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