Higher order resonances in the second fundamental mode of resonance B. Érdi Astronomy Department Eötvös University 5th Austrian-Hungarian Workshop on Celestial.

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Presentation transcript:

Higher order resonances in the second fundamental mode of resonance B. Érdi Astronomy Department Eötvös University 5th Austrian-Hungarian Workshop on Celestial Mechanics Wien, April 8-10, 2010

First model of resonance: the pendulum

The second fundamental mode of resonance: Henrard, Lemaitre (1983) Murray, Dermott (1995) capture into resonance passage through resonance Action, angle variables:

Transformation of the variables: Transformed Hamiltonian:

pericentric libration apocentric libration is possible First order resonances:

Second order resonances: pericentric and apocentric libration

Third order resonances:

Evolution of level curves depending on δ

Local extrema of the energy function:

Phase plane trajectories:

Restricted three-body problem:

7:4 resonance librational trajectories a= e=

Series of librational trajectories: the variation of e is small amplitude of libration: not much change

4:1 resonance: a= e=

amplitude of libration: case of exact resonance: libration exists for all e leaving the resonance, libration exist above certain values of e

Fourth order resonances: to be studied yet