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 trajectoriesa=0.688612

e=0.225657

Series of librational trajectories:

the variation of e is small

amplitude of libration:

not much change

4:1 resonance:

a=0.396851

e=0.899320

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