protected from node crossings, can be identified by means of the level lines of the averaged perturbing function: the libration centers correspond to minima of the averaged perturbing function, which can not occur on node crossing lines. There are also librations around maxima belonging to node crossing lines, with node crossings taking place in each cycle. Thus we have been able to prove  that some stable center of oscillation, either libration or circulation of !, always exists for every given value of the integrals a and p 1 \Gamma e 2 cos I , and this for an arbitrary number of perturbing planets. For objects which can cross the orbits of several planets, there are usually many such stable states, free from node crossings, with both symmetric and asymmetric librations. 1 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1.6 1.65 1.7 1.75 1.8 1.85 x 10 -4<F5.
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