A Comparison of the Dynamical Evolution of Planetary by Andrea Milani, Zoran Knežević (auth.), Rudolf Dvorak, Sylvio

By Andrea Milani, Zoran Knežević (auth.), Rudolf Dvorak, Sylvio Ferraz-Mello (eds.)

The papers during this quantity disguise quite a lot of topics overlaying the latest advancements in Celestial Mechanics from the theoretical aspect of nonlinear dynamical platforms to the appliance to actual difficulties. We emphasize the papers at the formation of planetary platforms, their balance and likewise the matter of liveable zones in extrasolar planetary platforms. a distinct subject is the soundness of Trojans in our planetary approach, the place a growing number of reasonable dynamical versions are used to give an explanation for their complicated motions: along with the real contribution from the theoretical standpoint, the result of a number of numerical experiments unraveled the constitution of the good sector round the librations issues.
This quantity could be of curiosity to astronomers and mathematicians attracted to Hamiltonian mechanics and within the dynamics of planetary systems.

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Additional info for A Comparison of the Dynamical Evolution of Planetary Systems: Proceedings of the Sixth Alexander von Humboldt Colloquium on Celestial Mechanics Bad Hofgastein (Austria), 21–27 March 2004

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2 eccentricity Figure 1. Distribution of all real L4 and L5 Trojans with respect to their inclination (top) and their eccentricity (bottom); L5 Trojans are depicted in light grey. 1984). g. , 2000). – On the other hand we used the program orbit9, developed by Milani2 (1999), a high order Runge Kutta method. g. Froeschle´, 1984). html 22 RUDOLF DVORAK AND RICHARD SCHWARZ Additionally to the LCEs we computed the maximum eccentricity during the evolution of an orbit to determine its stability and also the region of stable motion around the two equilibrium points.

C) The Poincare´ surface of section for the equations of motion under the full Hamiltonian of the planar circular restricted three-body problem. The section is given by the variables ðs; xÞ when the surface - À k0 ¼ 0 is crossed by an orbit at the positive sense. The Jacobi constant is taken equal to E ¼ À1:49948, which is very close to the Jacobi constant at L4, EL4 ¼ À1:49952. This corresponds to eccentricities e < 0:04. 44 CHRISTOS EFTHYMIOPOULOS It should be stressed that a large fraction of the KAM tori of Figure 1(a) are destroyed, when the eccentricity of Jupiter, or other secular perturbations, are turned on.

Thus, the remainder series is absolutely convergent. However, the bound (32) is far from optimal. A much better bound is found by applying the inequality (31) for n ¼ M. Thus we find 1 X ðNþ1Þ ðNþ1Þ ðgM;q Þk ; ð33Þ jjRðNþ1Þ jjq OjjRðNþ1;MÀ1Þ jjq þ jjUM jjq k¼0 where ðNþ1Þ gM;q ¼ Mq : rF ðM À N þ 1Þ ðNþ1Þ Let qà be such that gM;qà < 1. Then, for all q < qà we have ðNþ1Þ jjRðNþ1Þ jjq O jjRðNþ1;MÀ1Þ jjq þ jjUM jjq 1 ðNþ1Þ 1 À gM;qà : ð34Þ Define now the constant Bqà as ðNþ1Þ Bqà ¼ jjRðNþ1;MÀ1Þ jjqà þ jjM ðNþ1Þ jjqà 1=ð1 À gM;qÃ Þ jjUNþ1 jjqNþ1 à : ð35Þ Then, the following lemma holds: for all q < qà ; ðNþ1Þ jjRNþ1 jjq OBqà jjUNþ1 jjqNþ1 : ð36Þ FORMAL INTEGRALS AND NEKHOROSHEV STABILITY 39 Equation (36) determines a rigorous upper bound for the size of the remainder jjRNþ1 jjq in terms of the size of the leading term of the remainder ðNþ1Þ jjUNþ1 jj.

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