By Rudolf Dvorak, Sylvio Ferraz-Mello
Undesirable Hofgastein who made the very profitable Salzburger Abend with indi- nous track from Salzburg attainable. specific thank you additionally to the previous director of the Institute of Astronomy in Vienna, Prof. Paul Jackson for his beneficiant deepest donation. we must always now not put out of your mind our hosts Mr. and Mrs. Winkler and their staff from the lodge who made the remain really stress-free. None folks will fail to remember the final night, whilst the workers of kitchen lower than the le- ership of the cook dinner himself got here to supply us as farewell the well-known Salzburger Nockerln, a standard Austrian dessert. every person acquired loads of scienti?c enter through the lectures and the discussions and, to summarize, all of us had a spl- did week in Salzburg within the inn Winkler. all of us wish to come back back in 2008 to debate new effects and new views on a excessive point scienti?c commonplace within the Gasteinertal. Rudolf Dvorak and Sylvio Ferraz-Mello Celestial Mechanics and Dynamical Astronomy (2005) 92:1-18 (c) Springer 2005 DOI 10. 1007/s10569-005-3314-7 FROM ASTROMETRY TO CELESTIAL MECHANICS: ORBIT choice WITH VERY brief ARCS (Heinrich okay. Eichhorn Memorial Lecture) 1 2 ? ANDREA MILANI and ZORAN KNEZEVIC 1 division of arithmetic, college of Pisa, through Buonarroti 2, 56127 Pisa, Italy, e mail: milani@dm. unipi. it 2 Astronomical Observatory, Volgina 7, 11160 Belgrade seventy four, Serbia and Montenegro, e mail: zoran@aob. bg. a
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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
The mapping of Sa´ndor et al. reproduces the characteristics of the Poincare´ surface of section of the original Hamiltonian. In particular, the mapping has the same ﬁxed points and with the same stability as the Hamiltonian. In the case of zero eccentricity e ¼ 0, the mapping reads ! 1 ; xnþ1 ¼ xn þ 2pl sin sn 1 À ð2 À 2 cos sn Þ3=2 ! 1 À1 : ð51Þ snþ1 ¼ sn þ 2p ð1 þ xnþ1 Þ3 FORMAL INTEGRALS AND NEKHOROSHEV STABILITY 43 Figure 1(a) shows the phase portrait of the mapping (51). The invariant KAM curves of this mapping correspond to librations around L4 .
We conclude from this comparison that the eﬀect of diﬀerent chaoticity of the orbits for the Lagrangian points is an eﬀect which already appears when we include Saturn in the model. But, the diﬀerence in the chaotic behaviour of the L4 and the L5 orbits – found with the aid of the LCE – could be due to a biased choice of initial conditions for Trojans around the two equilibrium points. We therefore can not yet claim that there is in fact a diﬀerence in the dynamics between the two stable Lagrangian points for more realistic models including other planets.
The size of the stable region (number of stable orbits) was determined with a least square ﬁt: NðiÞ ¼ a Á i2 þ b Á i þ c with: a ¼ À0:0046 Æ 0:0007; b ¼ 0:0800 Æ 0:0352; 3 Alignment of Sun, Jupiter and the Trojan. STABILITY REGIONS OF THE TROJAN ASTEROIDS 23 Figure 2. Largeness of the stable regions around L4 and L5 in the dynamical model SJS synodic longitude versus initial semimajor axis (in AU). 2 AU. Points indicate stable orbits. c ¼ 9:13406 Æ 0:3782. N corresponds to the percentage of the stable orbits out of the grid in initial conditions speciﬁed above.
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 by Rudolf Dvorak, Sylvio Ferraz-Mello