By Rudolf Dvorak, Sylvio Ferraz-Mello

Undesirable Hofgastein who made the very winning Salzburger Abend with indi- nous track from Salzburg attainable. designated thank you additionally to the previous director of the Institute of Astronomy in Vienna, Prof. Paul Jackson for his beneficiant inner most donation. we should always no longer put out of your mind our hosts Mr. and Mrs. Winkler and their staff from the resort who made the remain rather relaxing. None folks will fail to remember the final night, whilst the employees of kitchen lower than the le- ership of the prepare dinner himself got here to provide us as farewell the recognized Salzburger Nockerln, a standard Austrian dessert. each person received loads of scienti?c enter throughout the lectures and the discussions and, to summarize, all of us had a spl- did week in Salzburg within the lodge Winkler. all of us wish to come back back in 2008 to debate new effects and new views on a excessive point scienti?c normal 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 decision 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, electronic mail: zoran@aob. bg. a

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**Extra 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**

**Sample text**

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 qÃ be such that gM;qÃ < 1. Then, for all q < qÃ we have ðNþ1Þ jjRðNþ1Þ jjq O jjRðNþ1;MÀ1Þ jjq þ jjUM jjq 1 ðNþ1Þ 1 À gM;qÃ : ð34Þ Deﬁne now the constant BqÃ as ðNþ1Þ BqÃ ¼ jjRðNþ1;MÀ1Þ jjqÃ þ jjM ðNþ1Þ jjqÃ 1=ð1 À gM;qÃ Þ jjUNþ1 jjqNþ1 Ã : ð35Þ Then, the following lemma holds: for all q < qÃ ; ðNþ1Þ jjRNþ1 jjq OBqÃ 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.

The sources of improvements were (a) the use of better variables, and (b) the measurement of the size of the remainder directly from the data of the calculated formal series. This eliminated the need for a priori estimates of this size, which were necessarily more pessimistic. The present paper improves the previous results by a factor 3. About 30% of real asteroids are included in the stability region found theoretically. The improvement is due partly to the choice of the variables given in Equation (49).

This algorithm is as follows. Starting the construction as previously with U2 ¼ zzÃ , yields a zero divisor in (22) with non-zero numerator for p ¼ m; q ¼ 0 or for p ¼ 0; q ¼ m, meaning that the coeﬃcients of the terms zm and zm Ã , of degree m, cannot be speciﬁed and the construction cannot proceed further. On the other hand, due to the resonant rotation angle x=2p ¼ 1=m, there is one more real-valued isolating integral of the linearized mapping z0 ¼ ei2p=m z. This is the phase integral S0;m ¼ zm þ zm Ã: ð23Þ Starting the construction with the terms S0;m yields a zero divisor in Equation , of degree m þ 2.