Download New Developments in the Dynamics of Planetary Systems: by William H. Jefferys (auth.), Rudolf Dvorak, Jacques Henrard PDF

By William H. Jefferys (auth.), Rudolf Dvorak, Jacques Henrard (eds.)

It is now a well-established culture that each 4 years, on the finish of wintry weather, a gaggle of 'celestial mechanicians' from world wide assemble within the Austrian Alps on the invitation of R. Dvorak. This time the colloquium was once held at Badhofgastein from March 19 to March 25, 2000 and was once dedicated to the 'New advancements within the Dynamics of Planetary Systems'. The papers lined a wide range of questions of present curiosity: t- oretical questions (resonances, KAM concept, delivery, ... ) and questions about numerical instruments (synthetic parts, symptoms of chaos, ... ) have been fairly good represented; in fact planetary theories and close to Earth gadgets have been additionally relatively well known. 3 designated lectures have been introduced in honor of deceased colleagues whom, to our dismay, we are going to now not meet on the 'Austrian Colloquia'. W. Jefferys introduced the Heinrich Eichhorn lecture on 'Statistics for the Twenty-first Century Astrometry', a subject on which Heinrich Eichhorn used to be a expert. A. Roy introduced a lecture honoring Victor Szehebely on 'Lifting the Darkness: technology within the 3rd Millenium', during which in wove anecdotes and remembrances of Victor which moved the viewers a great deal. A. Lemaitre spoke in honor of Michele Moons on 'Mech­ anism of catch in exterior Resonance'. the top of her speak was once dedicated to a brief and relocating biography of Michele illustrated via many slides.

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Extra info for New Developments in the Dynamics of Planetary Systems: Proceedings of the Fifth Alexander von Humboldt Colloquium on Celestial Mechanics held in Badhofgastein (Austria), 19–25 March 2000

Example text

The same as Figure 4, but for the data from the joint catalog. 001. Some questions could arise about the reliability of our estimates of the accuracy of proper elements for the asteroids whose orbits have been computed for 2 Myr only. Could some of these tum out to be more unstable over an extended time span? Out of:::::: 10, 000 cases, it would be unwise to make an absolute statement that this could never happen. However, the orbits selected for the extended integration have been the ones already showing instability of the proper elements and/or chaotic behavior, and the others are less likely to become unstable after having already been regular over 2 Myr.

Since KAM theory ensures us that the function E(w) is continuous on the sets r' and r", then there are two frequencies w' E r' and w" E r" corresponding to two invariant tori, say T' and T" respectively, such that they belong to the energy surface ~ related to the level E . Thus, it is enough to check that the initial data belong to the gap between T' and T" on the surface ~ in order to assure that the orbit will be trapped there forever. Applying the procedure above to our model we prove the following.

The Hamiltonian writes F=--1 2 (II_r_1_1 2{3 3 II 2{3 3) _ _1_2+T m 9 m Al +~ A:} f. (1) , where A 1 , A 2 are the action variables and A 1 , A. tJ = Q(mo + m 1 ) and {31 = m 1m 01(m 1 +mo) for j = 1,2, m 0 being the mass of the Sun and m 1 , m 2 the masses of the planets and g being the gravitational constant. Moreover, f. is the distance between the planets and T a term coming from the expression of the kinetic energy in heliocentric coordinates. We recall that the Poincare variables are ~J = 'IJ = fiAj J 1- J 1- e] cos w 1 , -fiAJj 1- }1- e] j = 1, 2 (2) sinw1, with the usual notations a;, e1 , z1 and w 1 for the semi-major axes, the eccentricities, the mean anomalies and the perihelion arguments, respectively.

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