The meeting began at 11:00 am with a brief address by outgoing president Burns highlighting the most relevant advances in Celestial Mechanics that occurred in the last 3 years.

AbstractWe investigate the secular evolution of non-resonant exoplanetary systems consisting of a central star and two co-planar planets using a semi-numerical averaging method of the first order in planetary masses (in this case equivalent to “averaging by scissors” or simply dropping the fast periodic terms). The resulting Hamiltonian level curves for different exoplanetary systems were compared to those obtained by direct numerical integration. Studying the dependence of the reliability of the averaging method (as well as chaoticity of numerically integrated trajectories) upon the initial conditions, we found that the averaging methods fails even for Hill stable systems. Based on the Hill stability criterion we introduced empirically a more restrictive stability condition, that enabled us to give an estimate for the region of validity of the averaging method in the plane of initial conditions.

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An adiabatic approximation for the non-planar, circular, restricted 3BP is presented for the external resonance 4/7. It can be used as a model for resonant Kuiper belt objects. The Hamiltonian is truncated at the fourth order in eccentricities and inclinations. After averaging, we have a system of two degrees of freedom with two frequencies. Numerical calculations show that the ratio of these frequencies is ~102. Having introduced suitable canonical variables, we used the adiabatic approach introduced by Wisdom in a different context. We left slow variables frozen and after solving the pendulum problem for fast variables, we used the averaged effect of fast variables on slow variables. In this way we obtained the guiding trajectories for slow variables as contour lines of adiabatic invariant. We discuss the existence of a chaotic region which is formed by trajectories crossing a critical curve which corresponds to the separatrix of fast pendulum motion, where the assumption of sharp division between fast and slow frequencies is not correct and the adiabatic theory fails. The model works well for e ~ 0.1 and can be used for finding the chaotic regions, but for e~ 0.17 it becomes unsatisfactory due to truncation and bad convergence of the Laplace expansion. Qualitatively it can, however, help us to understand how the protective mechanism works as the interplay of mean motion and Kozai–Lidov resonance.

1. Dynamics of Extrasolar Planets (C. Beaugé) The orbital fits of multi-planetary systems from radial velocity data has proved to be a complex task. In some cases, different orbital solutions provide similarly good fits, especially when two planets are near mean-motion resonances. Ferraz-Mello et al. (2005) and Goździewski et al. (2005) showed that the published best fits of systems HD82932 and HD160691 are dynamically unstable, and redetermined their orbital parameters with Monte Carlo and genetic algorithms. In both cases dynamically stable orbits were found with RMS similar to the published orbits. It was also shown that uncertainties in the stellar mass (FerrazMello et al. 2005) and the stellar jitter (Gozdziewski et al. 2005) can significantly affect the orbital determination. Ford (2005) used a Markov chain Monte Carlo technique to quantify the orbit uncertainties. For some planetary systems he found a strong correlation between the orbital elements and/or significant non-Gaussian error distribution in the parameter space. As a consequence, the actual uncertainties in the orbital fits can be much larger (or smaller) than those published. Multiple-planetary systems in mean-motion resonances are relevant for their complex dynamics, but also for the inferences on a past planetary migration. To date there are at least four confirmed resonant systems: GJ876, HD82943 and HD128311 in the 2/1 commensurability, and HD202206 in the 5/1. The two middle planets of 55Cnc seem to be in the 3/1 mean-motion resonance, although there is some doubt on the orbital fits and more observations are necessary. The orbital fits of all these candidate resonant systems place the planets in an Apsidal Corotation Resonance (ACR): both the resonant angle σ and the difference in longitudes of pericenter ∆$ oscillate around a stationary value. Snellgrove et al. (2001) found that an ACR configuration similar to the fit of the GJ876 planets could be explained via a smooth inward planetary migration from initially non-resonant circular orbits. Hadjidemetriou (2002), Hadjidemetriou & Psychoyos (2003), Beaugé et al. (2003) and Lee (2004) used numerical and analytical approaches to determine families of ACR in mean-motion resonances, particularly the 2/1 and 3/1, as a function of the planetary mass ratios, semimajor axes and eccentricities. Resonance capture under a wide range of migration mechanisms and the relationship between the ACR and migration has also been the subject of several studies (e.g. Lee & Peale 2002, Nelson & Papaloizou 2002, Papaloizou 2003, FerrazMello et al. 2003, Kley et al. 2005). For non resonant planetary systems, Goździewski (2002), Goździewski & Konacki (2004) and Goździewski et al. (2005) mapped the phase space near several systems (47 UMa, HD169830 and HD160691) and identified regions of stable and chaotic motion. Lee

A systematic study of the main asteroidal resonances of the third and fourth order is performed using mapping techniques. For each resonance one-parameter family of surfaces of section is presented together with a simple energy graph which helps to understand and predict the changes in the surfaces of section within the family. As the truncated Hamiltonian for the planar, elliptic, restricted three-body problem is used for the mapping, the method is expected to fail for high eccentricities. We compared, therefore, the surfaces of section with trajectories calculated by symplectic integrators of the fourth and six order employing the full Hamiltonian. We found a good agreement for small eccentricities but differences for the higher eccentricities (e ∼ 0.3).

The Laplace-Lagrange secular theory applied by Yokoyama et al. to the 3/1 and 2/1 asteroidal resonances is used to obtain formulae for the instability region of the first-order resonance p = (p + 1)/p. As this approach yields the instability region even for 3/2 resonance, where Hildas are observed, it is concluded that this calculation in itself cannot explain the origin of Kirkwood gaps and that more refined methods allowing to calculate the maximum eccentricity have to be applied.

Tables of coefficients of the disturbing function calculated at resonant value of semimajor axes are presented for resonances (p + q)/p, where (p + q) < 8. For given combination of angular variables and degrees of eccentricities e, e' and degrees of sin 1/2i, sin 1/2i' the corresponding coefficients of the Poisson series for the disturbing function are tabulated. These results were obtained by the program written in C language. The program can be employed for obtaining much more extensive tables taking into account higher degrees in eccentricities and inclinations. The highest degree of tabulated terms is 4. These coefficients are needed for mapping calculations.

A nonlinear theory of secular resonances is developed. Both terms corresponding to secular resonances ν5 and ν6 are taken into account in the Hamiltonian. The simple overlap criterion is applied and the condition for the overlap of these resonances is found. It is shown that in given approximation the value p = (1 - e2)1/2(1 - cosI) is an integral of motion, where the mean eccentricity e and mean inclination I are obtained by eliminating short-period perturbations as well as the nonresonant terms from the planets. The overlap criterion yields a critical value of parameter p depending on the semi-major axis a of the asteroid. For p greater than the critical value, resonance overlap occurs and chaotic motion has to be expected. A mapping is presented for fast calculation of the trajectories. The results are illustrated by level curves in surfaces of section method.

The motion of an asteroid located near the 5/2 resonance is investigated using previously obtained mapping (Šidlichovský, Melendo, 1986) based on the method of Wisdom. Two modes of motion with low and high eccentricity were found. The transition between these two modes is possible. The chaotic zone is investigated using Liapunov characteristic exponents (LCE). The reliability time for trajectories in the chaotic region, where the roundoff error propagates exponentially is estimated with LCE as 1E6 years. The transition between the modes is discussed in terms of the Wisdom zone of uncertainty. As the eccentricity in the high eccentricity mode goes to e=0.4, the evidence for collisional removal of asteroids from the gap is provided.

On presente une solution analytique des equations de l'evolution liee aux marees pour les orbites circulaires equatoriales d'un point materiel autour d'un corps triaxial deforme par les marees