Astronomy & Astrophysics

Free access article

Issue		A&A Volume 378, Number 2, November I 2001
Page(s)		569 - 586
Section		Formation and evolution of planetary systems
DOI		10.1051/0004-6361:20011189

A&A 378, 569-586 (2001)
DOI: 10.1051/0004-6361:20011189

Global dynamics of planetary systems with the MEGNO criterion

K. Gozdziewski^{1, 2}, E. Bois¹, A. J. Maciejewski³ and L. Kiseleva-Eggleton^{4, 5}

¹ Observatoire de Bordeaux, UMR/CNRS/INSU 5804, BP 89, 33270 Floirac, France
² Torun Centre for Astronomy, N. Copernicus University, Gagarina 11, 87-100 Torun, Poland
³ Institute of Astronomy, Zielona Góra University, Lubuska 2, 65-265 Zielona Góra, Poland
⁴ IGPP, LLNL, L-413, 7000 East Ave, Livermore, CA 94550, USA
⁵ UC Davis, Dept. of Physics, Davis, CA 95616, USA

(Received 11 June 2001 / Accepted 22 August 2001)

Abstract
In this paper we apply a new technique alternative to the numerically computed Lyapunov Characteristic Number (LCN) for studying the dynamical behaviour of planetary systems in the framework of the gravitational N-body problem. The method invented by P. Cincotta and C. Simó is called the Mean Exponential Growth of Nearby Orbits (MEGNO). It provides an efficient way for investigation of the fine structure of the phase space and its regular and chaotic components in any conservative Hamiltonian system. In this work we use it to study the dynamical behaviour of the multidimensional planetary systems. We investigate the recently discovered $\upsilon$ And planetary system, which consists of a star of $1.3 M_{\odot}$ and three Jupiter-size planets. The two outermost planets have eccentric orbits. This system appears to be one of the best candidates for dynamical studies. The mutual gravitational interaction between the two outermost planets is strong. Moreover the system can survive on a stellar evolutionary time scale as it is claimed by some authors (e.g., Rivera & Lissauer 2000b). As the main methodological result of this work, we confirm important properties of the MEGNO criterion such as its fast convergence, and short motion times (of the order of 10⁴ times the longest orbital period) required to distinguish between regular and chaotic behaviors. Using the MEGNO technique we found that the presence of the innermost planet may cause the whole system to become chaotic with the Lyapunov time scale of the order of 10³-10⁴ yr only. Chaos does not induce in this case visible irregular changes of the orbital elements, and therefore its presence can be overlooked by studying variations of the elements. We confirm explicitly the strong and sensitive dependence of the dynamical behaviour on the companion masses.

Key words: celestial mechanics, stellar dynamics -- methods: numerical, N-body simulations -- planetary systems -- stars: individual ( $\upsilon$ Andromedae)

Offprint request: K. Gozdziewski, chris@astri.uni.torun.pl

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