A simple model of the chaotic eccentricity of Mercury

¹ Centro de Astrofísica, Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal
² Astronomie et Systèmes Dynamiques, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMC, 77 Av. Denfert-Rochereau, 75014 Paris, France
³ Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA
e-mail: boue@oddjob.uchicago.edu

Received: 11 July 2012
Accepted: 17 October 2012

Abstract

Mercury’s eccentricity is chaotic and can increase so much that collisions with Venus or the Sun become possible. This chaotic behavior results from an intricate network of secular resonances, but in this paper, we show that a simple integrable model with only one degree of freedom is actually able to reproduce the large variations in Mercury’s eccentricity, with the correct amplitude and timescale. We show that this behavior occurs in the vicinity of the separatrices of the resonance g₁−g₅ between the precession frequencies of Mercury and Jupiter. However, the main contribution does not come from the direct interaction between these two planets. It is due to the excitation of Venus’ orbit at Jupiter’s precession frequency g₅. We use a multipolar model that is not expanded with respect to Mercury’s eccentricity, but because of the proximity of Mercury and Venus, the Hamiltonian is expanded up to order 20 and more in the ratio of semimajor axis. When the effects of Venus’ inclination are added, the system becomes nonintegrable and a chaotic zone appears in the vicinity of the separatrices. In that case, Mercury’s eccentricity can chaotically switch between two regimes characterized by either low-amplitude circulations or high-amplitude librations.