Abstract
The detection and identification of resonances is a key ingredient in the studies of stability of dynamical systems, with important applications for our Solar as well as for extrasolar systems. In this paper we study the detection of resonances and close encounters in the three-body problem using the fast Lyapunov indicator method. In order to investigate the close encounters, we needed to adapt the method to the model and its singularities. Our technical improvement lies in computation of the solutions of the singular variational equations by measuring the divergence of close initial conditions. We have used the Levi-Civita regularization for the integration of the equations of motion. As an application of the method, we show that it provides a correct detection of the tube manifolds related to the Lagrangian point L1 of the Sun–Jupiter system.
Highlights
Since the pioneering work of Henon and Heiles (Henon & Heiles 1964) wherein they studied the phase space with the method of Poincaresurface of section, global phase-space studies of dynamical systems have become a standard approach in many different problems
In the field of astronomy, global studies for the puzzling problem of the long-term stability of our Solar system can be found in, for example Nesvorny & Morbidelli (1998), Murray & Holman (1999), Robutel & Laskar (2001), Morbidelli (2002), Guzzo (2005, 2006), Robutel & Gabern (2006) and Hayes (2008); a picture of the global dynamics in galactic potential is provided by, for example Papaphilippou & Laskar (1998), Voglis, Tsoutsis & Efthymiopoulos (2006) and Namouni, Guzzo & Lega (2008); and studies of global dynamics related to cometary motion, close encounters and space mission design can be found in, for example Villac (2008), Koon et al (2001), Perozzi & Ferraz Mello (2010), Gomez et al (2004), Valsecchi (2005) and Koon et al (2008)
In this paper we propose a numerical computation of the solutions of the variational equations that works independently of the close encounter with the primary or with the secondary body
Summary
Since the pioneering work of Henon and Heiles (Henon & Heiles 1964) wherein they studied the phase space with the method of Poincaresurface of section, global phase-space studies of dynamical systems have become a standard approach in many different problems. In the field of astronomy, global studies for the puzzling problem of the long-term stability of our Solar system can be found in, for example Nesvorny & Morbidelli (1998), Murray & Holman (1999), Robutel & Laskar (2001), Morbidelli (2002), Guzzo (2005, 2006), Robutel & Gabern (2006) and Hayes (2008); a picture of the global dynamics in galactic potential is provided by, for example Papaphilippou & Laskar (1998), Voglis, Tsoutsis & Efthymiopoulos (2006) and Namouni, Guzzo & Lega (2008); and studies of global dynamics related to cometary motion, close encounters and space mission design can be found in, for example Villac (2008), Koon et al (2001), Perozzi & Ferraz Mello (2010), Gomez et al (2004), Valsecchi (2005) and Koon et al (2008) Many of these studies have been motivated by the celebrated KAM (Kolmogorov 1954; Arnold 1963; Moser 1958) and Nekhoroshev (Nekhoroshev 1977) theorems, providing fundamental information about the longterm stability of a Hamiltonian system from the global knowledge of phase space, from the distribution of resonances. We discuss in Appendix A the LC regularization in the three-body problem
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
