Nekhoroshev stability atL₄orL₅in the elliptic-restricted three-body problem - application to Trojan asteroids

Abstract

The problem of analytical determination of the stability domain around the points L4 or L5 of the Lagrangian equilateral configuration of the three‐body problem has served in the literature as a basic celestial mechanical model probing the predictive power of the so‐called Nekhoroshev theory of exponential stability in non‐linear Hamiltonian dynamical systems. All analytical investigations in this framework have so far been based on the circular restricted three‐body problem (CRTBP). In this work, we extend the analytical estimates of Nekhoroshev stability in the case of the planar elliptic‐restricted three‐body problem (ERTBP). To this end, we introduce an explicit symplectic mapping model for the planar ERTBP, obtained via Hadjidemetriou's method, which generalizes the family of mappings discussed in earlier papers. The mapping is based on an expansion of the disturbing function up to a sufficiently high order in the eccentricities and the variations of the semimajor axis, and it is given as a series around a period‐one fixed point of the system. Within the domain of the mapping's convergence, we then compute a Birkhoff normal form for 4D symplectic mappings as well as the associated approximate integrals of motion which can be expressed in terms of proper elements. The variations of the integrals predicted by the remainder of the normal form series provide a lower bound for the domain of Nekhoroshev stability for a time at least equal to the age of the Solar System. In the case of Jupiter's Trojans, the domain of the mapping's convergence lies entirely within the region of librational motion, in which the longitude of the perihelion of the asteroid librates around a fixed point value of ϖ. For most asteroids outside this domain macroscopic chaotic diffusion cannot be ruled out. The present analysis provides a physically relevant estimate of the domain of Nekhoroshev stability for proper librations (Dp < 100), and marginal for proper eccentricities (ep < 0.01). The formalism is developed in a general way allowing for applications in both, our Solar System and extrasolar system dynamics.

A parametric study of stability and resonances around L 4 in the elliptic restricted three-body problem

“…Most of the studies were devoted to the stability of Jupiter's Trojan asteroids (see e.g., Giorgilli and Skokos 1997;Levison et al 1997;Marzari and Scholl 2002;Efthymiopoulos and Sándor 2005;Robutel and Gabern 2006;Lhotka et al 2008), but other Trojans were also considered (Dvorak et al 2008b). Kinoshita and Nakai (2007) studied quasi-satellites of Jupiter in the 1:1 mean motion resonance.…”

The orbit of 2010 TK7: possible regions of stability for other Earth Trojan asteroids

“…the studies of Rabe (1967); Bien & Schubart (1984); Lhotka et al (2008) and Érdi et al (2009) as well as many others. Ever since, extensive numerical studies have been undertaken to find the extension of the stability regions around the equilibrium points of the planets, e.g.…”

The dynamics of inclined Neptune Trojans

“…The mapping is defined on the 4 dimensional Poincaré surface of section, but given in its implicit form. Former studies (Lhotka et al 2008) showed that the radius of convergence of the proposed mapping method becomes limited to the librational regime of the asteroid's motion, if we expand the generating function to make the mapping explicit by series reversion. Therefore we applied a semi-analytical approach and introduced a simple root-finding algorithm to iterate the mapping at each iteration step without expanding it into explicit form; so we can preserve all possible dynamical behaviour of the mapping.…”

“…A detailed description of the developement of the disturbing function in the 1:1 resonance with free parameters (m 1 , a , e , ω ) can be found e.g. in (Lhotka et al 2008), where an exponential stability estimate for the Trojan group of asteroids (e.g. Efthymiopoulos and Sándor 2005) was performed and generalized to be applicable also for different Solar system configurations.…”