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.