Optimized Nekhoroshev stability estimates for the Trojan asteroids with a symplectic mapping model of co-orbital motion

Abstract

This paper reports analytic estimates of the domain of Nekhoroshev stability for the orbits of Jupiter's Trojan asteroids calculated in the space of proper elements (Dp, ep), for a stability time exceeding the age of the Solar system (tstability= 1010 yr). The model used is a family of Hadjidemetriou mappings, for different values of the proper eccentricity ep, that represent the Poincaré sections of co-orbital motion in the Hamiltonian of the planar and circular restricted three-body problem. These explicit mappings are shown to reproduce accurately the dynamics that is implicitly induced by the corresponding Hamiltonian model. Optimal Nekhoroshev estimates are obtained by constructing the Birkhoff normal form for symplectic mappings. Our optimization is based on an ‘iterated remainder’ criterion. The asymptotic behaviour of the Birkhoff series is determined by a precise analysis of the accumulation of small divisors in the series terms at consecutive orders of normalization. About 35 per cent of asteroids from a recent catalogue (AstDys), with proper inclination Ip ≤ 5°, are shown to be Nekhoroshev-stable over the age of the Solar system. By calculating a resonant Birkhoff normal form, this percentage increases to 48 per cent.