Dynamics of the Jupiter Trojans with Saturn’s perturbation in the present configuration of the two planets

Abstract

The dynamics of the two Jupiter triangular libration points perturbed by Saturn is studied in this paper. Unlike some previous works that studied the same problem via the pure numerical approach, this study is done in a semianalytic way. Using a literal solution, we are able to explain the asymmetry of two orbits around the two libration points with symmetric initial conditions. The literal solution consists of many frequencies. The amplitudes of each frequency are the same for both libration points, but the initial phase angles are different. This difference causes a temporary spatial asymmetry in the motions around the two points, but this asymmetry gradually disappears when the time goes to infinity. The results show that the two Jupiter triangular libration points should have symmetric spatial stable regions in the present status of Jupiter and Saturn. As a test of the literal solution, we study the resonances that have been extensively studied in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006). The resonance structures predicted by our analytic theory agree well with those found in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006) via a numerical approach. Two kinds of chaotic orbits are discussed. They have different behaviors in the frequency map. The first kind of chaotic orbits (inner chaotic orbits) is of small to moderate amplitudes, while the second kind of chaotic orbits (outer chaotic orbits) is of relatively larger amplitudes. Using analytical theory, we qualitatively explain the transition process from the inner chaotic orbits to the outer chaotic orbits with increasing amplitudes. A critical value of the diffusion rate is given to separate them in the frequency map. In a forthcoming paper, we will study the same problem but keep the planets in migration. The time asymmetry, which is unimportant in this paper, may cause an observable difference in the two Jupiter Trojan groups during a very fast planet migration process.