Jumping mechanisms of Trojan asteroids in the planar restricted three- and four-body problems

Abstract

We explore minimal dynamical mechanisms for the transport of Trojan asteroids between the vicinities of the stable Lagrange points $L_{4}$ and $L_{5}$ within the framework of the planar restricted three- and four-body problems. This transport, called “jumping” of Trojan asteroids, has been observed numerically in the sophisticated Solar System models. However its dynamical mechanisms have not been fully explored yet. The present study shows that invariant manifolds emanating from an unstable periodic orbit around the unstable Lagrange point $L_{3}$ mediate the jumping of Trojan asteroids in the Sun–Jupiter planar restricted three-body problem. These invariant manifolds form homoclinic tangles and lobes when projected onto the configuration space through a discrete mapping. Thus the resulted lobe dynamics explains the mechanism for the jumping of Jupiter’s Trojan asteroids. In the Sun–Earth planar restricted three-body problem, on the other hand, invariant manifolds of an unstable periodic orbit around $L_{3}$ do not exhibit clear homoclinic tangles nor lobes, indicating that the jumping is very difficult to occur. It is then shown that the effect of perturbation of Venus is important for the onset of the jumping of Earth’s Trojan asteroids within the framework of the Sun–Earth–Venus planar restricted four-body problem. The results presented here could shed new insights into the transport mechanism as well as trajectory design associated with $L_{3}$, $L_{4}$, and $L_5$.