Peculiar trajectories around the Lagrangian equilateral points

In the present work we analyse the behaviour of a particle under the gravitational influence of two massive bodies and a particular dissipative force. The circular restricted three body problem, which describes the motion of this particle, has five equilibrium points in the frame which rotates with the same angular velocity as the massive bodies: two equilateral stable points (L4,L5L4,L5) and three colinear unstable points (L1,L2,L3L1,L2,L3). A particular solution for this problem is a stable orbital libration, called a tadpole orbit, around the equilateral points. The inclusion of a particular dissipative force can alter this configuration. We investigated the orbital behaviour of a particle initially located near L4L4 or L5L5 under the perturbation of a satellite and the Poynting–Robertson drag. This is an example of breakdown of quasi-periodic motion about an elliptic point of an area-preserving map under the action of dissipation. Our results show that the effect of this dissipative force is more pronounced when the mass of the satellite and/or the size of the particle decrease, leading to chaotic, although confined, orbits. From the maximum Lyapunov Characteristic Exponent a final value of γγ was computed after a time span of 106 orbital periods of the satellite. This result enables us to obtain a critical value of logγlogγ beyond which the orbit of the particle will be unstable, leaving the tadpole behaviour. For particles initially located near L4L4, the critical value of logγlogγ is −4.07 and for those particles located near L5L5 the critical value of logγlogγ is −3.96.