PERIODIC ORBITS IN THE VICINITY OF THE EQUILATERAL POINTS OF THE RESTRICTED FULL THREE-BODY PROBLEM

摘要

Periodic orbits are computed for a particle in the gravitational field of a binary system. First, the equations of motion are derived. One of the bodies is modeled as a sphere and the other as a constant density tri-axial ellipsoid. Previous work has investigated the dynamics and stability of relative equilibria of the binary system itself. The dynamics of a particle in the gravitational field of a binary system have also been studied. Equilibrium solutions near L_4, L_5 and their stability have been characterized. In the current model, the stability of these points are functions of the mass ratio, the distance between the bodies and the size parameters of the ellipsoid. In this paper, the binary system is in a non-synchronous motion where the ellipsoid spins about its maximum moment of inertia while being in mutual orbit about the sphere. In order to find periodic orbits, a Poincare map reduction method is used with the time periodic nonlinear equations of motion of the ellipsoid-sphere system. To provide more analytical insights on the behavior of this periodic system, a perturbation method is also applied. As an approximation, the gravity field of the ellipsoid body is expanded in terms of spherical harmonics. Numerical simulations show that stable periodic orbits exist for low values of the mass ratio. The ratio between the orbit period of the binary system and the rotation period of the primary is now an added free parameter of the system. We find it plays a strong role in determining the stability and size of the resulting periodic orbits. The stability region is reduced compared to known results of the Restricted Three-Body Problem and the synchronous case of the Restricted Full Three-Body Problem.