On Some Special Restricted Three Body Problems

Generally speaking, the Three-Body Problem is not integrable. Thus to simpli-fy the problem and to simulate real astronomical systems, we need to introduce some simplifications. Among them, the Restricted-Three-Body Problem is a great approxi-mation of many astronomical problems, e.g., a model of Sun+planet+massless test particles in studying the orbital evolution of satellites is often adopted and when tack-ling the evolution of main belt asteroids or Kuiper belt objects, we usually construct a model of Sun+Jupiter or Neptune+test particles; these Restricted-Three-Body Prob-lem models are verified to be excellent approximations.Particularly, in the hierarchical Three-Body system where it can be approximated as an inner orbit and an outer one and where the hierarchy ensures the stability, the problem can be further classified as inner restricted (where the test particle is on the inner orbit) and outer restricted (where the particle is on the outer orbit). We discuss these two kinds of systems in chapter 1 and 2, respectively.In treating the outer restricted hierarchical Three-Body problem, using the ex-panded perturbation function, we get an integrable system in the lowest order. We give the general description of the characteristics of the orbital evolution and it is found that, the outer longitude of ascending node may librate when the inclination is high. Then we introduce higher order terms and pay special attention to the evolution of the outer eccentricity. In cases where the two orbits are near coplanar, the eccentricity excitation is similar to the strictly coplanar situation. But for the near polar orbits, the eccentric-ity excitation can be diverse and this is because there may be two libration regions in the phase diagram, depending on initial inclination. Particularly, in the large excitation region, the eccentricity can evolve from about 0 to roughly 0.3, and this will signifi-cantly influence the stability of such orbits. Actually, with this eccentricity excitation mechanism, we can put close constraints on the stability boundary.When dealing with the inner restricted Three-Body problem, we put the emphasis on cases where the precession rate of the inner orbit and the rate of the revolution of the outer orbit are near. In this situation, the mean motion of the outer orbit cannot be eliminated. We derive the Hamiltonian including outer eccentricity in coplanar simplification and find out that, the instability boundary is in close relationship with the structures in the Poincare surface of section. The higher order Hamiltonian of inclined orbit is also presented here for the first time.