Limiting orbits around the equilateral centers of libration

Abstract

Publisher Summary This chapter describes the limiting orbits at the equilateral centers of libration. The planar restricted problem of three bodies is presented by the Hamiltonian function. For all these mass ratios, the limiting orbits have exhibited nontrivial characteristic exponents of the stable type. This result is somewhat puzzling. It has been suggested that the limiting orbits could perhaps be identified with that doubly asymptotic periodic orbit, which is the homoclinic solution common to the two manifolds of asymptotic solutions generated at the unstable equilibrium point. The fact that the limiting orbits have characteristic exponents of the stable type makes it embarrassing to accept this conjecture. Once the existence of limiting periodic orbits has been established, one can face the second part of Browns hypothesis. Each limiting orbit for a fixed value of the mass ratio μ should be the head of two families of periodic librations and for the first family, the period would decrease together with the Jacobi constant, whereas for the second family, the period would increase together with the Jacobi constant.