A new understanding of L₄ and L₅ axial orbits through the torus structure

In this paper, through the critical isosurface of the pseudo-Hamiltonian of co-orbital motions in the torus space, we provide a new understanding of L₄ and L₅ axial orbits and their invariant manifolds in the circular restricted three-body problem. The contact points on the critical isosurface of the pseudo-Hamiltonian correspond to the locations of L₄ and L₅ axial orbits in the torus space, and provide a set of good initial guesses of L₄ and L₅ axial orbits for the multiple shooting method. Furthermore, we calculate and analyse orbital behaviours of L₄ and L₅ axial orbit families. Based on the topological structure of the critical isosurface of the pseudo-Hamiltonian, compound dynamical motions of invariant manifolds associated with L₄ and L₅ axial orbits are discussed. We present an approximate estimation for libration amplitudes of different co-orbital portions of invariant manifolds. Results obtained from numerical integration demonstrate the validity of our semi-analytical approach in the torus space..