Halo orbits around the collinear points of the restricted three-body problem

Abstract

We perform an analytical study of the bifurcation of the halo orbits around the collinear points $L_1$, $L_2$, $L_3$ for the circular, spatial, restricted three--body problem. Following a standard procedure, we reduce to the center manifold constructing a normal form adapted to the synchronous resonance. Introducing a detuning, which measures the displacement from the resonance and expanding the energy in series of the detuning, we are able to evaluate the energy level at which the bifurcation takes place for arbitrary values of the mass ratio. In most cases, the analytical results thus obtained are in very good agreement with the numerical expectations, providing the bifurcation threshold with good accuracy. Care must be taken when dealing with $L_3$ for small values of the mass-ratio between the primaries; in that case, the model of the system is a singular perturbation problem and the normal form method is not particularly suited to evaluate the bifurcation threshold.

35 pages, 3 figures, updated version accepted for publication on Physica D

References in corpus (3)

• An energy-momentum map for the time-reversal symmetric 1:1 resonance with Z_2 X Z_2 symmetry
• Equivariant singularity analysis of the 2:2 resonance
• Normal forms for the epicyclic approximations of the perturbed Kepler problem

Cited by in corpus (7)

• On the analytical construction of halo orbits and halo tubes in the elliptic restricted three-body problem
• Transits close to the Lagrangian solutions $L_1,L_2$ in the Elliptic Restricted Three-body Problem
• Structure of the center manifold of the L1 and L2 collinear libration points in the restricted three-body problem
• Validated numerics for period-tupling and touch-and-go bifurcations of symmetric periodic orbits in reversible systems
• Bifurcation sequences in the symmetric 1:1 Hamiltonian resonance
• The dynamics around the collinear points of the elliptic three-body problem: A normal form approach
• Melnikov method for non-conservative perturbations of the three-body problem