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

• We consider halo orbits around the collinear Lagrangian–Eulerian points.
• An analytical estimate of the bifurcation threshold to halo orbits is obtained.
• The method is based on a normal form adapted to the synchronous resonance.
• A reduction to the central manifold is then performed.
• We make a comparison with available numerical data.

We perform an analytical study of the bifurcation of the halo orbits around the collinear points L1L1, L2L2, L3L3 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 L3L3 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.