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

TL;DR

This paper analytically studies the bifurcation of halo orbits around collinear points in the circular restricted three-body problem, providing accurate bifurcation thresholds and insights into the system's dynamics.

Contribution

It develops a normal form approach to determine bifurcation energies for halo orbits in the three-body problem, including cases with small mass ratios.

Findings

Analytical bifurcation thresholds align well with numerical results.

Normal form method effectively predicts bifurcation points for most mass ratios.

Singular perturbation issues arise for small mass ratios at L3, limiting the method's applicability.

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.