In this work an analytical study of the bifurcation of the halo orbits around the collinear points L1and L2for the circular, spatial, restricted three-body problem is presented. The energy level at which the bifurcation takes place, for arbitrary values of the mass ratio, is found by reducing the Hamiltonian of the problem into a synchronous resonant normal form bymeans of Lie Transformations. This naturally provides an integrable approximation the system, which yields to the reduction of the system to the center manifold. The bifurcation thresholds of the 1 W 1 resonant periodic orbit families are estimated, among which the ‘halo’ orbits. Analytical results are compared with the numerical ones existing in the literature. Initial conditions for generating halos are found inverting this analytical process.
Ceccaroni, M., Celletti, A., & Pucacco, G. (2016). Bifurcations thresholds of halo orbits. In Astrophysics and Space Science Proceedings (Vol. 44, pp. 35–44). Kluwer Academic Publishers. https://doi.org/10.1007/978-3-319-23986-6_3