Second-order matching in the restricted three-body problem (smallμ)

Elliptic orbits around the large primary are matched to hyperbolas, osculating at closest approach, around the small primary of the circular restricted three-body problem. The distance of closest approach to the small primary is assumed to be of the same order as the mass-ratio μ of small to large primary. The dependence of the hyperbola on initial conditions for the elliptic orbit is carried to second order jointly in μ and in the variations of the initial conditions, which are three-dimensional rather than two-dimensional.