Analytic theory for high-inclination orbits in the restricted three-body problem

This paper explores the analytical solution properties surrounding a hypothetical orbit in an invariant plane perpendicular to the line joining the two primaries in the circular restricted three-body problem. Assuming motion can be maintained in the plane, Jacobi's integral equation can be analytically integrated, yielding a closed-form expression for the period and path of the third body expressed with elliptic integral and elliptic function theory. In this case, the third body traverses a circular path with nonuniform speed. In a strict sense, the in-plane assumption cannot be maintained naturally. However, the hypothetical orbit is shown to satisfy Jacobi's integral equation and the tangential motion equation exactly and the other two motion equations approximately in bounded-averaged and banded sense. More important, the hypothetical solution can be used as the basis for an iterative analytical solution procedure for the three-dimensional trajectory where corrections are computable in closed form. In addition, the in-plane assumption can be strictly enforced with the application of a modulated thrust acceleration which is expressible in closed form. Presented methodology is primarily concentrated on halo-class orbits.