Iterated analytical solution to the restricted problem of three bodies

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 can not be maintained naturally. However, there may be cases where the assumption is approximately maintained over a finite time period. More importantly, 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. In this case, results are similar to those of Tsien for continuous radial thrust.