A New Transformation Invariant in the Orbital Boundary-Value Problem

Abstract

A fascinating corollary to Lambert's famous problem is developed. By applying this new property of two- body orbits, a simple reformulation of the two-point boundary-value problem is possible. This is accomplished by means of a geometrical transformation of the orbital foci which converts the original problem to one for which the initial point is an apsidal point. The elementary form of Kepler's equation then provides the analytic description of the time of flight. The elements of the original orbit are shown to be simply related to the corresponding elements of the transformed orbit. Finally, a simple iterative method of solving the transformed boundary-value problem using successive substitutions is developed. In most cases of interest, convergence is seen to be quite rapid. HERE are a variety1 of aesthetically appealing geometrical properties associated with the two-body, two-point orbital boundary-value problem. One of these, which is the basis for the main result of this paper, was discovered by Levine 2 over a decade ago in connection with an optical sighting problem for orbital navigation. Levine showed that the true anomaly of the point in an orbit where the velocity vector is parallel to the line of sight from the initial point to the terminal point is independent of the orbit. Until recently, this result was only of academic interest. Then, it was discovered that the eccentric anomaly of the point where the parallelism occurs is simply the arithmetic mean of the eccentric anomalies corresponding to the end points. This leads to a new corollary of Lambert's theorem in that the radial distance of this point is found to be a function of the same geometrical quantities as the flight time. The corollary,can be exploited by employing a geometrical transformation of the orbital foci which was suggested by Lagrange in a different context. By this means the original boundary value problem is shown to be equivalent to one for which the initial point is an apsidal radius. This radius, as well as the time of flight, are the invariants of the transformation. The paper concludes with a simple and elegant method for solving the transformed problem. The algorithm is analogous to the well-known method of successive substitutions frequently used to solve the elementary form of Kepler's equation. In this case, however, the technique is universal and applies equally well to elliptic, parabolic, and hyperbolic orbits. In most of the practical cases, the convergence is found to be quite rapid and almost independent of the initial trial solution.