Lambertian invariance and application to the problem of optimal fixed-time impulsive orbital transfer

Abstract

To develop an efficient technique for the numerical solution of the Lambert problem, a new Lambertian invariant, the ratio of two invariable lengths, is proposed to replace the Gauss ratio of two areas as the main iterative variable in the time equation, and iteration schemes are devised for fast convergence under various conditions. The problem of minimum fuel 2-impulse transfer between two coplanar circular orbits under fixed time of transfer is then analyzed and numerically solved by the technique developed. The use of multi-revolution to improve the solution in the long duration case is outlined and numerically illustrated; and the two cases, wherein the two circular orbits are in the same direction of motion (uni-rotating), or in oppositedirections (counter-rotating) are distinguished and compared. Finally, by extending the study from 2-impulse to 3-impulse transfer a global synthesis of the various possible types of fixed time optima under different transfer conditions is briefly presented.