An elegant Lambert algorithm

Of the many techniques extant for solving the two-body, two-point, time-constrained orbital boundary-value problem, commonly known today as Lambert's problem, none is more conceptually elegant than the classical method devised by Gauss. The simplicity of Gauss' method would certainly have been attractive to the modern astrodynamicist except for two major flaws — the method is singular for a transfer angle of 180 degrees and the convergence rate is extremely slow when that angle is not very small. In this paper a new algorithm is described which exactly parallels both the mechanics and the elegant simplicity of the classical one but is completely devoid of the two basic faults of the original. The equations of the new method are universal and not singular for the 180 degree transfer. (They are singular for a complete revolution through 360 degrees but this should not be cause for great alarm.) Furthermore, convergence is both remarkably rapid and almost uniform as well as being essentially independent of the initial guess. It should further be emphasized that all of the advantages of Gauss' method are inherent in the new method — notably, the preservation of numerical accuracy for small transfer angles (of the order of 2 or 3 degrees, for example).