The general Kepler equation and its solutions

Abstract

My father K. Stumpff (1947, 1949, 1951, 1959, 1962) developed a transcendental equation which replaces the original Kepler equation but is valid for all types of orbits. Other advantages over the classical methods are: a) the independent arguments of the equation follow from the vectors of position and velocity at any instant T_o, where T_o is not necessarily the perihelion time; b) an explicit knowledge of the classical orbital elements is not required; c) transformations of coordinate systems are avoided. The present paper discusses the properties of the general Kepler equation in a wide range of its independent arguments, and it is shown that analytic solutions, existing in special cases, can be used for the numerical solution of general cases. The theory is generalized insofar as it now can handle not only attracting forces but also repulsive ones. As a result of this investigation, FORTRAN subroutines were developed which can be used in connection with any two-body problem for the computation of position and velocity as function of time along any unperturbed orbit.