Extending Nacozy’s Approach to Correct All Orbital Elements for Each of Multiple Bodies

Abstract

For each object of an n-body problem in planetary dynamics, orbital elements except the mean anomaly are directly determined by five independently slow-varying quantities or quasi-integrals, which include the Keplerian energy, the three components of the angular momentum vector, and the z-component of the Laplace vector. The mean anomaly depends on the mean motion specified by the Keplerian energy. Decreasing integration errors of these quasi-integrals at every integration step means improving the accuracy of all the elements to a great extent. Because of this, we take reference values of these quantities in terms of the integral invariant relations as control sources of the errors and then give an extension of Nacozy's idea of manifold correction. The technique is almost the same as the linear transformation method of Fukushima in its explicit validity of correcting all elements, if the adopted basic integrators can give a necessary precision to the stabilizing sources considered. Especially it plays a more important role in significantly suppressing the growth of numerical errors in high eccentricities.