Efficient Orbit Integration by Dual Scaling for Consistency of Kepler Energy and Laplace Integral

Abstract

By adding the Laplace integral as the second auxiliary quantity to be integrated, we extend our scaling method to integrate quasi-Keplerian orbits numerically in order to suppress the growth of integration errors not only in the semimajor axis but also in the other orbital elements, especially in the eccentricity and in the longitude of pericenter. This time, the method simultaneously follows the time evolution of the Kepler energy and the Laplace integral in addition to integrating the usual equation of motion. By using a dual spatial scale transformation, it adjusts the position and velocity that are integrated in order to satisfy both the Kepler energy relation and another functional relation derived from the Laplace integral rigorously at each integration step. The scale factors for the position and for the velocity are set separately and are determined by solving a set of linear equations. Just as with the original scaling method, the new method is simple to implement, fast to compute, and applicable to a wide variety of integration methods, perturbation types, and problem complexities. With the exception of the J2 perturbation, the new method is superior to the original scaling method because it achieves significantly fewer integration errors for the physical properties such as the shape and the orientation of orbits at the cost of a negligibly small amount of additional computation.