Further Simplification of the Manifold Correction Method for Orbit Integration

Abstract

By introducing some quantities defined on the moving orbital plane, we have developed three ways to further reduce the number of variables in our two simplified methods of manifold correction, from nine to seven or six per celestial body. Among these, the simplest option uses a set of six variables consisting of the three-dimensional orbital angular momentum vector, the two independent components of the Laplace integral vector on the moving orbital plane, and a true orbital longitude measured from an origin solely determined from the orbital angular momentum vector. This scheme no longer requires any manifold correction, as does the standard method to integrate the Cartesian coordinates and velocity. However, the new method is much more precise than the standard method. For example, the longitude error of Mercury in a simultaneous integration of the Sun and nine major planets with a step size of 1.4 days exceeds 360 after a few thousand years when using the standard method but remains at the milliarcsecond level if the new method is used. In addition, the new method achieves better performance than any of the manifold correction methods we have developed, as long as round-off errors are negligible.