Velocity scaling method to correct individual Kepler energies

As variants of the velocity–position single scaling method of Fukushima, which extends Nacozy’s manifold correction scheme to monitor the integrated position and velocity by using the integral invariant relation and the same spatial scale transformation, a new velocity scaling method and a new position scaling method for correcting the varying Kepler energy of each body in an n-body problem of planetary dynamics are presented. Compared with Fukushima’s idea, the new schemes are simple to operate. Like other existing methods including the method of Fukushima and of Wu et al., the two new methods not only are almost the same effectiveness in significantly improving the orbital semi-major axis or mean anomaly at the epoch, but also can raise the accuracy of numerical integration by several orders. In particular, the new velocity scaling method as well as the method of Wu et al. is the most convenient in application.