The Long-term Error Estimation Method for the Numerical Integrations of Celestial Orbits

Abstract

Numerical methods have become a very important type of tool for celestial mechanics, especially in the study of planetary ephemerides. The errors generated during the computation are hard to know beforehand when applying a certain numerical integrator to solve a certain orbit. In that case, it is not easy to design a certain integrator for a certain celestial case when the requirement of accuracy were extremely high or the time-span of the integration were extremely large. Especially when a fixed-step method is applied, the caution and effort it takes would always be tremendous in finding a suitable time-step, because it is about whether the accuracy and time-cost of the final result are acceptable. Thus, finding the best balance between efficiency and accuracy with the least time cost appeared to be a major obstruction in the face of both numerical integrator designers and their users. To solve this problem, we investigate the variation pattern of truncation error and the pattern of rounding error distributions with time-step and time-span of the integration. According to those patterns, we promote an error estimation method that could predict the distribution of rounding errors and the total truncation errors with any time-step at any time-spot with little experimental cost, and test it with the Adams-Cowell method in the calculation of circular periodic orbits. This error estimation method is expected to be applied to the comparison of the performance of different numerical integrators, and also it can be of great help for finding the best solution to certain cases of complex celestial orbits calculations.