Successive Refinements in Long-term Integrations of Planetary Orbits

We report on accurate, long-term numerical simulations of the orbits of the major planets in our solar system. The equations of motion are directly integrated by a Störmer multi-step scheme, which is optimized to reduce round-off errors. The physical models are successively refined to include corrections due to general relativity and the finite size of the lunar orbit. In one case, the Earth– Moon system is resolved as two separate bodies and the results are compared to those based on analytically averaging the lunar orbit. Through this comparison, a better analytical model is obtained. The computed orbits are in good agreement with those of previous studies for the past five million years but not for earlier times. The inner planets exhibit chaotic behavior with a Lyapunov time of exponential separation of nearby orbits equal to about 4 million years.