Orbital Evolution of the Sun-Jupiter-Saturn-Uranus-Neptune Four-Planet System on Long-Time Scales

摘要

The four-planet problem is solved by constructing an averaged semi-analytical theory of secondorder motion by planetary masses. A discussion is given of the results obtained by numerical integration of the averaged equations of motion for the Sun-Jupiter-Saturn-Uranus-Neptune system over a time interval of 10 Gyr. The integration is based on high-order Runge-Kutta and Everhart methods. The motion of the planets is almost periodic in nature. The eccentricities and inclinations of the planetary orbits remain small. Short-period perturbations remain small over the entire interval of integration. Conclusions are drawn about the resonant properties of the motion. Estimates are given for the accuracy of the numerical integration.