Overlong simulations of the solar system dynamics with two alternating step-lengths

We compare the performance of four symplectic integration methods with leading order symplectic corrector in simulations of the Solar System. These simulations cover 10 Gyr. They are longer than the astrophysical predicted future of the present-day Solar System, thus this work is mainly a study of the integration methods. For the outer Solar System simulation, where the used stepsize was 100 days, the energy errors do not show any secular evolution. The maximum errors show a dependence on the method. The simulations of the full Solar System from Mercury, and including Pluto as a test particle, were calculated with a stepsize of 7 days. The energy errors behave somewhat differently having a small secular behavior. This may due to the short timestep and the short period of the planet Mercury or some small round off error produced by the code. Comparison of the eccentricity evolution’s within simulations show that some planets are dynamically strongly coupled. Venus and Earth form a dynamical pair, also Jupiter and Saturn form a dynamical pair. The FFT of the analysis of the simulations suggests that all the giant planets form a single dynamical quadruple system. The orbit of Mercury is possibly unstable. Each simulation is stopped when Mercury is expelled. All the methods show similar results for times less than 30 Myr in the way that the results for orbital elements are same within plotting precision. Inclusion of Mercury in simulations shortens the Solar System e-folding time to 3.3 Myr. It is clear that chaos has a strong effect in the evolution of orbital elements, especially eccentricities. This is easily seen in Mercury’s orbit when the simulation time exceeds at least 30 Myr. Our low-order simulations seem to match high-order methods over long timescales.