  Due to the chaotic nature of the Solar System, the question of its dynamic
long-term stability can only be answered in a statistical sense, e.g. based on
numerical ensemble integrations of nearby orbits. Destabilization, including
catastrophic encounters and/or collisions involving the Earth, has been
suggested to be initiated through a large increase in Mercury's eccentricity
(eM), with an estimated probability of ~1%. However, it has recently been shown
that the statistics of numerical Solar System integrations are sensitive to the
accuracy and type of numerical algorithm. Here I report results from
computationally demanding ensemble integrations (N=1,600 with slightly
different initial conditions) at unprecedented accuracy based on the full
equations of motion of the eight planets and Pluto over 5Gyr, including
contributions from general relativity. The standard symplectic algorithm
produced spurious results for highly eccentric orbits and during close
encounters, which were hence integrated with a suitable Bulirsch-Stoer
algorithm, specifically designed for these situations. The present study yields
odds for a large increase in Mercury's eccentricity that are less than previous
estimates. Strikingly, in two solutions Mercury continued on highly eccentric
orbits (after reaching eM values >0.93) for 80-100Myr before colliding with
Venus or the Sun. Most importantly, none of the 1,600 solutions led to a close
encounter involving the Earth or a destabilization of Earth's orbit in the
future. I conclude that Earth's orbit is dynamically highly stable for billions
of years, despite the chaotic behavior of the Solar System.
