which should be rigorously preserved up to machine-ε precision.
2.4.2 Error in planetary longitudes
Since the symplectic maps preserve total energy and total angular momentum of N-body dynamical systems inherently well， the degree of their preservation may not be a good measure of the accuracy of numerical integrations， especially as a measure of the positional error of planets， i.e. the error in planetary longitudes. To estimate the numerical error in the planetary longitudes， we performed the following procedures. We compared the result of our main long-term integrations with some test integrations， which span much shorter periods but with much higher accuracy than the main integrations. For this purpose， we performed a much more accurate integration with a stepsize of 0.125 d (1/64 of the main integrations) spanning 3 × 105 yr， starting with the same initial conditions as in the N1 integration. We consider that this test integration provides us with a ‘pseudo-true’ solution of planetary orbital evolution. Next， we compare the test integration with the main integration， N1. For the period of 3 × 105 yr， we see a difference in mean anomalies of the Earth between the two integrations of 0.52°(in the case of the N1 integration). This difference can be extrapolated to the value 8700°， about 25 rotations of Earth after 5 Gyr， since the error of longitudes increases linearly with time in the symplectic map. Similarly， the longitude error of Pluto can be estimated as 12°. This value for Pluto is much better than the result in Kinoshita & Nakai (1996) where the difference is estimated as 60°.
3 Numerical results – I. Glance at the raw data
In this section we briefly review the long-term stability of planetary orbital motion through some snapshots of raw numerical data. The orbital motion of planets indicates long-term stability in all of our numerical integrations: no orbital crossings nor close encounters between any pair of planets to