Although no output filtering was done when the numerical integrations were in process， we applied a low-pass filter to the raw orbital data after we had pleted all the calculations. See Section 4.1 for more detail.
2.4 Error estimation
2.4.1 Relative errors in total energy and angular momentum
According to one of the basic properties of symplectic integrators， which conserve the physically conservative quantities well (total orbital energy and angular momentum)， our long-term numerical integrations seem to have been performed with very small errors. The averaged relative errors of total energy (～10?9) and of total angular momentum (～10?11) have remained nearly constant throughout the integration period (Fig. 1). The special startup procedure， warm start， would have reduced the averaged relative error in total energy by about one order of magnitude or more.
Relative numerical error of the total angular momentum δA/A0 and the total energy δE/E0 in our numerical integrationsN± 1，2，3， where δE and δA are the absolute change of the total energy and total angular momentum， respectively， andE0andA0are their initial values. The horizontal unit is Gyr.
Note that different operating systems， different mathematical libraries， and different hardware architectures result in different numerical errors， through the variations in round-off error handling and numerical algorithms. In the upper panel of Fig. 1， we can recognize this situation in the secular numerical error in the total angular momentum， 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 pared 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 N?1 integration. We consider that this test integration provides us with a ‘pseudo-true’ solution of planetary orbital evolution. Next， we pare the test integration with the main integration， N?1. 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 N?1 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°.
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