.1 Relative errors in total energy and angular momentum
acc to one of the basic properties of symplectitegrators， which serve the physically servative quantities well {total orbital energy and angular momentum}， our long-term numerical iioo 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 stant throughout the iion 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 δaa0 and the total energy δee0 in our numerical iionsn± 1，2，3， where δe and δa are the absolute ge of the total energy and total angular momentum， respectively， ande0anda0are their initial values. the horizontal unit is gyr.
hat 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 reize this situation in the secular numerical error ial angular momentum， which should be rigorously preserved up to mae-ε precision.
2.4.2 error in plaary longitudes
sihe symplectic maps preserve total energy and total angular momentum of n-body dynamical systems ily well， the degree of their preservation may not be a good measure of the accuracy of numerical iions， especially as a measure of the positional error of plas， i.e. the error in plaary longitudes. to estimate the numerical error in the plaary longitudes， we performed the following procedures. we pared the result of our main long-term iions with some test iions， which span much shorter periods but with much higher accuracy than the main iions. For this purpose， we performed a much more accurate iion with a stepsize of 0.125 d {164 of the main iions} spanning 3 × 105 yr， starting with the same initial ditions as in the n?1 iion. we sider that this test iion provides us with a ‘pseudo-true’ solution of plaary orbital evolutio， we pare the test iion with the main iion， n?1. For the period of 3 × 105 yr， we see a differen mean anomalies of the earth betweewo iions of ～0.52°{in the case of the n?1 iion}. this difference be extrapolated to the value ～8700°， about 25 rotations of earth after 5 gyr， sihe error of longitudes increases linearly with time in the symplectic map. similarly， the longitude error of Pluto 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. gla the raw data
in this se we briefly review the long-term stability of plaary orbital motion through some snapshots of raw numerical data. the orbital motion of plas indicates long-term stability in all of our numerical iions: no orbital crossings nor close enters between any pair of plaook place.
3.1 general description of the stability of plaary orbits
First， we briefly look at the general character of the long-term stability of plaary orbits. our i here focuses particularly on the inner four terrestrial plas for which the orbital time-scales are much shorter than those of the outer five plas. as we see clearly from the planar orbital figurations shown in Figs 2 and 3， orbital positions of the terrestrial plas differ little between the initial and final part of eaumerical iion， which spans several gyr. the solid lines denoting the present orbits of the plas lie almost within the swarm of dots even in the final part of iions {b} and {d}. this indicates that throughout the eegration period the almular variations of plaary orbital motion remain nearly the same as they are at present.
Vertical view of the four inner plaary orbits {from the z -axis dire} at the initial and final parts of the iionsn±1. the axes units are au. the xy -plane is set to the invariant plane of solar system total angular momentu{a} the initial part ofn 1 { t 0 to 0.0547 × 10 9 yr}.{b} the final part ofn 1 { t 4.9339 × 10 8 to 4.9886 × 10 9 yr}.{c} the initial part of n?1 {t 0 to ?0.0547 × 109 yr}.{d} the final part ofn?1 { t ?3.9180 × 10 9 to ?3.9727 × 10 9 yr}. in each panel， a total of 23 684 points are plotted with an interval of about 2190 yr over 5.47 × 107 yr . solid lines in each panel dehe present orbits of the four terrestrial plas {taken from De245}.
the variation of etricities and orbital inations for the inner four plas in the initial and final part of the iion n 1 is shown in Fig. 4. as expected， the character of the variation of plaary orbital elements does not differ signifitly between the initial and final part of eategration， at least for Venu
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