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description of the quick results of the numerical integrations。 Very long…term stability of Solar system plaary motion is apparent both in plaary positions and orbital elements。 A rough estimation of numerical errors is also given。 Section 4 goes on to a discussion of the longest…term variation of plaary orbits using a low…pass filter and includes a discussion of angular momentum deficit。 In Section 5， we present a set of numerical integrations for the outer five plas that spans ± 5 × 1010 yr。 In Section 6 we also discuss the long…term stability of the plaary motion and its possible cause。
2 Description of the numerical integrations
（本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了起点也不一定能成功显示。）
2。3 Numerical method
We utilize a send…order Wisdom–Holman symplectic map as our main integration method （Wisdom &；amp； Holman 1991； Kinoshita， Yoshida &；amp； Nakai 1991） with a special start…up procedure to reduce the truncation error of angle variables，‘warm start’（Saha &；amp； Tremaine 1992， 1994）。
The stepsize for the numerical integrations is 8 d throughout all integrations of the nine plas （N±1，2，3）， which is about 1/11 of the orbital period of the innermost pla （Mercury）。 As for the determination of stepsize， we partly follow the previous numerical integration of all nine plas in Sussman &；amp； Wisdom （1988， 7。2 d） and Saha &；amp； Tremaine （1994， 225/32 d）。 We rounded the decimal part of the their stepsizes to 8 to make the stepsize a multiple of 2 in order to reduce the accumulation of round…off error in the putation processes。 In relation to this， Wisdom &；amp； Holman （1991） performed numerical integrations of the outer five plaary orbits using the symplectic map with a stepsize of 400 d， 1/10。83 of the orbital period of Jupiter。 Their result seems to be accurate enough， which partly justifies our method of determining the stepsize。 However， since the eccentricity of Jupiter （～0。05） is much smaller than that of Mercury （～0。2）， we need some care when we pare these integrations simply in terms of stepsizes。
In the integration of the outer five plas （F±）， we fixed the stepsize at 400 d。
We adopt Gauss' f and g functions in the symplectic map together with the third…order Halley method （Danby 1992） as a solver for Kepler equations。 The number of maximum iterations we set in Halley's method is 15， but they never reached the maximum in any of our integrations。
The interval of the data output is 200 000 d （～547 yr） for the calculations of all nine plas （N±1，2，3）， and about 8000 000 d （～21 903 yr） for the integration of the outer five plas （F±）。
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
Acrding to one of the basic properties of symplectic integrators， which nserve the physically nservative 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 nstant 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 regnize 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 plaary 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 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 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 nditions as in the N？1 integration。 We nsider that this test integration provides us with a ‘pseudo…true’ solution of plaary 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 &；amp； 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 plaary orbital motion through some snapshots of raw numerical data。 The orbital motion of plas indicates long…term stability in all of our numerical integrations： no orbital crossings nor close enunters between any pair of plas took 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 interest 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 can see clearly from the planar orbital nfigurations shown in Figs 2 and 3， orbital positions of the terrestrial plas differ little between the initial and final part of each numerical integration， 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 integrations （b） and （d）。 This indicates that throughout the entire integration period the almost regular 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 direction） at the initial and final parts of the integrationsN±1。 The axes units are au。 The xy …plane is set to the invariant plane of Solar system total angular momentum。（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 denote the present orbits of the four terrestrial plas （taken from DE245）。
The variation of eccentricities and orbital inclinations for the inner four plas in the initial and final part of the integration N＋1 is shown in Fig。 4。 As expected， the character of the variation of plaary orbital elements does not differ significantly between the initial and final part of each integration， at least for Venus， Earth and Mars。 The elements of Mercury， especially its eccentricity， seem to change to a significant extent。 This is partly because the o