resent value， the jovian plas remain stable over 1010 yr， or perhaps longer. Dun Lissauer also performed four similar experiments on the orbital motion of seven plas {Venus to une}， which cover a span of ～109 yr. their experiments on the seven plas are not yet prehensive， but it seems that the terrestrial plas also remain stable during the iion period， maintaining almular oscillations.
oher hand， in his accurate semi-analytical secular perturbation theory {Laskar 1988}， Laskar finds that large and irregular variations appear in the etricities and inations of the terrestrial plas， especially of mercury and mars on a time-scale of several 109 yr {Laskar 1996}. the results of Laskar“s secular perturbation theory should be firmed and iigated by fully numerical iions.
in this paper we present preliminary results of six long-term numerical iions on all nine plaary orbits， c a span of several 109 yr， and of two other iions c a span of ± 5 × 1010 yr. the total elapsed time for all iions is more than 5 yr， using several dedicated Pd workstations. one of the fual clusions of our long-term iions is that solar system plaary motioo be stable in terms of the hill stability mentioned above， at least over a time-span of ± 4 gyr. actually， in our numerical iions the system was far more stable than what is defined by the hill stability criterion: not only did no close enter happen during the iion period， but also all the plaary orbital elements have been fined in a narrion both in time and frequenain， though plaary motions are stochastic. sihe purpose of this paper is to exhibit and overview the results of our long-term numerical iions， we show typical example figures as evidence of the very long-term stability of solar system plaary motion. For readers who have more specifid deeper is in our numerical results， repared a webpage {access }， where we show raw orbital elements， their low-pass filtered results， variation of Delaunay elements and angular momentum deficit， as of our simple time–frequenalysis on all of our iions.
iion 2 we briefly explain our dynamical model， numerical method and initial ditions used in our iions. se 3 is devoted to a description of the quick results of the numerical iions. 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 giveion 4 goes on to a discussion of the loerm variation of plaary orbits using a low-pass filter and includes a discussion of angular momentum deficit. iion 5， we present a set of numerical iions for the outer five plahat spans ± 5 × 1010 yr. iion 6 we also discuss the long-term stability of the plaary motion and its possible cause.
2 Description of the numerical iions
{本部分涉及比較複雜的積分計算，作者君就不貼上來了，貼上來了起點也不一定能成功顯示。}
2.3 numerical method
we utilize a sed-order wisdom–holman symplectic map as our main iiohod {wisdom holman 1991; Kinoshita， Yoshida nakai 1991} with a special start-up procedure to reduce the truncation error of angle variables，‘warm start’{saha tremaine 1992， 1994}.
the stepsize for the numerical iions is 8 d throughout all iions of the nine plas {n±1，2，3}， which is about 111 of the orbital period of the innermost pla {mercury}. as for the determination of stepsize， we partly follow the previous numerical iion of all nine plas in sussman wisdom {1988， 7.2 d} and saha tremaine {1994， 22532 d}. we rouhe 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 iation processes. iion to this， wisdom holman {1991} performed numerical iions of the outer five plaary orbits using the symplectic map with a stepsize of 400 d， 110.83 of the orbital period of Jupiter. their result seems to be accurate enough， which partly justifies our method of determining the stepsize. however， sihe etricity of Jupiter {～0.05} is much smaller than that of mercury {～0.2}， we need some care when we pare these iions simply in terms of stepsizes.
iegration of the outer five plas {F±}， we fixed the stepsize at 400 d.
t gauss“ f and g funs in the symplectic map together with the third-order halley method {Danby 1992} as a solver for Kepler equations. the number of maximum iteratio in halley“s method is 15， but they never reached the maximum in any of our iions.
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 iion of the outer five plas {F±}.
although no output filtering was done when the numerical iions were in process， lied a low-pass filter to the raw orbital data after leted all the calculations. see se 4.1 for more detail.
2.4 error estimation
2.4
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