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Error estimation

Relative errors in total energy and angular momentum

Aording to one of the basic properties of syors， which conserve the physically conservative quantities well (total orbital energy and angular momentum)， our long-tero have been performed with very small errors. The averaged relative errors of total energy (～10?9) and of total angular momentum (～10?11) have restant 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 al energy δE/E0 in our nuionsN± 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 nual angular momentum， which should be rigorously preserved up to .

Error in plaary longitudes

Since the sy maps preserve total energy and total angular momentum of N-body dynaly well， the degree of their preservation may not be a good y of nuions， especially as a measure of the positional error of plas， . 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 ions with soions， which span much shorter periods but with y than the ions. For this purpose， we performed a ion with a stepsize of d (1/64 of the ions) spanning 3 × 105 yr， starting with the saditions as in the N?1 integration. We consider that this test integration provides us with a ‘pseudo-true’ solution of plaary orbital evolution. Next， we coion with the ion， N?1. For the period of 3 × 105 yr， we see a difference in mean anomalies of the Earth between the two integrations of ～°(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 ti 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-terary orbital motion through some snapshots of raw numerical data. The orbital s indicates long-term stability in all of our nuions: no orbital crossings nor close encounters between any pair of plas took place.

General description of the stability of plaary orbits

First， we briefly look at the general character of the long-terary 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 frofigurations shown in Figs 2 and 3， orbital positions of the terrestrial plas differ little between the initial and final part of each nuion， 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 (fro) 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 × 10 9 yr).(b) The final part ofN+1 ( t = × 10 8 to × 10 9 yr).(c) The initial part of N?1 (t= 0 to ? × 109 yr).(d) The final part ofN?1 ( t =? × 10 9 to ? × 10 9 yr). In each panel， a total of 23 684 points are plotted with an interval of about 2190 yr over × 107 yr . Solid lines in each panel denote the present orbits of the four terrestrial plas (taken from DE245).

The variation of eentricities 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 eentricity， seeo a significant extent. This is partly because the orbital ti is the shortest of all the plas， which leads to a more rapid orbital evolution than other plas; the innero instability. This result appears to be in some agreement with Laskar's (1994， 1996) expectations that large and irregular variations appear in the eentricities and inclinations of Mercury on a time-scale of several 109 yr. However， the effect of the possible instability of the orbit of Mercury may not fatally affect the global stability of the whole plaary system owing to the small mass of he long-term orbital evolution of ion 4 using low-pass filtered orbital elements.

The orbital motion of the outer five plas seems rigorously stable and quite regular over this ti 5).

Time–frequency maps

Although the plaary ability defined as the non-existence of close encounter events， the chaotic nature of plaary dynahe oscillatory period and aary orbital motion gradually over such long time-spans. Even such slight fluctuations of orbital variation in the frequency domain， particularly in the case of Earth， can potentially have a significant effect on its surface climate system through solar insolation variation (cf. Berger 1988).

To give an overview of the long-terge in periodicity in plaary orbital motion， we performed many fast Fourier transforime axis， and superposed the resulting periodgrams to draw two-dimensional time–frequency approach to drawing these time–frequency maps in this paper is very simple – much simpler than the wavelet analysis or Laskar's (1990， 1993) frequency analysis.

Divide the low-pass filtered orbital data into many fragments of the same length. The length of each data segment should be a o apply the FFT.

Each fragment of the data has a large overlapping part: for exah data begins from t=ti and ends at t=ti+T， the next data segment ranges from ti+δT≤ti+δT+T， where δT?T. We continue this division until we reach a certain nu+T reaches the total integration length.

We apply an FFT to each of the data fragments， and obtain n frequency diagrams.

In each frequency diagram obtained above， the strength of periodicity can be replaced by a grey-scale (or colour) chart.

We perform the replace all the grey-scale (or colour) charts into one graph for each integration. The horizontal axis of these new graphs should be the time， . the starting times of each fragment of data (ti， where i= 1，…， n). The vertical axis represents the period (or frequency) of the oscillation of orbital elements.

We have adopted an FFT because of its overwhelhe amount of nuposed into frequency cos is terribly huge (several tens of Gbytes).

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