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 compared 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 compare 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°.
3 Numerical results – I. Glance at the raw data
In this section we briefly review the long-term stability of planetary orbital motion through some snapshots of raw numerical data. The orbital motion of planets indicates long-term stability in all of our numerical integrations: no orbital crossings nor close encounters between any pair of planets took place.
3.1 General description of the stability of planetary orbits
First， we briefly look at the general character of the long-term stability of planetary orbits. Our interest here focuses particularly on the inner four terrestrial planets for which the orbital time-scales are much shorter than those of the outer five planets. As we can see clearly from the planar orbital configurations shown in Figs 2 and 3， orbital positions of the terrestrial planets 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 planets 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 planetary orbital motion remain nearly the same as they are at present.
Vertical view of the four inner planetary 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 planets (taken from DE245).
The variation of eccentricities and orbital inclinations for the inner four planets in the initial and final part of the integration N+1 is shown in Fig. 4. As expected， the character of the variation of planetary 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 orbital time-scale of the planet is the shortest of all the planets， which leads to a more rapid orbital evolution than other planets; the innermost planet may be nearest to instability. This result appears to be in some agreement with Laskar's (1994， 1996) expectations that large and irregular variations appear in the eccentricities 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 planetary system owing to the small mass of Mercury. We will mention briefly the long-term orbital evolution of Mercury later in Section 4 using low-pass filtered orbital elements.
The orbital motion of the outer five planets seems rigorously stable and quite regular over this time-span (see also Section 5).
3.2 Time–frequency maps
Although the planetary motion exhibits very long-term stability defined as the non-existence of close encounter events， the chaotic nature of planetary dynamics can change the oscillatory period and amplitude of planetary 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-term change in periodicity in planetary orbital motion， we performed many fast Fourier transformations (FFTs) along the time axis， and superposed the resulting periodgrams to draw two-dimensional time–frequency maps. The specific 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 multiple of 2 in order to apply the FFT.
Each fragment of the data has a large overlapping part: for example， when the ith 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 number N by which tn+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 replacement， and connect all the grey-scale (or colour) charts into one graph for each integration. The horizontal axis of these new graphs should be the time， i.e. 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 overwhelming speed， since the amount of numerical data to be decomposed into frequency components is terribly huge (several tens of Gbytes).
A typical example of the time–frequency map created by the above procedures is shown in a grey-scale diagram as Fig. 5， which shows the variation of periodicity in the eccentricity and inclination of Earth in N+2 integration. In Fig. 5， the dark area shows that at the time indicated by the value on the abscissa， the periodicity indicated by the ordinate is stronger than in the lighter area around it. We can recognize from this map that the periodicity of the eccentricity and inclination of Earth only changes slightly over the entire period covered by the N+2 integration. This nearly regular trend is qualitatively the same in other integrations and for other planets， although typical frequencies differ planet by planet and element by element.
4.2 Long-term exchange of orbital energy and angular momentum
We calculate very long-periodic variation and exchange of planetary orbital energy and angular momentum using filtered Delaunay elements L， G， H. G and H are equivalent to the planetary orbital angular momentum and its vertical component per unit mass. L is related to the planetary orbital energy E per unit mass as E=?μ2/2L2. If the system is completely linear， the orbital energy and the angular momentum in each frequency bin must be constant. Non-linearity in the planetary system can cause an exchange of energy and angular momentum in the frequency domain. The amplitude of the lowest-frequency oscillation should increase if the system is unstable and breaks down gradually. However， such a symptom of instability is not prominent in our long-term integrations.
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