第一百日（1）作屎的老猫（第3/6页）

（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 second-order Wisdom–Holman symplectic map as our main integration method (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 integrations is 8 d throughout all integrations of the nine planets (N±1，2，3)， which is about 1/11 of the orbital period of the innermost planet (Mercury). As for the determination of stepsize， we partly follow the previous numerical integration of all nine planets in Sussman & Wisdom (1988， 7.2 d) and Saha & 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 computation processes. In relation to this， Wisdom & Holman (1991) performed numerical integrations of the outer five planetary 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 compare these integrations simply in terms of stepsizes.

In the integration of the outer five planets (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 planets (N±1，2，3)， and about 8000 000 d (～21 903 yr) for the integration of the outer five planets (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 completed 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

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).