对火星轨道变化问题的最后解释

um 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