对火星轨道变化问题的最后解释

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.