s， earth and mars. the elements of mercury， especially its etricity， seem to ge to a signifit extent. this is partly because the orbital time-scale of the pla is the shortest of all the plas， which leads to a more rapid orbital evolution than other plas; the innermost pla may be o instability. this result appears to be in some agreement with Laskar“s {1994， 1996} expectations that large and irregular variations appear in the etricities and inations 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 mercury. we will mention briefly the long-term orbital evolution of mercury later iion 4 using low-pass filtered orbital elements.
the orbital motion of the outer five plas seems rigorously stable and quite regular over this time-span {see also se 5}.
3.2 time–frequency maps
although the plaary motion exhibits very long-term stability defined as the ence of close enter events， the chaotiature of plaary dynamics ge the oscillatory period and amplitude of plaary orbital motion gradually over such long time-spans. even such slight fluctuations of orbital variation in the frequenain， particularly in the case of earth， potentially have a signifit effe its surface climate system through solar insolation variation {cf. berger 1988}.
to give an overview of the long-term ge in periodicity in plaary 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} frequenalysis.
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， wheh data begins from tti and ends at tti t， the data segment ranges from ti δt≤ti δt t， where δt?t. we tihis division until we reach a certain number n by whi t reaches the total iioh.
ly an FFt to each of the data fragments， and obtain n frequency diagrams.
in each frequency diagram obtained above， the strength of periodicity be replaced by a grey-scale {or colour} chart.
we perform the replat， and ect all the grey-scale {or colour} charts into one graph for eategration. the horizontal axis of these nehs 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， sihe amount of numerical data to be deposed into frequency pos 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 etricity and ination of earth in n 2 iion. 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 strohan in the lighter area around it. we reize from this map that the periodicity of the etricity and ination of earth only ges slightly over the entire period covered by the n 2 iion. this nearly regular trend is qualitatively the same in other iions and for other plas， although typical frequencies differ pla by pla and element by element.
4.2 Long-term exge of orbital energy and angular momentum
we calculate very long-periodic variation and exge of plaary orbital energy and angular momentum using filtered Delaunay elements L， g， h. g and h are equivalent to the plaary orbital angular momentum and its vertical po per unit mass. L is related to the plaary orbital energy e per unit mass as e?μ22L2. if the system is pletely linear， the orbital energy and the angular momentum in each frequency bin must be stant. non-liy in the plaary system cause an exge of energy and angular momentum in the frequenain. 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 promi in our long-term iions.
in Fig. 7， the total orbital energy and angular momentum of the four inner plas and all nine plas are shown for iion n 2. the upper three panels show the long-periodic variation of total energy {denoted ase- e0}， total angular momentum { g- g0}， and the vertical po { h- h0} of the inner four plas calculated from the low-pass filtered Delaunay elements.e0，
本章尚未完結,請點擊下一頁繼續閱讀---->>>