对火星轨道变化问题的最后解释
The variation of etricities and orbital inations for the inner four plas in the initial and final part of the iion N+1 is shown in Fig. 4. As expected， the character of the variation of plaary orbital elements does not differ signifitly between the initial and final part of eabsp; iion， at least for Venus， 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， whibsp; leads to a more rapid orbital evolution than other plas; the innermost pla may be to 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 effebsp; of the possible instability of the orbit of Mercury may not fatally affebsp; 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 in Se 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–frequenbsp; maps
Although the plaary motion exhibits very long-term stability defined as the enbsp; of close enter events， the chaotibsp; nature of plaary dynamibsp; bsp; ge the oscillatory period and amplitude of plaary orbital motion gradually over subsp; long time-spans. Even subsp; slight fluctuations of orbital variation in the frequenbsp; domain， particularly in the case of Earth， bsp; potentially have a signifit effebsp; on its surfabsp; climate system through solar insolation variation (bsp; 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–frequenbsp; maps. The specifibsp; approabsp; to drawing these time–frequenbsp; maps in this paper is very simple – mubsp; simpler than the wavelet analysis or Laskar's (1990， 1993) frequenbsp; analysis.
Divide the low-pass filtered orbital data into many fragments of the same length. The length of eabsp; data segment should be a multiple of 2 in order to apply the FFT.
Eabsp; 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 data segment ranges from ti+δT≤ti+δT+T， where δT?T. We tinue this division until we reabsp; a certain number N by whibsp; tn+T reaches the total iion length.
We apply an FFT to eabsp; of the data fragments， and obtain n frequenbsp; diagrams.
In eabsp; frequenbsp; diagram obtained above， the strength of periodicity bsp; be replabsp; by a grey-scale (or colour) chart.
We perform the replat， and ebsp; all the grey-scale (or colour) charts into one graph for eabsp; iion. The horizontal axis of these new graphs should be the time， i.e. the starting times of eabsp; fragment of data (ti， where i= 1，…， n). The vertibsp; axis represents the period (or frequenbsp; of the oscillation of orbital elements.
We have adopted an FFT because of its overwhelming speed， sinbsp; the amount of numeribsp; data to be deposed into frequenbsp; pos is terribly huge (several tens of Gbytes).
A typibsp; example of the time–frequenbsp; map created by the above procedures is shown in a grey-scale diagram as Fig. 5， whibsp; 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 stronger than in the lighter area around it. We bsp; 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 typibsp; frequencies differ pla by pla and element by element.
4.2 Long-term exge of orbital energy and angular momentum
We calculate very long-periodibsp; 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 vertibsp; po per unit mass. L is related to the plaary orbital energy E per unit mass as E=?μ2/2L2. If the system is pletely linear， the orbital energy and the angular momentum in eabsp; frequenbsp; bin must be stant. Non-liy in the plaary system bsp; cause an exge of energy and angular momentum in the frequenbsp; domain. The amplitude of the lowest-frequenbsp; oscillation should increase if the system is unstable and breaks down gradually. However， subsp; 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-periodibsp; variation of total energy (denoted asE- E0)， total angular momentum ( G- G0)， and the vertibsp; po ( H- H0) of the inner four plas calculated from the low-pass filtered Delaunay elements.E0， G0， H0 denote the initial values of eabsp; quantity. The absolute differenbsp; from the initial values is plotted in the panels. The lower three panels in eabsp; figure showE-E0，G-G0 andH-H0 of the total of nine plas. The fluctuation shown in the lower panels is virtually entirely a result of the massive jovian plas.
paring the variations of energy and angular momentum of the inner four plas and all nine plas， it is apparent that the amplitudes of those of the inner plas are mubsp; smaller than those of all nine plas: the amplitudes of the outer five plas are mubsp; larger than those of the inner plas. This does not mean that the inner terrestrial plaary subsystem is more stable than the outer one: this is simply a result of the relative smallness of the masses of the four terrestrial plas pared with those of the outer jovian plas. Another thing we notibsp; is that the inner plaary subsystem may bee unstable more rapidly than the outer one because of its shorter orbital time-scales. This bsp; be seen in the panels denoted asinner 4 in Fig. 7 where the longer-periodibsp; and irregular oscillations are more apparent than in the panels denoted astotal 9. Actually， the fluctuations in theinner 4 panels are to a large extent as a result of the orbital variation of the Mercury. However， we ot the tribution from other terrestrial plas， as we will see in subsequent ses.
4.4 Long-term coupling of several neighb pla pairs
Let us see some individual variations of plaary orbital energy and angular momentum expressed by the low-pass filtered Delaunay elements. Figs 10 and 11 show long-term evolution of the orbital energy of eabsp; pla and the angular momentum in N+1 and N?2 iions. We notibsp; that some plas form apparent pairs in terms of orbital energy and angular momentum exge. In particular， Venus and Earth make a typibsp; pair. In the figures， they show ive correlations in exge of energy and positive correlations in exge of angular momentum. The ive correlation in exge of orbital energy means that the two plas form a closed dynamibsp; system in terms of the orbital energy. The positive correlation in exge of angular momentum means that the two plas are simultaneously under certain long-term perturbations. didates for perturbers are Jupiter and Saturn. Also in Fig. 11， we bsp; see that Mars shows a positive correlation in the angular momentum variation to the Veh system. Mercury exhibits certain ive correlations in the angular momentum versus the Veh system， whibsp; seems to be a rea caused by the servation of angular momentum in the terrestrial plaary subsystem.
It is not clear at the moment why the Veh pair exhibits a ive correlation in energy exge and a positive correlation in angular momentum exge. We may possibly explain this through the general fabsp; that there are no secular terms in plaary semimajor axes up to sed-order perturbation theories (bsp; Brouwer & Clemenbsp; 1961; Boccaletti & Pucabsp; 1998). This means that the plaary orbital energy (whibsp; is directly related to the semimajor axis a) might be mubsp; less affected by perturbing plas than is the angular momentum exge (whibsp; relates to e). Henbsp; the etricities of Venus and Earth bsp; be disturbed easily by Jupiter and Saturn， whibsp; results in a positive correlation in the angular momentum exge. On the other hand， the semimajor axes of Venus and Earth are less likely to be disturbed by the jovian plas. Thus the energy exge may be limited only within the Veh pair， whibsp; results in a ive correlation in the exge of orbital energy in the pair.
As for the outer jovian plaary subsystem， Jupiter–Saturn and Uranus–une seem to make dynamibsp; pairs. However， the strength of their coupling is not as strong pared with that of the Veh pair.
5 ± 5 × 1010-yr iions of outer plaary orbits
Sinbsp; the jovian plaary masses are mubsp; larger than the terrestrial plaary masses， we treat the jovian plaary system as an indepe plaary system in terms of the study of its dynamibsp; stability. Henbsp; we added a couple of trial iions that span ± 5 × 1010 yr， including only the outer five plas (the four jovian plas plus Pluto). The results exhibit the rigorous stability of the outer plaary system over this long time-span. Orbital figurations (Fig. 12)， and variation of etricities and inations (Fig. 13) show this very long-term stability of the outer five plas in both the time and the frequenbsp; domains. Although we do not show maps here， the typibsp; frequenbsp; of the orbital oscillation of Pluto and the other outer plas is almost stant during these very long-term iion periods， whibsp; is demonstrated in the time–frequenbsp; maps on our webpage.
In these two iions， the relative numeribsp; error in the total energy was ～10?6 and that of the total angular momentum was ～10?10.
5.1 Resonanbsp; in the une–Pluto system
Kinoshita & Nakai (1996) ied the outer five plaary orbits over ± 5.5 × 109 yr . They found that four major resonanbsp; between une and Pluto are maintained during the whole iion period， and that the resonanbsp; may be the main causes of the stability of the orbit of Pluto. The major four resonanbsp; found in previous researbsp; are as follows. In the following description，λ denotes the mean longitude，Ω is the longitude of the asding node and ? is the longitude of perihelion. Subscripts P and N denote Pluto and une.
Mean motion resonanbsp; between une and Pluto (3:2). The critibsp; argument θ1= 3 λP? 2 λN??P librates around 180° with an amplitude of about 80° and a libration period of about 2 × 104 yr.
The argument of perihelion of Pluto ωP=θ2=?P?ΩP librates around 90° with a period of about 3.8 × 106 yr. The dominant periodibsp; variations of the etricity and ination of Pluto are synized with the libration of its argument of perihelion. This is anticipated in the secular perturbation theory structed by Kozai (1962).
The longitude of the node of Pluto referred to the longitude of the node of une，θ3=ΩP?ΩN， circulates and the period of this circulation is equal to the period of θ2 libration. When θ3 bees zero， i.e. the longitudes of asding nodes of une and Pluto overlap， the ination of Pluto bees maximum， the etricity bees minimum and the argument of perihelion bees 90°. When θ3 bees 180°， the ination of Pluto bees minimum， the etricity bees maximum and the argument of perihelion bees 90° again. Williams & Benson (1971) anticipated this type of resonanbsp; later firmed by Milani， Nobili & Carpino (1989).
An argument θ4=?P??N+ 3 (ΩP?ΩN) librates around 180° with a long period，～ 5.7 × 108 yr.
In our numeribsp; iions， the resonanbsp; (i)–(iii) are well maintained， and variation of the critibsp; arguments θ1，θ2，θ3 remain similar during the whole iion period (Figs 14–16 ). However， the fourth resonanbsp; (iv) appears to be different: the critibsp; argument θ4 alternates libration and circulation over a 1010-yr time-scale (Fig. 17). This is an iing fabsp; that Kinoshita & Nakai's (1995， 1996) shorter iions were not able to disclose.
6 Discussion
What kind of dynamibsp; meism maintains this long-term stability of the plaary system? We bsp; immediately think of two major features that may be responsible for the long-term stability. First， there seem to be no signifit lower-order resonanbsp; (mean motion and secular) between any pair among the nine plas. Jupiter and Saturn are close to a 5:2 mean motion resonanbsp; (the famous ‘great inequality’)， but not just in the resonanbsp; zone. Higher-order resonanbsp; may cause the chaotibsp; nature of the plaary dynamibsp; motion， but they are not so strong as to destroy the stable plaary motion within the lifetime of the real Solar system. The sed feature， whibsp; we think is more important for the long-term stability of our plaary system， is the differenbsp; in dynamibsp; distanbsp; between terrestrial and jovian plaary subsystems (Ito & Tanikawa 1999， 2001). When we measure plaary separations by the mutual Hill radii (R_)， separations among terrestrial plas are greater than 26RH， whereas those among jovian plas are less than 14RH. This differenbsp; is directly related to the differenbsp; between dynamibsp; features of terrestrial and jovian plas. Terrestrial plas have smaller masses， shorter orbital periods and wider dynamibsp; separation. They are strongly perturbed by jovian plas that have larger masses， longer orbital periods and narrower dynamibsp; separation. Jovian plas are not perturbed by any other massive bodies.
The present terrestrial plaary system is still being disturbed by the massive jovian plas. However， the wide separation and mutual iion among the terrestrial plas renders the disturbanbsp; iive; the degree of disturbanbsp; by jovian plas is O(eJ)(order of magnitude of the etricity of Jupiter)， sinbsp; the disturbanbsp; caused by jovian plas is a forbsp; oscillation having an amplitude of O(eJ). Heightening of etricity， for example O(eJ)～0.05， is far from suffit to provoke instability in the terrestrial plas having subsp; a wide separation as 26RH. Thus we assume that the present wide dynamibsp; separation among terrestrial plas (> 26RH) is probably one of the most signifit ditions for maintaining the stability of the plaary system over a 109-yr time-span. Our detailed analysis of the relationship between dynamibsp; distanbsp; between plas and the instability time-scale of Solar system plaary motion is now on-going.
Although our numeribsp; iions span the lifetime of the Solar system， the number of iions is far from suffit to fill the initial phase spabsp; It is necessary to perform more and more numeribsp; iions to firm and examine in detail the long-term stability of our plaary dynamics.
——以上文段引自 Ito， T.& Tanikawa， K. Long-term iions and stability of plaary orbits in our Solar System. Mon. Not. R. Astron. Sobsp; 336， 483–500 (2002)
这只是作者君参考的一篇文章，关于太阳系的稳定性。
还有其他论文，不过也都是英文的，相关课题的中文文献很少，那些论文下载一篇要九美元（《Nature》真是暴利），作者君写这篇文章的时候已经回家，不在检测中心，所以没有数据库的使用权，下不起，就不贴上来了。