the variation of entricities and orbital inclinations for the inner four s in the initial and final part of the integration n+1 is shown in fig. 4. as expected， the character of the variation of ary orbital elements does not differ significantly between the initial and final part of each integration， at least for venus， earth and mars. the elements of mercury， especially its entricity， seem to change to a significant extent. this is partly because the orbital time-scale of the is the shortest of all the s， which leads to a more rapid orbital evolution than other s; the innermost may be nearest to instability. this result appears to be in some agreement withskar's (1994， 1996) expectations thatrge and irregr variations appear in the entricities and inclinations 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 ary system owing to the small mass of mercury. we will mention briefly the long-term orbital evolution of mercuryter in section 4 using low-pass filtered orbital elements.

the orbital motion of the outer five s seems rigorously stable and quite regr over this time-span (see also section 5).

3.2 time–frequency maps

although the ary motion exhibits very long-term stability defined as the non-existence of close encounter events， the chaotic nature of ary dynamics can change the oscitory period and amplitude of ary orbital motion gradually over such long time-spans. even such slight fluctuations of orbital variation in the frequency domain， particrly in the case of earth， can potentially have a significant effect on its surface climate system through sr instion variation (cf. berger 1988).

to give an overview of the long-term change in periodicity in ary 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 orskar's (1990， 1993) frequency analysis.

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 arge ovepping part: for example， when the ith data begins from t=ti and ends at t=ti+t， the next data segment ranges from ti+δt≤ti+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 reced by a grey-scale (or colour) chart.

we perform the recement， 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 oscition of orbital elements.

we have adopted an fft because of its overwhelming speed， since the amount of numerical data to be deposed into frequency ponents 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 entricity 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 entricity and inclination of earth only changes slightly over the entire period covered by the n+2 integration. this nearly regr trend is qualitatively the same in other integrations and for other s， although typical frequencies differ by and element by element.

4.2 long-term exchange of orbital energy and angr momentum

we calcte very long-periodic variation and exchange of ary orbital energy and angr momentum using filtered dunay elements l， g， h. g and h are equivalent to the ary orbital angr momentum and its vertical ponent per unit mass. l is rted to the ary orbital energy e per unit mass as e=?μ2/2l2. if the system is pletely linear， the orbital energy and the angr momentum in each frequency bin must be constant. non-linearity in the ary system can cause an exchange of energy and angr momentum in the frequency domain. the amplitude of the lowest-frequency oscition should increase if the system is unstable and breaks down gradually. however， such a symptom of instability is not prominent in our long-term integrations.

in fig. 7， the total orbital energy and angr momentum of the four inner s and all nine s are shown for integration n+2. the upper three panels show the long-periodic variation of total energy (denoted ase- e0)， total angr momentum ( g- g0)， and the vertical ponent ( h- h0) of the inner four s calcted from the low-pass filtered dunay elements.e0， g0， h0 denote the initial values of each quantity. the absolute difference from the initial values is plotted in the panels. the lower three panels in each figure showe-e0，g-g0 andh-h0 of the total of nine s. the fluctuation shown in the lower panels is virtually entirely a result of the massive jovian s.

paring the variations of energy and angr momentum of the inner four s and all nine s， it is apparent that the amplitudes of those of the inner s are much smaller than those of all nine s: the amplitudes of the outer five s are muchrger than those of the inner s. this does not mean that the inner terrestrial ary subsystem is more stable than the outer one: this is simply a result of the rtive smallness of the masses of the four terrestrial s pared with those of the outer jovian s. another thing we notice is that the inner ary subsystem may bee unstable more rapidly than the outer one because of its shorter orbital time-scales. this can be seen in the panels denoted asinner 4 in fig. 7 where the longer-periodic and irregr oscitions are more apparent than in the panels denoted astotal 9. actually， the fluctuations in theinner 4 panels are to arge extent as a result of the orbital variation of the mercury. however， we cannot neglect the contribution from other terrestrial s， as we will see in subsequent sections.

4.4 long-term coupling of several neighbouring pairs

let us see some individual variations of ary orbital energy and angr momentum expressed by the low-pass filtered dunay elements. figs 10 and 11 show long-term evolution of the orbital energy of each and the angr momentum in n+1 and n?2 integrations. we notice that some s form apparent pairs in terms of orbital energy and angr momentum exchange. in particr， venus and earth make a typical pair. in the figures， they show negative corrtions in exchange of energy and positive corrtions in exchange of angr momentum. the negative corrtion in exchange of orbital energy means that the two s form a closed dynamical system in terms of the orbital energy. the positive corrtion in exchange of angr momentum means that the two s are simultaneously under certain long-term perturbations. candidates for perturbers are jupiter and saturn. also in fig. 11， we can see that mars shows a positive corrtion in the angr momentum variation to the venus–earth system. mercury exhibits certain negative corrtions in the angr momentum versus the venus–earth system， which seems to be a reaction caused by the conservation of angr momentum in the terrestrial ary subsystem.

it is not clear at the moment why the venus–earth pair exhibits a negative corrtion in energy exchange and a positive corrtion in angr momentum exchange. we may possibly exin this through observing the general fact that there are no secr terms in ary semimajor axes up to second-order perturbation theories (cf. brouwer & clemence 1961; aletti & o 1998). this means that the ary orbital energy (which is directly rted to the semimajor axis a) might be much less affected by perturbing s than is the angr momentum exchange (which rtes to e). hence， the entricities of venus and earth can be disturbed easily by jupiter and saturn， which results in a positive corrtion in the angr momentum exchange. on the other hand， the semimajor axes of venus and earth are less likely to be disturbed by the jovian s. thus the energy exchange may be limited only within the venus–earth pair， which results in a negative corrtion in the exchange of orbital energy in the pair.

as for the outer jovian ary subsystem， jupiter–saturn and uranus–neptune seem to make dynamical pairs. however， the strength of their coupling is not as strong pared with that of the venus–earth pair.

5 ± 5 x 1010-yr integrations of outer ary orbits

since the jovian ary masses are muchrger than the terrestrial ary masses， we treat the jovian ary system as an independent ary system in terms of the study of its dynamical stability. hence， we added a couple of trial integrations that span ± 5 x 1010 yr， including only the outer five s (the four jovian s plus pluto). the results exhibit the rigorous stability of the outer ary system over this long time-span. orbital configurations (fig. 12)， and variation of entricities and inclinations (fig. 13) show this very long-term stability of the outer five s in both the time and the frequency domains. although we do not show maps here， the typical frequency of the orbital oscition of pluto and the other outer s is almost constant during these very long-term integration periods， which is demonstrated in the time–frequency maps on our webpage.

in these two integrations， the rtive numerical error in the total energy was ～10?6 and that of the total angr momentum was ～10?10.

5.1 resonances in the neptune–pluto system

kinoshita & nakai (1996) integrated the outer five ary orbits over ± 5.5 x 109 yr . they found that four major resonances between neptune and pluto are maintained during the whole integration period， and that the resonances may be the main causes of the stability of the orbit of pluto. the major four resonances found in previous research are as follows. in the following description，λ denotes the mean longitude，Ω is the longitude of the ascending node and ? is the longitude of perihelion. subscripts p and n denote pluto and neptune.

mean motion resonance between neptune and pluto (3:2). the critical argument θ1= 3 λp? 2 λn??p librates around 180° with an amplitude of about 80° and a libration period of about 2 x 104 yr.

the argument of perihelion of pluto wp=θ2=?p?Ωp librates around 90° with a period of about 3.8 x 106 yr. the dominant periodic variations of the entricity and inclination of pluto are synchronized with the libration of its argument of perihelion. this is anticipated in the secr perturbation theory constructed by kozai (1962).

the longitude of the node of pluto referred to the longitude of the node of neptune，θ3=Ωp?Ωn， circtes and the period of this cirction is equal to the period of θ2 libration. when θ3 bees zero， i.e. the longitudes of ascending nodes of neptune and pluto ovep， the inclination of pluto bees maximum， the entricity bees minimum and the argument of perihelion bees 90°. when θ3 bees 180°， the inclination of pluto bees minimum， the entricity bees maximum and the argument of perihelion bees 90° again. williams & benson (1971) anticipated this type of resonance，ter confirmed by mni， nobili & carpino (1989).

an argument θ4=?p??n+ 3 (Ωp?Ωn) librates around 180° with a long period，～ 5.7 x 108 yr.

in our numerical integrations， the resonances (i)–(iii) are well maintained， and variation of the critical arguments θ1，θ2，θ3 remain simr during the whole integration period (figs 14–16 ). however， the fourth resonance (iv) appears to be different: the critical argument θ4 alternates libration and cirction over a 1010-yr time-scale (fig. 17). this is an interesting fact that kinoshita & nakai's (1995， 1996) shorter integrations were not able to disclose.

6 discussion

what kind of dynamical mechanism maintains this long-term stability of the ary system? we can immediately think of two major features that may be responsible for the long-term stability. first， there seem to be no significant lower-order resonances (mean motion and secr) between any pair among the nine s. jupiter and saturn are close to a 5:2 mean motion resonance (the famous ‘great inequality’)， but not just in the resonance zone. higher-order resonances may cause the chaotic nature of the ary dynamical motion， but they are not so strong as to destroy the stable ary motion within the lifetime of the real sr system. the second feature， which we think is more important for the long-term stability of our ary system， is the difference in dynamical distance between terrestrial and jovian ary subsystems (ito & tanikawa 1999， 2001). when we measure ary separations by the mutual hill radii (r_)， separations among terrestrial s are greater than 26rh， whereas those among jovian s are less than 14rh. this difference is directly rted to the difference between dynamical features of terrestrial and jovian s. terrestrial s have smaller masses， shorter orbital periods and wider dynamical separation. they are strongly perturbed by jovian s that haverger masses， longer orbital periods and narrower dynamical separation. jovian s are not perturbed by any other massive bodies.

the present terrestrial ary system is still being disturbed by the massive jovian s. however， the wide separation and mutual interaction among the terrestrial s renders the disturbance ineffective; the degree of disturbance by jovian s is o(ej)(order of magnitude of the entricity of jupiter)， since the disturbance caused by jovian s is a forced oscition having an amplitude of o(ej). heightening of entricity， for example o(ej)～0.05， is far from sufficient to provoke instability in the terrestrial s having such a wide separation as 26rh. thus we assume that the present wide dynamical separation among terrestrial s (> 26rh) is probably one of the most significant conditions for maintaining the stability of the ary system over a 109-yr time-span. our detailed analysis of the rtionship between dynamical distance between s and the instability time-scale of sr system ary motion is now on-going.

although our numerical integrations span the lifetime of the sr system， the number of integrations is far from sufficient to fill the initial phase space. it is necessary to perform more and more numerical integrations to confirm and examine in detail the long-term stability of our ary dynamics.

——以上文段引自 ito， t.& tanikawa， k. long-term integrations and stability of ary orbits in our sr system. mon. not. r. astron. soc. 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。