the variationeccentricities and orbital inclinations for the inner four plasthe initial and final partthe integration n+1shown i expected， the characterthe variationplaary orbital elements does not differ significantly between the initial and final parteach integration，least for venus， earthelementsmercury， especially its eccentricity， seemchangea significanpartly because the orbital time-scalethe plathe shortestall the plas， which leadsa more rapid orbital evolution than other plas; the innermost pla maynearest result appearsbesome agreement with laskar''s (1994， 1996) expectaions that large and irregular variations appearthe eccentricities and inclinationsmercurya time-scaleseveral 109 yr. however， the effectthe possible instabilitythe orbitmercury may not fatally affect the global stabilitythe whole plaary system owingthe small mass o will mention briefly the long-term orbital evolutionmercury latersection 4 using low-pass filtered orbital elements.
the orbital motionthe outer five plas seems rigorously stable and quite regular over this time-span (see also section 5).
3.2 timefrequency maps
although the plaary motion exhibits very long-term stability definedthe non-existenceclose encounter events， the chaotic natureplaary dynamics can change the oscillatory period and amplitudeplaary orbital motion gradually over such lon such slight fluctuationsorbital variationthe frequency domain， particularlythe caseearth， can potentially have a significant effectits surface climate system through solar insolation variation (cf. berger 1988).
to giveoverviewthe long-term changeperiodicityplaary orbital motion，performed many fast fourier transformations (ffts) along the time axis， and superposed the resulting periodgramsdraw two-dimensional timefrequenc specific approachdrawing these timefrequency mapsthis papervery simple much simpler than the wavelet analysislaskar''s (1990， 1993) frequency analysis.
divide the low-pass filtered orbital data into many fragmentsthe sam lengtheach data segment shoulda multiple2orderapply the fft.
each fragmentthe data has a large overlapping part: for example， when the ith data begins from t=ti and endst=ti+t， the next data segment ranges from ti+δt≤ti+δt+t， where δt?t.continue this division untilreach a certain number nwhich tn+t reaches the total integration length.
we applyffteachthe data fragments， and obtain n frequency diagrams.
in each frequency diagram obtained above， the strengthperiodicity canreplaceda grey-scale (or colour) chart.
we perform the replacement， and connect all the grey-scale (or colour) charts into one graph for each horizontal axisthese new graphs shouldthe time， i.e. the starting timeseach fragmentdata (ti， where i= 1，…， n). the vertical axis represents the period (or frequency)the oscillationorbital elements.
we have adoptedfft becauseits overwhelming speed， since the amountnumerical databe deposed into frequency ponentsterribly huge (several tensgbytes).
a typical examplethe timefrequency map createdthe above proceduresshowna grey-scale diagram a， which shows the variationperiodicitythe eccentricity and inclinationearthn+2 fig. 5， the dark area shows thatthe time indicatedthe valuethe abscissa， the periodicity indicatedthe ordinatestronger thanthe lighter area around it.can recognize from this map that the periodicitythe eccentricity and inclinationearth only changes slightly over the entire period coveredthe n+2 nearly regular trendqualitatively the sameother integrations and for other plas， although typical frequencies differ plapla and elementelement.
4.2 long-term exchangeorbital energy and angular momentum
we calculate very long-periodic variation and exchangeplaary orbital energy and angular momentum using filtered delaunay elements l， g， h. g and h are equivalentthe plaary orbital angular momentum and its vertical ponent per uni relatedthe plaary orbital energy e per unit masse=μ2/2l2.the systempletely linear， the orbital energy and the angular momentumeach frequency bin must bthe plaary system can causeexchangeenergy and angular momentumthe frequenc amplitudethe lowest-frequency oscillation should increasethe systemunstable and breaks dow， such a symptominstabilitynot prominentour long-term integrations.
i， the total orbital energy and angular momentumthe four inner plas and all nine plas are shown for integration n+2. the upper three panels show the long-periodic variationtotal energy (denoted ase- e0)， total angular momentum ( g- g0)， and the vertical ponent ( h- h0)the inner four plas calculated from the low-pass filtered delaunay ， g0，denote the initial valueseac absolute difference from the initial valuesplottedth lower three panelseach figure showe-e0，g-g0 andh-h0the totalnin fluctuation shownthe lower panelsvirtually entirely a resultthe massive jovian plas.
paring the variationsenergy and angular momentumthe inner four plas and all nine plas，is apparent that the amplitudesthosethe inner plas are much smaller than thoseall nine plas: the amplitudesthe outer five plas are much larger than thosethe inne does not mean that the inner terrestrial plaary subsystemmore stable than the outer one: thissimply a resultthe relative smallnessthe massesthe four terrestrial plas pared with thosethe outer jovia thingnoticethat the inner plaary subsystem may bee unstable more rapidly than the outer one becauseits shorter orbital canseenthe panels denoted asinner 4 i the longer-periodic and irregular oscillations are more apparent thanthe panels denoted astotal 9. actually， the fluctuationstheinner 4 panels area large extenta resultthe orbital variationth，cannot neglect the contribution from other terrestrial plas，we will seesubsequent sections.
4.4 long-term couplingseveral neighbouring pla pairs
letsee some individual variationsplaary orbital energy and angular momentum expressedthe low-pass filtered delaunaandshow long-term evolutionthe orbital energyeach pla and the angular momentumn+1 andi notice that some plas form apparent pairstermsorbital energy and angular momentu particular， venus and earth make a typica the figures， they show negative correlationsexchangeenergy and positive correlationsexchangeangula negative correlationexchangeorbital energy means that the two plas form a closed dynamical systemtermsthe orbita positive correlationexchangeangular momentum means that the two plas are simultaneously under certain long-termfor perturbers are jupiteri，can see that mars shows a positive correlationthe angular momentum variationthe venuseart exhibits certain negative correlationsthe angular momentum versus the venusearth system， which seemsbe a reaction causedthe conservationangular momentumthe terrestrial plaary subsystem.
itnot clearthe moment why the venusearth pair exhibits a negative correlationenergy exchange and a positive correlationangular momentu may possibly explain this through observing the general fact that there aresecular termsplaary semimajor axesto second-order perturbation theories (cf. brouwer & clemence 1961; boccaletti & pucacco 1998). this means that the plaary orbital energy (whichdirectly relatedthe semimajor axis a) mightmuch less affectedperturbing plas thanthe angular momentum exchange (which relatese). hence， the eccentricitiesvenus and earth can bedisturbed easilyjupiter and saturn， which resultsa positive correlationthe angular momentu the other hand， the semimajor axesvenus and earth are less likelybe disturbedthe jovia the energy exchange maylimited only within the venusearth pair， which resultsa negative correlationthe exchangeorbital energythe pair.
as for the outer jovian plaary subsystem， jupitersaturn and uranusneptune seemmake dynamica， the strengththeir couplingnotstrong pared with thatthe venusearth pair.
5 ± 5 x 1010-yr integrationsouter plaary orbits
since the jovian plaary masses are much larger than the terrestrial plaary masses，treat the jovian plaary systeman independent plaary systemtermsthe studyits dynamica，added a coupletrial integrations that span ± 5 x 1010 yr， including only the outer five plas (the four jovian plas plus pluto). the results exhibit the rigorous stabilitythe outer plaary system over this lon configurations (fig. 12)， and variationeccentricities and inclinations (fig. 13) show this very long-term stabilitythe outer five plasboth the time and the frequencdo not show maps here， the typical frequencythe orbital oscillationpluto and the other outer plasalmost constant during these very long-term integration periods， whichdemonstratedthe timefrequency mapsour webpage.
in these two integrations， the relative numerical errorthe total energy was 106 and thatthe total angular momentum was 1010.
5.1 resonancesthe neptunepluto system
kinoshita & nakai (1996) integrated the outer five plaary orbits over ± 5.5 x 109. they found that four major resonances between neptune and pluto are maintained during the whole integration period， and that the resonances maythe main causesthe stabilitythe orbit o major four resonances foundprevious research are a the following description，λ denotes the mean longitude，Ωthe longitudethe ascending node and the longitude o p and n denote pluto and neptune.
mean motion resonance between neptune and pluto (3:2). the critical argument θ1= 32 λnp librates around 180° withamplitudeabout 80° and a libration periodabout 2 x 104 yr.
the argumentperihelionpluto wp=θ2=pΩp librates around 90° with a periodabout 3.8 x 106 yr. the dominant periodic variationsthe eccentricity and inclinationpluto are synchronized with the librationits argument oanticipatedthe secular perturbation theory constructedkozai (1962).
the longitudethe nodepluto referredthe longitudethe nodeneptune，θ3=ΩpΩn， circulates and the periodthis circulationequalthe periodθbees zero， i.e. the longitudesascending nodesneptune and pluto overlap， the inclinationpluto bees maximum， the eccentricity bees minimum and the argumentperihelion bees 90°. whenbees 180°， the inclinationpluto bees minimum， the eccentricity bees maximum and the argumentperihelion bees 90° & benson (1971) anticipated this typeresonance， later confirmedmilani， nobili & carpino (1989).
an argument θ4=pn+ 3 (ΩpΩn) librates around 180° with a long period， 5.7 x 108 yr.
in our numerical integrations， the resonances (i)(iii) are well maintained， and variationthe critical arguments θ1，θ2，θ3 remain similar during the whole integration period (figs 1416 ). however， the fourth resonance (iv) appearsbe different: the critical argumentalternates libration and circulation over a 1010-yr time-scale (fig. 17). thisan interesting fact that kinoshita & nakai''s (1995， 1996) shorter integrations were not abledisclose.
6 discussion
what kinddynamical mechanism maintains this long-term stabilitythe plaary system?can immediately thinktwo major features that mayresponsible for the long-ter， there seembesignificant lower-order resonances (mean motion and secular) between any pair among the nin and saturn are closea 5:2 mean motion resonance (the famous ‘great inequality’)， but not justthe resonanc resonances may cause the chaotic naturethe plaary dynamical motion， but they are notstrongto destroy the stable plaary motion within the lifetimethe real sola second feature， whichthinkmore important for the long-term stabilityour plaary system，the differencedynamical distance between terrestrial and jovian plaary subsystems (ito & tanikawa 1999， 2001). whenmeasure plaary separationsthe mutual hill radii (r_)， separations among terrestrial plas are greater than 26rh， whereas those among jovian plas are less tha differencedirectly relatedthe difference between dynamical featuresterrestrial and jovia plas have smaller masses， shorter orbital periods and wider dynamica are strongly perturbedjovian plas that have larger masses， longer orbital periods and narrower dynamica plas are not perturbedany other massive bodies.
the present terrestrial plaary systemstill being disturbedthe massive jovia， the wide separation and mutual interaction among the terrestrial plas renders the disturbance ineffective; the degreedisturbancejovian plaso(ej)(ordermagnitudethe eccentricityjupiter)， since the disturbance causedjovian plasa forced oscillation havingamplitudeo(ej). heighteningeccentricity， for example o(ej)0.05，far from sufficientprovoke instabilitythe terrestrial plas having such a wide separation aassume that the present wide dynamical separation among terrestrial plas (> 26rh)probably ohe most significant conditions for maintaining the stabilitythe plaary system over a 109-y detailed analysisthe relationship between dynamical distance between plas and the instability time-scalesolar system plaary motionnow on-going.
although our numerical integrations span the lifetimethe solar system， the numberintegrationsfar from sufficientfill the initial phasnecessaryperform more and more numerical integrationsconfirm and examinedetail the long-term stabilityour plaary dynamics.
以上文段引自 ito， t.& tanikawa， k. long-term integrations and stabilityplaary orbitsour sola， 483500 (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。