paring he variains f energy and angular enu f he inner fur planes and all nine planes， i is apparen ha he apliudes f hse f he inner planes are uh saller han hse f all nine planes: he apliudes f he uer five planes are uh larger han hse f he inner planes. his des n ean ha he inner erresrial planeary subsyse is re sable han he uer ne: his is siply a resul f he relaive sallness f he asses f he fur erresrial planes pared ih hse f he uer jvian planes. Anher hing e nie is ha he inner planeary subsyse ay bee unsable re rapidly han he uer ne beause f is shrer rbial ie-sales. his an be seen in he panels dened asinner 4 in Fig. 7 here he lnger-peridi and irregular sillains are re apparen han in he panels dened asal 9. Aually， he fluuains in heinner 4 panels are a large een as a resul f he rbial variain f he erury. Hever， e ann negle he nribuin fr her erresrial planes， as e ill see in subsequen seins.
4.4 Lng-er upling f several neighburing plane pairs
Le us see se individual variains f planeary rbial energy and angular enu epressed by he l-pass filered Delaunay eleens. Figs 10 and 11 sh lng-er evluin f he rbial energy f eah plane and he angular enu in N+1 and N? inegrains. e nie ha se planes fr apparen pairs in ers f rbial energy and angular enu ehange. In pariular， Venus and Earh ake a ypial pair. In he figures， hey sh negaive rrelains in ehange f energy and psiive rrelains in ehange f angular enu. he negaive rrelain in ehange f rbial energy eans ha he planes fr a lsed dynaial syse in ers f he rbial energy. he psiive rrelain in ehange f angular enu eans ha he planes are siulaneusly under erain lng-er perurbains. andidaes fr perurbers are Jupier and Saurn. Als in Fig. 11， e an see ha ars shs a psiive rrelain in he angular enu variain he Venus–Earh syse. erury ehibis erain negaive rrelains in he angular enu versus he Venus–Earh syse， hih sees be a reain aused by he nservain f angular enu in he erresrial planeary subsyse.
I is n lear a he en hy he Venus–Earh pair ehibis a negaive rrelain in energy ehange and a psiive rrelain in angular enu ehange. e ay pssibly eplain his hrugh bserving he general fa ha here are n seular ers in planeary seiajr aes up send-rder perurbain heries (f. Bruer ≈ap;ap; leene 1961; Balei ≈ap;ap; Pua 1998). his eans ha he planeary rbial energy (hih is direly relaed he seiajr ais a) igh be uh less affeed by perurbing planes han is he angular enu ehange (hih relaes e). Hene， he eenriiies f Venus and Earh an be disurbed easily by Jupier and Saurn， hih resuls in a psiive rrelain in he angular enu ehange. n he her hand， he seiajr aes f Venus and Earh are less likely be disurbed by he jvian planes. hus he energy ehange ay be liied nly ihin he Venus–Earh pair， hih resuls in a negaive rrelain in he ehange f rbial energy in he pair.
As fr he uer jvian planeary subsyse， Jupier–Saurn and Uranus–Nepune see ake dynaial pairs. Hever， he srengh f heir upling is n as srng pared ih ha f he Venus–Earh pair.
5 ± 5 × 1rains f uer planeary rbis
Sine he jvian planeary asses are uh larger han he erresrial planeary asses， e rea he jvian planeary syse as an independen planeary syse in ers f he sudy f is dynaial sabiliy. Hene， e added a uple f rial inegrains ha span ± 5 × 1010 yr， inluding nly he uer five planes (he fur jvian planes plus Plu). he resuls ehibi he rigrus sabiliy f he uer planeary syse ver his lng ie-span. rbial nfigurains (Fig. 1)， and variain f eenriiies and inlinains (Fig. 1) sh his very lng-er sabiliy f he uer five planes in bh he ie and he frequeny dains. Alhugh e d n sh aps here， he ypial frequeny f he rbial sillain f Plu and he her uer planes is als nsan during hese very lng-er inegrain perids， hih is densraed in he ie–frequeny aps n ur ebpage.
In hese inegrains， he relaive nuerial errr in he al energy as ～10?6 and ha f he al angular enu as ～10?10.
5.1 Resnanes in he Nepune–Plu syse
Kinshia ≈ap;ap; Nakai (1996) inegraed he uer five planeary rbis ver ± 5.5 × 109 yr . hey fund ha fur ajr resnanes beeen Nepune and Plu are ainained during he hle inegrain perid， and ha he resnanes ay be he ain auses f he sabiliy f he rbi f Plu. he ajr fur resnanes fund in previus researh are as flls. In he flling desripin，λ denes he ean lngiude，Ω is he lngiude f he asending nde and ? is he lngiude f perihelin. Subsrips P and N dene Plu and Nepune.
ean in resnane beeen Nepune and Plu (:). he riial arguen θ1= λP? λN??P libraes arund 180° ih an apliude f abu 80° and a librain perid f abu × 104 yr.
he arguen f perihelin f Plu ωP=θ=?P?ΩP libraes arund 90° ih a perid f abu .8 × 106 yr. he dinan peridi variains f he eenriiy and inlinain f Plu are synhrnized ih he librain f is arguen f perihelin. his is aniipaed in he seular perurbain hery nsrued by Kzai (196).
he lngiude f he nde f Plu referred he lngiude f he nde f Nepune，θ=ΩP?ΩN， irulaes and he perid f his irulain is equal he perid f θ librain. hen θ bees zer， i.e. he lngiudes f asending ndes f Nepune and Plu verlap， he inlinain f Plu bees aiu， he eenriiy bees iniu and he arguen f perihelin bees 90°. hen θ bees 180°， he inlinain f Plu bees iniu， he eenriiy bees aiu and he arguen f perihelin bees 90° again. illias ≈ap;ap; Bensn (1971) aniipaed his ype f resnane， laer nfired by ilani， Nbili ≈ap;ap; arpin (1989).
An arguen θ4=?P??N+ (ΩP?ΩN) libraes arund 180° ih a lng perid，～ 5.7 × 108 yr.
In ur nuerial inegrains， he resnanes (i)–(iii) are ell ainained， and variain f he riial arguens θ1，θ，θ reain siilar during he hle inegrain perid (Figs 14–16 ). Hever， he furh resnane (iv) appears be differen: he riial arguen θ4 alernaes librain and irulain ver a 1010-yr ie-sale (Fig. 17). his is an ineresing fa ha Kinshia ≈ap;ap; Nakai's (1995， 1996) shrer inegrains ere n able dislse.
6 Disussin
ha kind f dynaial ehanis ainains his lng-er sabiliy f he planeary syse? e an iediaely hink f ajr feaures ha ay be respnsible fr he lng-er sabiliy. Firs， here see be n signifian ler-rder resnanes (ean in and seular) beeen any pair ang he nine planes. Jupier and Saurn are lse a 5: ean in resnane (he faus ‘grea inequaliy’)， bu n jus in he resnane zne. Higher-rder resnanes ay ause he hai naure f he planeary dynaial in， bu hey are n s srng as desry he sable planeary in ihin he lifeie f he real Slar syse. he send feaure， hih e hink is re ipran fr he lng-er sabiliy f ur planeary syse， is he differene in dynaial disane beeen erresrial and jvian planeary subsyses (I ≈ap;ap; anikaa 1999， 001). hen e easure planeary separains by he uual Hill radii (R_)， separains ang erresrial planes are greaer han 6RH， hereas hse ang jvian planes are less han 14RH. his differene is direly relaed he differene beeen dynaial feaures f erresrial and jvian planes. erresrial planes have saller asses， shrer rbial perids and ider dynaial separain. hey are srngly perurbed by jvian planes ha have larger asses， lnger rbial perids and narrer dynaial separain. Jvian planes are n perurbed by any her assive bdies.
he presen erresrial planeary syse is sill being disurbed by he assive jvian planes. Hever， he ide separain and uual inerain ang he erresrial planes renders he disurbane ineffeive; he degree f disurbane by jvian planes is (eJ)(rder f agniude f he eenriiy f Jupier)， sine he disurbane aused by jvian planes is a fred sillain having an apliude f (eJ). Heighening f eenriiy， fr eaple (eJ)～0.05， is far fr suffiien prvke insabiliy in he erresrial planes having suh a ide separain as 6RH. hus e assue ha he presen ide dynaial separain ang erresrial planes (≈ap;g; 6RH) is prbably ne f he s signifian ndiins fr ainaining he sabiliy f he planeary syse ver a 109-yr ie-span. ur deailed analysis f he relainship beeen dynaial disane beeen planes and he insabiliy ie-sale f Slar syse planeary in is n n-ging.
Alhugh ur nuerial inegrains span he lifeie f he Slar syse， he nuber f inegrains is far fr suffiien fill he iniial phase spae. I is neessary perfr re and re nuerial inegrains nfir and eaine in deail he lng-er sabiliy f ur planeary dynais.
——以上文段引自 I， .≈ap; anikaa， K. Lng-er inegrains and sabiliy f planeary rbis in ur Slar Syse. n. N. R. Asrn. S. 6， 48–500 (00)
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