第76章 对火星轨道变化问题的最后解释
第76章 对火星轨道变化问题的最后解释
作者君在作品相关中已经解释过这个问题，并在此列出相关参考文献中的一篇开源论文。
以下是文章内容：
lng-er inegrans an sabiliy f pneary rbis in ur sr syse
absra
e presen he resuls f ery lng-er nuerial inegrans f pneary rbial ns er 109 -yr ie-spans inluing all nine pnes. a quik inspen f ur nuerial aa shs ha he pneary n， a leas in ur siple ynaial el， sees be quie sable een er his ery lng ie-span. a lser lk a he les-frequeny silns using a l-pass filer shs us he penially iffusie haraer f erresrial pneary n， espeially ha f erury. he behaur f he eenriiy f erury in ur inegrans is qualiaiely siir he resuls fr jaques skar's seur perurban hery (e.g. eax 0.35 er ± 4 gyr). heer， here are n apparen seur inreases f eenriiy r inlinan in any rbial eleens f he pnes， hih ay be reeale by sill lnr-er nuerial inegrans. e hae als perfre a uple f rial inegrans inluing ns f he uer fie pnes er he uran f ± 5 x 1010 yr. he resul iniaes ha he hree ajr resnanes in he nepune–plu syse hae been ainaine er he 1011-yr ie-span.
1 inrun
1.1efinin f he prble
he quesn f he sabiliy f ur sr syse has been ebae er seeral hunre years， sine he era f nen. he prble has arae any faus aheaiians er he years an has pye a enral rle in he eelpen f nn-linear ynais an has hery. heer， e n ye hae a efinie anser he quesn f heher ur sr syse is sable r n. his is parly a resul f he fa ha he efinin f he er ‘sabiliy’ is ague hen i is use in ren he prble f pneary n in he sr syse. aually i is n easy gie a lear， rigrus an physially eaningful efinin f he sabiliy f ur sr syse.
ang any efinins f sabiliy， here e ap he hill efinin (gan 1993): aually his is n a efinin f sabiliy， bu f insabiliy. e efine a syse as being unsable hen a lse enuner urs sehere in he syse， saring fr a erain iniial nfiguran (habers， eherill bss 1996; i anikaa 1999). a syse is efine as experiening a lse enuner hen bies apprah ne anher ihin an area f he rr hill raius. herise he syse is efine as being sable. henefrar e sae ha ur pneary syse is ynaially sable if n lse enuner happens uring he a f ur sr syse， abu ±5 gyr. inienally， his efinin ay be repe by ne in hih an urrene f any rbial rssing beeen eiher f a pair f pnes akes pe. his is beause e kn fr experiene ha an rbial rssing is ery likely lea a lse enuner in pneary an prpneary syses (yshinaga， kkub akin 1999). f urse his saeen ann be siply applie syses ih sable rbial resnanes suh as he nepune–plu syse.
1.2preus suies an ais f his researh
in ain he agueness f he nep f sabiliy， he pnes in ur sr syse sh a haraer ypial f ynaial has (sussan is 19八八， 1992). he ause f his hai behaur is n parly uners as being a resul f resnane erpping (urray hn 1999; lear， franklin hn 2001). heer， i ul require inegraing er an enseble f pneary syses inluing all nine pnes fr a per ering seeral 10 gyr hrughly unersan he lng-er elun f pneary rbis， sine hai ynaial syses are haraerie by heir srng epenene n iniial nins.
fr ha pin f ie， any f he preus lng-er nuerial inegrans inlue nly he uer fie pnes (sussan is 19八八; kinshia nakai 1996). his is beause he rbial pers f he uer pnes are s uh lnr han hse f he inner fur pnes ha i is uh easier fll he syse fr a gien inegran per. a presen， he lns nuerial inegrans publishe in jurnals are hse f unan lissauer (199八). alhugh heir ain ar as he effe f ps-ain-sequene sr ass lss n he sabiliy f pneary rbis， hey perfre any inegrans ering up 1011 yr f he rbial ns f he fur jian pnes. he iniial rbial eleens an asses f pnes are he sae as hse f ur sr syse in unan lissauer's paper， bu hey erease he ass f he sun graually in heir nuerial experiens. his is beause hey nsier he effe f ps-ain-sequene sr ass lss in he paper. nsequenly， hey fun ha he rssing ie-sale f pneary rbis， hih an be a ypial iniar f he insabiliy ie-sale， is quie sensiie he rae f ass erease f he sun. hen he ass f he sun is lse is presen alue， he jian pnes reain sable er 1010 yr， r perhaps lnr. unan lissauer als perfre fur siir experiens n he rbial n f seen pnes (enus nepune)， hih er a span f 109 yr. heir experiens n he seen pnes are n ye prehensie， bu i sees ha he erresrial pnes als reain sable uring he inegran per， ainaining als regur silns.
n he her han， in his aurae sei-analyial seur perurban hery (skar 19八八)， skar fins ha r an irregur arians an appear in he eenriiies an inlinans f he erresrial pnes， espeially f erury an ars n a ie-sale f seeral 109 yr (skar 1996). he resuls f skar's seur perurban hery l be nfire an inesigae by fully nuerial inegrans.
in his paper e presen preliinary resuls f six lng-er nuerial inegrans n all nine pneary rbis， ering a span f seeral 109 yr， an f her inegrans ering a span f ± 5 x 1010 yr. he al epse ie fr all inegrans is re han 5 yr， using seeral eiae ps an rksans. ne f he funaenal nlusns f ur lng-er inegrans is ha sr syse pneary n sees be sable in ers f he hill sabiliy enne abe， a leas er a ie-span f ± 4 gyr. aually， in ur nuerial inegrans he syse as far re sable han ha is efine by he hill sabiliy riern: n nly i n lse enuner happen uring he inegran per， bu als all he pneary rbial eleens hae been nfine in a narr regn bh in ie an frequeny ain， hugh pneary ns are shasi. sine he purpse f his paper is exhibi an erie he resuls f ur lng-er nuerial inegrans， e sh ypial exaple figures as eiene f he ery lng-er sabiliy f sr syse pneary n. fr reaers h hae re speifi an eeper ineress in ur nuerial resuls， e hae prepare a ebpa (aess )， here e sh ra rbial eleens， heir l-pass filere resuls， arian f eunay eleens an angur enu efii， an resuls f ur siple ie–frequeny analysis n all f ur inegrans.
in sen 2 e briefly expin ur ynaial el， nuerial eh an iniial nins use in ur inegrans. sen 3 is ee a en f he quik resuls f he nuerial inegrans. ery lng-er sabiliy f sr syse pneary n is apparen bh in pneary psins an rbial eleens. a rugh esian f nuerial errrs is als gien. sen 4 s n a isussn f he lns-er arian f pneary rbis using a l-pass filer an inlues a isussn f angur enu efii. in sen 5， e presen a se f nuerial inegrans fr he uer fie pnes ha spans ± 5 x 1010 yr. in sen 6 e als isuss he lng-er sabiliy f he pneary n an is pssible ause.
2 en f he nuerial inegrans
（本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了起点也不一定能成功显示。）
2.3 nuerial eh
e uilie a sen-rer is–hn syplei ap as ur ain inegran eh (is hn 1991; kinshia， yshia nakai 1991) ih a speial sar-up preure reue he runan errr f angle ariables，‘ar sar’(saha reaine 1992， 1994).
he sepsie fr he nuerial inegrans is 八 hrughu all inegrans f he nine pnes (n±1，2，3)， hih is abu 1/11 f he rbial per f he inners pne (erury). as fr he eerinan f sepsie， e parly fll he preus nuerial inegran f all nine pnes in sussan is (19八八， 7.2 ) an saha reaine (1994， 225/32 ). e rune he eial par f he heir sepsies 八 ake he sepsie a uliple f 2 in rer reue he auun f run-ff errr in he puan presses. in ren his， is hn (1991) perfre nuerial inegrans f he uer fie pneary rbis using he syplei ap ih a sepsie f 400 ， 1/10.八3 f he rbial per f jupier. heir resul sees be aurae enugh， hih parly jusifies ur eh f eerining he sepsie. heer， sine he eenriiy f jupier (0.05) is uh saller han ha f erury (0.2)， e nee se are hen e pare hese inegrans siply in ers f sepsies.
in he inegran f he uer fie pnes (f±)， e fixe he sepsie a 400 .
e ap gauss' f an g funns in he syplei ap her ih he hir-rer halley eh (anby 1992) as a sler fr kepler equans. he nuber f axiu ierans e se in halley's eh is 15， bu hey neer reahe he axiu in any f ur inegrans.
he ineral f he aa upu is 200 000 (547 yr) fr he aluns f all nine pnes (n±1，2，3)， an abu 八000 000 (21 903 yr) fr he inegran f he uer fie pnes (f±).
alhugh n upu filering as ne hen he nuerial inegrans ere in press， e applie a l-pass filer he ra rbial aa afer e ha plee all he aluns. see sen 4.1 fr re eail.
2.4 errr esian
2.4.1 reie errrs in al energy an angur enu
aring ne f he basi prperies f syplei inegrars， hih nsere he physially nseraie quaniies ell (al rbial energy an angur enu)， ur lng-er nuerial inegrans see hae been perfre ih ery sall errrs. he aera reie errrs f al energy (109) an f al angur enu (1011) hae reaine nearly nsan hrughu he inegran per (fig. 1). he speial sarup preure， ar sar， ul hae reue he aera reie errr in al energy by abu ne rer f agniue r re.
reie nuerial errr f he al angur enu δa/a0 an he al energy δe/e0 in ur nuerial inegransn± 1，2，3， here δe an δa are he abslue han f he al energy an al angur enu， respeiely， ane0ana0are heir iniial alues. he hrinal uni is gyr.
ne ha ifferen peraing syses， ifferen aheaial libraries， an ifferen harare arhieures resul in ifferen nuerial errrs， hrugh he arians in run-ff errr hanling an nuerial algrihs. in he upper panel f fig. 1， e an regnie his siuan in he seur nuerial errr in he al angur enu， hih l be rigrusly presere up ahine-e preisn.
2.4.2 errr in pneary lngiues
sine he syplei aps presere al energy an al angur enu f n-by ynaial syses inherenly ell， he egree f heir preseran ay n be a g easure f he auray f nuerial inegrans， espeially as a easure f he psinal errr f pnes， i.e. he errr in pneary lngiues. esiae he nuerial errr in he pneary lngiues， e perfre he flling preures. e pare he resul f ur ain lng-er inegrans ih se es inegrans， hih span uh shrer pers bu ih uh higher auray han he ain inegrans. fr his purpse， e perfre a uh re aurae inegran ih a sepsie f 0.125 (1/64 f he ain inegrans) spanning 3 x 105 yr， saring ih he sae iniial nins as in he n1 inegran. e nsier ha his es inegran pries us ih a ‘pseu-rue’ slun f pneary rbial elun. nex， e pare he es inegran ih he ain inegran， n1. fr he per f 3 x 105 yr， e see a ifferene in ean analies f he earh beeen he inegrans f 0.52°(in he ase f he n1 inegran). his ifferene an be exrape he alue 八700°， abu 25 rans f earh afer 5 gyr， sine he errr f lngiues inreases linearly ih ie in he syplei ap. siirly， he lngiue errr f plu an be esiae as 12°. his alue fr plu is uh beer han he resul in kinshia nakai (1996) here he ifferene is esiae as 60°.
3 nuerial resuls – i. gne a he ra aa
in his sen e briefly reie he lng-er sabiliy f pneary rbial n hrugh se snapshs f ra nuerial aa. he rbial n f pnes iniaes lng-er sabiliy in all f ur nuerial inegrans: n rbial rssings nr lse enuners beeen any pair f pnes k pe.
3.1 neral en f he sabiliy f pneary rbis
firs， e briefly lk a he neral haraer f he lng-er sabiliy f pneary rbis. ur ineres here fuses pariurly n he inner fur erresrial pnes fr hih he rbial ie-sales are uh shrer han hse f he uer fie pnes. as e an see learly fr he pnar rbial nfigurans shn in figs 2 an 3， rbial psins f he erresrial pnes iffer lile beeen he iniial an final par f eah nuerial inegran， hih spans seeral gyr. he sli lines ening he presen rbis f he pnes lie als ihin he sar f s een in he final par f inegrans (b) an (). his iniaes ha hrughu he enire inegran per he als regur arians f pneary rbial n reain nearly he sae as hey are a presen.
erial ie f he fur inner pneary rbis (fr he -axis iren) a he iniial an final pars f he inegransn±1. he axes unis are au. he xy -pne is se he inarian pne f sr syse al angur enu.(a) he iniial par fn+1 ( = 0 0.0547 x 10 9 yr).(b) he final par fn+1 ( = 4.9339 x 10 八 4.9八八6 x 10 9 yr). he iniial par f n1 (= 0 0.0547 x 109 yr).() he final par fn1 ( =3.91八0 x 10 9 3.9727 x 10 9 yr). in eah panel， a al f 6八4 pins are ple ih an ineral f abu 2190 yr er 5.47 x 107 yr . sli lines in eah panel ene he presen rbis f he fur erresrial pnes (aken fr e245).
he arian f eenriiies an rbial inlinans fr he inner fur pnes in he iniial an final par f he inegran n+1 is shn in fig. 4. as expee， he haraer f he arian f pneary rbial eleens es n iffer signifianly beeen he iniial an final par f eah inegran， a leas fr enus， earh an ars. he eleens f erury， espeially is eenriiy， see han a signifian exen. his is parly beause he rbial ie-sale f he pne is he shres f all he pnes， hih leas a re rapi rbial elun han her pnes; he inners pne ay be neares insabiliy. his resul appears be in se agreeen ih skar's (1994， 1996) expeans ha r an irregur arians appear in he eenriiies an inlinans f erury n a ie-sale f seeral 109 yr. heer， he effe f he pssible insabiliy f he rbi f erury ay n faally affe he glbal sabiliy f he hle pneary syse ing he sall ass f erury. e ill enn briefly he lng-er rbial elun f erury er in sen 4 using l-pass filere rbial eleens.
he rbial n f he uer fie pnes sees rigrusly sable an quie regur er his ie-span (see als sen 5).
3.2 ie–frequeny aps
alhugh he pneary n exhibis ery lng-er sabiliy efine as he nn-exisene f lse enuner eens， he hai naure f pneary ynais an han he silry per an apliue f pneary rbial n graually er suh lng ie-spans. een suh sligh fluuans f rbial arian in he frequeny ain， pariurly in he ase f earh， an penially hae a signifian effe n is surfae liae syse hrugh sr insn arian (f. berr 19八八).
gie an erie f he lng-er han in periiy in pneary rbial n， e perfre any fas furier ransfrans (ffs) alng he ie axis， an superpse he resuling pergras ra -iensnal ie–frequeny aps. he speifi apprah raing hese ie–frequeny aps in his paper is ery siple – uh sipler han he aele analysis r skar's (1990， 1993) frequeny analysis.
ie he l-pass filere rbial aa in any frans f he sae lengh. he lengh f eah aa sen l be a uliple f 2 in rer apply he ff.
eah fran f he aa has a r erpping par: fr exaple， hen he ih aa begins fr =i an ens a =i+， he nex aa sen rans fr i+δi+δ+， here δ?. e ninue his isn unil e reah a erain nuber n by hih n+ reahes he al inegran lengh.
e apply an ff eah f he aa frans， an bain n frequeny iagras.
in eah frequeny iagra baine abe， he srengh f periiy an be repe by a grey-sale (r lur) har.
e perfr he repeen， an nne all he grey-sale (r lur) hars in ne graph fr eah inegran. he hrinal axis f hese ne graphs l be he ie， i.e. he saring ies f eah fran f aa (i， here i= 1，…， n). he erial axis represens he per (r frequeny) f he siln f rbial eleens.
e hae ape an ff beause f is erheling spee， sine he aun f nuerial aa be epse in frequeny pnens is erribly hu (seeral ens f gbyes).
a ypial exaple f he ie–frequeny ap reae by he abe preures is shn in a grey-sale iagra as fig. 5， hih shs he arian f periiy in he eenriiy an inlinan f earh in n+2 inegran. in fig. 5， he ark area shs ha a he ie iniae by he alue n he absissa， he periiy iniae by he rinae is srnr han in he ligher area arun i. e an regnie fr his ap ha he periiy f he eenriiy an inlinan f earh nly hans slighly er he enire per ere by he n+2 inegran. his nearly regur ren is qualiaiely he sae in her inegrans an fr her pnes， alhugh ypial frequenies iffer pne by pne an eleen by eleen.
4.2 lng-er exhan f rbial energy an angur enu
e alue ery lng-peri arian an exhan f pneary rbial energy an angur enu using filere eunay eleens l， g， h. g an h are equialen he pneary rbial angur enu an is erial pnen per uni ass. l is ree he pneary rbial energy e per uni ass as e=μ2/2l2. if he syse is pleely linear， he rbial energy an he angur enu in eah frequeny bin us be nsan. nn-lineariy in he pneary syse an ause an exhan f energy an angur enu in he frequeny ain. he apliue f he les-frequeny siln l inrease if he syse is unsable an breaks n graually. heer， suh a syp f insabiliy is n prinen in ur lng-er inegrans.
in fig. 7， he al rbial energy an angur enu f he fur inner pnes an all nine pnes are shn fr inegran n+2. he upper hree panels sh he lng-peri arian f al energy (ene ase- e0)， al angur enu ( g- g0)， an he erial pnen ( h- h0) f he inner fur pnes alue fr he l-pass filere eunay eleens.e0， g0， h0 ene he iniial alues f eah quaniy. he abslue ifferene fr he iniial alues is ple in he panels. he ler hree panels in eah figure she-e0，g-g0 anh-h0 f he al f nine pnes. he fluuan shn in he ler panels is irually enirely a resul f he assie jian pnes.
paring he arians f energy an angur enu f he inner fur pnes an all nine pnes， i is apparen ha he apliues f hse f he inner pnes are uh saller han hse f all nine pnes: he apliues f he uer fie pnes are uh rr han hse f he inner pnes. his es n ean ha he inner erresrial pneary subsyse is re sable han he uer ne: his is siply a resul f he reie sallness f he asses f he fur erresrial pnes pare ih hse f he uer jian pnes. anher hing e nie is ha he inner pneary subsyse ay bee unsable re rapily han he uer ne beause f is shrer rbial ie-sales. his an be seen in he panels ene asinner 4 in fig. 7 here he lnr-peri an irregur silns are re apparen han in he panels ene asal 9. aually， he fluuans in heinner 4 panels are a r exen as a resul f he rbial arian f he erury. heer， e ann negle he nribun fr her erresrial pnes， as e ill see in subsequen sens.
4.4 lng-er upling f seeral neighburing pne pairs
le us see se iniual arians f pneary rbial energy an angur enu expresse by he l-pass filere eunay eleens. figs 10 an 11 sh lng-er elun f he rbial energy f eah pne an he angur enu in n+1 an n2 inegrans. e nie ha se pnes fr apparen pairs in ers f rbial energy an angur enu exhan. in pariur， enus an earh ake a ypial pair. in he figures， hey sh negaie rrens in exhan f energy an psiie rrens in exhan f angur enu. he negaie rren in exhan f rbial energy eans ha he pnes fr a lse ynaial syse in ers f he rbial energy. he psiie rren in exhan f angur enu eans ha he pnes are siulaneusly uner erain lng-er perurbans. aniaes fr perurbers are jupier an saurn. als in fig. 11， e an see ha ars shs a psiie rren in he angur enu arian he enus–earh syse. erury exhibis erain negaie rrens in he angur enu ersus he enus–earh syse， hih sees be a rean ause by he nseran f angur enu in he erresrial pneary subsyse.
i is n lear a he en hy he enus–earh pair exhibis a negaie rren in energy exhan an a psiie rren in angur enu exhan. e ay pssibly expin his hrugh bsering he neral fa ha here are n seur ers in pneary seiajr axes up sen-rer perurban heries (f. bruer leene 1961; balei pua 199八). his eans ha he pneary rbial energy (hih is irely ree he seiajr axis a) igh be uh less affee by perurbing pnes han is he angur enu exhan (hih rees e). hene， he eenriiies f enus an earh an be isurbe easily by jupier an saurn， hih resuls in a psiie rren in he angur enu exhan. n he her han， he seiajr axes f enus an earh are less likely be isurbe by he jian pnes. hus he energy exhan ay be liie nly ihin he enus–earh pair， hih resuls in a negaie rren in he exhan f rbial energy in he pair.
as fr he uer jian pneary subsyse， jupier–saurn an uranus–nepune see ake ynaial pairs. heer， he srengh f heir upling is n as srng pare ih ha f he enus–earh pair.
5 ± 5 x 1010-yr inegrans f uer pneary rbis
sine he jian pneary asses are uh rr han he erresrial pneary asses， e rea he jian pneary syse as an inepenen pneary syse in ers f he suy f is ynaial sabiliy. hene， e ae a uple f rial inegrans ha span ± 5 x 1010 yr， inluing nly he uer fie pnes (he fur jian pnes plus plu). he resuls exhibi he rigrus sabiliy f he uer pneary syse er his lng ie-span. rbial nfigurans (fig. 12)， an arian f eenriiies an inlinans (fig. 13) sh his ery lng-er sabiliy f he uer fie pnes in bh he ie an he frequeny ains. alhugh e n sh aps here， he ypial frequeny f he rbial siln f plu an he her uer pnes is als nsan uring hese ery lng-er inegran pers， hih is ensrae in he ie–frequeny aps n ur ebpa.
in hese inegrans， he reie nuerial errr in he al energy as 106 an ha f he al angur enu as 1010.
5.1 resnanes in he nepune–plu syse
kinshia nakai (1996) inegrae he uer fie pneary rbis er ± 5.5 x 109 yr . hey fun ha fur ajr resnanes beeen nepune an plu are ainaine uring he hle inegran per， an ha he resnanes ay be he ain auses f he sabiliy f he rbi f plu. he ajr fur resnanes fun in preus researh are as flls. in he flling en，λ enes he ean lngiue，Ω is he lngiue f he asening ne an?is he lngiue f periheln. subs p an n ene plu an nepune.
ean n resnane beeen nepune an plu (3:2). he riial arguen θ1= 3 λp 2 λnp libraes arun 1八0° ih an apliue f abu 八0° an a libran per f abu 2 x 104 yr.
he arguen f periheln f plu p=θ2=pΩp libraes arun 90° ih a per f abu 3.八 x 106 yr. he inan peri arians f he eenriiy an inlinan f plu are synhrnie ih he libran f is arguen f periheln. his is aniipae in he seur perurban hery nsrue by kai (1962).
he lngiue f he ne f plu referre he lngiue f he ne f nepune，θ3=ΩpΩn， irues an he per f his irun is equal he per f θ2 libran. hen θ3 bees er， i.e. he lngiues f asening nes f nepune an plu erp， he inlinan f plu bees axiu， he eenriiy bees iniu an he arguen f periheln bees 90°. hen θ3 bees 1八0°， he inlinan f plu bees iniu， he eenriiy bees axiu an he arguen f periheln bees 90° again. illias bensn (1971) aniipae his ype f resnane， er nfire by ini， nbili arpin (19八9).
an arguen θ4=pn+ 3 (ΩpΩn) libraes arun 1八0° ih a lng per， 5.7 x 10八 yr.
in ur nuerial inegrans， he resnanes (i)–(iii) are ell ainaine， an arian f he riial arguens θ1，θ2，θ3 reain siir uring he hle inegran per (figs 14–16 ). heer， he furh resnane (i) appears be ifferen: he riial arguen θ4 alernaes libran an irun er a 1010-yr ie-sale (fig. 17). his is an ineresing fa ha kinshia nakai's (1995， 1996) shrer inegrans ere n able islse.
6 isussn
ha kin f ynaial ehanis ainains his lng-er sabiliy f he pneary syse? e an ieiaely hink f ajr feaures ha ay be respnsible fr he lng-er sabiliy. firs， here see be n signifian ler-rer resnanes (ean n an seur) beeen any pair ang he nine pnes. jupier an saurn are lse a 5:2 ean n resnane (he faus ‘grea inequaliy’)， bu n jus in he resnane ne. higher-rer resnanes ay ause he hai naure f he pneary ynaial n， bu hey are n s srng as esry he sable pneary n ihin he lifeie f he real sr syse. he sen feaure， hih e hink is re ipran fr he lng-er sabiliy f ur pneary syse， is he ifferene in ynaial isane beeen erresrial an jian pneary subsyses (i anikaa 1999， 2001). hen e easure pneary separans by he uual hill raii (r_)， separans ang erresrial pnes are greaer han 26rh， hereas hse ang jian pnes are less han 14rh. his ifferene is irely ree he ifferene beeen ynaial feaures f erresrial an jian pnes. erresrial pnes hae saller asses， shrer rbial pers an ier ynaial separan. hey are srngly perurbe by jian pnes ha hae rr asses， lnr rbial pers an narrer ynaial separan. jian pnes are n perurbe by any her assie bies.
he presen erresrial pneary syse is sill being isurbe by he assie jian pnes. heer， he ie separan an uual ineran ang he erresrial pnes reners he isurbane ineffeie; he egree f isurbane by jian pnes is (ej)(rer f agniue f he eenriiy f jupier)， sine he isurbane ause by jian pnes is a fre siln haing an apliue f (ej). heighening f eenriiy， fr exaple (ej)0.05， is far fr suffiien prke insabiliy in he erresrial pnes haing suh a ie separan as 26rh. hus e assue ha he presen ie ynaial separan ang erresrial pnes (> 26rh) is prbably ne f he s signifian nins fr ainaining he sabiliy f he pneary syse er a 109-yr ie-span. ur eaile analysis f he renship beeen ynaial isane beeen pnes an he insabiliy ie-sale f sr syse pneary n is n n-ging.
alhugh ur nuerial inegrans span he lifeie f he sr syse， he nuber f inegrans is far fr suffiien fill he iniial phase spae. i is neessary perfr re an re nuerial inegrans nfir an exaine in eail he lng-er sabiliy f ur pneary ynais.
——以上文段引自 i， .& anikaa， k. lng-er inegrans an sabiliy f pneary rbis in ur sr syse. n. n. r. asrn. s. 336， 4八3–500 (2002)
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lng-er inegrans an sabiliy f pneary rbis in ur sr syse
absra
e presen he resuls f ery lng-er nuerial inegrans f pneary rbial ns er 109 -yr ie-spans inluing all nine pnes. a quik inspen f ur nuerial aa shs ha he pneary n， a leas in ur siple ynaial el， sees be quie sable een er his ery lng ie-span. a lser lk a he les-frequeny silns using a l-pass filer shs us he penially iffusie haraer f erresrial pneary n， espeially ha f erury. he behaur f he eenriiy f erury in ur inegrans is qualiaiely siir he resuls fr jaques skar's seur perurban hery (e.g. eax 0.35 er ± 4 gyr). heer， here are n apparen seur inreases f eenriiy r inlinan in any rbial eleens f he pnes， hih ay be reeale by sill lnr-er nuerial inegrans. e hae als perfre a uple f rial inegrans inluing ns f he uer fie pnes er he uran f ± 5 x 1010 yr. he resul iniaes ha he hree ajr resnanes in he nepune–plu syse hae been ainaine er he 1011-yr ie-span.
1 inrun
1.1efinin f he prble
he quesn f he sabiliy f ur sr syse has been ebae er seeral hunre years， sine he era f nen. he prble has arae any faus aheaiians er he years an has pye a enral rle in he eelpen f nn-linear ynais an has hery. heer， e n ye hae a efinie anser he quesn f heher ur sr syse is sable r n. his is parly a resul f he fa ha he efinin f he er ‘sabiliy’ is ague hen i is use in ren he prble f pneary n in he sr syse. aually i is n easy gie a lear， rigrus an physially eaningful efinin f he sabiliy f ur sr syse.
ang any efinins f sabiliy， here e ap he hill efinin (gan 1993): aually his is n a efinin f sabiliy， bu f insabiliy. e efine a syse as being unsable hen a lse enuner urs sehere in he syse， saring fr a erain iniial nfiguran (habers， eherill bss 1996; i anikaa 1999). a syse is efine as experiening a lse enuner hen bies apprah ne anher ihin an area f he rr hill raius. herise he syse is efine as being sable. henefrar e sae ha ur pneary syse is ynaially sable if n lse enuner happens uring he a f ur sr syse， abu ±5 gyr. inienally， his efinin ay be repe by ne in hih an urrene f any rbial rssing beeen eiher f a pair f pnes akes pe. his is beause e kn fr experiene ha an rbial rssing is ery likely lea a lse enuner in pneary an prpneary syses (yshinaga， kkub akin 1999). f urse his saeen ann be siply applie syses ih sable rbial resnanes suh as he nepune–plu syse.
1.2preus suies an ais f his researh
in ain he agueness f he nep f sabiliy， he pnes in ur sr syse sh a haraer ypial f ynaial has (sussan is 19八八， 1992). he ause f his hai behaur is n parly uners as being a resul f resnane erpping (urray hn 1999; lear， franklin hn 2001). heer， i ul require inegraing er an enseble f pneary syses inluing all nine pnes fr a per ering seeral 10 gyr hrughly unersan he lng-er elun f pneary rbis， sine hai ynaial syses are haraerie by heir srng epenene n iniial nins.
fr ha pin f ie， any f he preus lng-er nuerial inegrans inlue nly he uer fie pnes (sussan is 19八八; kinshia nakai 1996). his is beause he rbial pers f he uer pnes are s uh lnr han hse f he inner fur pnes ha i is uh easier fll he syse fr a gien inegran per. a presen， he lns nuerial inegrans publishe in jurnals are hse f unan lissauer (199八). alhugh heir ain ar as he effe f ps-ain-sequene sr ass lss n he sabiliy f pneary rbis， hey perfre any inegrans ering up 1011 yr f he rbial ns f he fur jian pnes. he iniial rbial eleens an asses f pnes are he sae as hse f ur sr syse in unan lissauer's paper， bu hey erease he ass f he sun graually in heir nuerial experiens. his is beause hey nsier he effe f ps-ain-sequene sr ass lss in he paper. nsequenly， hey fun ha he rssing ie-sale f pneary rbis， hih an be a ypial iniar f he insabiliy ie-sale， is quie sensiie he rae f ass erease f he sun. hen he ass f he sun is lse is presen alue， he jian pnes reain sable er 1010 yr， r perhaps lnr. unan lissauer als perfre fur siir experiens n he rbial n f seen pnes (enus nepune)， hih er a span f 109 yr. heir experiens n he seen pnes are n ye prehensie， bu i sees ha he erresrial pnes als reain sable uring he inegran per， ainaining als regur silns.
n he her han， in his aurae sei-analyial seur perurban hery (skar 19八八)， skar fins ha r an irregur arians an appear in he eenriiies an inlinans f he erresrial pnes， espeially f erury an ars n a ie-sale f seeral 109 yr (skar 1996). he resuls f skar's seur perurban hery l be nfire an inesigae by fully nuerial inegrans.
in his paper e presen preliinary resuls f six lng-er nuerial inegrans n all nine pneary rbis， ering a span f seeral 109 yr， an f her inegrans ering a span f ± 5 x 1010 yr. he al epse ie fr all inegrans is re han 5 yr， using seeral eiae ps an rksans. ne f he funaenal nlusns f ur lng-er inegrans is ha sr syse pneary n sees be sable in ers f he hill sabiliy enne abe， a leas er a ie-span f ± 4 gyr. aually， in ur nuerial inegrans he syse as far re sable han ha is efine by he hill sabiliy riern: n nly i n lse enuner happen uring he inegran per， bu als all he pneary rbial eleens hae been nfine in a narr regn bh in ie an frequeny ain， hugh pneary ns are shasi. sine he purpse f his paper is exhibi an erie he resuls f ur lng-er nuerial inegrans， e sh ypial exaple figures as eiene f he ery lng-er sabiliy f sr syse pneary n. fr reaers h hae re speifi an eeper ineress in ur nuerial resuls， e hae prepare a ebpa (aess )， here e sh ra rbial eleens， heir l-pass filere resuls， arian f eunay eleens an angur enu efii， an resuls f ur siple ie–frequeny analysis n all f ur inegrans.
in sen 2 e briefly expin ur ynaial el， nuerial eh an iniial nins use in ur inegrans. sen 3 is ee a en f he quik resuls f he nuerial inegrans. ery lng-er sabiliy f sr syse pneary n is apparen bh in pneary psins an rbial eleens. a rugh esian f nuerial errrs is als gien. sen 4 s n a isussn f he lns-er arian f pneary rbis using a l-pass filer an inlues a isussn f angur enu efii. in sen 5， e presen a se f nuerial inegrans fr he uer fie pnes ha spans ± 5 x 1010 yr. in sen 6 e als isuss he lng-er sabiliy f he pneary n an is pssible ause.
2 en f he nuerial inegrans
（本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了起点也不一定能成功显示。）
2.3 nuerial eh
e uilie a sen-rer is–hn syplei ap as ur ain inegran eh (is hn 1991; kinshia， yshia nakai 1991) ih a speial sar-up preure reue he runan errr f angle ariables，‘ar sar’(saha reaine 1992， 1994).
he sepsie fr he nuerial inegrans is 八 hrughu all inegrans f he nine pnes (n±1，2，3)， hih is abu 1/11 f he rbial per f he inners pne (erury). as fr he eerinan f sepsie， e parly fll he preus nuerial inegran f all nine pnes in sussan is (19八八， 7.2 ) an saha reaine (1994， 225/32 ). e rune he eial par f he heir sepsies 八 ake he sepsie a uliple f 2 in rer reue he auun f run-ff errr in he puan presses. in ren his， is hn (1991) perfre nuerial inegrans f he uer fie pneary rbis using he syplei ap ih a sepsie f 400 ， 1/10.八3 f he rbial per f jupier. heir resul sees be aurae enugh， hih parly jusifies ur eh f eerining he sepsie. heer， sine he eenriiy f jupier (0.05) is uh saller han ha f erury (0.2)， e nee se are hen e pare hese inegrans siply in ers f sepsies.
in he inegran f he uer fie pnes (f±)， e fixe he sepsie a 400 .
e ap gauss' f an g funns in he syplei ap her ih he hir-rer halley eh (anby 1992) as a sler fr kepler equans. he nuber f axiu ierans e se in halley's eh is 15， bu hey neer reahe he axiu in any f ur inegrans.
he ineral f he aa upu is 200 000 (547 yr) fr he aluns f all nine pnes (n±1，2，3)， an abu 八000 000 (21 903 yr) fr he inegran f he uer fie pnes (f±).
alhugh n upu filering as ne hen he nuerial inegrans ere in press， e applie a l-pass filer he ra rbial aa afer e ha plee all he aluns. see sen 4.1 fr re eail.
2.4 errr esian
2.4.1 reie errrs in al energy an angur enu
aring ne f he basi prperies f syplei inegrars， hih nsere he physially nseraie quaniies ell (al rbial energy an angur enu)， ur lng-er nuerial inegrans see hae been perfre ih ery sall errrs. he aera reie errrs f al energy (109) an f al angur enu (1011) hae reaine nearly nsan hrughu he inegran per (fig. 1). he speial sarup preure， ar sar， ul hae reue he aera reie errr in al energy by abu ne rer f agniue r re.
reie nuerial errr f he al angur enu δa/a0 an he al energy δe/e0 in ur nuerial inegransn± 1，2，3， here δe an δa are he abslue han f he al energy an al angur enu， respeiely， ane0ana0are heir iniial alues. he hrinal uni is gyr.
ne ha ifferen peraing syses， ifferen aheaial libraries， an ifferen harare arhieures resul in ifferen nuerial errrs， hrugh he arians in run-ff errr hanling an nuerial algrihs. in he upper panel f fig. 1， e an regnie his siuan in he seur nuerial errr in he al angur enu， hih l be rigrusly presere up ahine-e preisn.
2.4.2 errr in pneary lngiues
sine he syplei aps presere al energy an al angur enu f n-by ynaial syses inherenly ell， he egree f heir preseran ay n be a g easure f he auray f nuerial inegrans， espeially as a easure f he psinal errr f pnes， i.e. he errr in pneary lngiues. esiae he nuerial errr in he pneary lngiues， e perfre he flling preures. e pare he resul f ur ain lng-er inegrans ih se es inegrans， hih span uh shrer pers bu ih uh higher auray han he ain inegrans. fr his purpse， e perfre a uh re aurae inegran ih a sepsie f 0.125 (1/64 f he ain inegrans) spanning 3 x 105 yr， saring ih he sae iniial nins as in he n1 inegran. e nsier ha his es inegran pries us ih a ‘pseu-rue’ slun f pneary rbial elun. nex， e pare he es inegran ih he ain inegran， n1. fr he per f 3 x 105 yr， e see a ifferene in ean analies f he earh beeen he inegrans f 0.52°(in he ase f he n1 inegran). his ifferene an be exrape he alue 八700°， abu 25 rans f earh afer 5 gyr， sine he errr f lngiues inreases linearly ih ie in he syplei ap. siirly， he lngiue errr f plu an be esiae as 12°. his alue fr plu is uh beer han he resul in kinshia nakai (1996) here he ifferene is esiae as 60°.
3 nuerial resuls – i. gne a he ra aa
in his sen e briefly reie he lng-er sabiliy f pneary rbial n hrugh se snapshs f ra nuerial aa. he rbial n f pnes iniaes lng-er sabiliy in all f ur nuerial inegrans: n rbial rssings nr lse enuners beeen any pair f pnes k pe.
3.1 neral en f he sabiliy f pneary rbis
firs， e briefly lk a he neral haraer f he lng-er sabiliy f pneary rbis. ur ineres here fuses pariurly n he inner fur erresrial pnes fr hih he rbial ie-sales are uh shrer han hse f he uer fie pnes. as e an see learly fr he pnar rbial nfigurans shn in figs 2 an 3， rbial psins f he erresrial pnes iffer lile beeen he iniial an final par f eah nuerial inegran， hih spans seeral gyr. he sli lines ening he presen rbis f he pnes lie als ihin he sar f s een in he final par f inegrans (b) an (). his iniaes ha hrughu he enire inegran per he als regur arians f pneary rbial n reain nearly he sae as hey are a presen.
erial ie f he fur inner pneary rbis (fr he -axis iren) a he iniial an final pars f he inegransn±1. he axes unis are au. he xy -pne is se he inarian pne f sr syse al angur enu.(a) he iniial par fn+1 ( = 0 0.0547 x 10 9 yr).(b) he final par fn+1 ( = 4.9339 x 10 八 4.9八八6 x 10 9 yr). he iniial par f n1 (= 0 0.0547 x 109 yr).() he final par fn1 ( =3.91八0 x 10 9 3.9727 x 10 9 yr). in eah panel， a al f 6八4 pins are ple ih an ineral f abu 2190 yr er 5.47 x 107 yr . sli lines in eah panel ene he presen rbis f he fur erresrial pnes (aken fr e245).
he arian f eenriiies an rbial inlinans fr he inner fur pnes in he iniial an final par f he inegran n+1 is shn in fig. 4. as expee， he haraer f he arian f pneary rbial eleens es n iffer signifianly beeen he iniial an final par f eah inegran， a leas fr enus， earh an ars. he eleens f erury， espeially is eenriiy， see han a signifian exen. his is parly beause he rbial ie-sale f he pne is he shres f all he pnes， hih leas a re rapi rbial elun han her pnes; he inners pne ay be neares insabiliy. his resul appears be in se agreeen ih skar's (1994， 1996) expeans ha r an irregur arians appear in he eenriiies an inlinans f erury n a ie-sale f seeral 109 yr. heer， he effe f he pssible insabiliy f he rbi f erury ay n faally affe he glbal sabiliy f he hle pneary syse ing he sall ass f erury. e ill enn briefly he lng-er rbial elun f erury er in sen 4 using l-pass filere rbial eleens.
he rbial n f he uer fie pnes sees rigrusly sable an quie regur er his ie-span (see als sen 5).
3.2 ie–frequeny aps
alhugh he pneary n exhibis ery lng-er sabiliy efine as he nn-exisene f lse enuner eens， he hai naure f pneary ynais an han he silry per an apliue f pneary rbial n graually er suh lng ie-spans. een suh sligh fluuans f rbial arian in he frequeny ain， pariurly in he ase f earh， an penially hae a signifian effe n is surfae liae syse hrugh sr insn arian (f. berr 19八八).
gie an erie f he lng-er han in periiy in pneary rbial n， e perfre any fas furier ransfrans (ffs) alng he ie axis， an superpse he resuling pergras ra -iensnal ie–frequeny aps. he speifi apprah raing hese ie–frequeny aps in his paper is ery siple – uh sipler han he aele analysis r skar's (1990， 1993) frequeny analysis.
ie he l-pass filere rbial aa in any frans f he sae lengh. he lengh f eah aa sen l be a uliple f 2 in rer apply he ff.
eah fran f he aa has a r erpping par: fr exaple， hen he ih aa begins fr =i an ens a =i+， he nex aa sen rans fr i+δi+δ+， here δ?. e ninue his isn unil e reah a erain nuber n by hih n+ reahes he al inegran lengh.
e apply an ff eah f he aa frans， an bain n frequeny iagras.
in eah frequeny iagra baine abe， he srengh f periiy an be repe by a grey-sale (r lur) har.
e perfr he repeen， an nne all he grey-sale (r lur) hars in ne graph fr eah inegran. he hrinal axis f hese ne graphs l be he ie， i.e. he saring ies f eah fran f aa (i， here i= 1，…， n). he erial axis represens he per (r frequeny) f he siln f rbial eleens.
e hae ape an ff beause f is erheling spee， sine he aun f nuerial aa be epse in frequeny pnens is erribly hu (seeral ens f gbyes).
a ypial exaple f he ie–frequeny ap reae by he abe preures is shn in a grey-sale iagra as fig. 5， hih shs he arian f periiy in he eenriiy an inlinan f earh in n+2 inegran. in fig. 5， he ark area shs ha a he ie iniae by he alue n he absissa， he periiy iniae by he rinae is srnr han in he ligher area arun i. e an regnie fr his ap ha he periiy f he eenriiy an inlinan f earh nly hans slighly er he enire per ere by he n+2 inegran. his nearly regur ren is qualiaiely he sae in her inegrans an fr her pnes， alhugh ypial frequenies iffer pne by pne an eleen by eleen.
4.2 lng-er exhan f rbial energy an angur enu
e alue ery lng-peri arian an exhan f pneary rbial energy an angur enu using filere eunay eleens l， g， h. g an h are equialen he pneary rbial angur enu an is erial pnen per uni ass. l is ree he pneary rbial energy e per uni ass as e=μ2/2l2. if he syse is pleely linear， he rbial energy an he angur enu in eah frequeny bin us be nsan. nn-lineariy in he pneary syse an ause an exhan f energy an angur enu in he frequeny ain. he apliue f he les-frequeny siln l inrease if he syse is unsable an breaks n graually. heer， suh a syp f insabiliy is n prinen in ur lng-er inegrans.
in fig. 7， he al rbial energy an angur enu f he fur inner pnes an all nine pnes are shn fr inegran n+2. he upper hree panels sh he lng-peri arian f al energy (ene ase- e0)， al angur enu ( g- g0)， an he erial pnen ( h- h0) f he inner fur pnes alue fr he l-pass filere eunay eleens.e0， g0， h0 ene he iniial alues f eah quaniy. he abslue ifferene fr he iniial alues is ple in he panels. he ler hree panels in eah figure she-e0，g-g0 anh-h0 f he al f nine pnes. he fluuan shn in he ler panels is irually enirely a resul f he assie jian pnes.
paring he arians f energy an angur enu f he inner fur pnes an all nine pnes， i is apparen ha he apliues f hse f he inner pnes are uh saller han hse f all nine pnes: he apliues f he uer fie pnes are uh rr han hse f he inner pnes. his es n ean ha he inner erresrial pneary subsyse is re sable han he uer ne: his is siply a resul f he reie sallness f he asses f he fur erresrial pnes pare ih hse f he uer jian pnes. anher hing e nie is ha he inner pneary subsyse ay bee unsable re rapily han he uer ne beause f is shrer rbial ie-sales. his an be seen in he panels ene asinner 4 in fig. 7 here he lnr-peri an irregur silns are re apparen han in he panels ene asal 9. aually， he fluuans in heinner 4 panels are a r exen as a resul f he rbial arian f he erury. heer， e ann negle he nribun fr her erresrial pnes， as e ill see in subsequen sens.
4.4 lng-er upling f seeral neighburing pne pairs
le us see se iniual arians f pneary rbial energy an angur enu expresse by he l-pass filere eunay eleens. figs 10 an 11 sh lng-er elun f he rbial energy f eah pne an he angur enu in n+1 an n2 inegrans. e nie ha se pnes fr apparen pairs in ers f rbial energy an angur enu exhan. in pariur， enus an earh ake a ypial pair. in he figures， hey sh negaie rrens in exhan f energy an psiie rrens in exhan f angur enu. he negaie rren in exhan f rbial energy eans ha he pnes fr a lse ynaial syse in ers f he rbial energy. he psiie rren in exhan f angur enu eans ha he pnes are siulaneusly uner erain lng-er perurbans. aniaes fr perurbers are jupier an saurn. als in fig. 11， e an see ha ars shs a psiie rren in he angur enu arian he enus–earh syse. erury exhibis erain negaie rrens in he angur enu ersus he enus–earh syse， hih sees be a rean ause by he nseran f angur enu in he erresrial pneary subsyse.
i is n lear a he en hy he enus–earh pair exhibis a negaie rren in energy exhan an a psiie rren in angur enu exhan. e ay pssibly expin his hrugh bsering he neral fa ha here are n seur ers in pneary seiajr axes up sen-rer perurban heries (f. bruer leene 1961; balei pua 199八). his eans ha he pneary rbial energy (hih is irely ree he seiajr axis a) igh be uh less affee by perurbing pnes han is he angur enu exhan (hih rees e). hene， he eenriiies f enus an earh an be isurbe easily by jupier an saurn， hih resuls in a psiie rren in he angur enu exhan. n he her han， he seiajr axes f enus an earh are less likely be isurbe by he jian pnes. hus he energy exhan ay be liie nly ihin he enus–earh pair， hih resuls in a negaie rren in he exhan f rbial energy in he pair.
as fr he uer jian pneary subsyse， jupier–saurn an uranus–nepune see ake ynaial pairs. heer， he srengh f heir upling is n as srng pare ih ha f he enus–earh pair.
5 ± 5 x 1010-yr inegrans f uer pneary rbis
sine he jian pneary asses are uh rr han he erresrial pneary asses， e rea he jian pneary syse as an inepenen pneary syse in ers f he suy f is ynaial sabiliy. hene， e ae a uple f rial inegrans ha span ± 5 x 1010 yr， inluing nly he uer fie pnes (he fur jian pnes plus plu). he resuls exhibi he rigrus sabiliy f he uer pneary syse er his lng ie-span. rbial nfigurans (fig. 12)， an arian f eenriiies an inlinans (fig. 13) sh his ery lng-er sabiliy f he uer fie pnes in bh he ie an he frequeny ains. alhugh e n sh aps here， he ypial frequeny f he rbial siln f plu an he her uer pnes is als nsan uring hese ery lng-er inegran pers， hih is ensrae in he ie–frequeny aps n ur ebpa.
in hese inegrans， he reie nuerial errr in he al energy as 106 an ha f he al angur enu as 1010.
5.1 resnanes in he nepune–plu syse
kinshia nakai (1996) inegrae he uer fie pneary rbis er ± 5.5 x 109 yr . hey fun ha fur ajr resnanes beeen nepune an plu are ainaine uring he hle inegran per， an ha he resnanes ay be he ain auses f he sabiliy f he rbi f plu. he ajr fur resnanes fun in preus researh are as flls. in he flling en，λ enes he ean lngiue，Ω is he lngiue f he asening ne an?is he lngiue f periheln. subs p an n ene plu an nepune.
ean n resnane beeen nepune an plu (3:2). he riial arguen θ1= 3 λp 2 λnp libraes arun 1八0° ih an apliue f abu 八0° an a libran per f abu 2 x 104 yr.
he arguen f periheln f plu p=θ2=pΩp libraes arun 90° ih a per f abu 3.八 x 106 yr. he inan peri arians f he eenriiy an inlinan f plu are synhrnie ih he libran f is arguen f periheln. his is aniipae in he seur perurban hery nsrue by kai (1962).
he lngiue f he ne f plu referre he lngiue f he ne f nepune，θ3=ΩpΩn， irues an he per f his irun is equal he per f θ2 libran. hen θ3 bees er， i.e. he lngiues f asening nes f nepune an plu erp， he inlinan f plu bees axiu， he eenriiy bees iniu an he arguen f periheln bees 90°. hen θ3 bees 1八0°， he inlinan f plu bees iniu， he eenriiy bees axiu an he arguen f periheln bees 90° again. illias bensn (1971) aniipae his ype f resnane， er nfire by ini， nbili arpin (19八9).
an arguen θ4=pn+ 3 (ΩpΩn) libraes arun 1八0° ih a lng per， 5.7 x 10八 yr.
in ur nuerial inegrans， he resnanes (i)–(iii) are ell ainaine， an arian f he riial arguens θ1，θ2，θ3 reain siir uring he hle inegran per (figs 14–16 ). heer， he furh resnane (i) appears be ifferen: he riial arguen θ4 alernaes libran an irun er a 1010-yr ie-sale (fig. 17). his is an ineresing fa ha kinshia nakai's (1995， 1996) shrer inegrans ere n able islse.
6 isussn
ha kin f ynaial ehanis ainains his lng-er sabiliy f he pneary syse? e an ieiaely hink f ajr feaures ha ay be respnsible fr he lng-er sabiliy. firs， here see be n signifian ler-rer resnanes (ean n an seur) beeen any pair ang he nine pnes. jupier an saurn are lse a 5:2 ean n resnane (he faus ‘grea inequaliy’)， bu n jus in he resnane ne. higher-rer resnanes ay ause he hai naure f he pneary ynaial n， bu hey are n s srng as esry he sable pneary n ihin he lifeie f he real sr syse. he sen feaure， hih e hink is re ipran fr he lng-er sabiliy f ur pneary syse， is he ifferene in ynaial isane beeen erresrial an jian pneary subsyses (i anikaa 1999， 2001). hen e easure pneary separans by he uual hill raii (r_)， separans ang erresrial pnes are greaer han 26rh， hereas hse ang jian pnes are less han 14rh. his ifferene is irely ree he ifferene beeen ynaial feaures f erresrial an jian pnes. erresrial pnes hae saller asses， shrer rbial pers an ier ynaial separan. hey are srngly perurbe by jian pnes ha hae rr asses， lnr rbial pers an narrer ynaial separan. jian pnes are n perurbe by any her assie bies.
he presen erresrial pneary syse is sill being isurbe by he assie jian pnes. heer， he ie separan an uual ineran ang he erresrial pnes reners he isurbane ineffeie; he egree f isurbane by jian pnes is (ej)(rer f agniue f he eenriiy f jupier)， sine he isurbane ause by jian pnes is a fre siln haing an apliue f (ej). heighening f eenriiy， fr exaple (ej)0.05， is far fr suffiien prke insabiliy in he erresrial pnes haing suh a ie separan as 26rh. hus e assue ha he presen ie ynaial separan ang erresrial pnes (> 26rh) is prbably ne f he s signifian nins fr ainaining he sabiliy f he pneary syse er a 109-yr ie-span. ur eaile analysis f he renship beeen ynaial isane beeen pnes an he insabiliy ie-sale f sr syse pneary n is n n-ging.
alhugh ur nuerial inegrans span he lifeie f he sr syse， he nuber f inegrans is far fr suffiien fill he iniial phase spae. i is neessary perfr re an re nuerial inegrans nfir an exaine in eail he lng-er sabiliy f ur pneary ynais.
——以上文段引自 i， .& anikaa， k. lng-er inegrans an sabiliy f pneary rbis in ur sr syse. n. n. r. asrn. s. 336， 4八3–500 (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，相关课题的文文献很少，那些论文下载一篇要九美元（《naure》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。
