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以下是文章内容：

ongter tegrations and stabiitypary orbitsouyste

abstract

we present the resutsvery ongter nurica tegrationspary orbita otions over 109tispans cdg apuick spectionour nurica data shows that the pary otion，eastour sipe dynaica ode， seesbe quite stabe even over this very ong tispaoser ookthe owestfrequency oscition owpass fiter showsthe potentiay diffive characterterrestria pary otion， especiay thatrcury the behaviourthe eentricityrcuryour tegrationsquaitativey siirthe resuts fro jacques skar's secur perturbation theoryeax 035 overgyr however， there areapparent secur creaseseentricitycationany orbita eentsthe ps，be reveaedsti onrter nurica tegrationshaverforpetria tegrations cdg otionsthe outer fiveover the duration 5 x 1010the resut dicates that the three ajor renancesthe neptune–pto syste have been ataed over the 1011yr tispan

1 troduction

11defitionthe probe

the questionthe stabiityouyste has been debated over severa hundred years， sce the erane has attracted any fao atheaticians over the years and has pyeentra roethe deveopntnonear dynaics and chaos theory however，do not yet havefite answerthe questionwhether ouystestabenot thispartesutthe fact that the defitionthe ter stabiityvague whenis retionthe probepary otionthyste actuayis not easygivear， rigoro and physicay angfu defitionthe stabiityouyste

aong any defitionsstabiity， hereadopt thedefition gdan 1993: actuay thisnoefitionstabiity， butstabiitydefystebeg unstabe wheose enunter ours wherethe syste， startg frerta itia nfiguration chabers， pap boss 1996 ito apapapap tanikawa 1999 a systedefedexperiencose enunter wheodies approach one another withareathehi radi otherdefedbeg stabe henceforwardstate that our pary systedynaicay stabeno cose enunter happens durg the aouyste， about 5 gyr cidentay， this defitionbe repcedonewhichourrenceany orbita crossg beeen eithera pairps takes pce thisbecaeexperience thatorbita crossgvery ikeyeada cose enunterpary and proary systes yoshaga， kokubo apapapap ako 1999urse this statent cannotsipy appiedsystes with stabe orbita renances suchthe neptune–pto syste

12previo studies and aisthis research

additionthe vaguenessthe nceptstabiity， the ouyste shoharacter typicadynaica chaos ssan apapapap1988， 1992 the caethis chaotic behaviournow party understoodbeesutrenance overppg urray apapapap hoan 1999 ecar， frank apapapap hoan 2001 however，woud require tegratg overensebepary systes cdg aps foeriod verg severagyrthoroughy understand the ongter evotionpary orbits， sce chaotic dynaica systes are characterizedtheir strong dependenceitia nditions

fro that potvieanythe previo ongter nurica tegrations cded ony the outer fivessan apapapap1988 koshita apapapap nakai 1996 thisbecae the orbita periodsthe outerareuch onr than thosethe ner fourthatis uch easierfoo foiven tegration periodpresent， the onst nurica tegrations pubishedjournas are thoseduncan apapapap issauer 1998 athough theiart was the effectpostasequencss ossthe stabiitypary orbits， they perford any tegrations vergto 1011of the orbita otionsthe four jovianthe itia orbita eents and assesps are theas thoseouysteduncan apapapap issauer's paper， but they decrease the assthe sun graduaytheir nurica experints thisbecae they nsider the effectpostasequencss ossthe paper nsequenty， they found that the crossg tiscaepary orbits， which cana typica dicatorthe stabiity tiscae，quite sensitivethe rateass decreasethe sun assthe suncoseits present vae， the jovianrea stabe over 1010 yr，perhaps onr duncan apapapap issaueerford four siir experintsthe orbita otionsevenvenneptune， which vepan109their experintsthe sevenare not yet prehensive， butsees that the terrestriaa rea stabe durg the tegration period， atag aost regur oscitions

on the other hand，his aurate seianaytica secur perturbation theory skar 1988， skar fds thand irregur variations can appearthe eentricities and cationsthe terrestria ps， especiayrcury and arsa tiscaesevera 109skar 1996 the resutsskar's secur perturbation theory shoudnfird and vestigatedfuy nurica tegrations

this paper ary resutssix ongter nurica tegrationsapary orbits， verpansevera 109 yr， ando other tegrations verpan 5 x 1010the tota epsedfoegrationsore than 5 yr， g severa dedicated pcs and workstations onethe fundanta ncsionsour ongter tegrationsthayste pary otion seesbe stabetersthestabiity ntioned above，east oveispan 4 gyr actuay，our nurica tegrations the syste ore stabe than whatdefedthestabiity criterion: not ony didcose enunter happen durg the tegration period，the pary orbita eents have been nfednarrow region both and frequency doa， though pary otions are stochastic sce the purposethis paperto exhibit and overview the resutsour ongter nurica tegrations，shope figuresevidencethe very ongter stabiityr syste pary otion for readers ore specific and deeper terestsour nurica resuts，have prepareebpa aess ， whereshow raents， their owpass fitered resuts， variationdeunay eents and angur ontu deficit， and resutsour sipe ti–frequency anaysisaour tegrations

section 2

iefy exp our dynaica ode， nurica thod and itia nditions our tegrations section 3devoteda descriptionthe quick resutsthe nurica tegrations very ongter stabiityr syste pary otionapparent bothpary positions and orbita eentough estiationnurica errorsa given section 4 goestiscsionthe onstter variationpary orbit owpass fiter and cdeiscsionangur ontu deficitsection 5，presenetnurica tegrations for the outer fivethat spansx 1010 sectiondiscs the ongter stabiitythe pary otion and its possibe cae

2 descriptionthe nurica tegrations

本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了也不一定能成功显示。

23 nurica thod

we utiizendorder pecticas ouegration thooan 1991 koshita， yoshida apapapap nakai 1991 witpecia startup procedurereduce the truncation errorange variabes， startsaha apapapap treae 1992， 1994

the stepsize for the nurica tegrations8 d throughouegrationstheps n1，2，3， whichabout 111the orbita periodthe neroscuryfor the deterationstepsize，party foow the previo nurica tegrationapsssan apapapap1988，d and saha apapapap treae 1994， 22532 drounded the decia partthe their stepsizes8ake the stepsiztipe2orderreduce the auutionroundoff errorthe putation processesretionthis， p hoan 1991 perford nurica tegrationsthe outer five pary orbithe sypecticwittepsize400 d， 110the orbita periodjupiter their resut seesbe aurate enough， which party jtifies our thoddeterg the stepsize however， sce the eentricityjupiter 005uch saer than thatrcury 02， e care pare these tegrations sipytersstepsizes

the tegrationthe outer fivef，fixed the stepsize400 d

we adopt gas' f anunctionsthe sypectictother with the thirdorder haey thod danby 1992a ver for keper equations the nuberaxiu iterationssethaey's thod15， but they never reached the axiuanyour tegrations

the tervathe data output200 000 d 547for the cacutionsaps n1，2，3， and about 8000 000 d903for the tegrationthe outer fivef

athoughoutput fiterg was done erica tegrations wereprocess，appiewpass fiterthe raw orbita data after petehe cacutions see sectionfor ore detai

24 error estiation

241 retive errorstota energy and angur ontu

aordgonethe basic propertiessypectic tegrators， which nserve the physicay nservative quantitiestota orbita energy and angur ontu， our ongter nurica tegrations seehave been perforrrors the averad retive errorstota energy 109 andtota angur ontu 1011 have reaed neary nstant throughout the tegration period fig 1 the specia startup procedure，start， woud have reduced the averad retive errortota energyabout one orderagnitudeore

retive nurica errorthe tota angur ontu δaa0 and the tota energy δee0our nurica tegrationsn 1，2，3， whereandare the abte chanthe tota energy and tota angur ontu， respectivey， ande0anda0are their itia vaes the horizonta unitgyr

note that different operatg systes， different atheatica i

aries， and different hardware architectures resutdifferent nurica errors， through the variationsroundoff error handg and nurica agorithsthe upper panefig 1，can regnize this situationthe secur nurica errorthe tota angur ontu， which shoudrigoroy preservedto achee precision

242 errorpary ongitudes

sce the sypectic aps preserve tota energy and tota angur ontunbody dynaica systes herenty we， the degreetheir preservationnota good asurethe auracynurica tegrations， especiaya asurethe positiona errorps，the errorpary ongitudesestiate the nurica errorthe pary ongitudes，the foog procedures pared the resutoungter tegrationest tegrations， uch shorter periods but uch higher auracy than thegrations for this purpose，a uch ore aurate tegration wittepsize0125 d 164thegrations spanng 3 x 105 yr， starttia nditions thetegrationnsider that this test tegration provideswitseudotrue tionpary orbita evotion next， pare the test tegratioegration，for the period3 x 105 yr，seifference anoaiesthe earth beeen thegrations052° the casethetegration this difference canextrapotedthe vae 8700°， aboutrotationsearth after 5 gyr， sce the errorongitudes creases eary ethe sypecticsiiry， the ongitude errorpto canestiated12° this vae for ptouch better than the resutkoshita apapapap nakai 1996 where the differenceestiated60°

3 nurica resuts – i gncethe raw data

this section

iefy revie stabiitypary orbita otion throughsnapshotsraerica data the orbita otionps dicates ongter stabiityof our nurica tegrations:orbita crossgs nor cose enunters beeen any pairps took pce

31 nera descriptionthe stabiitypary orbits

first，

iefy ookthe nera characterthe ongter stabiitypary orbits our terest here foces particurythe ner four terrestriafor which the orbita tiscaes are uch shorter than thosethe outer fiveascan see ceary fro the pnar orbita nfigurations shownfigs 2 and 3， orbita positionsthe terrestriadiffer itte beeen the itia andparteach nurica tegration， which spans severa gyr thees denotg the present orbitstheie aostof dots eventheparttegrationnhis dicates that throughout the entire tegration period the aost regur variationspary orbita otion rea neary theas they arepresent

vertica viewthe four ner pary orbits fro thxis directionthe itia andpartsthe tegrationsn1 the axes units arethepnesetthe variant pner syste tota angur ontua the itia part ofn 1 000547yrb thepart ofn 1 49339 x849886yrc the itia partn1 t 000547 x 109 yrd thepart ofn139180 x939727each pane， a tota23 684 pots are potted withtervaabout 2190over 547 x 107eseach pane denote the present orbitsthe four terrestriataken fro de245

the variationeentricities and orbita cations for the ner four the itia andpartthe tegration n 1shownfig 4expected， the characterthe variationpary orbita eents does not differ significanty beeen the itia andparteach tegration，east for ven， earth and ars the eentsrcury， especiay its eentricity， seechana significant extent thisparty becae the orbita tiscaethe pthe shortesta the ps， which eadsa ore rapid orbita evotion than otherthe nerost pbe neareststabiity this resut appearsbe agreent with skar's 1994， 1996 expectations thand irregur variations appearthe eentricities and cationsrcurya tiscaesevera 109however， the effectthe possibe stabiitythe orbitrcurynot fatay affect the goba stabiitythe whoe pary systeto theassrcury ention

iefy the ongter orbita evotionrcury tersection 4 g owpass fitered orbita eents

the orbita otionthe outer fivesees rigoroite regur over this tispan seection 5

32 ti–frequency aps

athough the pary otion exhibits very ongter stabiity defedthe nonexistencecose enunter events， the chaotic naturepary dynaics can chan the oscitory period and apitudepary orbita otion graduay over such ong tispans even such sight fctuationsorbita variationthe frequency doa， particurythe caseearth， can potentiay havignificant effectits surface ciate syste througion variationberr 1988

to giveoverviewthe ongter chanperiodicitypary orbita otion， any fast fourier transforations ffts aong theaxis， and superposed the resutg periodgrasdraensiona ti–frequency aps the specific approachdrae–frequency apsthis papervery sipe – uch siper than the waveet anaysisskar's 1990， 1993 frequency anaysis

divide the owpass fitered orbita dataany fragntstheength the engtheach data segnt shouda utipe2orderappy the fft

each fragntthe dataoverppg part: for exape， when the ith data begs fro tti and endstti t， the next data segnt rans froδttit， where δttntue this division untireacerta nuber nwhicht reaches the tota tegration ength

we appyffteachthe data fragnts， and obtrequency diagras

each frequency diagra obtaed above， the strengthperiodicity canrepceda greyscaeour chart

the repcent， and nneche greyscaeour chartsone graph for each tegration the horizonta axisthese new graphs shoudthe ti，the startg tiseach fragntdata ti， where i 1，， n the vertica axis represents the periodfrequencythe oscitionorbita eents

we have adoptedfft becaeits overg speed， sce the aountnurica databe deposedfrequency ponentsterribysevera tensgbytes

a typica exapethe ti–frequencycreatedthe above proceduresshowngreyscae diagrafig 5， which shows the variationperiodicitythe eentricity and cationearth2 tegrationfig 5， the dark area shows thatthedicatedthe vaethe abscissa， the periodicity dicatedthe ordatestronr thanthe ighter area aroundwe can regnize fro thisthat the periodicitythe eentricity and cationearth ony chans sighty over the entire period veredthe n 2 tegration this neary regur trendquaitativey the other tegrations and for other ps， athough typica frequencies differand eenteent

42 ongter exchanorbita energy and angur ontu

we cacute very ongperiodic variation and exchanpary orbita energy and angur ontitered deunay eents ， g， nre equivaentthe pary orbita angur ontu and its vertica ponent per unit ass retedthe pary orbita energer unit asseμ222the systepetey ear， the orbita energy and the angur ontueach frequencnstant nonearitythe pary syste can caeexchanenergy and angur ontuthe frequency doa the apitudethe owestfrequency oscition shoud creasethe systeunstabe and

eaks down graduay however， sucyptostabiitynot proentour ongter tegrations

fig 7， the tota orbita energy and angur ontuthe four nerand aps are shown for tegration n 2 the upper three panes show the ongperiodic variationtota energy denoted ase e0， tota angur ontug0， and the vertica ponenth0the ner fourcacuted fro the owpass fitered deunay eentse0， g0，denote the itia vaeseach quantity the abte difference fro the itia vaespottedthe panes the ower three paneseach figure showee0，gg0 andhh0the totanethe fctuation shownthe ower panesvirtuay entireesutthe assive jovian ps

parg the variationsenergy and angur ontuthe ner fourand aps，is apparent that the apitudesthosethe nerare uch saer than thoseaps: the apitudesthe outer fiveare uchthan thosethe nerthis does notthat the ner terrestria pary subsysteore stabe than the outer one: thissipesutthe retive sanessthe assesthe four terrestriapared with thosethe outer joviananother thgnoticethat the ner pary subsystebee unstabe ore rapidy than the outer one becaeits shorter orbita tiscaes this canseenthe panes denoted asner 4fig 7 where the onrperiodic and irregur oscitions are ore apparent thanthe panes denoted astota 9 actuay， the fctuationsthener 4 panes arextenta resutthe orbita variationthe rcury however，cannot negect the ntribution fro other terrestria ps，weseesubsequent sections

44 ongter upgsevera neighbourairs

etseedividua variationspary orbita energy and angur ontu expressedthe owpass fitered deunay eents figsandsho evotionthe orbita energyeacnd the angur ontu1 andtegrationsnotice that for apparent pairstersorbita energy and angur ontu exchanparticur， ven and earth akypica pairthe figures， they show negative rretionsexchanenergy and positive rretionsexchanangur ontu the negative rretionexchanorbita energy ans thatcosed dynaica systetersthe orbita energy the positive rretionexchanangur ontu ans that the oare siutaneoy under certa ongter perturbations candidates for perturbers are jupiter and saturn afig 11，can see that ars showositive rretionthe angur ontu variationthe ven–earth syste rcury exhibits certa negative rretionsthe angur ontu vers the ven–earth syste， sbeaction caedthe nservationangur ontuthe terrestria pary subsyste

itnot cearthe ont why the ven–earth pair exhibitegative rretionenergy exchan anositive rretionangur ontu exchanpossiby exp this through observg the nera fact that there aresecur terspary seiajor axesto sendorder perturbation theories

oup cence 1961 boaetti apapapap pucao 1998 this ans that the pary orbita energy whichdirecty retedthe seiajor axightuch ess affectedperturbgthanthe angur ontu exchan which retese hence， the eentricitiesven and earth candisturbed easiyjupiter and saturn， which resutspositive rretionthe angur ontu exchanthe other hand， the seiajor axesven and earth are ess ikerbedthe jovianth the energy exchanbe iited ony with the ven–earth pair， which resutsnegative rretionthe exchanorbita energythe pair

as for the outer jovian pary subsyste， jupiter–saturn and uran–neptune seeake dynaica pairs however， the strengththeir upgnotstrong pared with thatthe ven–earth pair

5x 1010yr tegrationsouter pary orbits

sce the jovian pary asses are uchthan the terrestria pary asses，treat the jovian pary systean dependent pary systetersthe studyits dynaica stabiity hence，addepetria tegrations that spanx 1010 yr， cdg ony the outer fivethe four jovianps pto the resuts exhibit the rigoro stabiitythe outer pary syste over this ong tispan orbita nfigurations fig 12， and variationeentricities and cations figshow this very ongter stabiitythe outer five both theand the frequency doas athoughdo not shoaps here， the typica frequencythe orbita oscitionpto and the other outeris aost nstant durg these very ongter tegration periods， onstratedthe ti–frequency apsour webpa

thesegrations， the retive nurica errorthe tota energy was 106 and thatthe tota angur ontu was 1010

51 renancesthe neptune–pto syste

koshita apapapap nakai 1996 tegrated the outer five pary orbits over x 109 they found that four ajor renances beeen neptune and pto are ataed durg the whoe tegration period， and that the renancesbe thaesthe stabiitythe orbitpto the ajor four renances foundprevio research arefoowsthe foog description， denotes theongitude，the ongitudethe ascendg node and the ongitudeperiheion subscriptnenote pto and neptune

an otion renance beeen neptune and pto 3:2 the critica argunt3 p 2i

ates around 180° pitudeabout 80° an

ation periodabout 2 x 104 yr

the arguntperiheionpto wpθ2pp i

ates around 90° witeriodaboutx 106the doant periodic variationsthe eentricity and cationpto are synchronized with the i

ationits arguntperiheion thisanticipatedthe secur perturbation theory nstructedkozai 1962

the ongitudethe nodepto referredthe ongitudethe nodeneptune，θ3pn， circutes and the periodthis circutionequathe periodθ2 i

ationzero，the ongitudesascendg nodesneptune and pto overp， the cationpto bees axiu， the eentricity beesand the arguntperiheion bees 90°180°， the cationpto bees iu， the eentricity bees axiu and the arguntperiheion bees 90° aga pap benn 1971 anticipated this typerenance， ter nfirdini， nobii apapapap carpo 1989

an argunt θ4pn3i

ates around 180° witng period，x 108 yr

our nurica tegrations， the renances i–iii are ataed， and variationthe critica argunts θ1，θ2，θ3 rea siir durg the whoe tegration period figs 14–16however， the fourth renanceappearsbe different: the critica arguntaternates i

ation and circution over a 1010yr tiscae figthisan terestg fact that koshita apapapap nakai's 1995， 1996 shorter tegrations were not abediscose

6 discsion

whatof dynaica chanis atas this ongter stabiitythe pary syste ediatey thktajor features thatbe responsibe for the ongter stabiity first， there seebesignificant owerorder renancesotion and secur beeen any pair aong theps jupiter and saturn are cosea 5:2otion renance the fao great equaity， but not the renance zone higherorder renancescae the chaotic naturethe pary dynaica otion， but they are notstrongto destroy the stabe pary otion ethe reyste the send feature， which ore iportant for the ongter stabiityour pary syste，the differencedynaica distance beeen terrestria and jovian pary subsystes ito apapapap tanikawa 1999， 2001 easure pary separationsthe utuaradii r， separations aong terrestriaare greater than 26rh， ong jovianare ess than 14rh this differencedirecty retedthe difference betica featuresterrestria and jovianterrestriahave saer asses， shorter orbita periods and ica separation they are strongy perturbedjovianthat haveasses， onr orbita periods and narroica separation jovianare not perturbedany other assive bodies

the present terrestria pary systesti beg disturbedthe assive jovianhowever， the wide separation and utua teraction aong the terrestriarenders the disturbance effective the degreedisturbancejovianis oejorderagnitudethe eentricityjupiter， sce the disturbance caedjovianiorced oscition havgapitudeoej heightengeentricity， for exape oej005，far fro sufficientprovoke stabiitythe terrestriahavg sucide separation26rhe that the present ica separation aong terrestriaapapapgt 26rhprobaby onethe ost significant nditions for atag the stabiitythe pary syste over a 109yr tispan our detaied anaysisthe retionship betica distance beeenand the stabiity tiscaer syste pary otionnow ongog

athough our nurica tegrations span the ifetithyste， the nubertegrationsfar fro sufficientfi the itia phase spaceis necessaryperfor ore and ore nurica tegrationsnfir and exaedetai the ongter stabiityour pary dynaics

以上文段引自 ito， tapapap tanika tegrations and stabiitypary orbitsouystenostron c 336， 4–500 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。

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