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以下是文章内容：
long-terntegrationsandstabilityofplaaryorbitsinoursolarsystebr />
abstract
wepresenttheresultsofverylong-teruricalintegrationsofplaaryorbitaltionsover109-yrti-spansinc露dingallnineplas.aquickinspectionofournuricaldatashowsthattheplaarytion，atleastinoursiledynacaldel，seetobequitestableevenoverthisverylongti-span.acloserlookatthelowest-frequencyoscillationsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestrialplaarytion，especiallythatofrcury.entricityofrcuryinourintegrationsisqualitativelysilartotheresultsfroacqueslaskarssecularperturbationtheory(e.g.ex～0.35over～±4gyr).however，entricityorinclinationinanyorbitalelentsoftheplas，whichyberevealedbystilllonger-teruricalintegrations.wehavealsoperfordacoupleoftrialintegrationsinc露dingtionsoftheouterfiveplasoverthedurationof±5x1010yr.theresultindicatesthatthethreejorresonancesintheneptune–p露tosysteavebeenintainedoverthe1011-yrti-span.
1introduction
1.1definitionoftheproblebr />
thequestionofthestabilityofoursolarsysteasbeendebatedoverseveral混dredyears，sincetheeraofnewton.theprobleasattractednyfaustheticiansovertheyearsandhasplayedacentralroleinthedevelopntofnon-lineardynacsandchaostheory.however，wedonotyethaveadefiniteanswertothequestionofwhetheroursolarsystesstableornot.thisispartlyaresultofthefactthatthedefinitionoftheterstability’isvaguewhenitisusedinrelationtotheproblefplaarytioninthesolarsysteactuallyitisnoteasytogiveaclear，rigorousandphysicallyaningfuldefinitionofthestabilityofoursolarsyste
angnydefinitionsofstability，hereweadoptthehilldefinition(gladn1993):actuallythisisnotadefinitionofstability，butofinstability.urssowhereinthesystestarfrocertaininitialconfiguration(chaers，wetherill&a;a;a;a;boss1996;ito&a;a;a;a;tanikawa1999).asystesdefinedasexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinanareaofthelargerhillradius.otherwisethesystesdefinedasbeingstable.henceforwardwestatethatourplaarysystesdynacallystableifnocloseencounterhappensduringtheageofoursolarsysteabout±5gyr.incidentally，urrenceofanyorbitalcrossingbetweeneitherofapairofplastakesplace.thisisbecauseweknowfroxperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounterinplaaryandprotoplaarysyste(yoshinaga，kokubo&a;a;a;a;kino1999).ofcoursethisstatentcannotbesilyappliedtosystewithstableorbitalresonancessuchastheneptune–p露tosyste
1.2previousstudiesandaiofthisresearch
inadditiontothevaguenessoftheconceptofstability，theplasinoursolarsystehowacharactertypicalofdynacalchaos(sussn&a;a;a;a;wisdo988，1992).thecauseofthischaoticbehaviourisnowpartl玉nderstoodasbeingaresultofresonanceoverlapping(rray&a;a;a;a;holn1999;lecar，franklin&a;a;a;a;holn2001).however，itwouldrequireintegraoveranenseleofplaarysysteinc露dingallnineplasforaperiodcoveringseveral10gyrtothoroughl玉nderstandthelong-tervo露tionofplaaryorbits，sincechaoticdynacalsystearecharacterizedbytheirstrongdependenceoninitialconditions.
frohatpointofview，nyofthepreviouslong-teruricalintegrationsinc露dedonlytheouterfiveplas(sussn&a;a;a;a;wisdo988;kinoshita&a;a;a;a;nakai1996).thisisbecausetheorbitalperiodsoftheouterplasaresochlongerthanthoseoftheinnerfourplasthatitischeasiertofollowthesysteoragivenintegrationperiod.atpresent，thelongestnuricalintegrationspublishedinjournalsarethoseofduncan&a;a;a;a;lissauer(1998).althoughtheirintargetwastheeffectofpost-in-sequencesolarsslossonthestabilityofplaaryorbits，theyperfordnyintegrationscoveringupto～1011yroftheorbitaltionsofthefourjovianplas.theinitialorbitalelentsandssesofplasarethesaasthoseofoursolarsystenduncan&a;a;a;a;lissauerspaper，buttheydecreasethessofthesungraduallyintheirnuricalexperints.thisisbecausetheyconsidertheeffectofpost-in-sequencesolarsslossinthepaper.consequently，theyfoundthatthecrossingti-scaleofplaaryorbits，whichcanbeatypicalindicatoroftheinstabilityti-scale，isquitesensitivetotherateofssdecreaseofthesun.whenthessofthesunisclosetoitspresentva露e，thejovianplasreinstableover1010yr，orperhapslonger.duncan&a;a;a;a;lissaueralsoperfordfoursilarexperintsontheorbitaltionofsevenplas(venustoneptune)，whichcoveraspanof～109yr.theirexperintsonthesevenplasarenotyeprehensive，butitseethattheterrestrialplasalsoreinstableduringtheintegrationperiod，intainingalstregularoscillations.
ontheotherhand，uratese-analyticalsecularperturbationtheory(laskar1988)，entricitiesandinclinationsoftheterrestrialplas，especiallyofrcuryandrsonati-scaleofseveral109yr(laskar1996).theresultsoflaskarssecularperturbationtheoryshouldbeconfirdandinvestigatedbyfullynuricalintegrations.
inthispaperwepresentprelinaryresultsofsixlong-teruricalintegrationsonallnineplaaryorbits，coveringaspanofseveral109yr，andoftwootherintegrationscoveringaspanof±5x1010yr.thetotalelapsedtiforallintegrationsisrethan5yr，usingseveraldedicatedpcsandworkstations.oneofthefundantalconc露sionsofourlong-terntegrationsisthatsolarsystelaarytionseetobestableinterofthehillstabilityntionedabove，atleastoverati-spanof±4gyr.actually，inournuricalintegrationsthesysteasfarrestablethanwhatisdefinedbythehillstabilitycriterion:notonlydidnocloseencounterhappenduringtheintegrationperiod，butalsoalltheplaaryorbitalelentshavebeenconfinedinanarrowregionbothintiandfrequencydoin，thoughplaarytionsarestochastic.sincethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-teruricalintegrations，weshowtypicalexalefiguresasevidenceoftheverylong-tertabilityofsolarsystelaarytion.forreaderswhohaverespecificanddeeperinterestsinournuricalresults，wehavepreparedawebpage(aess)，whereweshowraworbitalelents，theirlow-passfilteredresults，variationofdelaunayelentsandangularntueficit，andresultsofoursileti–frequencyanalysisonallofourintegrations.
insection2webrieflyexplainourdynacaldel，nuricalthodandinitialconditionsusedinourintegrations.section3isdevotedtoadescriptionofthequickresultsofthenuricalintegrations.verylong-tertabilityofsolarsystelaarytionisapparentbothinplaarypositionsandorbitalelents.aroughestitionofnuricalerrorsisalsogiven.section4goesontoadiscussionofthelongest-terariationofplaaryorbitsusingalow-passfilterandinc露desadiscussionofangularntueficit.insection5，wepresentasetofnuricalintegrationsfortheouterfiveplasthatspans±5x1010yr.insection6wealsodiscussthelong-tertabilityoftheplaarytionanditspossiblecause.
2descriptionofthenuricalintegrations
（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。）
2.3nuricalthod
weutilizeasecond-orderwisdoholnsylecticpasourinintegrationthod(wisdoa;a;a;a;holn1991;kinoshita，yoshida&a;a;a;a;nakai1991)withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘wartart’(saha&a;a;a;a;treine1992，1994).
thestepsizeforthenuricalintegrationsis8dthroughoutallintegrationsofthenineplas(n±1，2，3)，whichisabout1/11oftheorbitalperiodoftheinnerstpla(rcury).asforthedeternationofstepsize，wepartlyfollowthepreviousnuricalintegrationofallnineplasinsussn&a;a;a;a;wisdo1988，7.2d)andsaha&a;a;a;a;treine(1994，225/32d).ulationofround-offerrorinthputationprocesses.inrelationtothis，wisdoa;a;a;a;holn(1991)perfordnuricalintegrationsoftheouterfiveplaaryorbitsusingthesylecticpwithastepsizeof400d，1/10.83oftheorbitalperiodofjupiter.urateenough，whichpartlyjustifiesourthodofdeterningthestepsize.however，entricityofjupiter(～0.05)ischsllerthanthatofrcury(～0.2)，weneedsocarewhenwparetheseintegrationssilyinterofstepsizes.
intheintegrationoftheouterfiveplas(f±)，wefixedthestepsizeat400d.
weadoptgaussfandgfunctionsinthesylecticptogetherwiththethird-orderhalleythod(danby1992)asasolverforkeplerequations.thenuerofxiterationswesetinhalleysthodis15，buttheyneverreachedthexinanyofourintegrations.
theintervalofthedataoutputis200000d(～547yr)forthecalculationsofallnineplas(n±1，2，3)，andabout8000000d(～21903yr)fortheintegrationoftheouterfiveplas(f±).
althoughnooutputfilteringwasdonewhenthenuricalintegrationswereinprocess，weappliedalow-passfiltertotheraworbitaldataafterwehapletedallthecalculations.seesection4.1forredetail.
2.4errorestition
2.4.1relativeerrorsintotalenergyandangularntubr />
aordingtooneofthebasicpropertiesofsylecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangularntu，ourlong-teruricalintegrationsseeohavebeenperfordwithverysllerrors.theaveragedrelativeerrorsoftotalenergy(～10?9)andoftotalangularntu～10?11)havereinednearlyconstantthroughouttheintegrationperiod(fig.1).thespecialstartupprocedure，wartart，wouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofgnitudeorre.
relativenuricalerrorofthetotalangularntua/a0andthetotalenergyδe/e0inournuricalintegrationsn±1，2，3，whereδeandδaaretheabso露techangeofthetotalenergyandtotalangularnturespectively，ande0anda0aretheirinitialva露es.thehorizonta露nitisgyr.
notethatdifferentoperasyste，differenttheticallibraries，anddifferenthardwarearchitecturesresultindifferentnuricalerrors，throughthevariationsinround-offerrorhandlingandnuricalalgorith.intheupperpaneloffig.1，wecanrecognizethissituationinthesecularnuricalerrorinthetotalangularntuwhichshouldberigorouslypreserveduptochine-eprecision.
2.4.2errorinplaarylongitudes
sincethesylecticpspreservetotalenergyandtotalangularntufn-bodydynacalsysteinherentlywell，uracyofnuricalintegrations，especiallyasaasureofthepositionalerrorofplas，i.e.theerrorinplaarylongitudes.toestitethenuricalerrorintheplaarylongitudes，weperfordthefollowingprocedures.wparedtheresultofourinlong-terntegrationswithsotestintegrations，uracythantheinintegrations.forthispurpose，urateintegrationwithastepsizeof0.125d(1/64oftheinintegrations)spanning3x105yr，starwiththesainitialconditionsasinthen?1integration.weconsiderthatthistestintegrationprovidesuswitha‘pseudo-true’so露tionofplaaryorbitalevo露tion.next，wparethetestintegrationwiththeinintegration，n?1.fortheperiodof3x105yr，weseeadifferenceinananoliesoftheearthbetweenthetwointegrationsof～0.52°(inthecaseofthen?1integration).thisdifferencecanbeextrapolatedtotheva露e～8700°，about25rotationsofearthafter5gyr，sincetheerroroflongitudesincreaseslinearlywithtiinthesylecticp.silarly，thelongitudeerrorofp露tocanbeestitedas～12°.thisva露eforp露toischbetterthantheresultinkinoshita&a;a;a;a;nakai(1996)wherethedifferenceisestitedas～60°.
3nuricalresults–i.glanceattherawdata
inthissectionwebrieflyreviewthelong-tertabilityofplaaryorbitaltionthroughsosnapshotsofrawnuricaldata.theorbitaltionofplasindicateslong-tertabilityinallofournuricalintegrations:noorbitalcrossingsnorcloseencountersbetweenanypairofplastookplace.
3.1generaldescriptionofthestabilityofplaaryorbits
first，webrieflylookatthegeneralcharacterofthelong-tertabilityofplaaryorbits.ourinterestherefocusesparticularlyontheinnerfourterrestrialplasforwhichtheorbitalti-scalesarechshorterthanthoseoftheouterfiveplas.aswecanseeclearlyfroheplanarorbitalconfigurationsshowninfigs2and3，orbitalpositionsoftheterrestrialplasdifferlittlebetweentheinitialandfinalpartofeachnuricalintegration，whichspansseveralgyr.thesolidlinesdenothepresentorbitsoftheplasliealstwithintheswarfdotseveninthefinalpartofintegrations(b)and(d).thisindicatesthatthroughouttheentireintegrationperiodthealstregularvariationsofplaaryorbitaltionreinnearlythesaastheyareatpresent.
verticalviewofthefourinnerplaaryorbits(frohez-axisdirection)attheinitialandfinalpartsoftheintegrationsn±1.theaxesunitsareau.thexy-planeissettotheinvariantplaneofsolarsysteotalangularntu(a)theinitialpartofn+1(t=0to0.0547x109yr).(b)thefinalpartofn+1(t=4.9339x108to4.9886x109yr).(c)theinitialpartofn?1(t=0to?0.0547x109yr).(d)thefinalpartofn?1(t=?3.9180x109to?3.9727x109yr).ineachpanel，atotalof23684pointsareplottedwithanintervalofabout2190yrover5.47x107yr.solidlinesineachpaneldenotethepresentorbitsofthefourterrestrialplas(takenfroe245).
entricitiesandorbitalinclinationsfortheinnerfourplasintheinitialandfinalpartoftheintegrationn+1isshowninfig.4.asexpected，thecharacterofthevariationofplaaryorbitalelentsdoesnotdiffersignificantlybetweentheinitialandfinalpartofeachintegration，atleastforvenus，earthandrs.theelentsofrcury，entricity，seeochangetoasignificantextent.thisispartlybecausetheorbitalti-scaleoftheplaistheshortestofalltheplas，whichleadstoarerapidorbitalevo露tionthanotherplas;theinnerstplaybenearesttoinstability.thisresultappearstobeinsoagreentwithlaskars(1994，1996)entricitiesandinclinationsofrcuryonati-scaleofseveral109yr.however，theeffectofthepossibleinstabilityoftheorbitofrcuryynotfatallyaffecttheglobalstabilityofthewholeplaarysystewingtothesllssofrcury.wewillntionbrieflythelong-terrbitalevo露tionofrcurylaterinsection4usinglow-passfilteredorbitalelents.
theorbitaltionoftheouterfiveplasseerigorouslystableandquiteregularoverthisti-span(seealsosection5).
3.2ti–frequencyps
althoughtheplaarytionexhibitsverylong-tertabilitydefinedasthenon-existenceofcloseencounterevents，thechaotatureofplaarydynacscanchangetheoscillatoryperiodandalitudeofplaaryorbitaltiongraduallyoversuchlongti-spans.evensuchslightf露ctuationsoforbitalvariationinthefrequencydoin，particularlyinthecaseofearth，canpotentiallyhaveasignificanteffectonitssurfaceclitesystehroughsolarinsolationvariation(cf.berger1988).
togiveanoverviewofthelong-terhangeinperiodicityinplaaryorbitaltion，weperfordnyfastfouriertransfortions(ffts)alongthetiaxis，andsuperposedtheresulperiodgratodrawtwo-dinsionalti–frequencyps.thespecificapproachtodrawingtheseti–frequencypsinthispaperisverysile–chsilerthanthewaveletanalysisorlaskars(1990，1993)frequencyanalysis.
dividethelow-passfilteredorbitaldataintonyfragntsofthesalength.thelengthofeachdatasegntshouldbealtipleof2inordertoapplythefft.
eachfragntofthedatahasalargeoverlappingpart:forexale，whentheithdatabeginsfro=tiandendsatt=ti+t，thenextdatasegntrangesfroi+δt≤ti+δt+t，whereδt?t.wecontinuethisdivisionuntilwereachacertainnuernbywhichtn+treachesthetotalintegrationlength.
weapplyanffttoeachofthedatafragnts，andobtainnfrequencydiagra.
ineachfrequencydiagrabtainedabove，thestrengthofperiodicitycanbereplacedbyagrey-scale(orcolour)chart.
weperforhereplacent，andconnectallthegrey-scale(orcolour)chartsintoonegraphforeachintegration.thehorizontalaxisofthesenewgraphsshouldbetheti，i.e.thestartisofeachfragntofdata(ti，wherei=1，…，n).theverticalaxisrepresentstheperiod(orfrequency)oftheoscillationoforbitalelents.
wehaveadoptedanfftbecauseofitsoverwhelngspeed，sincetheauntofnuricaldatatobedposedintofrequencponentsisterriblyhuge(severaltensofgbytes).
atypicalexaleoftheti–frequencypcreatedbytheaboveproceduresisshowninagrey-scalediagrasfig.5，entricityandinclinationofearthinn+2integration.infig.5，thedarkareashowsthatatthetiindicatedbytheva露eontheabscissa，theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.entricityandinclinationofearthonlychangesslightlyovertheentireperiodcoveredbythen+2integration.thisnearlyregulartrendisqualitativelythesainotherintegrationsandforotherplas，althoughtypicalfrequenciesdifferplabyplaandelentbyelent.
4.2long-terxchangeoforbitalenergyandangularntubr />
wecalculateverylong-periodicvariationandexchangeofplaaryorbitalenergyandangularntusingfiltereddelaunayelentsl，g，h.gandhareequivalenttotheplaaryorbitalangularntunditsverticaponentperunitss.lisrelatedtotheplaaryorbitalenergyeperunitssase=?μ2/2l2.ifthesystepletelylinear，theorbitalenergyandtheangularntuneachfrequencybinstbeconstant.non-linearityintheplaarysysteancauseanexchangeofenergyandangularntunthefrequencydoin.thealitudeofthelowest-frequencyoscillationshouldincreaseifthesystesunstableandbreaksdowngradually.however，suchasytofinstabilityisnotpronentinourlong-terntegrations.
infig.7，thetotalorbitalenergyandangularntufthefourinnerplasandallnineplasareshownforintegrationn+2.theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedase-e0)，totalangularntug-g0)，andtheverticaponent(h-h0)oftheinnerfourplascalculatedfrohelow-passfiltereddelaunayelents.e0，g0，h0denotetheinitialva露esofeachquantity.theabso露tedifferencefroheinitialva露esisplottedinthepanels.thelowerthreepanelsineachfigureshowe-e0，g-g0andh-h0ofthetotalofnineplas.thef露ctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthessivejovianplas.
paringthevariationsofenergyandangularntuftheinnerfourplasandallnineplas，itisapparentthatthealitudesofthoseoftheinnerplasarechsllerthanthoseofallnineplas:thealitudesoftheouterfiveplasarechlargerthanthoseoftheinnerplas.thisdoesnotanthattheinnerterrestrialplaarysubsystesrestablethantheouterone:thisissilyaresultoftherelativesllnessofthessesofthefourterrestrialplaparedwiththoseoftheouterjovianplas.anotherthingwenoticeisthattheinnerplaarysubsysteaybeunstablererapidlythantheouteronebecauseofitsshorterorbitalti-scales.thiscanbeseeninthepanelsdenotedasinner4infig.7wherethelonger-periodicandirregularoscillationsarereapparentthaninthepanelsdenotedastotal9.actually，thef露ctuationsintheinner4panelsaretoalargeextentasaresultoftheorbitalvariationofthercury.however，wecannotneglectthecontributionfrotherterrestrialplas，aswewillseeinsubsequentsections.
4.4long-terouplingofseveralneighbouringplapairs
letusseesoindividualvariationsofplaaryorbitalenergyandangularntuxpressedbythelow-passfiltereddelaunayelents.figs10and11showlong-tervo露tionoftheorbitalenergyofeachplaandtheangularntunn+1andn?2integrations.wenoticethatsoplasforpparentpairsinteroforbitalenergyandangularntuxchange.inparticular，venusandearthkeatypicalpair.inthefigures，theyshownegativecorrelationsinexchangeofenergyandpositivecorrelationsinexchangeofangularntuthenegativecorrelationinexchangeoforbitalenergyansthatthetwoplasforcloseddynacalsystenteroftheorbitalenergy.thepositivecorrelationinexchangeofangularntueansthatthetwoplasaresiltaneousl玉ndercertainlong-tererturbations.candidatesforperturbersarejupiterandsaturn.alsoinfig.11，wecanseethatrsshowsapositivecorrelationintheangularntuariationtothevenus–earthsystercuryexhibitscertainnegativecorrelationsintheangularntuersusthevenus–earthsystewhichseetobeareactioncausedbytheconservationofangularntuntheterrestrialplaarysubsyste
itisnotclearatthentwhythevenus–earthpairexhibitsanegativecorrelationinenergyexchangeandapositivecorrelationinangularntuxchange.weypossiblyexplainthisthroughobservingthegeneralfactthattherearenosecularterinplaarysejoraxesuptosecond-orderperturbationtheories(cf.brouwer&a;a;a;a;clence1961;aletti&a;a;a;a;o1998).thisansthattheplaaryorbitalenergy(whichisdirectlyrelatedtothesejoraxisa)ghtbechlessaffectedbyperturbingplasthanistheangularntuxchange(whichrelatestoe).hence，entricitiesofvenusandearthcanbedisturbedeasilybyjupiterandsaturn，whichresultsinapositivecorrelationintheangularntuxchange.ontheotherhand，thesejoraxesofvenusandeartharelesslikelytobedisturbedbythejovianplas.thustheenergyexchangeybelitedonlywithinthevenus–earthpair，whichresultsinanegativecorrelationintheexchangeoforbitalenergyinthepair.
asfortheouterjovianplaarysubsystejupiter–saturnanduranus–neptuneseeokedynacalpairs.however，thestrengthoftheircouplingisnotasstronparedwiththatofthevenus–earthpair.
5±5x1010-yrintegrationsofouterplaaryorbits
sincethejovianplaaryssesarechlargerthantheterrestrialplaarysses，wetreatthejovianplaarysystesanindependentplaarysystenterofthestudyofitsdynacalstability.hence，weaddedacoupleoftrialintegrationsthatspan±5x1010yr，inc露dingonlytheouterfiveplas(thefourjovianplasp露sp露to).theresultsexhibittherigorousstabilityoftheouterplaarysysteverthislongti-span.orbitalconfigurations(fig.12)，entricitiesandinclinations(fig.13)showthisverylong-tertabilityoftheouterfiveplasinboththetiandthefrequencydoins.althoughwedonotshowpshere，thetypicalfrequencyoftheorbitaloscillationofp露toandtheotherouterplasisalstconstantduringtheseverylong-terntegrationperiods，whichisdenstratedintheti–frequencypsonourwebpage.
inthesetwointegrations，therelativenuricalerrorinthetotalenergywas～10?6andthatofthetotalangularntuas～10?10.
5.1resonancesintheneptune–p露tosystebr />
kinoshita&a;a;a;a;nakai(1996)integratedtheouterfiveplaaryorbitsover±5.5x109yr.theyfoundthatfourjorresonancesbetweenneptuneandp露toareintainedduringthewholeintegrationperiod，andthattheresonancesybetheincausesofthestabilityoftheorbitofp露to.thejorfourresonancesfoundinpreviousresearchareasfollows.inthefollowingdescription，λdenotestheanlongitude，Ωisthelongitudeoftheascendingnodeand?isthelongitudeofperihelion.subscriptspandndenotep露toandneptune.
antionresonancebetweenneptuneandp露to(3:2).thecriticalarguntθ1=3λp?2λn??plibratesaround180°withanalitudeofabout80°andalibrationperiodofabout2x104yr.
thearguntofperihelionofp露towp=θ2=?p?Ωplibratesaround90°withaperiodofabout3.8x106yr.entricityandinclinationofp露toaresynchronizedwiththelibrationofitsarguntofperihelion.thisisanticipatedinthesecularperturbationtheoryconstructedbykozai(1962).
thelongitudeofthenodeofp露toreferredtothelongitudeofthenodeofneptune，θ3=Ωp?Ωn，circulatesandtheperiodofthiscirculationisequaltotheperiodofθ2libration.whenθ3beszero，i.e.thelongitudesofascendingnodesofneptuneandp露tooverlap，theinclinationofp露tobesxientricitybesnindthearguntofperihelionbes90°.whenθ3bes180°，theinclinationofp露tobesnientricitybesxindthearguntofperihelionbes90°again.willia&a;a;a;a;benson(1971)anticipatedthistypeofresonance，laterconfirdbylani，nobili&a;a;a;a;carpino(1989).
anarguntθ4=?p??n+3(Ωp?Ωn)libratesaround180°withalongperiod，～5.7x108yr.
inournuricalintegrations，theresonances(i)–(iii)arewellintained，andvariationofthecriticalarguntsθ1，θ2，θ3reinsilarduringthewholeintegrationperiod(figs14–16).however，thefourthresonance(iv)appearstobedifferent:thecriticalarguntθ4alternateslibrationandcirculationovera1010-yrti-scale(fig.17).thisisaninteresfactthatkinoshita&a;a;a;a;nakais(1995，1996)shorterintegrationswerenotabletodisclose.
6discussion
whatkindofdynacalchanisaintainsthislong-tertabilityoftheplaarysystewecaniediatelythinkoftwojorfeaturesthatyberesponsibleforthelong-tertability.first，thereseeobenosignificantlower-orderresonances(antionandsecular)betweenanypairangthenineplas.jupiterandsaturnareclosetoa5:2antionresonance(thefaus‘greatinequality’)，butnotjustintheresonancezone.higher-orderresonancesycausethechaotatureoftheplaarydynacaltion，buttheyarenotsostrongastodestroythestableplaarytionwithinthelifetioftherealsolarsystethesecondfeature，whichwethinkisreiortantforthelong-tertabilityofourplaarysysteisthedifferenceindynacaldistancebetweenterrestrialandjovianplaarysubsyste(ito&a;a;a;a;tanikawa1999，2001).whenweasureplaaryseparationsbythetualhillradii(r_)，separationsangterrestrialplasaregreaterthan26rh，whereasthoseangjovianplasarelessthan14rh.thisdifferenceisdirectlyrelatedtothedifferencebetweendynacalfeaturesofterrestrialandjovianplas.terrestrialplashavesllersses，shorterorbitalperiodsandwiderdynacalseparation.theyarestronglyperturbedbyjovianplasthathavelargersses，longerorbitalperiodsandnarrowerdynacalseparation.jovianplasarenotperturbedbyanyotherssivebodies.
thepresentterrestrialplaarysystesstillbeingdisturbedbythessivejovianplas.however，thewideseparationandtualinteractionangtheterrestrialplasrendersthedisturbanceineffective;thedegreeofdisturbancebyjovianplasiso(ej)(entricityofjupiter)，sincethedisturbancecausedbyjovianplasisaforcedoscillationhavinganalitudeofo(ej).entricity，forexaleo(ej)～0.05，isfarfroufficienttoprovokeinstabilityintheterrestrialplashavingsuchawideseparationas26rh.thusweassuthatthepresentwidedynacalseparationangterrestrialplas(&a;a;a;gt;26rh)isprobablyoneofthestsignificantconditionsforintainingthestabilityoftheplaarysystevera109-yrti-span.ourdetailedanalysisoftherelationshipbetweendynacaldistancebetweenplasandtheinstabilityti-scaleofsolarsystelaarytionisnowon-going.
althoughournuricalintegrationsspanthelifetiofthesolarsystethenuerofintegrationsisfarfroufficienttofilltheinitialphasespace.itisnecessarytoperforeandrenuricalintegrationstoconfirndexaneindetailthelong-tertabilityofourplaarydynacs.
——以上文段引自ito，t.&a;a;a;tanikawa，k.long-terntegrationsandstabilityofplaaryorbitsinoursolarsysten.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。
不过仍然有人质疑——“你说得太含糊了”，“火星轨道的变化比你想象要大得多！”
那好吧，既然作者君的简单解释不够有力，那咱们就看看严肃的东西，反正这本书写到现在，嚷嚷着本书bug一大堆，用初高中物理在书中挑刺的人也不少。
以下是文章内容：
long-terntegrationsandstabilityofplaaryorbitsinoursolarsystebr />
abstract
wepresenttheresultsofverylong-teruricalintegrationsofplaaryorbitaltionsover109-yrti-spansinc露dingallnineplas.aquickinspectionofournuricaldatashowsthattheplaarytion，atleastinoursiledynacaldel，seetobequitestableevenoverthisverylongti-span.acloserlookatthelowest-frequencyoscillationsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestrialplaarytion，especiallythatofrcury.entricityofrcuryinourintegrationsisqualitativelysilartotheresultsfroacqueslaskarssecularperturbationtheory(e.g.ex～0.35over～±4gyr).however，entricityorinclinationinanyorbitalelentsoftheplas，whichyberevealedbystilllonger-teruricalintegrations.wehavealsoperfordacoupleoftrialintegrationsinc露dingtionsoftheouterfiveplasoverthedurationof±5x1010yr.theresultindicatesthatthethreejorresonancesintheneptune–p露tosysteavebeenintainedoverthe1011-yrti-span.
1introduction
1.1definitionoftheproblebr />
thequestionofthestabilityofoursolarsysteasbeendebatedoverseveral混dredyears，sincetheeraofnewton.theprobleasattractednyfaustheticiansovertheyearsandhasplayedacentralroleinthedevelopntofnon-lineardynacsandchaostheory.however，wedonotyethaveadefiniteanswertothequestionofwhetheroursolarsystesstableornot.thisispartlyaresultofthefactthatthedefinitionoftheterstability’isvaguewhenitisusedinrelationtotheproblefplaarytioninthesolarsysteactuallyitisnoteasytogiveaclear，rigorousandphysicallyaningfuldefinitionofthestabilityofoursolarsyste
angnydefinitionsofstability，hereweadoptthehilldefinition(gladn1993):actuallythisisnotadefinitionofstability，butofinstability.urssowhereinthesystestarfrocertaininitialconfiguration(chaers，wetherill&a;a;a;a;boss1996;ito&a;a;a;a;tanikawa1999).asystesdefinedasexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinanareaofthelargerhillradius.otherwisethesystesdefinedasbeingstable.henceforwardwestatethatourplaarysystesdynacallystableifnocloseencounterhappensduringtheageofoursolarsysteabout±5gyr.incidentally，urrenceofanyorbitalcrossingbetweeneitherofapairofplastakesplace.thisisbecauseweknowfroxperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounterinplaaryandprotoplaarysyste(yoshinaga，kokubo&a;a;a;a;kino1999).ofcoursethisstatentcannotbesilyappliedtosystewithstableorbitalresonancessuchastheneptune–p露tosyste
1.2previousstudiesandaiofthisresearch
inadditiontothevaguenessoftheconceptofstability，theplasinoursolarsystehowacharactertypicalofdynacalchaos(sussn&a;a;a;a;wisdo988，1992).thecauseofthischaoticbehaviourisnowpartl玉nderstoodasbeingaresultofresonanceoverlapping(rray&a;a;a;a;holn1999;lecar，franklin&a;a;a;a;holn2001).however，itwouldrequireintegraoveranenseleofplaarysysteinc露dingallnineplasforaperiodcoveringseveral10gyrtothoroughl玉nderstandthelong-tervo露tionofplaaryorbits，sincechaoticdynacalsystearecharacterizedbytheirstrongdependenceoninitialconditions.
frohatpointofview，nyofthepreviouslong-teruricalintegrationsinc露dedonlytheouterfiveplas(sussn&a;a;a;a;wisdo988;kinoshita&a;a;a;a;nakai1996).thisisbecausetheorbitalperiodsoftheouterplasaresochlongerthanthoseoftheinnerfourplasthatitischeasiertofollowthesysteoragivenintegrationperiod.atpresent，thelongestnuricalintegrationspublishedinjournalsarethoseofduncan&a;a;a;a;lissauer(1998).althoughtheirintargetwastheeffectofpost-in-sequencesolarsslossonthestabilityofplaaryorbits，theyperfordnyintegrationscoveringupto～1011yroftheorbitaltionsofthefourjovianplas.theinitialorbitalelentsandssesofplasarethesaasthoseofoursolarsystenduncan&a;a;a;a;lissauerspaper，buttheydecreasethessofthesungraduallyintheirnuricalexperints.thisisbecausetheyconsidertheeffectofpost-in-sequencesolarsslossinthepaper.consequently，theyfoundthatthecrossingti-scaleofplaaryorbits，whichcanbeatypicalindicatoroftheinstabilityti-scale，isquitesensitivetotherateofssdecreaseofthesun.whenthessofthesunisclosetoitspresentva露e，thejovianplasreinstableover1010yr，orperhapslonger.duncan&a;a;a;a;lissaueralsoperfordfoursilarexperintsontheorbitaltionofsevenplas(venustoneptune)，whichcoveraspanof～109yr.theirexperintsonthesevenplasarenotyeprehensive，butitseethattheterrestrialplasalsoreinstableduringtheintegrationperiod，intainingalstregularoscillations.
ontheotherhand，uratese-analyticalsecularperturbationtheory(laskar1988)，entricitiesandinclinationsoftheterrestrialplas，especiallyofrcuryandrsonati-scaleofseveral109yr(laskar1996).theresultsoflaskarssecularperturbationtheoryshouldbeconfirdandinvestigatedbyfullynuricalintegrations.
inthispaperwepresentprelinaryresultsofsixlong-teruricalintegrationsonallnineplaaryorbits，coveringaspanofseveral109yr，andoftwootherintegrationscoveringaspanof±5x1010yr.thetotalelapsedtiforallintegrationsisrethan5yr，usingseveraldedicatedpcsandworkstations.oneofthefundantalconc露sionsofourlong-terntegrationsisthatsolarsystelaarytionseetobestableinterofthehillstabilityntionedabove，atleastoverati-spanof±4gyr.actually，inournuricalintegrationsthesysteasfarrestablethanwhatisdefinedbythehillstabilitycriterion:notonlydidnocloseencounterhappenduringtheintegrationperiod，butalsoalltheplaaryorbitalelentshavebeenconfinedinanarrowregionbothintiandfrequencydoin，thoughplaarytionsarestochastic.sincethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-teruricalintegrations，weshowtypicalexalefiguresasevidenceoftheverylong-tertabilityofsolarsystelaarytion.forreaderswhohaverespecificanddeeperinterestsinournuricalresults，wehavepreparedawebpage(aess)，whereweshowraworbitalelents，theirlow-passfilteredresults，variationofdelaunayelentsandangularntueficit，andresultsofoursileti–frequencyanalysisonallofourintegrations.
insection2webrieflyexplainourdynacaldel，nuricalthodandinitialconditionsusedinourintegrations.section3isdevotedtoadescriptionofthequickresultsofthenuricalintegrations.verylong-tertabilityofsolarsystelaarytionisapparentbothinplaarypositionsandorbitalelents.aroughestitionofnuricalerrorsisalsogiven.section4goesontoadiscussionofthelongest-terariationofplaaryorbitsusingalow-passfilterandinc露desadiscussionofangularntueficit.insection5，wepresentasetofnuricalintegrationsfortheouterfiveplasthatspans±5x1010yr.insection6wealsodiscussthelong-tertabilityoftheplaarytionanditspossiblecause.
2descriptionofthenuricalintegrations
（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。）
2.3nuricalthod
weutilizeasecond-orderwisdoholnsylecticpasourinintegrationthod(wisdoa;a;a;a;holn1991;kinoshita，yoshida&a;a;a;a;nakai1991)withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘wartart’(saha&a;a;a;a;treine1992，1994).
thestepsizeforthenuricalintegrationsis8dthroughoutallintegrationsofthenineplas(n±1，2，3)，whichisabout1/11oftheorbitalperiodoftheinnerstpla(rcury).asforthedeternationofstepsize，wepartlyfollowthepreviousnuricalintegrationofallnineplasinsussn&a;a;a;a;wisdo1988，7.2d)andsaha&a;a;a;a;treine(1994，225/32d).ulationofround-offerrorinthputationprocesses.inrelationtothis，wisdoa;a;a;a;holn(1991)perfordnuricalintegrationsoftheouterfiveplaaryorbitsusingthesylecticpwithastepsizeof400d，1/10.83oftheorbitalperiodofjupiter.urateenough，whichpartlyjustifiesourthodofdeterningthestepsize.however，entricityofjupiter(～0.05)ischsllerthanthatofrcury(～0.2)，weneedsocarewhenwparetheseintegrationssilyinterofstepsizes.
intheintegrationoftheouterfiveplas(f±)，wefixedthestepsizeat400d.
weadoptgaussfandgfunctionsinthesylecticptogetherwiththethird-orderhalleythod(danby1992)asasolverforkeplerequations.thenuerofxiterationswesetinhalleysthodis15，buttheyneverreachedthexinanyofourintegrations.
theintervalofthedataoutputis200000d(～547yr)forthecalculationsofallnineplas(n±1，2，3)，andabout8000000d(～21903yr)fortheintegrationoftheouterfiveplas(f±).
althoughnooutputfilteringwasdonewhenthenuricalintegrationswereinprocess，weappliedalow-passfiltertotheraworbitaldataafterwehapletedallthecalculations.seesection4.1forredetail.
2.4errorestition
2.4.1relativeerrorsintotalenergyandangularntubr />
aordingtooneofthebasicpropertiesofsylecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangularntu，ourlong-teruricalintegrationsseeohavebeenperfordwithverysllerrors.theaveragedrelativeerrorsoftotalenergy(～10?9)andoftotalangularntu～10?11)havereinednearlyconstantthroughouttheintegrationperiod(fig.1).thespecialstartupprocedure，wartart，wouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofgnitudeorre.
relativenuricalerrorofthetotalangularntua/a0andthetotalenergyδe/e0inournuricalintegrationsn±1，2，3，whereδeandδaaretheabso露techangeofthetotalenergyandtotalangularnturespectively，ande0anda0aretheirinitialva露es.thehorizonta露nitisgyr.
notethatdifferentoperasyste，differenttheticallibraries，anddifferenthardwarearchitecturesresultindifferentnuricalerrors，throughthevariationsinround-offerrorhandlingandnuricalalgorith.intheupperpaneloffig.1，wecanrecognizethissituationinthesecularnuricalerrorinthetotalangularntuwhichshouldberigorouslypreserveduptochine-eprecision.
2.4.2errorinplaarylongitudes
sincethesylecticpspreservetotalenergyandtotalangularntufn-bodydynacalsysteinherentlywell，uracyofnuricalintegrations，especiallyasaasureofthepositionalerrorofplas，i.e.theerrorinplaarylongitudes.toestitethenuricalerrorintheplaarylongitudes，weperfordthefollowingprocedures.wparedtheresultofourinlong-terntegrationswithsotestintegrations，uracythantheinintegrations.forthispurpose，urateintegrationwithastepsizeof0.125d(1/64oftheinintegrations)spanning3x105yr，starwiththesainitialconditionsasinthen?1integration.weconsiderthatthistestintegrationprovidesuswitha‘pseudo-true’so露tionofplaaryorbitalevo露tion.next，wparethetestintegrationwiththeinintegration，n?1.fortheperiodof3x105yr，weseeadifferenceinananoliesoftheearthbetweenthetwointegrationsof～0.52°(inthecaseofthen?1integration).thisdifferencecanbeextrapolatedtotheva露e～8700°，about25rotationsofearthafter5gyr，sincetheerroroflongitudesincreaseslinearlywithtiinthesylecticp.silarly，thelongitudeerrorofp露tocanbeestitedas～12°.thisva露eforp露toischbetterthantheresultinkinoshita&a;a;a;a;nakai(1996)wherethedifferenceisestitedas～60°.
3nuricalresults–i.glanceattherawdata
inthissectionwebrieflyreviewthelong-tertabilityofplaaryorbitaltionthroughsosnapshotsofrawnuricaldata.theorbitaltionofplasindicateslong-tertabilityinallofournuricalintegrations:noorbitalcrossingsnorcloseencountersbetweenanypairofplastookplace.
3.1generaldescriptionofthestabilityofplaaryorbits
first，webrieflylookatthegeneralcharacterofthelong-tertabilityofplaaryorbits.ourinterestherefocusesparticularlyontheinnerfourterrestrialplasforwhichtheorbitalti-scalesarechshorterthanthoseoftheouterfiveplas.aswecanseeclearlyfroheplanarorbitalconfigurationsshowninfigs2and3，orbitalpositionsoftheterrestrialplasdifferlittlebetweentheinitialandfinalpartofeachnuricalintegration，whichspansseveralgyr.thesolidlinesdenothepresentorbitsoftheplasliealstwithintheswarfdotseveninthefinalpartofintegrations(b)and(d).thisindicatesthatthroughouttheentireintegrationperiodthealstregularvariationsofplaaryorbitaltionreinnearlythesaastheyareatpresent.
verticalviewofthefourinnerplaaryorbits(frohez-axisdirection)attheinitialandfinalpartsoftheintegrationsn±1.theaxesunitsareau.thexy-planeissettotheinvariantplaneofsolarsysteotalangularntu(a)theinitialpartofn+1(t=0to0.0547x109yr).(b)thefinalpartofn+1(t=4.9339x108to4.9886x109yr).(c)theinitialpartofn?1(t=0to?0.0547x109yr).(d)thefinalpartofn?1(t=?3.9180x109to?3.9727x109yr).ineachpanel，atotalof23684pointsareplottedwithanintervalofabout2190yrover5.47x107yr.solidlinesineachpaneldenotethepresentorbitsofthefourterrestrialplas(takenfroe245).
entricitiesandorbitalinclinationsfortheinnerfourplasintheinitialandfinalpartoftheintegrationn+1isshowninfig.4.asexpected，thecharacterofthevariationofplaaryorbitalelentsdoesnotdiffersignificantlybetweentheinitialandfinalpartofeachintegration，atleastforvenus，earthandrs.theelentsofrcury，entricity，seeochangetoasignificantextent.thisispartlybecausetheorbitalti-scaleoftheplaistheshortestofalltheplas，whichleadstoarerapidorbitalevo露tionthanotherplas;theinnerstplaybenearesttoinstability.thisresultappearstobeinsoagreentwithlaskars(1994，1996)entricitiesandinclinationsofrcuryonati-scaleofseveral109yr.however，theeffectofthepossibleinstabilityoftheorbitofrcuryynotfatallyaffecttheglobalstabilityofthewholeplaarysystewingtothesllssofrcury.wewillntionbrieflythelong-terrbitalevo露tionofrcurylaterinsection4usinglow-passfilteredorbitalelents.
theorbitaltionoftheouterfiveplasseerigorouslystableandquiteregularoverthisti-span(seealsosection5).
3.2ti–frequencyps
althoughtheplaarytionexhibitsverylong-tertabilitydefinedasthenon-existenceofcloseencounterevents，thechaotatureofplaarydynacscanchangetheoscillatoryperiodandalitudeofplaaryorbitaltiongraduallyoversuchlongti-spans.evensuchslightf露ctuationsoforbitalvariationinthefrequencydoin，particularlyinthecaseofearth，canpotentiallyhaveasignificanteffectonitssurfaceclitesystehroughsolarinsolationvariation(cf.berger1988).
togiveanoverviewofthelong-terhangeinperiodicityinplaaryorbitaltion，weperfordnyfastfouriertransfortions(ffts)alongthetiaxis，andsuperposedtheresulperiodgratodrawtwo-dinsionalti–frequencyps.thespecificapproachtodrawingtheseti–frequencypsinthispaperisverysile–chsilerthanthewaveletanalysisorlaskars(1990，1993)frequencyanalysis.
dividethelow-passfilteredorbitaldataintonyfragntsofthesalength.thelengthofeachdatasegntshouldbealtipleof2inordertoapplythefft.
eachfragntofthedatahasalargeoverlappingpart:forexale，whentheithdatabeginsfro=tiandendsatt=ti+t，thenextdatasegntrangesfroi+δt≤ti+δt+t，whereδt?t.wecontinuethisdivisionuntilwereachacertainnuernbywhichtn+treachesthetotalintegrationlength.
weapplyanffttoeachofthedatafragnts，andobtainnfrequencydiagra.
ineachfrequencydiagrabtainedabove，thestrengthofperiodicitycanbereplacedbyagrey-scale(orcolour)chart.
weperforhereplacent，andconnectallthegrey-scale(orcolour)chartsintoonegraphforeachintegration.thehorizontalaxisofthesenewgraphsshouldbetheti，i.e.thestartisofeachfragntofdata(ti，wherei=1，…，n).theverticalaxisrepresentstheperiod(orfrequency)oftheoscillationoforbitalelents.
wehaveadoptedanfftbecauseofitsoverwhelngspeed，sincetheauntofnuricaldatatobedposedintofrequencponentsisterriblyhuge(severaltensofgbytes).
atypicalexaleoftheti–frequencypcreatedbytheaboveproceduresisshowninagrey-scalediagrasfig.5，entricityandinclinationofearthinn+2integration.infig.5，thedarkareashowsthatatthetiindicatedbytheva露eontheabscissa，theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.entricityandinclinationofearthonlychangesslightlyovertheentireperiodcoveredbythen+2integration.thisnearlyregulartrendisqualitativelythesainotherintegrationsandforotherplas，althoughtypicalfrequenciesdifferplabyplaandelentbyelent.
4.2long-terxchangeoforbitalenergyandangularntubr />
wecalculateverylong-periodicvariationandexchangeofplaaryorbitalenergyandangularntusingfiltereddelaunayelentsl，g，h.gandhareequivalenttotheplaaryorbitalangularntunditsverticaponentperunitss.lisrelatedtotheplaaryorbitalenergyeperunitssase=?μ2/2l2.ifthesystepletelylinear，theorbitalenergyandtheangularntuneachfrequencybinstbeconstant.non-linearityintheplaarysysteancauseanexchangeofenergyandangularntunthefrequencydoin.thealitudeofthelowest-frequencyoscillationshouldincreaseifthesystesunstableandbreaksdowngradually.however，suchasytofinstabilityisnotpronentinourlong-terntegrations.
infig.7，thetotalorbitalenergyandangularntufthefourinnerplasandallnineplasareshownforintegrationn+2.theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedase-e0)，totalangularntug-g0)，andtheverticaponent(h-h0)oftheinnerfourplascalculatedfrohelow-passfiltereddelaunayelents.e0，g0，h0denotetheinitialva露esofeachquantity.theabso露tedifferencefroheinitialva露esisplottedinthepanels.thelowerthreepanelsineachfigureshowe-e0，g-g0andh-h0ofthetotalofnineplas.thef露ctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthessivejovianplas.
paringthevariationsofenergyandangularntuftheinnerfourplasandallnineplas，itisapparentthatthealitudesofthoseoftheinnerplasarechsllerthanthoseofallnineplas:thealitudesoftheouterfiveplasarechlargerthanthoseoftheinnerplas.thisdoesnotanthattheinnerterrestrialplaarysubsystesrestablethantheouterone:thisissilyaresultoftherelativesllnessofthessesofthefourterrestrialplaparedwiththoseoftheouterjovianplas.anotherthingwenoticeisthattheinnerplaarysubsysteaybeunstablererapidlythantheouteronebecauseofitsshorterorbitalti-scales.thiscanbeseeninthepanelsdenotedasinner4infig.7wherethelonger-periodicandirregularoscillationsarereapparentthaninthepanelsdenotedastotal9.actually，thef露ctuationsintheinner4panelsaretoalargeextentasaresultoftheorbitalvariationofthercury.however，wecannotneglectthecontributionfrotherterrestrialplas，aswewillseeinsubsequentsections.
4.4long-terouplingofseveralneighbouringplapairs
letusseesoindividualvariationsofplaaryorbitalenergyandangularntuxpressedbythelow-passfiltereddelaunayelents.figs10and11showlong-tervo露tionoftheorbitalenergyofeachplaandtheangularntunn+1andn?2integrations.wenoticethatsoplasforpparentpairsinteroforbitalenergyandangularntuxchange.inparticular，venusandearthkeatypicalpair.inthefigures，theyshownegativecorrelationsinexchangeofenergyandpositivecorrelationsinexchangeofangularntuthenegativecorrelationinexchangeoforbitalenergyansthatthetwoplasforcloseddynacalsystenteroftheorbitalenergy.thepositivecorrelationinexchangeofangularntueansthatthetwoplasaresiltaneousl玉ndercertainlong-tererturbations.candidatesforperturbersarejupiterandsaturn.alsoinfig.11，wecanseethatrsshowsapositivecorrelationintheangularntuariationtothevenus–earthsystercuryexhibitscertainnegativecorrelationsintheangularntuersusthevenus–earthsystewhichseetobeareactioncausedbytheconservationofangularntuntheterrestrialplaarysubsyste
itisnotclearatthentwhythevenus–earthpairexhibitsanegativecorrelationinenergyexchangeandapositivecorrelationinangularntuxchange.weypossiblyexplainthisthroughobservingthegeneralfactthattherearenosecularterinplaarysejoraxesuptosecond-orderperturbationtheories(cf.brouwer&a;a;a;a;clence1961;aletti&a;a;a;a;o1998).thisansthattheplaaryorbitalenergy(whichisdirectlyrelatedtothesejoraxisa)ghtbechlessaffectedbyperturbingplasthanistheangularntuxchange(whichrelatestoe).hence，entricitiesofvenusandearthcanbedisturbedeasilybyjupiterandsaturn，whichresultsinapositivecorrelationintheangularntuxchange.ontheotherhand，thesejoraxesofvenusandeartharelesslikelytobedisturbedbythejovianplas.thustheenergyexchangeybelitedonlywithinthevenus–earthpair，whichresultsinanegativecorrelationintheexchangeoforbitalenergyinthepair.
asfortheouterjovianplaarysubsystejupiter–saturnanduranus–neptuneseeokedynacalpairs.however，thestrengthoftheircouplingisnotasstronparedwiththatofthevenus–earthpair.
5±5x1010-yrintegrationsofouterplaaryorbits
sincethejovianplaaryssesarechlargerthantheterrestrialplaarysses，wetreatthejovianplaarysystesanindependentplaarysystenterofthestudyofitsdynacalstability.hence，weaddedacoupleoftrialintegrationsthatspan±5x1010yr，inc露dingonlytheouterfiveplas(thefourjovianplasp露sp露to).theresultsexhibittherigorousstabilityoftheouterplaarysysteverthislongti-span.orbitalconfigurations(fig.12)，entricitiesandinclinations(fig.13)showthisverylong-tertabilityoftheouterfiveplasinboththetiandthefrequencydoins.althoughwedonotshowpshere，thetypicalfrequencyoftheorbitaloscillationofp露toandtheotherouterplasisalstconstantduringtheseverylong-terntegrationperiods，whichisdenstratedintheti–frequencypsonourwebpage.
inthesetwointegrations，therelativenuricalerrorinthetotalenergywas～10?6andthatofthetotalangularntuas～10?10.
5.1resonancesintheneptune–p露tosystebr />
kinoshita&a;a;a;a;nakai(1996)integratedtheouterfiveplaaryorbitsover±5.5x109yr.theyfoundthatfourjorresonancesbetweenneptuneandp露toareintainedduringthewholeintegrationperiod，andthattheresonancesybetheincausesofthestabilityoftheorbitofp露to.thejorfourresonancesfoundinpreviousresearchareasfollows.inthefollowingdescription，λdenotestheanlongitude，Ωisthelongitudeoftheascendingnodeand?isthelongitudeofperihelion.subscriptspandndenotep露toandneptune.
antionresonancebetweenneptuneandp露to(3:2).thecriticalarguntθ1=3λp?2λn??plibratesaround180°withanalitudeofabout80°andalibrationperiodofabout2x104yr.
thearguntofperihelionofp露towp=θ2=?p?Ωplibratesaround90°withaperiodofabout3.8x106yr.entricityandinclinationofp露toaresynchronizedwiththelibrationofitsarguntofperihelion.thisisanticipatedinthesecularperturbationtheoryconstructedbykozai(1962).
thelongitudeofthenodeofp露toreferredtothelongitudeofthenodeofneptune，θ3=Ωp?Ωn，circulatesandtheperiodofthiscirculationisequaltotheperiodofθ2libration.whenθ3beszero，i.e.thelongitudesofascendingnodesofneptuneandp露tooverlap，theinclinationofp露tobesxientricitybesnindthearguntofperihelionbes90°.whenθ3bes180°，theinclinationofp露tobesnientricitybesxindthearguntofperihelionbes90°again.willia&a;a;a;a;benson(1971)anticipatedthistypeofresonance，laterconfirdbylani，nobili&a;a;a;a;carpino(1989).
anarguntθ4=?p??n+3(Ωp?Ωn)libratesaround180°withalongperiod，～5.7x108yr.
inournuricalintegrations，theresonances(i)–(iii)arewellintained，andvariationofthecriticalarguntsθ1，θ2，θ3reinsilarduringthewholeintegrationperiod(figs14–16).however，thefourthresonance(iv)appearstobedifferent:thecriticalarguntθ4alternateslibrationandcirculationovera1010-yrti-scale(fig.17).thisisaninteresfactthatkinoshita&a;a;a;a;nakais(1995，1996)shorterintegrationswerenotabletodisclose.
6discussion
whatkindofdynacalchanisaintainsthislong-tertabilityoftheplaarysystewecaniediatelythinkoftwojorfeaturesthatyberesponsibleforthelong-tertability.first，thereseeobenosignificantlower-orderresonances(antionandsecular)betweenanypairangthenineplas.jupiterandsaturnareclosetoa5:2antionresonance(thefaus‘greatinequality’)，butnotjustintheresonancezone.higher-orderresonancesycausethechaotatureoftheplaarydynacaltion，buttheyarenotsostrongastodestroythestableplaarytionwithinthelifetioftherealsolarsystethesecondfeature，whichwethinkisreiortantforthelong-tertabilityofourplaarysysteisthedifferenceindynacaldistancebetweenterrestrialandjovianplaarysubsyste(ito&a;a;a;a;tanikawa1999，2001).whenweasureplaaryseparationsbythetualhillradii(r_)，separationsangterrestrialplasaregreaterthan26rh，whereasthoseangjovianplasarelessthan14rh.thisdifferenceisdirectlyrelatedtothedifferencebetweendynacalfeaturesofterrestrialandjovianplas.terrestrialplashavesllersses，shorterorbitalperiodsandwiderdynacalseparation.theyarestronglyperturbedbyjovianplasthathavelargersses，longerorbitalperiodsandnarrowerdynacalseparation.jovianplasarenotperturbedbyanyotherssivebodies.
thepresentterrestrialplaarysystesstillbeingdisturbedbythessivejovianplas.however，thewideseparationandtualinteractionangtheterrestrialplasrendersthedisturbanceineffective;thedegreeofdisturbancebyjovianplasiso(ej)(entricityofjupiter)，sincethedisturbancecausedbyjovianplasisaforcedoscillationhavinganalitudeofo(ej).entricity，forexaleo(ej)～0.05，isfarfroufficienttoprovokeinstabilityintheterrestrialplashavingsuchawideseparationas26rh.thusweassuthatthepresentwidedynacalseparationangterrestrialplas(&a;a;a;gt;26rh)isprobablyoneofthestsignificantconditionsforintainingthestabilityoftheplaarysystevera109-yrti-span.ourdetailedanalysisoftherelationshipbetweendynacaldistancebetweenplasandtheinstabilityti-scaleofsolarsystelaarytionisnowon-going.
althoughournuricalintegrationsspanthelifetiofthesolarsystethenuerofintegrationsisfarfroufficienttofilltheinitialphasespace.itisnecessarytoperforeandrenuricalintegrationstoconfirndexaneindetailthelong-tertabilityofourplaarydynacs.
——以上文段引自ito，t.&a;a;a;tanikawa，k.long-terntegrationsandstabilityofplaaryorbitsinoursolarsysten.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。