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long-ternbsp;integrationsandstabilityofparyorbitsinoursorsystebr>

abstract

wepresenttheresultsofverylong-ternbsp;nuricalintegrationsofparyorbitaltionsover109-yrti-spansincludingallninepsaquickinspectionofournuricaldatashowsthattheparytion，atleastinoursiledynacaldel，seetobequitestableevenoverthisverylongti-spanacloserlookatthelowest-frequencyosciltionsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestrialparytion，especiallythatofrcurythebehaviouroftheeentricityofrcuryinourintegrationsisqualitativelysirtotheresultsfronbsp;jacquesskar'ssecurperturbationtheoryhowever，therearenoarentsecurincreasesofeentricityorinclinationinanyorbitalelentsoftheps，whichyberevealedbystilllonger-ternbsp;nuricalintegrationswehavealsoperfordaupleoftrialintegrationsincludingtionsoftheouterfivepsoverthedurationof±5x1010yrtheresultindicatesthatthethreejorresonancesintheneptune–plutosystenbsp;havebeenintainedoverthe1011-yrti-span

1introduction

11definitionoftheproblebr>

thequestionofthestabilityofoursorsystenbsp;hasbeendebatedoverseveralhundredyears，sincetheeraofnewtontheproblenbsp;hasattractednyfaustheticiansovertheyearsandhaspyedacentralroleinthedevelopntofnon-lineardynacsandchaostheoryhowever，wedonotyethaveadefiniteanswertothequestionofwhetheroursorsystenbsp;isstableornotthisispartlyaresultofthefactthatthedefinitionoftheternbsp;‘stability’isvaguewhenitisusedinretiontotheproblenbsp;ofparytioninthesorsystenbsp;actuallyitisnoteasytogiveaclear，rigorousandphysicallyaningfuldefinitionofthestabilityofoursorsystebr>

angnydefinitionsofstability，hereweadoptthehilldefinitionbsp;actuallythisisnotadefinitionofstability，butofinstabilitywedefineasystenbsp;asbengunstablewhenacloseenunterourssowhereinthesystestartingfronbsp;acertaininitialnfigurationasystenbsp;isdefinedasexperiencingacloseenunterwhentwobodiesroachoneanotherwithinanareaofthergerhillradiusotherwisethesystenbsp;isdefinedasbeingstablehenceforwardwestatethatourparysystenbsp;isdynacallystableifnocloseenunterhensduringtheageofoursorsysteabout±5gyrincidentally，thisdefinitionyberepcedbyoneinwhichanourrenceofanyorbitalcrossingbetweeneitherofapairofpstakespcethisisbecauseweknowfronbsp;experiencethatanorbitalcrossingisverylikelytoleadtoacloseenunterinparyandprotoparysysteofursethisstatentcannotbesilyliedtosystewithstableorbitalresonancessuchastheneptune–plutosystebr>

12previousstudiesandaiofthisresearch

inadditiontothevaguenessofthenceptofstability，thepsinoursorsystenbsp;showacharactertypicalofdynacalchaosthecauseofthischaoticbehaviourisnowpartlyunderstoodasbeingaresultofresonanceoverlinghowever，itwouldrequireintegratingoveranenseleofparysysteincludingallninepsforaperiodveringseveral10gyrtothoroughlyunderstandthelong-ternbsp;evolutionofparyorbits，sincechaoticdynacalsystearecharacterizedbytheirstrongdependenceoninitialnditions

fronbsp;thatpointofview，nyofthepreviouslong-ternbsp;nuricalintegrationsincludedonlytheouterfivepsthisisbecausetheorbitalperiodsoftheouterpsaresochlongerthanthoseoftheinnerfourpsthatitischeasiertofollowthesystenbsp;foragivenintegrationperiodatpresent，thelongestnuricalintegrationspublishedinjournalsarethoseofduncan&a;a;a;a;lissaueralthoughtheirintargetwastheeffectofpost-in-sequencesorsslossonthestabilityofparyorbits，theyperfordnyintegrationsveringupto∼1011yroftheorbitaltionsofthefourjovianpstheinitialorbitalelentsandssesofpsarethesaasthoseofoursorsystenbsp;induncan&a;a;a;a;lissauer'spaper，buttheydecreasethessofthesungraduallyintheirnuricalexperintsthisisbecausetheynsidertheeffectofpost-in-sequencesorsslossinthepapernsequently，theyfoundthatthecrossingti-scaleofparyorbits，whichcanbeatypicalindicatoroftheinstabilityti-scale，isquitesensitivetotherateofssdecreaseofthesunwhenthessofthesunisclosetoitspresentvalue，thejovianpsreinstableover1010yr，orperhapslongerduncan&a;a;a;a;lissaueralsoperfordfoursirexperintsontheorbitaltionofsevenps，whichveraspanof∼109yrtheirexperintsonthesevenpsarenotyetrehensive，butitseethattheterrestrialpsalsoreinstableduringtheintegrationperiod，intainingalstregurosciltions

ontheotherhand，inhisauratese-analyticalsecurperturbationtheory，skarfindsthatrgeandirregurvariationscanearintheeentricitiesandinclinationsoftheterrestrialps，especiallyofrcuryandrsonati-scaleofseveral109yrtheresultsofskar'ssecurperturbationtheoryshouldbenfirdandinvestigatedbyfullynuricalintegrations

inthispaperwepresentprelinaryresultsofsixlong-ternbsp;nuricalintegrationsonallnineparyorbits，veringaspanofseveral109yr，andoftwootherintegrationsveringaspanof±5x1010yrthetotalepsedtiforallintegrationsisrethan5yr，usingseveraldedicatedpcsandworkstationsoneofthefundantalnclusionsofourlong-ternbsp;integrationsisthatsorsystenbsp;parytionseetobestableinterofthehillstabilityntionedabove，atleastoverati-spanof±4gyractually，inournuricalintegrationsthesystenbsp;wasfarrestablethanwhatisdefinedbythehillstabilitycriterionbsp;notonlydidnocloseenunterhenduringtheintegrationperiod，butalsoalltheparyorbitalelentshavebeennfinedinanarrowregionbothintiandfrequencydoin，thoughparytionsarestochasticsincethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-ternbsp;nuricalintegrations，weshowtypicalexalefiguresasevidenceoftheverylong-ternbsp;stabilityofsorsystenbsp;parytionforreaderswhohaverespecificanddeeperinterestsinournuricalresults，wehavepreparedawebpage，whereweshowraworbitalelents，theirlow-passfilteredresults，variationofdeunayelentsandangurntunbsp;deficit，andresultsofoursileti–frequencyanalysisonallofourintegrations

insection2webrieflyexpinourdynacaldel，nuricalthodandinitialnditionsusedinourintegrationssection3isdevotedtoadescriptionofthequickresultsofthenuricalintegrationsverylong-ternbsp;stabilityofsorsystenbsp;parytionisarentbothinparypositionsandorbitalelentsaroughestitionofnuricalerrorsisalsogivensection4goesontoadiscussionofthelongest-ternbsp;variationofparyorbitsusingalow-passfilterandincludesadiscussionofangurntunbsp;deficitinsection5，wepresentasetofnuricalintegrationsfortheouterfivepsthatspans±5x1010yrinsection6wealsodiscussthelong-ternbsp;stabilityoftheparytionanditspossiblecause

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。）

23nuricalthod

weutilizeasend-orderwisdoholnsylecticpasourinintegrationthodwithaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warnbsp;start’

thestepsizeforthenuricalintegrationsis8dthroughoutallintegrationsofthenineps，whichisabout111oftheorbitalperiodoftheinnerstpasforthedeternationofstepsize，wepartlyfollowthepreviousnuricalintegrationofallninepsinsussn&a;a;a;a;wisdonbsp;andsaha&a;a;a;a;treineweroundedthedecilpartofthetheirstepsizesto8tokethestepsizealtipleof2inordertoreducetheautionofround-offerrorintheutationprocessesinretiontothis，wisdonbsp;&a;a;a;a;holnperfordnuricalintegrationsoftheouterfiveparyorbitsusingthesylecticpwithastepsizeof400d，11083oftheorbitalperiodofjupitertheirresultseetobeaurateenough，whichpartlyjustifiesourthodofdeterningthestepsizehowever，sincetheeentricityofjupiterischsllerthanthatofrcury，weneedsocarewhenwearetheseintegrationssilyinterofstepsizes

intheintegrationoftheouterfiveps，wefixedthestepsizeat400d

weadoptgauss'fandgfunctionsinthesylecticptogetherwiththethird-orderhalleythodasasolverforkeplerequationsthenuerofxinbsp;iterationswesetinhalley'sthodis15，buttheyneverreachedthexinbsp;inanyofourintegrations

theintervalofthedataoutputis200000dforthecalcutionsofallnineps，andabout8000000dfortheintegrationoftheouterfiveps

althoughnooutputfilteringwasdonewhenthenuricalintegrationswereinprocess，weliedalow-passfiltertotheraworbitaldataafterwehadletedallthecalcutionsseesection41forredetail

24errorestition

241retiveerrorsintotalenergyandangurntubr>

aordingtooneofthebasicpropertiesofsylecticintegrators，whichnservethephysicallynservativequantitieswell，ourlong-ternbsp;nuricalintegrationsseenbsp;tohavebeenperfordwithverysllerrorstheaveragedretiveerrorsoftotalenergyandoftotalangurntunbsp;havereinednearlynstantthroughouttheintegrationperiodthespecialstartupprocedure，warnbsp;start，wouldhavereducedtheaveragedretiveerrorintotalenergybyaboutoneorderofgnitudeorre

retivenuricalerrorofthetotalangurntunbsp;δaa0andthetotalenergyδee0inournuricalintegrationsn±1，2，3，whereδeandδaaretheabsolutechangeofthetotalenergyandtotalangurnturespectively，ande0anda0aretheirinitialvaluesthehorizontalunitisgyr

notethatdifferentoperatingsyste，differenttheticallibraries，anddifferenthardwarearchitecturesresultindifferentnuricalerrors，throughthevariationsinround-offerrorhandlingandnuricalalgorithintheupperpaneloffig1，wecanregnizethissituationinthesecurnuricalerrorinthetotalangurntuwhichshouldberigorouslypreserveduptochine-eprecision

242errorinparylongitudes

sincethesylecticpspreservetotalenergyandtotalangurntunbsp;ofn-bodydynacalsysteinherentlywell，thedegreeoftheirpreservationynotbeagoodasureoftheauracyofnuricalintegrations，especiallyasaasureofthepositionalerrorofps，ietheerrorinparylongitudestoestitethenuricalerrorintheparylongitudes，weperfordthefollowingprocedureswearedtheresultofourinlong-ternbsp;integrationswithsotestintegrations，whichspanchshorterperiodsbutwithchhigherauracythantheinintegrationsforthispurpose，weperfordachreaurateintegrationwithastepsizeof0125dspanning3x105yr，startingwiththesainitialnditionsasinthen−1integrationwensiderthatthistestintegrationprovidesuswitha‘pseudo-true’solutionofparyorbitalevolutionnext，wearethetestintegrationwiththeinintegration，n−1fortheperiodof3x105yr，weseeadifferenceinananoliesoftheearthbetweenthetwointegrationsof∼052°thisdifferencecanbeextrapotedtothevalue∼8700°，about25rotationsofearthafter5gyr，sincetheerroroflongitudesincreaseslinearlywithtiinthesylecticpsirly，thelongitudeerrorofplutocanbeestitedas∼12°thisvalueforplutoischbetterthantheresultinkinoshita&a;a;a;a;nakaiwherethedifferenceisestitedas∼60°

3nuricalresults–ignceattherawdata

inthissectionwebrieflyreviewthelong-ternbsp;stabilityofparyorbitaltionthroughsosnapshotsofrawnuricaldatatheorbitaltionofpsindicateslong-ternbsp;stabilityinallofournuricalintegrationsbsp;noorbitalcrossingsnorcloseenuntersbetweenanypairofpstookpce

31generaldescriptionofthestabilityofparyorbits

first，webrieflylookatthegeneralcharacterofthelong-ternbsp;stabilityofparyorbitsourinterestherefocusesparticurlyontheinnerfourterrestrialpsforwhichtheorbitalti-scalesarechshorterthanthoseoftheouterfivepsaswecanseeclearlyfronbsp;thepnarorbitalnfigurationsshowninfigs2and3，orbitalpositionsoftheterrestrialpsdifferlittlebetweentheinitialandfinalpartofeachnuricalintegration，whichspansseveralgyrthesolidlinesdenotingthepresentorbitsofthepsliealstwithintheswarnbsp;ofdotseveninthefinalpartofintegrationsandthisindicatesthatthroughouttheentireintegrationperiodthealstregurvariationsofparyorbitaltionreinnearlythesaastheyareatpresent

verticalviewofthefourinnerparyorbitsattheinitialandfinalpartsoftheintegrationsn±1theaxesunitsareauthexy-pneissettotheinvariantpneofsorsystenbsp;totalangurntunbsp;theinitialpartofn+1thefinalpartofn+1theinitialpartofn−1thefinalpartofn−1ineachpanel，atotalof23684pointsareplottedwithanintervalofabout2190yrover547x107yrsolidlinesineachpaneldenotethepresentorbitsofthefourterrestrialps

thevariationofeentricitiesandorbitalinclinationsfortheinnerfourpsintheinitialandfinalpartoftheintegrationn+1isshowninfig4asexpected，thecharacterofthevariationofparyorbitalelentsdoesnotdiffersignificantlybetweentheinitialandfinalpartofeachintegration，atleastforvenus，earthandrstheelentsofrcury，especiallyitseentricity，seenbsp;tochangetoasignificantextentthisispartlybecausetheorbitalti-scaleofthepistheshortestofalltheps，whichleadstoarerapidorbitalevolutionthanotherps;theinnerstpybenearesttoinstabilitythisresultearstobeinsoagreentwithskar'sexpectationsthatrgeandirregurvariationsearintheeentricitiesandinclinationsofrcuryonati-scaleofseveral109yrhowever，theeffectofthepossibleinstabilityoftheorbitofrcuryynotfatallyaffecttheglobalstabilityofthewholeparysystenbsp;owingtothesllssofrcurywewillntionbrieflythelong-ternbsp;orbitalevolutionofrcuryterinsection4usinglow-passfilteredorbitalelents

theorbitaltionoftheouterfivepsseerigorouslystableandquitereguroverthisti-span

32ti–frequencyps

althoughtheparytionexhibitsverylong-ternbsp;stabilitydefinedasthenon-existenceofcloseenunterevents，thechaoticnatureofparydynacscanchangetheosciltoryperiodandalitudeofparyorbitaltiongraduallyoversuchlongti-spansevensuchslightfluctuationsoforbitalvariationinthefrequencydoin，particurlyinthecaseofearth，canpotentiallyhaveasignificanteffectonitssurfaceclitesystenbsp;throughsorinsotionvariation

togiveanoverviewofthelong-ternbsp;changeinperiodicityinparyorbitaltion，weperfordnyfastfouriertransfortionsalongthetiaxis，andsuperposedtheresultingperiodgratodrawtwo-dinsionalti–frequencypsthespecificroachtodrawingtheseti–frequencypsinthispaperisverysile–chsilerthanthewaveletanalysisorskar'sfrequencyanalysis

dividethelow-passfilteredorbitaldataintonyfragntsofthesalengththelengthofeachdatasegntshouldbealtipleof2inordertolythefft

eachfragntofthedatahasargeoverlingpartbsp;forexale，whentheithdatabeginsfronbsp;t=tiandendsatt=ti+t，thenextdatasegntrangesfronbsp;ti+δt≤ti+δt+t，whereδttwentinuethisdivisionuntilwereachacertainnuernbywhichtn+treachesthetotalintegrationlength

welyanffttoeachofthedatafragnts，andobtainnfrequencydiagra

ineachfrequencydiagranbsp;obtainedabove，thestrengthofperiodicitycanberepcedbyagrey-scalechart

weperfornbsp;therepcent，andnnectallthegrey-scalechartsintoonegraphforeachintegrationthehorizontalaxisofthesenewgraphsshouldbetheti，iethestartingtisofeachfragntofdatatheverticalaxisrepresentstheperiodoftheosciltionoforbitalelents

wehaveadoptedanfftbecauseofitsoverwhelngspeed，sincetheauntofnuricaldatatobedeosedintofrequencyonentsisterriblyhuge

atypicalexaleoftheti–frequencypcreatedbytheaboveproceduresisshowninagrey-scalediagranbsp;asfig5，whichshowsthevariationofperiodicityintheeentricityandinclinationofearthinn+2integrationinfig5，thedarkareashowsthatatthetiindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaarounditwecanregnizefronbsp;thispthattheperiodicityoftheeentricityandinclinationofearthonlychangesslightlyovertheentireperiodveredbythen+2integrationthisnearlyregurtrendisqualitativelythesainotherintegrationsandforotherps，althoughtypicalfrequenciesdifferpbypandelentbyelent

42long-ternbsp;exchangeoforbitalenergyandangurntubr>

wecalcuteverylong-periodicvariationandexchangeofparyorbitalenergyandangurntunbsp;usingfiltereddeunayelentsl，g，hgandhareequivalenttotheparyorbitalangurntunbsp;anditsverticalonentperunitsslisretedtotheparyorbitalenergyeperunitssase=−μ22l2ifthesystenbsp;isletelylinear，theorbitalenergyandtheangurntunbsp;ineachfrequencybinstbenstantnon-linearityintheparysystenbsp;cancauseanexchangeofenergyandangurntunbsp;inthefrequencydointhealitudeofthelowest-frequencyosciltionshouldincreaseifthesystenbsp;isunstableandbreaksdowngraduallyhowever，suchasytonbsp;ofinstabilityisnotpronentinourlong-ternbsp;integrations

infig7，thetotalorbitalenergyandangurntunbsp;ofthefourinnerpsandallninepsareshownforintegrationn+2theupperthreepanelsshowthelong-periodicvariationoftotalenergy，totalangurntunbsp;，andtheverticalonentoftheinnerfourpscalcutedfronbsp;thelow-passfiltereddeunayelentse0，g0，h0denotetheinitialvaluesofeachquantitytheabsolutedifferencefronbsp;theinitialvaluesisplottedinthepanelsthelowerthreepanelsineachfigureshowe-e0，g-g0andh-h0ofthetotalofninepsthefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthessivejovianps

aringthevariationsofenergyandangurntunbsp;oftheinnerfourpsandallnineps，itisarentthatthealitudesofthoseoftheinnerpsarechsllerthanthoseofallninepsbsp;thealitudesoftheouterfivepsarechrgerthanthoseoftheinnerpsthisdoesnotanthattheinnerterrestrialparysubsystenbsp;isrestablethantheouteronebsp;thisissilyaresultoftheretivesllnessofthessesofthefourterrestrialpsaredwiththoseoftheouterjovianpsanotherthingwenoticeisthattheinnerparysubsystenbsp;ybeunstablererapidlythantheouteronebecauseofitsshorterorbitalti-scalesthiscanbeseeninthepanelsdenotedasinner4infig7wherethelonger-periodicandirregurosciltionsarerearentthaninthepanelsdenotedastotal9actually，thefluctuationsintheinner4panelsaretoargeextentasaresultoftheorbitalvariationofthercuryhowever，wecannotneglectthentributionfronbsp;otherterrestrialps，aswewillseeinsubsequentsections

44long-ternbsp;uplingofseveralneighbouringppairs

letusseesoindividualvariationsofparyorbitalenergyandangurntunbsp;expressedbythelow-passfiltereddeunayelentsfigs10and11showlong-ternbsp;evolutionoftheorbitalenergyofeachpandtheangurntunbsp;inn+1andn−2integrationswenoticethatsopsfornbsp;arentpairsinteroforbitalenergyandangurntunbsp;exchangeinparticur，venusandearthkeatypicalpairinthefigures，theyshownegativerretionsinexchangeofenergyandpositiverretionsinexchangeofangurntunbsp;thenegativerretioninexchangeoforbitalenergyansthatthetwopsfornbsp;acloseddynacalsystenbsp;interoftheorbitalenergythepositiverretioninexchangeofangurntunbsp;ansthatthetwopsaresiltaneouslyundercertainlong-ternbsp;perturbationscandidatesforperturbersarejupiterandsaturnalsoinfig11，wecanseethatrsshowsapositiverretionintheangurntunbsp;variationtothevenus–earthsystenbsp;rcuryexhibitscertainnegativerretionsintheangurntunbsp;versusthevenus–earthsystewhichseetobeareactioncausedbythenservationofangurntunbsp;intheterrestrialparysubsystebr>

itisnotclearatthentwhythevenus–earthpairexhibitsanegativerretioninenergyexchangeandapositiverretioninangurntunbsp;exchangeweypossiblyexpinthisthroughobservingthegeneralfactthattherearenosecurterinparysejoraxesuptosend-orderperturbationtheoriesthisansthattheparyorbitalenergyghtbechlessaffectedbyperturbingpsthanistheangurntunbsp;exchangehence，theeentricitiesofvenusandearthcanbedisturbedeasilybyjupiterandsaturn，whichresultsinapositiverretionintheangurntunbsp;exchangeontheotherhand，thesejoraxesofvenusandeartharelesslikelytobedisturbedbythejovianpsthustheenergyexchangeybelitedonlywithinthevenus–earthpair，whichresultsinanegativerretionintheexchangeoforbitalenergyinthepair

asfortheouterjovianparysubsystejupiter–saturnanduranus–neptuneseenbsp;tokedynacalpairshowever，thestrengthoftheiruplingisnotasstrongaredwiththatofthevenus–earthpair

5±5x1010-yrintegrationsofouterparyorbits

sincethejovianparyssesarechrgerthantheterrestrialparysses，wetreatthejovianparysystenbsp;asanindependentparysystenbsp;interofthestudyofitsdynacalstabilityhence，weaddedaupleoftrialintegrationsthatspan±5x1010yr，includingonlytheouterfivepstheresultsexhibittherigorousstabilityoftheouterparysystenbsp;overthislongti-spanorbitalnfigurations，andvariationofeentricitiesandinclinationsshowthisverylong-ternbsp;stabilityoftheouterfivepsinboththetiandthefrequencydoinsalthoughwedonotshowpshere，thetypicalfrequencyoftheorbitalosciltionofplutoandtheotherouterpsisalstnstantduringtheseverylong-ternbsp;integrationperiods，whichisdenstratedintheti–frequencypsonourwebpage

inthesetwointegrations，theretivenuricalerrorinthetotalenergywas∼10−6andthatofthetotalangurntunbsp;was∼10−10

51resonancesintheneptune–plutosystebr>

kinoshita&a;a;a;a;nakaiintegratedtheouterfiveparyorbitsover±55x109yrtheyfoundthatfourjorresonancesbetweenneptuneandplutoareintainedduringthewholeintegrationperiod，andthattheresonancesybetheincausesofthestabilityoftheorbitofplutothejorfourresonancesfoundinpreviousresearchareasfollowsinthefollowingdescription，λdenotestheanlongitude，Ωisthelongitudeoftheascendingnodeandϖisthelongitudeofperihelionsubscriptspandndenoteplutoandneptune

antionresonancebetweenneptuneandplutothecriticalarguntθ1=3λp−2λn−ϖplibratesaround180°withanalitudeofabout80°andalibrationperiodofabout2x104yr

thearguntofperihelionofplutowp=θ2=ϖp−Ωplibratesaround90°withaperiodofabout38x106yrthedonantperiodicvariationsoftheeentricityandinclinationofplutoaresynchronizedwiththelibrationofitsarguntofperihelionthisisanticipatedinthesecurperturbationtheorynstructedbykozai

thelongitudeofthenodeofplutoreferredtothelongitudeofthenodeofneptune，θ3=Ωp−Ωn，circutesandtheperiodofthiscircutionisequaltotheperiodofθ2librationwhenθ3beszero，iethelongitudesofascendingnodesofneptuneandplutooverp，theinclinationofplutobesxitheeentricitybesninbsp;andthearguntofperihelionbes90°whenθ3bes180°，theinclinationofplutobesnitheeentricitybesxinbsp;andthearguntofperihelionbes90°againwillia&a;a;a;a;bensonanticipatedthistypeofresonance，ternfirdbyni，nobili&a;a;a;a;carpino

anarguntθ4=ϖp−ϖn+3libratesaround180°withalongperiod，∼57x108yr

inournuricalintegrations，theresonances–arewellintained，andvariationofthecriticalarguntsθ1，θ2，θ3reinsirduringthewholeintegrationperiodhowever，thefourthresonanceearstobedifferentbsp;thecriticalarguntθ4alternateslibrationandcircutionovera1010-yrti-scalethisisaninterestingfactthatkinoshita&a;a;a;a;nakai'sshorterintegrationswerenotabletodisclose

6discussion

whatkindofdynacalchanisnbsp;intainsthislong-ternbsp;stabilityoftheparysystenbsp;wecaniediatelythinkoftwojorfeaturesthatyberesponsibleforthelong-ternbsp;stabilityfirst，thereseenbsp;tobenosignificantlower-orderresonancesbetweenanypairangtheninepsjupiterandsaturnareclosetoa5nbsp;antionresonance，butnotjustintheresonancezonehigher-orderresonancesycausethechaoticnatureoftheparydynacaltion，buttheyarenotsostrongastodestroythestableparytionwithinthelifetioftherealsorsystenbsp;thesendfeature，whichwethinkisreiortantforthelong-ternbsp;stabilityofourparysysteisthedifferenceindynacaldistancebetweenterrestrialandjovianparysubsystewhenweasureparyseparationsbythetualhillradii，separationsangterrestrialpsaregreaterthan26rh，whereasthoseangjovianpsarelessthan14rhthisdifferenceisdirectlyretedtothedifferencebetweendynacalfeaturesofterrestrialandjovianpsterrestrialpshavesllersses，shorterorbitalperiodsandwiderdynacalseparationtheyarestronglyperturbedbyjovianpsthathavergersses，longerorbitalperiodsandnarrowerdynacalseparationjovianpsarenotperturbedbyanyotherssivebodies

thepresentterrestrialparysystenbsp;isstillbeingdisturbedbythessivejovianpshowever，thewideseparationandtualinteractionangtheterrestrialpsrendersthedisturbanceineffective;thedegreeofdisturbancebyjovianpsiso，sincethedisturbancecausedbyjovianpsisaforcedosciltionhavinganalitudeofoheighteningofeentricity，forexaleo∼005，isfarfronbsp;sufficienttoprovokeinstabilityintheterrestrialpshavingsuchawideseparationas26rhthusweassuthatthepresentwidedynacalseparationangterrestrialpsisprobablyoneofthestsignificantnditionsforintainingthestabilityoftheparysystenbsp;overa109-yrti-spanourdetailedanalysisoftheretionshipbetweendynacaldistancebetweenpsandtheinstabilityti-scaleofsorsystenbsp;parytionisnowon-going

althoughournuricalintegrationsspanthelifetiofthesorsystethenuerofintegrationsisfarfronbsp;sufficienttofilltheinitialphasespaceitisnecessarytoperfornbsp;reandrenuricalintegrationstonfirnbsp;andexaneindetailthelong-ternbsp;stabilityofourparydynacs

——以上文段引自ito，t&a;a;a;tanikawa，klong-ternbsp;integrationsandstabilityofparyorbitsinoursorsystenbsp;nnotrastronsoc336，483–500

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