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Long-termiionsandstabilityofparyorbitsinourSorsystem
Abstract
Wepresenttheresultsofverylong-termnumericaliionsofparyorbitalmotionsover109-yrtime-spansincludingallnineps.Aquispeofournumericaldatashowsthattheparymotion，atleastinoursimpledynamicalmodel，seemstobequitestableevehisverylongtime-span.Acloserlookatthelowest-frequencyosciltionsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestrialparymotion，especiallythatofMercury.ThebehaviouroftheetricityofMercuryinouriionsisqualitativelysimirtotheresultsfrJacquesLaskar'ssecurperturbationtheory(e.g.emax～0.35over～±4Gyr).However，therearenoapparentsecurincreasesofetricityorinationinanyorbitalelementsoftheps，whichmayberevealedbystillloermnumericaliions.Wehavealsoperformedacoupleoftrialiionsincludingmotionsoftheouterfivepsoverthedurationof±5×1010yr.Theresultindicatesthatthethreemajorresonancesintheune–Plutosystemhavebeenmaintainedoverthe1011-yrtime-span.
1Introdu
1.1Definitionoftheproblem
ThequestionofthestabilityofourSorsystemhasbeeedoverseveralhundredyears，siheeraofon.Theproblemhasattractedmanyfamousmathematisovertheyearsandhaspyedatralroleinthedevelopmentofnon-lineardynamidchaostheory.However，wedonotyethaveadefiniteaothequestionofwhetherourSorsystemisstableornot.Thisispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenitisusediiontotheproblemofparymotionintheSorsystem.Actuallyitisnoteasytogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofourSorsystem.
Amongmanydefinitionsofstability，herettheHilldefinition(Gdman1993):actuallythisisnotadefinitionofstability，butofinstability.Wedefineasystemasbegunstablewhenacloseenteroccurssewhereinthesystem，startingfracertaininitialfiguration(Chambers，Wetherill&Boss1996;Ito&Tanikawa1999).AsystemisdefinedasexperiengacloseenterwhentwobodiesapproaeanotherwithinanareaofthergerHillradius.Otherwisethesystemisdefinedasbeingstable.HenceforwardwestatethatourparysystemisdynamicallystableifnocloseenterhappensduringtheageofourSorsystem，about±5Gyr.Ially，thisdefinitionmayberepcedbyoneinwhioccurrenceofanyorbitalcrossiweeherofapairofpakespce.Thisisbecauseweknowfrexperiehatanorbitalcrossingisverylikelytoleadtoacloseenteriaryandprotoparysystems(Yoshinaga，Kokubo&Makino1999).Ofcoursethisstatementotbesimplyappliedtosystemswithstableorbitalresonancessuchastheune–Plutosystem.
1.2Previousstudiesandaimsofthisresearch
Inadditiontothevaguenessoftheceptofstability，thepsinourSorsystemshowacharactertypicalofdynamicalchaos(Sussman&Wisd1988，1992).Thecauseofthischaoticbehaviourisnowpartlyuoodasbeingaresultofresonanceoverpping(Murray&Holman1999;Lecar，Franklin&Holman2001).However，itwouldrequireiingoveranensembleofparysystemsincludingallninepsforaperiodcseveral10Gyrtothhlyuandthelong-termevolutionofparyorbits，sincechaotiamicalsystemsarecharacterizedbytheirstrongdependeninitialditions.
Frthatpointofview，manyofthepreviouslong-termnumericaliionsincludedonlytheouterfiveps(Sussman&Wisd1988;Kinoshita&Nakai1996).Thisisbecausetheorbitalperiodsoftheouterpsaresomuchlohanthoseoftheinnerfourphatitismucheasiertofollowthesystemfiveniionperiod.Atpresent，thelonumericaliionspublishedinjournalsarethoseofDun&Lissauer(1998).Althoughtheirmaintargetwastheeffectofpost-main-sequenasslossoabilityofparyorbits，theyperformedmanyiionscupto～1011yroftheorbitalmotionsofthefourjovias.TheinitialorbitalelementsandmassesofpsarethesameasthoseofourSorsysteminDun&Lissauer'spaper，buttheydecreasethemassoftheSungraduallyintheirnumericalexperiments.Thisisbecausetheysidertheeffectofpost-main-sequenasslossinthepaper.sequently，theyfoundthatthecrossingtime-scaleofparyorbits，whichbeatypicalindicatoroftheinstabilitytime-scale，isquitesensitivetotherateofmassdecreaseoftheSun.WhenthemassoftheSunisclosetoitspresentvalue，thejoviasremainstableover1010yr，orperhapslonger.Dun&Lissaueralsoperformedfoursimirexperimentsontheorbitalmotionofseves(Venustoune)，whichcoveraspanof～109yr.Theirexperimentsonthesevesarenotyetcprehensive，butitseemsthattheterrestrialpsalsoremainstableduringtheiionperiod，maintainingalmurosciltions.
Oherhand，inhisaccuratesemi-analyticalsecurperturbationtheory(Laskar1988)，Laskarfindsthatrgeandirregurvariationsappearintheetricitiesandinationsoftheterrestrialps，especiallyofMercuryandMarsonatime-scaleofseveral109yr(Laskar1996).TheresultsofLaskar'ssecurperturbationtheoryshouldbefirmedandiigatedbyfullynumericaliions.
Inthispaperwepresentpreliminaryresultsofsixlong-termnumericaliionsonallnineparyorbits，caspanofseveral109yr，andoftwootheriionscaspanof±5×1010yr.Thetotalepsedtimeforalliionsismorethan5yr，usingseveraldedicatedPdworkstations.Oneofthefualclusionsofourlong-termiionsisthatSorsystemparymotioobestableintermsoftheHillstabilitymentionedabove，atleastoveratime-spanof±4Gyr.Actually，inournumericaliionsthesystemwasfarmorestablethanwhatisdefinedbytheHillstabilitycriterion:notonlydidnocloseenterhappenduringtheiionperiod，butalsoalltheparyorbitalelementshavebeenfinedinanarrionbothintimeandfrequencydain，thoughparymotionsarestochastic.Sihepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-termnumericaliions，weshowtypicalexamplefiguresasevidenceoftheverylong-termstabilityofSorsystemparymotion.Forreaderswhohavemorespecifiddeeperisinournumericalresults，reparedawebpage(access)，whereweshowraworbitalelements，theirlow-passfilteredresults，variationofDeunayelementsandangurmentumdeficit，asofoursimpletime–frequenalysisonallofouriions.
Iion2webrieflyexpinourdynamicalmodel，numericalmethodandinitialditionsusedinouriions.Se3isdevotedtoadescriptionofthequickresultsofthenumericaliions.Verylong-termstabilityofSorsystemparymotionisapparentbothiarypositionsandorbitalelements.Aroughestimationofnumericalerrorsisalsogiveion4goesontoadiscussionoftheloermvariationofparyorbitsusingalow-passfilterandincludesadiscussionofangurmentumdeficit.Iion5，wepresentasetofnumericaliionsfortheouterfivephatspans±5×1010yr.Iion6wealsodiscussthelong-termstabilityoftheparymotionanditspossiblecause.
2Descriptionofthenumericaliions
（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。）
2.3Numericalmethod
Weutilizeased-orderWisd–Holmansymplecticmapasourmainiiohod(Wisd&Holman1991;Kinoshita，Yoshida&Nakai1991)withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warmstart’(Saha&Tremaine1992，1994).
Thestepsizeforthenumericaliionsis8dthroughoutalliionsofthenineps(N±1，2，3)，whichisabout1/11oftheorbitalperiodoftheinnermostp(Mercury).Asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericaliionofallninepsinSussman&Wisd(1988，7.2d)andSaha&Tremaine(1994，225/32d).Werouhedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheaccumutionofround-offerrorinthecputationprocesses.Iiontothis，Wisd&Holman(1991)performednumericaliionsoftheouterfiveparyorbitsusingthesymplecticmapwithastepsizeof400d，1/10.83oftheorbitalperiodofJupiter.Theirresultseemstobeaccurateenough，whichpartlyjustifiesourmethodofdeterminingthestepsize.However，siheetricityofJupiter(～0.05)ismuchsmallerthanthatofMercury(～0.2)，weneedsecarewhenwecparetheseiionssimplyintermsofstepsizes.
Iegrationoftheouterfiveps(F±)，wefixedthestepsizeat400d.
tGauss'fandgfunsinthesymplecticmaptogetherwiththethird-orderHalleymethod(Danby1992)asasolverforKeplerequations.ThenumberofmaximumiteratioinHalley'smethodis15，buttheyneverreachedthemaximuminanyofouriions.
Theintervalofthedataoutputis200000d(～547yr)forthecalcutionsofallnineps(N±1，2，3)，andabout8000000d(～21903yr)fortheiionoftheouterfiveps(F±).
Althoughnooutputfilteringwasdonewhenthenumericaliionswereinprocess，liedalow-passfiltertotheraworbitaldataafterletedallthecalcutions.SeeSe4.1formoredetail.
2.4Errorestimation
2.4.1Retiveerrorsintotalenergyandangurmentum
Acctooneofthebasicpropertiesofsymplectitegrators，whichservethephysicallyservativequantitieswell(totalorbitalenergyandangurmentum)，ourlong-termnumericaliioohavebeenperformedwithverysmallerrors.Theaveragedretiveerrorsoftotalenergy(～10?9)andoftotalangurmentum(～10?11)haveremainednearlystantthroughouttheiionperiod(Fig.1).Thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedretiveerrorintotalenergybyaboutoneorderofmagnitudeormore.
RetivenumericalerrorofthetotalangurmentumδA/A0andthetotalenergyδE/E0inournumericaliionsN±1，2，3，whereδEandδAaretheabsolutegeofthetotalenergyandtotalangurmentum，respectively，andE0andA0aretheirinitialvalues.ThehorizontalunitisGyr.
hatdifferentoperatingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-offerrorhandlingandnumericalalgorithms.IntheupperpanelofFig.1，wereizethissituationintheseumericalerrorialangurmentum，whichshouldberigorouslypreserveduptomae-εprecision.
2.4.2Erroriarylongitudes
SihesymplecticmapspreservetotalenergyandtotalangurmentumofN-bodydynamicalsystemsilywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuracyofnumericaliions，especiallyasameasureofthepositionalerrorofps，i.e.theerroriarylongitudes.Toestimatethenumericalerrorintheparylongitudes，weperformedthefollowingprocedures.Wecparedtheresultofourmainlong-termiionswithsetestiions，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemainiions.Forthispurpose，weperformedamuchmoreaccurateiionwithastepsizeof0.125d(1/64ofthemainiions)spanning3×105yr，startingwiththesameinitialditionsasintheN?1iion.Wesiderthatthistestiionprovidesuswitha‘pseudo-true’solutionofparyorbitalevolutio，wecparethetestiionwiththemainiion，N?1.Fortheperiodof3×105yr，weseeadifferenmeananaliesoftheEarthbetweewoiionsof～0.52°(inthecaseoftheN?1iion).Thisdifferencebeextrapotedtothevalue～8700°，about25rotationsofEarthafter5Gyr，siheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.Simirly，thelongitudeerrorofPlutobeestimatedas～12°.ThisvalueforPlutoismuchbetterthantheresultinKinoshita&Nakai(1996)wherethedifferenceisestimatedas～60°.
3Numericalresults–I.Gtherawdata
Inthissewebrieflyreviewthelong-termstabilityofparyorbitalmotionthroughsesnapshotsofrawnumericaldata.Theorbitalmotionofpsindicateslong-termstabilityinallofournumericaliions:noorbitalcrossingsnorcloseentersbetweenanypairofpookpce.
3.1Generaldescriptionofthestabilityofparyorbits
First，webrieflylookatthegeneralcharacterofthelong-termstabilityofparyorbits.Ouriherefocusesparticurlyontheinnerfourterrestrialpsforwhichtheorbitaltime-scalesaremuchshorterthanthoseoftheouterfiveps.AsweseeclearlyfrthepnarorbitalfigurationsshowninFigs2and3，orbitalpositionsoftheterrestrialpsdifferlittlebetweentheinitialandfinalpartofeaumericaliion，whichspansseveralGyr.Thesolidlinesdenotingthepresentorbitsofthepsliealmostwithintheswarmofdotseveninthefinalpartofiions(b)and(d).Thisindicatesthatthroughouttheeegrationperiodthealmurvariationsofparyorbitalmotionremainnearlythesameastheyareatpresent.
Verticalviewofthefourinnerparyorbits(frthez-axisdire)attheinitialandfinalpartsoftheiionsN±1.Theaxesunitsareau.Thexy-pneissettotheinvariantpneofSorsystemtotalangurmentum.(a)TheinitialpartofN+1(t=0to0.0547×109yr).(b)ThefinalpartofN+1(t=4.9339×108to4.9886×109yr).(c)TheinitialpartofN?1(t=0to?0.0547×109yr).(d)ThefinalpartofN?1(t=?3.9180×109to?3.9727×109yr).Ineachpanel，atotalof23684pointsareplottedwithanintervalofabout2190yrover5.47×107yr.Solidlinesineachpaneldehepresentorbitsofthefourterrestrialps(takenfrDE245).
ThevariationofetricitiesandorbitalinationsfortheinnerfourpsintheinitialandfinalpartoftheiionN+1isshowninFig.4.Asexpected，thecharacterofthevariationofparyorbitalelementsdoesnotdiffersignifitlybetweentheinitialandfinalpartofeategration，atleastforVenus，EarthandMars.TheelementsofMercury，especiallyitsetricity，seemtogetoasignifitextent.Thisispartlybecausetheorbitaltime-scaleofthepistheshortestofalltheps，whichleadstoamorerapidorbitalevolutionthanotherps;theinnermostpmaybeoinstability.ThisresultappearstobeinseagreementwithLaskar's(1994，1996)expectationsthatrgeandirregurvariationsappearintheetricitiesandinationsofMercuryonatime-scaleofseveral109yr.However，theeffectofthepossibleinstabilityoftheorbitofMercurymaynotfatallyaffecttheglobalstabilityofthewholeparysystemowingtothesmallmassofMercury.Wewillmentionbrieflythelong-termorbitalevolutionofMercuryteriion4usinglow-passfilteredorbitalelements.
Theorbitalmotionoftheouterfivepsseemsrigorouslystableandquitereguroverthistime-span(seealsoSe5).
3.2Time–frequencymaps
Althoughtheparymotionexhibitsverylong-termstabilitydefinedastheenceofcloseenterevents，thechaotiatureofparydynamicsgetheosciltoryperiodandamplitudeofparyorbitalmotiongraduallyoversuchlongtime-spans.Evensuchslightfluctuationsoforbitalvariationinthefrequencydain，particurlyinthecaseofEarth，potentiallyhaveasignifiteffeitssurfaceclimatesystemthroughsorinsotionvariation(cf.Berger1988).
Togiveanoverviewofthelong-termgeinperiodicityiaryorbitalmotion，weperformedmanyfastFouriertransformations(FFTs)alongthetimeaxis，andsuperposedtheresultingperiodgramstodrawtwo-dimensionaltime–frequencymaps.Thespecificapproachtodrawingthesetime–frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisorLaskar's(1990，1993)frequenalysis.
Dividethelow-passfilteredorbitaldataintomanyfragmentsofthesamelength.Thelengthofeachdatasegmentshouldbeamultipleof2inordertoapplytheFFT.
Eachfragmentofthedatahasargeoverppingpart:forexample，whehdatabeginsfrt=tiandendsatt=ti+T，thedatasegmentrangesfrti+δT≤ti+δT+T，whereδT?T.WetihisdivisionuntilwereachacertainnumberNbywhi+Treachesthetotaliioh.
lyanFFTtoeachofthedatafragments，andobtainnfrequencydiagrams.
Ineachfrequencydiagramobtainedabove，thestrengthofperiodicityberepcedbyagrey-scale(orcolour)chart.
Weperformtherept，andectallthegrey-scale(orcolour)chartsintoonegraphforeategration.Thehorizontalaxisofthesenehsshouldbethetime，i.e.thestartingtimesofeachfragmentofdata(ti，wherei=1，…，n).Theverticalaxisrepresentstheperiod(orfrequency)oftheosciltionoforbitalelements.
WehaveadoptedanFFTbecauseofitsoverwhelmingspeed，siheamountofnumericaldatatobedecposedintofrequencyentsisterriblyhuge(severaltensofGbytes).
Atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasFig.5，whichshowsthevariationofperiodicityintheetricityandinationofEarthinN+2iion.InFig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordinateisstrohaninthelighterareaaroundit.WereizefrthismapthattheperiodicityoftheetricityandinationofEarthonlygesslightlyovertheentireperiodcoveredbytheN+2iion.Thisnearlyregurtrendisqualitativelythesameinotheriionsandforotherps，althoughtypicalfrequenciesdifferpbypandelementbyelement.
4.2Long-termexgeoforbitalenergyandangurmentum
Wecalcuteverylong-periodicvariationandexgeofparyorbitalenergyandangurmentumusingfilteredDeunayelementsL，G，H.GandHareequivalenttotheparyorbitalangurmentumanditsverticalentperunitmass.LisretedtotheparyorbitalenergyEperunitmassasE=?μ2/2L2.Ifthesystemiscpletelylinear，theorbitalenergyandtheangurmentumineachfrequencybinmustbestant.Non-liyintheparysystemcauseanexgeofenergyandangurmentuminthefrequencydain.Theamplitudeofthelowest-frequencyosciltionshouldincreaseifthesystemisunstableandbreaksdowngradually.However，suchasymptofinstabilityisnotpriinourlong-termiions.
InFig.7，thetotalorbitalenergyandangurmentumofthefourinnerpsandallninepsareshownforiionN+2.Theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedasE-E0)，totalangurmentum(G-G0)，andtheverticalent(H-H0)oftheinnerfourpscalcutedfrthelow-passfilteredDeunayelements.E0，G0，H0deheinitialvaluesofeachquantity.Theabsolutedifferencefrtheinitialvaluesisplottedinthepanels.ThelowerthreepanelsineachfigureshowE-E0，G-G0andH-H0ofthetotalofnineps.Thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovias.
Cparingthevariationsofenergyandangurmentumoftheinnerfourpsandallnineps，itisapparentthattheamplitudesofthoseoftheinnerpsaremuchsmallerthanthoseofallnineps:theamplitudesoftheouterfivepsaremuchrgerthanthoseoftheinnerps.Thisdoesnotmeanthattheierrestrialparysubsystemismorestablethaerohisissimplyaresultoftheretivesmallnessofthemassesofthefourterrestrialpscparedwiththoseoftheouterjovias.Ahiiceisthattheinnerparysubsystemmaybestablemorerapidlythaeronebecauseofitsshorterorbitaltime-scales.Thisbeseeninthepanelsdenotedasinner4inFig.7wherethelonger-periodidirregurosciltionsaremoreapparentthaninthepanelsdenotedastotal9.Actually，thefluctuationsintheinner4panelsareteextentasaresultoftheorbitalvariationoftheMercury.However，weothetributionfrotherterrestrialps，aswewillseeinsubsequeions.
4.4Long-termcouplingofseveralneighbppairs
Letusseeseindividualvariationsofparyorbitalenergyandangurmentumexpressedbythelow-passfilteredDeunayelements.Figs10and11showlong-termevolutionoftheorbitalenergyofeaetandtheangurmentuminN+1andN?2iions.Wenoticethatsepsformapparentpairsintermsoforbitalenergyandangurmentumexge.Inparticur，Venusahmakeatypicalpair.Inthefigures，theyshowivecorretionsinexgeofenergyandpositivecorretionsinexgeofangurmentum.Theivecorretioninexgeoforbitalenergymeansthatthetwopsformacloseddynamicalsystemintermsoftheorbitalenergy.Thepositivecorretioninexgeofangurmentummeansthatthetwopsaresimultaneouslyundercertainlong-termperturbations.didatesforperturbersareJupiterandSaturn.AlsoinFig.11，weseethatMarsshoositivecorretionintheangurmentumvariationtotheVehsystem.MercuryexhibitscertaiivecorretionsintheangurmentumversustheVehsystem，whichseemstobeareacausedbytheservationofangurmentumierrestrialparysubsystem.
ItisnotclearatthementwhytheVehpairexhibitsaivecorretioninenergyexgeandapositivecorretioninangurmentumexge.ossiblyexpinthisthroughthegeneralfactthattherearenosecurtermsiarysemimajoraxesuptosed-orderperturbationtheories(cf.Brouwer&Clemence1961;Boccaletti&Pucacco1998).Thismeansthattheparyorbitalenergy(whichisdirectlyretedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbingphanistheangurmentumexge(whichretestoe).HeheetricitiesofVenusahbedisturbedeasilybyJupiterandSaturn，whichresultsinapositivecorretionintheangurmentumexge.Oherhand，thesemimajoraxesofVenusaharelesslikelytobedisturbedbythejovias.ThustheenergyexgemaybelimitedonlywithintheVehpair，whichresultsinaivecorretionintheexgeoforbitalenergyinthepair.
Asfortheouterjoviaarysubsystem，Jupiter–SaturnandUranus–uomakedynamicalpairs.However，thestrengthoftheircouplingisnotasstrongcparedwiththatoftheVehpair.
5±5×1010-yriionsofouterparyorbits
Sihejoviaarymassesaremuchrgerthaerrestrialparymasses，wetreatthejoviaarysystemasanindepeparysystemintermsofthestudyofitsdynamicalstability.Hence，weaddedacoupleoftrialiionsthatspan±5×1010yr，includingonlytheouterfiveps(thefourjoviasplusPluto).Theresultsexhibittherigorousstabilityoftheouterparysystemoverthislongtime-span.Orbitalfigurations(Fig.12)，andvariationofetricitiesandinations(Fig.13)showthisverylong-termstabilityoftheouterfivepsinboththetimeandthefrequencydains.Althoughwedonotshoshere，thetypicalfrequencyoftheorbitalosciltionofPlutoandtheotherouterpsisalmoststantduringtheseverylong-termiionperiods，whichisdemonstratediime–frequencymapsonourwebpage.
Iwoiions，theretivenumericalerrorialenergywas～10?6andthatofthetotalangurmentumwas～10?10.
5.1Resonancesintheune–Plutosystem
Kinoshita&Nakai(1996)iedtheouterfiveparyorbitsover±5.5×109yr.TheyfoundthatfourmajorresonancesbetweeuneandPlutoaremaintainedduringthewholeiionperiod，andthattheresonancesmaybethemaincausesofthestabilityoftheorbitofPluto.Themajorfourresonancesfoundinpreviousresearchareasfollows.Inthefollowingdescription，λdehemeanlongitude，Ωisthelongitudeoftheasdingnodeand?isthelongitudeofperihelion.SubscriptsPandePlutoaune.
MeanmotionresoweeuneandPluto(3:2).Thecriticalargumentθ1=3λP?2λN??Plibratesaround180°withanamplitudeofabout80°andalibrationperiodofabout2×104yr.
TheargumentofperihelionofPlutoωP=θ2=?P?ΩPlibratesaround90°withaperiodofabout3.8×106yr.ThedinantperiodicvariationsoftheetricityandinationofPlutoaresynizedwiththelibrationofitsargumentofperihelion.ThisisanticipatedinthesecurperturbationtheorystructedbyKozai(1962).
ThelongitudeofthenodeofPlutoreferredtothelongitudeofthenodeofune，θ3=ΩP?ΩN，circutesandtheperiodofthiscircutionisequaltotheperiodofθ2libration.Whenθ3beceszero，i.e.thelongitudesofasdingnodesofuneandPlutooverp，theinationofPlutobecesmaximum，theetricitybecesminimumandtheargumentofperihelionbeces90°.Whenθ3beces180°，theinationofPlutobecesminimum，theetricitybecesmaximumandtheargumentofperihelionbeces90°again.Williams&Benson(1971)anticipatedthistypeofresoerfirmedbyMini，Nobili&Carpino(1989).
Anargumentθ4=?P??N+3(ΩP?ΩN)libratesaround180°withalongperiod，～5.7×108yr.
Inournumericaliions，theresonances(i)–(iii)arewellmaintained，andvariationofthecriticalargumentsθ1，θ2，θ3remainsimirduringthewholeiionperiod(Figs14–16).However，thefourthresonance(iv)appearstobedifferent:thecriticalargumentθ4alternateslibrationandcircutionovera1010-yrtime-scale(Fig.17).ThisisaniingfactthatKinoshita&Nakai's(1995，1996)shorteriionswerenotabletodisclose.
6Discussion
Whatkindofdynamicalmeismmaintainsthislong-termstabilityoftheparysystem?Weimmediatelythinkoftwomajorfeaturesthatmayberesponsibleforthelong-termstability.First，thereseemtobenosignifitlower-orderresonances(meanmotionandsecur)betweenanypairamongthenineps.JupiterandSaturnareclosetoa5:2meanmotionresohefamous‘greatinequality’)，butnotjustintheresonancezone.Higher-orderresonancesmaycausethechaotiatureoftheparydynamicalmotion，buttheyarenotsastodestroythestableparymotionwithinthelifetimeoftherealSorsystem.Thesedfeature，whichwethinkismoreimportantforthelong-termstabilityofourparysystem，isthedifferendynamicaldistaweenterrestrialandjoviaarysubsystems(Ito&Tanikawa1999，2001).WhenwemeasureparyseparationsbythemutualHillradii(R_)，separationsamongterrestrialpsaregreaterthan26RH，whereasthoseamongjoviasarelessthan14RH.Thisdifferenceisdirectlyretedtothediffereweendynamicalfeaturesofterrestrialandjovias.Terrestrialpshavesmallermasses，shorterorbitalperiodsandwiderdynamicalseparation.Theyarestronglyperturbedbyjoviahathavergermasses，longerorbitalperiodsandnarrowerdynamicalseparation.Joviasarenotperturbedbyanyothermassivebodies.
Thepresentterrestrialparysystemisstillbeingdisturbedbythemassivejovias.However，thewideseparationandmutualiionamongtheterrestrialpsrehedisturbaneffective;thedegreeofdisturbancebyjoviasisO(eJ)(orderofmagnitudeoftheetricityofJupiter)，sihedisturbancecausedbyjoviasisaforcedosciltionhavinganamplitudeofO(eJ).Heighteningofetricity，forexampleO(eJ)～0.05，isfarfrsuffittoprovokeinstabilityierrestrialpshavingsuchawideseparationas26RH.Thusweassumethatthepresentwidedynamicalseparationamongterrestrialps(>26RH)isprobablyoneofthemostsignifitditionsformaintainingthestabilityoftheparysystemovera109-yrtime-span.Ourdetailedanalysisoftheretionshipbetweendynamicaldistaweesandtheinstabilitytime-scaleofSorsystemparymotionisnowon-going.
AlthoughournumericaliionsspanthelifetimeoftheSorsystem，thenumberofiionsisfarfrsuffittofilltheinitialphasespace.Itisnecessarytoperformmoreandmorenumericaliionstofirmandexamineiailthelong-termstabilityofourparydynamics.
——以上文段引自Ito，T.&Tanikawa，K.Long-termiionsandstabilityofparyorbitsinourSorSystem.Mon.Not.R.Astron.Soc.336，483–500(2002)
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