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以下是文章内容：
long-termiionsandstabilityofparyorbitsinoursorsystem
abstract
wepresenttheresultsofverylong-termnumeritegrationsofparyorbitalmotionsover109-yrtime-spansincludingalls.aquispeofournumericaldatashowsthattheparymotion，atleastinoursimpledynamicalmodel，seemstobequitestableevehisverylongtime-span.acloserlookatthelowest-frequencyosciltionsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestrialparymotion，especiallythatofmercury.thebehaviouroftheetriercuryinouriionsisqualitativelysimirtotheresultsfromjacquesskar'ssecurperturbationtheory(e.g.emax～0.35yr).however，therearenoapparentsecreasesofetricityorinationinanyorbitalelementsoftheps，whichmayberevealedbystillloermnumeritegrations.wehavealsoperformedacoupleoftrialiionsincludingmotionsoftheouterfivepsoverthedurationof±5x1010yr.theresultindicatesthatthethreemajorresoheune–plutosystemhavebeenmaintainedoverthe1011-yrtime-span.
1introdu
1.1definitionoftheproblem
thequestionofthestabilityofoursorsystemhasbeeedoverseveralhundredyears，siheeraofon.theproblemhasattractedmanyfamousmathematisovertheyearsandhaspyedatralroleinthedevelopmentofnon-lineardynamidchaostheory.however，wedohaveadefiniteahequestionofwhetheroursorsystemisstableornot.thisispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenitisusediiontotheproblemofparymotioninthesorsystem.actuallyitisogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofoursorsystem.
amongmanydefinitionsofstability，heretthehilldefinition(gdman1993):actuallythisisnotadefinitionofstability，butofinstability.wedefineasystemasbeingunstablewhenacloseenteroccurssomewhereiem，startingfromacertaininitialfiguration(chambers，wetherill&ito&&tanikawa1999).asystemisdefinedasexperiengacloseenterwhentwobodiesapproaeahinahergerhillradius.otherwisethesystemisdefinedasbeingstable.henceforwardwestatethatourparysystemisdynamicallystableifnocloseenterhappensduringtheageofoursorsystem，about±5gyr.ially，thisdefinitionmayberepcedbyoneinwhioccurrenceofanyorbitalcrossiweeherofapairofpakespce.thisisbecauseweknowfromexperieanorbitalcrossingisverylikelytoleadtoacloseenteriaryandprotoparysystems(yoshinaga，kokubo&&makino1999).ofcoursethisstatementotbesimplyappliedtosystemswithstableorbitalresonancessuchastheune–plutosystem.
1.2previousstudiesandaimsofthisresearch
inadditiontothevaguenessoftheceptofstability，thepsinoursorsystemshowacharactertypicalofdynamicalchaos(sussman&&wisdom1988，1992).thecauseofthischaoticbehaviourisnowpartlyuoodasbeiofresonanceoverpping(murray&lecar，franklin&&holman2001).however，itwouldrequireiingoveranensembleofparysystemsincludingallsforaperiodcseveral10gyrthlyuandthelong-termevolutionofparyorbits，sincechaotiamicalsystemsarecharacterizedbytheirstrongdependeninitialditions.
fromthatpointofview，manyofthepreviouslong-termnumeritegrationsincludedonlytheouterfiveps(sussman&kinoshita&&nakai1996).thisisbecausetheorbitalperiodsoftheouterpsaresomugerthanthoseoftheinnerfourphatitismucheasiertofollowthesystemfiveionperiod.atpresent，thelonumeritegrationspublishedinjournalsarethoseofdun&&lissauer(1998).althoughtheirmaintargetwastheeffectofpost-main-sequenasslossoabilityofparyorbits，theyperformedmanyiionscupto～1011yroftheorbitalmotionsofthefourjoviaheinitialorbitalelementsandmassesofpsarethesameasthoseofoursorsystemindun&&lissauer'spaper，buttheydecreasethemassofthesungraduallyintheirnumericalexperiments.thisisbecausetheysidertheeffectofpost-main-sequenasslossinthepaper.sequently，theyfoundthatthecrossingtime-scaleofparyorbits，whibeatypidicatoroftheinstabilitytime-scale，isquitesensitivetotherateofmassdecreaseofthesuhemassofthesunisclosetoitspresentvalue，thejoviasremainstableover1010yr，orperhapslonger.dun&&lissaueralsoperformedfoursimirexperimentsontheorbitalmotionofseves(venustoune)，whichcoveraspanof～109yr.theirexperimentsonthesevesarepreheitseemsthattheterrestrialpsalsoremainstableduriegrationperiod，maintainingalmurosciltions.
oherhand，inhisaccuratesemi-analyticalsecurperturbationtheory(skar1988)，skarfindsthatrgeandirregurvariationsappearintheetricitiesandinationsoftheterrestrialps，especiallyofmercuryandmarsonatime-scaleofseveral109yr(skar1996).theresultsofskar'ssecurperturbationtheoryshouldbefirmedandiigatedbyfullynumeritegrations.
inthispaperwepresentpreliminaryresultsofsixlong-termnumeritegrationsonallaryorbits，caspanofseveral109yr，andoftwootheriionscaspanof±5x1010yr.thetotalepsedtimeforalliionsismorethan5yr，usingseveraldedicatedpdworkstations.ohefualclusions-termiionsisthatsorsystemparymotioobestableintermsofthehillstabilitymentionedabove，atleastoveratime-spanof±4gyr.actually，inournumeritegratioemwasfarmorestablethanwhatisdefihehillstabilitycriterion:notonlydidnocloseenterhappenduriegrationperiod，butalsoalltheparyorbitalelementshavebeenfinedinanarrionbothintimeandfrequenain，thoughparymotioochasticethepurposeofthispaperistoexhibitandoverviewtheresults-termnumeritegrations，weshowtypicalexamplefiguresasevideheveryloabilityofsorsystemparymotion.forreaderswhohavemorespecifiddeeperisinournumericalresults，reparedawebpage(access)，whereweshowraworbitalelements，theirlow-passfilteredresults，variationofdeunayelementsandangurmomentumdeficit，asofoursimpletime–frequenalysisonallofouriions.
iion2webrieflyexpinourdynamicalmodel，numericalmethodandinitialditionsusedinouriioion3isdevotedtoadescriptionofthequickresultsofthenumeritegrations.veryloabilityofsorsystemparymotionisapparentbothiarypositionsandorbitalelements.aroughestimationofnumericalerrorsisalsogiveiooadiscussionoftheloermvariationofparyorbitsusingalow-passfilterandincludesadiscussionofangurmomentumdeficit.iion5，wepreseofnumeritegrationsfortheouterfivephatspans±5x1010yr.iion6wealsodiscusstheloabilityoftheparymotionanditspossiblecause.
2descriptionofthenumeritegrations
（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-orderwisdom–holmansymplecticmapasourmaiiohod(wisdom&kinoshita，yoshida&&nakai1991)ecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warmstart’(saha&&tremaine1992，1994).
thestepsizeforthenumeritegrationsis8dthroughoutalliionsofthes(n±1，2，3)，whichisabout111oftheorbitalperiodoftheinnermostp(mercury).asforthedeterminationofstepsize，wepartlyfollowthepreviousnumeritegrationofallsinsussman&&wisdom(1988，7.2d)andsaha&&tremaine(1994，22532d).werouhedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2ioreducetheaccumutionofround-offerrorintheputationprocesses.iiontothis，wisdom&&holman(1991)performednumeritegrationsoftheouterfiveparyorbitsusingthesymplecticmapsizeof400d，110.83oftheorbitalperiodofjupiter.theirresultseemstobeaccurateenough，whichpartlyjustifiesourmethodofdeterminiepsize.however，siheetricityofjupiter(～0.05)ismuchsmallerthanthatofmercury(～0.2)，weneedsomecarewhenweparetheseiionssimplyintermsofstepsizes.
iegrationoftheouterfiveps(f±)，wefixedthestepsizeat400d.
tgauss'fandgfunsinthesymplecticmaptogetherwiththethird-orderhalleymethod(danby1992)asasolverforkeplerequations.thenumberofmaximumiteratioinhalley'smethodis15，buttheyneverreachedthemaximuminanyofouriions.
theintervalofthedataoutputis200000d(～547yr)forthecalcutionsofalls(n±1，2，3)，andabout8000000d(～21903yr)fortheiionoftheouterfiveps(f±).
althoughnooutputfilteringwasdohenumeritegrationswereinprocess，liedalow-passfiltertotheraworbitaldataafterwehadpletedallthecalcutions.seese4.1formoredetail.
2.4errorestimation
2.4.1retiveerrorsintotalenergyandangurmomentum
acctoohebasicpropertiesofsymplectitegrators，whiservethephysicallyservativequantitieswell(totalorbitalenergyandangurmomentum)，-termnumeritegratioohavebeenperformedwithverysmallerrors.theaveragedretiveerrorsoftotalenergy(～10?9)andoftotangurmomentum(～10?11)haveremainednearlystantthroughouttheiionperiod(fig.1).thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedretiveerrorintotalenergybyaboutoneorderofmagnitudeormore.
retivenumericalerrorofthetotangurmomentumδaa0aalenergyδee0inournumeritegrationsn±1，2，3，whereδeandδaaretheabsolutegeofthetotalenergyandtotangurmomentum，respectively，ande0anda0aretheirinitialvalues.thehorizontalunitisgyr.
differeingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-offerrorhandlingandnumericalgorithms.intheupperpaneloffig.1，wereizethissituationintheseumericalerroriangurmomentum，whichshouldberigorouslypreserveduptomae-eprecision.
2.4.2erroriarylongitudes
sihesymplecticmapspreservetotalenergyandtotangurmomentumofn-bodydynamicalsystemsilywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuraeritegrations，especiallyasameasureofthepositionalerrorofps，i.e.theerroriarylongitudes.toestimatethenumericalerroriarylongitudes，weperformedthefollowingprocedures.weparedtheresultofourmainlong-termiionswithsometestiions，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemaiions.forthispurpose，weperformedamuchmoreaccurateiionsizeof0.125d(164ofthemaiions)spanning3x105yr，startingwiththesameinitialditionsasinthen?1iion.wesiderthatthistestiionprovidesusseudo-true’solutionofparyorbitalevolutio，weparethetestiionwiththemaiion，n?1.fortheperiodof3x105yr，weseeadifferenmeananomaliesoftheearthbetweewoiionsof～0.52°(inthecaseofthen?1iion).thisdifferenbeextrapotedtothevalue～8700°，about25rotatiohafter5gyr，siheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.simirly，thelongitudeerrorofplutobeestimatedas～12°.thisvalueforplutoismuchbetterthantheresultinkinoshita&&nakai(1996)wherethedifferenceisestimatedas～60°.
3numericalresults–i.gherawdata
inthissewebrieflyreviewtheloabilityofparyorbitalmotihsomesnapshotsofrawnumericaldata.theorbitalmotionofpsindicatesloabilityinallofournumeritegrations:noorbitalcrossingsnorcloseentersbetairofpookpce.
3.1generaldescriptionofthestabilityofparyorbits
first，webrieflylookatthegeneralcharacteroftheloabilityofparyorbits.ouriherefocusesparticurlyontheinnerfourterrestrialpsforwhichtheorbitaltime-scalesaremuchshorterthanthoseoftheouterfiveps.asweseeclearlyfromthepnarorbitalfigurationsshowninfigs2and3，orbitalpositionsoftheterrestrialpsdifferlittlebetweeiandfinalpartofeaumeritegration，whichspansseveralgyr.thesolidliingthepresentorbitsofthepsliealmostwithintheswarmofdotseveninthefinalpartofiions(b)and(d).thisindicatesthatthroughouttheeegrationperiodthealmurvariationsofparyorbitalmotionremainnearlythesameastheyareatpresent.
verticalviewofthefourinnerparyorbits(fromthez-axisdire)attheinitiandfinalpartsoftheiionsn±1.theaxesunitsareau.thexy-pneissettotheinvariantpneofsorsystemtotangurmomentum.(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).ineael，atotalof23684pointsareplottedwithanintervalofabout2190yrover5.47x107yr.solidlinesineaeldehepresentorbitsofthefourterrestrialpakenfromde245).
thevariationofetricitiesandorbitalinationsfortheinnerfourpsiiandfinalpartoftheiionn+1isshowninfig.4.asexpected，thecharacterofthevariationofparyorbitalelementsdoesnotdiffersignifitlybetweeiandfinalpartofeategration，atleastforvehandmars.theelementsofmercury，especiallyitsetricity，seemtogetoasignifitextent.thisispartlybecausetheorbitaltime-scaleofthepheshortestofalltheps，whichleadstoamorerapidorbitalevolutionthanotherpheinnermostpmaybeoinstability.thisresultappearstobeinsomeagreementwithskar's(1994，1996)expectationsthatrgeandirregurvariationsappearintheetricitiesandinationsofmercuryonatime-scaleofseveral109yr.however，theeffectofthepossibleinstabilityoftheorbitofmercurymaynotfatallyaffecttheglobalstabilityofthewholeparysystemowingtothesmallmassofmercury.wewillmentiohelong-termorbitalevolutionofmercuryteriion4usinglow-passfilteredorbitalelements.
theorbitalmotionoftheouterfivepsseemsrigorouslystableandquitereguroverthistime-span(seealsose5).
3.2time–frequencymaps
althoughtheparymotionexhibitsveryloabilitydefiheenceofcloseenterevents，thechaotiatureofparydynamigetheosciltoryperiodandamplitudeofparyorbitalmotiongraduallyoversugtime-spans.evensuchslightfluctuationsoforbitalvariationinthefrequenain，particurlyinthecaseofearth，potentiallyhaveasignifiteffeitssurfaceclimatesystemthroughsorinsotionvariation(cf.berger1988).
togiveanoverviewofthelong-termgeinperiodicityiaryorbitalmotion，weperformedmanyfastfouriertransformations(ffts)aloimeaxis，andsuperposedtheresultingperiodgramstodrawtwo-dimensionaltime–frequencymaps.thespecificapproachtodrawiime–frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisorskar's(1990，1993)frequenalysis.
dividethelow-passfilteredorbitaldataintomanyfragmentsofthesamelenh.thelenhofeachdatasegmentshouldbeamultipleof2ioapplythefft.
eachfragmentofthedatahasargeoverppingpart:forexample，whehdatabeginsfromt=tiandendsatt=ti+t，thedatasegmentrangesfromti+δt≤ti+δt+t，whereδt?t.wetihisdivisionuntilwereachacertainnumbernbywhi+treachesthetotaliionlenh.
lyanffttoeachofthedatafragments，andobtainnfrequencydiagrams.
ineachfrequencydiagramobtainedabove，thestrenhofperiodicityberepcedbyagrey-scale(orcolour)chart.
weperformtherept，andectallthegrey-scale(orcolour)chartsintoonegraphforeategration.thehorizontaxisofthesenehsshouldbethetime，i.e.thestartingtimesofeachfragmentofdata(ti，wherei=1，…，n).theverticaxisrepresentstheperiod(orfrequency)oftheosciltionoforbitalelements.
wehaveadoptedanfftbecauseofitsoverwhelmingspeed，siheamountofnumericaldatatobedeposedintofrequenposisterriblyhuge(severaltensofgbytes).
atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasfig.5，whichshowsthevariationofperiodicityintheetricityandinatiohinn+2iion.infig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordirohaninthelighterareaaroundit.wereizefromthismapthattheperiodicityoftheetricityandinatiohonlygesslightlyovertheentireperiodcoveredbythen+2iion.thisnearlyregurtrendisqualitativelythesameiegrationsandforotherps，althoughtypicalfrequenciesdifferpbypabyelement.
4.2long-termexgeoforbitalenergyandangurmomentum
wecalcuteverylong-periodicvariationandexgeofparyorbitalenergyandangurmomentumusingfiltereddeunayelementsl，g，h.gandhareequivalenttotheparyorbitangurmomentumanditsvertipoperunitmass.lisretedtotheparyorbitalenergyeperunitmassase=?μ22l2.ifthesystemispletelyliheorbitalenergyandtheangurmomentumineachfrequenmustbestant.non-liyiarysystemcauseanexgeofenergyandangurmomentuminthefrequenain.theamplitudeofthelowest-frequencyosciltionshouldincreaseifthesystemisunstableandbreaksdowngradually.however，suchasymptomofinstabilityisnotpromiin-termiions.
infig.7，thetotalorbitalenergyandangurmomentumofthefourinnerpsandallsareshownforiionn+2.theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedase-e0)，totangurmomentum(g-g0)，aipo(h-h0)oftheinnerfourpscalcutedfromthelow-passfiltereddeunayelements.e0，g0，h0deheinitialvaluesofeachquantity.theabsolutedifferentheinitialvaluesisplottedinthepahelowerthreepanelsineachfigureshowe-e0，g-g0andh-h0ofthetotalofs.thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovias.
paringthevariationsofenergyandangurmomentumoftheinnerfourpsandalls，itisapparentthattheamplitudesofthoseoftheinnerpsaremuchsmallerthanthoseofalls:theamplitudesoftheouterfivepsaremuchrgerthanthoseoftheinnerphisdoeshattheierrestrialparysubsystemismorestablethaerohisissimplyaresultoftheretivesmallnessofthemassesofthefourterrestrialpsparedwiththoseoftheouterjovias.ahiiceisthattheinnerparysubsystemmaybeeunstablemorerapidlythaeronebecauseofitsshorterorbitaltime-scales.thisbeseeninthepanelsdenotedasinner4infig.7wherethelonger-periodidirregurosciltionsaremoreapparentthaninthepanelsdeotal9.actually，thefluctuationsintheinner4panelsareteextentasaresultoftheorbitalvariationofthemercury.however，weothetributionfromotherterrestrialps，aswewillseeinsubsequeions.
4.4long-termcouplingofseveralneighbppairs
letusseesomeindividualvariationsofparyorbitalenergyandangurmomentumexpressedbythelow-passfiltereddeunayelements.figs10and11showlong-termevolutionoftheorbitalenergyofeaetandtheangurmomentuminn+1andn?2iioicethatsomepsformapparentpairsintermsoforbitalenergyandangurmomentumexge.inparticur，venusahmakeatypicalpair.inthefigures，theyshowivecorretionsinexgeofenergyandpositivecorretionsinexgeofangurmomentuhepositivecorretioninexgeofangurmomentummeansthatthetwopsaresimultaneouslyundercertainlong-termperturbations.didatesforperturbersarejupiterandsaturn.alsoinfig.11，weseethatmarsshows'itivecorretionintheangurmomentumvariationtothevehsystem.mercuryexhibitscertaiivecorretionsintheangurmomentumversusthevehsystem，whichseemstobeareacausedbytheservationofangurmomentumierrestrialparysubsystem.
itisnotclearatthemomentwhythevehpairexhibitsaivecorretioninenergyexgeand'itivecorretioninangurmomentumexge.ossiblyexpinthisthroughthegeneralfactthatthereareermsiarysemimajoraxesuptosed-orderperturbationtheories(cf.brou;boccaletti&&pucacco1998).thismeansthattheparyorbitalenergy(whichisdirectlyretedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbihanistheangurmomentumexge(whichretestoe).heheetricitiesofvenusahbedisturbedeasilybyjupiterandsaturn，whichresultsin'itivecorretionintheangurmomentumexge.oherhand，thesemimajoraxesofvenusaharelesslikelytobedisturbedbythejoviahustheenergyexgemaybelimitedonlywithinthevehpair，whichresultsiivecorretionintheexgeoforbitalenergyinthepair.
asfortheouterjoviaarysubsystem，jupiter–saturnanduranus–uomakedynamicalpairs.however，thestrenhoftheircouplingisnotasstrongparedwiththatofthevehpair.
5±5x1010-yriionsofouterparyorbits
sihejoviaarymassesaremuchrgerthaerrestrialparymasses，wetreatthejoviaarysystemasanindepeparysystemintermsofthestudyofitsdynamicalstability.hence，weaddedacoupleoftrialiionsthatspan±5x1010yr，includingonlytheouterfivephefourjoviaspluspluto).theresultsexhibittherigorousstabilityoftheouterparysystemoverthislongtime-span.orbitalfigurations(fig.12)，andvariationofetricitiesandinations(fig.13)showthisveryloabilityoftheouterfivepsinboththetimeandthefrequenains.althoughwedonotshoshere，thetypicalfrequencyoftheorbitalosciltionofplutoaherouterpsisalmoststantduringtheseverylong-termiionperiods，whichisdemonstratediime–frequencymapsonourwebpage.
iwoiions，theretivenumericalerrorialenergywas～10?6andthatofthetotangurmomentumwas～10?10.
5.1resoheune–plutosystem
kinoshita&&nakai(1996)iedtheouterfiveparyorbitsover±5.5x109yr.theyfoundthatfourmajorresoweeuneandplutoaremaintainedduringthewholeiionperiod，andthattheresonancesmaybethemaincausesofthestabilityoftheorbitofpluto.themajorfourresonancesfoundinpreviousresearchareasfollows.inthefollowingdescriptioesthemeanlongitude，Ωisthelongitudeoftheasdingnodeand?isthelongitudeofperihelion.subscriptspaeplutoaune.
meanmotionresoweeuneandpluto(3:2).thecriticargumentθ1=3λp?2λn??plibratesaround180°litudeofabout80°andalibrationperiodofabout2x104yr.
theargumentofperihelionofplutowp=θ2=?p?Ωplibratesaround90°eriodofabout3.8x106yr.thedominantperiodicvariationsoftheetricityandinationofplutoaresynizedwiththelibrationofitsargumentofperihelion.thisisanticipatedinthesecurperturbationtheorystructedbykozai(1962).
thelongitudeofthenodeofplutoreferredtothelongitudeofthenodeofune，θ3=Ωp?Ωn，circutesandtheperiodofthiscircutionisequaltotheperiodofθ2libration.whenθ3beeszero，i.e.thelongitudesofasdingnodesofuneandplutooverp，theinationofplutobeaximum，theetricitybeinimumandtheargumentofperihelionbees90°.whenθ3bees180°，theinationofplutobeinimum，theetricitybeaximumandtheargumentofperihelionbees90°again.williams&&benson(1971)anticipatedthistypeofresoerfirmedbymini，nobili&&carpino(1989).
anargumentθ4=?p??n+3(Ωp?Ωn)libratesaround180°withalongperiod，～5.7x108yr.
inournumeritegrations，theresonances(i)–(iii)arewellmaintained，andvariationofthecriticargumentsθ1，θ2，θ3remainsimirduringthewholeiionperiod(figs14–16).however，thefourthresonance(iv)appearstobedifferent:thecriticargumentθ4alternateslibrationandcircutionovera1010-yrtime-scale(fig.17).thisisaingfactthatkinoshita&&nakai's(1995，1996)shorteriionswerenotabletodisclose.
6discussion
whatkindofdynamicalmeismmaintainsthisloabilityoftheparysystem?weimmediatelythinkoftwomajorfeaturesthatmayberesponsiblefortheloability.first，thereseemtobenosignifitlower-orderresonances(meanmotionandsecur)betairamongthes.jupiterandsaturnareclosetoa5:2meanmotionresohefamous‘greatinequality’)，butnotjustintheresonane.higher-orderresonancesmaycausethechaotiatureoftheparydynamicalmotion，buttheyarenotsastodestroythestableparymotionwithinthelifetimeoftherealsorsystem.thesedfeature，whichwethinkismoreimportantfortheloabilityofourparysystem，isthedifferendynamicaldistaweenterrestriandjoviaarysubsystems(ito&&tanikawa1999，2001).whenwemeasureparyseparationsbythemutualhillradii(r_)，separationsamongterrestrialpsaregreaterthan26rh，whereasthoseamongjoviasarelessthan14rh.thisdifferenceisdirectlyretedtothediffereweendynamicalfeaturesofterrestriandjoviaerrestrialpshavesmallermasses，shorterorbitalperiodsandwiderdynamicalseparation.theyarestronglyperturbedbyjoviahathavergermasses，longerorbitalperiodsandnarrowerdynamicalseparation.joviasareurbedbyanyothermassivebodies.
thepresentterrestrialparysystemisstillbeingdisturbedbythemassivejovias.however，thewideseparationandmutualiionamoerrestrialpsrehedisturbaneffective;thedegreeofdisturbancebyjoviasiso(ej)(orderofmagnitudeoftheetricityofjupiter)，sihedisturbancecausedbyjoviasisaforcedosciltionhavinganamplitudeofo(ej).heighteningofetricity，forexampleo(ej)～0.05，isfarfromsuffittoprovokeinstabilityierrestrialpshavingsuchawideseparationas26rh.thusweassumethatthepresentwidedynamicalseparationamongterrestrialps(&;26rh)isprobablyohemostsignifitditionsformaintainiabilityoftheparysystemovera109-yrtime-span.ourdetailedanalysisoftheretionshipbetweendynamicaldistaweesaabilitytime-scaleofsorsystemparymotionisnowon-going.
althoughournumeritegrationsspanthelifetimeofthesorsystem，thenumberofiionsisfarfromsuffittofilltheinitialphasespace.itisnecessarytoperformmoreandmorenumeritegrationstofirmandexamineiailtheloabilityofourparydynamics.
——以上文段引自ito，t.&tanikawa，k.long-termiionsandstabilityofparyorbitsinoursorsyste.r.astron.soc.336，483–500(2002)
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