作者君在作品相关中其实已经解释过这个问题。
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-termintegrationsandstabilityoaryorbitsinoursrsystem
abstract
wepresenttheresultsofverylong-termnumericalintegrationsoaryorbitalmotionsover109-yrtime-spansincludingallnins.aquickinspectionofournumericaldatashowsthattharymotion，atleastinoursimpledynamicalmodel，seemstobequitestableevenoverthisverylongtime-span.acloserlookatthelowest-frequencyoscitionsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestriaarymotion，especiallythatofmercury.thebehaviouroftheentricityofmercuryinourintegrationsisqualitativelysimrtotheresultsfromjacqueskar'ssecrperturbationtheory(e.g.emax～0.35over～±4gyr).however，therearenoapparentsecrincreasesofentricityorinclinationinanyorbitalelementsofths，whichmayberevealedbystilllonger-termnumericalintegrations.wehavealsoperformedacoupleoftrialintegrationsincludingmotionsoftheouterfivsoverthedurationof±5x1010yr.theresultindicatesthatthethreemajorresonancesintheneptune–plutosystemhavebeenmaintainedoverthe1011-yrtime-span.
1introduction
1.1definitionoftheproblem
thequestionofthestabilityofoursrsystemhasbeendebatedoverseveralhundredyears，sincetheeraofnewton.theproblemhasattractedmanyfamousmathematiciansovertheyearsandhasyedacentralroleinthedevelopmentofnon-lineardynamicsandchaostheory.however，wedonotyethaveadefiniteanswertothequestionofwhetheroursrsystemisstableornot.thisispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenitisusedinrtiontotheproblemoarymotioninthesrsystem.actuallyitisnoteasytogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofoursrsystem.
amongmanydefinitionsofstability，hereweadoptthehilldefinition(dman1993):actuallythisisnotadefinitionofstability，butofinstability.wedefineasystemasbingunstablewhenacloseencounterurssomewhereinthesystem，startingfromacertaininitialconfiguration(chambers，wetherill&boss1996;ito&tanikawa1999).asystemisdefinedasexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinanareaofthrgerhillradius.otherwisethesystemisdefinedasbeingstable.henceforwardwestatethatouarysystemisdynamicallystableifnocloseencounterhappensduringtheageofoursrsystem，about±5gyr.incidentally，thisdefinitionmayberecedbyoneinwhichanurrenceofanyorbitalcrossingbetweeneitherofapairostakesce.thisisbecauseweknowfromexperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounteriaryandprotarysystems(yoshinaga，kokubo&makino1999).ofcoursethisstatementcannotbesimplyappliedtosystemswithstableorbitalresonancessuchastheneptune–plutosystem.
1.2previousstudiesandaimsofthisresearch
inadditiontothevaguenessoftheconceptofstability，thsinoursrsystemshowacharactertypicalofdynamicalchaos(sussman&wisdom1988，1992).thecauseofthischaoticbehaviourisnowpartlyunderstoodasbeingaresultofresonanceovepping(murray&holman1999;lecar，franklin&holman2001).however，itwouldrequireintegratingoveranensembleoarysystemsincludingallninsforaperiodcoveringseveral10gyrtothoroughlyunderstandthelong-termevolutionoaryorbits，sincechaoticdynamicalsystemsarecharacterizedbytheirstrongdependenceoninitialconditions.
fromthatpointofview，manyofthepreviouslong-termnumericalintegrationsincludedonlytheouterfivs(sussman&wisdom1988;kinoshita&nakai1996).thisisbecausetheorbitalperiodsoftheoutesaresomuchlongerthanthoseoftheinnerfousthatitismucheasiertofollowthesystemforagivenintegrationperiod.atpresent，thelongestnumericalintegrationspublishedinjournalsarethoseofduncan&lissauer(1998).althoughtheirmaintargetwastheeffectofpost-main-sequencesrmasslossonthestabilityoaryorbits，theyperformedmanyintegrationscoveringupto～1011yroftheorbitalmotionsofthefourjovias.theinitialorbitalelementsandmassesosarethesameasthoseofoursrsysteminduncan&lissauer'spaper，buttheydecreasethemassofthesungraduallyintheirnumericalexperiments.thisisbecausetheyconsidertheeffectofpost-main-sequencesrmasslossinthepaper.consequently，theyfoundthatthecrossingtime-scaleoaryorbits，whichcanbeatypicalindicatoroftheinstabilitytime-scale，isquitesensitivetotherateofmassdecreaseofthesun.whenthemassofthesunisclosetoitspresentvalue，thejoviasremainstableover1010yr，orperhapslonger.duncan&lissaueralsoperformedfoursimrexperimentsontheorbitalmotionofseves(venustoneptune)，whichcoveraspanof～109yr.theirexperimentsonthesevesarenotyeprehensive，butitseemsthattheterrestriasalsoremainstableduringtheintegrationperiod，maintainingalmostregroscitions.
ontheotherhand，inhisuratesemi-analyticalsecrperturbationtheoryskar1988)skarfindsthargeandirregrvariationscanappearintheentricitiesandinclinationsoftheterrestrias，especiallyofmercuryandmarsonatime-scaleofseveral109yrskar1996).theresultsoskar'ssecrperturbationtheoryshouldbeconfirmedandinvestigatedbyfullynumericalintegrations.
inthispaperwepresentpreliminaryresultsofsixlong-termnumericalintegrationsonallninaryorbits，coveringaspanofseveral109yr，andoftwootherintegrationscoveringaspanof±5x1010yr.thetotalpsedtimeforallintegrationsismorethan5yr，usingseveraldedicatedpcsandworkstations.oneofthefundamentalconclusionsofourlong-termintegrationsisthatsrsystearymotionseemstobestableintermsofthehillstabilitymentionedabove，atleastoveratime-spanof±4gyr.actually，inournumericalintegrationsthesystemwasfarmorestablethanwhatisdefinedbythehillstabilitycriterion:notonlydidnocloseencounterhappenduringtheintegrationperiod，butalsoalltharyorbitalelementshavebeenconfinedinanarrowregionbothintimeandfrequencydomain，thougarymotionsarestochastic.sincethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-termnumericalintegrations，weshowtypicalexamplefiguresasevidenceoftheverylong-termstabilityofsrsystearymotion.forreaderswhohavemorespecificanddeeperinterestsinournumericalresults，wehavepreparedawebpage(ess)，whereweshowraworbitalelements，theirlow-passfilteredresults，variationofdunayelementsandangrmomentumdeficit，andresultsofoursimpletime–frequencyanalysisonallofourintegrations.
insection2webrieflyexinourdynamicalmodel，numericalmethodandinitialconditionsusedinourintegrations.section3isdevotedtoadescriptionofthequickresultsofthenumericalintegrations.verylong-termstabilityofsrsystearymotionisapparentbothiarypositionsandorbitalelements.aroughestimationofnumericalerrorsisalsogiven.section4goesontoadiscussionofthelongest-termvariationoaryorbitsusingalow-passfilterandincludesadiscussionofangrmomentumdeficit.insection5，wepresentasetofnumericalintegrationsfortheouterfivsthatspans±5x1010yr.insection6wealsodiscussthelong-termstabilityoftharymotionanditspossiblecause.
2descriptionofthenumericalintegrations
（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
weutilizeasecond-orderwisdom–holmansymplecticmapasourmainintegrationmethod(wisdom&holman1991;kinoshita，yoshida&nakai1991)withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warmstart’(saha&tremaine1992，1994).
thestepsizeforthenumericalintegrationsis8dthroughoutallintegrationsofthenins(n±1，2，3)，whichisabout1/11oftheorbitalperiodoftheinnermos(mercury).asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericalintegrationofallninsinsussman&wisdom(1988，7.2d)andsaha&tremaine(1994，225/32d).weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheumtionofround-offerrorinthputationprocesses.inrtiontothis，wisdom&holman(1991)performednumericalintegrationsoftheouterfivaryorbitsusingthesymplecticmapwithastepsizeof400d，1/10.83oftheorbitalperiodofjupiter.theirresultseemstobeurateenough，whichpartlyjustifiesourmethodofdeterminingthestepsize.however，sincetheentricityofjupiter(～0.05)ismuchsmallerthanthatofmercury(～0.2)，weneedsomecarewhenwparetheseintegrationssimplyintermsofstepsizes.
intheintegrationoftheouterfivs(f±)，wefixedthestepsizeat400d.
weadoptgauss'fandgfunctionsinthesymplecticmaptogetherwiththethird-orderhalleymethod(danby1992)asasolverforkeplerequations.thenumberofmaximumiterationswesetinhalley'smethodis15，buttheyneverreachedthemaximuminanyofourintegrations.
theintervalofthedataoutputis200000d(～547yr)forthecalctionsofallnins(n±1，2，3)，andabout8000000d(～21903yr)fortheintegrationoftheouterfivs(f±).
althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalow-passfiltertotheraworbitaldataafterwehapletedallthecalctions.seesection4.1formoredetail.
2.4errorestimation
2.4.1rtiveerrorsintotalenergyandangrmomentum
ordingtooneofthebasicpropertiesofsymplecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangrmomentum)，ourlong-termnumericalintegrationsseemtohavebeenperformedwithverysmallerrors.theaveragedrtiveerrorsoftotalenergy(～10?9)andoftotngrmomentum(～10?11)haveremainednearlyconstantthroughouttheintegrationperiod(fig.1).thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedrtiveerrorintotalenergybyaboutoneorderofmagnitudeormore.
rtivenumericalerrorofthetotngrmomentumδa/a0andthetotalenergyδe/e0inournumericalintegrationsn±1，2，3，whereδeandδaaretheabsolutechangeofthetotalenergyandtotngrmomentum，respectively，ande0anda0aretheirinitialvalues.thehorizontalunitisgyr.
notethatdifferentoperatingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-offerrorhandlingandnumericlgorithms.intheupperpaneloffig.1，wecanrecognizethissituationinthesecrnumericalerrorinthetotngrmomentum，whichshouldberigorouslypreserveduptomachine-eprecision.
2.4.2erroriarylongitudes
sincethesymplecticmapspreservetotalenergyandtotngrmomentumofn-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheuracyofnumericalintegrations，especiallyasameasureofthepositionalerroros，i.e.theerroriarylongitudes.toestimatethenumericalerrorintharylongitudes，weperformedthefollowingprocedures.wparedtheresultofourmainlong-termintegrationswithsometestintegrations，whichspanmuchshorterperiodsbutwithmuchhigheruracythanthemainintegrations.forthispurpose，weperformedamuchmoreurateintegrationwithastepsizeof0.125d(1/64ofthemainintegrations)spanning3x105yr，startingwiththesameinitialconditionsasinthen?1integration.weconsiderthatthistestintegrationprovidesuswitha‘pseudo-true’solutionoaryorbitalevolution.next，wparethetestintegrationwiththemainintegration，n?1.fortheperiodof3x105yr，weseeadifferenceinmeananomaliesoftheearthbetweenthetwointegrationsof～0.52°(inthecaseofthen?1integration).thisdifferencecanbeextraptedtothevalue～8700°，about25rotationsofearthafter5gyr，sincetheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.simrly，thelongitudeerrorofplutocanbeestimatedas～12°.thisvalueforplutoismuchbetterthantheresultinkinoshita&nakai(1996)wherethedifferenceisestimatedas～60°.
3numericalresults–i.nceattherawdata
inthissectionwebrieflyreviewthelong-termstabilityoaryorbitalmotionthroughsomesnapshotsofrawnumericaldata.theorbitalmotionosindicateslong-termstabilityinallofournumericalintegrations:noorbitalcrossingsnorcloseencountersbetweenanypairostookce.
3.1generaldescriptionofthestabilityoaryorbits
first，webrieflylookatthegeneralcharacterofthelong-termstabilityoaryorbits.ourinterestherefocusesparticrlyontheinnerfourterrestriasforwhichtheorbitaltime-scalesaremuchshorterthanthoseoftheouterfivs.aswecanseeclearlyfromthenarorbitalconfigurationsshowninfigs2and3，orbitalpositionsoftheterrestriasdifferlittlebetweentheinitindfinalpartofeachnumericalintegration，whichspansseveralgyr.thesolidlinesdenotingthepresentorbitsofthsliealmostwithintheswarmofdotseveninthefinalpartofintegrations(b)and(d).thisindicatesthatthroughouttheentireintegrationperiodthealmostregrvariationsoaryorbitalmotionremainnearlythesameastheyareatpresent.
verticalviewofthefourinnearyorbits(fromthez-axisdirection)attheinitindfinalpartsoftheintegrationsn±1.theaxesunitsareau.thexy-neissettotheinvariantneofsrsystemtotngrmomentum.(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.solidlinesineachpaneldenotethepresentorbitsofthefourterrestrias(takenfromde245).
thevariationofentricitiesandorbitalinclinationsfortheinnerfousintheinitindfinalpartoftheintegrationn 1isshowninfig.4.asexpected，thecharacterofthevariationoaryorbitalelementsdoesnotdiffersignificantlybetweentheinitindfinalpartofeachintegration，atleastforvenus，earthandmars.theelementsofmercury，especiallyitsentricity，seemtochangetoasignificantextent.thisispartlybecausetheorbitaltime-scaleofthistheshortestofallths，whichleadstoamorerapidorbitalevolutionthanothes;theinnermosmaybenearesttoinstability.thisresultappearstobeinsomeagreementwitskar's(1994，1996)expectationsthargeandirregrvariationsappearintheentricitiesandinclinationsofmercuryonatime-scaleofseveral109yr.however，theeffectofthepossibleinstabilityoftheorbitofmercurymaynotfatallyaffecttheglobalstabilityofthewholarysystemowingtothesmallmassofmercury.wewillmentionbrieflythelong-termorbitalevolutionofmercurterinsection4usinglow-passfilteredorbitalelements.
theorbitalmotionoftheouterfivsseemsrigorouslystableandquiteregroverthistime-span(seealsosection5).
3.2time–frequencymaps
althoughtharymotionexhibitsverylong-termstabilitydefinedasthenon-existenceofcloseencounterevents，thechaotatureoarydynamicscanchangetheoscitoryperiodandamplitudeoaryorbitalmotiongraduallyoversuchlongtime-spans.evensuchslightfluctuationsoforbitalvariationinthefrequencydomain，particrlyinthecaseofearth，canpotentiallyhaveasignificanteffectonitssurfaceclimatesystemthroughsrinstionvariation(cf.berger1988).
togiveanoverviewofthelong-termchangeinperiodicityiaryorbitalmotion，weperformedmanyfastfouriertransformations(ffts)alongthetimeaxis，andsuperposedtheresultingperiodgramstodrawtwo-dimensionaltime–frequencymaps.thespecificapproachtodrawingthesetime–frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisoskar's(1990，1993)frequencyanalysis.
dividethelow-passfilteredorbitaldataintomanyfragmentsofthesamelength.thelengthofeachdatasegmentshouldbeamultipleof2inordertoapplythefft.
eachfragmentofthedatahasrgeoveppingpart:forexample，whentheithdatabeginsfromt=tiandendsatt=ti t，thenextdatasegmentrangesfromti δt≤ti δt t，whereδt?t.wecontinuethisdivisionuntilwereachacertainnumbernbywhichtn treachesthetotalintegrationlength.
weapplyanffttoeachofthedatafragments，andobtainnfrequencydiagrams.
ineachfrequencydiagramobtainedabove，thestrengthofperiodicitycanberecedbyagrey-scale(orcolour)chart.
weperformtherecement，andconnectallthegrey-scale(orcolour)chartsintoonegraphforeachintegration.thehorizontxisofthesenewgraphsshouldbethetime，i.e.thestartingtimesofeachfragmentofdata(ti，wherei=1，…，n).theverticxisrepresentstheperiod(orfrequency)oftheoscitionoforbitalelements.
wehaveadoptedanfftbecauseofitsoverwhelmingspeed，sincetheamountofnumericaldatatobedposedintofrequencponentsisterriblyhuge(severaltensofgbytes).
atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasfig.5，whichshowsthevariationofperiodicityintheentricityandinclinationofearthinn 2integration.infig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.wecanrecognizefromthismapthattheperiodicityoftheentricityandinclinationofearthonlychangesslightlyovertheentireperiodcoveredbythen 2integration.thisnearlyregrtrendisqualitativelythesameinotherintegrationsandforothes，althoughtypicalfrequenciesdiffebandelementbyelement.
4.2long-termexchangeoforbitalenergyandangrmomentum
wecalcteverylong-periodicvariationandexchangeoaryorbitalenergyandangrmomentumusingfiltereddunayelementsl，g，h.gandhareequivalenttotharyorbitngrmomentumanditsverticaponentperunitmass.lisrtedtotharyorbitalenergyeperunitmassase=?μ2/2l2.ifthesystemipletelylinear，theorbitalenergyandtheangrmomentumineachfrequencybinmustbeconstant.non-linearityintharysystemcancauseanexchangeofenergyandangrmomentuminthefrequencydomain.theamplitudeofthelowest-frequencyoscitionshouldincreaseifthesystemisunstableandbreaksdowngradually.however，suchasymptomofinstabilityisnotprominentinourlong-termintegrations.
infig.7，thetotalorbitalenergyandangrmomentumofthefourinnesandallninsareshownforintegrationn 2.theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedase-e0)，totngrmomentum(g-g0)，andtheverticaponent(h-h0)oftheinnerfouscalctedfromthelow-passfiltereddunayelements.e0，g0，h0denotetheinitialvaluesofeachquantity.theabsolutedifferencefromtheinitialvaluesisplottedinthepanels.thelowerthreepanelsineachfigureshowe-e0，g-g0andh-h0ofthetotalofnins.thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovias.
4.4long-termcouplingofseveralneighbourinpairs
letusseesomeindividualvariationsoaryorbitalenergyandangrmomentumexpressedbythelow-passfiltereddunayelements.figs10and11showlong-termevolutionoftheorbitalenergyofeacandtheangrmomentuminn 1andn?2integrations.wenoticethatsomsformapparentpairsintermsoforbitalenergyandangrmomentumexchange.inparticr，venusandearthmakeatypicalpair.inthefigures，theyshownegativecorrtionsinexchangeofenergyandpositivecorrtionsinexchangeofangrmomentum.thenegativecorrtioninexchangeoforbitalenergymeansthatthetwsformacloseddynamicalsystemintermsoftheorbitalenergy.thepositivecorrtioninexchangeofangrmomentummeansthatthetwsaresimultaneouslyundercertainlong-termperturbations.candidatesforperturbersarejupiterandsaturn.alsoinfig.11，wecanseethatmarsshowsapositivecorrtionintheangrmomentumvariationtothevenus–earthsystemrcuryexhibitscertainnegativecorrtionsintheangrmomentumversusthevenus–earthsystem，whichseemstobeareactioncausedbytheconservationofangrmomentumintheterrestriaarysubsystem.
itisnotclearatthemomentwhythevenus–earthpairexhibitsanegativecorrtioninenergyexchangeandapositivecorrtioninangrmomentumexchange.wemaypossiblyexinthisthroughobservingthegeneralfactthattherearenosecrtermsiarysemimajoraxesuptosecond-orderperturbationtheories(cf.brouwer&clemence1961;baletti&puco1998).thismeansthattharyorbitalenergy(whichisdirectlyrtedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbinsthanistheangrmomentumexchange(whichrtestoe).hence，theentricitiesofvenusandearthcanbedisturbedeasilybyjupiterandsaturn，whichresultsinapositivecorrtionintheangrmomentumexchange.ontheotherhand，thesemimajoraxesofvenusandeartharelesslikelytobedisturbedbythejovias.thustheenergyexchangemaybelimitedonlywithinthevenus–earthpair，whichresultsinanegativecorrtionintheexchangeoforbitalenergyinthepair.
asfortheouterjoviaarysubsystem，jupiter–saturnanduranus–neptuneseemtomakedynamicalpairs.however，thestrengthoftheircouplingisnotasstronparedwiththatofthevenus–earthpair.
5±5x1010-yrintegrationsofoutearyorbits
sincethejoviaarymassesaremucrgerthantheterrestriaarymasses，wetreatthejoviaarysystemasanindependenarysystemintermsofthestudyofitsdynamicalstability.hence，weaddedacoupleoftrialintegrationsthatspan±5x1010yr，includingonlytheouterfivs(thefourjoviaspluspluto).theresultsexhibittherigorousstabilityoftheoutearysystemoverthislongtime-span.orbitalconfigurations(fig.12)，andvariationofentricitiesandinclinations(fig.13)showthisverylong-termstabilityoftheouterfivsinboththetimeandthefrequencydomains.althoughwedonotshowmapshere，thetypicalfrequencyoftheorbitaloscitionofplutoandtheotheroutesisalmostconstantduringtheseverylong-termintegrationperiods，whichisdemonstratedinthetime–frequencymapsonourwebpage.
inthesetwointegrations，thertivenumericalerrorinthetotalenergywas～10?6andthatofthetotngrmomentumwas～10?10.
5.1resonancesintheneptune–plutosystem
kinoshita&nakai(1996)integratedtheouterfivaryorbitsover±5.5x109yr.theyfoundthatfourmajorresonancesbetweenneptuneandplutoaremaintainedduringthewholeintegrationperiod，andthattheresonancesmaybethemaincausesofthestabilityoftheorbitofpluto.themajorfourresonancesfoundinpreviousresearchareasfollows.inthefollowingdescription，λdenotesthemeanlongitude，Ωisthelongitudeoftheascendingnodeand?isthelongitudeofperihelion.subscriptspandndenoteplutoandneptune.
meanmotionresonancebetweenneptuneandpluto(3:2).thecriticrgumentθ1=3λp?2λn??plibratesaround180°withanamplitudeofabout80°andalibrationperiodofabout2x104yr.
theargumentofperihelionofplutowp=θ2=?p?Ωplibratesaround90°withaperiodofabout3.8x106yr.thedominantperiodicvariationsoftheentricityandinclinationofplutoaresynchronizedwiththelibrationofitsargumentofperihelion.thisisanticipatedinthesecrperturbationtheoryconstructedbykozai(1962).
thelongitudeofthenodeofplutoreferredtothelongitudeofthenodeofneptune，θ3=Ωp?Ωn，circtesandtheperiodofthiscirctionisequaltotheperiodofθ2libration.whenθ3beszero，i.e.thelongitudesofascendingnodesofneptuneandplutoovep，theinclinationofplutobesmaximum，theentricitybesminimumandtheargumentofperihelionbes90°.whenθ3bes180°，theinclinationofplutobesminimum，theentricitybesmaximumandtheargumentofperihelionbes90°again.williams&benson(1971)anticipatedthistypeofresonanceterconfirmedbymni，nobili&carpino(1989).
anargumentθ4=?p??n 3(Ωp?Ωn)libratesaround180°withalongperiod，～5.7x108yr.
inournumericalintegrations，theresonances(i)–(iii)arewellmaintained，andvariationofthecriticrgumentsθ1，θ2，θ3remainsimrduringthewholeintegrationperiod(figs14–16).however，thefourthresonance(iv)appearstobedifferent:thecriticrgumentθ4alternateslibrationandcirctionovera1010-yrtime-scale(fig.17).thisisaninterestingfactthatkinoshita&nakai's(1995，1996)shorterintegrationswerenotabletodisclose.
6discussion
whatkindofdynamicalmechanismmaintainsthislong-termstabilityoftharysystem?wecanimmediatelythinkoftwomajorfeaturesthatmayberesponsibleforthelong-termstability.first，thereseemtobenosignificantlower-orderresonances(meanmotionandsecr)betweenanypairamongthenins.jupiterandsaturnareclosetoa5:2meanmotionresonance(thefamous‘greatinequality’)，butnotjustintheresonancezone.higher-orderresonancesmaycausethechaotatureoftharydynamicalmotion，buttheyarenotsostrongastodestroythestablarymotionwithinthelifetimeoftherealsrsystem.thesecondfeature，whichwethinkismoreimportantforthelong-termstabilityofouarysystem，isthedifferenceindynamicaldistancebetweenterrestrindjoviaarysubsystems(ito&tanikawa1999，2001).whenwemeasuraryseparationsbythemutualhillradii(r_)，separationsamongterrestriasaregreaterthan26rh，whereasthoseamongjoviasarelessthan14rh.thisdifferenceisdirectlyrtedtothedifferencebetweendynamicalfeaturesofterrestrindjovias.terrestriashavesmallermasses，shorterorbitalperiodsandwiderdynamicalseparation.theyarestronglyperturbedbyjoviasthathavrgermasses，longerorbitalperiodsandnarrowerdynamicalseparation.joviasarenotperturbedbyanyothermassivebodies.
thepresentterrestriaarysystemisstillbeingdisturbedbythemassivejovias.however，thewideseparationandmutualinteractionamongtheterrestriasrendersthedisturbanceineffective;thedegreeofdisturbancebyjoviasiso(ej)(orderofmagnitudeoftheentricityofjupiter)，sincethedisturbancecausedbyjoviasisaforcedoscitionhavinganamplitudeofo(ej).heighteningofentricity，forexampleo(ej)～0.05，isfarfromsufficienttoprovokeinstabilityintheterrestriashavingsuchawideseparationas26rh.thusweassumethatthepresentwidedynamicalseparationamongterrestrias(>26rh)isprobablyoneofthemostsignificantconditionsformaintainingthestabilityoftharysystemovera109-yrtime-span.ourdetailedanalysisofthertionshipbetweendynamicaldistancebetweesandtheinstabilitytime-scaleofsrsystearymotionisnowon-going.
althoughournumericalintegrationsspanthelifetimeofthesrsystem，thenumberofintegrationsisfarfromsufficienttofilltheinitialphasespace.itisnecessarytoperformmoreandmorenumericalintegrationstoconfirmandexamineindetailthelong-termstabilityofouarydynamics.
——以上文段引自ito，t.&tanikawa，k.long-termintegrationsandstabilityoaryorbitsinoursrsystem.mon.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-termintegrationsandstabilityoaryorbitsinoursrsystem
abstract
wepresenttheresultsofverylong-termnumericalintegrationsoaryorbitalmotionsover109-yrtime-spansincludingallnins.aquickinspectionofournumericaldatashowsthattharymotion，atleastinoursimpledynamicalmodel，seemstobequitestableevenoverthisverylongtime-span.acloserlookatthelowest-frequencyoscitionsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestriaarymotion，especiallythatofmercury.thebehaviouroftheentricityofmercuryinourintegrationsisqualitativelysimrtotheresultsfromjacqueskar'ssecrperturbationtheory(e.g.emax～0.35over～±4gyr).however，therearenoapparentsecrincreasesofentricityorinclinationinanyorbitalelementsofths，whichmayberevealedbystilllonger-termnumericalintegrations.wehavealsoperformedacoupleoftrialintegrationsincludingmotionsoftheouterfivsoverthedurationof±5x1010yr.theresultindicatesthatthethreemajorresonancesintheneptune–plutosystemhavebeenmaintainedoverthe1011-yrtime-span.
1introduction
1.1definitionoftheproblem
thequestionofthestabilityofoursrsystemhasbeendebatedoverseveralhundredyears，sincetheeraofnewton.theproblemhasattractedmanyfamousmathematiciansovertheyearsandhasyedacentralroleinthedevelopmentofnon-lineardynamicsandchaostheory.however，wedonotyethaveadefiniteanswertothequestionofwhetheroursrsystemisstableornot.thisispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenitisusedinrtiontotheproblemoarymotioninthesrsystem.actuallyitisnoteasytogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofoursrsystem.
amongmanydefinitionsofstability，hereweadoptthehilldefinition(dman1993):actuallythisisnotadefinitionofstability，butofinstability.wedefineasystemasbingunstablewhenacloseencounterurssomewhereinthesystem，startingfromacertaininitialconfiguration(chambers，wetherill&boss1996;ito&tanikawa1999).asystemisdefinedasexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinanareaofthrgerhillradius.otherwisethesystemisdefinedasbeingstable.henceforwardwestatethatouarysystemisdynamicallystableifnocloseencounterhappensduringtheageofoursrsystem，about±5gyr.incidentally，thisdefinitionmayberecedbyoneinwhichanurrenceofanyorbitalcrossingbetweeneitherofapairostakesce.thisisbecauseweknowfromexperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounteriaryandprotarysystems(yoshinaga，kokubo&makino1999).ofcoursethisstatementcannotbesimplyappliedtosystemswithstableorbitalresonancessuchastheneptune–plutosystem.
1.2previousstudiesandaimsofthisresearch
inadditiontothevaguenessoftheconceptofstability，thsinoursrsystemshowacharactertypicalofdynamicalchaos(sussman&wisdom1988，1992).thecauseofthischaoticbehaviourisnowpartlyunderstoodasbeingaresultofresonanceovepping(murray&holman1999;lecar，franklin&holman2001).however，itwouldrequireintegratingoveranensembleoarysystemsincludingallninsforaperiodcoveringseveral10gyrtothoroughlyunderstandthelong-termevolutionoaryorbits，sincechaoticdynamicalsystemsarecharacterizedbytheirstrongdependenceoninitialconditions.
fromthatpointofview，manyofthepreviouslong-termnumericalintegrationsincludedonlytheouterfivs(sussman&wisdom1988;kinoshita&nakai1996).thisisbecausetheorbitalperiodsoftheoutesaresomuchlongerthanthoseoftheinnerfousthatitismucheasiertofollowthesystemforagivenintegrationperiod.atpresent，thelongestnumericalintegrationspublishedinjournalsarethoseofduncan&lissauer(1998).althoughtheirmaintargetwastheeffectofpost-main-sequencesrmasslossonthestabilityoaryorbits，theyperformedmanyintegrationscoveringupto～1011yroftheorbitalmotionsofthefourjovias.theinitialorbitalelementsandmassesosarethesameasthoseofoursrsysteminduncan&lissauer'spaper，buttheydecreasethemassofthesungraduallyintheirnumericalexperiments.thisisbecausetheyconsidertheeffectofpost-main-sequencesrmasslossinthepaper.consequently，theyfoundthatthecrossingtime-scaleoaryorbits，whichcanbeatypicalindicatoroftheinstabilitytime-scale，isquitesensitivetotherateofmassdecreaseofthesun.whenthemassofthesunisclosetoitspresentvalue，thejoviasremainstableover1010yr，orperhapslonger.duncan&lissaueralsoperformedfoursimrexperimentsontheorbitalmotionofseves(venustoneptune)，whichcoveraspanof～109yr.theirexperimentsonthesevesarenotyeprehensive，butitseemsthattheterrestriasalsoremainstableduringtheintegrationperiod，maintainingalmostregroscitions.
ontheotherhand，inhisuratesemi-analyticalsecrperturbationtheoryskar1988)skarfindsthargeandirregrvariationscanappearintheentricitiesandinclinationsoftheterrestrias，especiallyofmercuryandmarsonatime-scaleofseveral109yrskar1996).theresultsoskar'ssecrperturbationtheoryshouldbeconfirmedandinvestigatedbyfullynumericalintegrations.
inthispaperwepresentpreliminaryresultsofsixlong-termnumericalintegrationsonallninaryorbits，coveringaspanofseveral109yr，andoftwootherintegrationscoveringaspanof±5x1010yr.thetotalpsedtimeforallintegrationsismorethan5yr，usingseveraldedicatedpcsandworkstations.oneofthefundamentalconclusionsofourlong-termintegrationsisthatsrsystearymotionseemstobestableintermsofthehillstabilitymentionedabove，atleastoveratime-spanof±4gyr.actually，inournumericalintegrationsthesystemwasfarmorestablethanwhatisdefinedbythehillstabilitycriterion:notonlydidnocloseencounterhappenduringtheintegrationperiod，butalsoalltharyorbitalelementshavebeenconfinedinanarrowregionbothintimeandfrequencydomain，thougarymotionsarestochastic.sincethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-termnumericalintegrations，weshowtypicalexamplefiguresasevidenceoftheverylong-termstabilityofsrsystearymotion.forreaderswhohavemorespecificanddeeperinterestsinournumericalresults，wehavepreparedawebpage(ess)，whereweshowraworbitalelements，theirlow-passfilteredresults，variationofdunayelementsandangrmomentumdeficit，andresultsofoursimpletime–frequencyanalysisonallofourintegrations.
insection2webrieflyexinourdynamicalmodel，numericalmethodandinitialconditionsusedinourintegrations.section3isdevotedtoadescriptionofthequickresultsofthenumericalintegrations.verylong-termstabilityofsrsystearymotionisapparentbothiarypositionsandorbitalelements.aroughestimationofnumericalerrorsisalsogiven.section4goesontoadiscussionofthelongest-termvariationoaryorbitsusingalow-passfilterandincludesadiscussionofangrmomentumdeficit.insection5，wepresentasetofnumericalintegrationsfortheouterfivsthatspans±5x1010yr.insection6wealsodiscussthelong-termstabilityoftharymotionanditspossiblecause.
2descriptionofthenumericalintegrations
（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
weutilizeasecond-orderwisdom–holmansymplecticmapasourmainintegrationmethod(wisdom&holman1991;kinoshita，yoshida&nakai1991)withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warmstart’(saha&tremaine1992，1994).
thestepsizeforthenumericalintegrationsis8dthroughoutallintegrationsofthenins(n±1，2，3)，whichisabout1/11oftheorbitalperiodoftheinnermos(mercury).asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericalintegrationofallninsinsussman&wisdom(1988，7.2d)andsaha&tremaine(1994，225/32d).weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheumtionofround-offerrorinthputationprocesses.inrtiontothis，wisdom&holman(1991)performednumericalintegrationsoftheouterfivaryorbitsusingthesymplecticmapwithastepsizeof400d，1/10.83oftheorbitalperiodofjupiter.theirresultseemstobeurateenough，whichpartlyjustifiesourmethodofdeterminingthestepsize.however，sincetheentricityofjupiter(～0.05)ismuchsmallerthanthatofmercury(～0.2)，weneedsomecarewhenwparetheseintegrationssimplyintermsofstepsizes.
intheintegrationoftheouterfivs(f±)，wefixedthestepsizeat400d.
weadoptgauss'fandgfunctionsinthesymplecticmaptogetherwiththethird-orderhalleymethod(danby1992)asasolverforkeplerequations.thenumberofmaximumiterationswesetinhalley'smethodis15，buttheyneverreachedthemaximuminanyofourintegrations.
theintervalofthedataoutputis200000d(～547yr)forthecalctionsofallnins(n±1，2，3)，andabout8000000d(～21903yr)fortheintegrationoftheouterfivs(f±).
althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalow-passfiltertotheraworbitaldataafterwehapletedallthecalctions.seesection4.1formoredetail.
2.4errorestimation
2.4.1rtiveerrorsintotalenergyandangrmomentum
ordingtooneofthebasicpropertiesofsymplecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangrmomentum)，ourlong-termnumericalintegrationsseemtohavebeenperformedwithverysmallerrors.theaveragedrtiveerrorsoftotalenergy(～10?9)andoftotngrmomentum(～10?11)haveremainednearlyconstantthroughouttheintegrationperiod(fig.1).thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedrtiveerrorintotalenergybyaboutoneorderofmagnitudeormore.
rtivenumericalerrorofthetotngrmomentumδa/a0andthetotalenergyδe/e0inournumericalintegrationsn±1，2，3，whereδeandδaaretheabsolutechangeofthetotalenergyandtotngrmomentum，respectively，ande0anda0aretheirinitialvalues.thehorizontalunitisgyr.
notethatdifferentoperatingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-offerrorhandlingandnumericlgorithms.intheupperpaneloffig.1，wecanrecognizethissituationinthesecrnumericalerrorinthetotngrmomentum，whichshouldberigorouslypreserveduptomachine-eprecision.
2.4.2erroriarylongitudes
sincethesymplecticmapspreservetotalenergyandtotngrmomentumofn-bodydynamicalsystemsinherentlywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheuracyofnumericalintegrations，especiallyasameasureofthepositionalerroros，i.e.theerroriarylongitudes.toestimatethenumericalerrorintharylongitudes，weperformedthefollowingprocedures.wparedtheresultofourmainlong-termintegrationswithsometestintegrations，whichspanmuchshorterperiodsbutwithmuchhigheruracythanthemainintegrations.forthispurpose，weperformedamuchmoreurateintegrationwithastepsizeof0.125d(1/64ofthemainintegrations)spanning3x105yr，startingwiththesameinitialconditionsasinthen?1integration.weconsiderthatthistestintegrationprovidesuswitha‘pseudo-true’solutionoaryorbitalevolution.next，wparethetestintegrationwiththemainintegration，n?1.fortheperiodof3x105yr，weseeadifferenceinmeananomaliesoftheearthbetweenthetwointegrationsof～0.52°(inthecaseofthen?1integration).thisdifferencecanbeextraptedtothevalue～8700°，about25rotationsofearthafter5gyr，sincetheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.simrly，thelongitudeerrorofplutocanbeestimatedas～12°.thisvalueforplutoismuchbetterthantheresultinkinoshita&nakai(1996)wherethedifferenceisestimatedas～60°.
3numericalresults–i.nceattherawdata
inthissectionwebrieflyreviewthelong-termstabilityoaryorbitalmotionthroughsomesnapshotsofrawnumericaldata.theorbitalmotionosindicateslong-termstabilityinallofournumericalintegrations:noorbitalcrossingsnorcloseencountersbetweenanypairostookce.
3.1generaldescriptionofthestabilityoaryorbits
first，webrieflylookatthegeneralcharacterofthelong-termstabilityoaryorbits.ourinterestherefocusesparticrlyontheinnerfourterrestriasforwhichtheorbitaltime-scalesaremuchshorterthanthoseoftheouterfivs.aswecanseeclearlyfromthenarorbitalconfigurationsshowninfigs2and3，orbitalpositionsoftheterrestriasdifferlittlebetweentheinitindfinalpartofeachnumericalintegration，whichspansseveralgyr.thesolidlinesdenotingthepresentorbitsofthsliealmostwithintheswarmofdotseveninthefinalpartofintegrations(b)and(d).thisindicatesthatthroughouttheentireintegrationperiodthealmostregrvariationsoaryorbitalmotionremainnearlythesameastheyareatpresent.
verticalviewofthefourinnearyorbits(fromthez-axisdirection)attheinitindfinalpartsoftheintegrationsn±1.theaxesunitsareau.thexy-neissettotheinvariantneofsrsystemtotngrmomentum.(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.solidlinesineachpaneldenotethepresentorbitsofthefourterrestrias(takenfromde245).
thevariationofentricitiesandorbitalinclinationsfortheinnerfousintheinitindfinalpartoftheintegrationn 1isshowninfig.4.asexpected，thecharacterofthevariationoaryorbitalelementsdoesnotdiffersignificantlybetweentheinitindfinalpartofeachintegration，atleastforvenus，earthandmars.theelementsofmercury，especiallyitsentricity，seemtochangetoasignificantextent.thisispartlybecausetheorbitaltime-scaleofthistheshortestofallths，whichleadstoamorerapidorbitalevolutionthanothes;theinnermosmaybenearesttoinstability.thisresultappearstobeinsomeagreementwitskar's(1994，1996)expectationsthargeandirregrvariationsappearintheentricitiesandinclinationsofmercuryonatime-scaleofseveral109yr.however，theeffectofthepossibleinstabilityoftheorbitofmercurymaynotfatallyaffecttheglobalstabilityofthewholarysystemowingtothesmallmassofmercury.wewillmentionbrieflythelong-termorbitalevolutionofmercurterinsection4usinglow-passfilteredorbitalelements.
theorbitalmotionoftheouterfivsseemsrigorouslystableandquiteregroverthistime-span(seealsosection5).
3.2time–frequencymaps
althoughtharymotionexhibitsverylong-termstabilitydefinedasthenon-existenceofcloseencounterevents，thechaotatureoarydynamicscanchangetheoscitoryperiodandamplitudeoaryorbitalmotiongraduallyoversuchlongtime-spans.evensuchslightfluctuationsoforbitalvariationinthefrequencydomain，particrlyinthecaseofearth，canpotentiallyhaveasignificanteffectonitssurfaceclimatesystemthroughsrinstionvariation(cf.berger1988).
togiveanoverviewofthelong-termchangeinperiodicityiaryorbitalmotion，weperformedmanyfastfouriertransformations(ffts)alongthetimeaxis，andsuperposedtheresultingperiodgramstodrawtwo-dimensionaltime–frequencymaps.thespecificapproachtodrawingthesetime–frequencymapsinthispaperisverysimple–muchsimplerthanthewaveletanalysisoskar's(1990，1993)frequencyanalysis.
dividethelow-passfilteredorbitaldataintomanyfragmentsofthesamelength.thelengthofeachdatasegmentshouldbeamultipleof2inordertoapplythefft.
eachfragmentofthedatahasrgeoveppingpart:forexample，whentheithdatabeginsfromt=tiandendsatt=ti t，thenextdatasegmentrangesfromti δt≤ti δt t，whereδt?t.wecontinuethisdivisionuntilwereachacertainnumbernbywhichtn treachesthetotalintegrationlength.
weapplyanffttoeachofthedatafragments，andobtainnfrequencydiagrams.
ineachfrequencydiagramobtainedabove，thestrengthofperiodicitycanberecedbyagrey-scale(orcolour)chart.
weperformtherecement，andconnectallthegrey-scale(orcolour)chartsintoonegraphforeachintegration.thehorizontxisofthesenewgraphsshouldbethetime，i.e.thestartingtimesofeachfragmentofdata(ti，wherei=1，…，n).theverticxisrepresentstheperiod(orfrequency)oftheoscitionoforbitalelements.
wehaveadoptedanfftbecauseofitsoverwhelmingspeed，sincetheamountofnumericaldatatobedposedintofrequencponentsisterriblyhuge(severaltensofgbytes).
atypicalexampleofthetime–frequencymapcreatedbytheaboveproceduresisshowninagrey-scalediagramasfig.5，whichshowsthevariationofperiodicityintheentricityandinclinationofearthinn 2integration.infig.5，thedarkareashowsthatatthetimeindicatedbythevalueontheabscissa，theperiodicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.wecanrecognizefromthismapthattheperiodicityoftheentricityandinclinationofearthonlychangesslightlyovertheentireperiodcoveredbythen 2integration.thisnearlyregrtrendisqualitativelythesameinotherintegrationsandforothes，althoughtypicalfrequenciesdiffebandelementbyelement.
4.2long-termexchangeoforbitalenergyandangrmomentum
wecalcteverylong-periodicvariationandexchangeoaryorbitalenergyandangrmomentumusingfiltereddunayelementsl，g，h.gandhareequivalenttotharyorbitngrmomentumanditsverticaponentperunitmass.lisrtedtotharyorbitalenergyeperunitmassase=?μ2/2l2.ifthesystemipletelylinear，theorbitalenergyandtheangrmomentumineachfrequencybinmustbeconstant.non-linearityintharysystemcancauseanexchangeofenergyandangrmomentuminthefrequencydomain.theamplitudeofthelowest-frequencyoscitionshouldincreaseifthesystemisunstableandbreaksdowngradually.however，suchasymptomofinstabilityisnotprominentinourlong-termintegrations.
infig.7，thetotalorbitalenergyandangrmomentumofthefourinnesandallninsareshownforintegrationn 2.theupperthreepanelsshowthelong-periodicvariationoftotalenergy(denotedase-e0)，totngrmomentum(g-g0)，andtheverticaponent(h-h0)oftheinnerfouscalctedfromthelow-passfiltereddunayelements.e0，g0，h0denotetheinitialvaluesofeachquantity.theabsolutedifferencefromtheinitialvaluesisplottedinthepanels.thelowerthreepanelsineachfigureshowe-e0，g-g0andh-h0ofthetotalofnins.thefluctuationshowninthelowerpanelsisvirtuallyentirelyaresultofthemassivejovias.
4.4long-termcouplingofseveralneighbourinpairs
letusseesomeindividualvariationsoaryorbitalenergyandangrmomentumexpressedbythelow-passfiltereddunayelements.figs10and11showlong-termevolutionoftheorbitalenergyofeacandtheangrmomentuminn 1andn?2integrations.wenoticethatsomsformapparentpairsintermsoforbitalenergyandangrmomentumexchange.inparticr，venusandearthmakeatypicalpair.inthefigures，theyshownegativecorrtionsinexchangeofenergyandpositivecorrtionsinexchangeofangrmomentum.thenegativecorrtioninexchangeoforbitalenergymeansthatthetwsformacloseddynamicalsystemintermsoftheorbitalenergy.thepositivecorrtioninexchangeofangrmomentummeansthatthetwsaresimultaneouslyundercertainlong-termperturbations.candidatesforperturbersarejupiterandsaturn.alsoinfig.11，wecanseethatmarsshowsapositivecorrtionintheangrmomentumvariationtothevenus–earthsystemrcuryexhibitscertainnegativecorrtionsintheangrmomentumversusthevenus–earthsystem，whichseemstobeareactioncausedbytheconservationofangrmomentumintheterrestriaarysubsystem.
itisnotclearatthemomentwhythevenus–earthpairexhibitsanegativecorrtioninenergyexchangeandapositivecorrtioninangrmomentumexchange.wemaypossiblyexinthisthroughobservingthegeneralfactthattherearenosecrtermsiarysemimajoraxesuptosecond-orderperturbationtheories(cf.brouwer&clemence1961;baletti&puco1998).thismeansthattharyorbitalenergy(whichisdirectlyrtedtothesemimajoraxisa)mightbemuchlessaffectedbyperturbinsthanistheangrmomentumexchange(whichrtestoe).hence，theentricitiesofvenusandearthcanbedisturbedeasilybyjupiterandsaturn，whichresultsinapositivecorrtionintheangrmomentumexchange.ontheotherhand，thesemimajoraxesofvenusandeartharelesslikelytobedisturbedbythejovias.thustheenergyexchangemaybelimitedonlywithinthevenus–earthpair，whichresultsinanegativecorrtionintheexchangeoforbitalenergyinthepair.
asfortheouterjoviaarysubsystem，jupiter–saturnanduranus–neptuneseemtomakedynamicalpairs.however，thestrengthoftheircouplingisnotasstronparedwiththatofthevenus–earthpair.
5±5x1010-yrintegrationsofoutearyorbits
sincethejoviaarymassesaremucrgerthantheterrestriaarymasses，wetreatthejoviaarysystemasanindependenarysystemintermsofthestudyofitsdynamicalstability.hence，weaddedacoupleoftrialintegrationsthatspan±5x1010yr，includingonlytheouterfivs(thefourjoviaspluspluto).theresultsexhibittherigorousstabilityoftheoutearysystemoverthislongtime-span.orbitalconfigurations(fig.12)，andvariationofentricitiesandinclinations(fig.13)showthisverylong-termstabilityoftheouterfivsinboththetimeandthefrequencydomains.althoughwedonotshowmapshere，thetypicalfrequencyoftheorbitaloscitionofplutoandtheotheroutesisalmostconstantduringtheseverylong-termintegrationperiods，whichisdemonstratedinthetime–frequencymapsonourwebpage.
inthesetwointegrations，thertivenumericalerrorinthetotalenergywas～10?6andthatofthetotngrmomentumwas～10?10.
5.1resonancesintheneptune–plutosystem
kinoshita&nakai(1996)integratedtheouterfivaryorbitsover±5.5x109yr.theyfoundthatfourmajorresonancesbetweenneptuneandplutoaremaintainedduringthewholeintegrationperiod，andthattheresonancesmaybethemaincausesofthestabilityoftheorbitofpluto.themajorfourresonancesfoundinpreviousresearchareasfollows.inthefollowingdescription，λdenotesthemeanlongitude，Ωisthelongitudeoftheascendingnodeand?isthelongitudeofperihelion.subscriptspandndenoteplutoandneptune.
meanmotionresonancebetweenneptuneandpluto(3:2).thecriticrgumentθ1=3λp?2λn??plibratesaround180°withanamplitudeofabout80°andalibrationperiodofabout2x104yr.
theargumentofperihelionofplutowp=θ2=?p?Ωplibratesaround90°withaperiodofabout3.8x106yr.thedominantperiodicvariationsoftheentricityandinclinationofplutoaresynchronizedwiththelibrationofitsargumentofperihelion.thisisanticipatedinthesecrperturbationtheoryconstructedbykozai(1962).
thelongitudeofthenodeofplutoreferredtothelongitudeofthenodeofneptune，θ3=Ωp?Ωn，circtesandtheperiodofthiscirctionisequaltotheperiodofθ2libration.whenθ3beszero，i.e.thelongitudesofascendingnodesofneptuneandplutoovep，theinclinationofplutobesmaximum，theentricitybesminimumandtheargumentofperihelionbes90°.whenθ3bes180°，theinclinationofplutobesminimum，theentricitybesmaximumandtheargumentofperihelionbes90°again.williams&benson(1971)anticipatedthistypeofresonanceterconfirmedbymni，nobili&carpino(1989).
anargumentθ4=?p??n 3(Ωp?Ωn)libratesaround180°withalongperiod，～5.7x108yr.
inournumericalintegrations，theresonances(i)–(iii)arewellmaintained，andvariationofthecriticrgumentsθ1，θ2，θ3remainsimrduringthewholeintegrationperiod(figs14–16).however，thefourthresonance(iv)appearstobedifferent:thecriticrgumentθ4alternateslibrationandcirctionovera1010-yrtime-scale(fig.17).thisisaninterestingfactthatkinoshita&nakai's(1995，1996)shorterintegrationswerenotabletodisclose.
6discussion
whatkindofdynamicalmechanismmaintainsthislong-termstabilityoftharysystem?wecanimmediatelythinkoftwomajorfeaturesthatmayberesponsibleforthelong-termstability.first，thereseemtobenosignificantlower-orderresonances(meanmotionandsecr)betweenanypairamongthenins.jupiterandsaturnareclosetoa5:2meanmotionresonance(thefamous‘greatinequality’)，butnotjustintheresonancezone.higher-orderresonancesmaycausethechaotatureoftharydynamicalmotion，buttheyarenotsostrongastodestroythestablarymotionwithinthelifetimeoftherealsrsystem.thesecondfeature，whichwethinkismoreimportantforthelong-termstabilityofouarysystem，isthedifferenceindynamicaldistancebetweenterrestrindjoviaarysubsystems(ito&tanikawa1999，2001).whenwemeasuraryseparationsbythemutualhillradii(r_)，separationsamongterrestriasaregreaterthan26rh，whereasthoseamongjoviasarelessthan14rh.thisdifferenceisdirectlyrtedtothedifferencebetweendynamicalfeaturesofterrestrindjovias.terrestriashavesmallermasses，shorterorbitalperiodsandwiderdynamicalseparation.theyarestronglyperturbedbyjoviasthathavrgermasses，longerorbitalperiodsandnarrowerdynamicalseparation.joviasarenotperturbedbyanyothermassivebodies.
thepresentterrestriaarysystemisstillbeingdisturbedbythemassivejovias.however，thewideseparationandmutualinteractionamongtheterrestriasrendersthedisturbanceineffective;thedegreeofdisturbancebyjoviasiso(ej)(orderofmagnitudeoftheentricityofjupiter)，sincethedisturbancecausedbyjoviasisaforcedoscitionhavinganamplitudeofo(ej).heighteningofentricity，forexampleo(ej)～0.05，isfarfromsufficienttoprovokeinstabilityintheterrestriashavingsuchawideseparationas26rh.thusweassumethatthepresentwidedynamicalseparationamongterrestrias(>26rh)isprobablyoneofthemostsignificantconditionsformaintainingthestabilityoftharysystemovera109-yrtime-span.ourdetailedanalysisofthertionshipbetweendynamicaldistancebetweesandtheinstabilitytime-scaleofsrsystearymotionisnowon-going.
althoughournumericalintegrationsspanthelifetimeofthesrsystem，thenumberofintegrationsisfarfromsufficienttofilltheinitialphasespace.itisnecessarytoperformmoreandmorenumericalintegrationstoconfirmandexamineindetailthelong-termstabilityofouarydynamics.
——以上文段引自ito，t.&tanikawa，k.long-termintegrationsandstabilityoaryorbitsinoursrsystem.mon.not.r.astron.soc.336，483–500(2002)
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