对火星轨道变化问题的最后解释

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-termintegrationsandstabilityplaaryorbitsoursolarsystem
abstract
wepresenttheresultsverylong-termnumericalintegrationsplaaryorbitalmotionsover109-yrtime-spansincludingallnininspectionournumericaldatashowsthattheplaarymotion，leastoursimpledynamicalmodel，seemsbequitestableevenoverthisverylonlookthelowest-frequencyoscillationsusingalow-passfiltershowsthepotentiallydiffusivecharacterterrestrialplaarymotion，especiallythatobehaviourtheeccentricitymercuryourintegrationsqualitativelysimilartheresultsfromjacqueslaskar'ssecularperturbationtheory(e.g.emax0.35over±4gyr).however，thereareapparentsecularincreaseseccentricityinclinationanyorbitalelementstheplas，whichmayrevealedstilllonger-termnumericalihavealsoperformedacoupletrialintegrationsincludingmotionstheouterfiveplasovertheduration±5x1010yr.theresultindicatesthatthethreemajorresonancestheneptuneplutosystemhavebeenmaintainedoverthe1011-yrtime-span.
1introduction
1.1definitiontheproblem
thequestionthestabilityoursolarsystemhasbeendebatedoverseveralhundredyears，sincetheeraoproblemhasattractedmanyfamousmathematiciansovertheyearsandhasplayedacentralrolethedevelopmentnon-lineardynamicsandchao，donotyethaveadefiniteanswerthequestionwhetheroursolarsystemstableopartlyaresultthefactthatthedefinitiontheterm‘stability’vaguewhenisusedrelationtheproblemplaarymotionthesolaisnoteasygiveaclear，rigorousandphysicallymeaningfuldefinitionthestabilityoursolarsystem.
amongmanydefinitionsstability，hereadoptthehilldefinition(gladman1993):actuallythisnotadefinitionstability，butdefineasystembeingunstablewhenacloseencounteroccurssomewherethesystem，startingfromacertaininitialconfiguration(chambers，wetherillamp;boss1996;itoamp;tanikawa1999).asystemdefinedexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinareathelargerhilthesystemdefinedbeinstatethatourplaarysystemdynamicallystablenocloseencounterhappensduringtheageoursolarsystem，about±，thisdefinitionmayreplacedonewhichoccurrenceanyorbitalcrossingbetweeneitherapairplastakebecauseknowfromexperiencethatorbitalcrossingverylikelyleadacloseencounterplaaryandprotoplaarysystems(yoshinaga，kokuboamp;makino1999).coursethisstatementcannotsimplyappliedsystemswithstableorbitalresonancessuchtheneptuneplutosystem.
1.2previousstudiesandaimsthisresearch
inadditionthevaguenesstheconceptstability，theplasoursolarsystemshowacharactertypicaldynamicalchaos(sussmanamp;wisdom1988，1992).thecausethischaoticbehaviournowpartlyunderstoodbeingaresultresonanceoverlapping(murrayamp;holman1999;lecar，franklinamp;holman2001).however，wouldrequireintegratingoverensembleplaarysystemsincludingallnineplasforaperiodcoveringseveralgyrthoroughlyunderstandthelong-termevolutionplaaryorbits，sincechaoticdynamicalsystemsarecharacterizedtheirstrongdependenceinitialconditions.
fromthatpointview，manythepreviouslong-termnumericalintegrationsincludedonlytheouterfiveplas(sussmanamp;wisdom1988;kinoshitaamp;nakai1996).thisbecausetheorbitalperiodstheouterplasaremuchlongerthanthosetheinnerfourplasthatismucheasierfollowthesystemforagivenintegratiopresent，thelongestnumericalintegrationspublishedjournalsarethoseduncanamp;lissauer(1998).althoughtheirmaintargetwastheeffectpost-main-sequencesolarmasslossthestabilityplaaryorbits，theyperformedmanyintegrationscoveringto1011oftheorbitalmotionsthefourjoviainitialorbitalelementsandmassesplasarethesamethoseoursolarsystemduncanamp;lissauer'spaper，buttheydecreasethemassthesungraduallytheirnumericalbecausetheyconsidertheeffectpost-main-sequencesolarmasslossth，theyfoundthatthecrossingtime-scaleplaaryorbits，whichcanatypicalindicatortheinstabilitytime-scale，quitesensitivetheratemassdecreaseththemassthesuncloseitspresentvalue，thejovianplasremainstableover1010yr，perhapamp;lissaueralsoperformedfoursimilarexperimentstheorbitalmotionsevenplas(venusneptune)，whichcoveraspan109yr.theirexperimentsthesevenplasarenotyetprehensive，butseemsthattheterrestrialplasalsoremainstableduringtheintegrationperiod，maintainingalmostregularoscillations.
ontheotherhand，hisaccuratesemi-analyticalsecularperturbationtheory(laskar1988)，laskarfindsthatlargeandirregularvariationscanappeartheeccentricitiesandinclinationstheterrestrialplas，especiallymercuryandmarsatime-scaleseveral109(laskar1996).theresultslaskar'ssecularperturbationtheoryshouldconfirmedandinvestigatedfullynumericalintegrations.
inthispaperpresentpreliminaryresultssixlong-termnumericalintegrationsallnineplaaryorbits，coveringaspanseveral109yr，andtwootherintegrationscoveringaspan±5x1010yr.thetotalelapsedtimeforallintegrationsmorethan5yr，usingseveraldedicatedpcsandwthefundamentalconclusionsourlong-termintegrationsthatsolarsystemplaarymotionseemsbestabletermsthehillstabilitymentionedabove，leastoveratime-span±，ournumericalintegrationsthesystemwasfarmorestablethanwhatdefinedthehillstabilitycriterion:notonlydidcloseencounterhappenduringtheintegrationperiod，butalsoalltheplaaryorbitalelementshavebeenconfinedanarrowregionbothtimeandfrequencydomain，thoughplaarymotionsthepurposethispapertoexhibitandoverviewtheresultsourlong-termnumericalintegrations，showtypicalexamplefiguresevidencetheverylong-termstabilitysolarsystemplaarreaderswhohavemorespecificanddeeperinterestsournumericalresults，havepreparedawebpage(access)，whereshowraworbitalelements，theirlow-passfilteredresults，variationdelaunayelementsandangularmomentumdeficit，andresultsoursimpletimefrequencyanalysisallourintegrations.
insection2brieflyexplainourdynamicalmodel，numericalmethodandinitialconditionsusedouri3devotedadescriptionthequickresultsthenumericalilong-termstabilitysolarsystemplaarymotionapparentbothplaarypositionsandorbitaestimationnumericalerrorsals4goestoadiscussionthelongest-termvariationplaaryorbitsusingalow-passfilterandincludesadiscussionangularmometusection5，presentasetnumericalintegrationsfortheouterfiveplasthatspans±5x1010yr.section6alsodiscussthelong-termstabilitytheplaarymotionanditspossiblecause.
2descriptionthenumericalintegrations
（本部分涉及比较复杂的积分计算，作者君就不贴上来了。）
2.3numericalmethod
weutilizeasecond-orderwisdomholmansymplecticmapourmainintegrationmethod(wisdomamp;holman1991;kinoshita，yoshidaamp;nakai1991)withaspecialstart-upprocedurereducethetruncationerroranglevariables，‘warmstart’(sahaamp;tremaine1992，1994).
thestepsizeforthenumericalintegrations8dthroughoutallintegrationsthenineplas(n±1，2，3)，whichabout1/11theorbitalperiodtheinnermostpla(mercury).forthedeterminationstepsize，partlyfollowthepreviousnumericalintegrationallnineplassussmanamp;wisdom(1988，7.2d)andsahaamp;tremaine(1994，225/32d).roundedthedecimalpartthetheirstepsizes8makethestepsizeamultiple2orderreducetheaccumulationround-offerrortheputatiorelationthis，wisdomamp;holman(1991)performednumericalintegrationstheouterfiveplaaryorbitsusingthesymplecticmapwithastepsize400d，1/10.83theorbitalperiodoresultseemsbeaccurateenough，whichpartlyjustifiesourmethoddeterminingth，sincetheeccentricityjupiter(0.05)muchsmallerthanthatmercury(0.2)，needsomecarewhenparetheseintegrationssimplytermsstepsizes.
intheintegrationtheouterfiveplas(f±)，fixedthestepsize400d.
weadoptgauss'fandgfunctionsthesymplecticmaptogetherwiththethird-orderhalleymethod(danby1992)asolverforkeplenumbermaximumiterationssethalley'smethod15，buttheyneverreachedthemaximumanyourintegrations.
theintervalthedataoutput200000d(547yr)forthecalculationsallnineplas(n±1，2，3)，andabout8000000d(21903yr)fortheintegrationtheouterfiveplas(f±).
althoughoutputfilteringwasdonewhenthenumericalintegrationswereprocess，appliedalow-passfiltertheraworbitaldataafterhadpletedallthecsection4.1formoredetail.
2.4errorestimation
2.4.1relativeerrorstotalenergyandangularmomentum
accordingohebasicpropertiessymplecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangularmomentum)，ourlong-termnumericalintegrationsseemhavebeenperformedwithverysmalaveragedrelativeerrorstotalenergy(109)andtotalangularmomentum(1011)haveremainednearlyconstantthroughouttheintegrationperiod(fig.1).thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedrelativeerrortotalenergyaboutoneordermagnitudemore.
relativenumericalerrorthetotalangularmomentumδa/a0andthetotalenergyδe/e0ournumericalintegrationsn±1，2，3，whereandaretheabsolutechangethetotalenergyandtotalangularmomentum，respectively，ande0anda0aretheirinitiahorizontalunitgyr.
notethatdifferentoperatingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultdifferentnumericalerrors，throughthevariationsround-offerrorhandlingandnumericatheupperpanelo，canrecognizethissituationthesecularnumericalerrorthetotalangularmomentum，whichshouldrigorouslypreservedtomachine-eprecision.
2.4.2errorplaarylongitudes
sincethesymplecticmapspreservetotalenergyandtotalangularmomentumn-bodydynamicalsystemsinherentlywell，thedegreetheirpreservationmaynotagoodmeasuretheaccuracynumericalintegrations，especiallyameasurethepositionalerrorplas，i.e.theerrorplaarestimatethenumericalerrortheplaarylongitudes，performedthefollowinparedtheresultourmainlong-termintegrationswithsometestintegrations，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemainithispurpose，performedamuchmoreaccurateintegrationwithastepsize0.125d(1/64themainintegrations)spanning3x105yr，startingwiththesameinitialconditionsintheconsiderthatthistestintegrationprovideswitha‘pseudo-true’solutionplaaryorbita，parethetestintegrationwiththemainintegration，n1.fortheperiod3x105yr，seeadifferencemeananomaliestheearthbetweenthetwointegrations0.52°(inthecasetheintegration).thisdifferencecanextrapolatedthevalue8700°，aboutrotationsearthafter5gyr，sincetheerrorlongitudesincreaseslinearlywithtimethesymplecti，thelongitudeerrorplutocanestimated12°.thisvalueforplutomuchbetterthantheresultkinoshitaamp;nakai(1996)wherethedifferenceestimated60°.
3numericalresultsi.glancetherawdata
inthissectionbrieflyreviewthelong-termstabilityplaaryorbitalmotionthroughsomesnapshotsrawnumericaorbitalmotionplasindicateslong-termstabilityallournumericalintegrations:orbitalcrossingsnorcloseencountersbetweenanypairplastookplace.
3.1generaldescriptionthestabilityplaaryorbits
first，brieflylookthegeneralcharacterthelong-termstabilityplaarinterestherefocusesparticularlytheinnerfourterrestrialplasforwhichtheorbitaltime-scalesaremuchshorterthanthosetheouterfivcanseeclearlyfromtheplanarorbitalconfigurationsshownfigs2and3，orbitalpositionstheterrestrialplasdifferlittlebetweentheinitialandfinalparteachnumericalintegration，whichspansseverasolidlinesdenotingthepresentorbitstheplasliealmostwithintheswarmdotseventhefinalpartintegrations(b)and(d).thisindicatesthatthroughouttheentireintegrationperiodthealmostregularvariationsplaaryorbitalmotionremainnearlythesametheyarepresent.
verticalviewthefourinnerplaaryorbits(fromthez-axisdirection)theinitialandfinalpartstheintegrationsn±1.theaxesunitsareau.the-planesettheinvariantplanesolarsystemtotalangularmomentum.(a)theinitialpartofn+1(t=00.0547x9yr).(b)thefinalpartofn+1(t=4.9339x84.9886x9yr).(c)theinitialpartn1(t=00.0547x109yr).(d)thefinalpartofn1(t=3.9180x93.9727x9yr).eachpanel，atotal23684pointsareplottedwithintervalabout2190over5.47x107.solidlineseachpaneldenotethepresentorbitsthefourterrestrialplas(takenfromde245).
thevariationeccentricitiesandorbitalinclinationsfortheinnerfourplastheinitialandfinalparttheintegrationn+1showniexpected，thecharacterthevariationplaaryorbitalelementsdoesnotdiffersignificantlybetweentheinitialandfinalparteachintegration，leastforvenus，earthelementsmercury，especiallyitseccentricity，seemchangeasignificanpartlybecausetheorbitaltime-scaletheplatheshortestalltheplas，whichleadsamorerapidorbitalevolutionthanotherplas;theinnermostplamaynearestresultappearsbesomeagreementwithlaskar's(1994，1996)expectaionsthatlargeandirregularvariationsappeartheeccentricitiesandinclinationsmercuryatime-scaleseveral109yr.however，theeffectthepossibleinstabilitytheorbitmercurymaynotfatallyaffecttheglobalstabilitythewholeplaarysystemowingthesmallmassowillmentionbrieflythelong-termorbitalevolutionmercurylatersection4usinglow-passfilteredorbitalelements.
theorbitalmotiontheouterfiveplasseemsrigorouslystableandquiteregularoverthistime-span(seealsosection5).
3.2timefrequencymaps
althoughtheplaarymotionexhibitsverylong-termstabilitydefinedthenon-existencecloseencounterevents，thechaoticnatureplaarydynamicscanchangetheoscillatoryperiodandamplitudeplaaryorbitalmotiongraduallyoversuchlonsuchslightfluctuationsorbitalvariationthefrequencydomain，particularlythecaseearth，canpotentiallyhaveasignificanteffectitssurfaceclimatesystemthroughsolarinsolationvariation(cf.berger1988).
togiveoverviewthelong-termchangeperiodicityplaaryorbitalmotion，performedmanyfastfouriertransformations(ffts)alongthetimeaxis，andsuperposedtheresultingperiodgramsdrawtwo-dimensionaltimefrequencspecificapproachdrawingthesetimefrequencymapsthispaperverysimplemuchsimplerthanthewaveletanalysislaskar's(1990，1993)frequencyanalysis.
dividethelow-passfilteredorbitaldataintomanyfragmentsthesamlengtheachdatasegmentshouldamultiple2orderapplythefft.
eachfragmentthedatahasalargeoverlappingpart:forexample，whentheithdatabeginsfromt=tiandendst=ti+t，thenextdatasegmentrangesfromti+δt≤ti+δt+t，whereδt?t.continuethisdivisionuntilreachacertainnumbernwhichtn+treachesthetotalintegrationlength.
weapplyffteachthedatafragments，andobtainnfrequencydiagrams.
ineachfrequencydiagramobtainedabove，thestrengthperiodicitycanreplacedagrey-scale(orcolour)chart.
weperformthereplacement，andconnectallthegrey-scale(orcolour)chartsintoonegraphforeachhorizontalaxisthesenewgraphsshouldthetime，i.e.thestartingtimeseachfragmentdata(ti，wherei=1，…，n).theverticalaxisrepresentstheperiod(orfrequency)theoscillationorbitalelements.
wehaveadoptedfftbecauseitsoverwhelmingspeed，sincetheamountnumericaldatabedeposedintofrequencyponentsterriblyhuge(severaltensgbytes).
atypicalexamplethetimefrequencymapcreatedtheaboveproceduresshownagrey-scalediagrama，whichshowsthevariationperiodicitytheeccentricityandinclinationearthn+2fig.5，thedarkareashowsthatthetimeindicatedthevaluetheabscissa，theperiodicityindicatedtheordinatestrongerthanthelighterareaaroundit.canrecognizefromthismapthattheperiodicitytheeccentricityandinclinationearthonlychangesslightlyovertheentireperiodcoveredthen+2nearlyregulartrendqualitativelythesameotherintegrationsandforotherplas，althoughtypicalfrequenciesdifferplaplaandelementelement.
4.2long-termexchangeorbitalenergyandangularmomentum
wecalculateverylong-periodicvariationandexchangeplaaryorbitalenergyandangularmomentumusingfiltereddelaunayelementsl，g，h.gandhareequivalenttheplaaryorbitalangularmomentumanditsverticalponentperunirelatedtheplaaryorbitalenergyeperunitmasse=μ2/2l2.thesystempletelylinear，theorbitalenergyandtheangularmomentumeachfrequencybinmustbtheplaarysystemcancauseexchangeenergyandangularmomentumthefrequencamplitudethelowest-frequencyoscillationshouldincreasethesystemunstableandbreaksdow，suchasymptominstabilitynotprominentourlong-termintegrations.
i，thetotalorbitalenergyandangularmomentumthefourinnerplasandallnineplasareshownforintegrationn+2.theupperthreepanelsshowthelong-periodicvariationtotalenergy(denotedase-e0)，totalangularmomentum(g-g0)，andtheverticalponent(h-h0)theinnerfourplascalculatedfromthelow-passfiltereddelaunay，g0，denotetheinitialvalueseacabsolutedifferencefromtheinitialvaluesplottedthlowerthreepanelseachfigureshowe-e0，g-g0andh-h0thetotalninfluctuationshownthelowerpanelsvirtuallyentirelyaresultthemassivejovianplas.
paringthevariationsenergyandangularmomentumtheinnerfourplasandallnineplas，isapparentthattheamplitudesthosetheinnerplasaremuchsmallerthanthoseallnineplas:theamplitudestheouterfiveplasaremuchlargerthanthosetheinnedoesnotmeanthattheinnerterrestrialplaarysubsystemmorestablethantheouterone:thissimplyaresulttherelativesmallnessthemassesthefourterrestrialplasparedwiththosetheouterjoviathingnoticethattheinnerplaarysubsystemmaybeeunstablemorerapidlythantheouteronebecauseitsshorterorbitalcanseenthepanelsdenotedasinner4ithelonger-periodicandirregularoscillationsaremoreapparentthanthepanelsdenotedastotal9.actually，thefluctuationstheinner4panelsarealargeextentaresulttheorbitalvariationth，cannotneglectthecontributionfromotherterrestrialplas，wewillseesubsequentsections.
4.4long-termcouplingseveralneighbouringplapairs
letseesomeindividualvariationsplaaryorbitalenergyandangularmomentumexpressedthelow-passfiltereddelaunaandshowlong-termevolutiontheorbitalenergyeachplaandtheangularmomentumn+1andinoticethatsomeplasformapparentpairstermsorbitalenergyandangularmomentuparticular，venusandearthmakeatypicathefigures，theyshownegativecorrelationsexchangeenergyandpositivecorrelationsexchangeangulanegativecorrelationexchangeorbitalenergymeansthatthetwoplasformacloseddynamicalsystemtermstheorbitapositivecorrelationexchangeangularmomentummeansthatthetwoplasaresimultaneouslyundercertainlong-termforperturbersarejupiteri，canseethatmarsshowsapositivecorrelationtheangularmomentumvariationthevenuseartexhibitscertainnegativecorrelationstheangularmomentumversusthevenusearthsystem，whichseemsbeareactioncausedtheconservationangularmomentumtheterrestrialplaarysubsystem.
itnotclearthemomentwhythevenusearthpairexhibitsanegativecorrelationenergyexchangeandapositivecorrelationangularmomentumaypossiblyexplainthisthroughobservingthegeneralfactthatthereareseculartermsplaarysemimajoraxestosecond-orderperturbationtheories(cf.brouweramp;clemence1961;boccalettiamp;pucacco1998).thismeansthattheplaaryorbitalenergy(whichdirectlyrelatedthesemimajoraxisa)mightmuchlessaffectedperturbingplasthantheangularmomentumexchange(whichrelatese).hence，theeccentricitiesvenusandearthcanbedisturbedeasilyjupiterandsaturn，whichresultsapositivecorrelationtheangularmomentutheotherhand，thesemimajoraxesvenusandeartharelesslikelybedisturbedthejoviatheenergyexchangemaylimitedonlywithinthevenusearthpair，whichresultsanegativecorrelationtheexchangeorbitalenergythepair.
asfortheouterjovianplaarysubsystem，jupitersaturnanduranusneptuneseemmakedynamica，thestrengththeircouplingnotstrongparedwiththatthevenusearthpair.
5±5x1010-yrintegrationsouterplaaryorbits
sincethejovianplaarymassesaremuchlargerthantheterrestrialplaarymasses，treatthejovianplaarysystemanindependentplaarysystemtermsthestudyitsdynamica，addedacoupletrialintegrationsthatspan±5x1010yr，includingonlytheouterfiveplas(thefourjovianplaspluspluto).theresultsexhibittherigorousstabilitytheouterplaarysystemoverthislonconfigurations(fig.12)，andvariationeccentricitiesandinclinations(fig.13)showthisverylong-termstabilitytheouterfiveplasboththetimeandthefrequencdonotshowmapshere，thetypicalfrequencytheorbitaloscillationplutoandtheotherouterplasalmostconstantduringtheseverylong-termintegrationperiods，whichdemonstratedthetimefrequencymapsourwebpage.
inthesetwointegrations，therelativenumericalerrorthetotalenergywas106andthatthetotalangularmomentumwas1010.
5.1resonancestheneptuneplutosystem
kinoshitaamp;nakai(1996)integratedtheouterfiveplaaryorbitsover±5.5x109.theyfoundthatfourmajorresonancesbetweenneptuneandplutoaremaintainedduringthewholeintegrationperiod，andthattheresonancesmaythemaincausesthestabilitytheorbitomajorfourresonancesfoundpreviousresearchareathefollowingdescription，λdenotesthemeanlongitude，Ωthelongitudetheascendingnodeandthelongitudeopandndenoteplutoandneptune.
meanmotionresonancebetweenneptuneandpluto(3:2).thecriticalargumentθ1=32λnplibratesaround180°withamplitudeabout80°andalibrationperiodabout2x104yr.
theargumentperihelionplutowp=θ2=pΩplibratesaround90°withaperiodabout3.8x106yr.thedominantperiodicvariationstheeccentricityandinclinationplutoaresynchronizedwiththelibrationitsargumentoanticipatedthesecularperturbationtheoryconstructedkozai(1962).
thelongitudethenodeplutoreferredthelongitudethenodeneptune，θ3=ΩpΩn，circulatesandtheperiodthiscirculationequaltheperiodθbeeszero，i.e.thelongitudesascendingnodesneptuneandplutooverlap，theinclinationplutobeesmaximum，theeccentricitybeesminimumandtheargumentperihelionbees90°.whenbees180°，theinclinationplutobeesminimum，theeccentricitybeesmaximumandtheargumentperihelionbees90°amp;benson(1971)anticipatedthistyperesonance，laterconfirmedmilani，nobiliamp;carpino(1989).
anargumentθ4=pn+3(ΩpΩn)libratesaround180°withalongperiod，5.7x108yr.
inournumericalintegrations，theresonances(i)(iii)arewellmaintained，andvariationthecriticalargumentsθ1，θ2，θ3remainsimilarduringthewholeintegrationperiod(figs1416).however，thefourthresonance(iv)appearsbedifferent:thecriticalargumentalternateslibrationandcirculationovera1010-yrtime-scale(fig.17).thisaninterestingfactthatkinoshitaamp;nakai's(1995，1996)shorterintegrationswerenotabledisclose.
6discussion
whatkinddynamicalmechanismmaintainsthislong-termstabilitytheplaarysystem?canimmediatelythinktwomajorfeaturesthatmayresponsibleforthelong-ter，thereseembesignificantlower-orderresonances(meanmotionandsecular)betweenanypairamongtheninandsaturnareclosea5:2meanmotionresonance(thefamous‘greatinequality’)，butnotjusttheresonancresonancesmaycausethechaoticnaturetheplaarydynamicalmotion，buttheyarenotstrongtodestroythestableplaarymotionwithinthelifetimetherealsolasecondfeature，whichthinkmoreimportantforthelong-termstabilityourplaarysystem，thedifferencedynamicaldistancebetweenterrestrialandjovianplaarysubsystems(itoamp;tanikawa1999，2001).whenmeasureplaaryseparationsthemutualhillradii(r_)，separationsamongterrestrialplasaregreaterthan26rh，whereasthoseamongjovianplasarelessthadifferencedirectlyrelatedthedifferencebetweendynamicalfeaturesterrestrialandjoviaplashavesmallermasses，shorterorbitalperiodsandwiderdynamicaarestronglyperturbedjovianplasthathavelargermasses，longerorbitalperiodsandnarrowerdynamicaplasarenotperturbedanyothermassivebodies.
thepresentterrestrialplaarysystemstillbeingdisturbedthemassivejovia，thewideseparationandmutualinteractionamongtheterrestrialplasrendersthedisturbanceineffective;thedegreedisturbancejovianplaso(ej)(ordermagnitudetheeccentricityjupiter)，sincethedisturbancecausedjovianplasaforcedoscillationhavingamplitudeo(ej).heighteningeccentricity，forexampleo(ej)0.05，farfromsufficientprovokeinstabilitytheterrestrialplashavingsuchawideseparationaassumethatthepresentwidedynamicalseparationamongterrestrialplas(gt;26rh)probablyohemostsignificantconditionsformaintainingthestabilitytheplaarysystemovera109-ydetailedanalysistherelationshipbetweendynamicaldistancebetweenplasandtheinstabilitytime-scalesolarsystemplaarymotionnowon-going.
althoughournumericalintegrationsspanthelifetimethesolarsystem，thenumberintegrationsfarfromsufficientfilltheinitialphasnecessaryperformmoreandmorenumericalintegrationsconfirmandexaminedetailthelong-termstabilityourplaarydynamics.
以上文段引自ito，t.tanikawa，k.long-termintegrationsandstabilityplaaryorbitsoursola，483500(2002)
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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。