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    以下是文章內(nèi)容：

    long-termiionsandstabilityofplaaryorbitsinoursolarsystem

    abstract

    wepresenttheresultsofverylong-termnumeritegrationsofplaaryorbitalmotionsover109-yrtime-spansincludingallnineplas.aquispeofournumericaldatashowsthattheplaarymotion，atleastinoursimpledynamicalmodel，seemstobequitestableevehisverylongtime-span.acloserlookatthelowest-frequencyoscillationsusingalow-passfiltershowsusthepotentiallydiffusivecharacterofterrestrialplaarymotion，especiallythatofmercury.thebehaviouroftheetriercuryinouriionsisqualitativelysimilartotheresultsfromjacqueslaskar'ssecularperturbationtheory(e.g.emax～0.35yr).however，therearenoapparentsecularincreasesofetricityorinationinanyorbitalelementsoftheplas，whichmayberevealedbystillloermnumeritegrations.wehavealsoperformedacoupleoftrialiionsincludingmotionsoftheouterfiveplasoverthedurationof±5x1010yr.theresultindicatesthatthethreemajorresoheune–plutosystemhavebeenmaintainedoverthe1011-yrtime-span.

    1introdu

    1.1definitionoftheproblem

    thequestionofthestabilityofoursolarsystemhasbeeedoverseveralhundredyears，siheeraofon.theproblemhasattractedmanyfamousmathematisovertheyearsandhasplayedatralroleinthedevelopmentofnon-lineardynamidchaostheory.however，wedohaveadefiniteahequestionofwhetheroursolarsystemisstableornot.thisispartlyaresultofthefactthatthedefinitionoftheterm‘stability’isvaguewhenitisusediiontotheproblemofplaarymotioninthesolarsystem.actuallyitisogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofoursolarsystem.

    amongmanydefinitionsofstability，heretthehilldefinition(gladman1993):actuallythisisnotadefinitionofstability，butofinstability.wedefineasystemasbeingunstablewhenacloseenteroccurssomewhereiem，startingfromacertaininitialfiguration(chambers，wetherillitotanikawa1999).asystemisdefinedasexperiengacloseenterwhentwobodiesapproaeahinahelargerhillradius.otherwisethesystemisdefinedasbeingstable.henceforwardwestatethatourplaarysystemisdynamicallystableifnocloseenterhappensduringtheageofoursolarsystem，about±5gyr.ially，thisdefinitionmaybereplacedbyoneinwhioccurrenceofanyorbitalcrossiweeherofapairofplaakesplace.thisisbecauseweknowfromexperieanorbitalcrossingisverylikelytoleadtoacloseenterinplaaryandprotoplaarysystems(yoshinaga，kokubomakino1999).ofcoursethisstatementotbesimplyappliedtosystemswithstableorbitalresonancessuchastheune–plutosystem.

    1.2previousstudiesandaimsofthisresearch

    inadditiontothevaguenessoftheceptofstability，theplasinoursolarsystemshowacharactertypicalofdynamicalchaos(sussmanwisdom1988，1992).thecauseofthischaoticbehaviourisnowpartlyuoodasbeiofresonanceoverlapping(murraylecar，franklinholman2001).however，itwouldrequireiingoveranensembleofplaarysystemsincludingallnineplasforaperiodcseveral10gyrthlyuandthelong-termevolutionofplaaryorbits，sincechaotiamicalsystemsarecharacterizedbytheirstrongdependeninitialditions.

    fromthatpointofview，manyofthepreviouslong-termnumeritegrationsincludedonlytheouterfiveplas(sussmankinoshitanakai1996).thisisbecausetheorbitalperiodsoftheouterplasaresomugerthanthoseoftheinnerfourplahatitismucheasiertofollowthesystemfiveionperiod.atpresent，thelonumeritegrationspublishedinjournalsarethoseofdunlissauer(1998).althoughtheirmaintargetwastheeffectofpost-main-sequenasslossoabilityofplaaryorbits，theyperformedmanyiionscupto～1011yroftheorbitalmotionsofthefourjovianplaheinitialorbitalelementsandmassesofplasarethesameasthoseofoursolarsystemindunlissauer'spaper，buttheydecreasethemassofthesungraduallyintheirnumericalexperiments.thisisbecausetheysidertheeffectofpost-main-sequenasslossinthepaper.sequently，theyfoundthatthecrossingtime-scaleofplaaryorbits，whibeatypidicatoroftheinstabilitytime-scale，isquitesensitivetotherateofmassdecreaseofthesuhemassofthesunisclosetoitspresentvalue，thejovianplasremainstableover1010yr，orperhapslonger.dunlissaueralsoperformedfoursimilarexperimentsontheorbitalmotionofsevenplas(venustoune)，whichcoveraspanof～109yr.theirexperimentsonthesevenplasarepreheitseemsthattheterrestrialplasalsoremainstableduriegrationperiod，maintainingalmularoscillations.

    oherhand，inhisaccuratesemi-analyticalsecularperturbationtheory(laskar1988)，laskarfindsthatlargeandirregularvariationsappearintheetricitiesandinationsoftheterrestrialplas，especiallyofmercuryandmarsonatime-scaleofseveral109yr(laskar1996).theresultsoflaskar'ssecularperturbationtheoryshouldbefirmedandiigatedbyfullynumeritegrations.

    inthispaperwepresentpreliminaryresultsofsixlong-termnumeritegrationsonallnineplaaryorbits，caspanofseveral109yr，andoftwootheriionscaspanof±5x1010yr.thetotalelapsedtimeforalliionsismorethan5yr，usingseveraldedicatedpdworkstations.ohefualclusions-termiionsisthatsolarsystemplaarymotioobestableintermsofthehillstabilitymentionedabove，atleastoveratime-spanof±4gyr.actually，inournumeritegratioemwasfarmorestablethanwhatisdefihehillstabilitycriterion:notonlydidnocloseenterhappenduriegrationperiod，butalsoalltheplaaryorbitalelementshavebeenfinedinanarrionbothintimeandfrequenain，thoughplaarymotioochasticethepurposeofthispaperistoexhibitandoverviewtheresults-termnumeritegrations，weshowtypicalexamplefiguresasevideheveryloabilityofsolarsystemplaarymotion.forreaderswhohavemorespecifiddeeperisinournumericalresults，reparedawebpage(access)，whereweshowraworbitalelements，theirlow-passfilteredresults，variationofdelaunayelementsandangularmomentumdeficit，asofoursimpletime–frequenalysisonallofouriions.

    iion2webrieflyexplainourdynamicalmodel，numericalmethodandinitialditionsusedinouriioion3isdevotedtoadescriptionofthequickresultsofthenumeritegrations.veryloabilityofsolarsystemplaarymotionisapparentbothinplaarypositionsandorbitalelements.aroughestimationofnumericalerrorsisalsogiveiooadiscussionoftheloermvariationofplaaryorbitsusingalow-passfilterandincludesadiscussionofangularmomentumdeficit.iion5，wepreseofnumeritegrationsfortheouterfiveplahatspans±5x1010yr.iion6wealsodiscusstheloabilityoftheplaarymotionanditspossiblecause.

    2descriptionofthenumeritegrations

    （本部分涉及比較復(fù)雜的積分計算，作者君就不貼上來了，貼上來了起點(diǎn)也不一定能成功顯示。）

    2.3numericalmethod

    weutilizeased-orderwisdom–holmansymplecticmapasourmaiiohod(wisdomkinoshita，yoshidanakai1991)ecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warmstart’(sahatremaine1992，1994).

    thestepsizeforthenumeritegrationsis8dthroughoutalliionsofthenineplas(n±1，2，3)，whichisabout111oftheorbitalperiodoftheinnermostpla(mercury).asforthedeterminationofstepsize，wepartlyfollowthepreviousnumeritegrationofallnineplasinsussmanwisdom(1988，7.2d)andsahatremaine(1994，22532d).werouhedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2ioreducetheaccumulationofround-offerrorintheputationprocesses.iiontothis，wisdomholman(1991)performednumeritegrationsoftheouterfiveplaaryorbitsusingthesymplecticmapsizeof400d，110.83oftheorbitalperiodofjupiter.theirresultseemstobeaccurateenough，whichpartlyjustifiesourmethodofdeterminiepsize.however，siheetricityofjupiter(～0.05)ismuchsmallerthanthatofmercury(～0.2)，weneedsomecarewhenweparetheseiionssimplyintermsofstepsizes.

    iegrationoftheouterfiveplas(f±)，wefixedthestepsizeat400d.

    tgauss'fandgfunsinthesymplecticmaptogetherwiththethird-orderhalleymethod(danby1992)asasolverforkeplerequations.thenumberofmaximumiteratioinhalley'smethodis15，buttheyneverreachedthemaximuminanyofouriions.

    theintervalofthedataoutputis200000d(～547yr)forthecalculationsofallnineplas(n±1，2，3)，andabout8000000d(～21903yr)fortheiionoftheouterfiveplas(f±).

    althoughnooutputfilteringwasdohenumeritegrationswereinprocess，liedalow-passfiltertotheraworbitaldataafterwehadpletedallthecalculations.seese4.1formoredetail.

    2.4errorestimation

    2.4.1relativeerrorsintotalenergyandangularmomentum

    acctoohebasicpropertiesofsymplectitegrators，whiservethephysicallyservativequantitieswell(totalorbitalenergyandangularmomentum)，-termnumeritegratioohavebeenperformedwithverysmallerrors.theaveragedrelativeerrorsoftotalenergy(～10?9)andoftotalangularmomentum(～10?11)haveremainednearlystantthroughouttheiionperiod(fig.1).thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedrelativeerrorintotalenergybyaboutoneorderofmagnitudeormore.

    relativenumericalerrorofthetotalangularmomentumδaa0aalenergyδee0inournumeritegrationsn±1，2，3，whereδeandδaaretheabsolutegeofthetotalenergyandtotalangularmomentum，respectively，ande0anda0aretheirinitialvalues.thehorizontalunitisgyr.

    differeingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresultindifferentnumericalerrors，throughthevariationsinround-offerrorhandlingandnumericalalgorithms.intheupperpaneloffig.1，wereizethissituationinthesecularnumericalerrorialangularmomentum，whichshouldberigorouslypreserveduptomae-eprecision.

    2.4.2errorinplaarylongitudes

    sihesymplecticmapspreservetotalenergyandtotalangularmomentumofn-bodydynamicalsystemsilywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuraeritegrations，especiallyasameasureofthepositionalerrorofplas，i.e.theerrorinplaarylongitudes.toestimatethenumericalerrorintheplaarylongitudes，weperformedthefollowingprocedures.weparedtheresultofourmainlong-termiionswithsometestiions，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemaiions.forthispurpose，weperformedamuchmoreaccurateiionsizeof0.125d(164ofthemaiions)spanning3x105yr，startingwiththesameinitialditionsasinthen?1iion.wesiderthatthistestiionprovidesusseudo-true’solutionofplaaryorbitalevolutio，weparethetestiionwiththemaiion，n?1.fortheperiodof3x105yr，weseeadifferenmeananomaliesoftheearthbetweewoiionsof～0.52°(inthecaseofthen?1iion).thisdifferenbeextrapolatedtothevalue～8700°，about25rotatiohafter5gyr，siheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.similarly，thelongitudeerrorofplutobeestimatedas～12°.thisvalueforplutoismuchbetterthantheresultinkinoshitanakai(1996)wherethedifferenceisestimatedas～60°.

    3numericalresults–i.glaherawdata

    inthissewebrieflyreviewtheloabilityofplaaryorbitalmotihsomesnapshotsofrawnumericaldata.theorbitalmotionofplasindicatesloabilityinallofournumeritegrations:noorbitalcrossingsnorcloseentersbetairofplaookplace.

    3.1generaldescriptionofthestabilityofplaaryorbits

    first，webrieflylookatthegeneralcharacteroftheloabilityofplaaryorbits.ouriherefocusesparticularlyontheinnerfourterrestrialplasforwhichtheorbitaltime-scalesaremuchshorterthanthoseoftheouterfiveplas.asweseeclearlyfromtheplanarorbitalfigurationsshowninfigs2and3，orbitalpositionsoftheterrestrialplasdifferlittlebetweeialandfinalpartofeaumeritegration，whichspansseveralgyr.thesolidliingthepresentorbitsoftheplasliealmostwithintheswarmofdotseveninthefinalpartofiions(b)and(d).thisindicatesthatthroughouttheeegrationperiodthealmularvariationsofplaaryorbitalmotionremainnearlythesameastheyareatpresent.

    verticalviewofthefourinnerplaaryorbits(fromthez-axisdire)attheinitialandfinalpartsoftheiionsn±1.theaxesunitsareau.thexy-planeissettotheinvariantplaneofsolarsystemtotalangularmomentum.(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.solidlinesineaeldehepresentorbitsofthefourterrestrialplaakenfromde245).

    thevariationofetricitiesandorbitalinationsfortheinnerfourplasiialandfinalpartoftheiionn 1isshowninfig.4.asexpected，thecharacterofthevariationofplaaryorbitalelementsdoesnotdiffersignifitlybetweeialandfinalpartofeategration，atleastforvehandmars.theelementsofmercury，especiallyitsetricity，seemtoget