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Long-termiionsandstabilityofparyorbitsinourSorsystem
Abstract
Wepresenttheresusofverylong-termnumericaliionsofparyorbitalmotionsover109-yrtime-spansincludingallnineps.Aquispeofournumericaldatashowsthattheparymotion，atleastinoursimpledynamicalmodel，seemstobequitestableevehisverylongtime-span.Acloserlookatthelowest-frequencyosciionsusingalow-passfiershowsusthepotentiallydiffusivecharacterofterrestrialparymotion，especiallythatofMercury.ThebehaviouroftheetriercuryinouriionsisqualitativelysimirtotheresusfromJacquesLaskar''''''''ssecurperturbationthe.emax～0.35yr.However，therearenoapparentsecreasesofetricityorinationinanyorbitalelementsoftheps，whichmayberevealedbystillloermnumericaliions.Wehavealsoperformedacoupleoftrialiionsincludingmotionsoftheouterfivepsoverthedurationof±5×1010yr.Theresuindicatesthatthethreemajorresoheune–Plutosystemhavebeenmaintainedoverthe1011-yrtime-span.
1Introdu
1.1Definitionoftheproblem
ThequestionofthestabilityofourSorsystemhasbeeedoverseveralhundredyears，siheeraofon.Theproblemhasattractedmanyfamousmathematisovertheyearsandhaspyedatralroleinthedevelopmentofnon-lineardynamidchaostheory.However，wedohaveadefihequestionofwhetherourSorsystemisstableornot.Thisispartlyaresuofthefactthatthedefinitionoftheterm‘stability’isvaguewhenitisusediiontotheproblemofparymotionintheSorsystem.Actuallyitisogiveaclear，rigorousandphysicallymeaningfuldefinitionofthestabilityofourSorsystem.
Amongmanydefinitionsofstability，herettheHilldefinitionGdman1993:actuallythisisnotadefinitionofstability，butofinstability.Wedefineasystemasbeingunstablewhenacloseenteroccurssomewhereiem，startingfromacertaininitialfigurationChambers，Wetherill&Boss1996Ito&Tanikawa1999.AsystemisdefinedasexperiengacloseenterwhentwobodiesapproaeahinahergerHillradius.Otherwisethesystemisdefinedasbeingstable.HenceforwardwestatethatourparysystemisdynamicallystableifnocloseenterhappensduringtheageofourSorsystem，about±5Gyr.Ially，thisdefinitionmayberepcedbyoneinwhioccurrenceofanyorbitalcrossiweeherofapairofpakespce.ThisisbecauseweknowfromexperiehatanorbitalcrossingisverylikelytoleadtoacloseenteriaryandprotoparysystemsYoshinaga，Kokubo&Makino1999.Ofcoursethisstatementotbesimplyappliedtosystemswithstableorbitalresonancessuchastheune–Plutosystem.
内容未完，下一页继续阅读 </p><p style="font-size:16px">1.2Previousstudiesandaimsofthisresearch
Inadditiontothevaguenessoftheceptofstability，thepsinourSorsystemshowacharactertypicalofdynamicalchaosSussman&Wisdom1988，1992.ThecauseofthischaoticbehaviourisnowpartlyuoodasbeiofresonanceoverppingMurray&Holman1999Lecar，Franklin&Holman2001.However，itwouldrequireiingoveranensembleofparysystemsincludingallninepsforaperiodcseveral10Gyrtothhlyuandthelong-termevolutionofparyorbits，sinceicalsystemsarecharacterizedbytheirstrongdependeninitialditions.
Fromthatpointofview，manyofthepreviouslong-termnumericaliionsincludedonlytheouterfivepsSussman&Wisdom1988Kinoshita&Nakai1996.Thisisbecausetheorbitalperiodsoftheouterpsaresomuchlohanthoseoftheinnerfourphatitismucheasiertofollowthesystemfiveniionperiod.Atpresent，thelonumericaliionspublishedinjournalsarethoseofDun&Lissauer1998.Ahoughtheirmaintargetwastheeffectofpost-main-sequenasslossoabilityofparyorbits，theyperformedmanyiionscupto～1011yroftheorbitalmotionsofthefourjovias.TheinitialorbitalelementsandmassesofpsarethesameasthoseofourSorsysteminDun&Lissauer''''''''spaper，buttheydecreasethemassoftheSungraduallyintheirnumericalexperiments.Thisisbecausetheysidertheeffectofpost-main-sequenasslossinthepaper.sequently，theyfoundthatthecrossingtime-scaleofparyorbits，whibbeatypidicatoroftheinstabilitytime-scale，isquitesensitivetotherateofmassdecreaseoftheSuhemassoftheSunisclosetoitspresentvalue，thejoviasremainstableover1010yr，orperhapslonger.Dun&LissaueralsoperformedfoursimirexperimentsontheorbitalmotionofsevesVenustoune，whichcoveraspanof～109yr.Theirexperimentsonthesevesarepreheitseemsthattheterrestrialpsalsoremainstableduringtheiionperiod，maintainingalmurosciions.
Oherhand，inhisaccuratesemi-analyticalsecurperturbationtheoryLaskar1988，Laskarfindsthatrgeandirregurvariationsappeariricitiesandinationsoftheterrestrialps，especiallyofMercuryandMarsonatime-scaleofseveral109yrLaskar1996.TheresusofLaskar''''''''ssecurperturbationtheoryshouldbefirmedandiigatedbyfullynumericaliions.
Inthispaperwepresentpreliminaryresusofsixlong-termnumericaliionsonallnineparyorbits，caspanofseveral109yr，andoftwootheriionscaspanof±5×1010yr.Thetotalepsedtimeforalliionsismorethan5yr，usingseveraldedicatedPdworkstations.Ohefualclusions-termiionsisthatSorsystemparymotioobestableintermsoftheHillstabilitymentionedabove，atleastoveratime-spanof±4Gyr.Actually，inournumericaliioemwasfarmorestablethanwhatisdefiheHillstabilitycriterion:notonlydidnocloseenterhappenduringtheiionperiod，butalsoalheparyorbitalelementshavebeenfinedinanarrionbothintimeandfrequenain，thoughparymotioochastic.Sihepurposeofthispaperistoexhibitandoverviewtheresus-termnumericaliions，weshowtypicalexamplefiguresasevideheveryloabilityofSorsystemparymotion.Forreaderswhohavemorespecifiddeeperisinournumericalresus，reparedawebpageaccess，whereweshowraworbitalelements，theirlow-passfieredresus，variationofDeunayelementsandangurmomentumdeficit，asofoursimpletime–frequenalysisonallofouriions.
Iion2webrieflyexpinourdynamicalmodel，numericalmethodandinitialditionsusedinouriions.Se3isdevotedtoadescriptionofthequickresusofthenumericaliions.VeryloabilityofSorsystemparymotionisapparentbothiarypositionsandorbitalelements.Aroughestimationofnumericalerrorsisalsogiveiooadiscussionoftheloermvariationofparyorbitsusingalow-passfierandincludesadiscussionofangurmomentumdeficit.Iion5，wepreseofnumericaliionsfortheouterfivephatspans±5×1010yr.Iion6wealsodiscusstheloabilityoftheparymotionanditspossiblecause.
2Descriptionofthenumericaliions
本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了起点也不一定能成功显示。
2.3Numericalmethod
Weutilizeased-orderWisdom–HolmansymplecticmapasourmainiiohodWisdom&Holman1991Kinoshita，Yoshida&Nakai1991ecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warmstart’Saha&Tremaine1992，1994.
Thestepsizeforthenumericaliionsis8dthroughoutalliionsoftheninepsN±1，2，3，whichisabout1/11oftheorbitalperiodoftheinnermostpMercury.Asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericaliionofallninepsinSussman&Wisdom1988，7.2dandSaha&Tremaine1994，225/32d.Werouhedecimalpartofthetheirstepsizesto8tomakethestepsizeamuipleof2ioreducetheaccumutionofround-offerroriationprocesses.Iiontothis，Wisdom&Holman1991performednumericaliionsoftheouterfiveparyorbitsusingthesymplecticmapsizeof400d，1/10.83oftheorbitalperiodofJupiter.Theirresuseemstobeaccurateenough，whichpartlyjustifiesourmethodofdeterminiepsize.However，siheetricityofJupiter～0.05ismuchsmallerthanthatofMercury～0.2，weneedsomecarewhenweparetheseiionssimplyintermsofstepsizes.
内容未完，下一页继续阅读 </p><p style="font-size:16px">IegrationoftheouterfivepsF±，wefixedthestepsizeat400d.
tGauss''''''''fandgfunsinthesymplecticmaptogetherwiththethird-orderHalleymethodDanby1992asasolverforKeplerequations.ThenumberofmaximumiteratioinHalley''''''''smethodis15，buttheyneverreachedthemaximuminanyofouriions.
Theintervalofthedataoutputis200000d～547yrforthecalcutionsofallninepsN±1，2，3，andabout8000000d～21903yrfortheiionoftheouterfivepsF±.
Ahoughnooutputfieringwasdohenumericaliionswereinprocess，liedalow-passfiertotheraworbitaldataafterletedalhecalcutions.SeeSe4.1formoredetail.
2.4Errorestimation
2.4.1Retiveerrorsintotalenergyandangurmomentum
Aceofthebasicpropertiesofsymplectitegrators，whichservethephysicallyservativequantitieswelotalorbitalenergyandangurmomentum，-termnumericaliioohavebeenperformedwithverysmallerrors.Theaveragedretiveerrorsoftotalenergy～10?9andoftotangurmomentum～10?11haveremainednearlystantthroughouttheiionperiodFig.1.Thespecialstartupprocedure，warmstart，wouldhavereducedtheaveragedretiveerrorintotalenergybyaboutoneorderofmagnitudeormore.
RetivenumericalerrorofthetotangurmomentumδA/A0aalenergyδE/E0inournumericaliionsN±1，2，3，whereδEandδAaretheabsolutegeofthetotalenergyandtotangurmomentum，respectively，andE0andA0aretheirinitialvalues.ThehorizontalunitisGyr.
hatdiffereingsystems，differentmathematicallibraries，anddifferenthardwarearchitecturesresuindifferentnumericalerrors，throughthevariationsinround-offerrorhandlingandnumericalgorithms.IntheupperpanelofFig.1，wereizethissituationintheseumericalerrurmomentum，whichshouldberigorouslypreserveduptomae-εprecision.
2.4.2Erroriarylongitudes
SihesymplecticmapspreservetotalenergyandtotangurmomentumofN-bodydynamicalsystemsilywell，thedegreeoftheirpreservationmaynotbeagoodmeasureoftheaccuraericaliions，especiallyasameasureofthepositionalerrorofps，i.e.theerroriarylongitudes.Toestimatethenumericalerrorintheparylongitudes，weperformedthefollowingprocedures.Weparedtheresuofourmainlong-termiionswithsometestiions，whichspanmuchshorterperiodsbutwithmuchhigheraccuracythanthemainiions.Forthispurpose，weperformedamuchmoreaccurateiionsizeof0.125d1/64ofthemainiionsspanning3×105yr，startingwiththesameinitialditionsasintheN?1iion.Wesiderthatthistestiionprovidesusseudo-true’solutionofparyorbitalevolutio，weparethetestiionwiththemainiion，N?1.Fortheperiodof3×105yr，weseeadifferenmeananomaliesoftheEarthbetweewoiionsof～0.52°inthecaseoftheN?1iion.Thisdifferenbbeextrapotedtothevalue～8700°，about25rotatiohafter5Gyr，siheerroroflongitudesincreaseslinearlywithtimeinthesymplecticmap.Simirly，thelongitudeerrorofPlutobeestimatedas～12°.ThisvalueforPlutoismuchbetterthantheresuinKinoshita&Nakai1996wherethedifferenceisestimatedas～60°.
内容未完，下一页继续阅读 </p><p style="font-size:16px">3Numericalresus–I.Gtherawdata
Inthissewebrieflyreviewtheloabilityofparyorbitalmotihsomesnapshotsofrawnumericaldata.Theorbitalmotionofpsindicatesloabilityinallofournumericaliions:noorbitalcrossingsnorcloseentersbetairofpookpce.
3.1Generaldescriptionofthestabilityofparyorbits
First，webrieflylookatthegeneralcharacteroftheloabilityofparyorbits.Ouriherefocusesparticurlyontheinnerfourterrestrialpsforwhichtheorbitaime-scalesaremuchshorterthanthoseoftheouterfiveps.AsweseeclearlyfromthepnarorbitalfigurationsshowninFigs2and3，orbitalpositionsoftheterrestrialpsdifferlittlebetweeiandfinalpartofeaumericaliion，whichspansseveralGyr.Thesolidliingthepresentorbitsofthepsliealmostwithintheswarmofdotseveninthefinalpartofiionsbandd.Thisindicatesthatthroughouttheeegrationperiodthealmurvariationsofparyorbitalmotionremainnearlythesameastheyareatpresent.
Verticalviewofthefourinnerparyorbitsfromthez-axisdireattheinitiandfinalpartsoftheiionsN±1.Theaxesunitsareau.Thexy-pneissettotheinvariantpneofSorsystemtotangurmomentum.aTheinitialpartofN+1t=0to0.0547×109yr.bThefinalpartofN+1t=4.9339×108to4.9886×109yr.cTheinitialpartofN?1t=0to?0.0547×109yr.dThefinalpartofN?1t=?3.9180×109to?3.9727×109yr.Iotalof23684pointsareplottedwithanintervalofabout2190yrover5.47×107yr.SolidlinesineaeldehepresentorbitsofthefourterrestrialpstakenfromDE245.