1.2Previousstudiesandaimsofthisresearch
Inadditiontothevaguenessoftheconceptofstability,theplanetsinourSolarsystemshowacharactertypicalofdynamicalchaos(Sussman&&Wisdom1988,1992).Thecauseofthischaoticbehaviourisnowpartlyunderstoodasbeingaresultofresonanceoverlapping(Murray&Lecar,Franklin&&Holman2001).However,itwouldrequireintegratingoveranensembleofplanetarysystemsincludingallnineplanetsforaperiodcoveringseveral10Gyrtothoroughlyunderstandthelong-termevolutionofplanetaryorbits,sincechaoticdynamicalsystemsarecharacterizedbytheirstrongdependenceoninitialconditions.
Fromthatpointofview,manyofthepreviouslong-termnumericalintegrationsincludedonlytheouterfiveplanets(Sussman&Kinoshita&&Nakai1996).Thisisbecausetheorbitalperiodsoftheouterplanetsaresomuchlongerthanthoseoftheinnerfourplanetsthatitismucheasiertofollowthesystemforagivenintegrationperiod.Atpresent,thelongestnumericalintegrationspublishedinjournalsarethoseofDuncan&&Lissauer(1998).Althoughtheirmaintargetwastheeffectofpost-main-sequencesolarmasslossonthestabilityofplanetaryorbits,theyperformedmanyintegrationscoveringupto∼1011yroftheorbitalmotionsofthefourjovianplanets.TheinitialorbitalelementsandmassesofplanetsarethesameasthoseofourSolarsysteminDuncan&&Lissauer'spaper,buttheydecreasethemassoftheSungraduallyintheirnumericalexperiments.Thisisbecausetheyconsidertheeffectofpost-main-sequencesolarmasslossinthepaper.Consequently,theyfoundthatthecrossingtime-scaleofplanetaryorbits,whichcanbeatypicalindicatoroftheinstabilitytime-scale,isquitesensitivetotherateofmassdecreaseoftheSun.WhenthemassoftheSunisclosetoitspresentvalue,thejovianplanetsremainstableover1010yr,orperhapslonger.Duncan&&Lissaueralsoperformedfoursimilarexperimentsontheorbitalmotionofsevenplanets(VenustoNeptune),whichcoveraspanof∼109yr.Theirexperimentsonthesevenplanetsarenotyetcomprehensive,butitseemsthattheterrestrialplanetsalsoremainstableduringtheintegrationperiod,maintainingalmostregularoscillations.
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