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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