terfiveplahefourjovianplaspluspluto).theresultsexhibittherigorousstabilityoftheouterplaarysystemoverthislongtime-span.orbitalfigurations(fig.12)，andvariationofetricitiesandinations(fig.13)showthisveryloabilityoftheouterfiveplasinboththetimeandthefrequenains.althoughwedonotshoshere，thetypicalfrequencyoftheorbitaloscillationofplutoaherouterplasisalmoststantduringtheseverylong-termiionperiods，whichisdemonstratediime–frequencymapsonourwebpage.
iwoiions，therelativenumericalerrorialenergywas～10?6andthatofthetotalangularmomentumwas～10?10.
5.1resoheune–plutosystem
kinoshitanakai(1996)iedtheouterfiveplaaryorbitsover±5.5x109yr.theyfoundthatfourmajorresoweeuneandplutoaremaintainedduringthewholeiionperiod，andthattheresonancesmaybethemaincausesofthestabilityoftheorbitofpluto.themajorfourresonancesfoundinpreviousresearchareasfollows.inthefollowingdescriptioesthemeanlongitude，Ωisthelongitudeoftheasdingnodeand?isthelongitudeofperihelion.subscriptspaeplutoaune.
meanmotionresoweeuneandpluto(3:2).thecriticalargumentθ1=3λp?2λn??plibratesaround180°litudeofabout80°andalibrationperiodofabout2x104yr.
theargumentofperihelionofplutowp=θ2=?p?Ωplibratesaround90°eriodofabout3.8x106yr.thedominantperiodicvariationsoftheetricityandinationofplutoaresynizedwiththelibrationofitsargumentofperihelion.thisisanticipatedinthesecularperturbationtheorystructedbykozai(1962).
thelongitudeofthenodeofplutoreferredtothelongitudeofthenodeofune，θ3=Ωp?Ωn，circulatesandtheperiodofthiscirculationisequaltotheperiodofθ2libration.whenθ3beeszero，i.e.thelongitudesofasdingnodesofuneandplutooverlap，theinationofplutobeaximum，theetricitybeinimumandtheargumentofperihelionbees90°.whenθ3bees180°，theinationofplutobeinimum，theetricitybeaximumandtheargumentofperihelionbees90°again.williamsbenson(1971)anticipatedthistypeofresoerfirmedbymilani，nobilicarpino(1989).
anargumentθ4=?p??n 3(Ωp?Ωn)libratesaround180°withalongperiod，～5.7x108yr.
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