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long-terntegransandstabilityofryorbitsinoursrsystem
abstract
wepresenttheresultsofverylong-teruricalintegransofryorbitalnsover109-yrti-spansincludingallninepls.aquickinspecnofournuricaldatashowsthattheryn，atleastinoursiledynacaldel，seetobequitestableevenoverthisverylongti-span.acloserlookatthelowest-frequencyosciningalow-passfiltershowthepotentiallydifivecharacterofterrestrialryn，especiallythatofrcury.thebehauroftheentriercuryinourintegransisqualitativelysrtotheresultsfroacqueskar'ssecrperturbantheory(e.g.ex∼0.35over∼±4gyr).however，therearenoapparentsecrincreasesofentricityorinclinaninanyorbitalelentsofthepls，whichyberevealedbystilllonger-teruricalintegrans.wehavealsoperfordacoupleoftrialintegransincludingnsoftheouterfiveplsovertheduranof±5x1010yr.theresultindicatesthatthethreejorresonancesintheneptune–plutosysteavebeenintainedoverthe1011-yrti-span.
1introducn
1.1defininoftheproblem
thequesnofthestabilityofoursrsysteasbeendebatedoverseveralhundredyears，sincetheeraofnewton.theprobleasattractednyftheticiansovertheyearsandhasyedacentralroleinthedevelopntofnon-lineardynacsandchaostheory.however，wedonotyethaveadefiniteanswertothequesnofwhetheroursrsystesstableornot.thisispartlyaresultofthefactthatthedefininoftheterstability’isvaguewhenitiedinrntotheproblefryninthesrsysteactuallyitisnoteasytogiveaclear，rigorandphysicallyaningfuldefininofthestabilityofoursrsyste
angnydefininsofstability，hereweadoptthehilldefinin(dn1993):actuallythisisnotadefininofstability，butofinstability.wedefineasystesbeingunstablewhenacloseencounteroewhereinthesystestartingfrocertaininitialconfiguran(chaers，wetherill&ito&&tanikawa1999).asystesdefinedasexperiencingacloseencounterwhentwobodiesapproachoneanotherwithinanareaofthrgerhillrad.otherwisethesystesdefinedasbeingstable.henceforwardwestatethatourrysystesdynacallystableifnocloseencounterhappensduringtheageofoursrsysteabout±5gyr.incidentally，thisdefininyberecedbyoneinwhichanurrenceofanyorbitalcrossingbetweeneitherofapairofplstakesce.thisisbeceweknowfroxperiencethatanorbitalcrossingisverylikelytoleadtoacloseencounterinryandprotorysyste(yoshinaga，kokubo&&kino1999).ofcoursethisstatentplyappliedtosystewithstableorbitalresonancessuchastheneptune–plutosyste
1.2previstudiesandfthisresearch
inaddintothevaguenessoftheconceptofstability，theplsinoursrsystehowacharactertypiicalchaos(sn&&wisdo988，1992).theceofthischaoticbehaurisnowpartlyunderstoodasbeingaresultofresonanceovepping(rray&lecar，franklin&&holn2001).however，itwouldrequireintegratingoveranenseleofrysysteincludingallnineplsforapedcoveringseveral10gyrtothoroughlyunderstandthelong-tervolunofryorbits，sincechaoticdynacalsystearecharacterizedbytheirstrongdependenceoninitialcondins.
frohatpointofview，nyoftheprevilong-teruricalintegransincludedonlytheouterfivepls(sn&kinoshita&&nakai1996).thisisbecetheorbitalpedsoftheouterplsaresochlongerthanthoseoftheinnerfourplsthatitischeasiertofollowthesysteoragivenintegranped.atpresent，thelongestnuricalintegranspublishedinjournalsarethoseofduncan&&lissauer(1998).althoughtheirintargetwastheeffeain-sequenasslossonthestabilityofryorbits，theyperfordnyintegranscoveringupto∼1011yroftheorbitalnsofthefourjovianpls.theinitialorbitalelentsandssesofplsarethesaasthoseofoursrsystenduncan&&lissauer'spaper，buttheydecreasethessofthesungraduallyintheirnuricalexperints.thisisbecetheyconsidertheeffeain-sequenasslossinthepaper.consequently，theyfoundthatthecrossingti-scaleofryorbits，whichcanbeatypicalindicatoroftheinstabilityti-scale，isquitesensitivetotherateofssdecreaseofthesun.whenthessofthesunisclosetoitspresentvalue，thejovianplsreinstableover1010yr，orperhapslonger.duncan&&lissaueralsoperfordfoursrexperintsontheorbitalnofsevenpls(vetoneptune)，whichcoveraspanof∼109yr.theirexperintsonthesevenplsarenotyetprehensive，butitseethattheterrestrialplsalsoreinstableduringtheintegranped，intainingalstregroscins.
ontheotherhand，inhisuratese-analyticalsecrperturbantheoryskar1988)skarfindsthargeandirregrvarianscanappearintheentricitiesandinclinansoftheterrestrialpls，espeercuryandrsonati-scaleofseveral109yrskar1996).theresultsoskar'ssecrperturbantheoryshouldbeedandinvestigatedbyfullynuricalintegrans.
inthispaperwepresentprelinaryresultsofsixlong-teruricalintegransonallnineryorbits，coveringaspanofseveral109yr，andoftwootherintegranscoveringaspanof±5x1010yr.thetotalpsedtiforallintegransisrethan5yringseveraldedicatedpcsandworkstans.oneofthefundantalconnsofourlong-terntegransisthatsrsystrynseetobestableinterofthehillstabilitynnedabove，atleastoverati-spanof±4gyr.actually，inournuricalintegransthesysteasfarrestablethanwhatisdefinedbythehillstabilitycriten:notonlydidnocloseencounterhappenduringtheintegranped，butalsoalltheryorbitalelentshavebeenconfinedinanarrowrenbothintiandfrequenain，thoughrynsarestochastic.sincethepurposeofthispaperistoexhibitandoverviewtheresultsofourlong-teruricalintegrans，weshowtypicalexalefiguresasevidenceoftheverylong-tertabilityofsrsystryn.forreaderswhohaverespecificanddeeperinterestsinournuricalresults，wehavepreparedawebpage(ess)，raworbitalelents，theirlow-passfilteredresults，varianofdunayelentsandangrntueficit，andresultsofoursileti–frequencyanalysisonallofourintegrans.
insecn2webrieflyexinourdynacaldel，nuricalthodandinitialcondinedinourintegrans.secn3isdevotedtoadescripnofthequickresultsofthenuricalintegrans.verylong-tertabilityofsrsystrynisapparentbothinryposinsandorbitalelents.aroughestinofnuricalerrorsisalsogiven.secn4goesontoadisnofthelongest-terarianofryorbitingalow-passfilterandincludesadisnofangrntueficit.insecn5，wepresentasetofnuricalintegransfortheouterfiveplsthatspans±5x1010yr.insecn6wealsodissthelong-tertabilityoftherynanditspossiblece.
2descripnofthenuricalintegrans
（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。）
2.3nuricalthod
weutilizeasecond-orderwisdoholnsylectiainintegranthod(wisdokinoshita，yoshida&&nakai1991)withaspecialstart-upproceduretoreducethetruncanerrorofanglevariables，‘wartart’(saha&&treine1992，1994).
thestepsizeforthenuricalintegransis8dthroughoutallintegransoftheninepls(n±1，2，3)，whichisabout111oftheorbitalpedoftheinnerstpl(rcury).asforthedeternanofstepsize，wepartlyfollowtheprevinuricalintegranofallnineplsinsn&&wisdo1988，7.2d)andsaha&&treine(1994，22532d).weroundedthedecilpartofthetheirstepsizesto8tokethestepsizealtipleof2inordertoreducethenofround-offerrorintheputanprocesses.inrntothis，wisdo&holn(1991)perfordnuricalintegransoftheouterfiveryorbitingthesylecticpwithastepsizeof400d，110.83oftheorbitalpedofjupiter.theirresultseetobeurateenough，whichpartlytifiesourthodofdeterningthestepsize.however，sincetheentricityofjupiter(∼0.05)ischsllerthanthatofrcury(∼0.2)，weneedsocarewhenweparetheseintegranssilyinterofstepsizes.
intheintegranoftheouterfivepls(f±)，wefixedthestepsizeat400d.
weadoptgs'fandgfuncnsinthesylecticptogetherwiththethird-orderhalleythod(danby1992)asasolverforkeplerequans.thenuerofxiteranswesetinhalley'sthodis15，buttheyneverreachedthexinanyofourintegrans.
theintervalofthedataoutputis200000d(∼547yr)forthecalcnsofallninepls(n±1，2，3)，andabout8000000d(∼21903yr)fortheintegranoftheouterfivepls(f±).
althoughnooutputfilteringwasdonewhenthenuricalintegranswereinprocess，weappliedalow-passfiltertotheraworbitaldataafterwehadpletedallthecalcns.seesecn4.1forredetail.
2.4errorestin
2.4.1rtiveerrorsintotalenergyandangrntum
ordingtooneofthebasicpropertiesofsylecticintegrators，whichconservethephysicallyconservativequantitieswell(totalorbitalenergyandangrntu，ourlong-teruricalintegransseeohavebeenperfordwithverysllerrors.theaveragedrtiveerrorsoftotalenergy(∼10−9)andoftotngrntu∼10−11)havereinednearlyconstantthroughouttheintegranped(fig.1).thespecialstartupprocedure，wartart，wouldhavereducedtheaveragedrtiveerrorintotalenergybyaboutoneorderofgnitudeorre.
rtivenuricalerrorofthetotngrntuaa0andthetotalenergyδee0inournuricalintegransn±1，2，3，whereδeandδaaretheabsolutechangeofthetotalenergyandtotngrnturespectively，ande0anda0aretheirinitialvalues.thehorizontalunitisgyr.
notethatdifferentoperatingsyste，differenttheticallibraries，anddifferenthardwarearchitecturesresultindifferentnuricalerrors，throughthevariansinround-offerrorhandlingandnuris.intheupperpaneloffig.1，wecanrecognizethissituaninthesecrnuricalerrorinthetotngrntuwhichshouldberigorlypreserveduptochine-eprecin.
2.4.2errorinrylongitudes
sincethesylecticpspreservetotalenergyandtotngrntufn-bodydynacalsysteinherentlywell，thedegreeoftheirpreservanynotbeagoodasureoftheacericalintegrans，especiallyasaasureoftheposinalerrorofpls，i.e.theerrorinrylongitudes.toestitethenuricalerrorintherylongitudes，weperfordthefollowingprocedures.weparedtheresultofourinlong-terntegranswithsotestintegrans，whichspanchshorterpedsbutwithchhigheruracythantheinintegrans.forthispurpose，weperfordachreurateintegranwithastepsizeof0.125d(164oftheinintegrans)spanning3x105yr，startingwiththesainitialcondinsasinthen−1integran.weconsiderthatthistestintegranprovidewitha‘pseudo-true’solunofryorbitalevolun.next，weparethetestintegranwiththeinintegran，n−1.forthepedof3x105yr，weseeadifferenceinananoliesoftheearthbetweenthetwointegransof∼0.52°(inthecaseofthen−1integran).thisdifferencecanbeextraptedtothevalue∼8700°，about25rotansofearthafter5gyr，sincetheerroroflongitudesincreaseslinearlywithtiinthesylecticp.srly，thelongitudeerrorofplutocanbeestitedas∼12°.thisvalueforplutoischbetterthantheresultinkinoshita&&nakai(1996)wherethedifferenceisestitedas∼60°.
3nuricalresults–i.nceattherawdata
inthissecnwebrieflyreviewthelong-tertabilityofryorbitalnthroughsosnapshotsofrawnuricaldata.theorbitalnofplsindicateslong-tertabilityinallofournuricalintegrans:noorbitalcrossingsnorcloseencountersbetweenanypairofplstookce.
3.1generaldescripnofthestabilityofryorbits
first，webrieflylookatthegeneralcharacterofthelong-tertabilityofryorbits.ourinterestherefoesparticrlyontheinnerfourterrestrialplsforwhichtheorbitalti-scalesarechshorterthanthoseoftheouterfivepls.aswecanseeclearlyfrohenarorbitalconfiguransshowninfigs2and3，orbitalposinsoftheterrestrialplsdifferlittlebetweentheinitindfinalpartofeachnuricalintegran，whichspansseveralgyr.thesolidlinesdenotingthepresentorbitsoftheplsliealstwithintheswarfdotseveninthefinalpartofintegrans(b)and(d).thisindicatesthatthroughouttheentireintegranpedthealstregrvariansofryorbitalnreinnearlythesaastheyareatpresent.
verticalviewofthefourinnerryorbits(frohez-axisdirecn)attheinitindfinalpartsoftheintegransn±1.theaxesunitsareau.thexy-neissettotheinvariantneofsrsysteotngrntu(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).ineachpanel，atotalof23684tervalofabout2190yrover5.47x107yr.solidlinesineachpaneldenotethepresentorbitsofthefourterrestrialpls(takenfroe245).
thevarianofentricitiesandorbitalinclinansfortheinnerfourplsintheinitindfinalpartoftheintegrann+1isshowninfig.4.asexpected，thecharacterofthevarianofryorbitalelentsdoesnotdiffersignificantlybetweentheinitindfinalpartofeachintegran，atleastforve，earthandrs.theelentsofrcury，especiallyitsentricity，seeochangetoasignificantextent.thisispartlybecetheorbitalti-scaleoftheplistheshortestofallthepls，whichleadstoarerapidorbitalevolunthanotherpls;theinnerstplybenearesttoinstability.thisresultappearstobeinsoagreentwitskar's(1994，1996)expectansthargeandirregrvariansappearintheentricitiesandinclinansofre-scaleofseveral109yr.however，theeffectofthepossibleinstabilityoftheorbitofrcuryynotfatallyaffecttheglobalstabilityofthewholerysystewingtothesllssofrcury.wewillnnbrieflythelong-terrbitalevolunofrcurterinsecninglow-passfilteredorbitalelents.
theorbitalnoftheouterfiveplsseerigorlystableandquiteregroverthisti-span(seealsosecn5).
3.2ti–frequencyps
althoughtherynexhibitsverylong-tertabilitydefinedasthenon-existenceofcloseencounterevents，thechaotatureofrydynacscanchangetheoscitorypedandalitudeofryorbitalngraduallyoversue-spans.evensuchslightfluctuansoforbitalvarianinthefrequenain，particrlyinthecaseofearth，canpotentiallyhaveasignificanteffectonitssurfaceclitesystehroughsrinsnvarian(cf.berger1988).
togiveanoverviewofthelong-terhangeinpedicityinryorbitaln，weperfordnyfastfouriertransforns(ffts)alongthetiaxis，andsuperposedtheresultingpedgratodrawtwo-dinnalti–frequencyps.thespecificapproachtodrawingtheseti–frequencypsinthispaperisverysile–chsilerthanthewaveletanalysisoskar's(1990，1993)frequencyanalysis.
dividethelow-passfilteredorbitaldataintonyfragntsofthesalenh.thelenhofeachdatasegntshouldbealtipleof2inordertoapplythefft.
eachfragntofthedatahasrgeoveppingpart:forexale，whentheithdatabeginsfro=tiandendsatt=ti+t，thenextdatasegntrangesfroi+δt≤ti+δt+t，whereδt?t.wecontinuethisdivinuntilwereachacertainnuernbywhichtn+treachesthetotalintegranlenh.
weapplyanffttoeachofthedatafragnts，andobtainnfrequencydiagra.
ineachfrequencydiagrabtainedabove，thestrenhofpedicitycanberecedbyagrey-scale(orcolour)chart.
weperforherecent，andconnectallthegrey-scale(orcolour)chartsintoonegraphforeachintegran.thehorizontxisofthesenewgraphsshouldbetheti，i.e.thestartingtisofeachfragntofdata(ti，wherei=1，…，n).theverticxisrepresentstheped(orfrequency)oftheoscinoforbitalelents.
wehaveadoptedanfftbeceofitsoverwhelngspeed，sincetheauntofnuricaldatatobedeposedintofrequenponentsisterriblyhuge(severaltensofgbytes).
atypicalexaleoftheti–frequencypcreatedbytheaboveproceduresisshowninagrey-scalediagrasfig.5，whichshowsthevarianofpedicityintheentricityandinclinanofearthinn+2integran.infig.5，thedarkareashowsthatatthetiindicatedbythevalueontheabscissa，thepedicityindicatedbytheordinateisstrongerthaninthelighterareaaroundit.wecanrecognizefrohispthatthepedicityoftheentricityandinclinanofearthonlychangesslightlyovertheentirepedcoveredbythen+2integran.thisnearlyregrtrendisqualitativelythesainotherintegransandforotherpls，althoughtypicalfrequenciesdifferplbyndelentbyelent.
4.2long-terxchangeoforbitalenergyandangrntum
wecalcteverylong-pedicvarianandexchangeofryorbitalenergyandangrningfiltereddunayelentsl，g，h.gandhareequivalenttotheryorbitngrntunditsvertiponentperunitss.lisrtedtotheryorbitalenergyeperunitssase=−μ22l2.ifthesystespletelylinear，theorbitalenergyandtheangrntuneachfrequencybinstbeconstant.non-linearityintherysysteanceanexchangeofenergyandangrntunthefrequenain.thealitudeofthelowest-frequencyoscinshouldincreaseifthesystesunstableandbreaksdowngradually.however，suchasytofinstabilityisnotpronentinourlong-terntegrans.
infig.7，thetotalorbitalenergyandangrntufthefourinnerplsandallnineplsareshownforintegrann+2.theupperthreepanelsshowthelong-pedicvarianoftotalenergy(denotedase-e0)，totngrntug-g0)，andthevertiponent(h-h0)oftheinnerfourplscalctedfrohelow-passfiltereddunayelents.e0，g0，h0denotetheinitialvaluesofeachquantity.theabsolutedifferentheinitialvaluesisplottedinthepanels.thelowerthreepanelsineachfigureshowe-e0，g-g0andh-h0ofthetotalofninepls.thefluctuanshowninthelowerpanelsisvirtuallyentirelyaresultofthessivejovianpls.
paringthevariansofenergyandangrntuftheinnerfourplsandallninepls，itisapparentthatthealitudesofthoseoftheinnerplsarechsllerthanthoseofallninepls:thealitudesoftheouterfiveplsarecrgerthanthoseoftheinnerpls.thisdoesnotanthattheinnerterrestrialrysubsystesrestablethantheouterone:thisissilyaresultofthertivesllnessofthessesofthefourterrestrialplsparedwiththoseoftheouterjovianpls.anotherthingwenoticeisthattheinnerrysubsysteaybeeunstablererapidlythantheouteronebeceofitsshorterorbitalti-scales.thiscanbeseeninthepanelsdenotedasinner4infig.7wherethelonger-pedicandirregroscinsarereapparentthaninthepanelsdenotedastotal9.actually，thefluctuansintheinner4panelsaretorgeextentasaresultoftheorbitalvarianofthercury.however，wecannotneglectthecontribunfrotherterrestrialpls，aswewillseeinsubsequentsecns.
4.4long-terouplingofseveralneighbouringplpairs
leseesoindividualvariansofryorbitalenergyandangrntuxpressedbythelow-passfiltereddunayelents.figs10and11showlong-tervolunoftheorbitalenergyofeachndtheangrntunn+1andn−2integrans.wenoticethatsoplsforpparentpairsinteroforbitalenergyandangrntuxchange.inparticr，veandearthkeatypicalpair.inthefigures，theyshownegativecorrnsinexchangeofenergyandpositivecorrnsinexchangeofangrntuthenegativecorrninexchangeoforbitalenergyansthatthetwoplsforcloseddynacalsystenteroftheorbitalenergy.thepositivecorrninexchangeofangrntueansthatthetwoplsaresiltanelyundercertainlong-tererturbans.candidatesforperturbersarejupiterandsaturn.alsoinfig.11，wecanseethatrsshows'itivecorrnintheangrntuariantotheve–earthsystercuryexhibitscertainnegativecorrnsintheangrntuertheve–earthsystewhichseetobeareacncedbytheconservanofangrntuntheterrestrialrysubsyste
itisnotclearatthentwhytheve–earthpairexhibitsanegativecorrninenergyexchangeand'itivecorrninangrntuxchange.weypossiblyexinthisthroughobservingthegeneralfactthattherearenosecrterinrysejoraxesuptosecond-orderperturbantheories(cf.brouwer&baletti&&puco1998).thisansthattheryorbitalenergy(whichisdirectlyrtedtothesejoraxisa)ghtbechlessaffectedbyperturbingplsthanistheangrntuxchange(whichrtestoe).hence，theentricitiesofveandearthcanbedisturbedeasilybyjupiterandsaturn，whichresultsin'itivecorrnintheangrntuxchange.ontheotherhand，thesejoraxesofveandeartharelesslikelytobedisturbedbythejovianpls.ttheenergyexchangeybelitedonlywithintheve–earthpair，whichresultsinanegativecorrnintheexchangeoforbitalenergyinthepair.
asfortheouterjovianrysubsystejupiter–saturnandura–neptuneseeokedynacalpairs.however，thestrenhoftheircouplingisnotasstrongparedwiththatoftheve–earthpair.
5±5x1010-yrintegransofouterryorbits
sincethejovianryssesarecrgerthantheterrestrialrysses，wetreatthejovianrysystesanindependentrysystenterofthestudyofitsdynacalstability.hence，weaddedacoupleoftrialintegransthatspan±5x1010yr，includingonlytheouterfivepls(thefourjovianplsppluto).theresultsexhibittherigorstabilityoftheouterrysysteverthislongti-span.orbitalconfigurans(fig.12)，andvarianofentricitiesandinclinans(fig.13)showthisverylong-tertabilityoftheouterfiveplsinboththetiandthefrequenains.althoughwedonotshowpshere，thetypicalfrequencyoftheorbitaloscinofplutoandtheotherouterplsisalstconstantduringtheseverylong-terntegranpeds，whichisdenstratedintheti–frequencypsonourwebpage.
inthesetwointegrans，thertivenuricalerrorinthetotalenergywas∼10−6andthatofthetotngrntuas∼10−10.
5.1resonancesintheneptune–plutosystem
kinoshita&&nakai(1996)integratedtheouterfiveryorbitsover±5.5x109yr.theyfoundthatfourjorresonancesbetweenneptuneandplutoareintainedduringthewholeintegranped，andthattheresonancesybetheincesofthestabilityoftheorbitofpluto.thejorfourresonancesfoundinpreviresearchareasfollows.inthefollowingdescripn，λdenotestheanlongitude，Ωisthelongitudeoftheascendingnodeandϖisthelongitudeofperihen.subscriptspandndenoteplutoandneptune.
annresonancebetweenneptuneandpluto(3:2).thecriticrguntθ1=3λp−2λn−ϖplibratesaround180°withanalitudeofabout80°andalibranpedofabout2x104yr.
thearguntofperihenofplutowp=θ2=ϖp−Ωplibratesaround90°withapedofabout3.8x106yr.thedonantpedicvariansoftheentricityandinclinanofplutoaresynchronizedwiththelibranofitsarguntofperihen.thisisanticipatedinthesecrperturbantheoryconstructedbykozai(1962).
thelongitudeofthenodeofplutoreferredtothelongitudeofthenodeofneptune，θ3=Ωp−Ωn，circtesandthepedofthiscircnisequaltothepedofθ2libran.whenθ3beeszero，i.e.thelongitudesofascendingnodesofneptuneandplutoovep，theinclinanofplutobeaxitheentricitybeinindthearguntofperihenbees90°.whenθ3bees180°，theinclinanofplutobeinitheentricitybeaxindthearguntofperihenbees90°again.willia&&benson(1971)anticipatedthistypeofresonanceteredbni，nobili&&carpino(1989).
anarguntθ4=ϖp−ϖn+3(Ωp−Ωn)libratesaround180°withalongped，∼5.7x108yr.
inournuricalintegrans，theresonances(i)–(iii)arewellintained，andvarianofthecriticrguntsθ1，θ2，θ3reinsrduringthewholeintegranped(figs14–16).however，thefourthresonance(iv)appearstobedifferent:thecriticrguntθ4alternateslibranandcircnovera1010-yrti-scale(fig.17).thisisaninterestingfactthatkinoshita&&nakai's(1995，1996)shorterintegranswerenotabletodisclose.
6disn
whatkindofdynacalchanisaintainsthislong-tertabilityoftherysystewecaniediatelythinkoftwojorfeaturesthatyberesponsibleforthelong-tertability.first，thereseeobenosignificantlower-orderresonances(annandsecr)betweenanypairangtheninepls.jupiterandsaturnareeannresonance(thef‘greatinequality’)，butnottintheresonancezone.higher-orderresonancesycethechaotatureoftherydynacaln，buttheyarenotsostrongastodestroythestablerynwithinthelifeftherealsrsystethesecondfeature，whichwethinkisrrtantforthelong-tertabilityofourrysysteisthedifferenceindynacaldistancebetweenterrestrindjovianrysubsyste(ito&&tanikawa1999，2001).whenweasureryseparansbythetualhillradii(r_)，separansangterrestrialplsaregreaterthan26rh，whereasthoseangjovianplsarelessthan14rh.thisdifferenceisdirectlyrtedtothedifferencebetweendynacalfeaturesofterrestrindjovianpls.terrestrialplshavesllersses，shorterorbitalpedsandwiderdynacalseparan.theyarestronglyperturbedbyjovianplsthathavrgersses，longerorbitalpedsandnarrowerdynacalseparan.jovianplsarenotperturbedbyanyotherssivebodies.
thepresentterrestrialrysystesstillbeingdisturbedbythessivejovianpls.however，thewideseparanandtualinteraongtheterrestrialplsrendersthedisturbanceineffective;thedegreeofdisturbancebyjovianplsiso(ej)(orderofgnitudeoftheentricityofjupiter)，sincethedisturbancecedbyjovianplsisaforcedoscinhavinganalitudeofo(ej).heighteningofentripleo(ej)∼0.05，isfarfroufficienttoprovokeinstabilityintheterrestrialplshavingsuchawideseparanas26rh.tweassuthatthepresentwidedynacalseparanangterrestrialpls(&;26rh)isprobablyoneofthestsignificantcondinsforintainingthestabilityoftherysystevera109-yrti-span.ourdetailedanalysisofthernshipbetweendynacaldistancebetweenplsandtheinstabilityti-scaleofsrsystrynisnowon-going.
althoughournuricalintegransspanthelifefthesrsystethenuerofintegransisfarfroufficienttofilltheinitialphasespace.itisnecessarytoperfororeandrenuricalintegranstoandexaneindetailthelong-tertabilityofourrydynacs.
——以上文段引自ito，t.&tanikawa，k.long-terntegransandstabilityofryorbitsinoursrsysten.not.r.astron.soc.336，483–500(2002)
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还有其他论文，不过也都是英文的，相关课题的中文文献很少，那些论文下载一篇要九美元（《nature》真是暴利），作者君写这篇文章的时候已经回家，不在检测中心，所以没有数据库的使用权，下不起，就不贴上来了。
