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Extrasolar planetary dynamics with a generalized planar Laplace-Lagrange secular theory

The dynamical evolution of nearly half of the known extrasolar planets in multiple-planet systems may be dominated by secular perturbations. The commonly high eccentricities of the planetary orbits calls into question the utility of the traditional Laplace

ApJ,inpress

APreprinttypesetusingLTEXstyleemulateapjv.10/09/06

EXTRASOLARPLANETARYDYNAMICSWITHAGENERALIZEDPLANARLAPLACE-LAGRANGE

SECULARTHEORY

DimitriVeras1,2andPhilipJ.Armitage1,2

ApJ,inpress

arXiv:astro-ph/0703088v1 5 Mar 2007

ABSTRACT

Thedynamicalevolutionofnearlyhalfoftheknownextrasolarplanetsinmultiple-planetsystemsmaybedominatedbysecularperturbations.ThecommonlyhigheccentricitiesoftheplanetaryorbitscallsintoquestiontheutilityofthetraditionalLaplace-Lagrange(LL)seculartheoryinanalysesofthemotion.Weanalyticallygeneralizethistheorytofourth-orderintheeccentricities,comparetheresultwiththesecond-ordertheoryandoctupole-leveltheory,andapplythesetheoriestothelikelysecularly-dominatedHD12661,HD168443,HD38529andυAndmulti-planetsystems.Thefourth-orderschemeyieldsamultiply-branchedcriterionformaintainingapsidallibration,andimpliesthattheapsidalrateofasmallbodyisafunctionofitsinitialeccentricity,dependencieswhichareabsentfromthetraditionaltheory.Numericalresultsindicatethattheprimarydi erencethesecondandfourth-ordertheoriesrevealisanalterationinsecularperiodicities,http://www.wenkuxiazai.comparisonwithnumericalintegrationsindicatesthattheimprovementa ordedbythefourth-ordertheoryoverthesecond-ordertheorysometimesdwarfstheimprovementneededtoreproducetheactualdynamicalevolution.WeconcludethatLLseculartheory,toanyorder,generallyrepresentsapoorbarometerforpredictingseculardynamicsinextrasolarplanetarysystems,butdoesembodyausefultoolforextractinganaccuratelong-termdynamicaldescriptionofsystemswithsmallbodiesand/ornear-circularorbits.

Subjectheadings:planetsandsatellites:individual(HD12661,HD168443,HD190360,HD38529,HIP

14810)—planetsandsatellites:general—methods:analytical—celestialme-chanics

1.INTRODUCTION

Dynamicalsystemsofthreeormorebodiestypicallyincludephysicalphenomenaknownasmeanmotionres-onancesandsecularperturbations.Theformeroccurwhenpairsofbodieshaveorbitalperiodswhoseratiocanbeapproximatelyexpressedasaratiooftwosmallintegers.Otherwise,secularperturbationsdominatethesystem’slong-termevolution.Eachphenomenoncanbedescribedbyparticulartermsinagravitationalpoten-tialwhichhasbecomecommonlyknownasa“disturb-ingfunction”adynamicallyrichsegmentoftheHamiltonianofthesystem(Morbidelli2002).Laplace-Lagrange(LL)seculartheory,reviewedforexampleinMurray&Dermottrepresentsanelegantformulationinwhichthetimeevolutionofob-jects’orbitalparameterscanbedescribedanalyticallybyutilizingaselectfewtermsinthedisturbingfunctionuptosecond-orderineccentricitiesandinclinations.However,thetheoryhascomeunderrecentscrutinyforitsfailuretoreproducethephasespacestruc-tureofdynamicalsystemsofinterest,suchasextraso-larplanetarysystems.2006)showcasethelimitationsofthetheorybyusinganalternative,12th-orderexpansionineccentricity,andBeaug´e,Nesvorn´y,&Dones(2006)successfullyrepro-ducemotionsofirregularsatelliteswitheccentricitiesas07third-orderHoriaveragingmethod. ndsigni cantdiscrepancies

JILA,CampusBox440,UniversityofColorado,BoulderCO80309-0440;dimitri.veras@colorado.edu;pja@jilau1.colorado.edu2DepartmentofAstrophysicalandPlanetarySciences,Univer-sityofColorado,BoulderCO80309-0391

1

between rst-orderseculartheoryandN-bodysimula-´tions,anddemonstratethatspecial

caremustbegiventomodelcorrectlythesecularmotionofirregularsatellites.Christou&Murraygen-eralizeLaplace-Lagrangeseculartheorytosecond-orderinthemasses,andapplytheresulttothesecularly-dominatedUraniansatellitesystem.Severalextrasolarsystemsexhibit“hierarchical”behavior,and“octupole-level”seculartheoryhasbeendevelopedinordertode-(Krymolowski&Mazeh1999;2000;2003).Thesecularphaseiscomplex(F´ejozandtheiraccuratecharacterizationnecessitatesacomprehensivegeneralanalyticaltreatment.

Despitetheselimitations,classicalLLtheoryre-mainsuseful1977;Heppenheimer1980;2002;Wu&Murray2003;Zamaska&2004;2006;Namouni&Zhouinor-derto,qualitativelyatleast,describesecularmotion.TheformulationoftheresultsVeras&Armitageandallowsonetocircum-ventsingularitiespresentinLagrange’splanetaryequa-tions,whichexplicitlydescribethetimeevolutionofor-bitalelementsofasecondary.Hereweextendtheplanartheorytofourthorder,anddemonstratethephysicalcon-sequencesofthisextension.Indoingso,wehelpremoveamajorassumption(thatrequiringsmalleccentricities)ofthestandardtheory,andaddressthelimitsofthethe-ory’sapplicability,eveninitsmoregeneralstate.

Atleast20multi-planetextrasolarsystemsaround

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