Structure in narrow planetary rings 347

ABSTRACT

In this paper we review some open questions in the context of the structure observed in narrow planetary rings, and summarize some recent results of our work directed to answer them. Using the scattering approach to narrow rings we have succeeded to reproduce some of their structural properties in a qualitative sense, using unrealistic toy models as examples. We obtain narrow rings which are non-circular and display sharp edges. In addition, these rings may have multiple components which may entangle in a complicated dynamically evolving way forming a braided structure, or may display strongly azimuthal dependent features such as arcs. The appearance of these structural properties can be understood in terms of the underlying phase space.

Key words: narrow planetary rings, strands, arcs, scattering approach.

RESUMEN

En este artículo hacemos una revisión de algunas de las preguntas que permanecen abiertas en el contexto de la estructura observada en los anillos planetarios delgados, y resumimos algunos resultados recientes de nuestro trabajo encaminados a responderlas. Usando el enfoque de la dispersión en anillos delgados hemos reproducido algunas de sus propiedades estructurales de forma cualitativa, usando modelos de juguete irrealistas como ejemplos. Hemos obtenido anillos delgados no circulares que muestran bordes bien definidos. Además, estos anillos pueden tener componentes múltiples que se enredan de manera complicada y evolucionan dinámicamente formando estructuras trenzadas, o pueden exhibir estructuras con una fuerte dependencia azimutal como arcos. La aparición de estas estructuras se puede entender en términos del espacio fase subyacente.

Palabras clave: anillos planetarios delgados, componentes múltiples (hebras), arcos, enfoque de la dispersión.

Structureinnarrowplanetaryrings:Openquestionsandrecentresults

Luis Benet* and Olivier Merlo

Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México,Apdo. Postal 48-3, 62251 Cuernavaca, Morelos, Mexico.

* benet@fis.unam.mx

Benet,L.,Merlo,O.,2010,Structureinnarrowplanetaryrings:Openquestionsandrecentresults:RevistaMexicanadeCienciasGeológicas,v.27,núm.2,p.347-357.

INTRODUCTION: OBSERVATIONS

Saturnringsare,sincetheirdiscoverybyGalileoin1610,oneofthemostpuzzlingandbeautifulfeaturesoftheSolarSystem.Foralongtime,Saturnhadthespecialpositionbeingthe“ringedplanet”.In1977,thesomewhataccidentaldiscoveryofUranusringsbystellar-occultation

measurements(Elliotet al.,1977)changedthisview,andledtoarenewedinterestinringsystems.ThemainreasonforthiswasthattheUranianringsturnedtobeextremelydifferentfromthoseofSaturn:Theyarenarrow,opaque,sharp-edged,inclinedandeccentric(ElliotandNicholson,1984; Esposito, 2002). To quote some figures (Murray and Dermott,1999),thewidestringofUranus,theεring,is

Revista Mexicana de Ciencias Geológicas, v. 27, núm. 2, 2010, p. 347-357

Benet and Merlo348

Figure2.StructureinSaturn’sFring.ThesefourimagestakenbyCassiniofSaturn’sFringshowthe“knotted”structureindifferentlocations.(PIA07522.CourtesyNASA/JPL/SpaceScienceInstitute-Caltech).

andaroundtheoutermostεringofUranus;thesediscoveriesrepresented a confirmation of the theory. The shepherding confinement involves angular momentum transfer between theshepherdmoonsandtheringparticles,self-gravityandviscousdampingduetointer-particlecollisions(Borderieset al.,1983).Whilethefullscenarioforshepherdinghasnot

20−96kmwidewithanominalsemi−majoraxisat51,149km;incomparison,themainringsofSaturnareafewthousand kilometers wide. The Uranian rings were the first narrowplanetaryringsdiscovered,butarenottheonlyonesthatexist.ThePioneermissionandtheVoyagersuncoveredothernarrowringinSaturn,theFring,showinganamazingandpuzzlingstructure;occultationmeasurementspointedtheexistenceofringsaroundNeptune;Jupiter’sbroadringswerealsodiscovered(seeEsposito,2002foradetailedhistoricalaccount).Figures1-4arephotographsillustrat-ingsomeexamplesofthevarietyofstructurethatisfound.

Thesediscoveriesraisedanumberofnewques-tions,mostofwhichremainunanswered(Esposito,2002;Sicardy,2005).Forinstance,theeccentricityoftheεringis0.0079;Saturn'sFringhasaneccentricity0.0026.Aneccentricinclinednarrowringliketheεringisexpectedtocircularizeandspreadinrathershorttimescales,tmax~108years,whichis“considerablysmallerthantheageofthesolarsystem”(Esposito,2002).Thisestimatefollowsfrominter−particlecollisions,draganddifferentialprecession.Therefore,anefficientconfinementmechanismmustmaintainthesestructuralpropertiesoftheringoverlongertimescalesallowing,amongother,aneccentric ring(Esposito,2002).

ToexplainthestructuralfeaturesoftheUranianrings,new models were introduced where the confinement was inducedbynearbymoons.AmongthesemodelswementioninparticulartheshepherdingmodelintroducedbyGoldreichandTremaine(1979),wheretwomoonsaroundtheringwereproposedtoboundit.ThePioneerandVoyagermis-sionsdetectedtheshepherdmoonsaroundSaturn’sFring

Figure1.Uranus’ringsandthe“shepherd”satellitesoftheεringdiscov-eredbyVoyager2.TheimagewastakeninJanuary21of1986(PIA01976.CourtesyNASA/JPL/SpaceScienceInstitute-Caltech).

Structure in narrow planetary rings 349

eithertheshepherdsaretherebutaretoosmalltobede-tected,or“somephysicsisyettobeunderstood”(Sicardy,2005).Saturn’sFringturnedouttohaveaveryrichdy-namicalstructure(Smithet al.,1981;Smithet al., 1982;Murrayet al., 1997):besidesthenon-zeroeccentricity,itdisplaysmultiplecomponentsentangledinacomplicatedway,knownasstrandsandbraids,showingfurtherpuzzlingfeatureslikekinksandclumps.

Numericalsimulationshaveinvestigatedavarietyofphysicalinteractions,likethegravitationalperturbationsofshepherdmoonsoncircularandeccentricorbits(Giuliatti-Winteret al.,2000;ShowalterandBurns,1982),effectsduetotheactionofembeddedmoonlets(LissauerandPeale,1986),andringinter-particlecollisioneffects(Hänninen,1993;LewisandStewart,2000).Thecentralquestionsinvestigatedhavebeentheformationofstructure(strands,braids,clumps)andtheirshort-termstability.Whilethesestudieshaveledtointerestingpredictions,e.g.theformationofchannelsandstreamers(Giuliatti-Winteret al., 2000)whichhavebeenrecentlyobservedbyCassini(Murray et al.,2005),thereisnoself-consistentapproachfortheconfinement of narrow rings and their radial and azimuthal structure.TheFSaturnringremainsasthemostfascinatingandpuzzlingcase.

Thepresentpaperreviewssomeofourrecentworkonthispoint,namely,aself-consistentscenariofortheoccur-renceofnarrowringsandtheappearanceofstructure;wehave called it the scattering approach. The first section is devotedtodescribethebasicideasbehindourapproach.Thenextsectionweexemplifytheemergenceofstructuredringswithinscatteringapproachusinganunrealistictoymodel.Weobtainnon-circularnarrowringswithsharpedges,thatmaydisplaymultiplecomponentsandarcs.Ourresultsarequalitativesofar.Yet,theapproachisrobustandconsistent,andcanbeapplied–withtheintrinsiccomplicationsthatthisentails–tomorerealisticsituations.Afollowingsec-tionwedescribetherelevantphase–spacestructuresuponwhichourdynamicalapproachisbased.Thelastsectionisdevotedtoourconclusionsandoutlook.

THE SCATTERING APPROACH

Letusconsiderthe(N+1)-bodyfullHamiltonianwhichdescribesthemotionofacentralplanetofmassM0,surroundedbyNmmoonsandNrringparticles(N = Nm+Nr).Inaninertialframewehave

,(1)

= H Km+ V m-m+ H Kr+ V m-r+ V r-r . (2)

InEquation(1), P iisthelinearmomentumofthei-thparticle,withi = 0 representingthecentralplanet, Riisitspositionvector,Miisitsmass,andG isthegravita-tionalconstant.Hence,Ecuation(1)isthefullmany-body

= i=0

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beenfullyunderstood(Esposito,2002;Sicardy,2005),thepresenceofdissipationseemstobeessential;thisisactu-allyneededtoavoidcertainsingularities(Ogilvie,2007).Inaddition,theformulationassumesthattheringboundariesarelocatedatalower-orderresonance.

Yet, Saturn’s F ring does not fulfill the requirements toapplythistheory.Moreover,mostUranianringshavenoassociatedshepherdmoonsaroundthem(MurrayandThomason,1990),norsomenarroweccentricringsofSaturn,whichamongotherswouldprovideanexplanationfortheirsharpedges(MurrayandDermott,1999).Thus,

Figure3.RingletsintheEnckegap.(PIA08305.CourtesyNASA/JPL/SpaceScienceInstitute-Caltech).

Figure4.NeptuneringsandAdam’sarcsasseenbyVoyager2 inAugustof1989. (PIA01493.CourtesyNASA/JPL/SpaceScienceInstitute-Caltech).

Benet and Merlo350

problemwithgravitationalinteractionsandEquation(2)isaconvenientrearrangement:HKmistheHamiltonianforthekepleriantwo-bodyinteractionamongthemoonsandtheplanet,andH Kristhecorrespondingoneamongtheringparticlesandthecentralplanet.ThetermV m-mrepresentsthemoon-moongravitationalinteraction,V m-risthemoon-ringparticle interaction and, finally, V r-ristheringparticle–ringparticleinteraction.InEquation(1)wehaveconsideredpurelygravitationalinteractions;asweshallsee,ourap-proachisquitegeneralforconservativeinteractions.Theassumptionofpurelygravitationalinteractionsiscommonbecausetheinterestingstructuralpropertiesobservedarenotrelatedwithsub-micronsize(dust)particleswhereradiationforcesandelectromagneticinteractionsareindeedimpor-tant.Forexample,inSaturn’sFring,initialphotometricworkonVoyager’sdataindicatedthatthisringconsistsofacoreofcentimerter-sizeparticlessurroundedbymicronandsub-micronmaterial(Showalteret al.,1992);lateranalysisonthedatagatheredduringSaturn’sringplanecrossingin1995suggestedthattheringmaterialisdominatedbyapopulationlargerthan~10mmwithalowercutoffof0.3–0.5mm(Boshet al.,2002).

Intheplanetarycase,thereisaclearorderingofthemasses,M0 ≥ Mm ≥Mr,withMm/M0~10-8–10-4andMr/Mmevensmaller.Here,MmisatypicalmassforthemoonsandMr characterizesthemassoftheringparticles.Therefore,in a first order approximation, we may neglect the contribu-tionsfromV r-r,whichareofsecondorderinMr.Physically,thisamountstoignoreanyeffectsduetoringinter-particlecollisions.Moreover,duetothemassscalesoftheproblem,theeffectofindividualringparticlesinthemotionoftheplanetorthemoonscanbeneglected.Thissuggeststotreatthemotionoftheindividualringparticlesasarestrictedn-bodyproblem.Themotionofthecentralplanetandthemoonsissolvedconsistentlyinafullmany-bodycalcula-tion.Forthemotionoftheparticlesoftheringasolutionofthismany-bodyproblemisused,whichintroducesanexplicit timedependence.Therefore, thering-particleHamiltoniancanbewrittenas

(3)

whereV0referstothedominatinginteractionwiththecentralplanet(H Kr ),andVeffistheeffectiveinteractionduetotheplanetarymoons.TheexplicittimedependenceinEquation(3) is related to the specific solution of the full planet–moons problemused.Thisisusuallyakindofoscillatorymotionofthemoonsaroundthecentralplanet.Therefore,suchasolutionintroducesanintrinsicrotationinEquation(3),whichinthebestcaseisperiodicorquasi-periodic.Therestrictedthree-bodyproblemisanexampleofEquation(3),wheretheintrinsicrotationispreciselythecircularorellipticmotionofthetwo-bodyKeplerproblembetweentheplanetandthemoon.

WeconsidernowthedynamicsoftheHamiltonianEquation(3).Weshallbeinterestedinthosephase–space

H = 21 P 2 + V X t V X t0 eff( , ) ( , )+H = 21 P 2 + V X t V X t0 eff( , ) ( , )+

regionsthataredominatedbyscatteringtrajectories,i.e.,by trajectories that escape to infinity. Despite of the domi-nanceofunboundorbits,trappingisdynamicallypossible.Notice that scattering trajectories define a precise physical mechanismfortheparticlestoescapefromtheneighbor-hoodoftheplanet–moonssystem,whichmaythencreatestructureintheensembleofnon-escapingringparticles;hencethenameofscattering approach (BenetandMerlo,2004;MerloandBenet,2007).Forsimplicitywedescribethecaseoftwodegreesoffreedom,emphasizingthatthefollowingconsiderationscanbegeneralizedtomoredegreesoffreedom.

FortwodegreesoffreedomscatteringHamiltoniansystems,thedynamicscanbeunderstoodthroughtheperi-odicorbitsofthesystem,whicharetheorganizingcentersofthedynamics,andtheirassociatedinvariantstructuresinphase–space.Itisbeyondthescopeofthispresentationto summarize the theory of chaotic scattering. Suffice it to say that, first, phase–space is the natural object to analyze thedynamicsand,second,despiteofthedominantroleofunboundedtrajectories,strictlyboundedtrajectoriesmayformasetofpositivemeasureundercertainstabilityconditions.Thelatterholdsbecauseperiodicorbitsap-peargenericallythroughsaddle–centerbifurcations.Thatis,asaparameterofthesystemisvaried(e.g.,theJacobiintegralinthecircularrestrictedthree-bodyproblemisreduced),twonewperiodicorbitsappear,oneisstableandtheotherunstable.Theinvariantmanifoldsoftheunstableperiodicorbitboundaregioninphase–spacearoundthestableperiodicorbit,sotrajectoriescloseenoughtothestableperiodicorbitremainclosetoitforallfuturetimes.Thatis,testparticleswithinitialconditionsinsidethesephase–space regions will not escape to infinity along scat-teringtrajectories.FurtherreductionoftheJacobiintegralsetsinaperioddoublingbifurcationcascade,wheretheellipticpointbecomesinversehyperbolic;eventually,thehorseshoestructureislocallyhyperbolic,whichimpliesthatthesetoftrappedorbits(periodicandaperiodic)isofmeasurezero.TheseresultsaregenericforautonomoustwodegreesoffreedomscatteringHamiltonians(BenetandSeligman,2000;Benet,2001);forsystemswithmoredegreesoffreedomtherearetheoremswhichestablishtheconditionstohavesomeeffective stability,i.e.,trappedmotion is proven for very long but finite times (Jorba and Villanueva,1997a,1997b).

IntheintervalsoftheJacobiintegralwherethereisastableperiodicorbit,themanifoldsoftheunstableorbitdefine barriers which confine dynamically themotionoftheringparticles.SuchtrappingactuallytakesplacearoundthecentralplanetduetotheintrinsicrotationimplicitinEquation(3).Thetrappedorbitsthusremainclosetothestableperiodicorbit,notescapingirrespectivelyofwhethertheactualmotionisperiodic,quasi-periodicorevenchaotic.Asmentionedabove,thisholdsgenericallyfortime-inde-pendenttwodegreesoffreedomscatteringsystems;formoredegreesoffreedomeffectivedynamicaltrappinghasbeen

Structure in narrow planetary rings 351

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Figure 5. a) Example of a ring of non-interacting particles for the scattering billiard system on a circular orbit, in an inertial reference frame. The figure isobtainedbyprojectingthephasespacelocationofanensembleofparticles(insidethetrappingregion)ontotheX-Yspace,atagiventime.b)Detailofaregionofthering.Theblacklinesaretheanalyticalestimatesbasedonthestabilitypropertiesoftheorganizingperiodicorbit.

observed(MerloandBenet,2007;BenetandMerlo,2008).Genericallyhereimpliesthattheseresultsonlydependonthe local properties of phase–space and not on the specific interaction.Thismakesthewholescatteringapproachrobustinastrictmathematicalsense,whichphysicallyallowstoincludeothersmallconservativeperturbationsnotincludedinitiallyinEquation(1),asforinstancetheoblatenessoftheplanet.

WefocusonthewholeintervalofJacobiintegralwheresuchdynamicallytrappedmotiontakesplace.Weconsideranensembleofindependentringparticles,i.e.,non-interactingtestparticles,withessentiallyarbitraryinitialconditionsatt = 0 that belong to the specific interval wheretrappedmotionexists.Theparticleswhoseinitialconditionslieoutsidetheregionoftrappedmotion(forthespecific value of the Jacobi integral) will rapidly escape to infinity along a scattering trajectory. In contrast, those par-ticleswithintheregionoftrappedmotionstaydynamicallyconfined to trajectories close to the central periodic orbit. Thedistinctionamongthesetwotypesofinitialconditionsissharp.Therefore,lettingthesystemevolveforsometime,aringisobtainedbyprojectingintotheX-Yspaceatagiven(fixed) time the phase–space location of all the ring particles of the ensemble that are dynamically confined.

Fromtheseconsiderationssomeimportantstructuralpropertiesoftheringfollow.First,theringdisplayssharpedges,sincethedistinctionbetweentrappedparticlesandescapingparticlesisclearafterrathershorttimes.Second,theringsareingeneraleccentricsincethemotionofeachparticleoftheringisclosetotheorganizingperiodicorbit,whichtypicallydisplayssomeeccentricityinarotatingframe.Third,thenarrownessoftheringcanbeunderstoodfromthefactthattheregioninphase–spacecorrespondingtodynamicallyboundedmotionistypicallyverysmall(Benetet al., 1998),aswellastheintervalofvaluesofthe

parameterwherethereferenceperiodicorbitisstable(BenetandMerlo2004).

Weemphasizethat,despitethequalitativenatureoftheseresults,thesestructuralpropertiesareobservedinrealnarrowplanetaryrings,andsomeofthemarenotfullyunderstood(Esposito,2002;Sicardy,2005).

STRUCTURE IN NARROW RINGS USING THE SCATTERING APPROACH

Inthissectionwereviewsomeresultsonthestruc-tureoftheringsobtainedusingthescatteringapproachona specific toy model. Our toy model is a planar scattering billiardonaKeplerorbit,animpenetrablediskrotatingaroundagivenpointonacircularorellipticKeplerorbit.AthoroughdescriptionofthemodelcanbefoundinMerloandBenet(2007).Thismodelisunrealistic;yet,itisthemostsimplerealizationofEquation(3),itcanbestudiedtosomeextentanalyticallyand,mostimportant,thequalitativeresultsobtaineddisplayconsistentlymost–ifnotall–thestructuralpropertiesobservedinthenarrowplanetaryrings.Theringsweobtainarenarrow,non-circular,sharp-edged,maydisplayseveralcomponentswhicharebraided,andmayalsodisplaynon-continuousringsformedbyanumberofarcs.Itisencouragingtonotethatsuchanunrealisticsystem,whichonlyemphasizestheimportanceofconsideringscat-teringdynamics,candisplaysuchqualitativeresemblancetothestructuresobservedinrealplanetaryrings.Similarresultshavebeenobtainedinothersystems,includingthecircularrestrictedthreebodyproblem(MerloandBenet,2007) and in a consistent implementation using five bodies (Olmedo,2007).

Webeginwiththescatteringbilliardmovingonacircularorbit.Inaframerotatingwiththediskthenew

Benet and Merlo352

Figure6.DetailoftheringwhenthediskmovesonaneccentricKeplerorbitwitheccentricitya)ε=0.0001and(b)ε=0.00167.NotethechangeinthescalesamongtheframesandwithrespecttoFigure1b.

Hamiltonianbecomestimeindependentandisthereforeaconservedquantity,theJacobiintegral.InFigure5aweplotaring(greyregion)obtainedinthiscase;theshadedregionistheharddisk;Figure5bshowsanenlargementofa region of the ring. In these figures the continuous black linesareanalyticalestimatesgivenbythestabilitypropertiesofthecentralstableperiodicorbit(BenetandMerlo,2004);asshown,theygiveexcellentestimatesoftheboundariesofthering.Theringisnarrow,eccentricanddoesdisplaysharpedges.Inaninertialframe,theringrotatesaroundtheoriginmaintainingitsshape;thisisduetothecircularsym-metryoftheproblem.Notethatinthissystemthemotionofeachringparticleisstrictlyrectilinearbetweenconsecutiveencounters with the disk, where it is specularly reflected in alocalreferenceframe(Meyeret al., 1995).Noencounterswiththediskleadtoescapeoftheparticle.

Whenthekeplerianorbitofthediskhaseccentric-ityε ≠ 0, the time dependence of the problem cannot be

removedexceptbyextendingtheeffectivenumberofdegreesoffreedomintheusualway;then,thesystemhasmorethantwodegreesoffreedom.Figure6ashowsanenlargementofaregionoftheringobtainedwhenthediskmovesonakeplerianellipsewitheccentricityε=0.0001;Figure6bshowsadetailoftheringcorrespondingtoε=0.00167.First,weobservethattheringsarenarrowerthantheobtainedonewhenthediskmovesonacircularorbit.However,themoststrikingfeatureoftheringsdisplayedisthefactthattheyareactuallydividedintwoormoredistinctcomponents,knownasstrands.Thesestrandsareentangledalongtheazimuthalangle(measuredfromthecontactpointwiththedisk),formingabraidedstructure.Themotionoftheringparticlesissuchthat,aringparticlewhoseinitialconditionsbelongtoacertainringcomponentstaysinthatcomponentafterwards.Intermsofthephase–space,thisimpliesthateachcomponentbelongstoanindependentphase–space region; this interpretation will be confirmed

Structure in narrow planetary rings 353

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Thedynamicsofsuchmultiple-componentringsisrichandinteresting.InFigures7wepresentthewholeringatdistincttimesmeasuredasafractionoftheperiodofthedisk,Td=2p ,forε = 0.00167. In these figures, thehorizontalaxisrepresentsthepolaranglemeasuredanti-clockwisefromthepointincontactwiththedisk;theverticalaxiscorrespondstotheradialdisplacementwithrespecttotheaverageradialdistance.First,wenoteinFigures7theclearazimuthaldependenceontheradialdisplacement. Moreover, the figures show that each ring component,whichmaintainsitsindividuality,undergoeschangesintheirstructureindependentlyoftheothers.TheazimuthaldependencemanifestedinFigures7isaconsequenceofthebrokensymmetrywhichanon-zeroeccentricityinduces.

Figure6alsomanifestsasubtledependenceuponthevalueofε.ThisisfurtherillustratedinFigure8,whereweplotthecorrespondingenlargementofaregionoftheringobtainedforε = 0.00168. Comparing this figure with Figure6b,weobservethatoneoftheringcomponents,theinnermostinFigure8,hascompletelydisappearedandanon-continuousringhasappeared.Such“patches”orarcs,areactuallyfoundeverywhereinthering.TheirstructureobviouslyrecallussomeclumpybehaviorobservedinsomenarrowringsinSaturnandthefamousAdam’sarcsinNeptune.Thefactthatforε=0.00167wehadthreecomponentsandnowthereareonlytwoandthearcs,canmisleadinglybeinterpretedasabifurcationwhichbreaksoneringcomponentinmanyarcs.Yet,athoroughsearchwascarriedoutandourresultsindicate,asitwasshownalreadyinFigure6bbythedarkspots,thatthesearcsareindeedobservedforvaluesofεwherethethirdcomponentisstillpresent.Thesearcsappearfromaringcomponet;smallchangesonε preservesthem.Thisthereforerulesouttheideathatarcsfollowfromabifurcationthatdestroysindividual ring components. We finally observe that also in theringofFigure8therearearcsimmersedintheoutermostringcomponent.

Theaboveresultsindicatethat,usingthescatteringapproach,weindeedobtainringswhicharenarrow,non-circular,andhavesharpedges.Thesepropertiesfollownaturallyfromthephase–spacestructuresconsideredwithinthescatteringapproach,i.e.,thepropertiesoftheregionsoftrapped motion that appear literally as islands in the infinite oceanofescapingtrajectories.Inaddition,theseringsmaydisplaypropertieswhichshowanazimuthaldependence:Theymaydisplaymultiplecomponentswhichareentangledandformbraids,and/oranumberoflocalizedarcs.Thedynamicalbehaviorbecomesricherwithnewtimescaleswhicharemuchshorterthantheperiodoftherotatingpoten-tial.Thesestructuralpropertiesareobservedindistinctrealnarrowplanetaryrings.Whilealltheresultspresentedabovehavebeenillustratedusingatoymodel,asimplebilliardsystemrotatingonaKeplerorbit,thescatteringapproachisrobust,anditcanbeappliedtomorerealisticsituations(MerloandBenet,2007;Olmedo,2007).

latter.ItisworthmentioningherethatSaturn’sFringispreciselyanexampleofringwithmultiplecomponentswhich, in addition, does not fulfill the requirements of the sheperdingtheory(Esposito,2002).

Figure7.Wholeringrepresentedusingpolarcoordinates(seetext),whenthediskmovesonaneccentricKeplerorbitwithε=0.00167.Eachframerepresentsdifferenttimestexpressedasfractionsoftheorbitalperiod,Td = 2�. (a) t =Td/10;(b)t =3Td/10;(c)t =Td/2.Notehowthedifferentringcomponentsentangle,formingabraidedstructure.

Benet and Merlo354

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Figure9.Poincarésurfaceofsectiondisplayingthestructureofphasespaceforthescatteringbilliardonacircularorbit(twodegreeoffreedom).(a)Typicalstructurefoundafterthesaddle–centerbifurcation.Theblackandredcurvesarethemanifoldsoftheunstableperiodicorbit;theyboundaregion,which contains the stable periodic orbit, where trajectories cannot escape to infinity. (b) Phase-space structure when both periodic orbits are unstable. While there are orbits that never escape, the probability of finding them is zero.

PHASE–SPACE CONSIDERATIONS

Inthissectionweshalldescribethephase–spacestructuresuponwhichthescatteringapproachisbased.WeshallcharacterizetheunderlyingchangesthatcantakeplacewhentheHamiltonianhasmorethantwodegreesoffreedom,whicharethusresponsiblefortheappearanceofmultiplecomponentsandarcs.

Webeginwiththephase–spaceofatwodegreeoffreedomHamiltonianscatteringsystem,withandwithout

aregionoftrappedmotion.Asmentionedabove,agenericscenariofortheappearanceofperiodicorbitsinHamiltoniansystemsisthesocalledsaddle–centerbifurcation.Generichereimpliesthatthesamescenarioholdsforalargevarietyofinteractions;therelevantaspectsarethelocalproperties.Thesaddle–centerbifurcationoccurswhen,byvaryingaparameterofthesystem,twonewperiodicorbitsarecreated,oneofthemisstableandtheotherunstable.Insimpleterms,itoccurswhenthesolutionofaquadraticequationchangesfromhavingcomplexrootstoreal-valuedsolutions.Just

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Figure8.DetailoftheringwhenthediskmovesonaneccentricKeplerorbitwitheccentricityε=0.00168.TheinnermostringcomponentofFigure2bhasdisappeared,andadiscontinuousringformedbypatchesorarcsisnowapparent.Notethattheoutermostcomponentalsodisplaystheoccurrenceofothersetofarcs.

Structure in narrow planetary rings 355

afterthebifurcation,thestableandunstablemanifoldsoftheunstableperiodicorbitboundaregionaroundthestableperiodicorbit(Figure9a).Outofthisregionunstablemotiondominates and, if the unstable fixed point is the outermost inphase–space,thecorresponding(scattering)trajectoriesescape to infinity. On the contrary, trajectories whose initial conditionslieinsidecannotescapefromavicinityofthestable fixed point and are therefore dynamically trapped. Byfurthervaryingtheparameter,thecentralstableperiodicorbittypicallybecomesunstablethroughaperiod-doublingcascade.Thephase–spacechangestopologically(Figure9b),andmaybecomeeventuallycompletelyhyperbolic.FortwodegreesoffreedomHamiltoniansystemsthiscanbequantitativelycharacterizedbycomputingthetraceofthematrixthatdescribesthelinearizeddynamicsaroundaperiodicorbit.Whentheabsolutevalueofthetraceislessthan2,theperiodicorbitisstableandaregionofboundedmotionexists;otherwise,bothperiodicsolutionsareunstable.TheseelementaryfactswereusedtoconstructtheanalyticalestimatesdisplayedinFigure5(seeBenetandMerlo,2004).

Formorethantwodegreesoffreedom,thetopologi-calconstrainsthatimplythatthemanifoldsoftheunstableorbit define a bounded region around the stable one cease toapply.Arnolddiffusionfollowsfromthis.Thisispar-ticularlyimportantsincetheexplicittimedependenceinEquation (3) yields an effective phase–space of five dimen-sions.Yet,therearetheoremsthatprovideconditionsfortheexistenceofeffectiveboundedmotionaroundstabletori(JorbaandVillanueva,1997a,b).Inordertoobtainagraphicalrepresentationofthechangesinphase–space,weshalldescribetheparametricbehaviorofarelativemeasureofthephase–spacevolumewhichisoccupiedbytrappedtrajectories.This,asweshallshow,allowsustounderstandtheappearanceofstructureintherings.

Inordertounderstandtheappearanceofthestrandsorarcsweneedawayofcharacterizingthephase–spaceregionsoftrappedmotioninaglobalway.Aconvenientformofachievingthis,inparticularwhenthenumberofdegreesoffreedomismorethantwo,istoconsidertherelativephase–spacevolumeoccupiedbytheregionsoftrappedmotion(inthesenseofeffectivestability)intermsofaparameter.ForthescatteringbilliardinacircularorbitthisquantitycanbeparameterizedintermsoftheJacobiintegral.Yet,theJacobiintegralisnotconservedfornon-zeroeccentricity.Wehavethereforeoptedtousetheaveragetimebetweenconsecutivecollisionswiththedisk,⟨Δt⟩.Notethat this quantity is equivalent to the average first-return timetoaPoincarésection,whichmaybeusedinamoregeneralcontext.InFigure10wepresentthestructureoftherelativephase–spacevolumefor(a)ε=0and(b)ε=0.0001;thecorrespondingringsareillustratedinFigures5and6a,respectively.

In Figure 10a we observe at certain specific values of ⟨Δt⟩ thatthephase–spacevolumeoftheregionoftrappedmotionisreduceddrastically.Itcanbeshownthatthe

locationwheresuchabruptreductiontakesplaceisgivenbyaresonantconditiononthestabilityexponents,i.e.,inthe(complex)phaseoftheeigenvaluesofthelinearizeddynamics,andarenotrelatedtotheoccurrenceofrationalratiosamonganyrelevantorbitalperiods(BenetandMerlo,2008;BenetandMerlo,2009);thestructureofFigure10afortwodegreesoffreedomHamiltoniansisuniversal(Contopouloset al., 1999;Contopoulos et al.,2005).Thestructureofthehistogramuncoversimportantaspectsofthe dynamics; in particular, the fine scale “jumps” mark the destructionofcertaininvariantcurvesastheparameterischanged(SimóandVieiro,2009).Theseinvariantcurves,iftheyexistboundthemotionofsomeoutlyingchaoticregions;otherwisesuchtrajectoriesescapealongscatteringtrajectories.Thisoccursalsoaroundanysecondarysatelliteislands,whichaccountsfortheself-similarstructure(seeSimóandVieiro,2009).

InFigure10b,whichcorrespondstoε=0.0001,weobserveingeneralsomequalitativeresemblancewiththecaseε=0(Figure10a).However,insteadofalocalizeddrasticreductionofthephase–spacevolumeofthetrappedregionasdisplayedinFigure10a,inthepresentcaseatruegapisobservedinthehistogram.Thegapappearsaroundcertainstabilityresonancesoncetheeccentricityεisnon-vanishing,andisduetononlineareffects(BenetandMerlo,2008).Excitingsuchstabilityresonancedividestheregionsoftrappedmotionintwodisjointregions.Thesegapsareactuallyresponsiblefortheappearanceofmultiplestrands:Ifthegapiswideenough,theregionsoftrappedmotionare,inasense,distantinphase–space,andtheirprojec-tionontotheX-Yplaneyieldsaringwithtwoindependentstrands.Then,theappearanceofmultiple-componentringfollowsfromhigher-dimensionalandnonlineareffects.Forathoroughdescriptionofthedependenceofthehistogramson⟨Δt⟩ uponε seeBenetandMerlo(2009).

Thelastpointweshalladdresshereisrelatedtotheappearanceofarcs.Intuitively,weexpectthatarcsaretheresultofprojectingphase–spaceregionswhichresemblechainsofbubbles.Tounderstandtheirappear-ancewemustmentionthat,e.g.,forε=0.00167,thereareexactly149arcsalongthewholering.Furthermore,theexact configuration of the arcs in the X-Y space(labeledinanarbitraryway)isrepeatedafter229bounceswiththedisk.Theseobservationssuggestthattheappearanceofarcsislinkedwiththemeanmotionresonance149:229.Recentresultsconfirmthis,whichhasalsoservedtofind other occurrences of arcs (Benet and Merlo, 2009). Suchresonancesintroduceastrongazimuthaldependenceoncertaintrappingregionsinphase–space;thisseemstocreate,insuchahigherdimensionalphase–space,chainsofisolatedislandswheretrappingtakesplace,whichseemstobeanextensionofthePoincaré-Birkhofftheoremfortwo degrees of freedom systems. We shall finally note that Adam'sarcsinNeptuneareunderstoodthroughtheoccur-renceofeccentricityandinclinationresonances(NamouniandPorco,2002).

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CONCLUSIONS AND OUTLOOK

Inthispaperwehavereviewedsomeoftheopenproblemsrelatedtothestabilityandstructureofplanetarynarrowrings,anddescribedageneralandself-consistentapproachtounderstandtheseissues,thescatteringapproach.Ourapproachisbasedonthelocalstructureofphase–spacearoundstable(periodicorquasi-periodic)solutionsinregionswherescatteringdominatesthedynamics.Thecorrespondingstructureinphase–spaceallowsforasetofpositivemeasuretoexhibitdynamicallytrappedmotion,i.e. a non-zero probability to find a structure resembling aring.Thebasicideais,simply,whatweseeiswhatisdynamicallytrappedandthestructureispreciselyuncoveredbywhatithasescaped,i.e., it is created by the confining

mechanism.Wehaveillustratedourapproachusinganunrealistic

toymodel.Ourresults,whichhaveonlyaqualitativevalue,showtheoccurrenceofnarrownon-circularrings,withsharpedges,whichmaydisplaymultiplecomponentsandarcs.Theinterestshouldnotbetheexampleused,butthefactthatthescatteringapproachisrobust.Theresultsobtainedinsuchanunrealisticsystemareencouraging,preciselybecauseofthequalitativeagreementwiththeobservations.Thestructurethatappearsisunderstoodintermsofthelocalpropertiesofphase–spacewhichisdominatedbyscatteringtrajectories.ThisapproachcanbetakenoverusingmorerealisticHamiltonianmodelswhichcertainlyincludegravitationalinteractions.Weareworkingalongtheselines.

Figure10.HistogramsoftheaveragetimebetweenconsecutivecollisionswiththediskforanensembleofringparticlesforthescatteringbilliardonaKeplerorbitwith(a)ε=0and,(b)ε = 0.0001. The histograms give a measure of the phase−space volume occupied by trapped trajectories. The main gaps are related to the stability resonances, indicated as vertical dash−dotted lines. For non−zero ε,thestabilityresonancesseparatetheregionsoftrappedmotion.Thisyieldsmultipleringcomponents.

Structure in narrow planetary rings 357

ACKNOWLEDGMENTS

WewouldliketothanktoÁngelJorba,CarlesSimóandFathiNamounifordiscussionsandcomments.Weacknowledge financial support provided by the projects IN-110110(DGAPA-UNAM)and79988(CONACyT).OlivierMerlowasapostdoctoralfellowoftheSwissNationalFoundation(PBBS2-108932).

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Manuscriptreceived:February13,2008Correctedmanuscriptreceived:October6,2009Manuscriptaccepted:November17,2009