1987AJ 94.1350W 1350 Astron.J.94 (5),November1987 0004-6256/87/051350-11$00.90 © 1987Am.Astron. Soc.1350 turbed thatthe3/2resonance, theresonanceinwhich half revolutionsperorbit.This regionissostronglyper- from norotationatallinaninertial frametonearlytwoanda rion therotationalphasespace near synchronousrotationis dominated byalargechaoticzone. Thechaoticzoneextends of modemnonlineardynamicsshowed thatinfactforHype- orbit couplingisnotvalidforabodywithsuchlargeaspheri- friction andconsequentlytherotationrateisexpectedtobe city. Areanalysisofthespin-orbit problemusingtechniques short axis(Smithetal.1982).Thestandardtheoryofspin- of-round, withthelongaxisroughlytwiceas comparable totheorbitalmeanmotion(Peale1977). rotation ofHyperionhasbeensignificantlyaffectedbytidal general picture(Wisdom,Peale,andMignard1984).The Voyager picturesshowedHyperiontobesignificantlyout- inspired thiseleganttheory. commensurate rotationinthesolarsystemisonethat evolved satellitesareinsynchronousrotation.Itisrather nonsynchronous commensuratespin-orbitresonance.In has asignificantprobabilityofhavingbeencapturedinto unfortunate thattheonlyexampleofanonsynchronous those caseswheretherotationstateisknown,alltidally were showntobedynamicallystable,andinmanycases despinning issmallerthantheageofsolarsystem,none among thosenaturalsatellitesforwhichthetimescale nonsynchronous spin-orbitresonancesasitisencountered lution aswell.Theprobabilityofcaptureintoeachthese motion intheseresonancesisstableagainstfurthertidalevo- was estimated.However,Peale(1977)hasshownthat, equal tohalfanintegermultipleoftheorbitalmeanmotion too out-of-round,spin-orbitresonanceswithrotationrates fixed orbitwithnonzeroeccentricityandafigurethatisnot pling wasdeveloped(seeGoldreichandPeale1966).Fora evaluation ofthispicture,andthetheoryspin-orbitcou- established. TheresonantrotationofMercuryforcedare- sumed todeclinesteadilyuntilthesynchronouslockwas THE ASTRONOMICALJOURNAL Until thediscoveryofresonantrotationMercury smaller angularvelocities,theequilibriumobliquityiszero. velocity approachestwicethemeanorbitalmotion.For the equilibriumobliquitygoestozeroasspinangular tion, theequilibriumobliquitydecreases.Iforbitisfixed, value between0°and90°.Asthespinisslowedbytidalfric- the tidaltorquetendstodriveobliquityanequilibrium been understoodforover100yr(Darwin1879;seeGol- dreich andPeale1970).Ifthespinangularvelocityislarge, obliquities androtationratesofthenaturalsatelliteshave (Pettengill andDyce1965),therateofrotationwasas- Saturn’s satelliteHyperionisadramaticexceptiontothis © American Astronomical Society • Provided by the NASA Astrophysics Data System The basicmechanismsgoverningthetidalevolutionof Department ofEarth,Atmospheric,andPlanetarySciences,MassachusettsInstituteTechnology,Cambridge,02139 All irregularlyshapednaturalsatellitesmusttumblechaoticallybeforebeingcapturedintosynchronous rotation. ROTATIONAL DYNAMICSOFIRREGULARLYSHAPEDNATURALSATELLITES I. INTRODUCTION Received 23February1987;revised21July1987 VOLUME 94,NUMBER5 Jack Wisdom ABSTRACT candidate forchaotic tumbling,butinthiscase thereisno to thatofHyperion,Nereidcomes tomindasapossible its largeorbitaleccentricity(e^:0.75) andsizecomparable rotation; thesynchronousresonance isattitudestable.With the rotationisknown,thesesatellites areallinsynchronous each casetheorbitaleccentricity islow.Inthosecaseswhere natural satellitesthataresignificantly out-of-round,butin chaotically tumbling?Thereareactuallyanumberofother phase space.Shouldothernaturalsatellitesbeexpectedto counts forthelargechaoticzoneinHyperion’srotational gives astrongoverlapofprimaryresonances,andthusac- city parameteroforderunityandlargeorbitaleccentricity commensurability betweenTitanandHyperion.Anaspheri- primarily aforcedeccentricityduetothe4/3meanmotion has aratherlargeeccentricity,near0.1.Thiseccentricityis not onlyout-of-round(a—0.89),buttheorbitofHyperion proportional tosomepoweroftheeccentricity.Hyperionis the resonancesexceptforsynchronousresonanceare principal momentsofinertia.Inaddition,thewidthsall city parametera=y¡3(B—A)/C,whereAHyperion’srotationwasslowedbytidalfriction,there rently beinginthechaotic-tumblingstate(Goguen1983; this chaotic-tumblingstatethatareattitudestable:the2/1 Two spin-orbitresonancesareaccessibletoHyperionfrom calculation oftheLyapunovCharacteristicExponents orientation isunstable,Hyperionthenbegantotumble.The axis nearlyperpendiculartotheorbitplane.Sincethis tumble. Moreover,thechaoticzoneisalsoattitudeunstable. away fromtheorbitnormalandHyperionwouldbeginto under theslightestperturbationspinaxiswouldfall tion withthespinaxisperpendiculartoorbitalplane, other tidallyevolvednaturalsatellitesarefound,isattitude esting, thesynchronousspin-orbitresonance,inwhichall rotation ofMercuryislocked,notstable.Evenmoreinter- all observationstodateareconsistentwithHyperioncur- unstable. EvenifHyperionwereplacedinsynchronousrota- Hyperion mustbecapturedbyoneoftheseresonances,but 1986). Perhapsthemostconvincingevidenceforchaotic (LCEs) showsthatthetumblingmotionisfullychaotic. The presenceofthelargechaoticzonecanbeunderstood NOVEMBER 1987 1987AJ 94.1350W -43 1351 JACKWISDOM:IRREGULARLYSHAPEDSATELLITES ory fromwhichitwasdeveloped,thetheoryofHyperion’s most likelyspinstabilized.Liketheclassicalspin-orbitthe- chronous rotation(Peale1977).TheofNereidis reason tobelievethatNereidistidallyevolvedanearsyn- chaotic rotationapparentlyhasonlyasinglefollowertoday. fact, sinceallspin-orbitresonancesaresurroundedbychao- tate chaoticallyatsomepointintheirrotationhistories.In tic séparatrices(seeWisdomciö/.1984;Chirikov1979),itis nous rotationfromnonsynchronouscrossedachao- certain thatallsatelliteswerecapturedintosynchro- tic separatrix.Inmanycases,thischaoticseparatrixisex- to beoforder10timesthewidth3/2resonance surrounding the3/2resonanceofMercurymaybeestimated on theevolution.Forexample,widthofseparatrix ponentially smallandwouldhavehadnosignificanteffect bling satellitemaysignificantlyaffecttheorbitalevolution the chaoticseparatrixinasinglelibrationperiod.Inother estimate ofthewidthchaoticseparatrixforsyn- cases, thechaoticséparatricesarenotsomicroscopic.The ble chaoticseparatrix.Alloftheirregularlyshapednatural lously loweccentricityoftheorbitDeimosorperhaps orbital historyofanirregular-shapedsatellite(Sec.V).This and mustcertainlybetakenintoaccountinconsideringthe play thegeometryofphasespacetumblingmotion problem andsurfacesofsectioncanbecomputedthatdis- this casethemotionisreducibletoatwodegreeoffreedom tumbling atthepointofentryintosynchronousrotation.In lem evenfurther,itturnsoutthataprolateaxisymmet- tion wouldoccurevenforcircularorbits.Reducingtheprob- rad. Thisresultsuggeststhattheout-of-planetumblingmo- orbital eccentricityisanomalouslylow(e^0.0005),thean- sitive totheorbitaleccentricity.EvenforDeimos,where The magnitudeofthisdeflectionseemstoberelativelyinsen- chronous spin-orbitresonancegiveninWisdometal episode ofchaotictumblingmayhelpexplaintheanoma- orbit resonancewithoutpassingthroughtheattitude-unsta- ric bodyinacircularorbitcanbeattitudeunstabletochaotic gle fromthelongaxistoorbitalplanegetsaslarge0.86 orbital eccentricity.Thussatellitesthataresignificantlyout- the out-of-roundnessparametera,butonlylinearlyon of-round maybeexpectedtohavechaoticséparatricesof chaotic-tumbling state.Theenhanceddissipationinatum- satellites musthavespentacertainamountoftimeinthis significant sizeeveniftheorbitaleccentricityisrelatively “stretch marks”onPhobos(Sec.VI). chaotic separatrixengulfsboththe3/2and1/2reson- small. Forseveraloftheirregularlyshapedsatellites the chaoticzoneisattitudeunstable,andresultingchao- ances (Sec.II).Amoresurprisingresultisthatineverycase tic-tumbling motioncarriesthelongaxisofsatellite (Wisdom etal1984).TidalfrictionpullsMercuryacross through asignificantanglefromtheorbitalplane(Sec.Ill). natural toconsiderthisreduced problemsinceinthestan- This problemwasreviewedby Wisdometal(1984).Itis (Sec. IV).Itisnotpossibletoenterthesynchronousspin- (1984) (seealsoEq.(4)below)dependsexponentiallyon dard pictureoftidalevolutionthe spinaxisisdrivenperpen- spin-axis fixedperpendicularto theorbitplaneisconsidered. ments ofinertia A-/ = 0, 1351 (1) ) (3 1987AJ 94.1350W / shouldbetheaveragedresonanceHamiltonianforsyn- second orderine.)Whilethewidthdependsexponentially ing trajectoryintheintegrablee=Ocase.(Strictlyspeaking, where/ isthevalueofintegralevaluatedonseparat- on theparametera,itisonlylinearlydependentec- chronous resonance,butthisdiffersfrom/onlybytermsof the expectationthatwidthincreasesaseccentricity of thechaoticzoneiszeroforeccentricity,andverifies centricity. Theformulaobeystherequirementthatwidth 1352 JACKWISDOM:IRREGULARLYSHAPEDSATELLITES reliable wayofdeterminingtheactualextentchaotic estimate (4)forthewidthmaynotbeverygood.Theonly nominal positionofthe3/2resonance.Consequently, case. Thusthesynchronousislandevenextendsbeyond tic zones.Forthesatellitesconsideredhere,a>0.5inevery lites withlargeabutsmallemaystillhavesignificantchao- eccentricity (inthisapproximation)meansthatthosesatel- increases. Thefactthatthewidthdependslinearlyon the chaoticzoneismuchnarrowerthanfor tricity istakentobethecurrenteccentricityof0.015.While the shapeassumingauniformdensity,is0.86,andeccen- surface ofsectionforPhobos,wherea,asdeterminedfrom zone istocomputeasurfaceofsection.Figure1showsthe onance, the1/2resonance(theislandjustbelowcenter), Note thatasinglechaoticzoneengulfsthesynchronousres- Hyperion, itisstillamajorfeatureonthesurfaceofsection. a =0.56isverysimilartothesection inFig.1.Thechaotic forced librationofPhobosisonly0!8±0.2(Duxburyand chaotic zonenear0=0).Theobservedamplitudeofthe and the3/2resonance(thesmallislandincenterof integral/. Forthesynchronousspin-orbitresonancethises- primary resonancesintermsofthechaoticvariations the widthof fixed-axischaoticzone.Unfortunately, what largerforthesmallera. zones arejustaboutthesamesize, butthe3/2islandissome- mass distribution.However, the surfaceofsectionfor that theshapeofPhobosdoes not, accuratelyrepresentthe Callahan 1982).Thisimpliesanaofonly0.56.Itappears timate is size ofthechaoticseparatrixwhenthereisnooverlap pears. Wisdometah(1984)havegivenanestimateofthe chaotic zonessurroundingtwoneighboringislandsdisap- equivalently wherethelastinvariantcurveseparating new islandissuddenlyengulfedbythechaoticzone,or since therewillbecriticalvaluesoftheeccentricitywherea size ofthechaoticzonecannot,however,becontinuous, zone willincreasewiththeeccentricity.Thein It isareasonableexpectationthenthatthesizeofchaotic dicular totheorbitplaneunlesseccentricityisnonzero. s Janus Satellite Amalthea Deimos Phobos Epimetheus 1980S27 1980S26 3 The orbitaleccentricityisacritical factorindetermining (/ -/)//=(147re/a)e~^(4) © American Astronomical Society • Provided by the NASA Astrophysics Data System s ( 13.5±1)X10.50.7)(9.0 (70±8)X(57±8)X(50±5) (110±5)X(95±5)X(80±5) (135±5)X(85±5)X(75±5) (7.5 ±5)X(6.00.5)(5.0±1) (55 ±8)X(425)(33 (70±5)X(50±7)X(37±8) Shape (km) Table I.Dataforirregularlyshapednaturalsatellites. 5 be characterizedbytheextentofchaoticzoneindO/dt of2000 orbitperiodseach,isplottedversusorbitaleccentric- been crossedwhichcouldhaveincreasedtheeccentricity time ofcaptureintosynchronousrotation,thenPhobos tion. Synchronousrotationwasreachedessentiallyatthe near theunstableequilibrium(0=tt/2onsurfaceof have alwaysbeensmall.Thewidthofthechaoticzonemay the formationepoch.ThuseccentricityofPhobosmay after thelastresonancepassageratherthanaremnantfrom centricity isexplainedasatidalremnantoftheeccentricity than thecurrenteccentricity.Inthiscase,observedec- from anear-zeroinitialvaluetovaluesseveraltimeslarger history. Ontheotherhand,Yoder(1982)(seealsoMignard would certainlyhavehadaperiodofchaotictumblinginits time offormation.Iftheeccentricitywasindeedlargeat and atmosttenmillionyearsitsmoredistantinitialloca- The timescaleforthetidaldespinningofPhobosison zenave etal.(1981)allgivelargeeccentricity(e>0.6)to section). InFig.2,theextentofchaoticzoneforPhobos order of10yr(Peale1977)atitscurrentsemimajoraxis the orbitofPhobosneartimeitsformationorcapture. captured intothesynchronousresonanceisnotknown. Phobos; theorbitaleccentricityattimewhenPhoboswas there isconsiderableuncertaintyintheorbitalhistoryof Singer(1968), Lambeck(1979),Mignard(1981),andCa- (a =0.86),asdeterminedfromratherlimitedintegrations 1981) pointsoutthatanumberofresonancesmusthave a =yJ$(B—A)/C0.86,andtheorbital eccentricityis0.015.The rotation resonance. chaotic zoneengulfsthe3/2and1/2states aswellthesynchronous plotted versustheorientation at everyperiapse.Here perpendicular totheorbitplane.Therate ofchangetheorientationis Fig. 1.SurfaceofsectionforPhoboswith thespinaxisconstrainedtobe 0.86 0.89 0.78 0.66 0.81 0.99 1.14 0.009 0.003 0.0005 0.015 0.004 0.007 0.004 Phobos SurfaceofSection ,(±Æ.) \n dtJ 0.34 0.09 0.58 0.21 0.23 0.30 0.16 0.85 0.86 0.92 0.98 0.85 0.94 1.29 1352 1987AJ 94.1350W 1353 JACKWISDOM:IRREGULARLYSHAPEDSATELLITES ted. Forinstance,withe=0.015thesurfaceofsection ity. Becauseofthelimitednatureintegrations, ing theuncertaintiesinorbitalhistoriesofnatural larly shapednaturalsatellitesaregiveninTableI.Consider- of yearsinthechaoticzone. have spentmanyhundredsofthousands,probablymillions, case, thechaoticzoneisnevermicroscopic.Phobosmust carried outforFig.2therangeisroughly0.9-1.25.Inany shows thatthechaoticzoneextendsfromroughly0.65to chaotic zonemayextendsignificantlybeyondtherangeplot- small. EvenforDeimos(a=0.81),wheretheeccentricity zones fortheirregularlyshapedsatellitesarenotnegligibly tion. Whetherornottheeccentricitywasnearcurrent the eccentricityattimeofcaptureintosynchronousrota- each calculation.Itisplausiblethatthisrepresentativeof satellites, thecurrentorbitaleccentricityhasbeenusedin the orderof100millionyears(Peale1977),Deimosprob- since thetimescalefortidaldespinningofDeimosison eccentricity, thewidthsinTableIillustratethatchaotic than didPhobos. ably spentaconsiderablylongertimeinthechaoticzone tion leadstochaotictumbling. Such adeviationwillalways slightest deviationofthespinaxis fromtheequilibriumposi- chronous rotation.Uponentering thechaoticzone, plane bythetidaltorqueas spinisslowedtowardsyn- 1.25 near0=tt/2,whereasinthemorelimitedintegration bit plane.Thisisalsotrueofthechaoticzone;slightest system thespinaxisisbrought perpendiculartotheorbit natural outcomeoftidalevolution. Overtheageofsolar chaotic tumbling.The tumblingofHyperionisa deviation ofthespinaxisfromorbitnormalleadsto 1982), thechaoticzonecannotbeignored(Fig.3).Infact, (e^0.0005) isconsideredtobeanomalouslysmall(Yoder axis cannotmaintainanorientationperpendiculartotheor- nance isattitudeunstable(Wisdometal.1984).Thespin namics ofHyperionisthatthesynchronousspin-orbitreso- lower limit. ered only2000orbitperiods,sotheextentdisplayedmustbeviewedasa figure ofPhobos,butvaryingorbitaleccentricities.Theintegrationscov- Fig. 2.Theextentofthefixedspin-axischaoticzoneat0=tt/2with The widthsofthechaoticzonesforseveralotherirregu- © American Astronomical Society • Provided by the NASA Astrophysics Data System One ofthemostsurprisingfactsaboutrotationaldy- III. ATTITUDEINSTABILITY Extent ofChaoticZone positive. LCEsmayalsobeusedtodeterminetheattitude grees offreedomwithnointegrals.ThreeLCEsmaythenbe motion. Forthetime-dependentspin-orbitproblemwith the rangeofresultingout-of-planemotion. natural satellitesareattitudestableornot,and,ifwhatis the morenarrowchaoticzonesforotherirregularlyshaped tion inthechaoticzonewithspinaxisofreference stability ofmotioninthefixed-axischaoticzone.Forrota- spin axistakingarbitraryorientations,therearethreede- neighboring trajectorybeingdisplacedalongthedirectionof dence, oneofthesemustbezero,whichcorrespondstothe exponential growth.Therearenindependentnonnegative dom problemtherecan,ingeneral,bendistinctratesof chosen totheneighboringtrajectory.Foranndegreeoffree- rate ofexponentialdivergencecandependonthedirection The separationofnearbyquasiperiodicorbitsgrowslinear- rate ofdivergencenearbyorbits.Forchaoticorbitsthe dom etal.1984).TheLCEsmeasurethemeanexponential terms ofLyapunovCharacteristicExponents(LCEs)(Wis- perion’s obliquityissubject.Itnaturaltoaskthenwhether exist duetothenumerousminorperturbationswhichHy- trajectory fixedperpendiculartotheorbitplane,atleastone LCEs. IftheHamiltoniancontainsnoexplicittimedepen- ly, ontheaverage.Foraparticularreferencetrajectory, another will,ontheaverage,growexponentiallywithtime. distance betweentrajectoriesstartedsufficientlyclosetoone motion alongtheunstableeigendirectiongrowsexponential- for fixedpointsorFloquetanalysisperiodicorbitswhere is attitudeunstableiftwoormoreLCEsarepositive.Theuse chaotic natureofthereferencetrajectory.Thezone of theLyapunovexponentsmustbepositivetoreflect even forDeimoswithitssmall orbital eccentricity. normal issmall,ontheorderof 5-6 orbitperiods!Thisistrue for thegrowthofdeviations thespinaxisfromorbit Even moreremarkableisthatin everycasethee-foldingtime chaotic zoneisattitudeunstable, justasitisforHyperion. displaced fromtheorbitnormal. verified bycomputingatrajectorywithspinaxisslightly ly. Inanycase,theresultofanalysismaybedirectly natural generalizationoftheusuallinearstabilityanalysis of LCEstodefineattitudestabilityanonperiodicorbitis not microscopic. tricity of0.0005.Evenwithsuchasmalleccentricity,thechaoticzoneis Fig. 3.ThechaoticzoneforDeimos(a=0.81)withitscurrenteccen- Attitude stabilityofthechaoticzonemaybedefinedin The remarkableresultisthat in everycasestudiedthe Deimos SurfaceofSection 1353 1987AJ 94.1350W 4 towards theplanet,butthatatirregularintervalslong 4). Themaximumextentofthevariation\(p\overlonger verified tobeattitudestable,whilethe1/2resonanceisatti- Phobos thesynchronousand3/2spin-orbitresonancesare normal, atthecenterofattitudeunstable1/2island.For integrations of200orbitperiodsforPhobosaswellthe ers allvaluesofxp,but(pseemstohavealimitedrange(Fig. total of75orbitperiods.Theresultingtumblingmotioncov- tude unstable.Pointsareplotted100timesperorbit,fora started withtheaxisdisplaced10~radiansfromorbit dom etal.(1984).ThetrajectoryplottedinFigs.4and5was other irregularlyshapedsatellitesisgiveninTableI.Figure equations ofmotionaresimilartothosedescribedinWis- about thesmallestmomentofinertia(thelongaxis).The long axis)fromtheorbitplane;thisphysicallyimportant with thexaxisparalleltoplanet-to-satellitelinepointing aligned withthelongestaxisofbody.Theorientationis 5 showsthatthereisatendencyforthelongaxistopoint gles intheoriginalangularcoordinatesofWisdometal. angle isamorecomplicatedfunctionoftheorientationan- by theangle6aboutcaxis,(f)newb specified relativetoaninertialsetofaxesdefinedatperiapse, moments^ rbit>« dt 1 d(L)^Ldn orbitz (¿orbit )dt (¿z) 3dE (13) (12) (ID (10) -7 by numericalcomputation. appears tobenootherwaydetermineJ¥(e)than portant ifd¿?/deexceedsabout5000.Unfortunatelythere has amagnitudeoforderunityorgreater.ForDeimos,say, where e=0.0005and(L)/(¿)^10,thefactorisim- tant ifthequantity norable. Theanalogousfactorinthedenominatorisimpor- has acontributionfromthestandarddeviationofmean an estimateoftheerrortheseaverages.The were integratedfor5000orbitperiodseach.Thediamonds moments estimatedfromtheshapeofDeimos.Theresults The factorinvolving(L)/(Linthenumeratorisig- of eachindividualtrajectorywhichisdeterminedbyconsid- the orbitplaneforeacheccentricity.Theerrorbarsindicate indicate theaverageangularmomentumperpendicularto in thechaoticzonesurroundingsynchronousresonance are showninFig.8.Foreacheccentricitythreetrajectories pended inanattempttoestimateJf(e)fortheprincipal Substituting intoEq.(8)andrearrangingyields Thus, z z Dividing throughby(¿), zorbit zorhit z z (¿orbit )dt tion oftheorbitaleccentricity, e. mentum perpendiculartotheorbitplane, )/Cn,asafunc- Fig. 8.EstimatesforDeimosofthedimensionless averageangularmo- 2 z Considerable effort(about500VAXhr)hasbeenex- de_\ l~ef3(¿)\1dE}/ dt L2e\(L)Edt\l d{L) lz e (L)¿fde oth]t 1 q)dj? — _dn1dJ? z orbitz (L) dt~ndedt' z z z 1 d(¿orbit) dt 1 d(L)_\dndJ^de z \ e(¿orbit>^de)' (¿orbit >UItdeJt)' = £’C—+Cn Z 2 l-e (¿z)1dJ?\ z Average AngularMomentum dt dy, de de dt (17) (16) (15) (14) 1357 1987AJ 94.1350W 12 per orbitperiodis lar rotation.Theorderofmagnitudetheenergydissipated greater thanthatinanonsynchronoussatellitewithregu- pation mustbecomparabletoandisprobablysomewhat For achaoticallytumblingsatellite,therateofenergydissi- by thechaotictumblingitself.Dissipationofenergyina of thetideistimedependentforanonsynchronousrotation. body tidesraisedbytheplanet,anddeformationinduced ity, Ristheradiusofsatellite,Qspecificdissipation wherep isthemassdensity,corotationalangularveloc- synchronously rotatingsatellitesincethewholemagnitude tumbling satelliteissignificantlyenhancedoverthatina numerical coefficients havebeenconsistentlyignored inits function, andpistheshearmodulus(BurnsSafronov This dampingrateshouldnot betakentooliterallysince at arateoforder the eccentricityofachaotically tumblingsatelliteisdamped angles arealloforderunity,andhavebeenneglected.Thus orbital meanmotion«.Shapeparametersand“nutation” the chaotic-tumblingstate,coiswithinafactorof2 jectory exploresonthetidal-evolutiontimescale,and 1973 ).gisoforder100,andp10dyn/cm.For neglected. Theresultingrateofdecreasetheorbitaleccen- takes thesameformasforsynchronousrotation tricity whiletherotationisinchaotic-tumblingstatethen Fig. 8atfacevalue,thedenominatorinEq.(17)maybe in Fig.8representthebestestimateavailableoffunction chaotic zonenearsynchronousrotation.Thedatapresented to thoseperiodswhentherotationtrajectoryisinmain 1358 JACKWISDOM:IRREGULARLYSHAPEDSATELLITES ever, eveninthatcasetheestimatesthissectionwillapply considerations inthissectionwillnotbeapplicable.How- depend onwhichparticularpartoftheArnoldwebtra- is notwelldefined,thentheoutcomeoftidalevolutionwill freedom thechaoticzoneislimitedinextent,andthere duced spin-orbitproblemwithoneandone-halfdegreesof wander farfromtheprimarychaoticzone.Whileinre- dimensions (the“Arnoldweb”)canallowthetrajectoryto well definedinthethree-dimensionalproblem.Ifaverage average, itisnotclearthatsuchanaveragemathematically little doubtthattheangularmomentumhasawell-defined because anyparticulartrajectorymayspendconsiderable through them.Apparently,integrationsmuchlongerthan mean ofthemeansthreeseparatetrajectories.The ment andacontributionfromthestandarddeviationof larly acuteinthefullthree-dimensionaltumblingproblem average inanylimitedintegration.Thisproblemisparticu- time insmalltributariesofthechaoticzone,spoiling of theangularmomentum.Thedifficultyprobablyarises are onlymarginallyconsistentwithasmoothcurverunning determine. TheaveragesshowninFig.8arerathernoisyand ering each500orbit-periodsegmentasaseparateexperi- since theinterconnectednessofchaoticzoneinmany average angularmomentumwasunexpectedlydifficultto 5000 orbitperiodsarerequiredtoobtainconsistentaverages — =-Liil-lü..( Energy dissipationresultsfromthenonperiodicsolid- tÆ-pWRVyiQ, (19) dt 2eE 18) © American Astronomical Society • Provided by the NASA Astrophysics Data System e dt" (e).Thereisnoconvincingevidenceinthesedatathat 1 de z /de isgreaterthanabout100.Thustakingthedatain z 1 3 e^pQd pnR ' (20) 2 2 2 2 given byEq.(20),whichistwofactorsoftheorbitaleccen- bling didpersisttothepointofcompletelydampingec- R ,wheretheeccentricitytobeusedisthatincalculat- ity whileinsynchronousrotation. tricity shorterthanthetimescalefordampingeccentric- damping oftheorbitaleccentricityontimescaleroughly chaotic zonenearsynchronousrotationleadstoarapid rotational “Jacobiintegral,”J. somewhat fasterratethroughtheseculardampingof centricity, thechaotictumblingwouldthenbedampedata bration, whichisoforderCn.Comparingthetwodamping ing theeccentricitydampingrate.Thusifchaotictum- paratrix andtheJacobiintegralofdampedsynchronousli- rates, Eqs.(20)and21),thelatterislargerbyafactore/ needed tosignificantlyreducethechaoticzonerispresum- ably lessthanthedifferenceinJacobiintegralatse- taken tobeoforderCn.ThechangeintheJacobiintegral tion ofrotationalenergyandthehasbeen where Eq.(19)hasagainbeenusedfortherateofdissipa- the “Jacobiintegral”by/,dampingrateisoforder motion timestherotationalangularmomentum.Denoting Jacobi integralwillbedampedataratecomparabletothe cobi integralisequaltotherotationalenergyminusmean but de/dtmaychangesignassynchronousrotationisap- rate atwhichtherotationalenergyisdamped,sinceJa- zero. Oncetheeccentricityiszero,furtherevolutiondepends rotation trajectorymay“stick”tosomequasiperiodicisland proached, dependingonthetidalmodel. dt<0 anddE/dt<0with^<0arecompetingeffects.Gen- tricity. Thelargeeccentricityof Hyperion’sorbitsimplyex- inevitable stageintherotationhistories ofalltheirregularly on variationsofthe“Jacobiintegral.”Itseemslikelythat may remainchaoticuntiltheeccentricityhasdampedto when theeccentricityissmall,itmostlikelyfinalstate zone. Sincethesynchronousstatedominatesphasespace into thequasiperiodicislandbyweaktidaltorques.Such in thephasespacelongenoughthatrotationiscaptured tidal evolutionofasatelliteinchaoticrotation.First,the erally, forrapidrotationtheorbitaleccentricityincreases, predicted outcomewilldependonthetidalmodel,sincedL/ bling satellitemaybecontrastedwiththeorbitalevolutionof the orbitnormal bythetidaltorque,whilespin slowsand in anycase.Theregularprogression ofthespinaxisdrivento acerbates thechaotictumbling which wouldhaveoccurred shaped naturalsatellites,regardless oftheirorbitaleccen- isolated curiosity.Rather,chaotic tumblingisanaturaland momentum aremodifiedbythetidaltorques.Inthiscase ously rotatingsatellite,boththeorbitalenergyandangular a nonsynchronouslyrotatingsatellite.Fornonsynchron- otically tumblingsatellitethanforasynchronouslyrotating damping ofeccentricityistwofactorsesmallerforacha- “sticking” isawell-knownfeatureofthemotioninchaotic satellite. derivation. Theimportantpointisthatthetimescalefor (presuming itisattitudestable).Alternatively,therotation 32 In summary,therotationaltumblingmotioninlarge There aretwopossibilitiesfortheultimateoutcomeof This pictureoftheorbitalevolutionforachaoticallytum- The chaotictumblingofHyperionisnolongerjustan J dt~pQ 1 dJ_pnR, (21) VI. DISCUSSION 1358 1987AJ 94.1350W 9 1359 JACKWISDOM:IRREGULARLYSHAPEDSATELLITES bling satellitesarecapturedintoastatethatisstableto period ofchaotictumbling.Ultimately,chaoticallytum- state, isnowseetobeinterruptedbythebirthpangsofa is ultimatelycapturedintothefinalsynchronousrotation further tidaldissipation,whichineverycaseobservedisthe synchronous state.Evenifotherstatesarestable,withsuch small eccentricitiesthesynchronousstateisbyfarlargest be themostprobableendpoint.Hyperionisstillinchao- quasiperiodic islandandcouldconsequentlybeexpectedto tic zonetodaybecausethedespinningtimescaleissolong lites? Severalpossibilitiescometomind.ThefirstisMir- less, arethereanydirectlyobservableconsequencesofthis Unfortunately, itisnotyetpossibletoestimatethecapture and thesynchronousstateisattitudeunstableforHyperion. been suggested.SoterandHarris(1977)suggestedthe tion wouldhaveoccurredtooclosetothetimeofformation tumbling associatedwiththeentryintosynchronousrota- timescale fortidaldespinningtosynchronousrotationis as thesynchronousrotationstateisentered.ForMiranda While theenhancedtidaldissipationduringaperiodof of aperiodchaotictumbling?Thisappearsunlikely. anda. Isitpossiblethatitsexoticsurfacefeaturesarearesult episode ofchaotictumblingfortheirregularlyshapedsatel- tant initself;theworldworksasurprisingway.Neverthe- irregularly shapednaturalsatellitesisinterestingandimpor- remove itfromthechaoticzone(Wisdometal1984). to theborderforalongenoughtimethattidaltorquecan young. However,subsequentexaminationofthenumber of Phobosdecays.Thiswouldimplythatthegroovesare grooves werecausedbyincreasingtidalstressesastheorbit anda. to accountforthedisparateagesoffeaturesseenonMir- only 300000yr(Peale1977).Thustheepisodeofchaotic this musthaveoccurredsoonafteritsformation,sincethe source, theattitudeinstabilitydescribedaboveoccursonly chaotic tumblingcouldhaveprovidedasignificantheat zone insteadreliesonthetrajectorybeingtemporarilystuck cannot makeanysecularprogress.Capturefromthechaotic the chaoticzoneissoirregularthatweaktidaltorque probability orthetimespentinchaoticzone.Simulations more timeinthechaoticzonethantheywouldifthatpartof for Hyperionindicatedthattrajectoriesoftenspendmuch largest crater,Stickney,theymaybearesultoftheeventthat grooves areold(greaterthan10yr).Thomasetal.(1978) craters superimposedonthegroovesindicatedthat grooves. Anumberofpossiblecausesforthesegrooveshave suggested thatsincethegroovesaremostprominentnear the phasespacehadbeenregular.Qualitatively,motionin stead thattheeventcreatedStickneybrokepre- that sincesomeofthegroovesshowcrosscutting created thatcrater.Weidenshilling(1979)objectsthe ney becausethematerialwasweakened bytheimpact.The of grooves.Inthisscenario,more groovesappearnearStick- along planesofmaximumshearstress,producingthesystem viously establishedsynchronousrotationlock,andthetidal did notformsimultaneously.Weidenshillingproposesin- grooves bearnosimplerelationshiptothecrater,andclaims stresses fromthesubsequent“nutation”causedfaulting whatever eccentricityPhoboshad atthetimeofcaptureinto tumbling isanaturalpartofthe rotationhistoryofPhobos, synchronous rotation.Ifanutation thatwouldresultfrom study reportedinthispaperhas nowestablishedthatchaotic The newunderstandingoftherotationhistories The surfaceofPhobosismarkedbyaserieslinear © American Astronomical Society • Provided by the NASA Astrophysics Data System 9 23 the chaotictumblingthatPhobosunderwentasitentered an impactiscapableofcreatingthegrooves,thencertainly ney musthavealreadybeenpresentwhenthegrooveswere Weidenshilling, isthattheimpactcreatedStickney formed sincetheremustbesomereasonfortheassociationof synchronous rotationcouldhavedonethesame.Still,Stick- the grooveswithcratçr.Anotherpossibility,following drove Phobosintothechaoticzoneandstressesfrom the longaxisfromorbitplaneby20°caninitiatechaotic resulting chaotictumblingcreatedthegrooves.Displacing regular “nutation”insynchronousrotation(however,see tumbling. Thisplacesanupperlimitontheamplitudeofa Sec. IV).TheimpacthypothesisofThomasetal.isstillthe bling? Tidalstressesarisefromthegravitationalpotentialof they haveallexperiencedaperiodofchaotictumbling.Does may begroovesonotherirregularlyshapedsatellitessince tic tumblingasthesynchronousstatewasentered,thenthere simplest explanationforthegroovesonPhobos. chaotic tumbling. to makepredictionsregardinggroovesonothersatellites. the eventthatcreatedStickney.Inthiscaseitisnotpossible neighborhood of30.Perhapsthisdifferenceintidalstress been lessthan2.DeimosisroughlyhalfthesizeofPhobos. that thegroovesonPhoboswerecreatedbychaotictum- the lackofgroovesonDeimosthenruleoutpossibility Perhaps thegroovesaresimplyunrelatedtoepisodeof haps itisnecessarytoappealagainweaknessesinducedby could accountfortheabsenceofgroovesonDeimos.Per- The tidalstresseswouldthenhavedifferedbyafactorinthe between them.ThegroovesonPhobosareatleast10yrold, Mars aswellthecentrifugalpotentialofchaoticrota- tal historiesofPhobosandDeimosplacethesatellitescloser where Ristheradiusofsatelliteandasemimajor ly tumblingbodyisofthesameorderasorbitalmean Martian radii,wheretheratioofsemimajoraxesmusthave most orbitalhistorieswouldthenplacePhobosbeyond4 to oneanotherinthepastsincecorotationradiuslies axis. DeimosissmallerthanPhobosandfartherfromMars. motion, thetidalstressesareallproportionaltoRa~, number andrecallingthattherateofrotationachaotical- tion itself.UsingKelvin’sformulaforthedisplacementLove Currently, theratioofsemimajoraxesis2.5,butallorbi- before theresonanceencounters.Similaranalysisshowsthat irregularly shapedsatellitesconcernstheeccentricityofDei- tricity ofPhobos,evenassumingazeroinitialeccentricity iety ofresonancescouldcompletelyaccountfortheeccen- mos. Yoder(1982)hasshownthatpassagethroughavar- rotation islockedinsynchronous(Ureyetal.1959; in asatellitetendstodecreasetheorbitaleccentricityif ity ofDeimosislessthan0.0005.Dissipationenergywith- passage througha2/1meanmotionresonancewithPhobos Deimos shouldhaveaneccentricityoforder0.002,from larger tidaldissipationinDeimos thaninPhobos.Thesmall fore resonanceencounter,or there hasbeensignificantly cularized byanexceptionalimpact whichoccurredjustbe- solar system.Yoderconcludedthateithertheorbitwascir- eccentricity ofDeimosismuchgreaterthantheage on theorderoftwobillionyearsago.Instead,eccentric- an episodeofchaotictumbling. orbital eccentricityofDeimoscould alsobeaconsequenceof Goldreich 1963),butthetimescalefordampingof If thegroovesonPhoboswerearesultofperiodchao- Another possibleconsequenceofthechaotictumbling Evaluating expression(19)for Deimosusinganeccen- 1359 1987AJ 94.1350W tricity of0.002givesatimescale300millionyears.Now that theresonancewasencounteredabouttwobillionyears damp outaninitialeccentricityof0.002.Yoderestimated Deimos couldspendenoughtimeinthechaoticzoneto order 100millionyears(Peale1977).Itisthuspossiblethat the timescaleforDeimostoreachsynchronousrotationisof fect onitsorbitalevolution.Itcannotbeusedtosolvethe Apparently, thedissipationinHyperionhasanegligibleef- bling. TheevolutionoftheeccentricityDeimosdeserves ble. Perhapstheresonancepassageoccurredmuchearlier chaotic aftertheresonancepassage,butitiscertainlypossi- difficult tosaywhethertherotationislikelyhavebeen ago, butthereisconsiderableuncertaintyinthisestimate 1360 JACKWISDOM:IRREGULARLYSHAPEDSATELLITES Binzel, R.P.,Green,J.R.,andOpal,C.B.(1986).Nature320,511. Hyperion mean-motioncommensurability. timescale problemfortheestablishmentof4/3Titan- a timescalemuchgreaterthantheageofsolarsystem. when Deimoswasmorelikelytostillbechaoticallytum- Mignard, F.(1981).Mon.Not.R.Astron.Soc.194,365. Hénon, M.,andHeiles,C.(1964).Astron.J.69,73. Goldreich, P.,andPeale,S.J.(1970).Annu.Rev.Astron.Astrophys.6, Goldreich, P.,andPeale,S.J.(1966).Astron.71,425. Goldreich, P.(1963).Mon.Not.R.Astron.Soc.126,257. Duxbury, T.C,andCallahan,J.D.(1982).LunarPlanet.Abst.13,190. Darwin, G.(1879).Philos.Trans.PartI. Burns, J.A.,andSafronov,V.S.(1973).Mon.Not.R.Astron.Soc.165, further attention. Lambeck, K.(1979).J.Geophys.Res.84,5651. Goguen, J.(1983).PaperpresentedatIAUColloquiumNo.77,Natural Cazenave, A.,Dobrovolskis,andLago,B.(1981).Icarus44,730. This istobecomparedwiththetimescalereachsynchro- about fivemillionyears,usinganinitialeccentricityof0.02. Chirikov, B.V.(1979).Phys.Rep.52,263. nous rotationofafewmillionyears.Whetherornotthe the observedloweccentricityofDeimosafterpassage dissipation resultingfromthechaotictumblingcanexplain larization oftheorbitsbothPhobosandDeimos. through resonance,itmayhelpaccountfortheinitialcircu- (personal communication,Yoder1986).Consequently,itis 287. 403. Satellites, Ithaca,NY,July5-9. 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