Constraints to Uranus' Great Collision. IV. The Origin of Prospero

0.19 MB·English

by  Gabriela Parisi

Astronomy&Astrophysicsmanuscriptno.gparisi c ESO2008
(cid:13)
February2,2008
Constraints to Uranus’ Great Collision IV
The Origin of Prospero
M.GabrielaParisi1,2,3,⋆,GiovanniCarraro4,5,MicheleMaris6,andAdrianBrunini3
8
1 InstitutoArgentinodeRadioastronom´ıa(IAR),C.C.No5,1894VillaElisa,Argentina
0
e-mail:[email protected]
0
2 DepartamentodeAstronom´ıa,UniversidaddeChile,Casilla36-D,Santiago,Chile
2
e-mail:[email protected]

n 3 FacultaddeCienciasAstrono´micasyGeof´ısicas,UniversidadNacionaldeLaPlata,Argentina
a e-mail:gparisi,[email protected]
J 4 EuropeanSouthernObservatory(ESO),AlonsodeCordova3107,Vitacura,Santiago,Chile
0 e-mail:[email protected]
1 5 DipartamentodiAtronomia,Universita´diPadova,VicoloOsservatorio2,I-35122Padova,Italy
e-mail:[email protected]
] 6 INAF,OsservatorioAstronomicodiTrieste,ViaG.B.Tiepolo11,I-34131Trieste,Italy
h e-mail:[email protected]
p
- ReceivedJune2007;acceptedJanuary2008
o
r ABSTRACT
t
s
a Context. ItiswidelyacceptedthatthelargeobliquityofUranusistheresultofagreattangentialcollision(GC)withanEarthsize
[ proto-planet at the end of the accretion process. The impulse imparted by the GC had affected the Uranian satellitesystem. Very

recently,nineirregularsatellites(irregulars)havebeendiscoveredaroundUranus.Theirorbitalandphysicalproperties,inparticular
2
thoseoftheirregularProspero,setconstraintsontheGCscenario.
v
Aims. WeattempttosetconstraintsontheGCscenarioasthecauseofUranus’obliquityaswellasonthemechanismsabletogive
8
origintotheUranianirregulars.
5
Methods. Differentcapturemechanismsforirregularsoperateatdifferentstagesonthegiantplanetsformationprocess.Themech-
2
anismsabletocapturetheuranianirregularsbeforeandaftertheGCareanalysed.AssumingthattheywerecapturedbeforetheGC,
1
wecalculatetheorbitaltransferofthenineirregularsbytheimpulseimpartedbytheGC.Iftheirorbitaltransferresultsdynamically
.
1 implausible,theyshouldhaveoriginatedaftertheGC.Wetheninvestigateanddiscussthedissipativemechanisms abletooperate
0 later.
8 Results. Veryfewtransfersexistforfiveoftheirregulars,whichmakestheirexistencebeforetheGChardlyexpected.Inparticular
0 ProsperocouldnotexistatthetimeoftheGC.DifferentcapturemechanismsforProsperoaftertheGCareinvestigated.Gasdrag
: by Uranus’envelope and pull-down capture are not plausible mechanisms. Capture of Prospero through a collisionless interaction
v
seemstobedifficult.TheGCitselfprovidesamechanismofpermanentcapture.However,thecaptureofProsperobytheGCisalow
i
X probableevent.CatastrophiccollisionscouldbeapossiblemechanismforthebirthofProsperoandtheotherirregularsaftertheGC.
Orbitalandphysicalclusteringsshouldthenbeexpected.
r
a Conclusions. Either Prospero had to originate after the GC or the GC did not occur. In the former case, the mechanism for the
originofProsperoaftertheGCremainsanopenquestion.Anobservingprogramabletolookfordynamicalandphysicalfamiliesis
mandatory.Inthelattercase,anothertheorytoaccountforUranus’obliquityandtheformationoftheUranianregularsatelliteson
theequatorialplaneoftheplanetwouldbeneeded.
Keywords.Planetsandsatellites:general–Planetsandsatellites:formation–SolarSystem:general–SolarSystem:formation
1. Introduction
Very recently, rich systems of irregularsatellites (hereafter irregulars)of the giant planets have been discovered.Enabled by the
use of large-format digital images on ground-based telescopes, new observational data have increased the known population of
Jovianirregularsto55(Sheppardetal.2003),theSaturnianpopulationto38(Gladmanetal.2001,Sheppardetal.2005a,2006a)
andtheNeptunianpopulationto7(Holmanetal.2004,Sheppardetal.2006b).TheUraniansystemisofparticularinterestsincea
populationof9irregulars(namedCaliban,Sycorax,Prospero,Setebos,Stephano,Trinculo,S/2001U2:XXIVFerdinand,S/2001U3:
XXIIFranciscoandS/2003U3:XXIIIMargaret)hasbeendiscoveredaroundUranus(Gladmanetal.1998,2000,Kavelaarsetal.
2004,Sheppardetal.2005b).ThediscoveryoftheseobjectsprovidesauniquewindowonprocessesoperatingintheyoungSolar
System.IntheparticularcaseofUranus,theirexistencemaycastlightonthemechanismresponsibleforitspeculiarrotationaxis
(Parisi&Brunini1997,Bruninietal.2002(hereafterBP02)).
Sendoffprintrequeststo:M.GabrielaParisi
⋆ MemberoftheCarreradelInvestigadorCient´ıfico,ConsejoNacionaldeInvestigacionesCient´ıficasyTe´cnicas(CONICET),Argentina
2 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV
Irregularsofgiantplanetsarecharacterizedbyeccentric,highlytiltedwithrespectoftheparentplanetequatorialplane,andin
somecaseretrograde,orbits.Theseobjectscannothaveformedbycircumplanetaryaccretionastheregularsatellitesbuttheyare
likelyproductsofanearlycaptureofprimordialobjectsfromheliocentricorbits,probablyinassociationwithplanetformationitself
(Jewitt&Sheppard2005).Itispossibleforanobjectcirclingaboutthesuntobetemporarilytrappedbyaplanet.Intermsofthe
classicalthree-bodyproblemthistypeofcapturecanoccurwhentheobjectpassesthroughtheinteriorLagrangianpoint,L ,witha
2
verylowrelativevelocity.But,withoutanyothermechanism,suchacaptureisnotpermanentandtheobjectswilleventuallyreturn
toasolarorbitafterseveralorseveralhundredorbitalperiods.Toturnatemporarycaptureintoapermanentonerequiresasource
oforbitalenergydissipationandthatparticlescouldremaininsidetheHillspherelongenoughforthecapturetobeeffective.
Althoughcurrentlygiant planetshave no efficient mechanismof energydissipation for permanentcapture,at their formation
epochseveralmechanismsmayhaveoperated:1)gasdraginthesolarnebulaorinanextended,primordialplanetaryatmosphereor
inacircumplanetarydisk(Pollacketal.1979,Cuk&Burns2003),2)pull-downcapturecausedbythemassgrowthand/ororbital
expansionof the planetwhich expandsits Hill sphere(Brunini1995, Heppenheimer& Porco1977), 3) collisionlessinteractions
betweenamassiveplanetarysatelliteandguestbodies(Tsui1999)orbetweentheplanetandabinaryobject(Agnor&Hamilton
2006), and 4) collisional interaction between two planetesimals passing near the planet or between a planetesimal and a regular
satellite. This last mechanism, the so called break-up process, leads to the formation of dynamical groupings (e.g. Colombo &
Franklin 1971, Nesvorny et al. 2004). After a break-up the resulting fragments of each progenitor would form a population of
irregularswithsimilarsurfacecomposition,i.e.similarcolors,andirregularshapes,i.e.light-curvesofwideamplitude.Significant
fluctuationsinthelight-curvesofCaliban(Marisetal.2001)andProspero(Marisetal.2007a)andthetimedependenceobserved
inthespectrumofSycorax(Romonetal.2001)suggesttheideaofabreak-upprocessfortheoriginofthesebodies.
SeveraltheoriestoaccountforthelargeobliquityofUranushavebeenproposed.Kubo-Oka&Nakazawa(1995),investigated
the tidalevolutionof satellite orbitsandexaminedthe possibility thatthe orbitaldecayof a retrogradesatellite leads to the large
obliquityofUranus,butthelargemassrequiredforthehypotheticalsatellitemakesthispossibilityveryimplausible.Anasymmetric
infallortorquesfromnearbymassconcentrationsduringthecollapseofthemolecularcloudcoreleadingtotheformationofthe
SolarSystem,couldtwistthetotalangularmomentumvectoroftheplanetarysystem.Thistwistcouldgeneratetheobliquitiesof
theouterplanets(Tremaine1991).Thismodelhasthedisadvantagesthattheouterplanetsmustformbeforetheinfalliscomplete
andthattheconditionsfortheeventthatwouldproducethetwistareratherstrict.Themodelitselfisdifficulttobequantitatively
tested.Tsiganisetal.(2005)proposedthatthecurrentorbitalarchitectureoftheouterSolarSystemcouldhavebeenproducedfrom
aninitiallycompactconfigurationwithJupiterandSaturncrossingthe2:1orbitalresonancebydivergentmigration.Thecrossing
led to close encountersamong the giant planets, producinglarge orbital eccentricities and inclinations which were subsequently
dampedtothecurrentvaluebygravitationalinteractionswithplanetesimals.Theobliquitychangesduetothechangeintheorbital
inclinations. Since the inclinations are damped by planetesimals interactions on timescales much shorter than the timescales for
precessionduetothetorquesfromthe Sun,especiallyforUranusandNeptune,the obliquityreturnsto smallvaluesif itissmall
beforetheencounters(Hoietal.2007).
Large stochastic impacts at the last stage of the planetaryformationprocesshave been proposedas the possible cause of the
planetaryobliquities(e.g.Safronov1969). Thelargeobliquityof Uranus(98 ) isusuallyattributedto a greattangentialcollision
◦
(GC)betweentheplanetandanEarth-sizeplanetesimaloccurredattheendoftheepochofaccretion(e.g.,Parisi&Brunini1997,
Korycanskyetal.1990).ThecollisionimpartsanimpulsetoUranusandallowspreexistingsatellitesoftheplanettochangetheir
orbits. Irregularson orbits with too large semimajor axis escape from the system (Parisi & Brunini 1997), while irregulars with
a smallersemimajoraxismaybepushedto outerorinnerorbitsacquiringgreaterorlowereccentricitiesdependingonthe initial
orbitalelements,thegeometryoftheimpactandthesatellitepositionatthemomentofimpact.Theorbitsexcitedbythisperturbation
mustbeconsistentwiththepresentorbitalconfigurationoftheUranianirregulars(BP02).
InanattempttoclarifytheoriginofUranusobliquityandofitsirregulars,weareusinginthisstudythemostupdatedinformation
ontheirorbitalandphysicalproperties.
InSection2,weimprovethemodeldevelopedinBP02forthefiveUranianirregularsknownatthatepochandextendourstudy
tothenewfourUranianirregularsrecentlydiscoveredbyKavelaarsetal.(2004)andSheppardetal.(2005b).Theoriginofthese
objectsaftertheGCisdiscussedinSection3,whereseveralmechanismsfortheoriginofProsperoareinvestigated.Thediscussion
oftheresultsandtheconclusionsarepresentedinSection4.
2. Transferoftheirregularstotheircurrentorbits:
AssumingtheGCscenario,thetransfersofthenineknownirregularstotheircurrentorbitsarecomputedfollowingtheprocedure
developedinBP02forthefiveirregularsknownin2002.Wepresentimprovedcalculationsusingamorerealisticcodetocompute
theevolutionoftheirregularscurrentorbitaleccentricities.
IfthelargeobliquityofUranushasbeentheresultofagianttangentialimpact,theorbitsofpreexistingsatelliteschangeddue
totheimpulseimpartedtotheplanetbythecollision.TheangularmomentumandimpulsetransfertotheUraniansystematimpact
weremodeledusingtheUranuspresentdayrotationalandorbitalpropertiesasimputparameters(BP02).
JustbeforetheGC,thesquareoftheorbitalvelocityν ofapreexistingsatelliteofnegligiblemassisgivenby:
1
2 1
ν2 =Gm , (1)
1 U r − a !
1
M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV 3
rbeingthepositionofthesatelliteonitsorbitatthemomentoftheGC,a itsorbitalsemiaxisandm themassofUranusbefore
1 U
the impact.The impactormassis m andG is the gravitationalconstant. Afterthe GC, the satellite is transferredto anotherorbit
i
withsemiaxisa acquiringthefollowingsquareofthevelocity:
2
2 1
ν2 =G(m +m) . (2)
2 U i r − a !
2
Wesetν2 = A ν2 andν2 = B (1+m/m ) ν2,whereAandBarearbitrarycoefficients(0 < A 1, B > 0),ν beingtheescape
1 e 2 i U e ≤ e
velocityatrbeforetheGC.
Thesemiaxisofthesatelliteorbitbefore(a )andafter(a )theGCverifythefollowingsimplerelations:
1 2
r r
a = , a = . (3)
1 2
2(1 A) 2(1 B)
− −
If A < Bthena < a .Inthespecialcaseof B = 1,theorbitsareunboundfromthesystem.If A > Bthena > a ,theinitial
1 2 1 2
orbitistransferredtoaninnerorbit.WhenA= B,theorbitalsemiaxisremainsunchanged(a =a ).
1 2
Thepositionrofthesatelliteonitsorbitattheepochoftheimpactmaybeexpressedinthefollowingform:
2G m B A
r= U ′− 2, (4)
(∆V)2 "√AcosΨ √(B A)+Acos2Ψ#
± ′−
withB = B(1+m/m ).Sincestochasticprocessescanonlytakeplaceatverylatestagesinthehistoryofplanetaryaccretion(e.g.
′ i U
Lissauer&Safronov1991),theGCisassumedtooccurattheendofUranusformation(e.g.Korycanskyetal.1990).Themassof
UranusaftertheGC,(m +m ),istakenasUranus’presentmass.Ψistheanglebetweenν andtheorbitalvelocitychangeimparted
i U 1
toUranus∆V.Ananalyticalexpressionfor∆VisderivedinBP02assumingthattheimpactisinelastic(Korycanskyetal.1990)as
afunctionofm,theimpactparameterofthecollisionb,thepresentrotationangularvelocityofUranusΩ,thespinangularvelocity
i
whichUranuswouldhavetodayifthecollisionhadnotoccurredΩ ,andαwhichistheanglebetweenΩandΩ :
0 0
2R2 Ω2 2ΩΩ cosα 1/2
∆V = U Ω2+ 0 0 , (5)
5b " (1+ mi)2(1+ mi )4 − (1+ mi)(1+ mi )2#
mU 3mU mU 3mU
R being the present equatorial radius of Uranus. A collision with the core itself was necessary in order to impart the required
U
additionalmassandangularmomentum(Korycanskyetal.1990).Sincebisanunknownquantity,wetakeitsmostprobablevalue:
b=(2/3)R ,whereR isthecoreradiusofUranusatthemomentofcollisionassumedtobe1.8 104km(Korycanskyetal.1990,
C C
×
Bodenheimer&Pollack1986).Theresultshavea smoothdependencewith theimpactormasswhichallowsustotakem 1m
i
(Parisi&Brunini1997). ∼ ⊕
Theminimumeccentricityoftheorbitsbeforethecollisionisgivenby:
e =2(1 A) 1 if A 0.5 , e =1 2(1 A) if A>0.5, (6)
1min 1min
− − ≤ − −
whiletheminimumeccentricityoftheorbitsafterthecollisionis:
e =2(1 B) 1 if B 0.5 , e =1 2(1 B) if B>0.5. (7)
2min 2min
− − ≤ − −
Theminimumpossiblevalueof∆V (∆V )isobtainedfromEq.(5)foraninitialperiodT =20hrs(T =2π/Ω )andα=70o
min 0 0 0
(BP02).Therefore,althoughsimulationsofsolidaccretionproduceingeneralrandomspinorientations(e.g.Chambers2001),we
further assume T = 20 hrs and α=70o in order to set a maximum boundon Eq. (4). Upper boundsin a (a ) and a (a ) are
0 1 1M 2 2M
obtainedfromEq.(3)throughEqs.(4)and (5)taking∆V=∆V withΨ=180o,i.e.,assumingtheimpactinthedirectionopposite
min
totheorbitalmotionofthesatellitesandtakingthepositivesignofthesquarerootinEq.(4):
G m (B A)2 G m (B A)2
a = U ′− , a = U ′− . (8)
1M 2M
(∆V )2(1 A)(√B √A)2 (∆V )2(1 B)(√B √A)2
min − ′− min − ′−
ForeachA,wecalculatethevalueofB(B= B/(1+m/m ))correspondingto thetransfertoa = a,whereaisthepresent
′ i U 2M
orbitalsemiaxis of each one of the Uranian irregularsshown in the second columnof Table 1 in units of R . From Eq. (7), this
U
valueofBprovidestheminimumpossiblevalueofe ,e ,thattheorbitofeachirregularmayacquireatimpactforeachinitial
2min 2m
conditionAandeveryinitialconditionforT ,α,Ψandm,i.e.,ifatransferofagivenorbit(A,B)isnotpossibleforΨ=180o,T =
0 i 0
20hrsandα= 70o,thesametransfer(A,B)isnotpossibleforanyotherincidentdirectionoftheimpactorandforanyothervalue
ofT andαeither.
0
Since the orbits of the irregulars are time dependent, the orbital evolution of the five Uranian irregulars known in 2002 was
computed in PB02 by numerical integration of the equations of the elliptical restricted three body problem formed by the Sun,
Uranusandthesatellite.Inthispaper,wepresenttheorbitalevolutionofthenineknownUranianirregularsfor105 yrsusingthe
4 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV
Satellite r [km] a[R ] e a[R ] e (∆a)/a) (∆e/e)
s U mean i U i i i
Caliban 49 283 0.191 287.5 0.1973 1.602x10 2 3.289x10 2
− −
Sycorax 95 482 0.541 485 0.5436 6.307x10 3 4.795x10 3
− −
Prospero 15 645 0.432 648 0.4342 4.585x10 3 4.997x10 3
− −
Setebos 15 694 0.581 701.8 0.5853 1.123x10 2 7.366x10 3
− −
Stephano 10 314 0.251 333 0.2781 6.058x10 2 0.1080
−
Trinculo 5 336 0.218 361.6 0.2501 7.623x10 2 0.1475
−
Ferdinand 6 813 0.660 839.3 0.6701 3.241x10 2 1.533x10 2
− −
Francisco 6 169 0.142 280.6 0.3593 0.6604 1.530
Margaret 5.5 579 0.633 649 0.6705 0.1209 5.917x10 2
−
Table 1. PresentparametersoftheUranianirregularsandorbitaldampingdueto gasdragexertedbyUranusextendedenvelope.
r andaarethepresentphysicalradiusandthepresentorbitalsemiaxisoftheirregulars.e istheircalculatedmeaneccentricity
s mean
tabulatedinTable2.a ande aretheorbitalsemiaxisandeccentricityjustaftertheGC,while(∆a)/a)and(∆e/e)arethedamping
i i i i
oftheseorbitalelementssincetheepochoftheGCuntilthecontractionofUranusenvelope.
Satellite e e e
mean max min
Caliban 0.191 0.315 0.072
Sycorax 0.514 0.594 0.438
Prospero 0.432 0.571 0.305
Setebos 0.581 0.704 0.463
Stephano 0.251 0.381 0.121
Trinculo 0.218 0.237 0.200
Ferdinand 0.660 0.970 0.393
Francisco 0.142 0.187 0.093
Margaret 0.633 0.854 0.430
Table2.VariationoftheeccentricityoftheUranianirregularsduetoSolarandgiantplanetperturbationsoveraperiodof105yrs.
symplecticintegratorofWisdom&Holman(1991),wheretheperturbationsoftheSun,Jupiter,SaturnandNeptuneareincluded.
Themean(e ),maximum(e )andminimum(e )eccentricitiesareshowninTable2foralltheknownUranianirregulars.
mean max min
Thetransferofeachsatellite fromitsoriginalorbittothepresentoneispossibleonlyforthosevaluesof(A,B)whichsatisfy
theconditione <e .Asatellite didnotexistbeforetheimpactifithasnotransferandthesatelliteswiththewidestrangeof
2m max
transfersarethosewiththehighestprobabilityofexistingbeforetheimpact.
Thetransferswithinarangeof20R aroundeachpresentsatellitesemiaxis(a =a 20R ;atakenfromTable1)forallthe
U 2M U
±
Uranianirregularsare shown in Fig.1. Thereare few transfersforSetebos, Ferdinandand Margaret.Thismakesthe existenceof
thesesatellitesbeforecollisionlittleprobable.TheonlytransfersforTrinculoandProsperoareclosetothepericenterofaneccentric
initialouterorbit(e >0.58forProsperoande >0.62forTrinculo).Theminimumeccentricityaftercollisione forTrinculois
1m 1m 2m
intherange[0.16-0.23],veryclosetoe (0.237).ThisresultgivesaverylowprobabilityfortheexistenceofTrinculobeforethe
max
GC.ForProsperoe isintherange[0.52-0.57],e e (0.571).Thereforethissatellite couldnotexistbeforetheGC.Ifthe
2m 2m max
∼
presentlargeobliquityofUranuswascausedbyalargeimpactattheendofitsformation,Prosperohadtooriginateaftertheevent.
RelatingtheoriginoftheouterUraniansystemtoacommonformationprocess,alltheUranianirregularsprobablywereoriginated
aftertheGC.Thepossiblepost-GCoriginofProsperoandtheotherUranianirregularsisdiscussedinthefollowingsection.
3. OriginofProsperoaftertheGreatCollision
In this section, we analyze the possibility that Prospero have been capturedafter the GC. We investigatethe possible dissipative
mechanisms able to produce its permanentcapture taking into account that the giant impact is assumed to have occurred at late
stagesintheplanetaryaccretionprocess.
3.1.GasdragbyUranus’envelope
Bodenheimer&Pollack(1986)andPollacketal.(1996)studiedtheformationofthegiantplanetsbyaccretionofsolidsandgas.
Intheirmodel,thesocalledcoreinstabilityscenario,whenthemassofthecoreoftheplanethasgrownenoughagaseousenvelope
beginsto formaroundit. For Uranus, its envelopeextendeduntilits accretionradiuswhich was 500R at the end of Uranus’
U
∼
formation (Bodenheimer& Pollack 1986). The formationof Uranus is completed when there is no more nebular gas to accrete.
Otherwise, gas accretion by proto-Uranuswould have continued towards the runaway gas accretion phase and the planet would
have now a massive gaseousenvelope.Bodenheimer& Pollack (1986) obtained that after the end of accretion, the radius of the
envelopeofproto-Uranusremainedalmostconstant( 500R )overatimescaleof104 yrsandthencontractedrapidlyto 8R
U U
in105yrs.Thefinalcontractiontothepresent-daypla∼netaryradiusoccurredonaslowertimescaleof108yrs. ∼
Korycansky et al. (1990) carried out hydrodynamical calculations of the GC for a large set of initial conditions at the end
of accretion. They found a sharp transition between the cases where almost all the mass of the envelope of Uranus remained
after the impact and those where it was almost entirely dispersed by the impact. This implies that the impact should have not
M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV 5
Fig.1.ThetransferscapableofproducingthepresentorbitsoftheUranianIrregulars.A(B)isthesquareoftheratioofthesatellite’s
speed just before(after)the impactto the escape velocityat the satellite’s locationjust before(after)the impact.e (e ) is the
1m 2m
minimum eccentricity of the orbits before (after) collision. The full-black line A=B divides the upper region (the current orbits
arise frominnerorbits)andthe lowerregion(thecurrentorbitsarise fromouterorbits).Thevalueofe tabulatedin Table2 is
max
shownonthedashedlineforcomparationwithe .Trinculo:emptydowntriangles,Caliban:fullcircles,Sycorax:emptyrhombus,
2m
Ferdinand:emptysquares,Francisco:fulldowntriangles,Margaret:emptyuptriangles,Prospero:fulluptriangles,Setebos:empty
hexagons,Stephano:emptycircles.
dispersedtheenvelope,astherewouldhavebeennonebulargastore-accreteontheplanet.Theyshowedthattheenvelopereacts
hydrodynamicallyatimpactanditexpandsoutward.Aftertheshockthegasfalls backonthe coreovera timescale offewhours
beingthefinalresultareadjustmentinsteadofacatastrophictransformation.Thetimescaleforthishydrodynamicalprocessismuch
shorterthantheorbitalperiodoftheirregularswhichisoftheorderofyears.WemaythenassumethattheGCdidnotchangethe
envelopedensityprofile.
6 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV
TheextendedenvelopeofUranuscouldbeinprinciple,asourceofgasallowingthecaptureofProsperoandtheotherirregulars
aftertheGC.AssumingthattheGCdidnotchangetheenvelopeprofile,wefitfromFig.1ofKorycanskyetal.(1990)thedensity
profile of Uranus’ gaseous envelope before the GC, ρ = c R 4 g cm 3 with c = 1036 and R being measured in cm. It gives a
g te − − te
nebulardensityof 4 10 13gcm 3attheboundaryof500R inagreementwiththeminimummassnebulamodel.
− − U
∼ ×
As a firstapproximation,we computetheratio ofgasmasstraversedby a bodyofdensityρ andradiusr in a characteristic
s s
orbitalperiodP,tothemassofthebody.AssumingforthebodyacircularorbitofradiusR,wecalculatetheso-calledβparameter
(Pollacketal.1979):
P 2πRρ πr2 3πρ R
β= = g s = g , (9)
τ 43πρsr3s 2ρsrs
whereτisthecharacteristictimescaleforchanginganyoftheorbitalparameters.Forthepermanentcapturetooccurβcannotbe
verysmall,β 0.04(Pollacketal.1979).UsingEq.(9)andassuminganebuladensity10timesthatoftheminimumnebulamodel
(c =1037)for≥anobjectthesizeofProspero(ρ =1.5gcm 3)andatProspero’spericenter(R=278R ),β 9 10 5,whichistoo
te s − U −
∼ ×
smalltoaffecttheorbitofProspero.
FollowingBP02,wenowinvestigatewithmoredetailthepossibleeffectofgasdragontheUranianirregularsaftertheGCdue
toUranus’extendedenvelopebeforeitscontractiontoitspresentstate.FollowingtheprocedureofAdachietal.(1976),weobtain
thetimevariationsoftheeccentricityeandsemiaxisaofeachUranianirregular.Thedragforceperunitmassisexpressedinthe
form:
C πr2
F = Cρ v2 ,C = D s, (10)
− g rel 2m
wherev istherelativevelocityofthesatellitewithrespecttothegas.Incomputingthesatellitemassm,asatellitemeandensity
rel
ρ of1.5gcm 3 istakenforallthesatellites(http://ssd.jpl.nasa.gov/?sat phys par).ThedragcoefficientC is 1andr iseach
s − D s
∼
satelliteradiustakenforeachsatellitefromTable1.
Assuming that the orbital elements are constant within one Keplerian period (the variations of a and e are very small), we
considertheratesofchangeoftheelementsaveragedoveroneperiod,thatis:
hddati=−Cπ(G(mi+mU)a)21 Z 2π ρg(e2(1++1e+co2seθc)o2sθ)32dθ
0
de = C(1 e2) G(mi+mU) 21 2π ρg(e+cosθ)(e2+1+2ecosθ)12dθ. (11)
hdti −π − a ! Z (1+ecosθ)2
0
Sinceaftertheendofaccretionthegasdensityintheouterregionsoftheenvelopecontractsrapidly,wehaveintegratedEqs.(11)
back in time on 104 yrsfor the 9 Uranianirregulars.We have taken v as the satellite orbitalvelocity since we have assumed a
rel
nullgasvelocity.Thisassumptionmaximizestheorbitaldampingforretrogradesatelliteswhichallowsustosetupperboundsin
the damping effect for the orbital eccentricity and semiaxis of all the retrograde irregulars. The mean eccentricity e and the
mean
actual semiaxis a from Table 1 were taken as the initial conditions for the integrations. We take ρ = c R 4 g cm 3 with c =
g te − − te
1036 and R = a(1 e2)/(1+ecosθ), a beingmeasured in cm. The orbitaldampingis shown in Table 1, where a and e are the
i i
−
initialsemiaxisandeccentricityjustaftertheGCattheendofaccretionand∆a=(a a)and∆e=(e e ),arethedampingin
i i mean
− −
theorbitalsemiaxisandeccentricity.Stephano,TrinculoandMargarethadexperiencedlittleorbitalevolutionwhileFranciscohad
sufferedalargeorbitaldamping(seeTable1).ThepermittedtransfersofFig.1wouldincreaseforFranciscosincetheconditione
2m
<e shouldbethensatisfied.However,theorbitalevolutionofthelargersatellites,inparticularofProspero,resultsnegligible.Even
i
increasingthenebuladensitybyafactorof10(c =1037)thedampingoftheorbitalelementsofProsperoistoosmallwith∆a/a=
te i
0.046and∆e/e=0.048.ThissatellitehadnotexperiencedanyorbitalevolutionduetoUranus’extendedenvelopeandthencould
i
nothavebeencapturebygasdragaftertheGC.
The pressure forces acting on a body traveling through the gas not only decelerates it, but also subjects it to stresses. If the
stress isgreaterthan thestrengthof the body,the bodyisfracturedin fragmentsofdifferentsize. Thefragmentsmoveaway one
anothersincedragforcesvaryinverselywith sizeandactto separatethem.Theaveragepressureontheforwardhemisphereofa
non-rotating,sphericalbodyasitmovesthroughthegaswithrelativevelocityv isapproximatelyequaltothedynamicpressure,
rel
p = 1ρ v2 (Pollacketal.1979). Thebodywillfragmentinto piecesif p Q, where Q is thecompressivestrength.Values
dyn 2 g rel dyn ≥
ofQontheorderof3 106 dynecm 2 areneededtoshatterstrong(e.g.rock/ice)targets(whichis10timeslowerthanthevalue
−
adoptedforasteroids),×whilecompressivestrengthontheorderof3 104dynecm 2areappropriateforrelativelyweak(snow-like)
−
×
targets(Farinella&Davis1996,Stern1996).ForabodyoncircularorbitatthepresentpericenterofProspero(R=278R ),where
U
v isthecircularspeedaroundUranusatRandtakingc =1037forρ ,p 0.17dynecm 2,andatSycorax’spericenterp 1
rel te g dyn − dyn
∼ ∼
dynecm 2.Inbothcases p <<QandProsperocouldnothavebeenoriginatedbythedynamicalruptureofaparentobject.
− dyn
Acollisionmayfracturetheparentbodybutiftheenergyatimpactisnotsufficienttodispersethefragments,dragforcesmay
acttoseparatethemagainsttheirmutualattraction.Therelativeimportanceoftheseeffectsismeasuredbytheratioǫ ofthedrag
j
forceonagivenfragmentjtothegravitationalforceactingonjbytheotherfragmentsi(Pollacketal.1979):
F m
Dj j
ǫ = , (12)
j
F m
Xi,i,j Gij j
M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV 7
whereF isthegravitationalforcebetweentheparticlesiandj,andF isthedragforceontheparticlejofradiusr :
Gij Dj j
C ρ πr2v2
F = D g j , (13)
Dj 8πρ r3
3 s j
Thefragmentjisdispersedbythegasifǫ >1(Pollacketal.1979).Inordertoestimatetheorderofmagnitudeoftheeffect,we
j
computedEq.(12)forj=ProsperoatProspero’spericentertakingintoaccountthegravitationalattractionduetoanotherfragmentof
equalsize,andatSycoraxpericentertakingintoaccountthegravitationalforceofSycoraxonProspero.Inthefirstcase,weobtain
ǫ =10 5andinthesecondcaseatSycoraxpericenterǫ =0.04.Wethenconcludethatpressureforcesarenotstrongenoughneither
j − j
tofragmentapossibleparentobjectfromwhichProsperooriginated,nortodisperseitsfragments.InadditionProsperosufferedno
orbitalevolutionduetogasdragandcouldnothavebeencapturebyUranus’envelopeaftertheGC.
3.2.Pull-downcapture
Within the GC scenario, runaway of the cores of the planets occurred during the first stages of accretion but stopped for each
embryoafteritreachedasizeofabout1000Km.At10-35AUthefinalmassdistributioncontainedseveralhundredsofMars-size
(or larger) bodies dominating the mass of the residual disk. Beauge´ et al. (2002), investigated the effects of the post-formation
planetarymigrationonsatellitesorbits.Theyobtainedthatifthelarge-bodycomponent(composedofMars-sizebodies)dominated
themassoftheresidualdisk,thepresentlyacceptedchangeintheorbitofUranusof 3AU istoolargeanditisnotcompatible
∼
withtheobserveddistributionofitssatellites.Evenanorbitalchangeof 1.5AUalreadycausessufficientinstabilitiestoejectall
∼
theUranianirregulars.Pull-downcapturecausedbytheorbitalexpansionoftheplanetcouldthennotbeaplausiblemechanismfor
theoriginofProsperoandtheotherirregulars.Pull-downcapturecausedbythemassgrowthoftheplanetaftertheGCwouldnot
bepossiblegiventheimpactisassumedtohaveoccurredattheendoftheaccretionprocesswhentherewas nomoremasstobe
accretedbytheplanet.
3.3.Collisionlessinteractions
Withintheframeworkoftherestrictedthree-bodyproblem,acaptureisalwaysfollowedbyanescape.Toendupwithalongterm
capture,thesatellite hastodissipate energyinashorttime.Theentranceenergy∆E withinthe gravitationalfield oftheplanetis
(Tsui1999):
GM
∆E = 2.15µ2/3(1 δ) ⊙, (14)
− − a
p
µ= M /M andδ<<1,whereM anda arethemassandorbitalsemiaxisoftheplanet.
p p p
Tsui (⊙1999), suggested a permanentcapture mechanism where a guest satellite encounterssome existing inner orbit massive
planetary satellite causing its velocity vector to be deflected keeping the irregular in orbit around the planet. In this way, the
effectivetwo-bodypotentialwouldbe abouttwice the entranceenergy∆E ofthe guestsatellite. TheradiusR ofthe orbitofthe
1
guestsatelliteafterdeflectionisthengivenby:
0.23
R = µ1/3a . (15)
1 1 δ p
−
In the case of Uranus,for a minimumentranceenergyof δ=0,the minimumpermanentorbitalradiusof the guestsatellite is
R =955R .ThisvalueofR ismuchlargerthanthepresentsemiaxisofProspero(seeTable1),makingthecaptureofProsperoby
1 U 1
thismechanismimplausible.
Thefactthatbinarieshaverecentlybeendiscoveredinnearlyallthesolarsystem’ssmall-bodyreservoirssuggeststhatbinary-
planetgravitationalencounterscouldbring a possible mechanismfor irregularscapture (Agnor& Hamilton 2006). One possible
outcomeofgravitationalencountersbetweenabinarysystemandaplanetisanexchangereaction,whereonememberofthebinary
isexpelledandtheotherremainsboundtotheplanet.Tsui(1999)extendedthescenariooflargeanglesatellite-satellitescatteringto
theformationofthePluto-CharonpairassumingthatPlutowasasatelliteofNeptuneandthatCharonwasaguestsatellite.Through
Eqs.(14)and(15),theconditionsfortheescapeofthepairwasfound.Followingtheirscenario,letusconsiderthehypothesisthat
ProsperowasamemberofaguestbinaryenteringUranus’field,withenergydensity∆E ,abovetheminimumdensitygivenby
bin
Eq.(14) and δ=0. A close encounter with Uranus could result in disruption of the binary, leading to the ejection of one member
andcaptureoftheother.TheminimumsemiaxisR isgivenbyEq.(15).However,eventhisscenarioseemstobeunlikelysincethe
1
semiaxisofProsperoissmallerthan955R .
U
3.4.Collisionalinteractions:Break-upprocesses
Collisionalinteractionsbetweentwoplanetesimalspassingneartheplanetorbetweenaplanetesimalandaregularsatellite,theso
calledbreak-upprocess,leadstotheformationofdynamicalgroupings(e.g.Colombo&Franklin1971,Nesvornyetal.2004).The
resultingfragmentsofeachprogenitorbodyafterabreak-upwillformapopulationofirregularsexpectedtohavesimilarsurface
composition,i.e.similarcolors,andirregularshapes,i.e.largetemporalvariationsinthelightcurveastheseirregularbodiesrotate.
8 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV
Thecriticalrotationperiod(T )atwhichcentripetalaccelerationequalsgravitationalaccelerationforarotatingsphericalobject
c
is:
3π 1/2
T = , (16)
c Gρ !
ob
whereGisthegravitationalconstantandρ isthedensityoftheobject.Withρ =0.5,1,1.5and2gcm 3,T =4.7,3.3,2.7and2.3
ob ob − c
hrs,respectively.TherotationperiodofProsperoisabout4hrs(Marisetal.2007a),anditseemsunlikelythatlightanddarksurface
markingsonasphericalProsperocouldberesponsibleforitslightcurveamplitudeof0.2mag(Marisetal.2007a).Evenatlonger
periods,realbodieswillsuffercentripetaldeformationintoasphericalshapes.Foragivendensityandspecificangularmomentum
(H), the nature of the deformation depends on the strength of the object. In the limiting case of a strengthless (fluid) body, the
equilibriumshapeshavebeenwellstudied(Chandrasekhar1987).ForH 0.304[inunitsof(GM3R )1/2],whereM(kg)isthe
sphe
≤
mass of the objectand R is the radiusof an equal-volumesphere,the equilibriumshapes are the oblate Maclaurinspheroids.
sphe
For 0.304 H 0.390the equilibriumfiguresare triaxial Jacobiellipsoids. Strengthlessobjects with H > 0.390are rotationally
≤ ≤
unstabletofission.TheKuiperBeltobjects(KBOs),beingcomposedofsolidmatter,clearlycannotbestrengthless.However,itis
likelythattheinteriorstructuresofthesebodieshavebeenrepeatedlyfracturedbyimpacts,andthattheirmechanicalresponseto
appliedrotationalstressisapproximatelyfluid-like.Such“rubblepile”structurehaslongbeenstudiedintheasteroidbelt(Farinella
et al.1981) and in the Kuiper Disk (Sheppard& Jewitt 2002, Jewitt & Sheppard 2002, Romanishin & Tegler 1999). Farinella &
Davis(1996)obtainedthatKBOslargerthanabout100kmindiameteraremassiveenoughtosurvivecollisionaldisruptionover
theageofthesolarsystem,butmayneverthelesshavebeeninternallyfracturedintorubblepiles.
Whetheracollisionbetweenanimpactorandatargetresultsingrowthorerosiondependsprimarilyontheenergyoftheimpact
and the mass and strength of the target. If the mass of the impactor is small compared to the mass of the target m , the energy
s
requiredatimpacttoresultinabreak-upisgivenby:
1 3Gm2
m v2 m S + s, (17)
2 s col ≥ s 5γR
ms
wherev is the collisionspeed, S is the impactstrength,R is the radiusofthe targetand γ is a parameterwhichspecifies the
col ms
fractionofcollisionalkineticenergythatgoesintofragmentkineticenergyandisestimatedtobe 0.1(Farinella&Davis1996).
∼
The speed of the fragments is critical when the target has a gravity field. Fragments moving slower than the local escape speed
re-accumulatetoformrubberpilestructures.
WeattempttoinvestigatewhetherProsperocouldbeacollisionalfragmentorifacollisiononaprimaryProsperowouldresult
inarubberpilestructure.
IncomputingEq.(17),v2 =v2+v2 ,wherev isthescapespeedatthetargetsurfaceandv isthetypicalapproachvelocity
col e inf e inf
ofthetwoobjectsatadistancelargecomparedwiththeHillsphereofthetarget.FortwobodiescollidingintheKuiperdisk,v is
inf
givenby(Lissauer&Stewart1993):
5
v2 =v2 e2+i2 , (18)
inf k 4 !
wherev isthekeplerianvelocity,eisthemeanorbitaleccentricityandithemeanorbitalinclinationoftheKBOs.Wetake e =2 i
k
h i hi
(Stern1996).Eq.(17)wascomputedusingEq.(18)forvaluesofeintherange[0.01-0.1]andorbitalsemiaxesoftheKBOsinthe
range[30-60]AU.Incomputingthemassofthetargetm ,weconsidertheradiusR oftheKBOsintherange[10-500]kmand
s ms
densitiesbetween[0.5-2]gcm 3.Impactstrengthsintherange[3 104-3 106]ergcm 3weretaken.Weobtainthattargetswith
− −
× ×
R 210kmsufferdisruptionforallthevaluesoftheseparametersintheKuiperDisk.Prosperohasaradiusofjustonly15km.If
ms
≤
ProsperooriginatedfromtheKuiperBelt,itwouldhavebeenmorelikelyacollisionalfragmentratherthanprimarybody.Prospero
wouldpreservean irregularshape afterdisruptionsince it is suchsmall objectthat is unableto turnsphericalbecause its gravity
cannotovercomematerialstrength.Prosperocouldhavebeencapturedduringthebreak-upeventifthetwoKBOscollidedwithin
Uranus’Hillsphere,whichcouldbepossibleforaminimumorbitaleccentricityoftheoriginalKBOsof0.37.
We alsoconsiderthecaseinwhichthetargetisasatelliteofUranuswhichcollideswithaKBO thatenterstheHillsphereof
theplanet.Eq.(17)remainsvalidbutthefollowingexpressionofv isconsidered:
inf
v =v v , (19)
inf ip sp
−
wherev isthevelocityoftheKB0withrespecttoUranusandv isthesatelliteorbitalvelocity,attheepochofthevent.Weassume
ip sp
thatthesatelliteorbitalvelocityiscircular.Inordertogetboundsintherelativevelocity,wetaketwovaluesofv ,v =v v .
inf inf ip sp
±
Forv ,weassumethattheKBOdescribesahyperbolicorbitaroundUranusduringtheapproachgiving:
ip
2G(m +m ) GM
v2 = i U + ⊙, (20)
inf a a
s kb
whereweassumedGM /a astherelativevelocitybetweentheKBOandUranusfarfromtheencounter,(m +m )isthepresent
kb i U
massofUranusanda t⊙heorbitalsemiaxisofthesatellite.WecalculateEq.(17)usingEqs.(19)and(20)fora intherange[20,60]
s kb
AUanda [100-700]R forthesamevaluesofS,R anddensitieswehaveusedforthecollisionsamongKBOs.Weobtainthat
s U ms
M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV 9
forallthepossibleparameters,anyUranus’satellitewithradiusR 1000kmsuffersdisruptionifitcollideswithaKBOlarger
ms
≤
than10 km.Thisprocesswouldlead to the formationof two clustersofirregulars,one associatedto the preexistingsatellite and
theothertotheprimaryKBO.Thisprocesshasthedisadvantagethatitisunlikelythatthepreexistingsatellitewereformedfroma
circumplanetarydiskasregularsatellitesgiventhelargeorbitalsemiaxisrequiredforthisobject.
Break-upprocessespredictorbitalclustering.However,noobviousdynamicalgroupingsareobservedattheirregularsofUranus.
AfurtherintensivesearchofmorefaintirregularsaroundUranusisneededinordertolookfordynamicalandphysicalfamilies.
3.5.TheCGitselfasapossiblecapturemechanism
WenowturntothequestionofwhethertheGCitselfcouldhaveprovidedacapturemechanism(BP02).Sinceallthetransferswith
A> B leadtoamoreboundorbit,thisprocessmighttransformatemporarycaptureintoapermanentone(seeSection2andFig.1).
′
Moreover,apermanentcapturecouldevenoccurfromanheliocentricorbit(transferswithA=1).
ItisinterestingtoestimatethenumberofobjectsN inheliocentricorbitsatthetimeoftheGC,atdistancesfromUranusless
thanorequalto300R .AssumethattheGCoccurredwhenUranuswasalmostfullyformed,meaningthatitsfeedingzonewas
U
alreadydepleted of primordialplanetesimals. We assume that the objectspassing near Uranusat thattime were mainly escapees
fromthe Kuiperbelt. Using the impactrate onto Uranusandthe distributionsofvelocitiesanddiametersgivenbyLevisonet al.
(2000),andassumingthatthemassinthetransneptunianregionattheendoftheSolarSystemformationwas10timesitspresent
mass,a back-of-the-envelopecalculationgivesoneobjectofdiameterD 20kmpassingatadistanceR 300R fromUranus
U
≥ ≤
every6yrsattheendofaccretion(BP02).ThetypicalcrossingtimeT amongprotoplanetsintheouterSolarSystemislargerthan
C
onemillonyears(Zhouetal.2007).ThenumberofobjectspassingnearUranusduringatimescaleT isthen167000,whichgives
C
a probabilityof 6 10 6 for the capture of an object at about 300 R by the GC. This low rate of incoming objects, makes the
− U
×
possibilityofthecaptureofalltheirregularsfromheliocentricorbitsdifficult.Eventhecaptureofasingleobject,Prospero(note
thatProsperocouldnothaveanorbitboundtotheplanetbeforetheGC),turnsouttobelowprobable.Sincetemporarycapturecan
lengthenthetimewhichapassingbodycanspendneartheplanet,amoreplausiblesituationarisesifweassumethattheGCcould
produce the permanent capture of one or more parent objects which were orbiting temporarily around Uranus being the present
irregularstheresultofacollisionalbreak-upoccurringaftertheGC.
4. DiscussionandConclusions
ItisusuallybelievedthatthelargeobliquityofUranusistheresultofagreattangentialcollision(GC)withanEarth-sizedproto-
planetattheendofthe accretionprocess.We havecalculatedthetransferofangularmomentumandimpulseatimpactandhave
shownthatthe GChadstronglyaffectedtheorbitsofUraniansatellites. We calculatethetransferoftheorbitsofthenineknown
UranianirregularsbytheGC.Veryfewtransfersexistforfiveofthenineirregulars,makingtheirexistencebeforetheGChardly
expected.Inparticular,Prosperocouldnotexistatthetime oftheGC. Then,eitherProsperohadto originateaftertheGC orthe
GCdidnotoccur,inwhichcaseanothertheoryabletoexplainUranus’obliquityandtheformationoftheUranianregularsatellites
wouldbeneeded.ItisusuallybelievedthattheregularsatellitesofUranushaveaccretedfrommaterialplacedintoorbitbytheGC
(Stevensonetal.1986).
WithintheGCscenario,severalpossiblemechanismsforthecaptureofProsperoaftertheGCwereinvestigated.IftheUranian
irregulars belong to individual captures and relating the origin of the outer uranian system to a common formation process, gas
dragby Uranus’envelopeand pull-downcaptureseem to be implausible.Three-bodygravitationalencountersmightbe a source
of permanentcapture. However,we foundthat the minimum permanentorbitalradiusof a guestsatellite of Uranusis 955 R
U
∼
whilethecurrentsemiaxisofProsperois645R .TheGCitselfcouldprovideamechanismofpermanentcaptureandthecapture
U
ofProsperocouldhaveoccurredfromaheliocentricorbitasisrequiredwithintheGCscenario,butduetothelowrateofincoming
objectsit turnsoutto bedifficult.Break-upprocessescouldbe themechanismforthe originofProsperoandthe otherirregulars
in the frame of differentscenarios. Prospero might be a fragmentof a primary KBO fracturedby a collision with anotherKBO.
ThefragmentcouldhavebeencapturedbyUranusifthetwoKBOshadaminimumorbitaleccentricityof0.37.Prosperocouldbe
a secondarymemberofa collisionalfamilyoriginatedbythecollisionbetweenanothersatellite ofUranusanda KBO wherethe
parentsatelliteofProsperocouldhavebeencapturedbyanymechanismbeforeoraftertheGC.Thisprocesshasthedisadvantage
thatitisunlikelythatthepreexistingsatellitewereformedfromacircumplanetarydiskasregularsatellitesgiventhelargeorbital
semiaxis required for this object. Since collisional scenarios require in general high collision rates, perhaps the irregulars were
originally much more numerousthan now. Then, Prospero and also the other irregulars might be the result of mutual collisions
amonghypotheticalpreexistingirregulars(Nesvornyetal. 2003, 2007)whichcouldhavebeencapturedbyanyothermechanism
beforetheGC.
TheknowledgeofthesizeandshapedistributionofirregularsisimportanttoknowtheirrelationtotheprecursorKuiperBelt
population.It could bring valuable clues to investigate if they are collisional fragmentsfrom break-upprocesses occuring at the
Kuiper Belt and thus has nothing to do with how they were individually captured later by the planet, or if they are collisional
fragmentsproducedduringorafterthecaptureevent(Nesvornyetal.2003,2007).ThedifferentialsizedistributionoftheUranian
irregularsapproximatesapowerlawwithanexponentq=1.8(Sheppardetal.2005b).Ifweassumethatthesizedistributionofthe
nineirregularswithradiigreaterthan7kmextendsdowntoradiiofabout1km,wewouldexpectabout75irregularsofthissizeor
larger(Sheppardetal.2005b).
ThenucleiofJupiterfamilycometsarewidelyconsideredtobekilometer-sizedfragmentsproducedcollisionallyintheKuiper
Belt(Farinella&Davis1996).Jewittetal.(2003)comparedtheshapedistributionofcometarynucleiintheJupiterfamilywiththe
shapedistributionofsmallmain-beltasteroidsofsimilarsize(1km-10km)andwiththeshapedistributionoffragmentsproduced
10 M.GabrielaParisietal.:ConstraintstoUranus’GreatCollisionIV
in laboratoryimpactexperiments.They foundthat while the asteroidsand laboratoryimpactfragmentsshow similar distribution
of axis ratio ( b/a 0.7),cometarynucleiare moreelongated( b/a 0.6).Theypredictthatif cometsreflecttheir collisional
h i∼ h i∼
originintheKuiperBeltfollowedbysublimation-drivenmasslossonceinsidetheorbitofJupiter,smallKBOsshouldhaveaverage
shapesconsistentwith those ofcollisionally producedfragments(i.e., b/a 0.7).To date, constraintson the shapesof onlythe
h i∼
largestKBOsareavailable.Prosperobeingabitlargerthancometarynuclei,displaysavariabilityof0.21magintheRband(Maris
etal.2007a).Thiscorrespondstoanaxisratioprojectedintotheplaneofthesky,b/aof0.8.Theknowledgeofthesizeandshape
distribution of irregulars would shed light in the size and shape distribution of small KBOs as well as on the irregulars capture
mechanism.
Colors are an important diagnostic tool in attempting to unveil the physical status and the origin of the Uranian irregulars.
In particular it would be interesting to assess whether it is possible to define subclasses of irregulars just looking at colors, and
comparingcolorsofthesebodieswithcolorsofminorbodiesintheouterSolarSystem.Avalilableliteraturedatashowadispersion
in thepublishedvalueslargerthanquotederrorsforeachUranianirregular(Mariset al.2007a, andreferencestherein).We have
concludedinMarisetal.(2007a),thattheUranianirregularsareslightlyredbuttheyarenotasredasthereddestKBOs.
An intensivesearchforfainter irregularsanda longterm programof observationsableto recoverin a self consistentmanner
light-curves,colorsandphaseeffectsinformationsismandatory.
Acknowledgements. MGPresearchwassupportedbyInstitutoArgentinodeRadioastronom´ıa,IAR-CONICET,ArgentinaandbyCentrodeAstrof´ısica,Fondode
Investigacio´navanzadoenAreasPrioritarias,FONDAPnumber15010003,Chile.MMacknowledgesFONDAPforfinantialsupportduringavisittoUniversidad
deChile.PartoftheworkofMMhasbeensupportedbyINAFFFO-FondoRicercaLibera-2006.ABresearchwassupportedbyIALP-CONICET.Weappreciate
theusefulsuggestionsbytheReviewer,whichhavehelpedustogreatlyimprovethispaper.
References
Adachi,I.,Hayashi,C.&Nakazawa,K.1976,Prog.Theo.Phys.,56,1756
Agnor,C.B.&Hamilton,D.P.2006,Nature,441,192
Beauge´,C.,Roig,F.&Nesvorny´,D.2002,Icarus,158,438
Bodenheimer,P.&Pollack,J.B.1986,Icarus,67,391
Brunini,A.,Parisi,M.G.&Tancredi,G.2002,Icarus,159,166
BruniniA.1995,Earth,Moon,andPlanets,71,281
ChambersJ.E.2001,Icarus,152,205
ChandrasekharS.1987,EllipsoidalFiguresofEquilibrium(NewYork:Dover)
Colombo,G.&Franklin,F.A.1971,Icarus,15,186
Cuk,M.&Burns,J.A.2003,Icarus,167,369
Farinella,P.,Paolicchi,P.,Tedesco,E.F.&Zappala,V.1981,Icarus,46,114
Farinella,P.&Davis,D.R.1996,Science,273,938
Gladman,B.J.,Nicholson,P.D:,Burns,J.A.,Kavelaars,J.J.,Marsden,B.J.,WilliamsG.V.&Offutt,W.B.1998,Nature,392,897
Gladman,B.J.,Kavelaars,J.J.,Holman,M.,Petit,J-M,Scholl,H.,Nicholson,P.D.&Burns,J.A.2000,Icarus,147,320;erratum148,320,(2000)
Gladman,B.J.,Kavelaars,J.J.,Holman,M.,Nicholson,P.D.,Burns,J.A.,Hergenrother,C.W.,Petit,J-M,Marsden,B.J.,Jacobson,R.,Gray,W.&Grav,T.2001,
Nature,412,163
Grav,T.,Holman,M.J.&Fraser,W.2004,Astrophy.J.Letters,613,L77
Heppenheimer,T.A.&Porco,C.1977,Icarus,30,385
HoiLee,M.,Peale,S.J.,Pfahl,E.,Ward,W.R.2007,Icarus,inpress
Holman,M.,etal.2004,Nature,430,865
Jewitt,D.&Sheppard,S.2002,Astronom.J.,123,2110
Jewitt,D.&Sheppard,S.2005,SpaceScienceReviews,116,441
Jewitt,D.,Sheppard,S.&Fernandez,Y.2003,Astronom.J.,125,3366
Kavelaars,J.J.,etal.2004,Icarus,169,474
Korycansky,D.G.,Bodenheimer,P.,Cassen,P.&Pollak,J.B.1990,Icarus,84,528
Kubo-Oka,T.&Nakazawa,K.1995,Icarus,114,21
Levison,H.F.,Duncan,M.J.,Zahnle,K.,Holman,M.&Dones,L.2000,Icarus,143,415
Lissauer,J.J.&Safronov,V.1991,Icarus,93,288
Lissauer,J.J.&Stewart,G.R.1993,ProtostarsandplanetsIII,1061
Luu,J.&Jewitt,D.1996,Astronom.J.,112,2310
Maris,M.,Carraro,G.,Cremonese,G.&Fulle,M.2001,Astron.J.,121,2800
Maris,M.,Carraro,G.&Parisi,M.G.2007a,A&A,472,311
Maris,M.,Carraro,G.&Parisi, M.G.2007b,Proceedings oftheworkshop“Mutual EventsoftheUranian Satellites in2007-2008andfurtherobservations in
network”,Paris,2006,inpress
Nesvorny,D.,Alvarellos,J.L.A.,Dones,L.&Levison,H.2003,Astronom.J.,126,398
Nesvorny,D.,Beauge´,C.&Dones,L.2004,Astronom.J.,127,1768
Nesvorny,D.,Vokrouhlicky´,D.&Morbidelli,A.2007,Astronom.J.,133,1962
Parisi,M.G.&Brunini,A.1997,PlanetaryandSpaceScience,45,181
Pollack,J.B.,Burns,J.A.&Tauber,M.E.1979,Icarus,37,587
Pollack,J.B.,Hubickyj,O.,Bodenheimer,P.,Lissauer,J.J.,Podolak,M.&Geenzweig,Y.1996,Icarus,12462
Romanishin,W.&Tegler,S.C.1999,Nature,398,129
Romon,J.,deBergh,C.,Barucci,M.A.,Doressoundiram,A.,Cuby,J-G,LeBras,A.,Doute´,S.&Schmitt,B.2001,A&A,376,310
SafronovV.S.1969,EvolutionoftheProtoplanetaryCloudandFormationofTheEarthandthePlanets,NASATTF-677
Sheppard,S.S.&Jewitt,D.C.2002,Astronom.J.,124,1757
Sheppard,S.S.&Jewitt,D.C.2003,Nature,423,261
Sheppard,S.S.,Jewitt,D.C.&Marsden,B.G.2005a,MPEC2005-J13
Sheppard,S.S.,Jewitt,D.C.&Kleyna,J.2005b,Astron.J.,129,518
Sheppard,S.S.,Jewitt,D.C.&Kleyna,J.2006b,Astron.J.,132,171
Sheppard,S.S.,Jewitt,D.C.,Kleyna,J.&Marsden,B.G.2006a,IAUCir.No8727
Stevenson,D.J.,Harris,A.W.&Lunine,J.I.1986,InSatellites(Burns,J.A.&Matthews,D.M.S.,Eds.),Univ.ofArizonaPress,Tucson,39
SternS.A.1996,A&A,310,999