Multiple Components in Narrow Planetary Rings

3.2 MB·English

by  L. Benet

MultipleComponents inNarrow Planetary Rings
L. Benet∗ and O. Merlo†
InstitutodeCienciasF´ısicas,UniversidadNacionalAuto´nomadeMe´xico(UNAM),Cuernavaca, Mor.,Me´xico
(Dated:February6,2008)
Thephase-spacevolumeofregionsofregularortrappedmotion,forboundedorscatteringsystemswithtwo
degreesoffreedomrespectively, displaysuniversal properties. Inparticular, drasticreductionsinthevolume
(gaps)areobservedatspecificvaluesofacontrolparameter. Usingthestabilityresonancesweshowthatthey,
and not the mean-motion resonances, account for the position of these gaps. For more degrees of freedom,
excitingtheseresonancesdividestheregionsoftrappedmotion. Forplanetaryrings,wedemonstratethatthis
mechanismyieldsringswithmultiplecomponents.
8 PACSnumbers:05.45.-a,05.45.Jn,95.10.Ce,96.30.Wr
0
0
2 TheCassinimissionisprovidingunprecedentedamountsof scatteringdynamicsofa(ring)particleinaplanarsystemwith
data on the structureof Saturn rings[1]. As usual, surprises someintrinsicrotation(externalforce). Intheplanetarycase,
n
a and new questions emerge which call for theoretical under- this rotation comes from the orbital motion of one or sev-
J standing. Beyondtheintrinsicinterestonplanetaryrings[2], eral satellites. This rotation creates generically phase-space

9 thesearenowtheonlyexamplesofflatastrophysicalsystems regionsof dynamicallytrappedmotion, in what otherwise is
for which such detailed data is available. This makes their dominatedby unboundedtrajectories; “generic” implies that

]
analysis paradigmatic [2], which serves as analog for other it holds for a wide class of rotating potentials [14], includ-
D
systemslikegalaxiesandplanet-formingdisks. Afirstorder inggravitationalinteractions[9,15]. Forauniformlycircular
C
understandingofthedynamicsandprocessesinringshasbeen rotation, the system has two degrees of freedom (DOF) and
.
n obtained,butimportantquestionsremainunanswered[2]. In a constant of motion, the Jacobi integral. For these scatter-
i particular, narrow planetary rings are eccentric and display ingsystemstheorganizingperiodicorbitsgenericallyappear
l
n multiplecomponents(orstrands),kinks,clumpsandarcs(or in pairs throughsaddle-centerbifurcations,one is stable and
[
patches); the F ring of Saturn is a magnificent case [1, 2]. theotherunstable. Themanifoldsoftheunstableorbitinter-

2 Therefore, these narrow rings pose interesting questions in sect, isolating a region where trapped motion is of nonzero
v dynamicalastronomyandnonlineardynamicsrelatedtotheir measure if stable orbits exist. By changing the Jacobi inte-
9
stability,confinementandstructuralproperties. gral,thecentrallinearlystableperiodicorbitbecomesunsta-
3
0 Recentlyitbecameclearthatthephase-spacevolumeasso- bleandaperioddoublingcascadesetsin. Thisscenarioturns
2 ciatedtoregularorbitsforboundedsystemswithtwodegrees eventuallythehorseshoe(invariantset)intoahyperbolicone,
0 offreedom(DOF)displaysuniversalpropertiesasafunction thus destroyingany regionof trappedmotion [16]. Then, in
7 of a control parameter [3, 4]. For scattering systems such the Jacobi constant space, the regionsof trapped motion are
0
quantity is related to the trapped orbits. In particular, there bounded. Particles with initial conditions outside these re-
/
n arelocationswhereadrasticreductionofthephase-spacevol- gionsescaperapidlyalongscatteringtrajectories;thoseinside
li umeoccupiedbytheregularortrappedorbitsisobserved,i.e., remain dynamically trapped. Consider now an ensemble of
n gaps. Herewe showthatthesegapsarerelatedto theoccur- initialconditionsofindependentparticlesdistributedoverthe
:
v rence of stability resonances (SR), and not to mean-motion extendedphase space, which contains entirely one region of
Xi resonances(MMR).TheMMRcorrespondtolocationswhere trappedmotion.Then,duetotheintrinsicrotation,thepattern
the period of the particle is a rational of the orbital period formedbyprojectingthetrappedparticlesintotheX−Y plane
r
a of a satellite or planet. Examples include Kirkwood gaps in atagiventimeformsaring[14].Theringistypicallynarrow,
the asteroid main belt [5, 6] and the Cassini division in Sat- sharp-edgedandnoncircular;thisscenarioisgeneric[14].For
urn rings [7], where very few particles are observed. Other morethantwo DOF, e.g., a nonuniformrotationona Kepler
casesofMMRexistwhereparticlesaccumulatelocally,asthe ellipticorbit,theringdisplaysfurtherstructure[9].
Jupiter Trojans or the Hilda group [8]. The SR are defined The simplest example to illustrate the occurrence of rings
byaresonantconditioninthe(linear)stabilityexponentsofa is a planar scattering billiard: a disk on a Kepler orbit [10].
centralstableperiodicorbit. WeshowinthisLetterthatnon- Thissystemconsistsofapointparticlemovingfreelyunless
linear and higher-dimensionaleffectsrelated to the SR yield it collides with an impenetrable disk of radius d, which or-
ringswith two or morecomponents, or strands, as those ob- bitsaroundtheoriginonaKeplerorbitofsemi-majoraxisR
servedintheFring[1,2]. Weusethescatteringapproachto (d<R). Collisionswiththediskaretreatedasusual[10];no
narrowplanetaryrings[9]andadiskrotatingonaKeplerorbit collisionsleadtoescape. Forthediskonacircularorbit,the
asexample[10]. Ourresultscanbeusedinothercases,rang- simpleperiodicorbitsaretheradialcollisionorbits.Theyand
ingfromparticleaccelerators[11]togalacticdynamics[12], theirlinearstabilityaregivenby[9]
whereresonancesandtransportpropertiesareimportant[13].
The scattering approach to narrow rings [9] considers the J=2w 2(R−d)2(1+D f tanq )cos2q (D f )−2, (1)
d
2
TrDPJ=2+(cid:2)(D f )2(1−tan2q )−4(1+D f tanq )(cid:3)R/d. (2) )02 )32 )62 )92 )23 )53 )44
:3 :5 :7 :9 :1 :3 :9
1 1 1 1 2 2 2
( ( ( ( ( ( (
1 6 5 4 3 2
Equation(1)definesthevalueoftheJacobiintegralJ forthe 700 :1 :1 :1 :1 :1 :1
radial collision orbits in terms of q , the outgoing angle of a)
the particle’s velocity after a collision with the disk. Here, 600
D f =(2n−1)p +2q istheangulardisplacementofthedisk
500
betweenconsecutiveradialcollisionsandn=0,1,2,... isthe
number of full turns completed by the disk before the next
collision. Theperiodofthedisk’sorbitisTd =2p (w d =1). t>)400
Equation(2)providesthetraceofthelinearizedmatrixDPJ N(<D 300
aroundthe radialcollision orbits. Being a two DOF system,
periodicorbitsarelinearlystableiff|TrDP |≤2. 200
J
Tounderstandthephase-spacepropertiesthatdefinethedy-
100
namics, in particular, for many DOF, we consider a relative
measure of the phase-space volume occupied by the regions
0
4,08 4,1 4,12 4,14 4,16
oftrappedmotion[3,4]. Thisquantityisafunctionofsome
<D t>
controlparameterandtunesthehorseshoedevelopment[16].
ForthediskonacircularorbitagoodchoiceisJ. However,
thisquantityisnotconservedfornonzeroeccentricitye ,thus
beinguselessformorethantwoDOF.Aconvenientquantity 0.2
b)
is the average time between consecutive collisions with the
disk, hD ti. The average is defined, for a given initial condi-
tion(ringparticle),overthesuccessiveconsecutivecollisions 0.1
times; in addition,we consideran ensembleof them. In our
numericalcalculationsweconsideredanorbittobetrappedif
itdisplaysmorethan20000collisionswiththedisk;thenext Y 0
200000reflectionswereusedtoimprovethestatistics.
In Fig. 1(a) we show the histogram of hD ti for the n=0
trapped region; Fig. 1(b) shows a detail of the correspond- -0.1
ing ring [9]. We note that Fig 1(a) displays some structure,
namely, some rather well localized positions where there is
essentially no trapped motion in phase space or, at least, a -0.2
-0.34 -0.32 -0.3 -0.28
drasticreductionofitssize. Thisstructureissimilartothose X
inRef.[3],andisuniversal[3,4]. Then,universalityextends
to scattering systems in terms of the phase-space volume of
FIG. 1: (Color online) (a) Histogram of hD ti showing the relative
the regionsof trappedmotion. We note that the structure of
phase-spacevolumeoftheregionoftrappedmotionforthescatter-
Fig. 1(a) also resembles the structure of the population his-
ing billiard on a circular orbit. The continuous lines represent the
togramofasteroidsintermsoftheirsemimajoraxis[6],which
positionofthelowestMMR,indicatedinparentheses,thatoccurin
uncoverstheKirkwoodgapsandtheroleofMMR. theintervalofhD ti;dash-dottedlinescorrespondtothelower-order
Guidedbythesimilaritywiththeasteroids,weaskwhether SR. (b) Detail of the ring corresponding to this region of trapped
thegapsinFig.1(a)arerelatedtoMMR.Thequestioncanbe motion.
answered analytically for the radial collision periodic orbits
whicharetheorganizersofthedynamics.Theircollisiontime
istcol=D f /w d, hencetheMMRconditionistcol/Td = p/q, withthediskinarotatingframe(see[9]). Thus,theaverage
with pandqincommensurateintegers. InFig.1(a),wehave return time to the surface of section is precisely hD ti. Fig-
indicatedthe locationof the lowest mean-motionresonances ures2displaythesurfacesofsectionforthreevaluesoftheJa-
in the hD ti interval of interest; the actual resonances are in- cobiintegralaroundthelocationofthemaingapinFig.1(a).
dicatedinparentheses. We havealsoincludedthe29:44res- Satellite period-threeislands squeeze the centralstable fixed
onance because of its proximity to the position of the main point, reducing efficiently the region of trapped motion. At
gap. Whiletheseresonancesaredefinitelynotlow-orderres- some value of J there is a bifurcation with the central point
onances, we note that some may be very close to the gaps, being unstable. After this value, the whole structure reap-
whileothersarenot. Therefore,low-ordermean-motionres- pears with some symmetric inversion. This phenomenon is
onances are not related to the reduction of the phase-space intrinsicallynonlinearandisuniversal[17]. Notethattheex-
volume(gaps)oftheregionsoftrappedmotion. istenceoftheperiod-threeislandsinthesurfaceofsectionis
TounderstandthegapsofFig.1(a),weconsiderthephase- notequivalenttoa1:3MMR.Theperiod-threefixedpointsin-
space structuredefining a surface of section at the collisions volvedifferentcollisiontimes. Addingtheseindividualtimes
3
3,16 3,16 3,16
3,15 Wu 3,15 Wu 3,15 Wu
3,14 3,14 3,14
a a a
Ws Ws Ws
3,13 3,13 3,13
3,12 3,12 3,12
0,796 0,798 0,8 0,802 0,804 0,796 0,798 0,8 0,802 0,804 0,796 0,798 0,8 0,802 0,804
p p p
FIG.2: (Coloronline)SurfacesofsectionforthescatteringbilliardonacircularorbitfordecreasingvaluesofJ. Thesequenceillustratesthe
reductionofthephase-spacevolumeofthetrappedregionduetothe1:3SRcorrespondinginFig.1(a),fromlefttoright,tohD ti≈4.1382,
hD ti≈4.1409, andhD ti≈4.1445. Theblack/redcurvesarethestable/unstablemanifoldsoftheunstableperiodicorbit. Theblueregions
correspondtoregionsthatrepresentmorethan90%ofthephase-spacevolumeoccupiedbythetrappedorbits.
matches a multiple of the collision time of the central fixed ture remains. The histogram is divided in three distinct re-
point,whichingeneralisnotarationalofT . gions separated by those SR; e tunes the width of the gaps.
d
Thegapsin Fig.1(a)arethusrelatedtoa reductionofthe Weconstructtheringusingdifferentcolorsfortheinitialcon-
phase-spacevolume occupiedby trappedorbitsdue to satel- ditions in the different hD ti regions. The ring displays two
liteislands(seealso[18]).In [17]itisshownthatthepassage well–separatedcomponents.Eachoneisrelatedtoadifferent
throughsuch1:3resonances(Fig.2)leadsuniversallytoinsta- regionin phasespace; thisis a consequenceofthe largegap
bilities,whileresonancesoforderhigherthan4donotinduce opened by the 1:3 SR. These componentsare entangled and
instability;fortheresonanceoforder4thebehaviordepends formabraidedring[9].
uponwhichcontribution(resonantornonresonant)dominates We note in Fig. 3(b) that there are only two independent
thenormalform[17,18]. Theseresultsfollowfromthestruc- ringcomponentsinsteadofthree, asexpectedfromthethree
tureoftheresonantnormalform,i.e.,therelevant(nonlinear) disjointintervalsinhD tidefinedbythe1:3and1:6resonance.
contributionstothestabilityanalysisofacentrallinearlysta- Thisfollowsfromthefactthatthe1:6resonancedoesnotsep-
bleperiodicorbit.Wethusconsiderwhethertheseresonances arate enough the phase-space regions around it, so the pro-
arerelatedtothestructureofFig.1(a).Wewritetheeigenval- jections into the X−Y plane of these two intervals overlap
uesofthelinearizedmapDP aroundthestableradialcolli- and do not manifest two components. Yet, as observed in
J
sionorbitasl =e±ia . Weareinterestedinvaluesofa that Fig.3(b), thecorrespondingringcomponentdisplaysa clear
±
arerationalmultiplesof2p ,i.e.,a /(2p )= p/qwith pandq separation of the ring particles, except for a thin common
incommensurateintegers. The eigenvaluesl are related to strip,dependingonwhichsideofthe1:6resonancetheycome
±
eachotherbycomplexconjugation;byconsequence,the p:q from. For larger values of e we have observed rings with
resonance is related to the q−p:q resonance. Since these three strands [9]. Therefore, multiple ring components are
resonances involve the stability exponents, we refer to them obtained by exciting SR through nonzero eccentricity. Note
asstabilityresonances. thatthesepropertiesfollowfromthehigherdimensionalityof
Fromthedefinitionofthepurelyimaginaryeigenvaluesl , phasespaceandnonlineareffects.
±
|TrDP |≤2,wehavecosa =TrDP /2. Thiscanbewrit- Thefactthattheoverallstructureof Fig.3(a)is similar to
J J
tenintermsofthecollisiontimet =D f /w usingEq.(2). thetwoDOFcaseandtothepopulationhistogramofasteroids
col d
In Fig. 1(a), we have indicated the lower-order SR as dash- may be an indication of universality for higher dimensions.
dottedlines.Thecorrespondencewiththegapsisastonishing. Yet,ourresultsareinconclusiveforthisissue:Fig.3(a)repre-
Therefore,weattributethegapsinFig.1(a)tocrossingaSR. sentsthe firstresultsof thephase-spacevolumeoccupiedby
The SR, and, in particular, the 1:3, have interesting con- trappedorbitsforasystemofmorethantwoDOF.
sequences in the context of rings when we consider a small Tosummarize,wehavestudiedthephase-spacevolumeoc-
nonvanishing eccentricity of the disk’s orbit. Note that, for cupiedbytrappedorbitsusingascatteringbilliard,adiskro-
nonzeroe , the system is explicitlytime dependent, has two- tatingonaKeplerorbit,andsucceededinrelatingsomeofits
and-half DOF and cannot be handled with the usual tech- structuretotheoccurrenceofSR.FortwoDOFtheuniversal
niques. In Fig. 3(a) we illustrate the phase-space volume of structureof the phase-spacevolumeoccupiedby trappedor-
the trappedregionfor e =0.0001, indicating the location of bits is extendedto scattering systems [3, 4]. For more DOF
somee =0resonancesasa guide,andinFig. 3(b)we show thequalitativesimilarityofFig.3(a)andFig.1(a)isthefirst
a detailof the resultingring. In comparisonto Fig. 1(a), the indication that universality may hold also for higher dimen-
gaps at the 1:3 and 1:6 SR are wider, but the overall struc- sions; yet, this issue remains open. Stability resonances are
4
)0 )3 )6 )9 )2 )5 )4 yieldsaringwithtwoormorecomponentsorstrands,which
2 2 2 2 3 3 4
:31 :51 :71 :91 :12 :32 :92 may entangle and form a braidedring. This providesa sim-
( ( ( ( ( ( (
1 6 5 4 3 2
700 :1 :1 :1 :1 :1 :1 pleexplanationofrecentobservationsofplanetaryringswith
a) multiplecomponents[2]. Theseresultsshouldbeinteresting
600 beyondthecontextofplanetaryrings,insystemswherereso-
nancesandthephase-spacestructurearesignificant;examples
500
includeparticleaccelerators[11]andgalacticdynamics[12].
>)400
t
N(<D 300 We thank to R. Dvorak, T.H. Seligman and C. Simo´ for
usefuldiscussions,andthesupportbytheprojectsIN-111607
200 (DGAPA)and43375(CONACyT).O.Merloisapostdoctoral
fellowoftheSwissNationalFoundation(PA002-113177).
100
0
4,08 4,1 4,12 4,14 4,16
<D t>
∗ Electronicaddress:benet@fis.unam.mx
† Electronicaddress:merlo@fis.unam.mx
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