On the observability of resonant structures in planetesimal disks due to planetary migration

1.2 MB·English

by  Rémy Reche

Astronomy&Astrophysicsmanuscriptno.first (cid:13)c ESO2008
February2,2008
On the observability of resonant structures in planetesimal disks
due to planetary migration
R.Reche,H.Beust,J.-C.Augereau, andO.Absil
Laboratoired’AstrophysiquedeGrenoble,CNRS,UniversiteJoseph-Fourier,UMR5571,Grenoble,France
8
Received24May2007;accepted18December2007
0
0
ABSTRACT
2

n Context.Theobservedclumpystructuresindebrisdisksarecommonlyinterpretedasparticlestrappedinmean-motionresonances
a withan unseen exo-planet. Populating theresonances requires amigrating process of either theparticles (spiraling inward due to
J dragforces)ortheplanet(movingoutward).Becausethedragtime-scaleinresolveddebrisdisksisgenerallylongcomparedtothe

7 collisionaltime-scale,theplanetmigrationscenariomightbemorelikely,butthismodelhassofaronlybeeninvestigatedforplanets
1 oncircularorbits.
Aims.Wepresentathoroughstudyoftheimpactofamigratingplanetonaplanetesimaldisk,byexploringabroadrangeofmasses

] andeccentricitiesfortheplanet.Wediscussthesensitivityofthestructuresgeneratedindebrisdiskstothebasicplanetparameters.
h Methods.WeperformmanyN-bodynumericalsimulations,usingthesymplecticintegratorSWIFT,takingintoaccountthegravita-
p tionalinfluenceofthestarandtheplanetonmasslesstestparticles.Aconstantmigrationrateisassumedfortheplanet.
- Results.Theeffectof planetarymigrationonthetrappingof particlesinmeanmotion resonances isfoundtobeverysensitiveto
o
theinitialeccentricityoftheplanetandoftheplanetesimals.Aplanetaryeccentricityaslowas0.05isenoughtosmearoutallthe
r
t resonantstructures,exceptforthemostmassiveplanets.Theplanetesimalsalsoinitiallyhavetobeonorbitswithameaneccentricity
s
oflessthanthan0.1inordertokeeptheresonantclumpsvisible.
a
Conclusions.This numerical work extends previous analytical studies and provides a collection of disk images that may help in
[
interpretingtheobservationsofstructuresindebrisdisks.Overall,itshowsthatstringentconditionsmustbefulfilledtoobtainob-

1 servableresonantstructuresindebrisdisks.Theoreticalmodelsoftheoriginofplanetarymigrationwillthereforehavetoexplainhow
v planetarysystemsremaininasuitableconfigurationtoreproducetheobservedstructures.
1
Keywords.Method:N-bodysimulations–Celestialmechanics–(Stars:)planetarysystems–(Stars:individual:)Vega
9
6
2
. 1. Introduction MMRs formodelingasymmetricdisksis that, as explainedfor
1
examplein Murray&Dermott(2000), resonantobjectsare not
0
8 SincethefirstdirectimagingofadebrisdiskaroundβPictorisby uniformlydistributedinazimutharoundastar:rathertheygather
0 Smith&Terrile (1984), a dozen other optically thin dust disks at specific longitudesrelative to the perturbingplanet and sub-
: havebeenspatiallyresolvedaroundnearbymain-sequencestars sequently form clumps. This arises from properties specific to
v showing an infrared excess (Kalasetal. 2007; Schneideretal. MMRsas agivenparticletrappedina MMRwith a planetun-
i
X 2006, and references there in). The images often reveal asym- dergoesconjunctionswiththeplanetatspecificlocationsalong
metric structures and clumps, interpreted as the signature of itsorbit.Theparticlestendtogatheraroundthemoststableor-
r
a gravitationalperturbations.A planetimmersed in a debris disk bital configurations that ensure that the conjunctions occur at
usuallyproducesstructuressuchasagapalongitsorbit,byeject- themaximumrelativedistance.Theclumps,whicharetheresult
ing particles during close encounters, or density waves (e.g. a ofthecollectiveeffectsofresonantparticles,generallycorotate
one-armspiral),bymodifyingtheprecessionrateofthedustpar- withtheplanet(Kuchner&Holman2003),whileeachofthese
ticles (Wyatt 2005a). However, such structures cannot explain resonantbodieshasadifferentperiodfromthatoftheplanet(ex-
theobservationsofclumpy,non-axisymmetricdisks(Augereau ceptfor1:1resonantplanetesimals):hencethe motionof these
2004; Meyeretal. 2007), and resonant mechanisms with un- densitywavesdiffersfromtheorbitalmotionoftheresonantpar-
seen planets have been proposed to account for the observed ticles.However,MMRsareverythinradialstructuresthatusu-
asymmetries (Ozernoyetal. 2000; Quillen&Thorndike 2002; allytrapasmallnumberofparticlesinagivendisk.Therefore,
Kuchner&Holman 2003; Wyatt 2003). A particle belongs to any structure due to MMRs has a high chance of being totally
a mean motion resonance (MMR) when the particle to planet hidden by the emission of the non-resonant particles, as illus-
period ratio is a rational number, m : n with m and n in- tratedinFig.1.
tegers. An MMR is located at a semi-major axis a given by
ForclumpsduetoMMRstobeobserved,thepopulationof
a/a = (m/n)2/3, where a is the planet semi-major axis. In
p p resonant particles must be significantly enhanced by an addi-
theSolarSystem,forexample,about15%oftheknownKuiper
tional physicalprocess. Two mechanisms can account for this:
Belt objects, including Pluto, are trapped in the 3:2 resonance
Poynting-Robertson(P-R)dragandplanetmigration.Dustpar-
with Neptune (Chiangetal. 2007). The interesting property of
ticles that are too large to be ejected from the system by ra-
diation pressure can spiral inward into the star due to P-R
Send offprint requests to: R. Reche e-mail: drag and to some other minor forces like stellar wind drag
[email protected] (e.g. Augereau&Beust 2006). In the course of its inward mi-
2 R.Recheetal.:Ontheobservabilityofresonantstructuresinplanetesimaldisksduetoplanetarymigration
gration, a dust particle can be trapped into exterior MMRs
withaplanet,henceincreasingthecontrastoftheirasymmetric
patterns (Kuchner&Holman 2003; Deller&Maddison 2005).
Planet migration, on the other hand, involves particles of all
sizes, except those ejected by radiation pressure. Many parti-
cles can be trapped in MMRs by a planet migrating outward
in the disk. Each non-resonant particle crossing an MMR has
achancetrappedandsubsequentlymigrating,followingtheres-
onance(Wyatt2003).
SeveralauthorshavestudiedeithertheeffectofP-Rdrag,or
ofplanetmigration,ondisk structures,usingdifferentmethods
(analytic,semi-analyticornumerical)andvariousplanetparam-
eters(massandorbitaleccentricity).Asummaryofthemainpre-
viousstudiesisprovidedinTable1.TheP-Rdragscenariohas
been extensivelystudied fora wide rangeof parameters,while
the migrating planet scenario has been investigated only for a
planet on a circular orbit by Wyatt (2003). It is thus important
to better characterize the latter scenario in order to distinguish Fig.1. Exampleofa planetesimaldiskwithoutoutwardplanet
which of the two dominates the morphology of debris disks. migration, nor inward P-R drag migration of the test particles,
Moreover, a number of studies (LecavelierDesEtangsetal. accordingtoournumericalsimulationsas explainedinSection
1996; Wyatt 2005b; Krivovetal. 2007) have shown that colli- 2.Thestarandplanetlocations,projectedontotheorbitalplane
sions may prevent MMRs from being populated by P-R drag oftheplanet,arerepresentedbylargeredpoints,andtheplanet
since the collision timescale in massive debris disks might be orbitbyathingreenline.Theinitialplanetesimaldiskconsistsof
muchshorterthantheP-Rdragmigrationtimescale. 50000planetesimalsdistributedbetween40and75AU,withthe
Therefore,weproposeinthispapertoextensivelystudythe surfacedensitydistributionproportionalto r−1. Althoughsome
planetmigrationscenario,usingnumericalmodeling,bygener- planetesimalsaretrappedinMMRswiththeplanet,theyarenot
ating a synthetic catalog similar to what has been done for the sufficientlynumeroustogeneratespatialstructures(besidesthe
P-R drag scenario using analytical (Kuchner&Holman 2003) 1:1MMR).SeetheelectroniceditionoftheJournalforacolor
ornumericalstudies(Deller&Maddison2005). We extendthe versionofthisfigure.
pioneeringworkdonebyWyatt(2003)instudyingtheinfluence
of the planet eccentricity on the visibility of the resonant pat-
terns.InSection3,wediscussthecaseofaplanetmigratingon To perform our simulations we have used the symplectic
acircularorlow-eccentricityorbit.InSection4,weextendthis packageSWIFT (Wisdom&Holman1991; Levison&Duncan
studytoplanetsonorbitswitheccentricitiesupto0.7.Thisstudy 1994), to which we have added planetary migration. To do
isgeneralizedtovariousmigrationratesanddiskinitialstatesin this,wepluggedintheWyatt(2003)prescription.Thismethod
Section 5, and compared to previous studies in Section 6. The consists of adding an acceleration in the direction of the or-
limitationsofourapproacharediscussedinSection7. bital motion of the planet, with an intensity equal to: v˙p =
0.5a˙ GM /a3, whereG is the gravitationalconstant, M the
pq ∗ p ∗
2. Numericalmodel stellar mass and a˙ the variation rate of the planet semi-major
p
axisa .Thiscausesachangeintheplanetsemi-majoraxiswith-
p
We consideraplanetarysystemconsistingofastar surrounded
outmodifyingitseccentricity(foraplanetonalow-eccentricity
byaplanetandadebrisdisk.We addressthecaseoflargepar-
orbit;foraplanetonahighereccentricityorbitthechangeisnot
ticles,whichareinsensitivetopressureforces(radiation,stellar
significant)oritsinclination.We donotdiscussheretheorigin
windorgaspressure).Thesimulateddiskisthusratheraplan-
ofthemigration.Itcanbedueeithertothemigrationofalarge
etesimal disk than a debris disk and we only consider gravita-
internalplanet,ortothegravitationalinfluenceoftheplanetes-
tionalforces.Importantly,we also do nottake into accountthe
imals themselves. The most important here is that we keep a˙
p
gravitationalinteractionsbetweenplanetesimalsastheyareneg-
constantduringeachsimulation,generallyat0.5AUMyr−1,to
ligible, nor mutual collisions. Dynamically speaking, the plan-
matchtheWyatt(2003)modelfortheVegadisk.Wehaveused
etesimalsarethusconsideredastestparticles.Atypicalconfigu-
theRMVS3versionoftheSWIFTintegrator,inordertohavea
rationforthesimulationsisaVega-likecentralstar(2.5M )and
⊙ bettermodelingofthecloseencountersbetweentheplanetand
a planetorbitwith a 40AU pericenteratthe starting time.The
theplanetesimals.Inallthesimulations,thesystemevolutionis
initial planetesimal disk consists of 50000 planetesimals dis-
followedfor40Myr.
tributedbetween40and75AUoncircularorbits,withasurface
densitydistributionproportionaltor−1.Thediskmidplanecoin-
cideswiththeorbitalplaneoftheplanet,andtheinclinationsof
3. Planetsonlow-eccentricityorbits
theplanetesimalsarerandomlydistributedwithin±3◦.
Inthismodel,theplanetkeepsaKeplerianorbitaroundthe Thescenarioofa planetona low-eccentricityorbitisthe most
star, or migrates at a constant rate without modification of its studiedcase(seeTable1),forseveralreasons.First,planetswere
eccentricity. This basic model is easy to implement and to an- originallyexpectedtobeonalmostcircularorbitsbecause,dur-
alyze, and corresponds to the case described by Wyatt (2003). ingtheprotoplanetaryphase,circumstellargasforcestheplanets
The goal of this paper is to extend this initial work to a wider toremainonveryloweccentricityorbits.Aplanetonsuchanor-
rangeofplaneteccentricitiesbya numericalstudy.We startby bitthereforecorrespondstothe“standardscenario”.Also,alow
studying planets on low-eccentricityorbits (e < 0.1), and then or zero planetary eccentricity simplifies an analytical analysis
extendourworktolargereccentricities. (Kuchner&Holman 2003). Nevertheless,it mustbe noted that
R.Recheetal.:Ontheobservabilityofresonantstructuresinplanetesimaldisksduetoplanetarymigration 3
Table1.SummaryofrecentpapersonparticlestrappinginMMRswithaplanet.
Authors Method Planetparameters Migration Notes
Massratioa eccentricity
Wyatt(2003) semi-analytic 0.0003to3 0 planet Forcedmigration
Wyatt(2006) semi-analytic none Thisworkextendsthepreviousonetosmaller
particlessensitivetoradiationpressure.
Kuchner&Holman(2003) analytic 0.005to15 0to0.6 particles Onlyresonantparticlestrappedduringmigra-
tionduetoP-Rdrag.
Deller&Maddison(2005) numerical 0.01to3 0to0.7 particles Particles migrate due to P-R drag and solar
wind.
Moro-Mart´ın&Malhotra(2002) numerical 0.05 0 particles StudyoftheKuiperBelt.
Thispaper numerical 0.001to3 0to0.7 planet Onlyplanetesimalsdisk,forcedmigration
a PlanetarymassinJovianmassdividedbystellarmassinsolarmass.
manyoftheextrasolarplanetsdetectedsofar havehigheccen- spatial separation, while resonancesnear the planet are too
tricities1,andwewillthereforeextendourstudytohigheccen- closetoeachothertotraplargepopulations.Thisshowsthat
tricitiesinSection4. anymodelingofstructuregenerationwiththisprocessmust
Although this standard scenario has already been well be done globally, as the variousMMRs compete with each
studied, all its aspects have not yet been investigated. Wyatt othertobefilled.
(2003) studied the case of an outward migrating planet on – ThelongtermevolutionofplanetesimalstrappedinMMRs
a strictly circular orbit, while Kuchner&Holman (2003) and globallyresultsinanincreaseoftheireccentricity.Ifthisec-
Deller&Maddison (2005) studied the case of planets on fixed centricitybecomeshigh enough,planetesimalscan become
low-eccentricityorbits,consideringonlyinwarddustmigration planet-crossing.Thankstotheresonance,theyareneverthe-
due to P-R drag. We propose in this section to numerically lessphase-protectedagainstcloseencounterswiththeplanet.
study a migrating planet on a circular or low-eccentricity orbit Butiftheeccentricitygrowstoohigh,thisphase-protection
tosearchforpossibledifferenceswithrespecttopreviousstud- does not hold any longer,due to strong modulationsof the
ies. angular velocity. Hence MMRs are limited in eccentricity
(Sicardyetal.1993),withe =(2/5m )1/2.Thisconcerns
max p
morespecifically the MMRs that are close to the planet,as
3.1.Generaltrends
the planetesimals may easily become planet crossing. We
Figures 2 to 5 show examples of results obtained with our thus expect the closest MMRs to lose planetesimals when
numerical model. It appears that, with a planet on a low- theyreachagiveneccentricitylimit.ThisisillustratedinFig.
eccentricity orbit, the planetesimals trapped in MMRs are nu- 3wherethelimitsineccentricityoftheresonantpopulations
merousanddominatetheshapeofthedisk.Fourimportantfac- areclearlyvisible.
torsmustbe takeninto accountto determinewhichresonances
governtheaspectofthestructuresinthedisk: Inconclusion,onlyafewresonances,namelythemostexter-
nalofthefirstorsecondorderresonances(4:3,3:2,5:3and2:1
– External MMRs at large a/a are weaker and less resonance)capturemostoftheplanetesimals.However,depend-
p
able to trap numerous planetesimals than closer MMRs ingontheplanetmass,theplanetesimalsthatarenottrappedin
(Kuchner&Holman2003).First-orderresonances(i.e.m = resonancescan,ornot,changetheshapeofthedisk.Inoursim-
n + 1) with large m values are the closest and hence the ulations,planetswitha massabove1Jupitermassejectalmost
strongest MMRs. However, MMRs near the planet are lo- all the non-resonant planetesimals: the MMR structures there-
cated close to each other and compete. A resonance over- fore appear clearly. Below 1 Jupiter mass, planets cannot eject
lap criterion (Wisdom 1980; Duncanetal. 1989) predicts all the non-resonant planetesimals, which can then partly hide
thatfirst-orderresonancesbecomecompletelychaoticwhen theMMRstructures.
m > 0.45µ−2/7 +1, where µ = m /M . This places a limit
p ∗
onthenearestresonancethatcanbepopulated.IntheSolar
3.2.Circularorbits
System, the first completely overlappedMMR is the 17:18
fortheEarthandthe4:5inthecaseofJupiter. In the circular orbit case (Fig. 2 and 3), differencesappear be-
– Wyatt (2003) showed that the planetary mass acts as a tweenthesimulations,dependingontheplanetmass:
threshold for the probability of capture in first-order exter-
nalMMRs:fora fixedmigrationrate,theprobabilitydrops – For an Earth mass planet, non resonant planetesimals are
quicklyto0belowacertainplanetarymass,whileabovethis stillboundtothesystemandhidealltheresonantstructures.
masstheprobabilitygrowsquicklyto1.ForMMRsofhigher However,asmallholeattheplanetlocationcanbeobserved.
order,thetransitionislesssharp. – ForaNeptunemassplanet,twoclumpsofequaldensityare
– An MMR that traps all the planetesimals at a given semi- generatedbythe3:2resonanceandtheyarelocatedinoppo-
major axis a stops the growth of any other resonance that sitionwithrespecttothestar.Italsoappearsthattheplanet
r
reachesthisa afterward.Resonanceslike3:2or5:3canthus isnotmassiveenoughtocapturemanyplanetesimalsinthe
r
trapalargenumberofplanetesimalsthankstoalargeenough 2:1resonanceandthusdoesnotgeneratethinringsatlarge
distanceslikemoremassiveplanets.Non-resonantplanetes-
1 www.extrasolar.eu imalscreatearinginsidetheplanetorbit.
4 R.Recheetal.:Ontheobservabilityofresonantstructuresinplanetesimaldisksduetoplanetarymigration
Fig.2.Spatialdistributionofplanetesimalsforaplanetonastrictlycircularorbit.Thestarandplanetlocations,projectedontothe
orbitalplane of the planet, are representedby larger red points, and the planetorbitby a thin green line. The 4 rows correspond
respectivelyto Earth mass, Neptunemass, Saturn mass and3 Jupitermass planets, fromtop to bottom.The 3 columnsshow the
diskafter5,15and40Myr.Theinitialplanetesimaldiskconsistsof50000planetesimalsdistributedbetween40and75AU,with
thesurfacedensitydistributionproportionaltor−1.SeetheelectroniceditionoftheJournalforacolorversionofthisfigure.
– For a Saturn mass planet, the 2:1 resonant pattern appears than 180◦ in longitude. The change in the 2:1 pattern is
inadditiontothe3:2one.The2:1MMRproducesonlyone discussed in several papers (Chiang&Jordan 2002; Wyatt
clump near one of the two generatedby the 3:2 resonance: 2003;Murray-Clay&Chiang2005):thisresonancehastwo
theglobalstructurethusbecomesasymmetric.But,asinthe librationcentersbuttheydonothavethesametrappingprob-
previous case, the non-resonant planetesimals are still nu- abilityandonlymassiveplanetscanpopulatethesecondli-
merousintheinnerpartofthediskandpartlyhidethereso- brationcenter.
nantstructures.
– Fora3Jupitermassplanet,the2:1resonantpatternchanges
and generates two clumps of equal density near those of 3.3.Low-eccentricityorbits
the 3:2 resonance. The global structure is symmetric but
MostofthestructuresdiscussedinSection3.2disappearasthe
the two clumpsare no longerin opposition,because the li-
planeteccentricityincreases(Fig.4).Thelibrationamplitudeof
bration centers of the 2:1 resonance are separated by less
resonantplanetesimalsindeedincreases,smoothingthe density
R.Recheetal.:Ontheobservabilityofresonantstructuresinplanetesimaldisksduetoplanetarymigration 5
Fig.3. Same simulationsas in Fig. 2, with the planetesimalsrepresentedin a (semi-majoraxis, eccentricity)plane.The plotting
conventionsarethesameasinFig.2.SeetheelectroniceditionoftheJournalforacolorversionofthisfigure.
waves along the orbit. For eccentricities between 0.05 and 0.1, 4. Planetsoneccentricorbits
thedisklookslikearingwithaholeatthelocationoftheplanet,
Ateccentricitieslargerthan0.1,planetscansignificantlymodify
or no longer shows structures for the lower mass planets. The
thediskgeometryandproduceadynamicallywarmdiskwhere
rings are not only due to the 1:1 resonant planetesimals coro-
all planetesimals may reach high eccentricities. Analytical de-
tating with the planet, but are also populated by other major
velopments are more complex, but are still feasible as shown
resonances (e.g., 2:1,3:2). Massive planets (last row of Fig. 4)
byKuchner&Holman(2003)forresonantplanetesimalsandby
arelesssensitivetothiseffectbecausetheycanmoreefficiently
Wyattetal. (1999) fornon-resonantplanetesimals.So far,only
eject planetesimalsduringclose encounters,even if they are in
theP-Rdragmigrationscenariohasbeentestedforsuchplane-
theMMRs.Onlyresonantplanetesimalswithlowlibrationam-
taryorbits.Weexploreheretheplanetmigrationscenariointhe
plitudes can survive and the disk remains structured as in the
caseofeccentricorbits(Fig.6and7).
strictlycircularcase.
According to previous works, three phenomena should
change in our simulations with respect to the low-eccentricity
orbit case. First, close encounters between the planet and the
planetesimalsaremorefrequent,increasingthedepletionrateof
thedisk.Asthe probabilityofejectionincreaseswiththe plan-
6 R.Recheetal.:Ontheobservabilityofresonantstructuresinplanetesimaldisksduetoplanetarymigration
Fig.4. SameasFig.2,forsimilarplanets,butonalow-eccentricityorbit(e =0.05).SeetheelectroniceditionoftheJournalfora
p
colorversionofthisfigure.
etarymass, themore massivethe planet,the moredepletedthe The depletion rate is so efficient that the disk is almost en-
disk.Second,eventhe non-resonantplanetesimalssee theirec- tirelydepletedduringour40Myrsimulations,exceptinthecase
centricitiesrisesignificantlyduetothegravitationalperturbation of an Earth mass or Neptune mass planet on moderate eccen-
of the planet. Finally, the trapping probability is also modified tric orbits (below 0.2 or 0.3). An Earth mass planet on a more
(Quillen 2006): increasing the planetary eccentricity decreases eccentricorbit(above0.5)canalsoproducetransientcollective
thetrappingprobabilityofthefirstorderresonancesbutcanin- non-resonant effects, spatially fixed, in the azimuthal distribu-
creaseitforhigherorderresonances.Withaplanetonaneccen- tionoftheplanetesimals(Fig.9)beforethediskisdepleted,as
tricorbit,theresonantplanetesimalsarealsonotwellprotected explainedbelow.
against close encounters, which limits even more the number
ofplanetesimalsinMMRs.Themigrationthuscannotpopulate
theMMRsenoughtogeneratedetectablepatterns:forstructures 4.1.Non-resonantstructures
generatedby a planeton a moderateor high eccentricity orbit,
The theoretical background for the dynamics of the planetesi-
theMMRs,andthusthemigrationoftheplanet,isnolongeran
malsinoursimulationsistherestrictedthree-bodyproblem,i.e.,
important factor whatever the planet mass as the non-resonant
a problemwherea mass-less test particleorbitinga star isper-
dynamicsdominatestheshapeofthedisk.
turbedbyaplanetorbitingthestaronanunperturbedKeplerian
R.Recheetal.:Ontheobservabilityofresonantstructuresinplanetesimaldisksduetoplanetarymigration 7
Fig.5..SameasFig.3,forsimilarplanets,butonalow-eccentricityorbit(e =0.05).SeetheelectroniceditionoftheJournalfor
p
acolorversionofthisfigure.
orbit. We restrict ourselvesto the planar case for simplicity. In coupledwiththemeananomalythathasbeenremovedfromthe
this framework,the Hamiltonianof the problemis (see, for in- Hamiltonianbytheaveragingprocess.ThesecularHamiltonian
stanceMorbidelli&Moons1993) oftheplanarproblemturnsouttohaveonlyonedegreeoffree-
dom.Itdependsforinstanceonlyontheeccentricityoftheplan-
GM 1 r·r
H =− ∗ −Gm  − p , (1) etesimaleandofitslongitudeofperiastronνwithrespecttothat
0 2a p(cid:12)(cid:12)r−rp(cid:12)(cid:12) r3p  oecfctehnetrpilcaintyete.F,owreancayngdivraewnfiaxpehdavsealpuoerstrfaoirta(i.aen.,dlefvoerlthcuerpvleasnoeft
whereaistheosculatin(cid:12)gsemi(cid:12)-majoraxisoftheorbitoftheplan- p
Hamiltonian)ofthedynamicsinan(ν,e)plane.
etesimal,M isthemassofthestar,m thatoftheplanet,Gisthe
∗ p
constantofgravitation,andrandr aretheheliocentricposition Twoexamplesfore =0.1ande =0.5areshowninFig.8.
p p p
vectorsoftheplanetesimalandoftheplanetrespectively.Ifthe They both correspond to a = 1.3a (a is the semi-major axis
p p
testparticleisnotlockedintoamean-motionresonancewiththe oftheplanet).Thewholedynamicalproblemobviouslysimply
planet,itssecularmotionisinvestigatedbyperformingadouble scales with a , so that a only needs to be given in units of a .
p p
temporalaveragingofH overtheorbitalmotionsoftheplanet Thepresentcase (a = 1.3a ) correspondsto a planetesimalor-
0 p
andoftheplanetesimal(Beust&Dutrey2006).Inthiscontext, biting outside the planet’s orbit, like those we are simulating.
the semi-majoraxis a is a secular constant, as it is canonically NotealsothatthetopologyoftheHamiltonianisindependentof
8 R.Recheetal.:Ontheobservabilityofresonantstructuresinplanetesimaldisksduetoplanetarymigration
Fig.6. SameasFig.2,forsimilarplanets,butonamoderateeccentricityorbit(e =0.3).SeetheelectroniceditionoftheJournal
p
foracolorversionofthisfigure.
themassm oftheplanet,becausethenon-constantpartofH is to the apoastron of their orbits, as due to Kepler’s second law,
p 0
proportionaltom .HencetheplotsinFig.8holdforanyplane- theplanetesimalsspendmostoftheirtimenearapoastron.This
p
tarymass.Theplanetarymassm onlyaffectsthespeedatwhich is the origin of the clumps we obtain in our simulations with
p
theplanetesimalmovesalongtheHamiltonianlevelcurves(the low-massplanets(Fig.9).
speedis∝mp). So, why does this not hold for more massive planets? The
secular dynamics described above is valid as long as the plan-
The plotsin Fig. 8 holdfor a = 1.3a , but forothervalues
p etesimaldoesnotundergoanyclose encounterwith the planet.
of a, we have similar plots. Conversely,the shape of the phase
Inthecaseofacloseencounter,theorbitoftheplanetesimalis
portrait depends critically on e . We see that for a small e , a
p p suddenlychanged,anditisoftenejected.ManyregionsinFig.8
planetesimalwithasmallinitialewillkeepesmallforever.For
correspond to a planet crossing orbit. The probability of hav-
a largee however,any planetesimalwith a small initial e will
p ingacloseencounterwiththeplanetwithinagiventimespanis
bedriventohighevaluesandν≃0.Startingfromapopulation
higher if the planet is more massive. It scales as m2/3, because
of planetesimalswith negligibleeccentricities, we end up after p
a certain delay with many highly eccentric planetesimals with theHillradiusrH oftheplanetscalesasm1p/3,andtheencounter
theirlinesofapsidesmoreorlessalignedwiththatoftheplanet cross-sectionisexpectedtoscaleasr2.Themassratiobetween
H
ν ≃ 0. Thisnaturally generatesa clump of planetesimals close a 3 Jupiter mass planet and an Earth-sized planet is ∼ 1000.
R.Recheetal.:Ontheobservabilityofresonantstructuresinplanetesimaldisksduetoplanetarymigration 9
Fig.7. SameasFig.3,forsimilarplanets,butonamoderateeccentricityorbit(e = 0.3).SeetheelectroniceditionoftheJournal
p
foracolorversionofthisfigure.
We thus expect a planetesimal to undergo 100 times more en- encounterswiththeplanet,whichfinallydepletemostofthedisk
counterswiththefirstplanetthanwiththesecond.Finally,with after35Myr,inoursimulations.
massiveplanets,mostoftheplanetesimalsaresubjecttoaclose
encounterwiththeplanetwithinthetimespanofthesimulation
described in Fig.6. This is why the disk appearsso depleted at 5. Generalization
the end. Conversely, for low-mass planets, the close encounter
As explained in the previoussections, we have run many sim-
probabilityissolowthatmanyplanetesimalskeepfollowingthe
ulations in order to sample correctly the parameter space of
seculardynamicsuntiltheendof therun.Therefore,theyhave
planetary eccentricity versus planetary mass. We can thus ad-
enoughtimetogenerateastrongasymmetricclump.
dress the question of the visibility of asymmetric structures in
Westressherethatthisclumpisnotduetoanymean-motion a disk. Asalreadymentioned,theshape ofa disk is dominated
resonance.Thereisthusnoneedforplanetmigrationinthiscase eitherbyresonantorbynon-resonantplanetesimals.Theregion
and this may appear as an alternative scenario to mean-motion whereMMRsdominatethediskshapecorrespondtoplanetson
resonance for generating transient clumps. Nevertheless, even low-eccentricityorbits.Inthisregion,twosituationscanoccur:
withanEarthmassplanet,theseclumpsdonotlastaslongasthe planetscangenerateclearresonantpatternswithseveralvisible
resonant clumps. Planetesimals are not protected against close clumpsin the disk (generallywhile on circularor verylow ec-
10 R.Recheetal.:Ontheobservabilityofresonantstructuresinplanetesimaldisksduetoplanetarymigration
Fig.8. Phase portraits(Hamiltonianlevelcurvesin an (ν,e) plane) of the secular non-resonantdynamicsin the planarrestricted
threebodyproblem,forafixedsemi-majoraxisa = 1.3a ,andplaneteccentricitiese = 0.5(leftplot)ande = 0.1(rightplot).
p p p
Theredlineseparatesregionswheretheplanetesimalorbitdoesnotcrossthatoftheplanet(loweccentricitiesfore = 0.1anda
p
smallislandaroundν=0fore =0.5)fromregionswherebothorbitscrosseachother.SeetheelectroniceditionoftheJournalfor
p
acolorversionofthisfigure.
simulation,initialdistributionoftheplanetesimals...)whilethe
limitsofthethreezonesarequiteindependent.
However,wehaveassumedforthesesimulationsaconstant
migrationrateof0.5AUMyr−1andadiskofplanetesimalsini-
tiallyoncircularorbits.Withdifferentassumptionstheoutcomes
ofthesimulationscouldbechanged:we havethusinvestigated
these two parameters in order to discuss the robustness of our
conclusions.Foragivenplanetarymassandeccentricity,weex-
pectthestructurestochange,asthetrappingprobabilitydepends
onthemigrationrateandonthe planetesimaleccentricity.But,
again,ourmainfocusistodeterminateiftheresonantstructures
arevisibleornot.Forinstance,inthelow-eccentricityorbitcase,
NeptunemassandJupitermassplanetsdonotproducethesame
structuresbuttheyhavethesamesensitivitytotheplaneteccen-
tricity. Here, we investigate if the migration rate or the initial
planetesimaleccentricitychangethese conclusions.The results
oftheseadditionalsimulationsaresummarizedinTables3 and
Fig.9. An Earth mass planet on a very eccentric orbit (ep = 4,inthesamemannerasinTable2forthenominalcase.Table3
0.6).TheplottingconventionsarethesamethatFig.1.Thedisk correspondstosimulationswithdifferentmigrationratesbutun-
structure does not rotate with the planet and is spatially fixed. excitedinitialplanetesimaldisks.Table4describessimulations
See the electronic edition of the Journal for a color version of withinitiallyexciteddisks.
thisfigure.
5.1.Migrationrate
centric orbits) or produce smooth patterns, with only a hole at Inourwork,themigrationrateparameterwaschosentomatch
the planet location as the visible structure (generally while on the best fit obtained by Wyatt (2003) for the Vega debris disk
anorbitwithaneccentricitybetween0.05and0.1).Concerning to allow direct comparison. In his paper, Wyatt discussed the
thenon-resonantplanetesimals,Earthmassplanetsoneccentric impactofthemigrationrateintherestrictivecaseofaplaneton
orbits can generate observable structures by secular perturba- a circular orbit, and has shown that the trapping probability in
tions.Outsidetheseregions,thedisksdonotshowanyobserv- anMMRincreaseswithdecreasingmigrationrates(orwhenthe
able structures, when they are not totally depleted. The results starislessmassive).
forallthesesimulationsaresummarizedinTable2andFig.10. Wehavethusperformedsimulationswithalower(0.05AU
Threemainregionscanbeidentified.InzonesIandII,observ- Myr−1) and a higher (5 AU Myr−1) migration rate than previ-
ablestructuresinthediskaregeneratedbyMMRswhileinzone ously(Table3).Overall,ourresultsareingoodagreementwith
III, transient structures are generated by non-resonant mecha- those of Wyatt (2003), even for eccentric orbits: the trapping
nisms.Intheremainingregion,thediskdoesnotshowanystruc- probabilityincreaseswhenthemigrationratedecreases.Forex-
ture.InzoneI,MMRscreateclumpydiskswhileinzoneIIthey ample, with a low migration rate, a Neptune mass planet traps
generate a smooth disk with a hole at the planet location. This planetesimalsinthe2:1MMR,whileaSaturnmassplanetpop-
figurealsoshowsthefractionofplanetesimalsstillboundtothe ulatesthesecondlibrationcenterofthe2:1resonance.
systemafter40Myr(backgroundcolor).However,thisquantity Although the disk resonant shape is modified because the
is sensitive to several parameters (stellar mass, duration of the populated MMRs change with the migration rate, the depen-