Outer edges of debris discs how sharp is sharp

cESO2008󰀎

Outeredgesofdebrisdiscs

howsharpissharp?

P.Th´ebault1,2,Y.Wu3

1

arXiv:0801.3724v3 [astro-ph] 4 Feb 200823

StockholmObservatory,AlbanovaUniversitetcentrum,SE-10691Stockholm,Sweden

ObservatoiredeParis,SectiondeMeudon,F-92195MeudonPrincipalCedex,France

DepartmentofAstronomyandAstrophysics,UniversityofToronto,50St.GeorgeStreet,Toronto,ONM5S3H4,Canada

Received;accepted

ABSTRACT

Context.Ringsorannulus-likefeatureshavebeenobservedinmostimageddebrisdiscs.Outsidethemainring,whilesomesystems(e.g.,βPictorisandAUMic)exhibitsmoothsurfacebrightnessprofiles(SB)thatfalloffroughlyas∼r−3.5,others(e.g.HR4796AandHD1396)displaylargedropsinluminosityatthering’souteredgeandsteeperradialluminosityprofiles.

Aims.Weseektounderstandthisdiversityofouteredgeprofilesunderthe“natural”collisionalevolutionofthesystem,withoutinvokingexternalagentssuchasplanetsorgas.

Methods.Weuseamulti-annulusstatisticalcodetofollowtheevolutionofacollisionalpopulation,ranginginsizefromdustgrainstoplanetesimalsandinitiallyconfinedwithinabelt(the”birthring”).Thecrucialeffectofradiationpressureonthedynamicsandspatialdistributionofthesmallestgrainsistakenintoaccount.Weexplorethedependenceoftheresultingdiscsurfacebrightnessprofileonvariousparameters.

Results.Thedisctypicallyevolvestowarda“standard”steadystate,wheretheradialsurfacebrightnessprofilesmoothlydecreaseswithradiusasr−3.5outsidethebirthring.Thisconfirmsandextendsthesemi-analyticalstudyofStrubbe&Chiang(2006)andprovidesafirmbasisforinterpretingobserveddiscs.Deviationsfromthistypicalprofile,intheformofasharpouteredgeandasteeperfall-off,occurfortwo”extreme”cases:1)Whenthebirthringissomassivethatitbecomesradiallyopticallythickforthesmallestgrains.However,therequireddiscmassisprobablytoohighheretoberealistic.2)Whenthedynamicalexcitationofthedust-producingplanetesimalsissolow(and≤0.01)thatthesmallestgrains,whichotherwisedominatetheopticaldepthofthesystem,arepreferentiallydepleted.Thislow-excitationcase,althoughpossiblynotgeneric,cannotberuledoutbyobservationsformostsystems,.

Conclusions.Our“standard”profileprovidesasatisfactoryexplanationforalargegroupofdebrisdiscsthatshowsmoothouteredgesandSB∝r−3.5.Systemswithsharperouteredges,barringotherconfiningagents,couldstillbeexplainedby“natural”collisionalevolutioniftheirdynamicalexcitationisverylow.Weshowthatsuchadynamically-coldcaseprovidesasatisfactoryfittothespecificHR4796Aring.

Keywords.stars:circumstellarmatter–stars:individual:βPictoris–stars:individual:HR4796A–planetarysystems:formation

1.Introduction

1.1.theubiquityofring-likefeaturesDustydebrisdiscshavebeendetectedbytheirinfraredex-cessaround∼15%ofnearbymainsequencestars(e.g.Backman&Paresce1993).Morethanadozenofthesediscshavealsobeenimaged,mainlyinscatteredlight,sincetheinitialobservationoftheβPictorissystembySmith&Terrile(1984).Oneunexpectedresultfromtheseimagesisthatalmostnosys-temdisplaysasmoothextendedradialprofile:theusualmor-phologyisthepresenceofrings(orannuli)wherethebulkofthedustpopulationislocated.ThisringmorphologyisinfactsocommonthatStrubbe&Chiang(2006)pointedoutthatthedebrisdiscphenomenoncouldmoreappropriatelyberenameddebris“ring”phenomenon.Evenforthearchetypaldebris“disc”βPictoris,whichhasbeenimagedfrom5toafewthousandAU,thebulkofthedustisprobablyconcentratedinarathernarrowregionbetween80and120AU(e.g.Augereauetal.2001).Oneofthefewsystemsactuallyresemblinganextended“smooth”

2Th´ebaultandWu:OuterEdgesofdebrisdiscs

Table1.Geometryandsurfacebrightnessprofileforaselectionofdebrisdiscsystemsresolvedinscatteredlightimages.

Systema

Orientation

DetectedRadialExtentb

SurfaceBrightnessSB∝rα

Reference

Formanyofthesesystems,manyadditionalfeatures,i.e.,warps,clumps,etc.,havebeenobserved(itisespeciallytrueforβPictoris)butwe

focushereonthemainissueofaverageradialprofilesb

Theradialextentsandsurfacebrightnessprofilesaregivenforregionsbeyondthelikely“birthring”–theregionwherescatteringluminositypeaksandwheremostparentbodiesarebelievedtoreside(seeSec.2)

a

Thepresenceofringfeaturesiscommonlyattributedtosomesculptingmechanismsincluding,inparticular,thepresenceofamassiveplanet(see,forinstance,Quillen(2007a)forFomalhaut,Freistetteretal.(2007)forβPictoris,Wyattetal.(1999)forHR4796A,orthemoregeneralstudyofMoro-Martin&Malhotra(2005)).Thegravitationaleffectofsuchaplanetcantruncateorcreategapsinthedisc,eitherdi-rectlyonthedustparticlesorindirectlyontheplanetesimalsthatproducetheseparticles.1.2.outeredgesNevertheless,whileplanetsmaybeanaturalandeasyexpla-nationforsculptingtheinneredges,outeredgesareadifferentproblem.Theyaredifficulttoexplaininthelightofonewellestablishedfactaboutdebrisdiscs,i.e.thattheobserveddustisnotprimordialbutsteadilyproducedbycollisions,throughacol-lisionalcascadestartingatmuchlargerparentbodies,maybeintheplanetesimalsizerange(e.g.Lagrangeetal.2000).Inthisre-spect,evenifthereisasharpouteredgefortheparentbodypop-ulation,collisionswouldconstantlyproduceverysmallgrainsthatwouldbelaunchedbyradiationpressureontoeccentricorevenunboundorbits,thuspopulatingtheregionbeyondtheouteredgeanderasingtheappearanceofanarrowringovertimescaleswhichmightbeshorterthanthoseforgravitationalsculptingbyaplanet.Thisissueisacriticalone,sincethesesmallgrainsdom-inatethetotalgeometriccrosssectionandthusthefluxinscat-teredlight(seethediscussioninTh´ebault&Augereau2007).Onepossibleconfiningmechanismforthesharpouteredgeisgas.Fordiscstransitingbetweengas-dominatedphase(proto-planetarydiscs)todust-dominatedphase(debrisdiscs),narrowdustringsmayarise(Klahr&Lin2005).Besla&Wu(2007)furtherdemonstratethatthereexistsaninstabilitywithwhichtheresidualgascollectsgrainsofvarioussizes(eventhosesubjecttoradiationpressure)intoanarrowbelt.However,thismechanismrequiresatleastcomparableamountofgasanddust.Thismaybedifficulttojustifyformostdebrisdiscs,whichareevolvedsys-temswheretheamountofgasisprobablytoolowtopreventthesmallestgrainstobelaunchedontoveryeccentricorbitssmooth-ingoutanysharpouteredge.

“Razorsharp”outeredgesarethusverydifficulttoexplaininthepresenceofthisunavoidableoutwardlaunchingofsmallgrains.However,althoughnoperfectabruptouteredgehasin-

deedbeenobserved1(unlikeinneredges,whichareinsomecases,likeFomalhaut,almostrazorsharp),agreatvarietyofouteredgeprofilesdoesexist,fromrelativelysmoothtoverysteep(seeTab.1).

Inthisstudy,weaddresstheissueofhowthesedifferentprofilescanbephysicallyachieved:canthisdiversitybeex-plainedbythesole“natural”evolutionofacollisionalactivediscsteadilyproducingsmall,radiation-pressureaffectedgrains,oris(are)additionalmechanism(s)needed?Weconsiderinitialcon-ditionswhichareapriorithemostfavourableforcreatingsharpedges,byassumingapopulationoflargeparentbodiesconfinedwithinanannuluswithanabruptcutoffatitsouteredge.Howthisinitialconfinementmayhavecomeaboutisitselfanin-terestingquestionbutisnotthefocusofthecurrentpaper(seehoweverthediscussioninSec.5.2).Theoutcomeweconsiderasareferenceforourinvestigationistheradialsurfacebrightness(SB)profileinscatteredlight,sincethisisanobservablewhichisreasonablywellconstrainedformostimageddebrisdiscs(eitherdirectlyobservedorobtainedbyde-projection).Weconsiderthenominalcaseofadiscseenedge-on,butresultscaneasilybeextrapolatedtoface-onsystems,sinceSBprofilesforbothori-entationstendtowardsthesameradialdependencefarfromthebirthring(giventhesameradialdustdistribution).

Inseveralpreviousstudies,alladdressingthespecificβPiccase(Lecavelieretal.1996;Augereauetal.2001;Th´ebault&Augereau2005),ithasbeenarguedthatthe“nat-ural”SBprofileoutsidethecollisionalyactiveparentbodiesbelt,or“birthring”,fallsoffas∼r−5.ThisisbasedontheassumptionthatallparticlesproducedinthebirthringhaveasizedistributionwhichscalesasdN/ds∝s−3.5,(asexpectedforanidealizedinfinitecollisionalcascadeatequilibrium,seeDohnanyi1969),downtotheradiationblow–outlimits=s0.5(wheretheratioofradiationpressuretogravityβ=0.5).Thesmallestradiationpressure-affectedgrains,whichdominatethelightreceivingarea,arethendilutedalongtheireccentricor-bit.ThisgeometricalspreadresultsinSB∝r−5.However,Strubbe&Chiang(2006)arguedthat,sincehigh-βgrainsspendalongtimeinthecollisionalyinactiveregionbeyondthebirthring,thedN/ds∝s−3.5collisionalequilibriumlawshouldonlyapplytothesmallfractionfofthesegrainswhicharepresent

Th´ebaultandWu:OuterEdgesofdebrisdiscs3

inthecollisionalyactivebirthring.Thisresultsina1/fexcess

ofthedisc-integratednumberofsmall.5grains,whichinturnre-sultsinaflatterSBprofileinr−3.TheyappliedtheirtheorytotheAUMicdisc2andreproducedtheobservedSBprofile,spectralenergydistributionanddisccolour.Strubbe&Chiang(2006)furtherarguedthattheobservedSBprofiledependsonlyweaklyontheradialandsizedistributionsofgrainswithinthebirthring.Thediscswhichexhibitafall-offsharperthanr−3.5arethuspuzzlinginthefaceofthistheory.

TheinnovativemodelofStrubbe&Chiang(2006)isbuildonanalyticalderivationsandMonte-Carlomodelingwhichdidnotactuallytreatthecollisionalevolutionofthesystemandre-liesonseveralsimplifyingassumptions.ThemainoneisthatthesizedistributionisfixedandisassumedtofollowtheidealizeddN/ds∝s−3.5scaling(correctedbythefraction1/f),whereasseveralstudieshaveshownthatthislawcannotholdinrealsys-tems(seeTh´ebault&Augereau2007,andreferencestherein)3.Anotherissueisthatwhenevaluatingcollisionallife-times,onlytheverticalvelocityofthegrainswastakenintoaccount,thusne-glectingtheirradialmovementwhichcanbeappreciable,ifnotdominantforthesmallestgrains.Finally,thespecificdynamicsofthesmallradiation-pressure-affectedgrains,inparticularthefactthattheysuffermuchmorefrequentcollisionsandatmuchhighervelocities,isnottakenintoaccount.

2.OurApproach

Were-addresstheseissuesusinganumericalapproachquantita-tivelyfollowingthecollisionalevolutionofthefullsystem.Westartwithabirthringofparentbodiesinaperfectlyconfinedannulusandletitcollisionallyevolve.Thetemporalaswellasspatialevolutionofthesizedistributionarefollowed,takingintoaccounttheradialexcursionsofhigh–βparticles.Aspreviouslymentioned,wederiveforeachsimulationthesurfacebrightnessprofileinscatteredlight.2.1.numericalmodelWeuseastatisticalparticle–in–a–boxmodeltofollowtheevolu-tioninsizeandspatialdistributionofapopulationofcollision-allyinteractingbodies.Thiscodehasinitiallybeendeveloped,initssingle–annulusversion,forthestudyoftheinnerβPicdisc(Th´ebaultetal.2003),andlaterupgradedtoamulti–annulusversion(i.e.,with1-Dradialresolution)forthestudyofcolli-sionalprocessesinextendeddebrisdiscs(Th´ebault&Augereau2007).Thedetaileddescriptionofthecodecanbefoundinthesetwopapers,andherewerecallsomeofitsmaincharacteristics.Theentiresystemisspatiallydividedintondialannuli.Withineachannulusofindexia,theaconcentricra-solidbodypop-ulationisdividedbysizeintonrangespanningfromkilometretosbinsthatcoverabroadsizemicron.Withastandardlog(2)sizeincrement,thisrequiresns∼100.Evolutionoftheparticlenumberwithinone(ia,i)bin(ibeingthesizedistributionindex)iscontributedbyalldestructiveimpactsbetween(ia,i)objectsandbodiesfromother(ia′,i′)binsaswellasallimpactsbetweenothersizebinsproducingnew(ia,i)objects.Collisionratesareestimatedstatistically.Theserates,aswellascollisionoutcomes,

1/2

4

󰀏e2󰀐+󰀏i2󰀐󰀊󰀏vkep(ia)󰀐

(1)

wherevForkepthe(ia)istheKeplerianvelocityatradialdistancersmallestparticles,theeffectofradiationpressure,ia.

whichplacesobjectsonhighlyeccentricorbits,istakenintoac-count.Interannuliinteractions,inducedbythesignificantradialexcursionofthesebodiesareconsidered,and∆vN–bodyruns.

(ia,i,ia′,i′)arede-rivedthroughseparatedeterministicCollisionoutcomesaredividedinto2types,crateringandfragmentation,dependingontheratiobetweenthespecificim-pactingkineticenergyEcolandthespecificshaterringenergyQ∗,whichdependsonobjectsizesandcomposition.Inbothregimes,thesizedistributionsofthenewlyproducedfrag-mentsarederivedthroughdetailedenergyscalingprescrip-tions,whicharepresentedatlengthinTh´ebaultetal.(2003)andTh´ebault&Augereau(2007).Possiblereaccumulationontotheimpactingobjectsisalsoaccountedfor.2.2.SetUpOurnumericalmodelrequiresthefollowinginputs:thering’sav-eragedistancefromthestar,rBR,itsradialwidth∆re󰀐andinclinations󰀏i󰀐oftheparentBR,theaver-agefreeeccentricities󰀏bodies(nonaffectedbyradiationpressure)andtheinitialparticles’sizedistributionandtotalmass.WechosetoparameterizethelatterbyMdust,thetotalmassofobjectswithsizessDuetothenumericalcostofthedetailedsizedistributionevolutionprocedure,󰀏e󰀐isassumedfixed,independentofposi-tionandtime.Thefirstindependencyisjustifiedaswearecon-sideringarelativelynarrowbirthring.Thetimeindependencyisanacceptablesimplificationwhenconsideringthattheeccen-tricitydispersionisimposedbythelargestbodiespresentinthesystem(e.g.Quillen2007b)andthat,accordingtoourcurrentunderstandingofdebrisdiscs,i.e.systemsinwhichthebulkoftheplanetesimalaccretionprocessisalreadyover,thesebodiesshouldbelargeplanetaryembryosthataretoobigandisolatedtobesignificantlyaffectedbycollisionalerosionoverthe∼107−108yrstimescaleconsideredhere(see,e.g.Wyattetal.2007;L¨ohneetal.2007).Thisisalsowhythesedynamics-imposingembryosareleftoutofthecollisionalcascadenumericallystud-iedhere(formoreonthesubject,seeTh´ebault&Augereau2007,aswellasthediscussioninSec.5.1.2).

Forthesakeofclarity,weconsideranominalcase(Tab.2)withset-upmatchingascloselyaspossibletheβPictorissystem,i.e.Mdust=0.1M⊕and󰀏e󰀐=0.1(e.g.Augereauetal.2001).Fortheradiationpressureblowoutsize,wetakes0.5=5µm,thevaluederivedforcompactsilicatesaroundaβPiclikeA5Vstar.Thesystemisdividedintotworadialzones:

–Theparentbodyzone,or“birthring”,locatedbetweenrandrthecenteratrinwhereoutwithwefollowthecollisionalBRanddividedinto6annuli,evolutionofthewholesolid

4Th´ebaultandWu:OuterEdgesofdebrisdiscs

Table2.Nominalcasesetup.Thefieldsmarkedbya∗areex-ploredasfreeparametersinthesimulations.

bodypopulation,fromamaximumsizesmax=10kmthat

sitsintheplanetesimalsizerangetoaminimumsizes2µmbelowtheradiationblow-outlimit.Theinitialsizemin=dis-tributionintheentirerangeisassumedtofollowtheideal-izedcollisional“equilibrium”distribution,dN(s)0∝s−3.5ds(Dohnanyi1969).Ourrunsshowthatthischoiceisnotcru-cial:intherelevantdust–sizerange,thesizedistributionisquicklyrelaxedtowardanewsteadystatewithaprofilethatisindependentoftheinitialchoiceandthatdeviatessignif-icantlyfromaDohnanyi-likepowerlaw(seesection4andFig.9).

–Theouterzone,whichisdevoidofparticlesatthebeginningoftherunsandgetsprogressivelypopulatedbysmallgrainscomingfromthebirthring.Consequently,weonlyheregrainswiths≤s∗max,wheres∗

follow

tothebiggestgrainsabletoleavemaxroughlycorrespondstheparentbodyregionandistakenconservativelytohavearadiation–to–gravity–

ratioofβ=0.1.Spatially,thisregionisdividedinto3annulijust(s∗

max)

outsidethemainringplusoneadditional,in-finitelyextended“buffer”annulus.Withinthelatterzone,nocollisionalevolutionismodeled,andonlytheorbitalevolu-tionofthegrainsisconsidered:eitherescapeofthesystemforunboundgrainsorprogressiontotheapoastronandre-turntotheinnerannulifortheboundones.Theradialex-tentoftheouterzone(notincludingthe“buffer”annulus)issetto∼4rBR,whichistypicallytheextentofthe“outer”re-gionconsideredforthe2mostfamousbirth-ring/outer-zonesystems,i.e.,βPicandAUMic(e.g.Augereauetal.2001;Strubbe&Chiang2006).

Theradialsurfacebrightnessprofileinscatteredlightisthensyntheticallycomputedusingthedustsizeandradialdistribu-tions,assuminganr−2dilutionofthestellarfluxandisotropicscattering(althoughdifferentscatteringpropertiesarealsoex-plored).

Weexplorearoundthenominalset-up(seeTab.2),especiallyforthetwofundamentalparameterswhichareMdustandthedy-namicalexcitationofthesystem(asparameterizedby󰀏e󰀐).

3.NumericalResults

3.1.nominalcase:smoothedgeandr−3.5profileFig.1presentsresultsobtainedforourβPic–likenominalcase.Ascanbeclearlyseen,thesystemrapidly(∼105yrs)reachesasteadystateafterwhichtheSBprofilenolongerevolvessignifi-cantly.Westoptheintegrationat107yrs.This∼105yrstimescaleisthetimeittakesforthesmallgrainsthatfilltheouterradialzonetoreachanequilibriumbetweencollisionalproductionanddestruction.Adirectconsequenceofthisfastevolutionisthattheinitialsharpouteredgeoftheparentbodyregionisquickly

Fig.1.Evolutionofthemidplaneluminosityprofileinscatteredlight.Nominalcase:parentbodyring[80,120]AU,M0.1M,󰀏e󰀐=0.1.Eachmidplaneluminosityhasbeenrenormal-dust=⊕izedtoitsvalueat50AU.Thedarkgreyarearepresentsthepar-entbodyregionwhereallmass−3.5isinitiallylocated.Thenarrowlightgreyarearepresentsarslopereconstructedbackwardsfromthefinalluminosityvalueat400AUwitha±15%width.

Fig.2.Nominalcaseatt=107yrs.Respectivecontributionsofdifferentgrainpopulations(parameterizedbytheirβvalue)tothetotalscatteredflux,asfunctionsoftheradialdistance.

smoothedout.Oncethesteadystateisreached,theprofileinther>routregionliesveryclosetoSB(r)∝r−3.5.

Whenlookingattherespectivecontributionsfromdifferentdustpopulationstothetotalprofile,itappearsthatthescatteredfluxis,inthe120–400AUregion,dominatedbyhigh–βgrainsinthe0.25<β<0.4range(Fig.2).Thesegrainshaveor-bitaleccentricitiesinthe0.33–0.75rangeandapoastroninthe240–700AUregionandthusspendalargefractionoftheirorbitsinthedomainlocatedbetweenrhigherβ(closeto0.5)onlyweaklyoutand4rcontributeBR.Grainswitheventothefluxintheroftenoutto4ratseveralBRregionbecausetheyhaveorbitswhoseapoastronis1000AUandwillthusspendmostoftheirtimeoutsidethe≤400AUregionconsideredhere.

Th´ebaultandWu:OuterEdgesofdebrisdiscs5

Fig.3.SameasFig.1,butwithMdust=0.001M⊕,allotherpa-rameters8beingthesame.Notethatthefinaltimescaleishere

t=10yrs.

Table3.Resultsobtainedfordifferentvaluesofthesystem’sdynamicalexcitation󰀏e󰀐(=2󰀏i󰀐)andwidthofthebirthring∆rBRa.AlsoshownareresultsforthenominalrunbutwiththeSBcomputedusingadifferentscatteringphasefunction(seetextfordetails)

Run

OuteredgesharpnessbSBprofilecSB(rout)/SB(rout+10)

slopeα

scatteringanisotropyg=0.52.22-3.68scatteringanisotropyg=0.82.22-3.98

2πrH

(2)

wheredNia/ds(r)isthedifferentialnumberofs–sizedparticlesinaradialannulusofindexiacenteredatradialdistancer,Histheverticalheightofthediscatthatdistanceandi0istheannulusindexcorrespondingtotheradialdistancersimplytheopticalthicknessofthedisctostellar0.τphotons.radisofcourseSinceparticleorbitsareneverstraightradiallines,thisquantityisonlyafirstapproximationoftheirrealin-plane(orhorizontal)collisionalprobability.However,forthesmallestgrainsthisisarelativelygoodfirst-orderapproximation(seethemorethoroughdiscussiononhorizontalandverticalcollisionprobabilitiesinSec.5.1).

Fig.5showsthatτrad≥1overmostofthesourcering4intheearlyepoch,sothatfewgrainscanescapethebirthringwithout

6Th´ebaultandWu:OuterEdgesofdebrisdiscs

Fig.5.Evolutionwithtimeoftheradialopticaldepth(definedas

ineq.[2])fortheveryhighdust–masscaseMdust=10M⊕.

Fig.6.SameasFig.1,butforadynamically“verycold”systemwith󰀏e󰀐=0.001.

collidingwithanothergrain,hencethedensity(andluminosity)depletioninther>routregions.However,thisradialopticaldepthsteadilydecreasesovertime,duetorapidmasserosionbyenergeticcollisionswithinthebirthring.Itdropsbelowunityaf-ter∼105yrs,andis∼0.6bythetime(∼106yrs)thesharpouteredgeissmoothedout(seeFig.4).Afterthat,thesystembehaveslikethenominalcase.Were-examinethisradiallyopticallythickcaseinmoredetailinSec.5.1.3.2.2.dynamicallycoldsystem

AnotherwayofobtainingadeparturefromthestandardSBpro-fileistodecreasethedynamicalexcitationofthesystemtoaverylowvalue,typically󰀏e󰀐≤0.01.Fig.6showstheresultsforthecase󰀏e󰀐=0.001(=2󰀏i󰀐),whereasharpouteredgeismaintainedoutsidethebirthringfortheentiredurationofthesimulation(107yrs).

Theabruptluminosityfall-offattheouteredgeishereduetoaglobaldepletionofsmallgrainsdirectlyresultingfromanim-

Fig.7.Geometricalcrosssectionperlogarithmicsizerange,in-tegratedonlyforgrainsinsidethebirthring,forsystemswithdifferentdynamicalexcitation.Thesecurvesatobtainedattheendoftheintegration(107yrs)andarenormalizedtohavethesamevalueatthelargegrainsize.

balancebetweenthecollisionalproductionanddestructionratesofsmallandlargeparticlesinsuchadynamicallycolddisc.Thiscanbeunderstoodinthefollowingway:

Foragivendustmass,alowerdynamicalexcitationdoesnotchangethecollisionratebetweenlargeobjectsnotaffectedbyradiationpressure.Indeed,becauseoftheequipartition󰀏e󰀐=2󰀏i󰀐thedecreasein󰀏∆v󰀐isexactlycompensatedbytheincreaseoftheparticlenumberdensityduetothereducedthickness.However,thelower󰀏∆v󰀐valuesmeanthatcollisionswillbelessdestructiveandproducelesssmallerfragments.Thismeansthattherateatwhichsmallgrains(theonessignificantlyaffectedbyradiationpressure)areproducedissignificantlyreduced.Onthecontrary,therateatwhichthesesmallgrainsaredestroyedishigher.Indeed,collisionsvelocitiesforimpactsinvolvingthemarenotsignificantlyreducedbythesmall󰀏e󰀐forparentbodies,sincesmallgraindynamicsispredominantlyimposedbyradia-tionpressure.Furthermore,therateatwhichsuchimpactsoccurisincreased,comparedtothenominalcase,becauseofthein-creasedradialopticaldepth(seeSec.5.1.2foramoredetailedanalysis).

As󰀏e󰀐decreases,thisimbalancebecomesmoresevereandthedepletionofthesmallestgrainsismoreacute.Thisisillus-tratedinFig.7displaying,fordifferentvaluesofthesystem’sdy-namicalexcitation,therespectivecontributionsofdifferentgrainsizestothetotalgeometricalcrosssectionσ.Forthenominalcase(󰀏e󰀐=0.1),weobtainthestandardresultthatσisdom-inatedbythesmallestgrainsclosetothecut–offsize.As󰀏e󰀐getssmaller,however,thecontributionofthesesmallergrainsprogressivelydecreases.Belowthelimitingvalue󰀏e󰀐∼0.01,thiseffectissopronouncedthatthesystem’sopticaldepthisnolongerdominatedbythesmallestgrains,butbymuchbiggerparticlesinthe100-1000s0.5range.Theseparticleshavetheiror-bitslargelyconfinedwithinthebirthring.Thisexplainswhyasharpluminositydecreaseisobservedattheouteredgeofthebirthring.

Notethatcontrarytotheveryhighmasscase,thesharpouteredgedoesnotsmoothoutwithtimebutpersistthroughoutthe107yrsofthesimulation.

Th´ebaultandWu:OuterEdgesofdebrisdiscs7

4.Analyticalderivation:The’universal’r−3.5profile

Ournumericalexplorationhasshownthatther−3.5surface

brightnessprofilebeyondtheouteredgeseemstobethemostgenericoutcomeforacollisionalringsystemundertheactionofstellarradiationpressure.Wereinvestigatethisissuefromananalyticalpointofviewandderivesimplifiedformulaeconfirm-ingthisresult.WetakehereasabasistheanalyticalapproachofStrubbe&Chiang(2006)andextendittomoregeneralcasesregardinggrainsizeandspatialdistributions.

Wefirstlyassumethat,withinthebirthring,particlesfollowapower-lawsizedistributionofindexq(insteadoffixingq=−3.5foraDohnanyiequilibrium):dNBR∝sqds.

(3)

Sincethesegrainsspendmostoftheirorbitsintheemptyre-gionoutsidethebirthring,theirtotalnumberintegratedoverthewholesystem(Ntot)willbeboostedbyafactor1/f(e),wheref(e)isthefractionofanorbitalperiodabodyofeccentricitye∼β/(1−β)spendswithinthebirthring(Strubbe&Chiang2006),sothatdNtot=

1

f(e)

sqds.

(4)

WealsofollowStrubbe&Chiang(2006)inmakingthesim-plifyingbutreasonableassumptionthatallhigh-βgrainsareonaveragemostlyseenneartheirapoastron.Thus,ateachgivendistancerfromthestar,theopticaldepthisdominatedbyparti-clesofsize

s1

dom(r)=

s0.5.(5)r

Sinceweareinterestedintheregionr≤4rtheirapoastron

BR,onlygrainswitha(1+e)=1−β󰀊

rBR

1−β

rBR=

e

󰀉

1−

rBR+∆rBR

dNtot(sdom)r

1f(e)

dr

s2dom(r).

(9)

Usingtheradialdependenceofsdomgivenbyeq.5,weobtaindNBR

(sdom

)

󰀈−q−2rBR

r

1−

rBR

f(e)

󰀇r3

(11)

wheref(e)isgivenbysolvingEqu.7.

Asatest,wefirstconsiderthesamefiducialcaseasStrubbe&Chiang(2006),i.e.theDohnanyiq=−3.5value.Theradialprofileofτ⊥(r)obtainedthiswayscalesasr−1.5(Fig.8).Departuresfromthisslopearerelativelylimited,eveninthe1-1.5rBRregion,whereτ⊥(r)hasa∼r−1.75dependence.Ifweapplytheusualruleofthumbthatforanopticaldepthprofileτ⊥(r)∝rα,themidplaneSBscalesasrα−2(e.g.Nakano1990)5,weobtainSB(r)∝r−3.5inmostoftheouterregion,confirm-ingtheresultofStrubbe&Chiang(2006).Forcomparison,wealsoplotonthegraphtheτ⊥(r)profilederivedwhen(incor-rectly)assumingthattheDohnanyilawappliestotheentiresys-tem,i.e.,dNtot∝s−3.5ds(aswasdoneinLecavelieretal.1996;Augereauetal.2001;Th´ebault&Augereau2005).Notsurpris-ingly,werecovertheasymptoticdependenceτ⊥(r)∝r−3,corre-spondingtotheSB(r)∝r−5slopederivedinthesepaststudies.

However,ashasbeenoftenpointedout(e.g.Th´ebaultetal.2003),sizedistributionsfollowapowerlawwithindexq=−3.5onlyinunrealisticallyidealizedsystems,withaninfinitesizerangeandnosizedependenceforcollisionalprocesses.Inoursimulations,thesteadystatesizedistributionwithinthebirthringsignificantlydepartsfromthisvalue(seeFig.9).Thisde-parturefromtheDohnanyilaw,especiallyinthecrucialsizedo-mainofthesmallestgrains,isawellknowresultswhichhasbeenobtainedanddiscussedinseveralpreviousstudies(e.g.CampoBagatinetal.1994;Th´ebaultetal.2003;Krivovetal.2006;Th´ebault&Augereau2007;L¨ohneetal.2007).Themainreasonforthisbehaviouristhatradiationpressureintroducesanaturalminimumcut-offs0.5inthesizedistribution,sothatbodiesofsizes1justaboves0.5areoverabundantbecauseofthelackofsmalls8Th´ebaultandWu:OuterEdgesofdebrisdiscs

Fig.9.Grainsizedistributioninsidethebirthring(dNSec3.ThesizerangeherecoversBR/ds),for

allcasespresentedinthatofgrainsdominatingtheopticaldepthintheouterregion.Inordertofacilitatecomparison,allcurveshavebeenrenormalizedtotheirpeakvalueats=1.2sDohnanyiprofiledN0.5.ThethickgreylineshowsthetheoreticalBR∝s−3.5ds.

Fig.10.SameasFig.8butfordifferentsizedistributionsindexq.Allsizedistributionsleadtoτ⊥∝r−1.5atlargedistances,butsteepersizedistributionsapproachingthisasymptoteearlier.

byimpactswiths1ones,areunderabundant,whichinturnleadstoanoverabundanceofobjectsofsizesdistributiondisplaysapronounced3>swavy-pattern2,etc.TheresultingsizeobservablealsoinFig.9.Forthesizerangewhichishereofspecialinterest,i.e.grainsinthe0.15<β<0.4(1.25sthedistributionissteeperthanaDohnanyi0.5Fig.11.Ratiosbetweenthehorizontalandverticalopticaldepthplottedasafunctionofgrainsizefordifferentvaluesofdynami-calexcitation.Thestraightlinesassociatedwitheachcasemarkthevaluesofτrad/τ⊥(seeeq.2).ThegrainsareassumedtobelaunchedwithlocalKepleriancircularvelocityatthecentreofthering.Thefractionalringwidth∆r0.3.DecreasingthisvaluewillBR/rmoveBRtakesthenominalvalueofthepeakstowardstheright.

5.Discussion

5.1.Howtoescapetheuniversalr−3.5profileOurnumericalexplorationsshowthatthereareonly2waysnottoend5upwiththestandardresultwithnosharpedgeandSB∝r−3.:averymassive,radiallyopticallythickdiscorady-namicallycoldsystem.Forthese2“extreme”cases,theanalyti-calderivationoftheprevioussectionbecomesinvalid.Althoughthereasonswhytheanalyticalstudynolongerholdsaredifferentforeachcase,theyareneverthelessforbothcasesdistinctcon-sequencesofthesamecrucialcharacteristicsofthesystem’sdy-namicswhenpushedtoitslimits:thestrongimbalancebetweenthecollisionratesandvelocitiesoflargeandsmallparticles.Therefore,beforediscussingthese2casesinmoredetail,letuspresentthemechanismsatplaybehindthislarge/smallparticledichotomy.

Forlargebodies,wehaveseenthatcollisionvelocitiescanbederivedusingthestandardexpressionfortheirdynamicalrmsexcitation(Eq.1).Sincefortheseobjectsequipartitionbetweenin-planeandoff-planemotionsresultsinafixed󰀏e󰀐/󰀏i󰀐ratio,itfollowsthatcollisionvelocitiesaredirectlyproportionaltothesystem’sinclinationwhereascollisionratesareindependentof󰀏i󰀐.Thatistosaythatthecollisionalopticaldepthisdirectlyproportionaltotheverticalopticaldepthτ⊥(whichdoesnotvarywith󰀏i󰀐).Thisisnolongerthecaseforsmallgrainspushedoneccentricorbitsbyradiationpressure,whichcansamplethe

Th´ebaultandWu:OuterEdgesofdebrisdiscs9

wholeradialextentofthebirthring.Radiationpressureinduced

eccentricitiesareequalto

e󰀉

(1−2aβ/r)(1−󰀏e2󰀐)

β=1−

1rBR

6

TheanalyticalderivationofSec.4isinvalidhere,sinceitimplicitlyassumesthatsmallhigh-βgrainsdominatethetotalopticaldepth

10Th´ebaultandWu:OuterEdgesofdebrisdiscs

tor10uncertaintyintheirratio(seeforexampletheattemptat

connectingthesetwoquantifiesforthespecificcaseofAUMicperformedbyAugereau&Beust2006),sothatourresultishereprobablynotveryconstraining.

Thesecondandprobablycrucialissueishowlikelyitistofindsuchdynamicallycolddiscs,with󰀏e󰀐and󰀏i󰀐lowerthan≃0.01.AsnotedbyTh´ebault&Augereau(2007):“theonlyob-servationalconstraint[onthedisc’sdynamicalexcitation]comesfrommeasuringthedisc’sverticalthicknessandderivingesti-matesoforbitalinclinations,butsuchconstraintsarescarce”.Edge-ondiscsrepresentthemostfavourablecasessinceH/rcanbedirectlymeasured.However,evenforthetwomoststudieddiscs,onlypartialinformationisavailable.Kristetal.(2005)findH/r≤0.04forAUMic(andH/r≤0.02closetothepo-sitionofmaximumsurfacedensity).TheβPictorisdiscappearsthickerwithH/rratiosaslargeas≃0.1(Golimowskietal.2006).7

.Themodellingandinversionofscatteredlightbrightnesspro-filesofinclined,ring-shapeddiscsdonotprovidemanymoreconstraints.TheHD181327ringsforexample,mighthaveH/rratioaslargeasabout0.1atthepositionsofmaximumsur-facedensity,buttheactualratioscouldbetwotimessmaller(Schneideretal.2006).

Therearethuslargeuncertainties,butitseemshoweverthatthe0.01to0.1rangeisthemostrealisticonefor󰀏i󰀐(andthus󰀏e󰀐ifassumingequipartition).This0.01-0.1rangedoesalsomakesensewhenconsideringsimpletheoreticalargumentsregardingthesizesofthebiggestobjectswithinthedisc.Considerindeedabeltofplanetesimalssittingat∼100AUfromthecentralstar.Ifthemaximumsizeislimitedto,say,∼10km,thenexcitationbymutualviscousstirringleadstovaluesoftheorderoftheirsurfaceescapevelocity(∼10m/s),andcollisionsamongthemgenerateparticleswithdispersionvelocitythatdoesnotexceedthesamevelocity.TheresultingH/R∼vesc/vkep∼0.003andisdynamicallycold.Ontheotherhand,ifthemaximumsizeis∼500km,thenH/R∼0.15andtheresultingdebrisdiscisdy-namicallyhot.Thisdynamicallyhotcaseprobablymakesmoresensewithintheframeofthestandardplanetformationscenario,inwhichdebrisdiscscorrespondtosystemsinwhichthebulkofplanetesimalaccretionprocessisalreadyoverandlargeplane-taryembryosarepresent(e.g.Kenyon&Bromley2005).5.2.ApplicationtorealsystemsAsaconsequence,weexpectournominalresult,validinthe󰀏e󰀐≥0.01range,tocorrespondtothe“natural”collisionalevolutionofmostdiscswhenlefttothemselves.Thisnominalmodel,withnosharpedgeandluminosityfallingasr−3.5com-pareswellagainsttheouterregionSBprofiles(beyond≃40AUand≃120AUrespectively,seeTable1)forthetwoperhapsmostemblematicdebrisdiscs:βPicandAUMic.Forthesetwosys-tems,whichareamongstthefewonesforwhich(partial)discthicknessestimatesareavailable,thisresultisingoodagreementwiththeobserved󰀏i󰀐,whichisforbothdiscsinthe≥0.01range,andthusinthedynamically”hot”regimedisplayedinFig.1.8ForAUMic,thisconclusionconfirmstheinitialresultsobtainedbyStrubbe&Chiang(2006)withamoresimplifiedapproach.

Th´ebaultandWu:OuterEdgesofdebrisdiscs11

5.3.fittingasharpedgering:HR4796AAsformostdebrisdiscs,thereexistsnofullyreliableobserva-tionalconstraintsontheverticalheightandthereforedynamical

excitationoftheHR4796Adisc.FrommodelfittingofscatteredlightandIRimages,Augereauetal.(1999)arguethatthescaleheightat70AU(thelocationofthemainring)hastobelessthan7-8AU,or2󰀏i󰀐∼󰀏e󰀐≤0.1.However,fromimagefittingandtheoreticalconstraints,Kenyonetal.(1999)findthatthemostlikelyvalueisH∼0.5AU,i.e.2󰀏i󰀐∼󰀏e󰀐≤0.007.Itisthusplausiblethatthissystemfallsintothedynamicallycoldcate-gory.

WenumericallyexploreaseriesofdynamicallycoldHR4796-likesystems,fixingthering’scenterat70AUandtak-ingtheradialextentofthebirthring∆rBRaswellas󰀏e󰀐asfreeparameters.TheobservedprofilecomesfromHST/NICMOS(Schneideretal.1999)andisdisplayedinFig.6ofWahhajetal.(2005).Fig.12showsthebestfitobtainedinourparameterexploration.Itcorrespondstoadynamicallycoldcasewith󰀏e󰀐=0.0035andaradialwidthofthebirthring∆r17AUderivedfromobservationsBR=16AU,whichisclosetothe≃13−(Schneideretal.1999;Schneider2001)10.Asacomparison,wealsodisplaya“nominal”profileobtainedfor󰀏e󰀐=0.1.Thelat-terisfullyincompatiblewiththeobservedprofileoverthe80-120AUdomainbyalmost2magnitudesinbrightness.Thebestfit(dynamicallycoldcase)compareswellwiththedatainthe70-110AUregionanddepartsfromtheobservedSB(r)profileonlybetween110and120AU,wherethenumericalprofilebecomestooflat.Itisdifficulttosayhowsignificantthisdiscrepancyintheseoutermost10AUis,astherearenoerrorbarsgivenintheWahhajetal.(2005)plot.Itispossiblethatconfusionfromtheskybackgroundentersatthesedistances.

Wemustthusremaincarefulbutitseemsthatthereisatleastapossibilityforourdynamicallycoldmodeltoprovideanexpla-nationforsharpouteredgediscslikeHR4796A.Thequestionofhowrealisticthisexplanationcanbeisanotherissue.WehaveseeninSec.5.1.2thatthe󰀏i󰀐<0.01condition,althoughprobablynotgenericwithrespecttoplanet-formationscenarios,cannot,inmostcases,beexplicitlyruledoutbyobservations.

Theremayhoweverbeapredictionmadebythecold-discscenariowhichcouldbeobservationallychecked,i.e.,theun-derabundanceofgrainsintheµmtosub-millimetrerange11.ForHR4796,forinstance,Augereauetal.(1999)haveper-formeddetailedfitsoftheSEDaswellasofthermalandscat-teredlightimages.Theirbestfitimpliedthatmostofthegeomet-ricalcrosssectionwascontainedingrainsclosetotheminimumvaluefortheirsizedistribution,i.e.smin∼10µm.However,thisfitwasobtainedassuminganimposedsizedistributionins−3.5,sothattheseresultscannotbeusedtoruleoutthepossibilityofhighersminvaluesforalternatesizedistributions.Wahhajetal.(2005)performedsimilarfitswithpartlymorerecentdataandfoundthattheeffectivesizeforgrainswithintheringis∼50µm.Thiswouldbeinrelativelygoodagreementwithourdynami-callycoldcase.However,thisvaluewasobtainedassuminga

12

ThisseemstobearecurrentissuewhichisnotlimitedtoHR4796A:vastpopulationsofunboundparticlescouldalsodominatethescatteredluminosityofHD141569A(Augereau&Papaloizou2004),andmightbepresentintheouterβPictorisdisc(Augereauetal.2001)

12Th´ebaultandWu:OuterEdgesofdebrisdiscs

withtypically󰀏e󰀐=2󰀏i󰀐≤0.01.Inthiscase,smallgrains

aredestroyedmuchmoreefficientlythantheyarecreated,leadingtoadepletionofthispopulation.Thesystem’sopti-caldepthandluminosityarethendominatedbylargegrainswhichdonotleavethemainbirthring,leadingtoasharpouteredge.Evenifthiscasemightnotcorrespondtothemostgenericdebrisdiscconfiguration,itcannotberuledoutbyobservationsandisthusapossibleexplanationtosomeoftheobservedsystems.

Tonumericallyinvestigatetheapplicabilityofthedynami-callycoldcasetorealsharp-edgesystems,weconsiderthespe-cificcaseofHR4796A.Wefindareasonablygoodfitofthissystem’souterregionluminosityprofilewithadynamicalex-citation󰀏e󰀐∼0.0035.Thereisthusthepossibilitythatsuchasharpouteredgecouldbeexplainedbythenaturalcollisionalevolutionofaconfineddiscoflargeparentbodies.

Acknowledgements.PTwishestothankJean-CharlesAugereauandAlexanderKrivovforfruitfuldiscussions.

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Th´ebaultandWu:OuterEdgesofdebrisdiscs13

ListofObjects

‘βPictoris’onpage1

‘βPictoris’onpage2‘βPictoris’onpage11‘βPictoris’onpage11