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Astronomy&Astrophysicsmanuscriptno.sminpap-final-lang (cid:13)c ESO2016 February10,2016 Dust production in debris discs: constraints on the smallest grains P.Thebault1 LESIA-ObservatoiredeParis,UPMCUniv.Paris06,Univ.Paris-Diderot,France Received;accepted ABSTRACT Context.Thesurfaceenergyconstraint putsalimitonthesmallestfragment s thatcanbeproduced afteracollision.Basedon 6 surf analyticalconsiderations,thismechanismhasbeenrecentlyidentifiedasbeinghavingthepotentialtopreventtheproductionofsmall 1 dustgrainsindebrisdiscsandtocutofftheirsizedistributionatsizeslargerthantheblow-outsize. 0 Aims.We numerically investigate the importance of this effect to find out under which conditions it can leave a signature in the 2 small-sizeendofadisc’sparticlesizedistribution(PSD).Animportantpartofthisworkistomapout,inadiscatsteady-state,what b isthemostlikelycollisionaloriginforµm-sizeddustgrains,intermsofthesizesoftheircollisionalprogenitors. e Methods.Forthefirsttime,weimplementthesurfaceenergyconstraintintoacollisionalevolutioncode.Weconsideratypicaldebris F discextendingfrom50to100auandtwodifferentstellartypes:sun-likeandAstar.Wealsoconsidertwolevelsofstirringinthedisc: dynamically’hot’(<e>=0.075)and’cold’(<e>=0.01).Inallcases,wederives mapsasafunctionoftargetandprojectilesizes, 9 surf s ands ,andcomparethemtoequivalentmapsforthedust-productionrate.Wethencomputedisc-integratedprofilesofthePSDand t p estimatetheimprintofthesurfaceenergyconstraint. ] P Results.Wefindthatthe(sp,st)regionsofhighssurf valuesdonotcoincidewiththoseofhighdustproductionrates.Asaconsequence, E thesurfaceenergyconstraintgenerallyhasaweakeffectonthesystem’sPSD.Themaximum ssurf-induceddepletionofµm-sized grainsis∼ 30%andisobtainedforasun-likestarandadynamically’hot’case.Forthee=0.01cases,thesurfaceenergyeffectis . h negligiblecomparedtothemassivesmallgraindepletionthatisinducedbyanothermechanism:the’natural’imbalancebetweendust p productionanddestructionratesinlow-stirringdiscsidentifiedbyThebault&Wu(2008) - o Keywords.planetarysystem–debrisdiscs–circumstellarmatter r t s a 1. Introduction rangefrom∼ 0.5µmforsolar-typestarsto2-10µmforlate-type [ A stars (Krivov, 2007) 1. As a consequence, in scattered light, Circumstellar debris discs have been detected around main- 4 disc luminosities should always be dominated by grains close v sequence stars because of the excess luminosity produced by to s , and, for A stars, s grainsshould also dominate the blow blow 7 µm- to mm-sized dust. Because of its relatively short lifetime, thermalfluxuptomid-IRwavelengths. 0 this dust cannotbe primordialand is thoughtto be steadily re- However,observationshavesometimeschallengedthisbasic 9 plenishedbyachainoferosivecollisionsstartingfroma reser- view.Somediscshave,forexample,beenfoundtocontainim- 0 voiroflarge,planetesimal-like,parentbodies(e.g.Wyatt,2008; 0 Krivov,2010).Inanidealizedcase,thesizedistributionofbod- portantquantitiesofsub-microngrainsbelowsblow(Ardilaetal., . 2004;Fitzgeraldetal., 2007;Johnsonetal.,2012).Whilethere 1 ieswithinsuchacollisionalcascadeshouldsettletowardapower doesnotseemtobe onesinglestraightforwardscenarioforthe 0 law of the form dN ∝ sqds, where the index q is close to -3.5 presenceofsuchtinyparticles,somepossibleexplanationshave 6 (Dohnanyi, 1969). Such a power law has the interesting char- been suggested. One of them is that, at very small sizes, the 1 acteristicsthat,whilemostofthedisc’smassiscontainedinthe β(s) curve starts to decrease with decreasing s before reach- : biggestobjectsofthecascade,mostofthegeometricalcrosssec- v ing a constant value that can be < 0.5 for K, G or F stars, Xi tmioenansshothualdt,baetaclolnwtaaivneeldeningtthhseλsm.a2llπesstgr,atihnesdoifscsi’zselusmmiinn.oTshitiys thusplacingsub-microngrainsonboundorbits(Johnsonetal., min 2012). Another possibility is that we are witnessing a power- r shouldbedominatedbythesesmallestgrains. a ful collisional chain-reaction, called an ’avalanche’, of high-β Determining s is consequently of crucial importance. min particles that are passing through a dense ring of larger grains Luckily,formoststars,stellarradiationpressureimposesa’nat- (Grigorievaetal.,2007). ural’minimumcut-offsizethatis,inprinciple,easytoestimate. Indeed, because this pressure is both ∝ 1/s and, like stellar gravity, ∝ 1/r2 (r being the radial distance to the star), there 1.1.Discswith"toolarge"minimumdustsizes is a minimum size s below which radiation pressure over- blow comesgravityandgrainsarequicklyblownoutfromthesystem. Here we will focus on the opposite problem, i.e., the systems Becausesmallgrainsareprobablyproducedfromparentbodies forwhichtheobservationallyderived smin hasbeenfoundtobe onKeplerianorbits,the criteriaforestimating sblow is givenby larger than sblow. The existence of such systems was first un- β(s ) = F /F = 0.5(forparentbodiesoncircular ambiguouslydeterminedbyPawelleketal.(2014),whoconsid- blow Rad.Press. grav. orbits). For typical astro-silicates (Draine, 2003), values for β 1 Forsubsolarstars,radiationpressureistooweaktoovercomegrav- Sendoffprintrequeststo:P.Thebault ity,butstellarwindcouldplayasimilarroleforactiveMstarslikeAU Correspondenceto:[email protected] Mic(Augereau&Beust,2006;Schüppleretal.,2015) 2 Thebault:smallestgrainsindebrisdiscs eredasampleof34resolveddiscs2 andfoundthat,whilediscs to s , the size of the largestfragment,andthe impactor’ssize lfr around A-star have s ∼ s , the s /s ratio increased s : min blow min blow 0 towards low-mass stars and could reach ∼ 10 for solar-type 2 24γs stars.TheseresultswerelaterconfirmedbyPawellek&Krivov s = 0 s−1 (1) n(2o0t1d5e)p,ewnhdoosnhmowateedritahlactotmheposssmitinio/nsbsloowrtgrreanidnipsorroobsuitsyt.anddoes surf ηρs0v2rel+24γ lfr where v is the impact velocity, γ is the surface energy per These results seem to be in contradiction with the afore- rel unit surface, and η is the fraction of the kinetic energy that is mentioned behaviour expected in discs where a size distribu- usedtocreateanewsurface.Interestingly,thisformulagivesthe tion in dN ∝ sqds holds down to the blow-out size s , blow counter-intuitive result that, for a given v and a fixed s /s for which luminosities should be dominated by grains close rel lfr 0 ratio(i.e.,forself-similarimpacts),thesizeofthesmallestfrag- to the blow-out size. Interestingly, numerical investigations of ment increases with decreasing s sizes. For low-velocity im- the collisional evolutions of debris discs have shown that, at 0 pacts, low values of η and for material with high γ (like ice), least in the small-size domain, size distributions can in fact s can be significantly higher than s for s . 1m (see significantly depart from an idealized dN ∝ sqds power- surf blow 0 Fig.1ofKrijt&Kama,2014). law. The simulations of Thebaultetal. (2003); Krivovetal. However, while deriving estimates of s is analytically (2006); Thebault&Augereau (2007) have indeed shown that surf possible for one given collision, giving global estimates for a size-distributionscanexhibitpronounced’wavy’patternsinthe . 100s domain, triggered by the absence of potential de- whole debris disc is a much more problematic task, since the blow valuesof s will varygreatly dependingon the absolute and structiveprojectilesbelow s .However,thiswavinesscannot surf blow relativesizesoftheimpactingobjects.Intheirpioneeringstudy, deprive the system from grains close to s , and, generally, blow it even has just the opposite effect, inducing a density peak of Krijt&Kama (2014) attempt to derive a disc-integrated ssurf grainsinthe1.5−2s region(Thebault&Augereau,2007). byonlyconsideringimpactsbetweenequal-sizeds0objectsand blow byrestrictingthemselvesto"barelycatastrophic"impactswhere Apossiblecauseofsmall-graindepletioncouldbethewell- each impactor is split into two identical fragments. This sim- knownPoynting-Robertsondrag,whichcausessmallparticlesto plifyingchoicereliedontwomainassumptions:1)Foragiven slowlyspiralstarwards.However,thiseffectislikelytobeonly targetsize,andif s isfixed,itiswithaprojectileofthesame noticeablefortenuousdiscsforwhichthe collisionaltimescale lfr size that the smallest "smallest fragments" should be obtained, ofsmallgrainsbecomeslargerthant (Wyatt,2005).Thefirst PR hencethechoiceofequal-sizedimpactsasthemostconstraining plausible mechanism for small-grain depletion that could also ones, and 2) in a collisional cascade, small grains should pref- work for bright discs was proposed by Thebault&Wu (2008), erentiallyoriginatefromcollisionsthatinvolveparticlesthatare whoshowedthat,indiscswithalowdynamicalexcitation(low barelylargerthanthemselves;sothatevenifcollisionsamongst orbitaleccentricitieseandinclinationsi),thereisanimbalance largeobjetshaveverysmall s values(because s ∝ s−1 if between the production and destruction rates of small grains. surf surf 0 s /s is fixed) the amount of small dust they produce cannot Indeed,while their productionrate is controlledby the erosion lfr 0 compensate for the amount of small dust "not produced" (be- of larger particles and is thus low because of these particles’ cause of their larger s ) by collisions amongst small grains, low <e>, their destructionrate, whichis controlledby impacts surf hence the focuson the smallest "barely catastrophic"impacts . involving the small grains themselves, is much higher because Underthesesimplifyinghypotheses,thedisc-integrated sDD is thesegrainsareplacedonhigh-eorbitsbyradiationpressurere- surf gardlessofthedynamicalexcitationintherestofthesystem.As thenthessurf obtainedforthesmallests0objectthatcanbesplit aresult,dependingonthevaluefor<e>,therecouldbeastrong intwobyanimpactatvrel depletionofgrainsuptosub-mmsizes.Themainissuewiththis scenarioisthatitrequiresvaluesof<e>thatcouldpossiblybe sDD r L −1 f −2 η −1 γ unrealisticallylow(seeSec.4). surf =2.4 ∗ (2) sblow (cid:18)5AU(cid:19) L⊙! 10−2! (cid:18)10−2(cid:19) (cid:18)0.1J.m−2(cid:19) 1.2.Maximumsurfaceenergyandminimumfragmentsize where f =(1.25e2+i2)0.5. Another, potentially more generic scenario has recently been proposed by Krijt&Kama (2014). It is based on an energy- 1.3.Needforanumericalapproach conservation criteria, related to the physics of collisions themselves, which, remarkably enough, had never before been in- As rightfully stated by Krijt&Kama (2014) themselves, Eq.2 voked in the context of debris discs. The argument is that a should only be taken as an order of magnitude estimate, as it destructivecollision cannotproducea size distribution of frag- only takes into accounta very limited range of collision types. ments that reaches an infinitely small value, as this would re- But even this role as an order-of-magnitude indication should quireaninfiniteamountofenergy.Thisisbecausecreatingnew betakenwithcaution,becausethesimplifyingassumptionsthat fragmentsmeansincreasingtheamountofexposedsurface,and justify the predominant role of equal-sized barely-catastrophic this requires energy.There is thus a minimum size s in the impacts might not hold in a realistic disc. The first problem is surf fragmentdistribution,whichisgivenbytherequirementthatthe thatthispredominantrolehasbeeninferredbyassumingthatslfr total surface energy of all fragments cannot exceed the kinetic isfixed,whichisfarfrombeingthecaseinrealisticconditions, energyoftheimpact.Forthespecificcaseofafullydestructive where slfr strongly depends on the two impactors’ mass ratio impactbetweentwoequal-sizedbodiesandforaq = −3.5size andonvrel(e.g.Leinhardt&Stewart,2012).Anotherproblemis distribution,Krijt&Kama(2014)derivedarelationlinkings that it was derived assuming that a q = −3.5 power law holds surf forthewholesizedistribution,whichseveraldebrisdiscstudies 2 There is an inherent degeneracy between grain size and disc ra- have shown to be erroneous, especially in the small grain-size dius that can only be broken for resolved systems (see discussion in domain. Last but not least, it neglects the contribution of cra- Pawellek&Krivov,2015) teringimpacts,whichmighthaveadominantroleintheglobal Thebault:smallestgrainsindebrisdiscs 3 dustproductionanddestructionbalance(Thebault&Augereau, Table 1. Set-up for the numerical simulations. Here, γ is the 2007;Kobayashi&Tanaka,2010)3. surfaceenergypersurfaceunitofmaterialandηisthefraction Pawellek&Krivov(2015)improvedonEq.2bytakinginto ofthekineticenergythatisusedforcreatinganewsurface. accountthefactthatvelocitiesofimpactsinvolvingsmallgrains might have higher values because radiation pressure places smin−num sblow small grains on highly eccentric or even parabolic orbits. This smax−num 50m significantly reduces the values of sDD /s with respect to Initialmass 0.01M⊕ surf blow r 50AU Eq.2. Interestingly, this did not, however, improve the fit to min r 100AU the observationally-derived sDsuDrf/sblow, for which the original Lm∗a/xL⊙ 1or9 Krijt&Kama (2014) formula seems to provide a better match <e> 0.075or0.01 (seeFig.14ofPawellek&Krivov,2015).However,eventheim- Q∗prescription Benz&Asphaug(1999)(forbasalt) proved formula used by Pawellek&Krivov (2015) is a disc- ssurf AppendixofKrijt&Kama(2014) averagedanalyticalestimatethatstillreliesonthesamesimpli- γ 0.74J.m−2 η 0.01 fyingassumptionsmentionedearlier. Weproposeheretotakethesestudiesastepfurtherbyincor- porating,forthe firsttime, the surface-energyconstraintwithin a numerical collisional evolution code, estimating s (i,j) for smallestsizebinconsideredinthecode),wecutthepost-impact surf allpairsofimpactingbodiesofsizes si and sj andtakingthese fragmentdistributionatssurf(i,j)insteadofsmin−num. valuesintoaccountinthecollision-outcomeprescriptionofour Weestimatethevaluesofssurf(i,j)forbothfragmentingand code.We describeour numericalmodeland the set-upsforthe crateringimpacts,usingtheequationspresentedintheAppendix differentcasesthatweexploreinSec.2.Resultsarepresentedin ofKrijt&Kama(2014): Sec.3, where we first compare the s (i,j) maps to equivalent surf (si,sj)mapsforthelevelofdustproductioninthedisc.Wethen 6γ s3+s3 2 2 daelnilsepcrolganyysictdhoeenrsestdrteacaiandstyebss,ytaacnteodmpinapvrateriscintlieggasttihezeetshedeiPssitSgrDinbasuttutioorentsohfootsbheteaoisnbuetradfiancfeoedr- ssurf =ηρv(cid:16)2reil(sisjj)(cid:17)3 s−lf1r for fragmentation (3) forcontrolrunswithoutthe s criteria.InSec.4,wethendis- surf 6γκ 2 cuss the importanceof the surface-energyconstraint, in partic- s = s−1 for cratering (4) ularwithrespecttotheconcurrent"low-stirring"dust-depletion surf ηρv2  lfr mechanismidentifiedbyThebault&Wu(2008). whereweassumethatiistrheeltargetand jtheprojectile(s ≥ s ). i j Forthecrateringcase,κistheratiom /m ,wherem isthe crat j crat total mass excavated from the target i. The crucial point here 2. Model is that, instead of having to assume values for the largest frag- We use the statistical collisional model developed ment’ssizeslfr andtheratioκ,wecanretrievebothquantitiesin by Thebaultetal. (2003) and later upgraded by aself-consistentwayfromourcollision-outcomeprescription. Thebault&Augereau (2007). This has a "particle-in-a- box" structure, where particles are sorted into logarithmic 2.1.Set-up size bins separated by a factor 2 in mass. It also has a 1D spatialresolution,beingdividedintoradiallyconcentricannuli. WedonotconsiderthefullrangeofpossibleL∗/L exploredby ⊙ Collision rates between all size bins are computed using an Pawellek&Krivov (2015), but restrict ourselves to the two il- estimate of the average orbital eccentricity and inclinations in lustrativecasesofasun-likeL∗ = L starandaβ-Pic-likeA5V ⊙ each size bin. Crucially, the code takes into account the in- starwithL∗ = 9L .Weconsiderareferencedebrisdiscextend- ⊙ creasedeccentricities,andthusimpactvelocitiesofsmallgrains ing from r = 50au to r = 100au. The disc’s total initial min max whose orbits are affected by stellar radiation pressure, as well mass is M = 0.5M , distributed between s = 50m disc ⊕ max−num asthe factthatthese grainsare able tocrossseveralconcentric ands = s ,whichcorrespondsto∼0.01M of<1mm min−num blow ⊕ annuli(seeAppendixofThebault&Augereau,2007).Collision dust4.Thevalueof s isequalto0.5µmforthesun-likecase blow outcomes are then divided into two categories, cratering and (with80sizebinsfroms tos )and4µmfortheAstarone max blow fragmentation,dependingon the ratio betweenthe specific im- (71sizebins).Asforthedynamicalstateofthedisc,weconsider pactkineticenergyandthespecificshatteringenergyQ∗,which onedynamically’hot’casewith<e>=0.075andonedynami- depends on object sizes and composition. In both regimes, the cally’cold’casewith<e >=0.01.ForeachL∗ and< e>case, size of the largest fragment and the size distributions of the werunbothasimulationtakingintoaccountthe s prescrip- surf other debris are derived through the detailed energy-scaling tion and a reference case with no s . We let the runs evolve surf prescriptions that are presented in Thebaultetal. (2003) and for107years,whichisenoughtoreachacollisionalsteady-state Thebault&Augereau (2007). inthedustsizedomain(<1cm). Themainupgradeforthepresentrunsisthatweimplementa As for the free parametersof the s prescriptions (Eqs.3 surf prescriptionforssurf.Wederivessurf(i,j)forallpossiblecollid- and 4), we adopt a conservative approach and chose, amongst ingpairsofsizessiandsjandimplementthisparameterintoour theηandγvaluesconsideredbyKrijt&Kama(2014),theones collision-outcomeprescription:if,foragivencollisionbetween that are in principle the most favourable to yielding large s surf thebins"i" and"j", ssurf(i,j) > smin−num (where smin−num is the values.We thustakeη = 0.01(i.e.,1%ofthekineticenergyis 3 To their credit, Krijt&Kama (2014) briefly investigate the role 4 The value for M is not a crucial parameter, since it will only disc of cratering impacts in their Appendix, but again assuming a fixed affectthetimescaleforthediscevolutionwithoutaffectingtheresults s /s ratio(seeFig.B.1.ofthatpaper) (s (i,j)anddustproductionmaps,PSDs)obtainedatsteady-state lfr target surf 4 Thebault:smallestgrainsindebrisdiscs Table 2. Main results for all four collisional runs. sDD /s dustproduction,whichcorresponds,hereagain,tofragmenting surf blow i’assnmdtah∆lel-M’s4aimsnbldopwl≤’imsfi≤ee2d0dsibulaowmnaa’l-ryseitzitcehadeldfvruaascltu,tieroensgapilevceetxnivceebslyys,(aEosqrc.2do.emp∆pleaMtrieos≤dn2)tsoboloawf (iismFinpgoa.2tcbtfs)u.bllNyyesnvee∼grtlhsigbelilobewlsesp,,rstohojeetchctoailtnetwsrioebnuetx5iopsbnelcoowtft.htheesmh.iagx2hi0m-ssubslmuorwfstruaerrgfgaieoctnes controlsimulationwithnoconstraintons . energycriteriatohaveatleastsomeeffectonthesystem’sevo- surf lution. L∗/L <e> s sDD /s ∆M ∆M ⊙ blow surf blow s≤2sblow 4sblow≤s≤20sblow 9 0.075 4µm 0.4 -0.038 +0.074 9 0.01 4µm 19.8 -0.049 +0.199 3.1.2. Solar-typestar 1 0.075 0.5µm 3.1 -0.277 +0.410 For a Sun-like star, orbital velocities are lower and collisions 1 0.01 0.5µm 153 -0.127 +0.145 less energeticthan for an A-star for equivalentorbitalparame- ters.Thes /s valuesarethuslogicallyhigher,asisclearly surf blow usedforcreatingnewsurface)andγ = 0.74J.m−2,whichisthe seeninFig.1candd.The ssurf/sblow > 1domainismuchmore valueforwaterice5. extendedthanforanAstar,withpeakvaluesreaching∼ 25for the <e>=0.075 case and even ≥ 100 for the dynamically cold Allmainparametersfortheset-uparesummarizedinTable1. case. As was already the case for the A-star runs, these peak s valuesarelocated,inthe(s,s )map,onalinecorrespond- surf i j 3. Results ingtothetransitionfromcrateringtofragmentingimpacts6.For the<e>=0.075run,thistransitionalwaysoccursfor s ≤ 0.5s, j i 3.1.Dustprogenitorsandssurf maps witha s /s ratiothatisdecreasingtowardslargersizesfollow- j i For each of the consideredset-ups, Fig.1 presentsthe ssurf(i,j) inganapproximatelawin s2/s1 ∼ 0.4−0.1log(s2/10µm)7.For valueforeverypairofimpactingtargetandprojectileofsizes si <e>=0.01,the peak ssurf line still correspondsto the fragmen- and sj. We also show, as a useful comparisontool, the equiva- tation/crateringtransition,butisnowmuchclosertothe si = sj lent(s,s ) mapsofthe amountofµm-sized dustproducedasa diagonal,whichreflectsthefactthat,fortheselowerimpactve- i j functionoftargetandprojectilesizes(Fig.2). locities,ittakesalargerprojectiletofragmentagiventarget. We also note that, for both the <e>=0.075 and <e>=0.01 cases,therearenohighs /s valuesforimpactors.10µm. 3.1.1. A-star surf blow This is because such small grainshave more energeticimpacts For the dynamically’hot’(<e>= 0.075)A-star case, Eq.2 pre- (able to producesmaller fragments)because of their radiation- dicts a low sDD /s of only ∼ 0.4, and our s (i,j) map is pressure-affected orbits. Incidentally, this s . 10µm region is surf blow surf the one where the µm-dust production is the highest (Figs.2c ingoodagreementwiththisprediction,sincealmostall(i,j)im- and d). This means that a large fraction of the dust-generating pacts result in s /s < 1 (Fig.1a) . There is, admittedly, surf blow a very narrow range of impacts for which s /s ≥ 1, but impactsare,infact,abletoproducefragmentsdowntotheblow- surf blow out size. The regions of high s /s >5 only contribute to even there the ratio does not exceed 2. Moreover, this limited surf blow approximately20%ofthe≤ 10µmdustproductionforboththe family of s > s collisions corresponds to a region that surf blow <e>=0.075and<e>=0.01cases. hasonlyaverylimitedcontributiontothetotaldustproduction. Thedustproductionisinsteaddominatedbyimpactsthatinvolve s ∼ s = 5µmprojectilesandlarger5s . s . 20s tar- blow blow blow 3.2.Sizedistribution gets(seeFig.2a).Thisisbecause s ∼ s grainsareplacedon blow high-eorbitsbyradiationpressureandimpactallothergrainsat Figures 3 and 4 present the particle size-distribution (PSD), in veryhighvelocities.Theyarethusveryefficientatfragmenting terms of the differential distribution of geometric cross section targets and producingdusty debris over a wide range of target dΣ/ds, at collisional steady-state8, for all four of the explored sizes. set-ups,aswellasforthecontrolrunswherethes constraint surf Thesituationisslightlydifferentforthedynamically’cold’ is ignored. Visualizing the dΣ/ds distribution allows us to im- A-starcase,forwhichthefractionofs >s impactsismore surf blow mediately identify which particle sizes dominate the system’s extended and the maximum value for s (i,j) reaches ∼ 5 surf opticaldepth,andthusitsluminosity. (Fig.1b).This valueis, however,wellbelow the analyticalpre- dictionsDD ∼20s ofEq.2.Wenotethatthesehigh-s (i,j) surf blow surf impactsarelocatedclosethe s = s diagonal,whichcouldap- 6 This relatively sharp transition at the cratering/fragmentation i j peartocontradictFig.B.1ofKrijt&Kama(2014),whichshows boundary is due to the discontinuity in the ssurf prescriptions of Krijt&Kama (2014) (Eqs.3 and 4) at this boundary (s = 2−1/3s ). insteadaminimumofthe s curveforequal-sizedimpactors. lfr 1 surf More refined and self-consistent s prescriptions should probably But this is because this analytically-derivedcurveis onlyvalid surf be derived in the future, but we chose to stick to the laws given by for fragmenting impacts, whereas here, with <e>= 0.01, the Krijt&Kama(2014)forthisexploratorywork. high-collisional-energyrequirementforfragmentationcanonly 7 This negative slope reflects the fact that, in the strength regime, bemetwhenprojectileshavesizescomparabletothetarget.All the specific shattering energy Q∗ decreases with increasing sizes other cases result in cratering, for which ssurf are, in general, (Benz&Asphaug,1999) muchsmaller.Aswasthecaseforthe<e>= 0.075run,the(i,j) 8 Or,moreexactly,whensteady-stateisatleastreachedwhereitmat- regionofhigh ssurf valuesdoesnotmatchtheregionofhighest tershere,thatisinthes<1cmdomain.Inthelarge-sizedomain,thereis adifferenceofPSDslopesbetweenthedynamicallyhotandcolddiscs 5 Weareawarethatthisvalueisnotself-consistentwithourvalues thatisduetothe<e>=0.01systemsnot yethavinghadtimetoreach of s , which are derived for astro-silicates. Nevertheless, this is in collisionalsteady-statefortheselargerobjects.Collisionalsteady-state blow linewithourchoiceofconsideringthemaximumpossibleeffectforthe doesindeedworkitswayupthePSDwithtime,ataratethatstrongly surface-energyconstraint increaseswithdecreasingdynamicalstirring(e.g.,Löhneetal.,2008). Thebault:smallestgrainsindebrisdiscs 5 Fig.1.Disc-integratedmeanvaluesofthes /s ratioforallpossibletarget-projectilepairsofsizess ands .Upperleftpanel: surf blow i j A-star and <e>=0.075.Upper right: A-star and <e>=0.01.Bottom left: solar-type star and <e>=0.075.Bottom right: solar-type starand<e>=0.01.Thedashedwhitelinedelineatesthes = s diagonalofequal-sizedimpactors. i j 3.2.1. Genericfeatures curveat s. 50s ,thanforAstars,wherethePSDisroughly blow flatinthesames.50s domain.Nevertheless,forbothcases Beforeassessingtheadditionaleffectofthe s constraint,let blow surf thedepletionofsmallgrainsisverystrong,exceedingoneorder usfirstemphasisesomecharacteristicsoftheobtainedsizedis- ofmagnitudeascomparedtoadynamically’hot’system. tributions that are well known generic features of steady-state collisional discs, and that also affect the small-size domain of thePSDs. A first ’classic’ result is that, for all cases, the size distri- 3.2.2. Effectofthe s constraint surf butiondisplaysa clearwavystructureinthe. 100µmdomain. This waviness is more pronounced than in Pawellek&Krivov (2015) because we consider a wider disc, so that small grains, As expected from the s (i,j) and dust-production maps (see surf placed on high-e orbits par radiation pressure, will be able to Sec.3.1), the effect of the s constraint is very weak for the surf impacttargetparticlesatahigherimpactangleandthushigher A-star runs. For the <e>=0.075case, the PSD is almost indis- v .Thiswillreinforcetheirshatteringpowerandthustheam- tinguishablefromthecontrolrunwiths switchedoff(Fig.3). rel surf plitudeofthesizedistributionwave(seeThebault&Augereau, Some differencesbetween the with- and without-s runs are surf 2007,foradetaileddiscussionon"wave-enhancing"factors). visible for the <e>=0.01 case, but they remain relatively lim- Anothercrucialresultisthestrongdepletionofsmallparti- ited. Interestingly, they are not so much visible as a dearth of cles for all <e>=0.01cases. This is the signatureof the mech- grainsinthe s≤10−20µmdomainwherethefew s ≥ s surf blow anism identified by Thebault&Wu (2008), i.e., the strong im- valueslie(seeFig.1b),butratherasanexcessoflargerparticles balancebetweenthesmall-dustproductionanddestructionrates inthe∼30-50µmrange.Thisisadirectconsequenceofthespe- for dynamically cold systems (see Sect.1.1). As underlined by cificdynamics,andthusdestructiveefficiency,ofgrainscloseto Pawellek&Krivov (2015), this effect is more pronounced for s .Inthe"s -free"case,itisindeedthishighdestructiveef- blow surf solar-type stars, for which we find a clear drop of the dΣ/ds ficiencythatisresponsibleforthestrongdepletionof30−50µm 6 Thebault:smallestgrainsindebrisdiscs Fig.2.Disc-integratedandnormalizedproductionratesofs≤10µmdustasafunctionoftargetandprojectilesizes.Thesemapsare estimated,atcollisionalsteady-state,forthe’controlruns’wherethes criteriaisturnedoff.Thisallowsforaneasycomparison surf withFig.1bydirectlyshowingwhichhighs (i,j)regionswillhaveasignificantdampingeffectonthesmall-dustproductionrate. surf Upperleftpanel:A-starprimaryand<e>=0.075.Upperright:A-starprimaryand<e>=0.01.Bottomleft:solar-typeprimaryand <e>=0.075.Bottomright:solar-typeprimaryand<e>=0.01. particlesinthePSD(thefirstdipinthe"wave")9.Soremovinga of s ≤ 2s grains is close to 30%, and the related excess of blow smallnumberofsmalldestructiveimpactorswillnecessarilyin- 4s ≤ s ≤ 20s particles exceeds 40% (Table 2). For the blow blow creasethenumberof30−50µmgrains.Andgiventhatwestart <e>=0.01 run, these excesses and depletions are only ∼ 12% from a strongly depleted population of 30−50µm particles in and ∼ 14%, respectively. This is a rather counter-intuitive re- thes -freecase,theirrelativeincreaseinthes runswillbe sult, as Eq.2 predicts a sDD /s ratio that is 50 times higher surf surf surf blow higherthantherelativedecreaseof s ≤ 10−20µmgrains.And for <e>=0.01 than for <e>=0.075, and even the more accu- thisisexactlywhatweobserve(seeTable2):a∼ 5%depletion rate Figs.1c and d show peak s (i,j) values that are still 6-7 surf of s ∼ sblow grains that causes a ∼ 20% excess of s ∼ 30µm times higher for the dynamically cold case. However, as dis- particles,whichisthesizeofthebiggestobjectsthatcanbede- cussed in Sec.3.1.2, the crucial point is that these (s,s ) re- i j stroyedbyprojectilesclosetosblow.However,forthis<e>=0.01 gionsofhigh ssurf valuesdonot matchthoseof highdustpro- case,the ssurf-induceddepletionsandexcessesremainmarginal duction. Most of the dust is indeed created by collisions on whencomparedtothemuchmoresignificantglobaldepletionof s ≤ 10µm(. 20s )targets(Fig.2),forwhich s rarelyex- blow surf s.50sblowgrainscausedbythelowdynamicalexcitationofthe ceedssblow(Fig.1).Anotherreasonforwhichthessurf constraint system(seeDiscussion). onlyleavesaweaksignatureonthe<e>=0.01systemisthat,in The imprint of the ssurf limit is, logically, much more vis- thes≤100µmrange,thePSDisalreadymassivelydepletedbe- ible around a Sun-like star (Fig.4). Interestingly, this is espe- causeofthedustproductionimbalanceinherenttodynamically cially true of the dynamicallyhotdisc, forwhich the depletion ’cold’discs.ThiseffectisevenstrongerthanfortheA-starcase, withadepletionthatreachesalmosttwoordersofmagnitudefor 9 Note that the presence of a wave does not depend on the high-e s∼ s grains(Fig.4). blow orbitsofsmallgrains,butthesehigh-eorbits,andthushighimpactve- locities,stronglyincreasethewave’samplitude Thebault:smallestgrainsindebrisdiscs 7 not create a sharp cut-off in the size distribution. This limited amplitude, combined to the aforementionedexcess of particles in the ∼ 3s to ∼20-30s range, rendersthe PSD plateau- blow blow likeatsizessmallerthan∼10s .Itisthusdifficulttodefinea blow proper s forthesizedistribution,but,atleastfromaqualita- min tivepointofview,we confirmthat,forthiscase,themaximum energycriteriadoeshaveavisibleeffectonthelower-endofthe PSD. Thesituation is radicallydifferentforthe dynamicallycold cases, for which the simulated size distributions are much less affected by the s constraint than would be expected from surf simple analytical estimates, which predict sDD /s values in surf blow excessof20,oreven150(seeTable2).Eventhoughwedofind some (s,s ) target-projectile configurations that result in large i j s comparabletothesevalues(Fig.1),thedecisivepointisthat surf these collisions only have a marginalcontributionto the disc’s totalsmall-dustproduction.Asaresult,thedifferenceswiththe reference s -free cases’ PSDs remain very limited: for nei- surf ther the A-star northe Sun-like cases do we obtain a depletion Fig.3. A-star case. Geometrical cross-section as a function of particle size, at t = 107years, integrated over the whole 50- that exceeds ∼ 10% in the small grain domain close to sblow. Crucially,the s constraintis neverstrongenoughto change 100au disc, for both the dynamically ’hot’ (<e>=0.075) and surf thesizeofgrainsthatdominatethesystem’sgeometricalcross- ’cold’(<e>=0.01)cases.Foreachcase,areferencerunwiththe section,andthusitsluminosity. s constraintswitched-offis also presented.Thetwo vertical surf Moreimportantly,weconfirmthatthemostefficientwayof dottedlinesrepresenttheanalyticalvaluesgivenbyEq.2. depletingacollisionaldebrisdiscfromitssmallgrainsis,byfar, to reduce its dynamical excitation. The production/destruction imbalance mechanism identified by Thebault&Wu (2008) for low-stirring discs has an effect that exceeds, by more than one orderofmagnitude,thatof the s limit. Thisis trueboth for surf theamplitudeofthesmall-graindepletionandforthesize-range thatis affected.Forlow <e>values,thesize s belowwhich cold thePSDisdepletedis,toafirstorder,givenbytherelation β(s) =<e> (5) 1−β(s) whichtranslatesinto 1+<e> s = s (6) cold 2<e> blow leading to s ∼ 50s for our <e>=0.01 case, a value that cold blow roughlyagreeswiththeoneobtainedinthesimulations(Figs.3 and4).Itshouldbenotedthatthisdominanceofthelow-stirring imbalance effectover the surface energyconstraintis probably evenstrongerthanthatwitnessedinFigs3and4,sincewehave taken, for our s prescription, the parameters that were the Fig.4.SameasforFig.3,butforasolar-typestar. surf mostlikelytoleadtohighs values(seeSec.2.1).Onarelated surf note,choosingdiscconfigurationsthatenhancetheamplitudeof 4. Discussionandconclusion the surface-energyconstraint, by, for instance, decreasing <e> or increasing [r ,r ], would not change this dominance ei- min max Figs.3 and 4 seems to indicate that the two dynamically ’hot’ ther,becausetheseconfigurationswouldalsoenhancethedust- casesbearsomesimilaritieswiththepredictionofEq.2regard- production-imbalance effect by further decreasing the level of ingtheimprintofthe ssurf constraintonthePSD.IntheA-star stirringinthedisc. case, this similarity is, of course, simply that the surface en- We emphasise that we did not attempt to fit observed ergy constraint has no effect on the PSD, but in the <e>=0.01 s /s trendswithanexhaustiveparameterexploration(<e>, min blow case, we do indeed find a depletionof grains in the s ≤ 2sblow Mdisc, [rmin,rmax], etc.) in the spirit of the Pawellek&Krivov domain, which is roughly consistent with the sDsuDrf/sblow ∼ 3 (2015)study.Ourgoalwasheretoquantify,inaself-consistent value given by Eq.2. However, this quantitative agreement is way, the relative effects of the two potential dust-depletion probably largely a coincidence, firstly because the depletion mechanismsthatarethesurfaceenergyconstraintandthelow- of s ≤ s grains does not come from the "smallest-barely- stirringdust-productionimbalance.Forthesakeofclarity,andto surf catastrophic-equal-size-impactors"collisions considered in de- clearlyidentifythemechanismsatplay,werestrictedourstudy riving Eq.2 (see Figs1c and 2c), and secondly because this de- to a reference wide disc and to the two illustrative reference pletionstopsat s∼2s largelybecauseofthe’natural’wavi- casesofasun-likeandanA-typestar.Testrunswithanarrower, blow nessofthePSD,combinedwiththeexcessof≥3s particles. ring-like disc, have, however,been performed.They gave rela- blow We also note that this depletion is limited to ∼ 30% and does tively similar results as to the relative imprint of the s and surf 8 Thebault:smallestgrainsindebrisdiscs low-stirringeffectsonthePSD,althoughwithalesspronounced – At such low-stirring levels, the system’s PSD in the small waviness in the PSD’s shape, which was to be expected, since sizedomainis,instead,totallydominatedbyanothermech- smallgrainsclosetos willimpactlargertargetswithinanar- anism:theimbalancebetweensmalldustproductionandde- blow rowringatalowervelocitythantheywouldhaveinanextended structionratesidentifiedbyThebault&Wu(2008)forlow-e disc. discs.Thisimbalancecreatesadepletionofsmallgrainsthat Despitethislimitedparameterexploration,wenotehowever isatleastoneorderofmagnitudemorepronouncedthanthat that our conclusions seem to agree with the numerical inves- causedbythesurface-energyconstraint,anditaffectsgrains tigations of Pawellek&Krivov (2015), who were able to find overamuchwidersizerange. a reasonable fit to the s /s trend observed on their 34- min blow Evenifitseffectisnotasdecisiveascouldbeanalyticallyex- stars sample with the low-stirring-induced mechanism alone. pected,wedo,however,recommendimplementingthesurface- The main assumption for this fit to work is that, while the dynamical stirring of A stars should be of the order of e ∼ 0.1, energyconstraintincollisional-evolutioncodes,asitmightleave visible signatures in the low-s end of PSDs for some star-disc it should decrease towards lower mass stars and be as low as ∼ 0.01 for solar-type objects. Such values might appear unre- configurations. alistically lowif we assume theclassicalview thatdebrisdiscs arestirredbylargelunar-to-Mars-sizedlargeplanetesimals(e.g., References Thebault,2009).However,thelevelandthecauseofstirringis stillanopenissueinpresentdebrisdiscsstudies(seediscussion Ardila, D. R.;Golimowski, D. A.; Krist, J.E.; Clampin, M.; Williams, J. P.; inPawellek&Krivov,2015).Moreover,weknowofatleasttwo Blakeslee,J.P.;Ford,H.C.;Hartig,G.F.;Illingworth,G.D.,2004,ApJ,617, 147 discs,aroundtheG0V-starHD207129(Löhneetal.,2012),and Augereau,J.C.,Lagrange,A.M.,Mouillet,D.,Papaloizou,J.C.B.,&Grorod, theK2V-starHIP17429(Schüppleretal.,2014),forwhichcol- P.A.1999,A&A,348,557 lisionalmodellingpredicts<e>barelylargerthan0.01. Augereau,J.-C.,Beust,H.,2006,A&A,455,987 Benz,W.,&Asphaug,E.1999,Icarus,142,5 DohnanyiJ.S.,1969,JGR74,2531 5. Summary Draine,B.T.2003,ARA&A,41,241 Fitzgerald,MichaelP.;Kalas,PaulG.;Graham,JamesR.,2007,ApJ,670,557 Based on analytical considerations, the pioneering study of Fujiwara,A.,Kamimoto,G.,&Tsukamoto,A.1977,Icarus,31,277 Grigorieva,A.,Artymowicz,P.,Thebault,P.,2007,A&A,461,537 Krijt&Kama (2014) finds that the surface energy constraint, Johnson,B.C.,Lisse,C.M.,Chen,C.H.,etal.2012,ApJ,761,45 which limits the size of the smallest fragment produced after Kobayashi,H.,Tanaka,H.,2010,Icarus,206,723 destructive collisions, could potentially affect the evolution of Krijt,S.,Kama,M.,2014,A&A,566,2 debris discs by limiting the smallest sizes of observablegrains Krivov,A.V.,Löhne,T.,Sremcevic´,M.,2006,A&A,455,509 Krivov,A.V.2007,in3dustinPlanetarySystems",Krueger.,H.,&Graps,A. toavalues largerthantheblow-outsize s . surf blow Editors,p123. Herewenumericallyquantifytheimportanceofthismecha- Krivov,A.V.2010,ResearchinAstronomyandAstrophysics,10,383 nismbyincorporating,forthefirsttime,thesurfaceenergycon- Leinhardt,Z.M.,&Stewart,S.T.2012,ApJ,745,79 straint into a statistical code that follows the collisional evolu- Lisse,C.M.,Chen,C.H.,Wyatt,M.C.,etal.2009,ApJ,701,2019 tionofadebrisdisc.Insteadofrelyingonananalyticalsystem- Löhne,T.,Krivov,A.V.,&Rodmann,J.2008,ApJ,673,1123 Löhne,T.,Augereau,J.-C.;Ertel,S.;Marshall,J.P.;Eiroa,C.;Mora,A.;Absil, averaged value, we compute s (i,j) for all impacting target- surf O.;Stapelfeldt,K.;Thebault,P.;Bayo,A,J.,and11co-authors,2012,A&A, projectile pairs of sizes si and sj in the disc. We consider two 537,110 stellartypes,sun-likeandAV5,andtwolevelsofstirringforthe Pawellek, Nicole; Krivov, Alexander, Marshall, Jonathan P.; Montesinos, disc, <e>=0.075 and <e>=0.01. Our main results and conclu- Benjamin; Abrahám, Pter; Moór, Attila; Bryden, Geoffrey; Eiroa, Carlos, 2014,ApJ,792,65 sionsareasfollows: Pawellek,Nicole;Krivov,Alexander,2015,MNRAS,454,320 Schüppler, Ch., Löhne, T.,Krivov, A.V.; Ertel, S.;Marshall, J.P.;Eiroa, C., – Weconfirmthat,forallconsideredset-ups,thereisafraction 2014,A&A,567,127 ofthe(si,sj)spaceforwhichssurf ≥ sblow,especiallyforlow Schüppler, Ch., Löhne, T., Krivov, A. V.; Ertel, S.; Marshall, J. P.; Wolf, S.; <e>and/orlow-massstars. Wyatt,M.C.;Augereau,J.-C.;Metchev,S.A.,2015,A&A,581,97 – However, the (s,s ) regions of high s do not coincide Thebault,P.,Augereau,J.C.,Beust,H.,2003,A&A,408,775 i j surf Thebault,P., 2009,A&A,505,1269 withtheregionsofhighdustproduction.Indeed,mostofthe Thebault,P.,Augereau,J.C.,2007,A&A,472,169 µm-sized dustis producedbyimpactsinvolving∼3-20s blow Thebault,P.,Wu,Y.,2008,A&A,481,713 targetsandsmall.2-3sblow projectiles,forwhich ssurf is,in Wyatt,M.C.2005,A&A,433,1007 general, smaller than the blow-outsize. This is mainly due Wyatt,M.C.2008,ARA&A,46,339 tothefactthatimpactsinvolvingsmallgrainsareveryener- getic,becausetheseparticlesareplacedonhigh-eorbitsby radiationpressure – Because of this discrepancy between high-s and dust- surf producingcollisions,theglobaleffectofthesurfaceenergy constraintisgenerallyrelativelylimited.Theonlyset-upfor whichithasavisiblesignatureontheparticlesizedistribu- tion(PSD)isforadynamically’hot’discaroundasolar-type star.Buteventhere,thedepletionofsmalldustdoesnotex- ceed30%. – Forthelow<e>=0.01cases,whichshouldinprinciplebethe mostfavourabletostrongsurface-energyconstraints,thede- pletionofsmallgrainsneverexceeds12%andisneverable to change the sizes of particles that dominate the system’s geometricalcross-section.