How Many Rubble Piles Are in the Asteroid Belt?

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How Many Rubble Piles Are in the Asteroid Belt?

A Campo Bagatin, Jean-Marc Petit

To cite this version:

A Campo Bagatin, Jean-Marc Petit. How Many Rubble Piles Are in the Asteroid Belt?. Icarus,

Elsevier, 2001, 149 (1), pp.198-209. �10.1006/icar.2000.6531�. �hal-03003257�

A. Campo Bagatin, J{M. Petit Observatoire de laC^oted'Azur,Ni e, Fran e

and

P. Farinella 

Dipartimento diAstronomia, Universitadi Trieste,Italy

petitobs-ni e.fr

Submitted to I arus April 11, 2000

length:

40 manus riptpages 1 table

9 gures



Died onMar h 25th,2000, from theresultof heart disease. Hewastheonewhointrodu ed ustothephysi sof ollisionsandthe ollisionalevolutionofAsteroids. Thispaperisdedi atedto hismemory.

Send proofs to:

Jean-Mar Petit

Observatoire de la C^ote d'Azur B.P. 4229

F-06304Ni e Cedex 4Fran e

Tel: +33 492 003089 Fax: +33 492 0030 33 Internet: petitobs-ni e.fr

Wehavedevelopedanewversionofthe odebuiltbyCampoBagatinetal. ,1994a,b and Campo Bagatin, 1998 to modelthe ollisionalevolutionof the asteroidsize distribution. The new ode distinguishesbetween \inta t", un-fra turedasteroidsthatdidnotundergo atastrophi ollisions,andasteroids onvertedbyenergeti ollisionsintorea umulatedbodies,or\rubble{piles". The distin tion an also be made on a physi alground, by assigning dier-ent ollisionalparameters to the two kinds of obje ts, with the obje tive of simulatingthe dierent responsesto energeti impa tsthatrubble{pilesmay have{duetotheirdierentstru ture{in omparisontounshatteredbodies. Rubble{piles abundan eturns out to be generally higherwhen su h targets aresupposedtotransfer lesskineti energyto thefragments thanmonolithi asteroids.

We have runa number of simulations of the ollisionalevolution pro ess toassess thesizerangewhererea umulatedbodiesshouldbeexpe tedtobe abundant in the main asteroid belt. We nd that this diameter range goes fromabout10to 100km,butmayextendto smallerorlargerbodies depend-ing onthe prevailing ollisionalresponseparameters, su h asthe strength of the material, the strength s aling law, the fra tion of kineti energy of the impa t transferedto thefragmentsand therea umulation model.

Boththe sizerangeand the resultingfra tion ofrubble{pilesvary widely dependingon theinput parameters. This re e tthe large un ertaintiesstill present in the modelisationof high velo ityimpa t out omes. In parti ular, the simulationsthat take into a ount the derived \hydro ode" s aling laws (Davis et al. ,1994)showthatnearly 100% ofthemainbelt asteroidslarger than a few kms should be rea umulated obje ts. On the other hand the present ode shows that thes aling{law re ently proposed byDurda et al. , 1998produ esalmostnorubble{pile. Thiss aling{lawwasproposedtomat h the a tual populationof asteroids whi h it fails to do if ollisional pro esses area ountedforina self{ onsistent way.

Keywords: Minor planets, asteroids { Collisions { Rea umulation { Collisional evolution

I Introdu tion

Due to the large number of obje ts and the non{negligible orbital e entri ities and in linations, the asteroid belt population forms a ollisional system. Typi al ollision velo ities are of the order of 5 km/s (Farinella and Davis, 1992, Bottke et al. , 1994, Vedder, 1997) in the main belt, and at these speeds a wide range of ollisionalout omes ispossible (Davis et al. , 1989, Petit and Farinella,1993). We referto Campo Bagatinet al. , 1994a,band espe iallytoCampoBagatin, 1998 for anintrodu tionand a review of asteroidbeltphysi aland ollisionalissues.

In monolithi asteroidal impa ts, there is an energy threshold beyond whi h lo alized target damage asso iated with ratering events gives way to global shat-tering and targetbreakup. In the latter ase, the eje tion velo ity of the fragments

with independent helio entri orbits, or onversely may be so low that most (or all) of the fragments fall ba k and are rea umulated into a gravitationally bound \rubble{pile".

Re entyears' spa eprobesand radarobservationsof asteroidssu hasMathilde and Eros (NEAR probe), Gaspra and Ida (Galileo probe), Castalia, Toutatis and 1999JM8 (radar observations) are throwing new light into the previously poorly known stru tures of asteroids. The hara teristi s of some of these bodies suggest that they an be rea umulated bodies; pe uliar features su h as the presen e of enormous raterbasins,abouthalfthesizeofthebodyitself,inthe aseofMathilde, suggestthatthe ollisionalresponsetoenergeti impa tsmaybedierent ompared to monolithi obje ts. In fa t, rubble{piles may be less eÆ ient than unshattered obje tsindeliveringkineti energyof impa tstoexternalfragments(Asphaugetal. , 1998),due tosome reasons that willbe explainedin Se . III.

The rea umulation pro ess may be ompli ated by the existen e of a orrela-tion between velo ity and mass of the eje ted fragments. Some eviden e of su h a orrelation has been found in the past from laboratory experiments (Nakamura & Fujiwara, 1991; Nakamura et al. , 1992; Giblin et al. , 1994; Giblin, 1998). The experimental results a tually show a large dispersion in the data, and the slope of the tentative lineart mat hing them has tobe onsidered with autionwhen on-sidering a s hemati mathemati al orrelation to display the phenomenon. In fa t, simplifying the \dispersed" mass{velo ity orrelation, i.e. velo ity spread over a large rangeofvaluesfor anygiven mass,with awell-dened, straightpower{law re-lationshipmayyieldinsome asetomisleadingresults,aswillbeshown inSe .IV.A and IV.B.

Anyway, this is potentially an important nding, be ause su h a orrelation would strongly ae t the extent to whi h rea umulation is ee tive in reating a \shoulder" in the ollisionally evolved size distribution (Campo Bagatin et al. , 1994b) or determining the internal stru ture of the resulting rubble{pile obje ts (Wilson et al. , 1999) and the amount of material falling ba k after a shattering event. Re ent data, however, indi ate that the velo ity{mass orrelation may be weaker or stronger|depending on the material properties|than assumed earlier, and possibly dependingonthe spe i impa t onditionsorthe experimentalsetup (Giblin, 1998).

Anotherpossible ompli atingfa toristhatnon{disruptive rateringeventsmay also reateadeeplayerofregolith(fragmentedmaterial)andthusfavourthe trans-formationofinta tasteroidsintorubble{piles. However, quantitativeestimates(e.g. Farinella et al. , 1993) show that the rater eje ta a umulated over an asteroid's lifetimetypi ally a ount foronly asmall fra tionof the total mass.

Here, we shall report the numeri alresults that we have obtained on the abun-dan eof rubble{pileasteroids inthe main beltpopulation, asafun tionof size and of some poorly known ollisional response parameters. Hopefully, future data on the out omes of hypervelo ity impa ts,the internal stru ture of asteroidsand their size distribution as afun tion of materialtype willfurther onstrain our models.

havebeenused,andofthemodelsofrea umulation onsideredinthefragmentation ode. The denition of \rubble-pile" is given in Se . III, and a dis ussion on the possible physi al dieren es existing between these kind of bodies and monolithi bodies isalso in luded. The resultsof varioussimulations are shown and explained in Se . IV. Finally,some on lusions are drawn fromthe results of the whole set of simulationsperformedin this work and are presented inSe . V.

II Numeri al Model

Theaimofthepresentworkistogiveanestimateoftheabundan eofrea umulated asteroidsinthe mainbeltasa fun tionoftheir size, takingintoa ount the depen-den e on various physi al ee ts and modelling assumptions. For this purpose, we havedeveloped anew version ofthe numeri alevolutionmodeldes ribed inCampo Bagatin et al. , 1994a,b. This ode simulates the ollisionalhistory of asteroidsby evolvingintimethepopulationsofobje tsresidinginaset ofN

bin

dis retesize bins (usually60), integrating numeri allyasetof N

bin

non{linear, rst{orderdierential equations. From onebin tothe next one,the massofthe bodies hangesby afa tor of 2, i.e. the diameter hanges by a fa tor of 2

1=3

. In the original version of the ode, atany time step the expe ted numberof ollisionsbetween bodies belonging to any pair of bins is omputed, and in ea h ase a ollisional out ome algorithm provides the size distribution of the obje ts resulting from the impa ts, whi h are thenredistributedintothesizebins. Inthe urrentversion,the ollisionalalgorithm also re ords the kind of out ome o urring in ea h ase ( ratering, rea umulation or disruption, with the latter term denoting breakup without rea umulation) and separates the resulting bodies into rea umulated obje ts and \single" fragments. This allows us to ompute, in any size bin, the fra tion of rea umulated bodies over the total population, and how this fra tion evolveswith time (we assume that at the beginningof the evolution itis zero).

The ollisionalout ome algorithmworksinthe followingway. Forea h olliding bodythe availableimpa tenergyE perunitvolumeV is omparedtoa fragmenta-tionthresholdS (theimpa tstrength),whi hisassumedtos ale withsizea ording totwodierentee ts: agravitationalself{ ompressionee tassuggestedbyDavis et al. , 1985 and Housen et al. , 1991, and a strain{rate ee t (Housen and Hol-sapple, 1990). The latter ee t de reases the strength for in reasing sizes, until the former takes overand makes large bodies stronger and stronger (Davis et al. , 1994). In absen e of well established data, and to make things simple, we negle t any dependen e of the imap tstrength on the impa t velo ity. When the available energyex eedsthe threshold,this orresponds toashatteringevent, wherethe mass of the largest inta t fragment is less than half the mass of the target. Otherwise, it isa rateringevent. We have onsideredthree main s alinglaws, basi ally the ones summarizedinDaviset al. ,1994,that wereport herefor thesakeof ompleteness.

S =S 0 + 4 G R 15 (1)

Strain rate s aling:

S =S 0  R 10 m  0:24 " 1+2:1410 11   R 10 m  1:89 # (2) Hydro ode{basalt: S =S 0  R 10 m  0:43 " 1+1:0710 17   R 10 m  3:07 # (3) S 0

isthematerialstrength,asmeasuredbylaboratoryexperiments,istheso{ alled self{ ompression oeÆ ient, Gthe gravitational onstant,  the materialdensity,R the target radius. Figure 1 shows the variation of S as a fun tion of the radius of the target R . Here, as well as in all our simulations, we have assumed a value of the self{ ompression oeÆ ient =100. Forthe energy s alingand the strain{rate s aling, S 0 =3
10 6 J/m 3

, whilefor the hydro ode s aling, S 0 =8:22
10 6 J/m 3 . Notethatsomeauthorsprefer to onsider thespe i energy|thatisthe energy perunitmass(Q

 S

,orsimplyQ 

)|forshattering,insteadofthestrength; it'sworth re alling that these two quantities are related by the following simple relationship: Q



=S=.

The behaviour of the three s aling laws written above are displayed together with the s aling law proposed by Durdaet al. , 1998, inFig. 1.

The energy{s aling for the strength was formerly proposed by Davis et al. , 1985, who just a ounted for gravitational self{ ompression, without any strain{ rateee t. As anbeseen inFig. 1,foragivenmaterialthestrength is onstantfor any body ofsize smaller(strengthregime)thanthe size atwhi hthe gravitys aling in reases the value of the strength. After a transitionrange, the strength keeps on steadily in reasing with size (gravityregime).

In the ase of strain{rate s aling, we used two dierent exponents: the nominal value 0.24, that an be found in Eq. 2, and 0.33. The rst one refers to what Housen et al. , 1991 report, the se ond one is to follow more re ent al ulations (Holsapple, 1994). This exponent has been given dierent values in the last two de ades. Fujiwara, 1980, and independently, Farinella et al. , 1982 suggested the 0.5 value, that meant a marked dependen e of strain{rate ee ts on size, large obje ts{belowthegravityregime{shouldhavethenbeenveryweak. Laters aling theoriesperformedby Housenet al. ,1991suggestedamu hshallowervalueforthis dependen e (0.24). In re ent years, both hydro odes and revisited s aling theories are suggesting again a higher value for this exponent (0.33, Holsapple, 1994; 0.43, Davis et al. , 1994;0.59, 0.667, Housen and Holsapple, 1999).

The hydro ode{s aling instead is a re ently derived s aling law, inferred from hydro ode simulationsof fragmentationpro esses (Davis et al. , 1994)),it depends on the properties of dierent materials and has a strong dependen e of strain{rate

25 km.

We also report here a re ently proposed s aling law(Durda et al. , 1998), that wasarguedtomat hthe a tualdistributionofasteroidsinthemainbelt,ifinserted inthepresentfragmentationand ollisionalevolution odes. We he kedthats aling lawwithourself{ onsistentfragmentation odeandweanti ipateherethatthenal size distribution does not t the observations, mainly be ause of in onsisten ies in Durdaet al. model (see Se . IV.F).

On ethe omparisonbetweentheimpa tenergyperunitvolumeandthevalueof S ismade,theimpa tmayhappentobeeithera rateringevent,orafragmentation event. Inboth ases thesize distributionofthe reated fragmentsis al ulated. See Petit and Farinella, 1993 and Campo Bagatin, 1998 for a detailed presentation of the fragmentation model. The riti al quantity that dis riminates ratering from shattering is the mass fra tion between the largest remnant (M

LR

) and the target (M

T

),whi h isgiven by:

f LR = M LR M T =0:5 " SM T (E K =2) # 1:24 (4)

in the ase of shattering f LR

0:5.

As for the rea umulation pro ess, we have followed Campo Bagatin et al. , 1994b inadoptingtwo dierent models.

a)A\mass{velo ity"model(PetitandFarinella,1993),PFmodelinwhatfolows, that assumes that there is a weak power{law orrelation between the mass and ve-lo ityoffragmentseje ted ina atastrophi ollision,a ordingtothe experimental results mentionedin Se . I. The general aspe t of the relationship between mass m and velo ity V that we adopted in our nominal ase is

V /M r

: (5)

Thevalueofexponentr hasbeenfoundtobeabout1=6byNakamuraandFujiwara, 1991,ortobe inbetween 0and1=6, witha meanvalue of1=13(Giblin,1998), with a onsiderable dispersion of data around the proposed slopes. Any fragment with velo itylargerthanthe es apevelo ityV

es

,derived fromthegravitationalpotential ofthetwo ollidingbodies,willes ape,whilethoseslowerthanV

es

willrea umulate onthelargestremnant. Inthis model,agivenmass orresponds toasinglevelo ity. b) A so{ alled \ umulative" model, in whi h no orrelation between mass and velo ityoffragmentsis onsidered. Inthis ase,thevelo itydistributionisthesame for allfragmentmass and the fra tion of mass with a velo ity larger than V is

f(>V) M(>v) M T =  V V min  k ; (6)

where k is a given exponent, larger than 2 (to allow for onservation of energy), and generally assumed to be k 9=4, and V

min

is alower uto for the velo ity of fragments. By integrating over v between v

min

fr V min = s k 2 k E fr M T : (7)

The kineti energy of the fragments isa given fra tionof the impa t kineti energy deposited inthe target E

K =2: E fr =f KE E K 2 : (8)

This modelassumesthat in ea h mass bin, there are fragments withvelo ity larger than V

es

, whi h would es ape. The fra tion of su h fragments is f(> V es

). The other ones are rea umulated onthe largestremnant.

The above dened f KE

is a poorly known parameter in ollisional pro esses. Laboratoryexperimentssuggestvaluesofthe orderof0.01,whileinordertobeable to reprodu e the formation of asteroid families this value is onstrainted to be of the order of 0.1. It may even vary with size and probably with impa t speed. In thesimulationpresented inSe .IV.F,wedenef

KE

asafun tionofthe targetsize. The situation seems to be more omplex in the ase of rea umulated obje ts, as dis ussed in Se . III.

The three s aling laws previously des ribed have been derived from physi al onsiderations. We also tested the s aling law proposed by Durda et al. , 1998, whi h was derived by tting the observed size distribution of asteroids with the results of their model. We used the parameters from their t 1. The quantity derived by Durda et al. is Q

 D

, the spe i energy for dispersal, i.e. M T

Q  D

is the minimumkineti energy of the proje tilerequiredto disperseatleast half the mass of thetarget,a ounting forpotentialrea umulation. From theirnumbers, we get:

Q  D =1:1445
10 7 D 1:028 e  5:932
10 6 p 8:89510 12 +6:58610 10 ln(D)+2:46610 11 ln(D) 2  ; (9)

where D = 2R . In our ollisional model, we onsider the spe i energy required to shatter the target, Q

 S

. Rea umulation is a ounted for subsequently, by the spe i ation of the fra tion of kineti energy given to the fragments (f

KE

) and the velo ity distribution (PF or umulative). How are Q

 D and Q  S related? As a rst step, let us derive the relationship between Q

 D

and f KE

. Here we assume a umulative velo ity distribution (Eq. 6). Sin e dispersal is dened by f(>v

es

)>1=2,fromEq.6weobtainthatforthe riti al ollision orrespondingto Q  D , v min =2 1=k v es . Were allthat Q  D

is theminimumproje tileenergy required fortargetdispersal,assumingthat1=2oftheproje tileenergyE

K

isdeliveredtothe targetandthatafra tionf

KE

ofthisispartitionedintokineti energyoffragments. Thus, for the riti al ollision orrespondingto Q

 D we have: f KE = 2E fr M T Q  D = k k 2 2 2=k v 2 es Q  D ; (10)

to disperse agiven targetas the sum of the energy needed to shatterthe body and the energy required to disperse the fragments:

E K =Q  D M T = Q  S M T f SH + 1 f KE 0:411 2GM 2 T D : (11) Here, we introdu ef SH

whi h isthe fra tionof kineti energy of theproje tileused to shatter the target, assumed to be 1/2 in our work, while Durda and Dermott used f

SH = f

KE

. In Eq. 10, for onsisten y with Eq. 11 and Durda and Dermott, we use: V 2 es =0:822 4GM T D : (12) This yields: Q  S =f SH " Q  D 1 f KE 0:411 2GM T D # =f SH Q  D " 1 k 2 k 2 2=k 1 4 # : (13)

Figure1shows the valueofS Durda =Q  S asafun tionofR ,using =2500kg/m 3 , as inall our simulations.

On e the out ome of the impa ts between any given pair of asteroids is al- ulated, the results are plugged into the ollisonal evolution model, as mentioned above.

Campo Bagatin et al. , 1994a, and Campo Bagatin, 1998, showed that in a ollisionalsystemwithasharp utoatsmallsizes,aperturbationtothestationary distributions arises in form of \waves", that is os illations about the steady state power law distribution found by Dohnanyi, 1969. To avoid this \wave" ee t, we donot evolvethe smallermass bins that would dire tly be in uen ed by the uto with the general algorithm. Instead, we for e those bins (typi ally the 15 smaller mass bins) tofollowa power law distribution,the exponent of whi h is the average powerlawexponent ofthe next 10bins. In this way, the bins thatevolve a ording to the general algorithmalways have an appropriate set of proje tiles and then do not exhibit the wave pattern. If the number of onstrainted bins is de reased, we may see a wave develop. If it is in reased, nothing hanges, but this in rease the omputationalrequirement. Duetothisfor ing ofthe numberofbodiesinthesmall size bins, there is a hange inmass. But it isnegligeable due tothe rather shallow size distrbution at smallsizes, the mass being in the large bodies.

III About \Rubble{Piles"

The existen e of asteroid rubble{piles is both possible and probable. It is possible be ause from a theoreti al point of view it is lear that su h obje ts may form; on the other hand, many authors have suggested su h bodies to explain features of omets and asteroids ompositions (Weissman, 1986, Asphaug and Benz, 1994, Melosh and Ryan, 1997, Whipple, 1998, Wilson et al. , 1999). And it is now also probable that rubble{piles exist; observations of the late omet Shoemaker{Levy 9

lowdensitiesandporous onstitutionsduetotheirstru ture (Chapmanetal.,1999, Cheng etal., 1999,Davis etal., 1999).

Many arbitrary denitions may be given in general of what a \rea umulated body"is,dependingontheamountofmass ontributedbytheminorfragmentswith respe ttothatofthelargestremnantresultingfromthebreakup. Stri tlyspeaking, we ould onsider asa rea umulatedbodyany fragment omingfroma shattering event onwhi hasingle pebble hadsoftly landedattra tedby itsgravitationaleld. Of ourse,that wouldprovidelittleinformationaboutthe stru tureof theasteroids wandering in the asteroid beltand of the Near Earth Asteroids. Among the many possiblearbitrarydenitions,wehavesele tedthefollowingsimpleone: we onsider a body to be rea umulated if the total mass of the rea reted minor fragments is greater than the mass of the largest single fragment M

LR , that isif M LR < 1 2 M; (14)

M being the mass of the whole rea umulated body. Of ourse inreality the tran-sition is probably not so sharp: in many instan es the impa t out ome will be a singlelargebody oatedwithalayerofsmallerrea retedbodiesa ountingforless than half of the total mass, but stillforming adeep \megaregolith",in other ases the result will be the formation of an aggregate of a few huge (2{3) fragments, as seems tobethe ase of Toutatis. However, the denitiongiven aboveappears tous suitableas a rst approximation.

There are a few points about rea umulated bodies that should be stressed in order todistinguish them from monolithi ,unshattered obje ts.

Some laboratory experiments have been performed by D.R.Davis and E.Ryan by impa tingpre{shattered targets, that is obje ts that have been previously frag-mented by a primary impa t and then glued together with a spe ial, weak glue, simulating gravitational binding. These targets were then re{impa ted. Many in-formations about the physi al hara teristi s of the target | su h as the impa t strength S

0

| were obtained from the size distribution of the resulting fragments. In thiswaythey estimated that thisparameter doesnot hangeinasigni antway with respe t to the ase of non{pre{shattered material. No signi ant orrelation between mass and velo ity of fragmentshas been found in this ase. On the other hand,noinformationonotherphysi alparameterslikef

KE

wasavailablefromthese experiments. Apart from this attempt for looking for dieren es between the re-sponse toimpa tsonrea umulatedand monolithi bodies,thereare somephysi al reasons that suggestin prin iplea dierent behaviour.

Some authors have studied the propagation of sho k waves in porous (non{ homogeneous)bodiesbothbyanalysingexperimentalout omes (Loveet al. ,1993), and by means of hydro ode numeri al simulations (Asphaug et al. , 1998), with similar on lusions pointing towards diÆ ulties in the propagation itself.

When a monolithi body is shattered by an energeti ollision, the sho k wave generated at the onta t surfa e travels a ross the body ex iting the pre{existing aws, that oales e and giverise tomanydierentfragments;the sho k wave even-tuallyreboundsattheoppositesideofthebodyandsubsequentlyextinguishes(e.g.:

ergy that is delivered to the fragments (this fra tion an be hara terized by f KE

) toworkagainstgravitationalbindingenergy. Thefragmentsthat areaggregatedby self{gravity after eje tion, fall upon ea h other in a random way forming an irreg-ular stru ture with many voids, eventually with a layer of regolith formed by ne debrisoverthe surfa e. Ifsu hanobje thappenstoundergoasubsequent energeti ollision we expe t that a large amount of damage is produ ed in the proximity of the impa t point, with the appearan e of fast eje ta. The generated sho k wave is destined to extinguish quite qui kly, in fa t it shall rebound on the surfa e of the fragmentsinwhi hitdevelopsand shallnotbeabletopropagatefurtherinthe tar-get: this is a main dieren ein the response to impa tsbetween monolithi bodies and rubble{piles. (Notethat this maynotbetrueif theproje tileislargerthan the fragmentsformingthe rubble-pile). Thefa tthat thesho kwave isabortedimplies thatthe fragmentsformingthestru ture ofarubble{pile|ex ludingtheonesvery losetothe impa tpoint|shouldnotbeseriouslydamaged,inagreementwith the experimentsbyRyanet al. ,1991, sotheirmass distributionshouldbemore orless the same as before the impa t. A large part of the impa t energy is then expe ted to go into a very large number of highlyinelasti ollisions between the fragments forming the rubble-pile, and into rotations of the fragments themselves that shall s rambletheobje t hangingsomehowitsshape. Asaresultofthispro ess, alotof energy is dissipated, and littlekineti energy is going to rea h external fragments, whi h thenwill not be able to es ape the bindingenergy of the rubble{pile. Ifthis iswhathappens inthis kindofimpa ts,thedispersionofrubble{pilesrequiresmore energeti impa tsthanthedispersionofunshatteredbodies,a on lusionsupported also by Love et al. , 1993. Following this onje ture, we have hosen to make two dierentkindsof simulations. Even ifthe s enariodes ribed abovelooksreasonable (we are stillworking ona quantitative estimate of this kind of kineti pro ess), we onservatively hose to onsider|at least for the nominal ase|all bodies, both rubble{pilesandunshatteredbodies,torespondtoimpa tsinthesame way (f

KE = 0.1), and we investigated the ee t of onsidering rubble{pilesas havingadierent response, s hemati allysummarized by imposing f

KE

= 0.01.

IV Results

Allthesimulationsreportedbelowhavebeenrun onsideringthefulldiameterrange from1mto1000km,althoughwehaveplottedonlytherangefrom100mto1000km (anyway, norea umulationwasfoundtobenoti eable atsmallersizes). The mean relative ollisionalvelo ityassumed isV

rel

=5:85km=se . The simulatedtimespan forthe ollisionalevolutionofthe asteroidbeltis4:510

9

yr,namelythe ageof the SolarSystem. InTable1,wegivethes alinglawsandparametersforea h ollisional model weused. Asdes ribedinSe . II, ea hmodel, ex eptDurda's s alinglaw, an havetwo dierent velo ity distributions: the PF distribution, hara terised by \r", and the umulative distribution, hara terised by \k". Hen e a ollisional model willbe designated by its numberand a subs ript (PF or um) dening the velo ity

intrinsi ollision probabilities (Bottke et al. , 1994). For the initial onditions, we have assumed the same moderate-mass population that we had adopted in our earlier works (CampoBagatin et al. , 1994a,b). Werefer tothose papers, toDavis et al. , 1985, 1989, 1994 and to Campo Bagatin, 1998 for a detailed dis ussion of the general features of the ollisional evolution pro ess and its dependen e on the starting onditions and the assumed impa t response parameters. It is interesting to note that some onstraints on the a priori unknown parameters an be derived by the observed spin rate distribution of asteroids, besides their size distribution (Davis et al. ,1989; Farinella et al. ,1985, 1992).

IV.A Case 1: the Strain{Rate s aling law

We ompare here the ee t of the two velo ity distributions. Figure 2presents the fra tion of rubble-piles in the ase of strain{rate s aling law ( ase 1),as a fun tion ofdiameterofthe target. Thesolidlineshows the umulativevelo itydistributions, whilethe dashedlineshows PF velo itydistributions. The mainfeatureisthe large rubble-pile fra tion for bodies from a few hundred kilometers down to 10 or 1 km. The upper uto is due to the large strength of bodies of that size. In order to have a rubble-pile, a ording to our denition, the mass of the largest fragment afterthe ollisionand beforerea umulationmust be less thanhalfthe target mass (the rea umulated body needs tobe at least twi e as massiveas the largest inta t fragment), orrespondingtowhatis alledafragmentation. Hen ethekineti energy of the proje tile must be larger than 4=3R

3

S, implying a minimum size for the proje tile(R p R p;min =R (2S=(V 2 rel )) 1=3

). Giventhe largevalueofS,the number of available proje tiles is quite small, and the probability of shattering the largest targetsover the ageof the solar system isquitelow. In addition,the kineti energy deposited by a shattering ollision grows with target size faster than the binding gravitationalpotential,hen e the kineti energy available forthe fragmentsis large enoughtoallowthemtoes apethe gravitationalpotential. Thusthela kof rubble-piles of large size isdue to a de ien y of both fragmentation and rea umulation. Atthesmall-size uto,onthe ontrary,thevelo itydistributionplaysanimportant role.

IV.A.1 Cumulative velo ity distribution

We rst onsider the umulative velo ity distribution ase. In Fig. 3a, we present theratioofrea umulatedmassovermassofthelargestinta tfragmentfora riti al mass proje tile|that is for a proje tile large enough to shatter the target|(solid line) and the maximum ratio over dierent proje tile sizes (dashed line). The hor-izontal dotted line sets the limit for what we all a rubble-pile. A tually, rea u-mulation to forma rubble-pile an o ur onlyfor targets larger than about 20 km, whi h orresponds to the smallest size of rubble-piles in Fig. 2. Note that when rubble-piles an form (dashed line above the horizontal dotted line) then impa ts by thesmallestproje tiles apableof shatteringthetargets an already reatethem

line). These small proje tiles are the most numerous, and hen e this explains the ratherlargefra tionofrubble-pilesobtainedinthe ollisionalevolution. Inaddition, fortargetssmallerthan100 km,therubble{pileformed ontainsmorethan halfthe mass of the original target. Hen e, it belongs to the same bin than the target, so very few of the formed rubble-piles belong to the bins smaller than 20 km. The rubble-piles in these bins, when hit by a larger than riti al proje tile, would not rea umulate, and hen e would disappear fromthe population.

IV.A.2 PF velo itydistribution

Figure 3b presents the same quantities in the ase of PF velo ity distribution. In this ase, targets down to 1.4 km an be shattered and yet produ e a rubble-pile. However,oneneedsaproje tilelargerthanthe riti alsizetoobtainarubblepile. In Fig.4,wedisplaythesizeofthe riti alproje tile(solidline),thesizeofthesmallest proje tile that yieldsa rubble-pile (short-dashed line) and the size of the proje tile giving the largest rea umulation ratio (long-dashed line), as a fun tion of target size. We an see that down to 10-20 km,the size of the smallestimpa torthat an reatearubble-pileisjustslightlylargerthanthe riti alshatteringimpa tor(whi h is just in the next size bin). Hen e the number of rea umulation events is quite signi ant ompared tothe numberof disruptions. Forsmaller sizes,the numberof proje tiles that an reate a rubble-pile is less and less signi ant ompared to the numberof disruptive proje tiles, and the mass of the rea umulated body be omes a smaller fra tion of the target mass. Hen e the fra tion of rubble-piles de reases below 10 km, and remains non-zero even at sizes smaller than 1.4 km, a tually down to 500 m.

Another major dieren e with the umulative velo ity distribution is the large gap in size between the rea umulated body and the largest es aping fragments. In the PF model, the largest fragments are the slowest, and therefore they are the ones that will rea umulate. Only the rather small fragments an es ape. In the umulative velo ity distribution, on the ontrary, fragments of every size have the same velo ity distribution. Thus for any given fragment size, there is the same fra tions of es aping fragments and of rea umulated fragments. It results then that the size distribution of es aping fragments shows a smaller gap between the rea umulated body and the largest es aping fragment.

IV.B Case 2: a shallower mass-velo ity relationship

ThePFand umulativevelo itydistributions anbeviewedasthetwoextremesofa widespe trumof velo itydistributions. Forthe umulativedistribution,weassume no relation between mass and velo ity. For the PF distribution, there is a stri t relation between the mass and the velo ity of the fragments. Up to now, we have studied the ase were the mass-velo ity relationship is as steep as the experiments allow. We now onsider a PF velo ity distribution with a shallower dependan e, i.e. a smaller value of the exponent r: r = 1=13. In this ase, the small fragments

not hange the other parameters, there is an ex ess of kineti energy to be shared betweenthelargestfragments. Sothelargestfragmentstendtohavealargervelo ity thanbefore,and mostof themwouldes apethegravitationalbindingpotential. As a result, we expe t tohave less rubble-piles with to this small value of r than with the former larger value, as an be seen in Fig.5.

IV.C Case 3: more dissipative rubble-piles

We onsider now the ase inwhi h rubble-pileshave alowereÆ ien y in delivering kineti energy to fragments. As explained in Se . III, rubble-piles are very likely to be at least as resistant to shattering than monolithi obje ts, but they might be mu h more resistant to dispersion. The only hange in fragmentation modelin this ase, ompared to ase 1,is in the f

KE

oeÆ ient, set to 0.01 for rubble-piles, onsidered now tobe more dissipative.

Figure 6 represents the fra tion of rubble-piles. The main dieren e with the previous ase(Fig.2)isalargerfra tionofrubble-pilesatsizessmallerthan200km. The upper uto is the same as in ase 1, for the same reasons. In the main size range forwhi hrubble-piles are important,theirfra tion hanges from40-60%in ase1to60-90%inthepresent ase,dependingontherea umulationmodel. Due to the smallamount of kineti energy that the fragments an get, when a ollision o urs most of them would rea umulate on the largest one. While monolithi bodies ontinuetobedisruptedbysome riti al ollisions,rubble-pilesinsteadwould survive thesame ollisions;a tually,they areshattered bythose ollisions,but then they rea umulate to form a body of essentially the same size. In this way their populations grows with time.

At the lowest uto size, the umulative velo ity distribution ase allows for rubble{pilesdownto4km,thatistosizesvetimessmallerthanfor ase1(20km). Rea umulation and reationof rubble-piles is then possible for targets assmall as 4 km. At the beginning of the simulation, there is no rubble-pile in the range 4{ 14 km. Very few rubble-piles are reated in this size range by ollisions on targets largerthan20kmbylargerthan riti alproje tiles. Theserubble-piles,ontheother hand, are mostly shattered and rea umulated to roughly the same size; all these pro esses result ina smallpopulationof rubble-piles atthis size range.

For the PF velo ity distribution ase, rubble-piles as small as 500 m an be shatteredandrea umulated. However,hereagain|asexplainedinSe .IV.A.2|for sizessmallerthan10km,thisrequiresproje tileslargerthanthe riti alshattering size. Sin e these proje tiles are less numerous than riti alones, the most ommon out ome of a shattering ollision would be the reation of monolithi fragments, smallerthan halfthetarget'smass. Thenthe fra tionof rubble-pilesismoreorless the same|in this size range|for the PF velo ity distribution ase, in the ase of \standard" and low values of f

KE .

We onsider now \weaker" bodies, for whi h the strain{rate s aling lawwith S 0 = 3
10 5 J/m 3

and the hydro ode s aling lawrepresentative ofbasalt with S 0 =8:22
10 6 J/m 3

are onsidered. The main dieren ebetween the PF and the umulative velo ity distribution that we have seen so far, i.e. a noti eable fra tion of rubble-pilesthat extendstosmallersizes intheformer ase thaninthelatter,holdsfor the ases of weaker bodies that we have tested. In these ases, we will study only the umulativevelo ity distribution, sin e it gives smooth nal populations. In the PF ases, rubble-piles are simplyalsopresent atsizes an orderof magnitude smaller.

The fra tions of rubble-piles for both ases 4 and 7 are shown on Fig. 7. This fra tionislargerthanfor ase1(Fig.2),and rubble-pilesarepresentatsmallerand largersizes. Atallsizes,targets an beshattered bysmallerproje tilesthan in ase 1, thatis with less kineti energy. Hen e the fragments reated inthe ollisionsare less likely to es ape, even for ollisions by riti al proje tiles. Targets larger than 200 km an therefore be shattered and rea umulated, as well as the ones smaller than10km. Thesmallersizeof riti alproje tilesin reasestheirnumberforagiven target size, thus favouring the reation of rubble-piles between 10 and 200 km, as ompared to ase 1. Here, the strength in the strain{rate s aling law is 10 times smallerthan the one displayed onFig. 1. Soit be omessmaller thanfor hydro ode s alingatsizessmallerthanabout 5km. Itfollowsthatrubble-pilesexistonlydown to 4 km inthe ase of hydro ode s aling, while they existdown to 2 km in the ase of strain{rate s aling.

IV.E Cases 5 and 6: Strong bodies

We onsider now the two ases in whi h we assume very \strong" targets. One an see on Fig. 1 that multiplying the strain{rate s aling strength by 10 makes it very lose to the energy s alingstrength. Sowe expe t to get rathersimilar results in both ases. In the present ases, as opposed to the previous ones, it is rather diÆ ultto reaterubble-piles. Theenergyrequiredtoshatteragiventargetis10to 100 times largerthan before, andthe riti al impa tsdelivera lotof kineti energy to the fragments, while the largest fragments are rather large, making it diÆ ult to rea umulate enough mass toform what we alla rubble-pile. A tually, for the umulativedistributionvelo ity ases,thereare essentiallynorubble-piles(fra tion less than3
10

3

)at any size.

In thePF velo ity distribution ases,rubble-piles anstillbe reated, albeitina narrowersize-range, as anbeseeninFig.8. Thelargestrubble-pilesare abouthalf the size as those in ase 1, while the smallestones are about twi e as large,around 1 km. The maximum fra tion of rubble-piles is about 40%, between 20and 50 km.

IV.F Case 8: Durda et al. s aling law

We nally onsider the s aling law proposed by Durda et al. , 1998 that they obtained by tting the \debiased" observed size distribution of asteroidsfor bodies

in Se . II, they derived a s aling law for the riti al spe i energy for dispersal, while our model requires the riti al energy for shattering. This led us to dene a varying f

KE

as a fun tion of the size of the target (Eq. 10). Se ond, the details of fragmentationmodels are dierent (see Durda and Dermott, 1997 and Petit and Farinella,1993). For onsisten yreasons, weonly onsideredthe umulativevelo ity distribution ase, with k = 9=4. In Fig. 9, we have plotted Eq. 10 with this value of k, and Q

 D

given by Eq. 9 and V es

by Eq. 12. This urve is hara terized by a maximumof0.6around6-7kmand signi antde reases forbothsmallerandlarger targets.

For the reationof rubble-piles,we must onsider three dierent regimes. First, for the large targets,larger than 200 km,the strength is very large, so large that almost no proje tile an shatter targets of that size. In ase shattering o urs, the kineti energy available is so large that enough fragments es ape toavoid reating a rubble-pile, despite the de rease of f

KE

from 0.1 to 0.015. For intermediate size targets,between 1and 200 km,the strength isin thesame range asinthe previous ases. However, f

KE

is large at that size, and the kineti energy of the fragments allowthemtoes ape. Finally,forsmaller targets,even thoughthe strength isquite small and f

KE

is small also, the gravitational binding energy de reases too fast to allowformu hrea retion. In addition,the ollisionallifetimeof bodies ofthatsize is quite small, due to the small strength, and most of the fragments are a tually regenerated by fragmentation of larger targets. These freshly reated fragments appear atrst asmonolithi bodies.

As notedabove,f KE

rea hes unrealisti allylowvaluesatsmallsizes,and maybe even atlarge sizes. Thisresults fromthe requirementthat onlyhalfthemass of the target has a kineti energy large enough to es ape (Eq. 10). However, it may turn out that the amount of kineti energy of the fragment is larger than the minimum required by f(>V

es

)>1=2. This is the ase, for example, for smalltargets where the riti al shattering energy is quite large ompared to the gravitational binding energy, thusyielding to V

min >V

es

. So amore realisti denition of f KE ould be: f KE =max k k 2 2 2=k v 2 es Q  D ;0:1 ! : (15) In this ase, Q  S

is omputed fromthe rst part of Eq. 13. However, the ratio

1 f KE 0:411 2GM T D Q  D

neverex eeds 0.05,even whenf KE

has nolowerlimit,and it anbemu hsmaller thanthat ifwe useEq.15. Hen e the strengthismostly un hanged, andthereisno rea umulation of rubble-piles (a ording to our denition). In all the simulations performed using Durda et al. s aling law, the maximum rea umulation is about 20-25% ofregolith ontop of the largestfragment.

Here we give a brief explanation as why we think that Durda et al. , 1998 is not totally self{ onsistent. The ollisional evolution model used is that work

whi h seems to suer from some unphysi al assumptions. D& D use a power-law dierential size distribution for the fragments resulting from a ollision between a target and a proje tilelarger than D

min

determined by their eq. (9)

dN =BD p

dD: (16)

They make a onfusion in the denition of p. They rst give its value as being slightlylargerthan2.5(eq. (11),page149),andthenasbeingslightlylargerthan3.5 (page155). The latervaluea tually orresponds tothe dierentialsize distribution exponent, while the former orresponds to the umulative size distribution. Their eq. (12)relatesthedierentialsizedistributionexponenttothefra tionalsizeofthe largestremnantb. Sotheappropriaterangeofvaluesforpisreally3.5to4. Intheir simulations, they x p, hen e xing b. So the size of the largest fragment is xed withrespe ttothesizeofthetarget,regardlessofthekineti energyofthe ollision. Finally,inD&D,therea umulationisnegle ted. A tually,theydon't onsider the rea umulationtobeadierentpro essfromshattering. Inashaterringfollowedby rea umulation,onewouldrsthaveapower-lawsize distributionoffragments,and then, some of them (whi h ones depend on the mass-velo ity relationship) would rea rete on the largest fragment. Hen e the distribution would onsist of a very large fragment(the rubblepile) and a power-lawdistribution of smaller fragments, with a gap in between. In Durda et al. , 1998, the only hange is to dene the minimum kineti energy required as:

E min

=MQ 

: (17)

The largestremnant an thereforebeeither amonolithi fragment orarubble pile. But thesize distributionisalways apower-lawwithaxed exponent,startingfrom that largestremnant. Inaddition,the values they use forp(2.82 and 2.47) seemto be umulative size distributionexponents,whi his not learly stated, and ontrary to the expe ted dierential size distribution exponent.

IV.G Benz and Asphaug, 1999 s aling law

During the review pro ess, we have learnt about the new s aling laws proposed by BenzandAsphaug,1999and whi hwasnotyetpublishedatthe timeweperformed allthe al ulationspresented inthe present paper. Here again, asin Durdaet al. , 1998,the authorsgiveQ

 D

asafun tionof thesize ofthe target. Inaddition,allthe other assumptions that we make about the out ome of a ollision are not present inthat work,neither expli itlynor impli itly. Therefore, it is not reallyrelevantto use Benzand Asphaug s alinglawin our model. However, we run asimulation for a basalti target, and dening Q

 S

a ording to Eq. 13,f KE

being given by Eq. 15. Forreasonssimilartothosepresented inDurdaet al. ase,we foundnorubble{pile satisfyingour dention.

Ourresultsshowthatrea umulationisprobably ommonpla e(albeitnot \univer-sal")formain{beltasteroidsintheintermediatediameterrangefrom10to200km. In this range the fra tion of rubble-piles goes from about 30% to100%, depending on physi al parameters. This may extend to smaller or larger sizes, namely from 0.5-1to500 km,dependingonthe ollisionalresponseparameters. Theupper uto varies mainlywith the assumed s aling lawfor the impa t strength S, allowing for larger targets to rea umulate after ollisions if they are weak. Rubble-piles an existin anoti eable fra tion down to0.5-1 kmif we assumea xed mass-velo ity relationship su h as the one in Eq. 5. For a probably more realisti velo ity distri-bution, su h as the one given in Eq. 6, the smallest rubble-piles have a diameter of 3 to 15 km, depending on the strength of the targets. When assuming weak targets,su h as with the hydro ode s alinglaw, orthe strain{rates aling law with S 0 = 3
10 5 J/m 3

, the fra tion of rubble-piles ex eeds 90% from 5 to 300 km. On the opposite, for strong targets, the formation of rubble-piles be omes more diÆ- ult. A tually, no rubble-pile is formed with the umulative velo ity distribution, for the energy s aling and the strain{rate s aling with S

0 = 3
10 7 J/m 3 , and it doesnotex eed 40%forthePF velo itydistribution. Whenapplyingour algorithm to Durda et al. , 1998 s aling law, and hanging from Q

 D to Q  S , we obtain no rubble{pile a ording toour denition (largest fragment less than half the mass of the rea umulated body).

The simulationsperformedinthisworkshowthat themass-velo ity dependen e is a main feature of the physi s of ollisions at high velo ity. Caution has to be taken when onsidering this relationship in a very s hemati way, espe ially for smallvalues of the exponent of the power law. In fa t,negle ting the dispersion of data a tuallypresentinalllaboratoryexperiments onthis issue,may leadtowrong on lusions. We nd a very strong dependen e of the rea umulation fra tion, at any given size, on the mass{velo ity relationship. Considering very shallow mass{ velo ity relationships, leads to very small fra tions of rubble{piles, below 20% for r=1=13 around100km (Fig.5),and negligibleatother size ranges; loseto0% at allsizesforr=0. Webelievethatthisisonlyanee toftheway themass{velo ity relationshipismodelled. Experimentalresultsforthemassandvelo ityoffragments show that data are widely dispersed, and that they are loosely distributed about somepower{lawrelationship. Ifwes hemati allyassumeamathemati alexpression likethe one given inthe textfor that relationship,we are for ing every fragment of mass m

i

tohave a given velo ity v i

. In the ase inwhi h the relationship between these two quantities is very shallow, that is when r is very smallor,as an extreme ase, when it is zero, then all fragments would have almost the same speed. That speed may besmaller orlarger than the es apevelo ity from the body. Inthe ase with r = 1=13 we nd that most of the fragments have velo ities larger than the es ape velo ity, and we obviously end up with almost no rea umulation. In this way we introdu e an arti ial bias into the pro ess, that shows up to be relevant at very small values of the exponent r. More laboratory experiments are indeed needed tobetterboundthis relationship. Itis learnowthathavingnorelationship

at all the same thing as having a mass{velo ity relationship with a null exponent. It seems obvious that both PF and umulative models are two \extreme" ways to model therea umulationpro ess; inthe former ase the mass-velo ityrelationship is taken into a ount \too seriously",while inthe latter itis ompletely negle ted. Is the truth somewhere in between? That is probably the ase, and we intend to rene the modelling of the phenomenon in future work.

Another interesting result of our simulations is that if we wish to model in a realisti way the ollisional evolution of asteroidal systems, we should not negle t the physi al dieren es between rea umulated and unshattered bodies. We have summarized this dieren es just fo using on the response to impa ts in terms of the dierent fra tion of kineti energy f

KE , (f

KE

=0.1 for unshatteredobje ts, and f

KE

=0.01forrubble{piles)deliveredtothefragmentsinthe two ases, andwehave found that this implies adete table dieren e in the populationof asteroidsat the end of the ollisional evolution, and that the fra tion of rea umulated asteroids is noti eably higher.

Both the fa ts that rubble-piles may be harder to disrupt ompared to unshat-tered obje ts, and that they may not be unusual also at km-sizes, may have dire t onsequen esonfuture deviation/destru tionstrategiesregardingtheriskofimpa t events on Earth by asteroids.

A knowledgments

A. Campo Bagatin worked on this proje t in 1998{1999 while staying at the Ob-servatoirede laC^ote d'Azur atNi e,thanks tothe \Henry Poin ares" fellowship of the Centre National de la Re her he S ientique.

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Figure 1: Strength versus diameter of the target a ording to the dierent s aling laws onsidered in the simulations. The strength is the riti al energy per unit volumeneeded toshattera targetso that the largestfragmenthas half the mass of the inital target.

Figure 2: Fra tion of rubble-piles (rea umulated asteroids with mass larger than twi e themassofthe largestfragment)versus diameterattheendof ollisional evo-lution. Thisplot orrespondstothe ase1,strain{rates aling,forboth ummulative (solid line) and PF (dashed line) velo ity distribution. The urves are somewhat irregular due to the sto hasti nature of the ollisional pro ess. Depending on the size of the riti al proje tile ompared to the enter of the orresponding bin, the largest fragment an vary a lot,and alsoits abilityto rea rete material.

Figure3: Ratioofmassofrea umulatedfragmentsovermassofthelargestfragment aftershattering,asafun tionoftargetsize,for ase1. Solidline orrespondstoanat least riti alproje tile(belongingtothe smallestbinthat islarge enoughtoshatter the target), and dashed line to the maximum rea umulation when varying the proje tile mass. (a)Cumulative velo ity distribution. (b) PF velo ity distribution.

Figure4: Proje tilediameter versus targetdiameterforthe riti alproje tile(solid line), the smallest proje tile apable of reating a rubble-pile (short-dashed line) and for the maximumrea umulation (long-dashedline) for ase1 and PF velo ity distribution, asafun tion of targetsize. The two dashed urves stop whenthere is no more rea umulation. The roughness of the urve is due to the dis reteness of the size bins (size ratio of 2

1=3

), preventing to nd the exa t riti al values.

Figure 5: Same asFig. 2, but for PF velo ity distribution and r=1=13.

Figure 6: Same asFig. 2, but for more dissipative rubble-piles, i.e. f KE

= 0.01 for rubble-piles.

Figure 7: Same asFig. 2, but for ases 4 (solidline) and 7(dashed line).

Figure 8: Same asFig. 2, but for ases 5 (solidline) and 6(dashed line).

Figure 9: f KE

as a fun tion of target diameter, as derived from Durda et al. best tfor Q

 D

Case S alinglaw S 0 (J/m ) f KE r k 1 Eq. 2 3 10 6 0.1 1/6 2.25 2 Eq. 2 3 10 6 0.1 1/13 -3 Eq. 2 3 10 6 0.1/0.01 1/6 2.25 4 Eq. 2 3 10 5 0.1 1/6 2.25 5 Eq. 2 3 10 7 0.1 1/6 2.25 6 Eq. 1 3 10 6 0.1 1/6 2.25 7 Eq. 3 8.22 10 6 0.1 1/6 2.25 8 Eq. 13 - Eq. 10 - 2.25

Table1: Thephysi alparametersusedinthesimulationsaresummarizedhere. The materialdensity = 2500 kg/m

3

and the self{ ompression oeÆ ient  =100. For ea h asewelistherethes alinglaw,thematerialstrengthS

0

,anelasti ity oeÆ ient f

KE

, mass{velo ity exponent r, and umulative velo ity distribution exponent k, when appli able.