New data on the velocity–mass relation in catastrophic disruption

\

Pergamon

Planet[ Space Sci[\ Vol[ 35\ No[ 7\ pp[ 810Ð817\ 0887 Þ 0887 Elsevier Science Ltd All rights reserved 9921Ð9522:87:,*see front matter PII ] S921Ð9522"87#99905Ð5

New data on the velocityÐmass relation in catastrophic disruption Ian Giblin Dipartimento di Matematica\ Universita di Pisa\ via Buonarroti 1\ I!45016 Pisa\ Italy Received 0 September 0885 ^ revised 18 December 0886 ^ accepted 5 January 0887

Abstract[ The relationship between fragment velocity and mass following a disruptive impact is of great importance when modelling populations of small bod! ies such as the asteroid main belt or the more recently observed Edgeworth!Kuiper belt where mutual col! lisions play an important role in their dynamic evol! ution[ The velocity!mass relation following these mut! ual collisions strongly a}ects not only the collisional lifetime of large "gravity dominated# asteroids\ but also the rate at which mass is ejected from the belt into resonances\ providing a source of resupply for the Earth!crossing asteroid population and\ in the case of the Edgeworth!Kuiper Belt\ the short period comets[ Although considerable work has been done on the sub! ject of the relationship between velocity and mass of fragments from cratering and catastrophic disruption events\ it has recently become apparent that there may not be a valid general relationship between these quan! tities[ In this paper I present a summary of size!velocity data obtained from single! and twin!camera _lms of hypervelocity\ highly catastrophic impacts into spheri! cal 10 cm targets of arti_cial rock with strength and density similar to basalt[ The 1D velocities of at least 840 fragments larger than approximately 09 mm have so far been measured in 7 similar experiments[ Of these\ 58 have been studied in 2D in the recent experiments using two cameras at 59>[ The data collected here sug! gest that in general there is only a weak correlation between mass and velocity\ and that the best!_tting exponent varies between 9 and −0:5 with an average value of approximately −0:02[ Þ 0887 Elsevier Science Ltd[ All rights reserved

velocity and mass of each ejected fragment[ A good knowl! edge of this relationship is crucial in modelling collisional systems such as the asteroid main belt*where disruptive impacts are believed to be the dominant evolutionary force[ A substantial amount of data has been published on crater ejecta\ e[g[ Gault and Heitowit "0852#\ Vickery "0876#\ but the available data on catastrophic frag! mentation of _nite targets is generally less extensive and usually constrained by selection e}ects and data reduction problems\ e[g[ Fujiwara and Tsukamoto "0879#\ Davis and Ryan "0889#\ Martelli et al[ "0883#[ The problem of measuring the velocity of fragments from an impact is essentially one of the instrumentation[ Recent advances in the available technology "speci_cally high!speed _lm analysis and computerised image pro! cessing# for analysing fragmentation experiments\ tog! ether with systematic studies of standard experimental setups\ have allowed researchers to study these phenom! ena in greater depth\ e[g[ Nakamura et al[ "0881#\ Giblin et al[ "0883a#[ The remainder of this paper is organised as follows ] section 1 brie~y describes the experiments ^ sec! tion 2 covers the analysis of high!speed _lm and a review of the commonly used techniques for 2D velocity deter! mination\ with an overview of the system used to carry out a fairly detailed study of fragment velocity from a number of experiments ^ section 3 discusses some of the notable features of this analysis process as applied to the 0881 data ^ section 4 describes the results of the analysis in the context of previous work\ alongside a discussion of selection e}ects which have in various ways biased the data presented here\ and follows with a summary of evi! dence for a general mass!velocity relation in these data ^ section 5 concludes\ with some suggestions for future work[

Introduction One of the most signi_cant areas in the study of frag! mentation experiments is the relationship between the Correspondence to ] I[ Giblin[ Beechwood\ Filmer Laneoti 1\ Sevenoaks\ TN03 4AG ^ Tel[ ] 9933 0621 346007 ^ Fax ] 9933 0621 631542 ^ E!mail ] ianÝBRL[u[net[com

Experiments The experiments considered here constitute part of a series of fragmentation studies carried out over recent years

811

I[ Giblin ] New data on the velocity!mass relation in catastrophic disruption

under the guidance of the late Prof[ G[ Martelli "former head of the Space Physics Group at the University of Sussex# in collaboration with the Planetology Group at the Turin Astronomical Observatory\ the Space Mech! anics Group at the University of Pisa and the CNR Insti! tute of Space Physics in Rome\ Italy[ Designed to simulate collisions between asteroids in the main belt\ these experiments involve the use of a contact explosive charge\ buried at an appropriate depth into a target of arti_cial rock[ The use of a contact charge has been shown "Holsapple\ 0879^ Ryan and Melosh\ 0887# to provide an impact closely similar to that of a solid projectile initially travelling at the detonation velocity of the explosive "in this case 5[0 km s−0#\ given that certain conditions regarding the size\ shape and burial depth of the explosive charge are met[ In this case the explosive is placed in a cylindrical cavity "in the base of the spherical target# 14 mm wide and 15 mm deep with a conical end as produced by a masonry drill[ The equivalent impact speci_c energy in these experiments was of the order of some units times 096 erg g−0\ su.cient to produce cata! strophic fragmentation of rocky targets "Fujiwara et al[\ 0878#[ The contact charge technique is described in more detail in Giblin et al[ "0883b# alongside a discussion of its usefulness in experiments of this type[ A total of 01 similar experiments upon 10 cm targets of alumina cement have been carried out ] four in September 0878 and a further eight in September 0881[ The cured cement has a static compressive strength of 06×092 J kg−0 and a density of 0[66 g cm−2[ Where a harder core was present "shots 0878!1 and 0878!2# this had a density of 3[19 g cm−2[ This gave target masses of 7613 g\ 7531 g\ 7521 g and 7407 g for shots 0 to 3 in 0878 and 7009 g\ 7909 g\ 8219 g\ 7629 g for shots 4 to 7 in 0881 "shots 0 to 3 from the 0881 programme are not considered here#[ The experiments were performed in the open in order to minimise secondary fragmentation against the walls of a target chamber and to record the uninterrupted ballistic trajectories of the fragments[ The experiments were _lmed using single "in 0878# or twin "in 0881# NAC E!09 high! speed cameras running at 699 and 399 frames per second\ respectively[ The impact was made from below\ with the impact direction vertically aligned in the image plane[ Where two cameras were used\ the mutual angle was 59 degrees[ A more detailed description of the experimental technique and parameters can be found in Giblin et al[ "0883a\b^ 0887#[ Following both sets of experiments\ the 05 mm cine _lm was transcribed to VHS video and digitised frame!by! frame using an Acorn Risc PC[ Using software specially written for this project\ the digital video sequence can be replayed full!screen "in the case of the 0878 single camera experiments# or in a split screen format "for the 0881 twin camera sequences#\ allowing individual ejected fragments to be identi_ed and tracked[ Once tracked\ fragments can be {{cut|| from each image and viewed individually at a variable rate up to 099 frames per second[ The software shifts the images in order to place a speci_c fragment in the centre of the view*this\ combined with the enhanced frame rate\ allows us to build a good mental picture of the fragment while in ~ight and thus easily measure the size\ shape and rotation rate in a large number of cases[

Measurement of fra`ment velocities from the experimental _lms Given a _lmed record of a disruption experiment\ the measurement of 1D fragment velocities in the image plane is essentially straightforward once any perspective and other geometric e}ects are taken into account[ If a large number of fragments have been tracked in two dimen! sions\ it is possible to estimate an upper limit to the 2D velocity distribution from the upper limit of the 1D vel! ocities "Nakamura and Fujiwara\ 0880#\ as well as the mean true ejection velocity for a given experiment "Davis and Ryan\ 0889#[ Finding the true 2D velocity of a speci_c fragment is a lot harder[ A variety of techniques have been used to estimate the velocities of individual fragments[ Three recently used methods are brie~y described below[ Method A[ Fujiwara and Tsukamoto "0879# and Ryan "0881# have assumed that fragments are launched radially outward from the surface of the target ^ this assumption coupled with the fragments| initial positions in the target allows the three!dimensional trajectory to be calculated from the 1D view on the _lm[ The main drawback of this technique is that it introduces selection e}ects*only surface fragments can be easily studied\ and then only those for which their initial positions can be determined by matching the _lm records to a reconstructed target[ Method B[ A second technique\ used previously by Nak! amura and Fujiwara "0880#\ is to assume that the fastest measured 1D velocities really are the 2D velocities of the speci_c fragments*this is a valid assumption if "a# an almost complete analysis is made of the size and 1D vel! ocity of the outer fragment envelope\ and "b# the dis! ruption is assumed to be symmetrical[ Method C[ The third technique which can be used to determine 2D velocities is that of identifying a given frag! ment in two or more synchronised _lms from di}erent directions[ Once a fragment has been tracked in both views it is a simple procedure to calculate the actual 2D position and velocity in each frame[ This approach* which is the technique used in the analysis reported here* has previously been used in conjunction with method A by Nakamura and Fujiwara "0880# and Nakamura "0882# in the analysis of basalt and alumina experiments similar to those reported here[ This method has the main advan! tage that it is accurate*there is very little error in the resulting 2D velocity measurements*and the main dis! advantage that it is extremely di.cult to carry out[ Gen! erally one has to accept only being able to study a few easily matched fragments this way[ In our twin!camera analysis software\ the problem of matching fragments between the two views\ in order to measure their true 2D velocities\ has been solved by what could be called a {{brute force|| application of method C[ First\ the size "major axis in all cases\ plus minor axes where possible#\ 1D velocity\ rotation rate and axial orien! tation in ~ight of as many fragments as possible are mea! sured[ Fragments tend to rotate such that their top "in an impact from below the target# moves toward the impact point\ as noted by Fujiwara and Tsukamoto "0879#[ Fur! thermore\ very few fragments are found to be rotating around their longest axis[ With the exception of a few strongly tumbling fragments "that is\ fragments not ro! tating around a principal axis and precessing su.ciently

I[ Giblin ] New data on the velocity!mass relation in catastrophic disruption

strongly as to hinder measurement#\ this means that reliable measurement of the fragment major axis length and rotation rate is usually possible from almost any single!camera view orthogonal to the impact direction[ The study of tumbling fragments in these experiments is discussed in Giblin and Farinella "0886#[ When all possible data have been collected from the separate 1D images\ the computer attempts to match frag! ments based upon several simple constraints*such as the fact that the rotation rate will be the same in each view* together with some more subtle considerations which include the expected fragment axial orientation in a given view\ the period during which the fragment is visible in each _lm sequence\ and so on[ The program then provides a map of {{candidate matches|| which generally need to be followed up by direct study[ Nevertheless\ this is nine tenths of the way there*by viewing the two cut!out mini! animations one beside the other\ it is possible to immedi! ately con_rm or rule out the majority of possible matches[ As will be explained in the next section\ the combination of software development and the analysis of some very highly catastrophic experiments "that is to say\ a very large number of fragments# have meant that only a few experiments have so far been analysed in this way[ The system can be usefully employed even when only a small number of fragments have been properly studied\ but is then more susceptible to selection e}ects[

Some comments on the analysis of the 0881 data Since there was no direct control over the lighting con! ditions in our open air experiments\ the di}erent _lms exhibit considerable variation in exposure\ as well as the time of day a}ecting the position and size of shadows which can hinder analysis[ Further but less signi_cant degradation occurs in the telecine "transcription from cine to video# process and again when the images are _nally digitised for use on the computer[ The _rst shot to be analysed was shot 0881Ð5\ partly because it has "by a small margin# the best quality images and a minimum of shadowing in the _eld of view[ This _rst analysis was carried out while developing the software and before any experience of the 2D matching problem had been gained[ As a result it represents the most unbiased data set[ Once the software had reached a reasonably stable version cap! able of some level of automatic 2D matching\ the approach was changed[ Shot 4 is a transitional case since it has been well analysed in 1D with the latter parts of the analysis devoted to _nding new 2D fragment tracks "that is to say\ by direct inspection of the images in order to choose which fragments to check#[ This newer approach yields more 2D matches per unit time "of work# but is subject to selection e}ects in that it is biased toward {{inter! esting|| fragments[ For a fragment to be interesting it usually needs to be either distinctly rotating or distinctly not rotating "both of which allow it to be spotted more easily in the second view#\ or perhaps an unusual shape "elongated\ ~at\ unusually round\ there was even one easily identi_ed fragment shaped like Snoopy|s head#[ Some fragments could be matched in 2D just by being visible at a time when the _eld of view is particularly uncluttered\

812

Table 0[ 1D Velocity data Shot 0878Ð0 0878Ð1 0878Ð2 0878Ð3 0881Ð4 0881Ð4 0881Ð5 0881Ð5 0881Ð6 0881Ð6 0881Ð7 0881Ð7

cam[ fl 0 0 0 0 0 1 0 1 0 1 0 1

9[922 9[981 9[077 9[940 9[05 9[92 9[03 9[90

n

nant "ms−0#

039 8[4 022 7[2 003 09[9 018 7[7 096 5[83 45 6[39 191 8[16 078 8[21 82 6[65 04 7[32 22 8[91 08 7[72

nmax "ms−0#

n¹ 1D "ms−0#

24 18 15 28 17[8 02[4 21[0 29[4 16[4 07[3 07[3 01[3

03[3 00[0 00[2 09[5 7[27 6[14 09[3 8[25 8[40 8[31 7[67 7[44

namely at the start or end of the dispersion process[ Shots 0881Ð6 and 0881Ð7 continue the trend of more selective analysis ^ the actual percentages of 1D tracks which have also been identi_ed in 2D "in camera 1# are 49)\ 7[1)\ 59) and 36) for shots 4\ 5\ 6\ 7 respectively[ This very well illustrates the point[ In terms of errors\ it is hard to assess whether or not these factors cast doubt onto the less studied shots[ There is no known bias introduced either by tracking certain shaped fragments\ or by picking fragments which are visible during certain times "this in principle constrains velocity but the e}ect is minimal because the _eld of view is large in these experiments#[ If anything these e}ects will tend to yield most data at the ends of the mass\ velocity and rotation rate distributions while neglecting the median fragments with average sizes\ shapes and rotation rates[

Results and discussion Table 0 summarises the measured 1D velocity data from each studied experiment from 0878 and 0881 and for each camera\ with the number of fragments n\ the measured antipodal velocity nant\ maximum measured velocity nmax\ and mean 1D velocity n¹ 1D[ Also included is fl\ the mass fraction remaining in the largest remnant[ Asterisks indi! cate that fl has been estimated from the experimental _lms rather than measured in!hand\ in cases where the largest remnants broke after impact[ From a statistical viewpoint\ an interesting problem is presented by the twin!camera data[ If a given shot has "say# 399 visible fragments of measurable size\ and we study one fragment in each view\ there is only a 0:399 chance that this is the same fragment and the data can reasonably be treated as two independent observations[ As the number of tracked fragments increases\ the chances that a fragment has been tracked in both views "and the observations are therefore not independent# increases also[ Thus for a study such as that of shot 0881Ð5 in Table 0 where 191 and 078 fragments have been tracked and studied in the two camera views\ we cannot treat the two data sets as independent[ If selection e}ects are similar between the two cameras\ as described in the previous

813

I[ Giblin ] New data on the velocity!mass relation in catastrophic disruption Table 2[ Comparison between 1D and 2D mean velocity data for {{2D fragments||

Table 1[ 2D Velocity data Shot

n

0881Ð4 0881Ð5 0881Ð6 0881Ð7

17 12 8 8

nant "ms−0# nmax "ms−0# n¹ 2D "ms−0# 6[39 8[29 6[80 7[57

04[8 15[5 08[5 01[3

6[28 8[17 8[79 8[05

Shot

n n¹ 1D!0 n¹ 1D!1 n¹ 2D "ms−0# "ms−0# "ms−0#

0881Ð4 0881Ð5 0881Ð6 0881Ð7

17 12 8 8

5[08 6[60 7[88 7[87

5[25 7[91 8[23 8[09

6[28 8[17 8[79 8[05

Mean ratio 0[07 0[07 0[96 0[90

section\ then the problem becomes still more acute[ A total of 840 fragments have been tracked in the _rst view across all shots reported here\ with a further 168 tracked in the second view[ Of this latter set\ 58 are known to overlap with the _rst camera since they have been matched between the views and thus their velocities identi_ed in three dimensions[ Some statistics of these 58 fragments are reported in Table 1[ Where an intact antipodal cap "in the shape of a spheri! cal section# is not observed\ antipodal velocity is estimated as the velocity of the largest visible fragment ejected in the antipodal direction[ Since the impact "hence antipodal# direction is in the image plane in all cases\ we would expect the measured 1D antipodal velocities to be similar between the two views "where two cameras have been used# and each to be close to their true 2D values[ This is indeed the case*the variation between cameras is 6)\ ³0)\ 8) and 1) in shots 0881Ð4 to 0881Ð7 respectively[

in 2D than small fast ones[ So\ whilst even the smallest\ fastest fragments can be followed in 1D\ they will not be present in the 2D data[ Thus we should look at only those fragments which have been tracked in both 1D and 2D[ Table 2 summarises this subset of the data[ As shown in Table 2\ when we consider only fragments which have been tracked both in 1D and 2D\ we _nd a mean 2D ] mean 1D velocity ratio greater than one "it would be very worrying if this were not the case;# but this ratio is far from the 2:1 used previously[ Again\ though\ this is a result of a selection e}ect[ For a fragment to be {{trackable|| in 2D it must be visible*and preferably look similar*in both views[ Thus the most likely place for a 2D track to be found is in the impact direction\ which lies in the image plane for both cameras[ These fragments\ since they are almost travelling perpendicular to both views\ have very similar 1D and 2D velocities[

Mean and maximum velocities

Antipodal velocities

One of the most puzzling aspects of laboratory frag! mentation experiments is the fact that the typical velocities are so low[ Analysis of asteroid families "Zappala et al[\ 0873# indicates that fragment ejection velocities for aster! oid!sized bodies must have been of the order of 099 m s−0 to form the currently observed families[ This contrasts with many laboratory results "Fujiwara and Tsukamoto\ 0879 ^ Davis and Ryan\ 0889 ^ Takagi et al[\ 0880#\ which have yielded a mean velocity an order of magnitude lower for rocky targets undergoing hypervelocity impact[ Mean laboratory ejection velocities are typically around 9[0) to 9[4) of the impactor velocity\ i[e[ of the order of 09 ms−0 for most hypervelocity "km s−0# impacts[ The data presented here are consistent with previous experimental work in terms of mean ejection velocities[ The conundrum regarding ejection velocities may possibly be solved by determining a suitable scaling law for fragment ejection velocities as a function of target size "Ryan and Melosh\ 0887#[ An interesting point regarding the data in Tables 0 and 1 for 1D and 2D velocities is that the scale factor used by some previous authors "Davis and Ryan\ 0889\ used a factor of 2:1# to estimate the mean 2D velocity from the mean 1D velocity "based upon geometrical arguments# does not work here[ By taking the average 1D velocity observed in each of the cameras in the 0881 studies\ the mean ratio between mean 2D and 1D velocities is 9[88; The explanation for this is simple selection e}ects\ such that large\ slow fragments are more likely to be tracked

Fujiwara and Tsukamoto "0879# arrived at the following expression for the characteristic speed in m s−0 of anti! podal fragments in catastrophic disruption of rocky tar! gets "note that here the units have been changed to MKS with a corresponding change in the numerical constant# ] nant  2[4×09−1

9[65

0 1 E Mt

"0#

where E is the total impact energy in Joules and Mt the mass of the target in kg[ In the experiments described here\ the contact charge technique is a source of some ambiguity as to the total energy E[ Based upon a ballistic pendulum measurement "Giblin et al[\ 0883a#\ in our experiments E:Mt ¼ 3×092 J kg−0\ whilst a consideration of the mass and speci_c energy of explosive "Housen et al[\ 0880# yields an upper limit "assuming perfect coupling of the explosive to the target# of E:Mt ¼ 0×093 J kg−0[ Substitution into eqn "0# gives a range of nant from ½08 to ½27 m s−0[ Thus there is a factor of less than three di}erence between the predicted lower limit and our mean measured antipodal 2D velocity of nant  7[2 m s−0 "s  9[7 ms−0#[ The slightly lower observed velocity may be attributable to imperfect coupling of the contact charge to the target\ or simply a failure in the assumed explosive charge:impactor equivalence[ Nakamura "0882# reports that alumina targets used in experiments largely similar to these exhibited some degree of local shattering and spallation in the antipodal region due to geometric focus!

I[ Giblin ] New data on the velocity!mass relation in catastrophic disruption

ing of the shock wave\ thus confusing the measurements of antipodal fragment velocity[ This phenomenon was not observed in the experiments reported here*in fact\ partial reconstruction of the target indicated a minimum of frag! mentation or spallation in the antipodal region[ There is a possibility that this is a result of our use of the contact charge technique rather than a real impactor[ VelocityÐmass relation Is fragment velocity related to mass< Results of detailed crater ejecta studies "Gault and Heitowit 0852 ^ Vickery 0876# have indicated a power!law relationship between fragment velocity and mass in cratering of rock!like materials at high velocity[ Subsequently\ laboratory dis! ruption experiments "Fujiwara and Tsukamoto 0879 ^ Nakamura et al[\ 0881 ^ Nakamura 0882# have suggested that this general power law\ of the form nm−k

"1#

holds to some extent for disruption of _nite macroscopic targets[ Nakamura and Fujiwara "0880# found a value of k ¼ 05 in the high velocity "km s−0# regime\ in reasonable agreement with the results of Gault and Heitowit "0852# regarding cratering[ However\ results of other studies "Ryan 0881\ Takagi et al[\ 0885# have indicated little or

814

no correlation between velocity and mass for broadly similar experimental parameters[ In all cases authors have pointed out the lack of complete data as well as the prob! lem of selection e}ects in the analysis of laboratory frag! mentation experiments[ Figures 0 and 1 show logÐlog plots of the estimated mass and measured 1D velocities for the 0878 and 0881 data respectively[ Figure 2 shows the equivalent plots from the 0881 2D data[ It should be noted that the {{mass|| quoted here is a very simple estimate based upon the cube of the major axis length of the fragment[ This provides a strict upper bound to the mass while introducing sub! stantial errors for non!spherical fragments[ This has been used because "i# the major axis length is available for all studied fragments and "ii# no marked dependence of shape upon size has been found in these experiments\ so the errors should be evenly distributed[ Error bars therefore indicate a factor of 3 "possibly an underestimate# in the negative direction only\ and curve _ts have been applied to the nominal data points[ Table 3 summarises the best _t estimates of k in eqn "1# based upon both the unmodi! _ed 1D and more accurate 2D velocity data[ Fractional forms are shown alongside the actual estimate in order to compare these data to the {{standard|| result of 0:5[ Recalling the discussion in Section 3 of the selection e}ects in "most notably# shots 0881Ð6 and 0881Ð7\ the estimated mass is likely to be further from the real values

Fig[ 0[ 1D velocity and estimated mass of fragments measured in shots 0878Ð0 to 0878Ð3[ Mass is normalised to the target mass[ Velocity is normalised to the antipodal velocity

815

I[ Giblin ] New data on the velocity!mass relation in catastrophic disruption

Fig[ 1[ 1D velocity and estimated mass of all measured fragments in shots 0881Ð4 to 0881Ð7[ Mass is normalised to the target mass[ Velocity is normalised to the antipodal velocity

Fig[ 2[ 2D velocity and estimated mass from shots 4\ 5\ 6 and 7 in the 0881 run[ Mass is normalised to the target mass[ Velocity is normalised to the antipodal velocity[ The _tted lines indicate a power law of the form nm−k as described in the text

I[ Giblin ] New data on the velocity!mass relation in catastrophic disruption Table 3[ Estimates of k Shot

k1D!0

k1D!1

k2D

0878Ð0 9[03 "¼0:6# * * 0878Ð1 9[9944 "¼0:079# * * 0878Ð2 9[915 "¼0:32# * * 0878Ð3 9[932 "¼0:12# * * 0881Ð4 9[03 "¼0:6# 9[02 "¼0:7# −9[925 "¼−0:17# 0881Ð5 9[01 "¼0:7# 9[930 "¼0:13# 9[978 "¼0:00# 0881Ð6 9[984 "¼0:00# 9[04 "¼0:6# 9[06 "¼0:5# 0881Ð7 9[03 "¼0:6# 9[97 "¼0:02# 9[983 "¼0:00# mean ] 9[977 "¼0:00# 9[09 "0:09# 9[968 "¼0:02#

in these shots than in other cases[ This however is not limited to either large or small fragments and applies across the whole size range[ It could be argued that the very largest fragments do not need to be oddly!shaped to be easily identi_ed ^ if this is the case the mass of smaller fragments is more likely to have been overestimated and the gradient of the logÐlog plots increased[ So\ we are probably dealing with something closer to the limit "most positive or most negative slope in Figs[ 0 and 2# in these cases[ It is interesting to note that only the steepest slope in Fig[ 2 agrees with the experimental results of Nakamura and Fujiwara "0880#\ and that this data is known to be subject to strong selection e}ects in the case of both 1D and 2D results[ It may be the case that these selection e}ects are similar to those present in previous work[

Conclusions The analysis of these hypervelocity impact experiments has yielded a substantial amount of new data on sizes and velocities of fragments from strongly disrupted rocky targets[ The general properties of the ejecta velocity _eld "antipodal velocity\ mean velocity# are in good agreement with the results of previous studies by other authors[ Although subject to numerous selection e}ects which are di.cult to assess quantitatively\ the data presented here suggest a weak negative correlation between velocity and mass\ and signi_cant variability from experiment to experiment\ with an average slope of 0:02[ The conclusion is reinforced by the fact that the least biased data set "of shot 0881Ð5# exhibits a slope of 0:00\ less than 19) from the average value[ This slope is shallower than that found by some previous researchers "Nakamura and Fujiwara\ 0880#[ In order to better assess the error margins associ! ated with these estimates of the exponent k in the velocity! mass power law it is obviously necessary to carry out more extensive comparisons between data sets both previously published and current[ This will hopefully be covered in a forthcoming paper\ once more size\ shape\ velocity and other data have been collected for the experiments dis! cussed here as well as forthcoming experimental pro! grammes[ Acknowled`ements[ I would like to thank Paolo Paolicchi and Akiko Nakamura for their comments as referees\ and also Paolo Farinella and Don Davis for helpful discussions[ I was supported

816

for this work by ESA fellowship PERS:id:857[ Partial support by the Italian Space Agency "ASI# and NATO grant 7594:32 are also gratefully acknowledged[

References Davis\ D[\ Weidenschilling\ S[ J[\ Farinella\ P[\ Paolicchi\ P[ and Binzel\ R[ P[ "0878# Asteroid Collisional History ] E}ects on Sizes and Spins[ In Asteroids II\ ed[ R[ P[ Binzel\ T[ Gehrels and M[ S[ Matthews\ pp[ 794Ð715[ Univ[ of Arizona Press[ Davis\ D[ and Ryan\ E[ "0889# On collisional disruption ] Exper! imental results and scaling laws[ Icarus 72\ 045Ð071[ Farinella\ P[\ Froeschle\ Cl[ and Gonczi\ R[ "0883# Meteorite delivery and transport\ Asteroids\ Comets\ Meteors 0882\ Kluwer\ Dordrecht\ pp[ 194Ð111[ Fujiwara\ A[ and Tsukamoto\ A[ "0879# Experimental study on the velocity of fragments in collisional breakup[ Icarus 33\ 031Ð042[ Fujiwara\ A[\ Cerroni\ P[\ Davis\ D[\ Ryan\ E[\ Di Martino\ M[\ Holsapple\ K[ and Housen\ K[ "0878# Experiments and scaling laws for catastrophic disruptions[ In Asteroids II\ ed R[ P[ Binzel\ T[ Gehrels and M[ S[ Matthews\ pp[ 139Ð154\ Univ[ of Arizona Press[ Gault\ D[ E[ and Heitowit\ E[ D[ "0852# The partition of energy for hypervelocity impact craters formed in rock[ Proc[ 5th Hypervelocity Impact Symp[ Vol[ 1\ Firestone\ Cleveland\ pp[ 308Ð345[ Giblin\ I[ and Farinella\ P[ "0886# Tumbling fragments from experiments simulating asteroidal catastrophic disruption[ Icarus\ 016\ 313Ð329[ Giblin\ I[\ Martelli\ G[\ Smith\ P[ N[\ Cellino\ A[\ Di Martino\ M[\ Zappala\ V[\ Farinella\ P[ and Paolicchi\ P[ "0883a# Field fragmentation of macroscopic targets simulating asteroidal catastrophic collisions[ Icarus 009\ 192Ð113[ Giblin\ I[\ Martelli\ G[\ Smith\ P[ N[ and Di Martino\ M[ "0883b# Simulation of hypervelocity impacts using a contact charge[ Planetary and Space Science 31\ 0916Ð0929[ Giblin\ I[ Martelli\ G[\ Farinella\ P[\ Paolicchi\ P[\ Di Martino\ M[ and Smith\ P[ N[ "0887# The properties of fragments from catastrophic disruption events[ Icarus\ in press[ Holsapple\ K[ A[ "0879# The equivalent depth of burst for impact cratering[ J[ Geochem[ Soc[ Meteor[ Soc[ Suppl[ 03\ 1260Ð 1390[ Housen\ K[ R[\ Schmidt\ R[ M[ and Holsapple\ K[ A[ "0880# Laboratory simulations of large scale fragmentation event[ Icarus 83\ 079Ð089[ Martelli\ G[\ Ryan\ E[ V[\ Nakamura\ A[ N[ and Giblin\ L[ "0883# Catastrophic Disruption Experiments ] Recent Results[ Planet[ Space Sci[ 31\ 0902Ð0915[ Nakamura\ A[ "0882# Laboratory simulation on the velocity of fragments from impact disruptions[ Institute of Space and Astronautical Science "Kanagawa\ Japan# Report 540[ Nakamura\ A[ M[ and Fujiwara\ A[ "0880# Velocity distribution of fragments formed in a simulated collisional disruption[ Icarus 81\ 021Ð035[ Nakamura\ A[ M[\ Suguiyama\ K[ and Fujiwara\ A[ "0881# Velocity and spin of fragments from impact disruptions 0[ An experimental approach to a general law between velocity and mass[ Icarus 099\ 016Ð024[ Petit\ J[!M[ and Farinella\ P[ "0882# Modelling the outcomes of high!velocity impacts between small solar system bodies[ Celest[ Mech[ 46\ 0Ð17[ Ryan\ E[ V[ "0881# Catastrophic collisions ] laboratory impact experiments\ hydrocode simulations and the scaling problem[ Ph[D[ Dissertation\ Univ[ of Arizona\ Tucson\ Arizona[ Ryan\ E[ V[ and Melosh\ H[ J[ "0887# Impact fragmentation ] from the laboratory to asteroids[ Icarus\ 022\ 0Ð13[ Takagi\ Y[\ Kato\ M[ and Mizutani\ H[ "0880# Velocity dis!

817

I[ Giblin ] New data on the velocity!mass relation in catastrophic disruption

tribution of fragments of catastrophic impacts[ Asteroids\ Comets\ Meteors 0880\ LPI\ Houston\ pp[ 486Ð599[ Takagi\ Y[\ Nakamura\ A[ M[ and Fujiwara\ A[ "0885# Fragment velocity dependence on fragment mass impact fragmentation phenomena[ Asteroids\ Comets\ Meteors 0885 "abstracts#\ LPI\ Houston\ p[00[

Vickery\ A[ M[ "0876# Variation in ejecta size with ejecta velocity[ Geophys[ Res[ Letters 03\ 615Ð618[ Zappala\ V[\ Farinella\ P[\ Knez³evic and Paolicchi\ P[ "0873# Collisional origin of asteroid families ] mass and velocity distributions[ Icarus 48\ 150Ð174[