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Asteroid collisional evolution: results from current scaling algorithms
Planet. Space Sci.,
Vol. 42, No. 8, pp. 599-610, 1994 Copyright 0 1994Elsevier Science Ltd Printed in Great Britain. All rights reserved 0032-0633/94 $7.00+0.00
Pergamon
0032-0633(94)EOO61-T
Asteroid collisional evolution : results from current scaling algorithms Donald R. Davis,’ Eileen V. Ryan’ and Paolo Farinella’ ’ Planetary Science Institute, 620 North 6th Avenue, Tucson, AZ 85705-8331,U.S.A. ’ Dipartimento di Matematica, Universita di Pisa, Via Buonarroti 2, 56127 Pisa, Italy, and Observatoire de la C&e d’Azur, Dept. Cassini, B.P. 229,06304 Nice Cedex 4, France
Received 22 September 1993; revised 25 February 1994; accepted 25 February 1994
1. Introduction
Correspondence
to
: D. R. Davis
Unraveling the complex collisional history of the asteroids requires understanding how large-scale collisional events have acted to produce many of the observed characteristics of these objects. The sizes and spins of most asteroids are the result of over 4 Byr of collisional history (Davis et al., 1979, 1989). The formation of Vesta and its family, unique among the asteroids, needed a large-scale thermal alteration of Vesta, followed by just the right amount of collisions: one or a few large cratering impact(s) that ejected fragments up to 10 km in size from the basaltic crust to form the family (Binzel and Xu, 1993), but not impacts so large and frequent as to fracture the crust and thoroughly mix it with the underlying olivinerich mantle material (Davis et al., 1985). Other asteroids, though, such as 16 Psyche, have apparently experienced much more energetic collisions than did Vesta. Psyche is believed to be the metal-rich core of a differentiated Vestalike parent body that had its crust and mantle stripped away by such large impacts (Chapman, 1986). Recent development of a numerical hydrocode includ-
600 ing finite strength effects (Melosh et al., 1992) and the application of this code to problems of the collisional fra~entation of asteroids has provided a new tool to estimate the specific energy needed to break up asteroids of different sizes. This work predicts a more rapid weakening of asteroids with size in the strength regime than did the earlier scaling theory of Holsapple and Housen (1990), which was based on dimensional analysis but employed many of the same assumptions as did the hydrocode algorithm. The hydrocode scaling result does not, however, reflect both shattering and dispersal of fragments from the target body, only shattering. Specifically, the hydrocode predicts that after the body is shattered, most of the fragments do not achieve escape velocities (Ryan, 1993), due to low fraction fKEof the projectile’s kinetic energy being partitioned into fragment kinetic energy. This is equivalent to say that the targets are much more resistant to disruption (with fragment dispersal) than to shattering, as indicated by the value of the threshold specific energy Q*. More work to compute the scaling of fKEby hydrocode simulations of largescale impact break-up events is in progress (Ryan and Melosh, 1994), and results quoted here using the Q* scaling relationship are only preliminary. One important test for a scaling law for collisional disruption is whether it produces results in collisional modeling programs that are consistent with observations about the real asteroid population. For example, do plausible collisional scenarios lead to the preservation of Vesta’s basaltic crust over the age of the solar system? Does the size distribution match that of the present asteroid belt? In order to test models generated using different scaling laws, we have incorporated them into our program for studying asteroid collisional evolution. Cases were then run using the same initial population of asteroids and collisional parameters, varying only the choice of the scaling law between the runs. We have explored different starting populations and choices of the critical collisional parameters, including some results derived from the hydrocode calculations. In the remainder of this paper, we will summarize these results and discuss implications of the different scaling laws for the collisional history of asteroids, based on our simulations of asteroid collisional evolution.
2. Scaling laws for the fragmentation threshold An understanding of how collisions have acted to produce the observed characteristics of asteroids depends critically upon having a good scaling theory by which to extend laboratory impact experimental results, involving targets typically 10 cm in size, to large scale asteroidal collisions, involving bodies that can be several hundred km in size, For asteroids Iarger than several tens of km in size, gravity plays a dominant role in dete~ining the specific energy (Q*) needed to fracture a body. For small asteroids and laboratory experiments, it is the material strength of the target that is important for determining Q*; this is the strength regime. For large asteroids, though, there is a
D. R. Davis et al. : Asteroid collisional evolution
large pressure due to gravitational self-compression that acts throughout the volume of the body (Davis et al., 1985). In the gravity regime, it is this pressure that largely determines the specific energy required to shatter a body. The transition region occurs where the energy density needed to rupture the material bonds is comparable to that needed to overcome the gravitational compression. The size regime in which this transition occurs depends on the material from which the body is made. For weak bodies, the transition occurs at smaller sizes than it does for bodies made of strong material. Early studies of asteroidal collisions assumed a simple energy scaling in the strength regime, i.e. the specific energy required to fracture the material bonds of the body was thought to be independent of the size of the body. Fujiwara (1980) and Farinella et al. (1982) first provided independent physical arguments suggesting that Q* may depend on target size-actually, be proportional to R-'12. The first rigorous effort to develop a scaling theory was the dimensional analysis model by Housen and Holsapple (1990), which incorporated a strain-rate dependence on the specific energy needed for shattering a body. For a fixed impact speed, the strain-rate model with “nominal” values of the material parameters gave a Q* that varies as R-o.24 in the strength regime. As we recalled above, Davis et al. (1985) showed that gravitational self-compression of the target’s interior gives rise to a gravity regime, where Q* is mainly determined by the self-compressional effect, in addition to the strength regime where only the material properties are relevant. In the gravity regime, Q* becomes proportional to R2 with the energy scaling and to R'.65 with the Housen and Holsapple scaling, reflecting the dependence on both projectile energy and velocity. Recently, Melosh et al. (1992) have incorporated the Grady-Kipp fragmentation model, including both a size and a strain-rate dependence, into a 2-D hydrocode program and used this code to study collisional fragmentation. Then Ryan and Melosh (1994) have used this tool to calculate Q* as a function of target size for different types of materials. Generally, the hydrocode calculates that large bodies are significantly weaker (Q* is much smaller) than does strain-rate theory for the same impact event. Figure 1 compares results for these different algorithms. Which of these, then, shall we adopt as the preferred one for carrying out studies of asteroid collisional evolution? To provide some insight as to how these different scaling algorithms affect collisional evolution scenarios, we undertook to compare the outcome of identical cases of collisional evolution, changing the scaling laws. We consider four of the scaling relationships shown in Fig. I : (1) simple energy scaling with self-compression added; (2) strain-rate scaling ; (3) hydrocode scaling for basalt and (4) hydrocode scaling for weak mortar. For the hydrocode cases, basalt and weak mortar are chosen to represent a broad range of possible asteroid material properties. These four cases span the range of currently proposed scaling relationships. In addition to the results in the strength regime, there is the question of how to include gravitational compression effects. In our collisional code, the effectiveness of self-compression is specified by a
D. R. Davis et al.: Asteroid collisional evolution
0
WEAK CEMENT
_ V = 5.5
loo
601
MORTAR
km/a
10’
102
103 TARGET
10’
IO5
DIAMETER
lo6
IO’
1oa
(cm)
Fig. 1. Critical specific energy Q* as a function of target size, as calculated from four scaling relations : (1) simple energy scaling with self-compressional term added (solid line) ; (2) strain rate scaling (dashed line) ; (3) hydrocode computations for basalt (filled circles) ; and (4) for weak mortar (open squares). Impact velocity is 5.8 km s-l, and is assumed to be constant over the range of target sizes. Full triangles represent experimental results by Housen et al. (1991), with the corresponding largest fragment to target mass ratios
model parameter, k (see Davis et al., 1985, 1989). Preliminary but sketchy data indicated that a value k = 1 was adequate to represent the limited experimental information then available for the Davis et al. work. More recently, an elegant set of experiments by Housen et al. (1991) measured the explosive energy needed to disrupt bodies at different ambient pressures. These experiments suggested that much higher values of k are needed in our model ; in fact k values between 10 and 100 are required to make the model results match the experimental results (more exactly, the implied value of k is of the order of 10 if one assumes the energy scaling, of 100 in the strain-rate case). Specifically, we used the following formulae, whose label numbers correspond to those given above for the different scaling laws. Energy scaling : dimensionless
s=so+
4rckGp2R2 15 .
(1)
Strain rate scaling : 1+2.14x s = so(&~o~z4[
W1k($---~~*9].
(2) Hydrocode-basalt
:
S= So(&)u~43[1+1.07X Hydrocode-weak S=
lo-l7 ($07].
(3)
mortar :
So(&)o~6’[l+2.00X
Here S is the impact strength defined, e.g., in Davis et al. (1989) that is Q* times the density p of the material (always assumed to be 2.5 g crnw3), So is the value of S referring to the typical size of a laboratory target (20 cm diameter). The results of Ryan and Melosh (1993, plotted in Fig. 1) have been scaled from an impact velocity of V = 1.65 km SC’to the average asteroidal value of 5.8 km SC’ (see Farinella and Davis, 1992) assuming that S is proportional to V”.63,as indicated by the hydrocode results for basalt. We have just added the terms referring to the strength and gravity regimes to determine the value of S for any given size body, since the transition range is narrow and over most of the asteroidal size range either of the two terms dominates. Note that the hydrocode results were consistent with S, values of 8.22 x lo7 and 3.62 x lo7 erg cm -3 for the basalt and weak mortar cases, respectively ; however, in our test runs we have kept So as a free parameter even when the hydrocode scaling laws were actually used. As for the k parameter defined above, we stress that its value is tied to experiments. All scaling theory and hydrocode calculations have parameters of this type and their values are ultimately derived from experiments. The physics of fragmentation and the physical state of the target and projectile are sufficiently complex that we cannot deduce the outcome of such collisions from first principles. Nor is it likely that we will be able to do so for a long time.
3. Collisional evolution of the size distribution IO-17
(&r2y.
(4)
In order to simulate the collisional evolution of the asteroid size distribution we used the code described in detail
602 in Davis et al. (1989) and Farinella et al. (1992), which evolves in time the numbers of objects residing in 23 discrete logarithmic size bins, whose central values span the range 6.2-1000 km in diameter in such a way that there is always a factor 2 in mass (1.26 in size) between two neighboring bins. We refer to the two papers quoted above also for a rigorous definition and discussion of the input parameters of the code, most of which describe the response of asteroids to cratering and shattering impacts. We carried out a series of numerical experiments using the four different scaling algorithms described in section 2. The parameters of our model which were used for our standard case are given in Table 1. While our principal interest is in the effects of scaling laws, we did change the model parameters in a few cases. The labels appearing in the figures specify which parameter values have been changed, besides the scaling law adopted in each case. Note that our starting populations were built by calculating the number of bodies present in each of the 23 diameter bins using two power law distributions with a transition diameter of 100 km. We assumed that a total of 400 bodies exceeded this size and were distributed according to N( > L>)KXD-~Z+’ ’ , with an incremental population index bz = 4.5). The four largest size bins, containing Ceres, Pallas and Vesta, however, were assumed to start with 1, 0,O and 2 bodies, respectively. For D < 100 km, we used a standard population N( >D)GcD~~J+‘, with b, = 4. This initial population is close to those used in previous work (Davis et al., 1989 ; Farinella et af., 1992) and derived from the requirements that: (i) the initial total mass of the asteroid belt is not much larger than the current one ; (ii) larger and larger asteroids have been less and less depleted by impact disruption. Given these constraints, as we shall see, the final population is not very sensitive to the detailed shape of the initial size distribution. We based the size distribution for the current belt, to be compared with the output of the code after 4.5 x lo9 yr of evolution, on IRAS results augmented by groundbased observations when IRAS data were lacking (see Cellino Table 1. Parameters used for the numerical runs Total evolution time : 4.5 x IO9yr Number of size bins : 23 Central diameter of first bin : 6.2007 km Size ratio between two adjacent bins : 1.25992 Maximum fractional change allowed in a time step : 0.1 Average impact velocity : 5.8 km SK’ Sigma of impact velocity distribution : 1.9 km SK’ Nominal value for impact strength : 3 x lo7 erg cmm3 Nominal value for self-compressional parameter k : 1.0 Energy partitioning coefficient_&, (both craters and break-up) : 0.1 Exponent of ejecta mass vs velocity power law (both craters and break-up) : - 2.25 Crater excavation coefficient : 1.Ox lo-’ g erg-’ Material density : 2.5 g cm-’ Initial population transition diameter : 100 km Initial population N(D > DfransltioJ: 400 Nominal power law exponent I) < Dtransition : 4.0 Nominal power law exponent D > Dlramition : 4.5 Populations of four largest bins : 2, 0, 0, 1
D. R. Davis et al. : Asteroid collisional evolution
et al., 1991). We used the size distribution derived by Cellino et al. for diameters larger than 44 km. As shown in that paper, this size distribution is essentially identical with that derived by Chapman (as reported in Gradie et aE., 1989) for the entire asteroid population. At smalter sizes our knowledge of the asteroid population is incomplete, i.e. we do not have albedo measurements or diameters for all asteroids smaller than 44 km. So, we adopted two techniques to estimate the small-size asteroid distribution : first, we used the results of the Palomar--Leiden Survey (PLS, Van Houten et al., 1970) for the slope of the asteroid population and anchored this power law distribution at D = 44 km. The second approach consisted of extrapolating the Cellino et al. power law dist~bution for each of their five zones of the main belt and then summing up the results. The PLS slope used was -2.95 (incremental diameter distributions are used hereinafter) which was the fit by Van Houten et al. to their absolute magnitude distribution. The conversion to diameters assumed a constant albedo across the size range. We chose this value rather than the commonly quoted slope of - 3.5 found by Dohnanyi (1969), in agreement with his analytical theory for the equilibrium size distribution of a collisionally evolving population. The extrapolation based on the Cellino et al. results used their “corrected random model” to find albedos for asteroids that have no measured values. This model assumes that all asteroids in a semimajor axis zone have the same distribution of albedos, so the IRAS measured albedo distribution, corrected approximately for the flux overestimation problem (Veeder et al., 1989), was randomly sampled to find the albedos for small asteroids. (The assumptions that one makes regarding the albedo distribution can significantly affect the diameter distribution derived from a measured magnitude distribution, as shown by Cellino et al.). The two models that we assumed covered the plausible range of albedo distributions, ranging from the constant albedo model used for the PLS data to the random model for the Cellino et al. extrapolation. The results from these two approaches yield different estimates for the small size population of asteroids. We adopted the geometric mean of these two estimates as the population for sizes down to 5.5 km, the lower size bound in our numerical simulation. We estimate the uncertainty in the population at different sizes as follows : for bodies larger than 250 km diameter, the uncertainty is fixed at rtO.5, since the sizes of asteroids in that diameter range are well determined. For bodies between 50 and 250 km diameter, the uncertainty is just the statistical uncertainty of +JN (N being the total number of bodies residing in the bin), while for diameters smaller than 25 km, we used an uncertainty of -&log 2 in log N. This uncertainty is only an estimate based on the range of the population of small asteroids calculated by different workers (see Farinella and Davis, 1994). A smooth fit between the JN regime and the smaller sizes is constructed between the 25 and 50 km sizes. Our adopted size distribution and its uncertainty, derived from the above procedures, are shown in Figs 27. The results of the collisional evolution code using energy scaling are shown in Fig. 2 : part (a) gives what we
D. R. Davis et al. : Asteroid collisional evolution
ld
I
k=lO
‘”
f
a
10'
ii!
= 10~ It
4 2
10'
*
Fig. 2. Results using energy scaling : Part (a) is our so-calIed standard case derived from earlier work and uses the widely quoted laboratory strength of basalt and a small initial mass pop~ation. In part (b), we increase the impact strength to 8 x IO’erg cme3. In part (c), we have increased the parameter k, which describes how &ectiveIy self-compression increases Q*, to match the laboratory results of Housen et al. (1991). This case gives our best fit to the overall asteroid size distribution. Part (d) repeats the case of part (c), but with a different random number sequence term a standard case and uses the nominal parameters that we have employed in earlier studies of asteroid collisional evolution (see Table 1). This case, though, shows significant depletion relative to the observed population at sizes smaller than 40 km diameter. However, increasing the laboratory impact strength to the value reported by Nakamura and Fujiwara (199 l), larger by nearly a factor of 3 than earlier reported values, produces much better agreement between the model results and the observed population, with only a few bins disagreeing (Fig. 2(b)). increasing the strength of the gravitational self-compression term, to match the laboratory experiments of Housen et al. (1991), i.e. setting k = 10, produces the best agreement between our model results and the observed asteroid population (Fig. 2(c) and (d)). We next changed to strain-rate scaling and reran the cases of Fig. 2(a) and 2(c) ; these results are shown in Fig. 3. For the 2(c) case, we had to use k = 100 in order to have the calculated Q* in the gravity regime match the Housen et al. ex~~ments. As seen from Fig. 3(a) and
(b), the calculated size distribution does not match the observed one, although the results using the larger selfcompression effect (Fig. 3(b)) are markedly superior. In order to see if changing the starting population could produce an improved match to the observed size distribution, we reran the case of Fig. 3(b) with both a shallower initial small size population (Fig. 3(c)) and a steeper one (3(d)). The final results were relatively unaffected by these changes in the starting population. The scaling algorithms produced by fitting many runs of the hydrocode were next incorporated and the above described cases were rerun using these scalings ; results are shown in Fig. 4 for the scaling using basalt. In general, this algorithm does not give a particularly good match to the observed population. In all cases, the small size population is depleted relative to that observed. This is due to the weak impact strength of bodies smaller than about 10-20 km in size. This scaling algorithm predicts a small size slope that is very shallow relative to the observed estimated slope in this size range among the asteroids.
D. R. Davis et al. : Asteroid collisional evolution
Diemeter Cc)
Diameter(km) (d)
scaling : Part (a) is the standard case described above, changing only the scaling law. Here the population distribution does not match the observed one and Vesta is very likely to have been shattered due to its low strength. Part (b) gives results for strain-rate scaling with k = 100, the value needed to match the experimental results of Housen et al. (1991). In Fig. 3(c) and (d), we vary the small diameter starting population Fig. 3. Results using strain-rate
Changing the hydrocode algorithm to that based on weak mortar accentuated the trend observed for basalt material : small asteroids are strongly depleted and have
too shallow a slope to match the observed population (Fig. 5). This result is obviously a consequence of the steep fall off in strength with increasing size, cf. Fig. 1 and equation (4). It is interesting to note that the various scaling algorithms-strain-rate, hydrocode-basalt and hydrocodeweak mortar-can be simulated quite well by using energy scaling with the appropriate choice of S, to calculate the size distribution of asteroids. This is because Q* is approximately constant in the size range from the smallest observable asteroids up to the size where self-compression becomes important, typically between about 10 and 100 km in diameter. We illustrate this approximation in Fig. 6, which uses energy scaling with .I$ of 2 x 106, 7 x 10’ and 4 x IO4erg cm-3 in order to simulate runs with strain-rate scaling, hydrocode-basalt and hydrocode-weak mortar, respectively. Comparing these results with those shown
in Figs 3(a), 4(a) and 5(a), respectively, indicates the similarity between these cases. Thus as far as the evidence from the asteroidal size distribution is concerned, energy scaling is indistinguishable from the various scaling algorithms that have been put forth in recent years, provided that the appropriate value of S,, is selected to represent each scaling law. Laboratory impact experiment data (Fujiwara et al., 1989) as well as hydrocode calculations to date (Ryan, 1993) suggest thatf,, is considerably lower than the 10% value we adopted, based on the requirement to reproduce the observed properties of asteroid families (see discussion in Davis et al., 1989). So we reran the case for hydrocodebasalt scaling usingf,, = 0.02. This case (Fig. 7) still does not match the real asteroid population: asteroids in the 20-60 km size range are overabundant relative to the real population, because they are resistant to disruption due to the lowf,,. Also, the small diameter population has a shallower slope than does the observed population. The same features are found with the hydrocode-weak mortar scaling.
605
Material:basalt, b, = 4.
Fig. 4. Results from hydrocode calculations using basalt : part (a) shows the standard case with this scaling. Parts (b) and (c) illustrate the standard case but using S, = 8 x 10’ erg cmW3for two different random number sequences. In Fig. 4(d) and (e) we show the effect on the final population of varying the initial small diameter population
4. Constraints from V&a
There are other constraints on the collisional history of asteroids besides the size distribution. The fact that Vesta
has a basaltic crust which has survived, except for one or two large cratering events, since the formation of the solid surface provides a powerful constraint on the collisional history of this body. The relevant information comes from
D. R. Davis et al. : Asteroid collisional evolution
Materiel: weak mcilar
-
+
Final population
Diameter (km)
(a)
d
Y
-ng
Finalpopuistion
Real populaikln with error bars
+ g &. * 6.x x ff\, * zi\.& ‘>,
.‘-‘-‘-
1
I
Iniual PqNlalion
/ IIl$/il
3 I III,/,/ 10’
t to2
d&. -B
4 m*li,,I f0’
Di@neter (kw Cc)
Fig. 5. Results from hydrocode calculations using weak mortar : part (a) again shows the standard case. Parts (b) and (c) illustrate the effect of varying the starting small diameter population
the recent observational work of Binzei and Xu (1993), who showed that 12 members of the Vesta family (see Zappala et al., 1990, 1994) have a basaltic spectral signature essentially identical to that of Vesta, and from the rotationally resolved spectra of Gaffey (1983), which established that Vesta has a uniform basaltic surface except for a limited area, where the data suggest that a mixture of diogenite is added to the basalt. This can be interpreted to be a large crater on Vesta which punched through the basaltic crust and excavated part of the olivinerich upper mantle, mixing this mantle material with the basaltic surface. Lightcurve photometry, speckle interferometry and polarimetry data (see Cethno et al., 1987, 1989) also support this interpretation. One can construct a plausible scenario for the origin of the eucrites, a type of basaltic achondrite meteorites with spectral features closely resembling those of Vesta and its smaller family members. The eucrites have radiometric formation ages of around 4.5-4.6 Byr ago (Drake, 1979), show little evidence of shock and have cosmic ray exposure ages in the range from 1 to 70 Myr. Making the reasonable assumption that the eucrites came from Vesta,
one that is consistent with current meteorite delivery models (Farinella et al., 1993), then the basaltic surface was formed very early in solar system history, and collisions have not mixed the crust and upper mantle sufficiently to alter the basaltic signature, except for the large cratering event noted by Gaffey. Exactly when this cratering event took place, we cannot say, since there was no shock resetting of the age of the meteorites. However, the cosmic ray exposure ages place a probable lower bound of 70 Myr on the age of the impact (unless all the specimens that we have were pre-existing on the surface of Vesta, a very unlikely possibility). This is a lower bound because, almost certainly, the meteorites that we have are multigenerational fragments from the cratering impact. Another indicator is the likely age of the Vesta family, as estimated from the models of its post-formation collisional evolution developed by Marzari et ai. (1994). They find it unlikely that the family is older than s lByr, since its largest memers are only 5-10 km in diameter, and only a small fraction of them would survive impact breakup longer than this time (unless their impact strength is unreasonably high, as judged by current scaling laws).
D. R. Davis et al. : Asteroid collisional evolution
607
Oimnmef (km) Fig. 7. Results from rerunning case 4(b) but withy,, = 0.02, a value more consistent with hydrocode calculations than thef,, of 0.10 that was our nominal value
(4
#
FMpopubt&il
.‘-.-‘-
1
klilialpopllaaon
10'
10*
10'
-(km)
(4 Fig. 6. Results using energy scaling to simulate strain-rate scaling (a), and hydrocodebasalt (b) and hydrocode-weak mortar results (c)
Also, if the three near-Earth V-type asteroids studied by Cruikshank et al. (1991) are fragments from the same event which generated the crater and the family, the limited lifetime of such bodies against impact on Earth
(x lo8 yr, see Milani et al., 1990) also suggests that Vesta’s giant crater is relatively young. The volume of the largest possible crater existing on Vesta’s surface-some 250 km in diameter, based on observations that the largest sustainable crater is about 50% of the target body size--can be estimated to be some 2 or 3% of the volume of Vesta. Using the cratering gravity-scaling equations of Schmidt and Housen (1987), one finds that a projectile about 35 km in diameter hitting Vesta at 5.3 km s-l, the mean impact speed onto Vesta, is needed to form this crater. With the current population estimated as described in section 3, there are x 1700 asteroids in the 20-35 km diameter range, so we can calculateusing the methodology described in Farinella and Davis (1992)-that one impact from projectiles in this size range occurs on average every 3 x 10’ years. Taking into account that the projectile population in this size range is uncertain by plus or minus a factor of 2, it is not unreasonable to assume that the last real impact took place some 500-1000 Myr ago, which would be consistent with the constraints summarized above. Using our model runs, we have used the Vesta surface constraint in a conservative fashion to further test which of the scaling relationships described above are consistent with the astronomical evidence regarding the collisional history of asteroids. We require Vesta not be shattered over solar system history, which means that the largest impactor onto Vesta fractured less than 50% of its mass, a much larger fraction than the few percent that we estimate have been fractured by the observed crater. Table 2 summarizes the effect of collisions on the two asteroids residing in the bin centered at 500 km diameter, namely Vesta and Pallas, for each of the 20 cases that we ran. In column 2, we show the number N of bodies remaining in the bin at the end of the simulations. In five cases, those with < 1 body, one of the bodies was disrupted, while in seven other cases there was a large amount of collisional erosion (those with 1.9 > N > 1.0). The diameter of the minimum projectile needed to shatter a 500-km target body for the average 5.8 km s-’ impact velocity is given in column 3-this should be compared with the 261 km
D. R. Davis et al. : Asteroid collisional evolution
608 Table 2. Summary of shattering impacts onto the Vesta-Pallas bin No. and case name
1 STANDARD 2 STKlO 3 STKlOB 4 STS8 5 SR 6 SRKlOO 7 SRKlOOB13.5 8 SRKlOOB14.5 9 HYDRS3 10 HYDRSS 11 HYDRSIB 12 HYDRS8B13.5 13 HYDRS8B14.5 14 HYDR2 15 HYDR2B13.5 16 HYDR2B14.5 17 STS0.2 18 STS0.07 19 STS0.004 20 HYDRS8F02
Final no.
Diam smlst
Diam lgst
Shat eng of lgst
Total no. of shat imp
1.95 1.99 1.50 1.99 0.98 1.99 1.99 1.99 0.99 1.63 1.36 1.97 0.97 1.88 1.73 0.96 1.65 1.49 0.99 1.99
17 159 159 82 35 161 161 161 36 50 50 50 50 44 44 44 74 73 73 50
19 99 157 99 198 157 99 125 250 125 157 157 79 79 125 62 158 125 198 79
1.3 0.2 1.2 0.9 219 0.7 0.1 0.45 348 24 41 2.5 81 7.0 9.5 3.3 9.0 12.0 36.0 2.1
1.0 0 1.0 0 2.0 0.5 0 0 1.6 1.5 0.5 0.5 2.6 5.3 2.0 0.5 0.5
6.6 10.0
2.0
4.3
Cum shat energy
2.3 0.17 0.99 1.05 &8 0.18 0.42 21 39 3.1 11 24
Small D slope
-3.8 -3.5 -3.4 -3.6 -3.2 -3.2 -3.4 -3.0 -2.6 -2.8 -2.8 -2.8 -2.8 -2.7 -2.6 -2.4 -3.1 -2.8 -2.6 -2.1
The columns give the case name, the number of bodies at the end of the simulation in the bin, the diameter of the smallest projectile that can just shatter the target body at an impact speed of 5.8 km s-r (the average impact speed among mainbelt asteroids), the diameter of the largest projectile that hit either target body in the simulation, the total collisional energy delivered by the largest impactor normalized by the energy needed to barely shatter the target body, the cumulative number of impactors capable of shattering a target body, the total collisional energy deposited by the shattering impactors and the slope (incremental diameter) of the small diameter end of the population at the end of the simulation. The case names describe the scaling used (ST = standard energy scaling, SR = strain-rate, HYDR = hydrocode), KlO means that the self-compression coefficient k was set to 10, etc., S gives the value of S,, in units of lo7 erg cm -3, i.e. S3 means that 3 x lo7 was used. The B at the end means that a different random number seed was chosen and F02 indicates thatf,, was reset to 0.02. B13.5 and B14.5 indicate changes in the starting size distribution of small asteroids. A projectile 261 km in diameter is needed to disperse 50% of the target body’s mass at an impact speed of 5.8 km s-‘. Because of random variation in the impact speed, a smaller projectile than that listed can shatter or disperse a given sized target body.
diameter required to disperse half the target mass against self-gravity ; note, however, that because of random variations in the impact speed a smaller projectile than that listed can shatter or disperse a given target body. The largest impactor on either Pallas or Vesta ranges from 80 to 200 km in diameter (see column 4), for cases that start with the same initial population. This rather wide range reflects the different possible collisional histories of the asteroid population, and includes a random effect since the number of impacts is calculated using Poisson statistics when small numbers are involved. Although Vesta may have been comparatively “lucky’‘-i.e. the few largest projectiles to involve the 500-km bin may have hit Pallas-we see from this table that only energy scaling and strain-rate scaling predict that Vesta has escaped being shattered over solar system history, i.e. that the specific energy of the largest projectile and the cumulative shattering energy normalized to Q* (column 7) are less than unity. Also, the strong gravitational compressive strengthening term of Housen et al. (1991) is required to preserve the 500-km sized asteroids against shattering. The slope of the small diameter (5-50 km) “tail” of the final asteroid population is given in column 8, and can be compared with estimates of the slope for asteroids in that size range.
5. Discussion and conclusions Our initial results using four scaling algorithms for the variation of Q* with target size that have been developed in recent years show that the Housen and Holsapple algorithm and simple energy scaling provide the best match with the observed asteroid size distribution and the existence of the basaltic crust of Vesta. Scaling algorithms based on the hydrocode of Melosh et al. (1992) fail to reproduce either of the above conditions, even when the energy partitioning coefficientf,, is varied from 10 to 2%. However, these results have been generated assuming that JKE is constant with changing target size. This may not be the case, and work is currently under way using the hydrocode to calculate this quantity over a wide range of impact conditions. We do find that the degree of strengthening of Q* due to gravitational self-compression that is indicated by the experiments of Housen et al. (1991) is essential to reproducing the properties of the asteroids that were examined here (Ryan, 1992). The equivalence of these constantpressure experiments to disruption under gravitational self-compression has been verified by hydrocode cal-
culations,
such that the largest fragment
masses are
D. R. Davis et al. : Asteroid collisional evolution
roughly the same for each case. This removes any lingering suspicions that theorists might have had, and must be regarded as the best estimate of how Q* varies in the gravity regime. The inability of the current hydrocode algorithm to reproduce major features of the asteroids may result from our incomplete understanding of how asteroids respond to large impacts. For example, we assume that if a large impact fractures much of the volume of a body, the fragments are given a large enough velocity so that the material which does not achieve escape speed becomes thoroughly mixed. Thus a large crater on Vesta which fractures much of its mantle would excavate a significant amount of diogenite from beneath the basaltic crust and distribute this mantle material over the surface. But if the fragment velocities are low as suggested by Asphaug and Melosh (1993 ; some 100 m s-’ when scaled from Phobos’ to Vesta’s size), i.e. a very lowf,, value applies in terms of our model parameters, then the degree of mixing may be much less and a body such as Vesta could withstand large impacts without showing a significant signature of underlying material. However, this is hardly consistent with the ejection of 5-10 km-sized fragments at speeds of several hundred m s-’ to form the observed Vesta family. Exactly what the dynamics of ejecta fragments is in such an impact remains to be worked out, and this appears essential to reconcile the hydrocode results with asteroid observations. The divergence of the small diameter slope between the observed asteroids and the hydrocode results could also be reduced if& were shown to scale with target size. In this case, varying degrees of fragment dispersal/ reaccumulation might cause the slope of the small diameter end of the calculated distribution to be different. Exactly how much different awaits results from hydrocode calculations onf& and its scaling We fiud that the quest for a scaling algorithm that will explain the observed properties of asteroids is not yet ended and that further work is required to understand the implications of these algorithms for the collisional evolution of asteroids. This paper serves to point out areas of difficulty that need further work, and to demonstrate how experimental validation of all aspects of the scaling algorithms is essential for understanding the observed asteroid belt. Acknowledgments. P. F. has worked at this project while staying at the Observatoire de la C&e d’Azur (Nice, France) thanks to the “G. Colombo” fe~ows~p of the European Space Agency. This project has been partially supported by NATO collaborative research grant 0137/89. This is PSI. Contribution No. 308. Planetary Science Institute is a non-profit division of Science Applications International Corporation. We thank E. Asphaug, K. Housen and P. Paolicchi for helpful discussions and comments.
References Asphaug, E. and Melosb, H. J., The Stickney impact on Phobos : A dynamical model. Icarus 101,144164,1993. Binzel, R. P. and Xu, S., Chips off 4 Vesta: Evidence for the
parent body of basaltic achondrite meteorites. Science 268, 186-191, f993. Cell&, A., ZappalP, V., Di Martino, M., Farm&a, P. and Pat+ lie&i, P., Flattening, pole and albedo features of 4 Vesta from photometric data. Icarus 76,546-5651987. CeBiuo, A., Di Martino, M., Drummond, J., Farin&, P., Paolicchi, P. and Zappal& V., Vesta’s shape, density and albedo features. As&on. Astrophys. 219,32O-321, 1989. Cellino, A., ZappalP, V. and Farm&a, P., The asteroid size distribution from IRAS data. Mon. Not. R. a.vtr. Sot. 253, 561-574,199l. Chapman, C. R., Implications of the inferred compositions of asteroids for their collisional evolution. Mem. S.A.It. 57, 103-122,1986. Cr~~ha~, D. P., Tholen, D. J., Hartmann, W. K,, Bell, J. F. and Brown, R. H., Three basaltic. ~-approaching asteroids and the source of basaltic meteorites. Icarus 89, 1-13, 1991. Davis, D. R., Chapman, C. R., Greenberg, R., Weideuachilllng, S. J. and Harris, A. W. Collisional evolution of asteroids: populations, rotations and velocities. In Asteroids (edited by T. Gehrels), pp. 528-557. Univ. of Arizona Press, Tucson, 1979. Davis, D. R., Chapman, C. R., WeidenschiBing, S. J. and Greenberg, R., Collisional history of asteroids: Evidence from Vesta and the Hirayama families. Zcurw 62, 30-53, 1985. Davis, D. R., Weide~~ng, S. J., FarineRa, P., Paolieehi, P. and Binzel, R. P., Asteroid ~ol~sional history : effects on sizes and spins. In Asteroids II (edited by R. P. Binzel, T. Gehrels and M. S. Matthews) pp. 805826. Univ. of Arizona Press, Tucson, 1989. Dobnanyi, J. W., Collisional model of asteroids and their debris. J. geophys. Res. 74,2531-2554, 1969. Drake, M. J., Geochemical evolution of the eucrite parent body : Possible nature and evolution of asteroid 4 Vesta. In Asteroi& (edited by T. Gehrels), pp. 765-782. Univ. of Arizona Press, Tucson, 1979. Farlnella, P. and Davis, D. R., Collision probabilities and impact velocities in the main asteroid belt. Icarus 97, 111-123, 1992. Farinella, P, and Davis, D. R., Will the real asteroid size distribution please step forward. Lunar Planet Sci. Co& XXV, 1994,365366,1994. Farmella, P., PaoRechi, P. and Zappala, V., The asteroids as outcomes of catastrophic collisions. Icarus 52,409-433,1982. Farmella, P., Davis, D. R., Paolicchi, P., CeRino, A. and Zappal&, V., Asteroid collisional evolution : An integrated model for the evolution of asteroid rotation rates. Astron. Astrophys. 253,604614, 1992. Farlnella, P., Gonczi, R., FroeschR, Ch. and FroeschlC, C., The injection of asteroid fragments into resonances. Icarus 101, 174187,1993. Fujiwara, A., On the mechanism of catastrophic destruction of minor planets by high-velocity impact. Icarus 41, 356-364, 1980. Fujiwara, A., Cerroni, P., Davis, D., Di Mar&o, M., Holsapple, K., Heusen, K. and Ryan, E., Experiments and scaling laws for catastrophic collisions. Asteroids ZZ,240-265, 1989. Gaffey, M. J., The asteroid (4) Vesta : Rotational spectral variations, surface material heterogeneity, and implications for the origin of the basaltic achondrites. Lunar Planet. Sci. XIV, 23 l-232, 1983. Gradle, J. C., Chapman, C. R. and Tedesco, E. F. Distribution of taxonomic classes and the compositional structure of the asteroid belt. In Asteroids ZZ (edited by R. P. Binzel, T. Gehrels and M. S. Matthews), pp. 316-335, Univ. ofArizona Press, Tucson, 1989. Hoisapple, K. A., The scaling of impact processes in planetary sciences, Ann. Rev. Earth Planet. Sci. 21, 333-373, 1993. Heusen, K. and Holsappie, K., On the fra~en~tion of asteroids and planetary satellites. Icarus 84,226-253, 1990.
610 Housen, K. R., Schmidt, R. M. and Holsapple, K. A., Laboratory simulations of large scale fragmentation events. Icarus 94, 180-190, 1991. Marzari, F., Davis, D. R., Vanzaui, V., Cellino, A., Zappala, V. and Farinella, P., Origin and evolution of the Vesta family. 1994, in preparation. Melosh, H. J., Ryan, E. V. and Asphaug, E., Dynamic fragmentation in impacts : Hydrocode simulation of laboratory impacts. J. geophys. Res. 97, 14735-14759, 1992. Milani, A., Carpino, M. and Marzari, F., Statistics of close approaches between asteroids and planets: Project SPACEGUARD. Icarus 88,292-335, 1990. Nakamura, A. and Fujiwara, A., Velocity distribution of fragments formed in a simulated collisional disruption. Icarus 92, 132-146, 1991. Ryan, E. V., Catastrophic collisions : Laboratory impact experiments, hydrocode simulations, and the scaling problem. Ph.D. thesis, Univ. of Arizona, 1992. Ryan, E. V., Asteroid collisions : Target size effects and resultant velocity distributions. Lunar Planet. Sci. Conf. XXIV, 12271228,1993.
D. R. Davis et al. : Asteroid collisional evolution Ryan, E. V. and Melosh, H. J., Hydrocode simulations of largescale impacts : A size dependence for fragmentation. 1994, in preparation. Schmidt, R. M. and Housen, K. R., Some recent advances in the scaling of impact and explosion cratering. ht. J. Impact Eng. 5,543-560,1987. Van Houten, C. J., Van Houten-Groeneveld, I., Herget, P. and Gehrels, T., The Palomar-Leiden survey of faint minor planets. Astron. Astrophys. Suppl. 2, 339-448, 1970. Veeder, G. J., Tedesco, E. F. and Matson, D. L., Asteroid results from the IRAS survey. In Asteroids ZZ(edited by R. P. Binzel, T. Gehrels and M. S. Matthews), pp. 282-289. Univ. of Arizona Press, Tucson, 1989. Zappala, V., Cellino, A., Farinella, P. and KneZeviC,Z., Asteroid families. I. Identification by hierarchical clustering and reliability assessment. Astron. J. 100, 203&2046, 1990. Zappala, V., Cellino, A., Farinella, P. and Milani, A., Asteroid families. II. Extension to unnumbered multi-opposition asteroids. Astron. J., 772-801, 1994.
Vol. 42, No. 8, pp. 599-610, 1994 Copyright 0 1994Elsevier Science Ltd Printed in Great Britain. All rights reserved 0032-0633/94 $7.00+0.00
Pergamon
0032-0633(94)EOO61-T
Asteroid collisional evolution : results from current scaling algorithms Donald R. Davis,’ Eileen V. Ryan’ and Paolo Farinella’ ’ Planetary Science Institute, 620 North 6th Avenue, Tucson, AZ 85705-8331,U.S.A. ’ Dipartimento di Matematica, Universita di Pisa, Via Buonarroti 2, 56127 Pisa, Italy, and Observatoire de la C&e d’Azur, Dept. Cassini, B.P. 229,06304 Nice Cedex 4, France
Received 22 September 1993; revised 25 February 1994; accepted 25 February 1994
1. Introduction
Correspondence
to
: D. R. Davis
Unraveling the complex collisional history of the asteroids requires understanding how large-scale collisional events have acted to produce many of the observed characteristics of these objects. The sizes and spins of most asteroids are the result of over 4 Byr of collisional history (Davis et al., 1979, 1989). The formation of Vesta and its family, unique among the asteroids, needed a large-scale thermal alteration of Vesta, followed by just the right amount of collisions: one or a few large cratering impact(s) that ejected fragments up to 10 km in size from the basaltic crust to form the family (Binzel and Xu, 1993), but not impacts so large and frequent as to fracture the crust and thoroughly mix it with the underlying olivinerich mantle material (Davis et al., 1985). Other asteroids, though, such as 16 Psyche, have apparently experienced much more energetic collisions than did Vesta. Psyche is believed to be the metal-rich core of a differentiated Vestalike parent body that had its crust and mantle stripped away by such large impacts (Chapman, 1986). Recent development of a numerical hydrocode includ-
600 ing finite strength effects (Melosh et al., 1992) and the application of this code to problems of the collisional fra~entation of asteroids has provided a new tool to estimate the specific energy needed to break up asteroids of different sizes. This work predicts a more rapid weakening of asteroids with size in the strength regime than did the earlier scaling theory of Holsapple and Housen (1990), which was based on dimensional analysis but employed many of the same assumptions as did the hydrocode algorithm. The hydrocode scaling result does not, however, reflect both shattering and dispersal of fragments from the target body, only shattering. Specifically, the hydrocode predicts that after the body is shattered, most of the fragments do not achieve escape velocities (Ryan, 1993), due to low fraction fKEof the projectile’s kinetic energy being partitioned into fragment kinetic energy. This is equivalent to say that the targets are much more resistant to disruption (with fragment dispersal) than to shattering, as indicated by the value of the threshold specific energy Q*. More work to compute the scaling of fKEby hydrocode simulations of largescale impact break-up events is in progress (Ryan and Melosh, 1994), and results quoted here using the Q* scaling relationship are only preliminary. One important test for a scaling law for collisional disruption is whether it produces results in collisional modeling programs that are consistent with observations about the real asteroid population. For example, do plausible collisional scenarios lead to the preservation of Vesta’s basaltic crust over the age of the solar system? Does the size distribution match that of the present asteroid belt? In order to test models generated using different scaling laws, we have incorporated them into our program for studying asteroid collisional evolution. Cases were then run using the same initial population of asteroids and collisional parameters, varying only the choice of the scaling law between the runs. We have explored different starting populations and choices of the critical collisional parameters, including some results derived from the hydrocode calculations. In the remainder of this paper, we will summarize these results and discuss implications of the different scaling laws for the collisional history of asteroids, based on our simulations of asteroid collisional evolution.
2. Scaling laws for the fragmentation threshold An understanding of how collisions have acted to produce the observed characteristics of asteroids depends critically upon having a good scaling theory by which to extend laboratory impact experimental results, involving targets typically 10 cm in size, to large scale asteroidal collisions, involving bodies that can be several hundred km in size, For asteroids Iarger than several tens of km in size, gravity plays a dominant role in dete~ining the specific energy (Q*) needed to fracture a body. For small asteroids and laboratory experiments, it is the material strength of the target that is important for determining Q*; this is the strength regime. For large asteroids, though, there is a
D. R. Davis et al. : Asteroid collisional evolution
large pressure due to gravitational self-compression that acts throughout the volume of the body (Davis et al., 1985). In the gravity regime, it is this pressure that largely determines the specific energy required to shatter a body. The transition region occurs where the energy density needed to rupture the material bonds is comparable to that needed to overcome the gravitational compression. The size regime in which this transition occurs depends on the material from which the body is made. For weak bodies, the transition occurs at smaller sizes than it does for bodies made of strong material. Early studies of asteroidal collisions assumed a simple energy scaling in the strength regime, i.e. the specific energy required to fracture the material bonds of the body was thought to be independent of the size of the body. Fujiwara (1980) and Farinella et al. (1982) first provided independent physical arguments suggesting that Q* may depend on target size-actually, be proportional to R-'12. The first rigorous effort to develop a scaling theory was the dimensional analysis model by Housen and Holsapple (1990), which incorporated a strain-rate dependence on the specific energy needed for shattering a body. For a fixed impact speed, the strain-rate model with “nominal” values of the material parameters gave a Q* that varies as R-o.24 in the strength regime. As we recalled above, Davis et al. (1985) showed that gravitational self-compression of the target’s interior gives rise to a gravity regime, where Q* is mainly determined by the self-compressional effect, in addition to the strength regime where only the material properties are relevant. In the gravity regime, Q* becomes proportional to R2 with the energy scaling and to R'.65 with the Housen and Holsapple scaling, reflecting the dependence on both projectile energy and velocity. Recently, Melosh et al. (1992) have incorporated the Grady-Kipp fragmentation model, including both a size and a strain-rate dependence, into a 2-D hydrocode program and used this code to study collisional fragmentation. Then Ryan and Melosh (1994) have used this tool to calculate Q* as a function of target size for different types of materials. Generally, the hydrocode calculates that large bodies are significantly weaker (Q* is much smaller) than does strain-rate theory for the same impact event. Figure 1 compares results for these different algorithms. Which of these, then, shall we adopt as the preferred one for carrying out studies of asteroid collisional evolution? To provide some insight as to how these different scaling algorithms affect collisional evolution scenarios, we undertook to compare the outcome of identical cases of collisional evolution, changing the scaling laws. We consider four of the scaling relationships shown in Fig. I : (1) simple energy scaling with self-compression added; (2) strain-rate scaling ; (3) hydrocode scaling for basalt and (4) hydrocode scaling for weak mortar. For the hydrocode cases, basalt and weak mortar are chosen to represent a broad range of possible asteroid material properties. These four cases span the range of currently proposed scaling relationships. In addition to the results in the strength regime, there is the question of how to include gravitational compression effects. In our collisional code, the effectiveness of self-compression is specified by a
D. R. Davis et al.: Asteroid collisional evolution
0
WEAK CEMENT
_ V = 5.5
loo
601
MORTAR
km/a
10’
102
103 TARGET
10’
IO5
DIAMETER
lo6
IO’
1oa
(cm)
Fig. 1. Critical specific energy Q* as a function of target size, as calculated from four scaling relations : (1) simple energy scaling with self-compressional term added (solid line) ; (2) strain rate scaling (dashed line) ; (3) hydrocode computations for basalt (filled circles) ; and (4) for weak mortar (open squares). Impact velocity is 5.8 km s-l, and is assumed to be constant over the range of target sizes. Full triangles represent experimental results by Housen et al. (1991), with the corresponding largest fragment to target mass ratios
model parameter, k (see Davis et al., 1985, 1989). Preliminary but sketchy data indicated that a value k = 1 was adequate to represent the limited experimental information then available for the Davis et al. work. More recently, an elegant set of experiments by Housen et al. (1991) measured the explosive energy needed to disrupt bodies at different ambient pressures. These experiments suggested that much higher values of k are needed in our model ; in fact k values between 10 and 100 are required to make the model results match the experimental results (more exactly, the implied value of k is of the order of 10 if one assumes the energy scaling, of 100 in the strain-rate case). Specifically, we used the following formulae, whose label numbers correspond to those given above for the different scaling laws. Energy scaling : dimensionless
s=so+
4rckGp2R2 15 .
(1)
Strain rate scaling : 1+2.14x s = so(&~o~z4[
W1k($---~~*9].
(2) Hydrocode-basalt
:
S= So(&)u~43[1+1.07X Hydrocode-weak S=
lo-l7 ($07].
(3)
mortar :
So(&)o~6’[l+2.00X
Here S is the impact strength defined, e.g., in Davis et al. (1989) that is Q* times the density p of the material (always assumed to be 2.5 g crnw3), So is the value of S referring to the typical size of a laboratory target (20 cm diameter). The results of Ryan and Melosh (1993, plotted in Fig. 1) have been scaled from an impact velocity of V = 1.65 km SC’to the average asteroidal value of 5.8 km SC’ (see Farinella and Davis, 1992) assuming that S is proportional to V”.63,as indicated by the hydrocode results for basalt. We have just added the terms referring to the strength and gravity regimes to determine the value of S for any given size body, since the transition range is narrow and over most of the asteroidal size range either of the two terms dominates. Note that the hydrocode results were consistent with S, values of 8.22 x lo7 and 3.62 x lo7 erg cm -3 for the basalt and weak mortar cases, respectively ; however, in our test runs we have kept So as a free parameter even when the hydrocode scaling laws were actually used. As for the k parameter defined above, we stress that its value is tied to experiments. All scaling theory and hydrocode calculations have parameters of this type and their values are ultimately derived from experiments. The physics of fragmentation and the physical state of the target and projectile are sufficiently complex that we cannot deduce the outcome of such collisions from first principles. Nor is it likely that we will be able to do so for a long time.
3. Collisional evolution of the size distribution IO-17
(&r2y.
(4)
In order to simulate the collisional evolution of the asteroid size distribution we used the code described in detail
602 in Davis et al. (1989) and Farinella et al. (1992), which evolves in time the numbers of objects residing in 23 discrete logarithmic size bins, whose central values span the range 6.2-1000 km in diameter in such a way that there is always a factor 2 in mass (1.26 in size) between two neighboring bins. We refer to the two papers quoted above also for a rigorous definition and discussion of the input parameters of the code, most of which describe the response of asteroids to cratering and shattering impacts. We carried out a series of numerical experiments using the four different scaling algorithms described in section 2. The parameters of our model which were used for our standard case are given in Table 1. While our principal interest is in the effects of scaling laws, we did change the model parameters in a few cases. The labels appearing in the figures specify which parameter values have been changed, besides the scaling law adopted in each case. Note that our starting populations were built by calculating the number of bodies present in each of the 23 diameter bins using two power law distributions with a transition diameter of 100 km. We assumed that a total of 400 bodies exceeded this size and were distributed according to N( > L>)KXD-~Z+’ ’ , with an incremental population index bz = 4.5). The four largest size bins, containing Ceres, Pallas and Vesta, however, were assumed to start with 1, 0,O and 2 bodies, respectively. For D < 100 km, we used a standard population N( >D)GcD~~J+‘, with b, = 4. This initial population is close to those used in previous work (Davis et al., 1989 ; Farinella et af., 1992) and derived from the requirements that: (i) the initial total mass of the asteroid belt is not much larger than the current one ; (ii) larger and larger asteroids have been less and less depleted by impact disruption. Given these constraints, as we shall see, the final population is not very sensitive to the detailed shape of the initial size distribution. We based the size distribution for the current belt, to be compared with the output of the code after 4.5 x lo9 yr of evolution, on IRAS results augmented by groundbased observations when IRAS data were lacking (see Cellino Table 1. Parameters used for the numerical runs Total evolution time : 4.5 x IO9yr Number of size bins : 23 Central diameter of first bin : 6.2007 km Size ratio between two adjacent bins : 1.25992 Maximum fractional change allowed in a time step : 0.1 Average impact velocity : 5.8 km SK’ Sigma of impact velocity distribution : 1.9 km SK’ Nominal value for impact strength : 3 x lo7 erg cmm3 Nominal value for self-compressional parameter k : 1.0 Energy partitioning coefficient_&, (both craters and break-up) : 0.1 Exponent of ejecta mass vs velocity power law (both craters and break-up) : - 2.25 Crater excavation coefficient : 1.Ox lo-’ g erg-’ Material density : 2.5 g cm-’ Initial population transition diameter : 100 km Initial population N(D > DfransltioJ: 400 Nominal power law exponent I) < Dtransition : 4.0 Nominal power law exponent D > Dlramition : 4.5 Populations of four largest bins : 2, 0, 0, 1
D. R. Davis et al. : Asteroid collisional evolution
et al., 1991). We used the size distribution derived by Cellino et al. for diameters larger than 44 km. As shown in that paper, this size distribution is essentially identical with that derived by Chapman (as reported in Gradie et aE., 1989) for the entire asteroid population. At smalter sizes our knowledge of the asteroid population is incomplete, i.e. we do not have albedo measurements or diameters for all asteroids smaller than 44 km. So, we adopted two techniques to estimate the small-size asteroid distribution : first, we used the results of the Palomar--Leiden Survey (PLS, Van Houten et al., 1970) for the slope of the asteroid population and anchored this power law distribution at D = 44 km. The second approach consisted of extrapolating the Cellino et al. power law dist~bution for each of their five zones of the main belt and then summing up the results. The PLS slope used was -2.95 (incremental diameter distributions are used hereinafter) which was the fit by Van Houten et al. to their absolute magnitude distribution. The conversion to diameters assumed a constant albedo across the size range. We chose this value rather than the commonly quoted slope of - 3.5 found by Dohnanyi (1969), in agreement with his analytical theory for the equilibrium size distribution of a collisionally evolving population. The extrapolation based on the Cellino et al. results used their “corrected random model” to find albedos for asteroids that have no measured values. This model assumes that all asteroids in a semimajor axis zone have the same distribution of albedos, so the IRAS measured albedo distribution, corrected approximately for the flux overestimation problem (Veeder et al., 1989), was randomly sampled to find the albedos for small asteroids. (The assumptions that one makes regarding the albedo distribution can significantly affect the diameter distribution derived from a measured magnitude distribution, as shown by Cellino et al.). The two models that we assumed covered the plausible range of albedo distributions, ranging from the constant albedo model used for the PLS data to the random model for the Cellino et al. extrapolation. The results from these two approaches yield different estimates for the small size population of asteroids. We adopted the geometric mean of these two estimates as the population for sizes down to 5.5 km, the lower size bound in our numerical simulation. We estimate the uncertainty in the population at different sizes as follows : for bodies larger than 250 km diameter, the uncertainty is fixed at rtO.5, since the sizes of asteroids in that diameter range are well determined. For bodies between 50 and 250 km diameter, the uncertainty is just the statistical uncertainty of +JN (N being the total number of bodies residing in the bin), while for diameters smaller than 25 km, we used an uncertainty of -&log 2 in log N. This uncertainty is only an estimate based on the range of the population of small asteroids calculated by different workers (see Farinella and Davis, 1994). A smooth fit between the JN regime and the smaller sizes is constructed between the 25 and 50 km sizes. Our adopted size distribution and its uncertainty, derived from the above procedures, are shown in Figs 27. The results of the collisional evolution code using energy scaling are shown in Fig. 2 : part (a) gives what we
D. R. Davis et al. : Asteroid collisional evolution
ld
I
k=lO
‘”
f
a
10'
ii!
= 10~ It
4 2
10'
*
Fig. 2. Results using energy scaling : Part (a) is our so-calIed standard case derived from earlier work and uses the widely quoted laboratory strength of basalt and a small initial mass pop~ation. In part (b), we increase the impact strength to 8 x IO’erg cme3. In part (c), we have increased the parameter k, which describes how &ectiveIy self-compression increases Q*, to match the laboratory results of Housen et al. (1991). This case gives our best fit to the overall asteroid size distribution. Part (d) repeats the case of part (c), but with a different random number sequence term a standard case and uses the nominal parameters that we have employed in earlier studies of asteroid collisional evolution (see Table 1). This case, though, shows significant depletion relative to the observed population at sizes smaller than 40 km diameter. However, increasing the laboratory impact strength to the value reported by Nakamura and Fujiwara (199 l), larger by nearly a factor of 3 than earlier reported values, produces much better agreement between the model results and the observed population, with only a few bins disagreeing (Fig. 2(b)). increasing the strength of the gravitational self-compression term, to match the laboratory experiments of Housen et al. (1991), i.e. setting k = 10, produces the best agreement between our model results and the observed asteroid population (Fig. 2(c) and (d)). We next changed to strain-rate scaling and reran the cases of Fig. 2(a) and 2(c) ; these results are shown in Fig. 3. For the 2(c) case, we had to use k = 100 in order to have the calculated Q* in the gravity regime match the Housen et al. ex~~ments. As seen from Fig. 3(a) and
(b), the calculated size distribution does not match the observed one, although the results using the larger selfcompression effect (Fig. 3(b)) are markedly superior. In order to see if changing the starting population could produce an improved match to the observed size distribution, we reran the case of Fig. 3(b) with both a shallower initial small size population (Fig. 3(c)) and a steeper one (3(d)). The final results were relatively unaffected by these changes in the starting population. The scaling algorithms produced by fitting many runs of the hydrocode were next incorporated and the above described cases were rerun using these scalings ; results are shown in Fig. 4 for the scaling using basalt. In general, this algorithm does not give a particularly good match to the observed population. In all cases, the small size population is depleted relative to that observed. This is due to the weak impact strength of bodies smaller than about 10-20 km in size. This scaling algorithm predicts a small size slope that is very shallow relative to the observed estimated slope in this size range among the asteroids.
D. R. Davis et al. : Asteroid collisional evolution
Diemeter Cc)
Diameter(km) (d)
scaling : Part (a) is the standard case described above, changing only the scaling law. Here the population distribution does not match the observed one and Vesta is very likely to have been shattered due to its low strength. Part (b) gives results for strain-rate scaling with k = 100, the value needed to match the experimental results of Housen et al. (1991). In Fig. 3(c) and (d), we vary the small diameter starting population Fig. 3. Results using strain-rate
Changing the hydrocode algorithm to that based on weak mortar accentuated the trend observed for basalt material : small asteroids are strongly depleted and have
too shallow a slope to match the observed population (Fig. 5). This result is obviously a consequence of the steep fall off in strength with increasing size, cf. Fig. 1 and equation (4). It is interesting to note that the various scaling algorithms-strain-rate, hydrocode-basalt and hydrocodeweak mortar-can be simulated quite well by using energy scaling with the appropriate choice of S, to calculate the size distribution of asteroids. This is because Q* is approximately constant in the size range from the smallest observable asteroids up to the size where self-compression becomes important, typically between about 10 and 100 km in diameter. We illustrate this approximation in Fig. 6, which uses energy scaling with .I$ of 2 x 106, 7 x 10’ and 4 x IO4erg cm-3 in order to simulate runs with strain-rate scaling, hydrocode-basalt and hydrocode-weak mortar, respectively. Comparing these results with those shown
in Figs 3(a), 4(a) and 5(a), respectively, indicates the similarity between these cases. Thus as far as the evidence from the asteroidal size distribution is concerned, energy scaling is indistinguishable from the various scaling algorithms that have been put forth in recent years, provided that the appropriate value of S,, is selected to represent each scaling law. Laboratory impact experiment data (Fujiwara et al., 1989) as well as hydrocode calculations to date (Ryan, 1993) suggest thatf,, is considerably lower than the 10% value we adopted, based on the requirement to reproduce the observed properties of asteroid families (see discussion in Davis et al., 1989). So we reran the case for hydrocodebasalt scaling usingf,, = 0.02. This case (Fig. 7) still does not match the real asteroid population: asteroids in the 20-60 km size range are overabundant relative to the real population, because they are resistant to disruption due to the lowf,,. Also, the small diameter population has a shallower slope than does the observed population. The same features are found with the hydrocode-weak mortar scaling.
605
Material:basalt, b, = 4.
Fig. 4. Results from hydrocode calculations using basalt : part (a) shows the standard case with this scaling. Parts (b) and (c) illustrate the standard case but using S, = 8 x 10’ erg cmW3for two different random number sequences. In Fig. 4(d) and (e) we show the effect on the final population of varying the initial small diameter population
4. Constraints from V&a
There are other constraints on the collisional history of asteroids besides the size distribution. The fact that Vesta
has a basaltic crust which has survived, except for one or two large cratering events, since the formation of the solid surface provides a powerful constraint on the collisional history of this body. The relevant information comes from
D. R. Davis et al. : Asteroid collisional evolution
Materiel: weak mcilar
-
+
Final population
Diameter (km)
(a)
d
Y
-ng
Finalpopuistion
Real populaikln with error bars
+ g &. * 6.x x ff\, * zi\.& ‘>,
.‘-‘-‘-
1
I
Iniual PqNlalion
/ IIl$/il
3 I III,/,/ 10’
t to2
d&. -B
4 m*li,,I f0’
Di@neter (kw Cc)
Fig. 5. Results from hydrocode calculations using weak mortar : part (a) again shows the standard case. Parts (b) and (c) illustrate the effect of varying the starting small diameter population
the recent observational work of Binzei and Xu (1993), who showed that 12 members of the Vesta family (see Zappala et al., 1990, 1994) have a basaltic spectral signature essentially identical to that of Vesta, and from the rotationally resolved spectra of Gaffey (1983), which established that Vesta has a uniform basaltic surface except for a limited area, where the data suggest that a mixture of diogenite is added to the basalt. This can be interpreted to be a large crater on Vesta which punched through the basaltic crust and excavated part of the olivinerich upper mantle, mixing this mantle material with the basaltic surface. Lightcurve photometry, speckle interferometry and polarimetry data (see Cethno et al., 1987, 1989) also support this interpretation. One can construct a plausible scenario for the origin of the eucrites, a type of basaltic achondrite meteorites with spectral features closely resembling those of Vesta and its smaller family members. The eucrites have radiometric formation ages of around 4.5-4.6 Byr ago (Drake, 1979), show little evidence of shock and have cosmic ray exposure ages in the range from 1 to 70 Myr. Making the reasonable assumption that the eucrites came from Vesta,
one that is consistent with current meteorite delivery models (Farinella et al., 1993), then the basaltic surface was formed very early in solar system history, and collisions have not mixed the crust and upper mantle sufficiently to alter the basaltic signature, except for the large cratering event noted by Gaffey. Exactly when this cratering event took place, we cannot say, since there was no shock resetting of the age of the meteorites. However, the cosmic ray exposure ages place a probable lower bound of 70 Myr on the age of the impact (unless all the specimens that we have were pre-existing on the surface of Vesta, a very unlikely possibility). This is a lower bound because, almost certainly, the meteorites that we have are multigenerational fragments from the cratering impact. Another indicator is the likely age of the Vesta family, as estimated from the models of its post-formation collisional evolution developed by Marzari et ai. (1994). They find it unlikely that the family is older than s lByr, since its largest memers are only 5-10 km in diameter, and only a small fraction of them would survive impact breakup longer than this time (unless their impact strength is unreasonably high, as judged by current scaling laws).
D. R. Davis et al. : Asteroid collisional evolution
607
Oimnmef (km) Fig. 7. Results from rerunning case 4(b) but withy,, = 0.02, a value more consistent with hydrocode calculations than thef,, of 0.10 that was our nominal value
(4
#
FMpopubt&il
.‘-.-‘-
1
klilialpopllaaon
10'
10*
10'
-(km)
(4 Fig. 6. Results using energy scaling to simulate strain-rate scaling (a), and hydrocodebasalt (b) and hydrocode-weak mortar results (c)
Also, if the three near-Earth V-type asteroids studied by Cruikshank et al. (1991) are fragments from the same event which generated the crater and the family, the limited lifetime of such bodies against impact on Earth
(x lo8 yr, see Milani et al., 1990) also suggests that Vesta’s giant crater is relatively young. The volume of the largest possible crater existing on Vesta’s surface-some 250 km in diameter, based on observations that the largest sustainable crater is about 50% of the target body size--can be estimated to be some 2 or 3% of the volume of Vesta. Using the cratering gravity-scaling equations of Schmidt and Housen (1987), one finds that a projectile about 35 km in diameter hitting Vesta at 5.3 km s-l, the mean impact speed onto Vesta, is needed to form this crater. With the current population estimated as described in section 3, there are x 1700 asteroids in the 20-35 km diameter range, so we can calculateusing the methodology described in Farinella and Davis (1992)-that one impact from projectiles in this size range occurs on average every 3 x 10’ years. Taking into account that the projectile population in this size range is uncertain by plus or minus a factor of 2, it is not unreasonable to assume that the last real impact took place some 500-1000 Myr ago, which would be consistent with the constraints summarized above. Using our model runs, we have used the Vesta surface constraint in a conservative fashion to further test which of the scaling relationships described above are consistent with the astronomical evidence regarding the collisional history of asteroids. We require Vesta not be shattered over solar system history, which means that the largest impactor onto Vesta fractured less than 50% of its mass, a much larger fraction than the few percent that we estimate have been fractured by the observed crater. Table 2 summarizes the effect of collisions on the two asteroids residing in the bin centered at 500 km diameter, namely Vesta and Pallas, for each of the 20 cases that we ran. In column 2, we show the number N of bodies remaining in the bin at the end of the simulations. In five cases, those with < 1 body, one of the bodies was disrupted, while in seven other cases there was a large amount of collisional erosion (those with 1.9 > N > 1.0). The diameter of the minimum projectile needed to shatter a 500-km target body for the average 5.8 km s-’ impact velocity is given in column 3-this should be compared with the 261 km
D. R. Davis et al. : Asteroid collisional evolution
608 Table 2. Summary of shattering impacts onto the Vesta-Pallas bin No. and case name
1 STANDARD 2 STKlO 3 STKlOB 4 STS8 5 SR 6 SRKlOO 7 SRKlOOB13.5 8 SRKlOOB14.5 9 HYDRS3 10 HYDRSS 11 HYDRSIB 12 HYDRS8B13.5 13 HYDRS8B14.5 14 HYDR2 15 HYDR2B13.5 16 HYDR2B14.5 17 STS0.2 18 STS0.07 19 STS0.004 20 HYDRS8F02
Final no.
Diam smlst
Diam lgst
Shat eng of lgst
Total no. of shat imp
1.95 1.99 1.50 1.99 0.98 1.99 1.99 1.99 0.99 1.63 1.36 1.97 0.97 1.88 1.73 0.96 1.65 1.49 0.99 1.99
17 159 159 82 35 161 161 161 36 50 50 50 50 44 44 44 74 73 73 50
19 99 157 99 198 157 99 125 250 125 157 157 79 79 125 62 158 125 198 79
1.3 0.2 1.2 0.9 219 0.7 0.1 0.45 348 24 41 2.5 81 7.0 9.5 3.3 9.0 12.0 36.0 2.1
1.0 0 1.0 0 2.0 0.5 0 0 1.6 1.5 0.5 0.5 2.6 5.3 2.0 0.5 0.5
6.6 10.0
2.0
4.3
Cum shat energy
2.3 0.17 0.99 1.05 &8 0.18 0.42 21 39 3.1 11 24
Small D slope
-3.8 -3.5 -3.4 -3.6 -3.2 -3.2 -3.4 -3.0 -2.6 -2.8 -2.8 -2.8 -2.8 -2.7 -2.6 -2.4 -3.1 -2.8 -2.6 -2.1
The columns give the case name, the number of bodies at the end of the simulation in the bin, the diameter of the smallest projectile that can just shatter the target body at an impact speed of 5.8 km s-r (the average impact speed among mainbelt asteroids), the diameter of the largest projectile that hit either target body in the simulation, the total collisional energy delivered by the largest impactor normalized by the energy needed to barely shatter the target body, the cumulative number of impactors capable of shattering a target body, the total collisional energy deposited by the shattering impactors and the slope (incremental diameter) of the small diameter end of the population at the end of the simulation. The case names describe the scaling used (ST = standard energy scaling, SR = strain-rate, HYDR = hydrocode), KlO means that the self-compression coefficient k was set to 10, etc., S gives the value of S,, in units of lo7 erg cm -3, i.e. S3 means that 3 x lo7 was used. The B at the end means that a different random number seed was chosen and F02 indicates thatf,, was reset to 0.02. B13.5 and B14.5 indicate changes in the starting size distribution of small asteroids. A projectile 261 km in diameter is needed to disperse 50% of the target body’s mass at an impact speed of 5.8 km s-‘. Because of random variation in the impact speed, a smaller projectile than that listed can shatter or disperse a given sized target body.
diameter required to disperse half the target mass against self-gravity ; note, however, that because of random variations in the impact speed a smaller projectile than that listed can shatter or disperse a given target body. The largest impactor on either Pallas or Vesta ranges from 80 to 200 km in diameter (see column 4), for cases that start with the same initial population. This rather wide range reflects the different possible collisional histories of the asteroid population, and includes a random effect since the number of impacts is calculated using Poisson statistics when small numbers are involved. Although Vesta may have been comparatively “lucky’‘-i.e. the few largest projectiles to involve the 500-km bin may have hit Pallas-we see from this table that only energy scaling and strain-rate scaling predict that Vesta has escaped being shattered over solar system history, i.e. that the specific energy of the largest projectile and the cumulative shattering energy normalized to Q* (column 7) are less than unity. Also, the strong gravitational compressive strengthening term of Housen et al. (1991) is required to preserve the 500-km sized asteroids against shattering. The slope of the small diameter (5-50 km) “tail” of the final asteroid population is given in column 8, and can be compared with estimates of the slope for asteroids in that size range.
5. Discussion and conclusions Our initial results using four scaling algorithms for the variation of Q* with target size that have been developed in recent years show that the Housen and Holsapple algorithm and simple energy scaling provide the best match with the observed asteroid size distribution and the existence of the basaltic crust of Vesta. Scaling algorithms based on the hydrocode of Melosh et al. (1992) fail to reproduce either of the above conditions, even when the energy partitioning coefficientf,, is varied from 10 to 2%. However, these results have been generated assuming that JKE is constant with changing target size. This may not be the case, and work is currently under way using the hydrocode to calculate this quantity over a wide range of impact conditions. We do find that the degree of strengthening of Q* due to gravitational self-compression that is indicated by the experiments of Housen et al. (1991) is essential to reproducing the properties of the asteroids that were examined here (Ryan, 1992). The equivalence of these constantpressure experiments to disruption under gravitational self-compression has been verified by hydrocode cal-
culations,
such that the largest fragment
masses are
D. R. Davis et al. : Asteroid collisional evolution
roughly the same for each case. This removes any lingering suspicions that theorists might have had, and must be regarded as the best estimate of how Q* varies in the gravity regime. The inability of the current hydrocode algorithm to reproduce major features of the asteroids may result from our incomplete understanding of how asteroids respond to large impacts. For example, we assume that if a large impact fractures much of the volume of a body, the fragments are given a large enough velocity so that the material which does not achieve escape speed becomes thoroughly mixed. Thus a large crater on Vesta which fractures much of its mantle would excavate a significant amount of diogenite from beneath the basaltic crust and distribute this mantle material over the surface. But if the fragment velocities are low as suggested by Asphaug and Melosh (1993 ; some 100 m s-’ when scaled from Phobos’ to Vesta’s size), i.e. a very lowf,, value applies in terms of our model parameters, then the degree of mixing may be much less and a body such as Vesta could withstand large impacts without showing a significant signature of underlying material. However, this is hardly consistent with the ejection of 5-10 km-sized fragments at speeds of several hundred m s-’ to form the observed Vesta family. Exactly what the dynamics of ejecta fragments is in such an impact remains to be worked out, and this appears essential to reconcile the hydrocode results with asteroid observations. The divergence of the small diameter slope between the observed asteroids and the hydrocode results could also be reduced if& were shown to scale with target size. In this case, varying degrees of fragment dispersal/ reaccumulation might cause the slope of the small diameter end of the calculated distribution to be different. Exactly how much different awaits results from hydrocode calculations onf& and its scaling We fiud that the quest for a scaling algorithm that will explain the observed properties of asteroids is not yet ended and that further work is required to understand the implications of these algorithms for the collisional evolution of asteroids. This paper serves to point out areas of difficulty that need further work, and to demonstrate how experimental validation of all aspects of the scaling algorithms is essential for understanding the observed asteroid belt. Acknowledgments. P. F. has worked at this project while staying at the Observatoire de la C&e d’Azur (Nice, France) thanks to the “G. Colombo” fe~ows~p of the European Space Agency. This project has been partially supported by NATO collaborative research grant 0137/89. This is PSI. Contribution No. 308. Planetary Science Institute is a non-profit division of Science Applications International Corporation. We thank E. Asphaug, K. Housen and P. Paolicchi for helpful discussions and comments.
References Asphaug, E. and Melosb, H. J., The Stickney impact on Phobos : A dynamical model. Icarus 101,144164,1993. Binzel, R. P. and Xu, S., Chips off 4 Vesta: Evidence for the
parent body of basaltic achondrite meteorites. Science 268, 186-191, f993. Cell&, A., ZappalP, V., Di Martino, M., Farm&a, P. and Pat+ lie&i, P., Flattening, pole and albedo features of 4 Vesta from photometric data. Icarus 76,546-5651987. CeBiuo, A., Di Martino, M., Drummond, J., Farin&, P., Paolicchi, P. and Zappal& V., Vesta’s shape, density and albedo features. As&on. Astrophys. 219,32O-321, 1989. Cellino, A., ZappalP, V. and Farm&a, P., The asteroid size distribution from IRAS data. Mon. Not. R. a.vtr. Sot. 253, 561-574,199l. Chapman, C. R., Implications of the inferred compositions of asteroids for their collisional evolution. Mem. S.A.It. 57, 103-122,1986. Cr~~ha~, D. P., Tholen, D. J., Hartmann, W. K,, Bell, J. F. and Brown, R. H., Three basaltic. ~-approaching asteroids and the source of basaltic meteorites. Icarus 89, 1-13, 1991. Davis, D. R., Chapman, C. R., Greenberg, R., Weideuachilllng, S. J. and Harris, A. W. Collisional evolution of asteroids: populations, rotations and velocities. In Asteroids (edited by T. Gehrels), pp. 528-557. Univ. of Arizona Press, Tucson, 1979. Davis, D. R., Chapman, C. R., WeidenschiBing, S. J. and Greenberg, R., Collisional history of asteroids: Evidence from Vesta and the Hirayama families. Zcurw 62, 30-53, 1985. Davis, D. R., Weide~~ng, S. J., FarineRa, P., Paolieehi, P. and Binzel, R. P., Asteroid ~ol~sional history : effects on sizes and spins. In Asteroids II (edited by R. P. Binzel, T. Gehrels and M. S. Matthews) pp. 805826. Univ. of Arizona Press, Tucson, 1989. Dobnanyi, J. W., Collisional model of asteroids and their debris. J. geophys. Res. 74,2531-2554, 1969. Drake, M. J., Geochemical evolution of the eucrite parent body : Possible nature and evolution of asteroid 4 Vesta. In Asteroi& (edited by T. Gehrels), pp. 765-782. Univ. of Arizona Press, Tucson, 1979. Farlnella, P. and Davis, D. R., Collision probabilities and impact velocities in the main asteroid belt. Icarus 97, 111-123, 1992. Farinella, P, and Davis, D. R., Will the real asteroid size distribution please step forward. Lunar Planet Sci. Co& XXV, 1994,365366,1994. Farmella, P., PaoRechi, P. and Zappala, V., The asteroids as outcomes of catastrophic collisions. Icarus 52,409-433,1982. Farmella, P., Davis, D. R., Paolicchi, P., CeRino, A. and Zappal&, V., Asteroid collisional evolution : An integrated model for the evolution of asteroid rotation rates. Astron. Astrophys. 253,604614, 1992. Farlnella, P., Gonczi, R., FroeschR, Ch. and FroeschlC, C., The injection of asteroid fragments into resonances. Icarus 101, 174187,1993. Fujiwara, A., On the mechanism of catastrophic destruction of minor planets by high-velocity impact. Icarus 41, 356-364, 1980. Fujiwara, A., Cerroni, P., Davis, D., Di Mar&o, M., Holsapple, K., Heusen, K. and Ryan, E., Experiments and scaling laws for catastrophic collisions. Asteroids ZZ,240-265, 1989. Gaffey, M. J., The asteroid (4) Vesta : Rotational spectral variations, surface material heterogeneity, and implications for the origin of the basaltic achondrites. Lunar Planet. Sci. XIV, 23 l-232, 1983. Gradle, J. C., Chapman, C. R. and Tedesco, E. F. Distribution of taxonomic classes and the compositional structure of the asteroid belt. In Asteroids ZZ (edited by R. P. Binzel, T. Gehrels and M. S. Matthews), pp. 316-335, Univ. ofArizona Press, Tucson, 1989. Hoisapple, K. A., The scaling of impact processes in planetary sciences, Ann. Rev. Earth Planet. Sci. 21, 333-373, 1993. Heusen, K. and Holsappie, K., On the fra~en~tion of asteroids and planetary satellites. Icarus 84,226-253, 1990.
610 Housen, K. R., Schmidt, R. M. and Holsapple, K. A., Laboratory simulations of large scale fragmentation events. Icarus 94, 180-190, 1991. Marzari, F., Davis, D. R., Vanzaui, V., Cellino, A., Zappala, V. and Farinella, P., Origin and evolution of the Vesta family. 1994, in preparation. Melosh, H. J., Ryan, E. V. and Asphaug, E., Dynamic fragmentation in impacts : Hydrocode simulation of laboratory impacts. J. geophys. Res. 97, 14735-14759, 1992. Milani, A., Carpino, M. and Marzari, F., Statistics of close approaches between asteroids and planets: Project SPACEGUARD. Icarus 88,292-335, 1990. Nakamura, A. and Fujiwara, A., Velocity distribution of fragments formed in a simulated collisional disruption. Icarus 92, 132-146, 1991. Ryan, E. V., Catastrophic collisions : Laboratory impact experiments, hydrocode simulations, and the scaling problem. Ph.D. thesis, Univ. of Arizona, 1992. Ryan, E. V., Asteroid collisions : Target size effects and resultant velocity distributions. Lunar Planet. Sci. Conf. XXIV, 12271228,1993.
D. R. Davis et al. : Asteroid collisional evolution Ryan, E. V. and Melosh, H. J., Hydrocode simulations of largescale impacts : A size dependence for fragmentation. 1994, in preparation. Schmidt, R. M. and Housen, K. R., Some recent advances in the scaling of impact and explosion cratering. ht. J. Impact Eng. 5,543-560,1987. Van Houten, C. J., Van Houten-Groeneveld, I., Herget, P. and Gehrels, T., The Palomar-Leiden survey of faint minor planets. Astron. Astrophys. Suppl. 2, 339-448, 1970. Veeder, G. J., Tedesco, E. F. and Matson, D. L., Asteroid results from the IRAS survey. In Asteroids ZZ(edited by R. P. Binzel, T. Gehrels and M. S. Matthews), pp. 282-289. Univ. of Arizona Press, Tucson, 1989. Zappala, V., Cellino, A., Farinella, P. and KneZeviC,Z., Asteroid families. I. Identification by hierarchical clustering and reliability assessment. Astron. J. 100, 203&2046, 1990. Zappala, V., Cellino, A., Farinella, P. and Milani, A., Asteroid families. II. Extension to unnumbered multi-opposition asteroids. Astron. J., 772-801, 1994.