Collisional Evolution of Small-Body Populations 545

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Davis et al.: Collisional Evolution of Small-Body Populations 545

Collisional Evolution of Small-Body Populations

Donald R. Davis Planetary Science Institute

Daniel D. Durda Southwest Research Institute

Francesco Marzari Università di Padova

Adriano Campo Bagatin Universidad de Alicante

Ricardo Gil-Hutton Félix Aguilar Observatory

Asteroid collisional evolution studies are aimed at understanding how collisions have shaped observed features of the asteroid population in order to further our understanding of the formation and evolution of our solar system. We review progress in developing more realistic collisional scaling laws, the effects of relaxing over-simplifying assumptions used in earlier collisional evolution studies, and the implications of including observables, such as collisionally produced families, on constraining the collisional history of main-belt asteroids. Also, collisional studies are extended to include Jupiter Trojans and the Hilda population. Results from collisional evolution models strongly suggest that the mass of main-belt asteroids was only modestly larger, by up to a factor of 5 or so, at the time that the present collisionally erosive environment was established, presumably early in solar system history. Major problems remain in identifying the appropriate scaling algorithm for determining the threshold for catastrophic disruption as well as understanding the resulting size and velocity distribution of fragments. Dynamical effects need to be combined with collisional simulations in order to understand the structure of the small asteroid size distribution.

1. INTRODUCTION mental to understanding asteroid collisional history; however, a consensus has not yet developed among workers as The importance of collisions in shaping the present aster- to the appropriate methodology to go from laboratory-scale oid belt has been recognized for nearly 50 years, following experiments to giant collisions involving bodies to hundreds the pioneering work of Piotrowski (1953) on the frequency of kilometers in size. Gravity becomes significant in the of asteroidal collisions. Since that time, increasingly sophis- asteroid size range between subkilometer-sized bodies and ticated tools have been developed for tracing the collisional the ~850-km-diameter Ceres and thus adds to the complex- history of asteroid populations and understanding the role ity of understanding collisional outcomes. The effects of of collisions in producing observed features of various small- size-dependent collisional outcomes as well as boundary body populations. Here we review the present state of un- conditions on the overall population distribution are active derstanding of asteroid collisional evolution, concentrating topics of research. As discussed in section 3, the existence on progress since the Asteroids II volume. of a small size cutoff in a collisionally interacting popula- Observations relevant to collisional processes are de- tion can produce a wavelike structure, a feature not recog- scribed in section 2, including recent work on extending the nized in earlier work on collisional evolution. asteroid size distribution to subkilometer sizes. Determin- Collisional evolution of Trojans and Hildas has been ing the size distribution is a key constraint for collisional studied in depth in recent years, as reviewed in section 3. models and is a major discriminant among the various scal- These populations are particularly interesting to study be- ing laws proposed for describing collisional outcomes. In cause they are dynamically isolated from much of the main section 3 we summarize work on scaling laws that describes asteroid belt, although they do collisionally interact with the outcome of collisions involving bodies of various sizes each other. Additionally, these populations are dynamically hitting at speeds of multikilometers per second. Understand- stable, but collisions can eject fragments into unstable or- ing collisional outcomes for asteroid-sized bodies is funda- bits that are then lost from the system. Comparing the col-

545 546 Asteroids III lisional evolution of these populations with each other pro- for clarity in some cases; however, the population index is vides insight to features common to the collisional evolution numerically the same in these two formulations. An excel- of small-body populations. Another collisionally evolved lent discussion of the various power laws used to describe population discovered recently is the Kuiper Belt, a vast small-body populations is given by Colwell (1994).] It is population of remnant planetesimals orbiting beyond Nep- widely believed that the largest asteroids (although where tune. Kuiper Belt objects are thought to share many features “largest” begins is a debated question) are primordial ob- with main-belt asteroids, including a significant role for col- jects whose sizes have not been significantly altered by collisions in shaping the present Kuiper Belt. However, this lisions; only the small size end of the population is colli- chapter will be restricted to the role of collisions on aster- sionally evolved. oids in the inner solar system, and the interested reader is The so-called “bump” in the main-belt asteroid size dis- referred to Farinella et al. (2000) for a comprehensive re- tribution centered at about 100 km diameter has been ex- view of the role of collisions in the Kuiper Belt. plained in three different ways: (1) Anders (1965) proposed Finally, section 5 gives our perspective on the major out- that this was the signature of collisional modification of an standing problems in the field of asteroid collisional studies. initial Gaussian-like planetesimal size distribution. (2) Davis We expect that by the time of the publication of Aster- et al. (1979, 1984) found that this feature marked the transi- oids IV, most of these will be solved and an equally chal- tion from the gravity strength regime to the strength dominant lenging set of questions will be posed for the next decade. size regime and hence was a product of collisional physics. (3) Durda et al. (1998) argued that this feature was a 2. OBSERVATIONAL CONSTRAINTS secondary “bump” produced by the wave from the strength- gravity transition that occurs at much smaller sizes (see The litmus test for models of the collisional history of section 3). asteroids is how well they match the observed characteris- While there is structure in the small asteroid size distribu- tics of the present asteroid belt, i.e., those features of the tion, a power-law fit to the cumulative absolute magnitude belt that are the product of 4.5 b.y. of collisional history. distribution of the Palomar Leiden Survey (PLS) (Kresak, So, what are the collisionally produced observables in the 1977; Van Houten et al., 1970) gives a slope of 1.95 over the asteroid belt and other small body populations? First is the size range larger than a few kilometers (D > 2–5 km). Fig- size-frequency distribution; Fig. 1 shows the inferred size- ure 2 shows various recent estimates of the small end of the frequency distribution for (a) main-belt asteroids, (b) Tro- main-belt size small distribution (see also Jedicke et al., 2002). jans, and (c) Hildas. [Throughout this chapter we use the It is worth remarking that several lines of evidence point cumulative diameter distribution, i.e., the number of bodies to the existence of a variable index in the size distribution larger than diameter D, to describe the size distribution of of the small asteroids. For main-belt asteroids smaller than populations. This relationship is expressed mathematically about 20 km, the PLS (Van Houten et al., 1970) found vari- as N(>D) = K D–b where b is the cumulative diameter popu- able indexes. Also, Cellino et al. (1991) analyzed IRAS data lation index. We also use the incremental size distribution for different zones of the asteroid belt and different size ranges (but always exceeding a few tens of kilometers, to avoid discovery biases) and found indexes ranging from ~–2 1011 to ~–4. Similar variability occurs for objects with orbits 1010 crossing those of the inner planets, as suggested by the lunar 9 Asteroids: ~0.1 Lunar Mass 10 and martian cratering record (Grün et al., 1985; Strom et al., 108 Trojans: ~0.001 Lunar Mass 1992), the in situ dust detection experiments carried out on 107 Hildas: ~0.0005 Lunar Mass spacecraft (Grün et al., 1985; Love and Brownlee, 1993), 106 and telescopic searches for small Earth-approaching aster- 105 oids (Ceplecha, 1992; Rabinowitz, 1993). Analyses of the 104 103 cratering record on the surface of 951 Gaspra, imaged in 102 1991 by the Galileo probe, have indicated that a cumulative

Number per Diameter Bin 101 size distribution index of –2.7 ± 0.5 over the size range 0.4– 100 1.5 km (Belton et al., 1992), while for Ida, the index varies 100 101 102 103 from ~–3.3 to –3.5 for crater diameters >1 km (Belton et al., 1994). The craters were produced by projectiles (~20 to 200 m Diameter (km) across) that have cratered the surfaces of Gaspra and Ida. A powerful constraint on the asteroid collisional history Fig. 1. Comparison of the size-frequency distribution for main- is the existence today of the basaltic crust of Vesta, which, belt asteroids (MBA), Trojans, and Hildas to show the relative if it is the ultimate source of the eucrite meteorites as is abundance as a function of size in the three populations. Note that while the Trojan population is numerically 10–25% of the main- widely believed, dates to the earliest era of the solar system belt population at sizes <100 km diameter, the total mass of the some 4.54 b.y. ago. Any collisional model must preserve this Trojans is only about 1% of the MBA. Most of the mass for these 25–40-km-thick crust (Gaffey et al., 1993) during the colli- populations is in the largest bodies. Each diameter bin spans a sional bombardment over solar system history (Davis et al., factor of 2 in mass. 1984). Recent Hubble Space Telescope (HST) observations Davis et al.: Collisional Evolution of Small-Body Populations 547

1012 ing to the proposed “W” (or wet) taxonomic class (Rivkin X 11 Farinella et al. (1992) (1) 10 X Farinella et al. (1992) (2) et al., 2000). However, the M-class asteroid 16 Psyche ap- 10 X Farinella et al. (1992) (3) 10 X X Farinella et al. (1992) (4) pears to be the collisionally exposed core of a parent body 109 Galileo Team X Davis et al. (1994) that was virtually identical to Vesta. How is it possible to 8 X 10 Durda et al. (1998) Model X disrupt the Psyche parent body and remove all traces of this 107 X SAM99 Model X SDSS (2001) 6 X giant collision (there is no family associated with Psyche 10 X Small size bump 5 today) except for Psyche itself, while still preserving the 10 X X 4 10 X Large size crust of Vesta (Chapman, 1985)? This problem was investi- X bump 103 gated by Davis et al. (1999), who concluded that it was very 2 Cumulative Number > D Cumulative 10 difficult, but not impossible, based on collisional modeling, 101 to create Psyche and preserve Vesta’s crust, but other expla- 100 10–2 10–1 100 101 102 103 nations for Psyche should be sought. Furthermore, recent work suggests a mean density for Psyche of 1.8 ± 0.6 g/cc D (km) (Viateau, 2000), which is inconceivably low for the Fe core of a differentiated body. The more general problem for col- Fig. 2. Nine estimates of the main-belt asteroid size distribution. lisional effects on asteroid materials is how to disrupt the The four models from Farinella et al. (1992b) are based on assum- dozens of parent bodies needed to explain the iron meteor- ing N(>50 km) = 700, a cumulative diameter slope of –1.0 be- ite collection while eliminating from the asteroid belt most tween 50 km > D > Dtr and the Dohnanyi (1969) slope (cumulative of the volumetrically more abundant dunite mantles of these diameter) slope of –2.5 for D < Dtr. The values of Dtr are taken to bodies. Collisional grinding of this mantle material has been be 10 km, 25 km, 1 km, and 40 km, respectively. The model la- suggested, but such a process would need to be extremely beled “Galileo” is the adopted asteroid population for calculating Bell the projectile flux onto Gaspra. They used the PLS slope (–1.95) efficient [see the section on the great dunite shortage in for D > 0.175 km and the Dohnanyi slope for D < 0.175. The et al. (1989); see also Burbine et al. (1996)]. Davis et al. (1994) model derives from extrapolating the known Asteroid families are another observable that is the prod- population (assumed to be complete) at D = 44 km using the PLS uct of collisional evolution. There are approximately 25 reli- slope and the Cellino et al. (1991) “corrected random” albedo able families and over 60 statistically significant clusters model down to D = 5.5 km. The Durda et al. (1998) model is the identified in asteroid proper elements [see Zappalà et al. fit to the asteroid size distribution determined by Jedicke and (2002) and Bendjoya and Zappalà (2002) for detailed dis- Metcalfe (1998), who used Spacewatch data to estimate the small cussions of asteroid families]. size main-belt distribution. The SAM99 model is the Statistical Another constraint on the asteroid small size population Asteroid Model of Tedesco et al. (2002), who combined the ob- comes from the cosmic-ray-exposure (CRE) ages of stony served size distribution of the main asteroid families with that of meteorites, which are measured to be around 20 m.y. The the background population into a single unified population estimate. The small size distribution is obtained by extrapolating the CRE age measures the time that the meteorite was in meter- observed large size trends. Finally, the SDSS 2001 model is the sized bodies (or near the surface of a larger body); however, fits to the Sloan Digital Sky Survey data by Ivezic et al. (2001), the collisional lifetime of meter-sized objects is calculated to who find that a broken power law well represents their data. For be around 1 m.y. (O’Brien and Greenberg, 1999). Any suc- 5 km < D < 40 km, the slope is –3.0, while for 0.4 km < D < 5 km, cessful asteroid collisional history must be consistent with a –1.3 slope is a good fit to their data. the observed CRE ages. Collisions have modified asteroid rotation rates over solar system history, so information about the collisional history of the belt is embedded in the spin rates of asteroids of Vesta revealed the existence of a ~450-km-diameter basin of all sizes. However, as discussed by Davis et al. (1989) thought to be formed by the impact of a ~40-km-diameter and Farinella et al. (1992a), the uncertainties in modeling projectile (Marzari et al., 1996; Asphaug, 1997). This im- how collisions alter rotation rates are so large that any de- pact, presumably the largest since Vesta’s crust formed, pro- finitive work on this topic is yet to be done. However, recent vides a very specific constraint on Vesta’s collisional history. progress has been made on understanding fragment rota- Yet this event, which apparently excavated through the crust tion rates from catastrophic disruption events using numeri- to some degree, was not energetic enough to shatter Vesta cal hydrocodes by Love and Ahrens (1997) and Asphaug and destroy its basaltic crust. The fundamental reason that and Scheeres (1999). These investigations provided deeper the large impact did not destroy Vesta’s crust is that it occurs insights into fragmental spin rates as a function of fragment in the gravity — rather than the strength — regime, as size as well as providing data on critical parameters needed shown by detailed numerical simulations (Asphaug, 1997). to model the collisional evolution of asteroid spin rates. At the same time, collisions are argued to have produced In summary, any successful collisional evolution sce- the parent bodies of iron meteorites, widely interpreted to nario for main-belt asteroids must satisfy the observable be the metallic cores of disrupted differentiated asteroids. constraints imposed by the size distribution, the number of These metallic cores were identified with the M-class aster- families, and the existence of an old basalt crust on Vesta; oids; however, this interpretation has been challenged by the the CRE ages of strong meteorites; and the cratering pro- discovery of water of hydration on many M asteroids, lead- jectile flux onto asteroids. Other observables, such as aster- 548 Asteroids III

TABLE 1. Intrinsic probability of collision and mean impact velocity for different populations of minor bodies.

Intrinsic Probability Impact Velocity Asteroid Populations (10–18 yr–1 km–2) (km s–1) Reference Main Belt (MB) MB-MB 2.85 ± 0.66 5.81 ± 1.88 Farinella and Davis (1992) MB-MB 3.97 — Yoshikawa and Nakamura (1994) MB-MB 2.86 5.3 Bottke et al. (1994) MB-MB 4.38 4.22 Vedder (1996, 1998) MB-SPC 1.51 10.86 Gil-Hutton (2000) MB-UN 1.08 14.0 Gil-Hutton and Brunini (1999)

Hildas (Hil) Hil-Hil 2.31 ± 0.10 3.09 ± 1.47 Dahlgren (1998) Hil-Hil 1.93 3.36 ± 1.52 Dell’Oro et al. (2001) Hil-Tro 0.24 ± 0.06 4.59 ± 1.71 Dahlgren (1998) Hil-Tro 0.27 4.49 ± 1.68 Dell’Oro et al. (2001) Hil-MB 0.62 ± 0.04 4.78 ± 1.78 Dahlgren (1998) Hil-MB 0.66 4.80 ± 1.77 Dell’Oro et al. (2001) Hil-UN 6.5 10.98 Gil-Hutton and Brunini (1999)

Trojans (Tro)

L4-L4 6.46 ± 0.09 4.90 ± 0.07 Marzari et al. (1996) L4-L4 7.79 ± 0.67 4.66 Dell’Oro et al. (1998) L5-L5 5.30 ± 0.10 4.89 ± 0.10 Marzari et al. (1996) L5-L5 6.68 ± 0.18 4.51 Dell’Oro et al. (1998) L4-SPC 0.33 6.78 ± 2.71 Dell’Oro et al. (2001) L5-SPC 0.35 6.67 ± 2.59 Dell’Oro et al. (2001) Tro-UN 0.50 8.19 Gil-Hutton and Brunini (1999) For each population the values calculated by various authors with the relative errors, when available, are reported. The more recent values are usually derived from a larger sample of orbits and are probably more accurate. Colli- sion rates with external populations are given for (a) short-period comets (SPC), and (b) Uranus/Neptune scattered planetesimals (UN).

∆ oid rotation rates, dust production, etc., are altered by, or are dius Rt by projectiles of radius Rp within a time ( T) is then the product of, collisional processes. However, our under- given by standing of how collisions affect them is not sufficiently ma- 〈 〉 2 ∆ ture at this time to allow imposition of additional constraints. Ncoll = Pi (Rt + Rp) V Np T

3. SCALING LAWS, BOUNDARY CONDITIONS, where V is the mean impact speed and Np is the spatial den- AND COLLISIONAL EVOLUTION sity of projectiles. 〈 〉 For each population several estimates of Pi and V are 3.1. Basic Collisional Parameters: Collision given in Table 1. Different authors improved the Öpik/ Rates and Impact Speeds Wetherill analytical formulation with corrections related to specific features of the orbital distribution of the population The frequency of collisions and the collisional impact under study or to statistical mechanics (Greenberg, 1982; speeds are fundamental quantities for collisional evolution Namiki and Binzel, 1991; Farinella and Davis, 1992; Bottke studies. While the basic theory for calculating collision rates and Greenberg, 1993; Bottke et al., 1994; Vedder, 1996, and speeds was developed by Öpik (1951) and Wetherill 1998; Dell’Oro and Paolicchi, 1998; Dell’Oro et al., 1998, (1967) and applied by various workers in subsequent years, 2001). These methods also allow fast computations of up- 〈 〉 there has been recent work to refine such calculations and dated values of Pi and V whenever the discovery of new to extend them to both Trojans and Hildas. The intrinsic objects significantly enriches the known population. Alter- collision probability, Pi, gives the collision rate per unit natively, different authors have resorted to a direct numeri- cross-section area of target and projectile per unit time, cal approach based on the integration of the orbits of the while the actual number of collisions onto a target of ra- asteroids over a sufficiently long timespan. The derived dis- Davis et al.: Collisional Evolution of Small-Body Populations 549

tribution of close encounters and mutual speeds recorded largest body, Ceres in the asteroid belt) but no lower cut- during the integration can be extrapolated to infer the colli- off. These workers analytically found a stationary solution sion probability and characteristic impact speed (Yoshikawa to the governing differential equations for collisional evo- and Nakamura, 1994; Marzari et al., 1996; Dahlgren, 1998). lution. To trace the detailed time history of collisional evo- It is interesting to note that Pi and V for each of the three lution and to include more realistic collisional physics re- populations treated here are rather similar, differing by at quires numerical models. The first such models led to new most a factor of 4 in Pi and a little more than a factor of 2 insights regarding asteroid collisional history (Chapman and in V. The collision rates between populations, though (e.g., Davis, 1975; Davis et al., 1979) and numerical models have Hildas with main-belt asteroids or Hildas with Trojans), can been used by nearly all workers subsequently. be an order of magnitude smaller than the intrapopulation The most important result from Dohnanyi’s work is that collision rates. However, the collision speeds for the inter- as the collisional process gives rise to a cascade of frag- population collisions are similar to those within a population. ments shifting mass toward smaller and smaller sizes, a sim- While various authors have attempted to infer the pri- ple power-law equilibrium size distribution is approached, mordial population of asteroids by integrating backward in which has a cumulative diameter slope of –2.5. Other au- time from the present belt, the inherent instability of the thors have confirmed this result, both analytically and nu- collisional problem prevents such results from converging. merically (e.g., Paolicchi, 1994; Williams and Wetherill, Indeed, the earliest result of Dohnanyi (1969), which showed 1994; Tanaka et al., 1996). In the real world, however, the that under specific circumstances the asteroid population basic assumptions of Dohnanyi’s theory are not fulfilled. It was collisionally relaxed and independent of the starting was realized early on that gravitational binding, a mass de- population, meant that it would be impossible to uniquely pendent process, plays a dominant role in holding large as- infer the initial asteroid population by going back in time teroids together, hence the assumption that self-similar colli- from the present belt. All recent models of asteroid collisional outcomes could not hold in the gravity regime (Davis sional evolution have assumed various hypothetical initial et al., 1979). Below we separately explore progress made populations and propagated these forward in time using a in recent years in understanding the consequences of relax- variety of scaling laws, and have compared the terminal ing the above assumptions 2 (collisional parameters are size- model population with the present observed belt. independent) and 3 (no lower cutoff in the size distribution). Relaxing assumption 1 (on collisional cross-sections) has only 3.2. Scaling Laws and Collisional Evolution a minor effect on the overall collisional evolution. Recent collisional models replace Dohnanyi assump- The modeling of the collisional evolution of populations tion 2 with a scaling law describing how the collisional out- of small bodies is generally performed in two main stages: comes vary with target and impactor size. In its most gen- (1) the simulation of the outcomes of single collisions, and eral form, a scaling law is simply an algorithm for extrapo- (2) the integration of the equations of evolution, assuming lating the outcomes of collisions for which we have direct, given boundary conditions. A set of simplifying assump- physical experimental experience to the outcomes of colli- tions is usually made in these kind of models: Bodies are sional events at size scales much larger or smaller than can considered to be spherical and homogeneous, and they move be practically treated under laboratory conditions. One im- and collide within a finite volume around the Sun with rela- portant component of scaling laws is determining the criti- * tive impact speeds determined by the orbits of the target cal impact specific energy, QD, the energy per unit target and projectile. Simulations are performed by numerically mass delivered by the projectile required for catastrophic integrating a given set of first-order, nonlinear, differential disruption of the target, i.e., such that the largest resulting equations, taking into account the rate of variation of the object has a mass one-half that of the original body. Notice population of asteroids of any given mass due to the num- that this object may be formed by partial reaccumulation ber of fragments produced or destroyed by collisions, and of fragments due to self-gravity. Sometimes, as in laboratory to the removal of bodies by nongravitational effects (e.g., experiments, the shattering impact specific energy is given Campo Bagatin et al., 1994a). in terms of the energy per unit target mass required for the * A useful mathematical analysis for the collisional evolu- catastrophic shattering of the target, QS (regardless of reaccu- tion of the asteroid belt was introduced some 30 years ago mulation of fragments), such that the largest fragment pro- by Dohnanyi (1969, 1971) and independently by Hellyer duced has a mass of one-half that of the original target. The (1970, 1971). Three crucial assumptions of these models two definitions only differ in the regime in which the effect are: (1) the interacting bodies are modeled as spheres of of gravity is much larger than that of solid-state forces. equal density, and their collisional cross-section is simply A number of experimental and analytical studies have * proportional to the squared sum of the radii; (2) all the focused on quantifying just how QD scales with target size collisional response parameters are size-independent, imply- in the strength-dominated and gravity-dominated regimes ing that fragmentation occurs for a fixed projectile-to-target (see Durda et al., 1998, for a recent review) and at what mass ratio and no self-gravitational effect is taken into ac- sizes the transition from strength-scaling to gravity-scaling count; (3) the population has an upper cutoff in mass (a takes place. Dimensional analyses (Farinella et al., 1982; 550 Asteroids III

Housen and Holsapple, 1990), impact experiments (Hols- 107 apple, 1993; Housen and Holsapple, 1999), and hydrocode 6 studies (Ryan, 1992; Benz and Asphaug, 1999) show that 10

* –0.24 5 QD for strength-dominated targets scales as roughly D (J/kg) 10 to D–0.61, where D is the target diameter. On the other hand, D * 104 determining the size dependence of QD in the gravity-dominated regime has proven to be more difficult, particularly 103 SL2 due to the lack of direct experimental experience in the disruption of asteroid-sized targets. Numerous analytical 102 SL1b arguments and hydrocode studies (e.g., Davis et al., 1985; 1 Specific Energy, Q* Specific Energy, 10 SL1a Housen and Holsapple, 1990; Love and Ahrens, 1996; Melosh and Ryan, 1997; Benz and Asphaug, 1999) yield 100 0.001 0.010 0.100 1.000 10.000 100.000 1000.000 * 1.13 2 size-dependent QD that scale as D to D . Recent hydrocode studies (Benz and Asphaug, 1999) and numerical colli- Diameter (km) sional models (Durda et al., 1998) indicate that the transition from the strength-dominated to the gravity-dominated Fig. 3. Three hypothetical scaling laws used to explore the be- regime occurs at target diameters of about 100–300 m, havior of evolved size distributions. while many earlier studies generally place the transition in the ~1–10-kilometer size range. Observations of asteroid spin rates (Pravec and Harris, 2000) confirm these theoreti- cal findings, showing a lack of objects rotating with peri- Figure 3 shows three hypothetical scaling laws used to ods less than 2.2 h among asteroids with absolute magnitude explore the behavior of evolved size distributions (Durda H < 22, evidence that asteroids larger than a few hundred et al., 1998). For scaling law SL1a, at small sizes we as- meters across are indeed mostly gravity-dominated aggre- sume a strength-scaling law with a D–0.4 size dependence. gates with negligible tensile strength (“rubble piles”; see At large sizes a gravity-scaling relationship with a D1.1 size Richardson et al., 2002). dependence is similar to the predictions of hydrocode mod- Durda (1993) and Durda and Dermott (1997) examined els. Scaling law SL1b is similar to SL1a with the exception * the influence of QD on the shape of an evolving size distri- of a more gradual transition between the strength and grav- bution by showing that the power-law slope index of a pop- ity-scaling regimes. Scaling law SL2 is identical in shape ulation in collisional equilibrium is a function of the size to SL1a, but is 100× stronger everywhere. * dependence of QD. In the case of pure energy scaling (size- Figure 4 shows the resulting evolved size distributions * independent QD), the slope index of the size distribution is for colliding populations with the properties of disruption * –2.5, as Dohnanyi (1969) showed. When QD decreases with specific energy governed by scaling laws SL1a, SL1b, and increasing target size, as is the case in the strength-scaling SL2. In all three cases, the transition from strength- to grav- regime, the slope index of the equilibrium size distribution ity-scaling at D = 100 m results in a “bump” in the size dis- * is steeper than Dohnanyi’s; instead, when QD increases, as tribution for objects just larger than this size. This bump in the gravity-scaling regime, the resulting slope index is results from the increased disruption lifetimes of objects in shallower than –2.5. this size range over what they would have been had gravity- * Given the wide range of QD proposed by different scal- scaling not taken over to strengthen larger objects. Objects ing laws, the question arises as to which one best describes in this bump represent an excess supply of projectiles that the real world? Experiments involving asteroid-sized bodies can disrupt targets roughly an order of magnitude larger are unfortunately as yet impossible to arrange, so the next than themselves, resulting in a wavelike perturbation to the best test is to compare model results using different scaling size distribution at larger sizes, as described in further de- laws with asteroid collisional observables (section 2). Davis tail below. The abruptness of the transition from strength- et al. (1994) tested several proposed scaling laws and found to gravity-scaling can significantly influence the shape of that strain-rate scaling of Housen et al. (1991) and simple the resulting size distribution, with more gradual transitions energy scaling best satisfied all constraints. Further work (SL1b vs. SL1a) resulting in smaller amplitude waves. Greater testing scaling laws was carried out by Durda et al. (1998), strengths at all sizes (SL2 vs. SL1a) result in higher fre- who presented results from numerical experiments illustrating quency waves, since the critical projectile size for target dis- in a more systematic fashion the sensitivity of an evolved ruption is closer to the size of the targets themselves, and size distribution on the shape of the strength-scaling law. the mechanism driving the wave occurs over a smaller range * A linear relationship between logQD and logD yields a of sizes. power-law size distribution; if there is nonlinearity in this These results demonstrate that evolved size distributions relationship (due to a transition from strength- to gravity- generated by collisional models depend strongly (and un- scaling, for instance), then structure is introduced into the derstandably) upon the shape of the size-strength scaling evolving size distribution due to a nonlinearity in collision law. With this in mind, Durda et al. (1998) adopted a least- lifetimes of the colliding objects. squares approach and adjusted the strength law for asteroi- Davis et al.: Collisional Evolution of Small-Body Populations 551

1010 belt asteroids. Durda et al. (1998) interpret the bump in the (a) size distribution between ~3 and 30 km (see Fig. 2) as a 108 primary bump due to the transition from strength scaling

T0 to gravity scaling for asteroids larger than ~150 m; the well- known bump at ~50–200 km is a secondary feature result- 106 ing from a wavelike structure induced in the size distribution by the ~3–30-km primary bump. 104 Evolution for hypothetical Q*D vs. D relation SL1a 3.3. Boundary Conditions and Incremental Number 102 Collisional Evolution

100 Even more severe departures from simple power-law 0.001 0.010 0.100 1.000 10.000 100.000 1000.000 distributions result from relaxation of Dohnanyi assumption 3 (no lower cutoff in the size distribution). When a 1010 (b) small-size cutoff in the initial distribution is imposed, a remarkable feature appearing in the simulations is that the 8 10 final distribution is not a fixed-exponent power-law, but T 0 rather displays a wavy pattern of variable wavelength and 106 amplitude, superimposed on an “average” power law of slope close to Dohnanyi’s canonical value. 104 Evolution for hypothetical The wavy structure is produced as follows. We divide Q*D vs. D relation SL1b the asteroid mass range into a series of discrete mass bins,

Incremental Number 102 so the smallest one in the distribution has particles removed only in two ways: collisions within the bin itself or by its particles hitting larger targets. Thus there is only a small 100 0.001 0.010 0.100 1.000 10.000 100.000 1000.000 number of bodies removed from this bin. The next larger bin, however, has bodies removed not only by the above 1010 processes, but also due to impacts by projectiles from the (c) smallest bin. This leads to an enhanced removal rate from 108 this bin, and its population is more rapidly depleted than is

T0 that of the smallest bin. As we proceed to ever larger sizes,

106 the depletion rate increases relative to that of smaller bins due to the ever-increasing number of projectiles per target body. Thus the population is increasingly depleted, and 104 Evolution for hypothetical Q*D vs. D relation SL2 develops a steeper slope than Dohnanyi’s equilibrium value, up to the size bin for which the smallest available projec- Incremental Number 2 10 tile can shatter its members. Beyond this size, larger target bins no longer have an enhanced depletion rate, but rather 100 0.001 0.010 0.100 1.000 10.000 100.000 1000.000 have a somewhat reduced removal rate due to the rapid decrease in the number of available projectiles with increas- Diameter (km) ing size. This leads to a flattening of the size distribution and to a slope lower than the equilibrium one — thus a Fig. 4. Resulting evolved size distributions for colliding popula- wavelike pattern arises. This pattern propagates itself to tions with strength properties governed by scaling laws: (a) SL1a, larger and larger sizes due to the rise and fall of the rela- (b) SL1b, and (c) SL2. Solid lines give the population at 1, 2, 3, tive number of projectiles capable of shattering larger bod- 4, and 4.5 b.y. ies. It is interesting to note that this shift propagates in such a way that the final distribution at sizes between 1 and 100 km is substantially changed when the cutoff is changed. dal bodies to obtain a best fit to the actual asteroid size Figure 5, for instance, shows the effect of shifting the cut- distribution determined from the cataloged asteroids and off size from 1 cm to 1 mm. Thus, the behavior of very Spacewatch data. Their derived strength scaling law, with small particles can in principle affect the size distribution a strength minimum at ~150 m diameter as independently of the observable asteroids in a very significant way. Only suggested by hydrocode models (e.g., Benz and Asphaug, when averaged over several orders of magnitude in size 1999) and observed rotation rates (e.g., Pravec and Harris, does the overall size distribution still approximately match 2000), naturally gives rise to an evolved main-belt asteroid the canonical Dohnanyi equilibrium slope. size distribution in good agreement with the two-bump Is such a small-particle cutoff actually present in the structure observed in the actual size distribution of main- actual asteroid population? The answer to this question is 552 Asteroids III

1028 pend on a number of poorly known critical parameters that ...... Starting population govern the mass distribution and the reaccumulation of frag- Evolved population (80 bins) ments after their formation (see Campo Bagatin et al., Evolved population (90 bins) 1021 1994b). Numerical models generally calculate the fragmen- * tal size distribution, assumed to be a power law, from QS and the collisional energy. The inelasticity parameter, f , 1014 KE which determines what fraction of the collisional kinetic energy goes into the fragments, is used to calculate the

107 ejecta fraction that escapes the bodies’ gravity. Its value ...... depends on the composition, internal structure, and size of Number per Diameter Bin the bodies, and it is often taken between 1% and 10%. On 100 the other hand, f is implicitly embedded in Q* in that a 10 –6 10 –5 10 –4 10 –3 10 –2 10 –1 100 101 102 103 KE D * * large QD implies a small fKE, while a small QD goes with a Diameter (km) larger fKE (Campo Bagatin et al., 2001). The fragment mass-speed relation is critical to calculating Fig. 5. Propagation of the wave effect with different small-size collisional outcomes and relationship of the form V(m) = cutoffs (solid line: 1 cm; dashed line: 1 mm). Cm–r (Nakamura et al., 1992; Giblin, 1998), with r ranging from 0 (no mass-velocity dependence) to 1/6, as found by Giblin (1998). The effects of this dependence have been studied by Petit and Farinella (1993) and Campo Bagatin not obvious, but many nongravitational forces do indeed et al. (1994b). Even a shallow dependence of ejection speed act on interplanetary matter and are efficient at different on fragment mass makes a significant difference in the sizes: the solar wind, Poynting-Robertson drag, and the amount of mass that can be reaccumulated by objects in Yarkovsky effect. There is strong observational evidence the range 1 to 100 km (Campo Bagatin et al., 2001). Also, that most of the micrometeoroid mass is concentrated at the size at which a significant ratio of aggregate rubble-pile particles sizes around 100 µm, and that the corresponding asteroids is present seems to depend strongly on the reaccu- mass distribution flattens considerably for smaller sizes mulation model. For example, assuming a mass-velocity re- (Grün et al., 1985; Love and Brownlee, 1993). Thus the real lationship with r = 1/6 provides some 10% rubble-pile aster- low-mass cutoff of the asteroid population probably lies at oids for 2-km-sized bodies, while r = 0 prevents reaccumu- sizes of micrometers or smaller. Solar radiation pressure on lated objects for sizes below some 10 km (Fig. 6). It is thus particles of size ~1 µm (Burns et al., 1979) converts them clear that further efforts are necessary to understand the de- into hyperbolic β meteoroids, which, according to Grün et al. pendence of ejection speeds of fragments on their masses. (1985), provide the main loss mechanism for the meteor- On the other hand, recent work using hydrocodes to model itic complex. Durda (1993) and Durda and Dermott (1997) collisional fragmentation (Benz and Asphaug, 1999; Michel imposed a small-mass cutoff in their collisional model that et al., 2001; see also section 5) suggests that essentially all empirically matched the Grün et al. (1985) data and found asteroids from ~1 to 100 km diameter are reaccumulated it too gradual to induce significant wave structure in the rubble piles. evolved population. Models that attempt to simulate the removal of small particles from the colliding population through radiation forces (Campo Bagatin et al., 1994a) do 1 show a noticeable wave structure only when Poynting Robertson drag is artificially enhanced; the Yarkovsky ef- 0.8 fect — much more effective than the Poynting-Robertson — might instead be responsible for such a pattern. Clearly, 0.6 more work is needed in this area to explicitly treat within collisional models the various nongravitational forces that 0.4 act to remove small objects from the asteroid population

and to determine how strong and abrupt a small-mass cut- Fraction Rubble-pile 0.2 off must be in order for wavelike features to appear in the evolved size distribution. 0 0.1 1 10 100 1000 3.4. Model Parameters and Diameter (km) Collisional Evolution Fig. 6. Fraction of rubble piles in the asteroid belt as a function * * Besides the scaling laws for QD or QS, which affect the of size, according to different reaccumulation models (solid line: disruption/fragmentation threshold and the number of frag- mass-velocity dependence with r = 1/6; dashed line: no mass- ments produced in any collision, fragmentation models de- velocity relation, i.e., r = 0), and keeping fKE = 0.01 (see text). Davis et al.: Collisional Evolution of Small-Body Populations 553

Recent models of asteroid collisional evolution consider ise for understanding the asteroid size distribution from the two different kinds of objects (Campo Bagatin et al., 2001): largest bodies down to subdecameter-sized asteroids. monoliths (high fKE), and rubble-pile reaccumulated objects (low fKE), usually called rubble-piles, or aggregate aster- 4. COLLISIONAL EVOLUTION OF oids. The number of aggregate asteroids present in the main TROJANS AND HILDAS asteroid belt can be estimated through collisional modeling to range from 30% to 100% in the range 10 to 200 km, 4.1. Trojans depending on the adopted scaling law, the exponent r, and the choice of other collisional parameters, especially the Jovian Trojan asteroids have a peculiar dynamical be- inelasticity parameter, fKE. havior: They librate around the Lagrangian equilibrium A characteristic pattern of collisional evolution outcomes points L4 and L5 located at the maxima of the “pseudo- is that the final size distribution of bodies appears to be potential” defined within the frame of the restricted three- fairly insensitive to the initial conditions, provided most of body problem (Greenberg and Davis, 1978). The resonant the mass is in bodies that can be collisionally disrupted at link with Jupiter confines them in a limited volume of space impact speeds characterizing the population. For any popu- and, notwithstanding that they have an average high incli- lation characterized by an impact speed, V, there is a larg- nation on the ecliptic, their collisional activity is compa- est-sized body that can be collisionally disrupted even if rable to that of main-belt asteroids. Collisions for Trojans hit by an equally large body. As long as most of the popu- are a relevant evolutionary mechanism not only because lation mass is in bodies smaller than this critical size, the they grind down the primordial size distribution, but also final distribution is essentially independent of the starting because they significantly alter the parameters of the libra- distribution. Even for the extreme initial distribution with- tional motion. The velocity imparted to fragments in a col- out bodies smaller than 1 km, the subkilometer size range lision generate new orbital parameters that, for librating is populated by fragments within a few million years, and bodies, may even drive them out of the resonance. The pres- the stationary distribution is attained. ent Trojan population has almost surely lost memory of the primordial distribution of libration amplitude and proper 3.5. Asteroid Observables and eccentricity. Collisional Evolution The first step to quantitatively model the collisional evolution of Trojans is to derive precise estimates of the typi- The previous three sections tested asteroid collisional cal collisional frequencies and impact velocities for the two evolution models against the observed size distribution of swarms. Since the librational motion sets kinematical con- asteroids, but as noted in section 2, there are other colli- straints to the values of the orbital angles respective to those sionally produced observables: of Jupiter, the standard statistical approaches based on Families. Marzari et al. (1999) used the ~85 recognized Wetherill’s (1967) theory, even in more recent formulations families and statistically significant grouping recognized in (Bottke and Greenberg, 1993; Vedder, 1996, 1998), cannot the present belt as a constraint on the overall asteroid colli- be applied. Critical assumption of these models is that the sional history. Their work showed that a small mass initial perihelion argument and longitude of nodes of the bodies belt best reproduced the observed number and types of fami- precess regularly, while this condition is violated for Tro- lies produced by disruption of parent bodies larger than jans. More sophisticated analytical models like that of 100 km diameter. Dell’Oro et al. (1998) or, alternatively, a direct numerical Vesta’s crust. Preservation of the basaltic crust of Vesta approach (Marzari et al., 1996) have been employed to de- requires that the mass of the asteroid belt was at most sev- termine the collisional parameters (Table 1). eral times the mass of the present belt at the time that the Armed with improved values of the impact parameters, present dynamical environment was established. This state- Marzari et al. (1997) numerically simulated the collisional ment assumes a power-law size distribution for the early evolution of Trojan asteroids over the age of the solar sys- asteroid population; it would be possible for the early aster- tem. They found that the slope of a primordial planetesimal- oid belt to contain significantly more mass than it does today like initial population truncated below 10 km in diameter if such mass were in a few massive bodies that were re- and with a steep slope (–5.5 incremental) gives a good match moved from the belt by dynamical, rather than collisional, to the present population (Fig. 7a). An initial distribution of processes. Work by Davis et al. (1984) and Davis et al. (1994) this kind would have been collisionally very active since it showed the range of initial asteroid populations that were is far from Dohnanyi’s equilibrium slope and, as a conse- consistent with the preservation of Vesta’s crust. quence, we would expect a large number of asteroidal fami- Cosmic-ray exposure ages of meteorites and the Gaspra/ lies to be produced over solar system history. Some of these Ida cratering record. Reconciling these constraints with families have been identified, even with a sample of only the overall asteroid size distribution remains a challenging 174 Trojan orbits by Milani (1993, 1994), and by Beaugé problem (O’Brien and Greenberg, 2001). Collisional models and Roig (2001), based on a set of 533 Trojans. Another that include Poynting-Robertson drag and the Yarkovsky ef- important aspect of the collisional evolution is the large flux fect, along with hydrocode-derived scaling laws, offer prom- of breakup fragments out of the Trojan swarms. As pointed 554 Asteroids III

107 (a)

106 108

5 (b) Known Population 10 107 Final Population 106 Initial Population 104 6× Current Mass 105 103 104

3 102 10

Number per Diameter Bin 102 101 Number per Diameter Bin 101

1 100 1 101 102 100 101 102 103 Diameter (km) Diameter (km)

Fig. 7. (a) Numerical simulation of the collisional history of Trojans. The continuous line is the starting population, the dotted line the evolved population, the open squares show the inferred population by Shoemaker et al. (1989), and the filled circles the known distribution of Trojans in 1997. The initial population is computed as a power law with a –5.5 slope from 250 to 20 km in diameter, then it has a rapid decline to 0 between 20 and 10 km. It is a planetesimal-like initial population where the accretion process produced bodies as large as 250 km from 10- to 20-km-sized primordial planetesimals. After 4.5 b.y. the evolved population reproduces well the present population. (b) Numerical simulation of the collisional evolution of Hildas where the collisional process with a planetesimal population scattered from the Uranus-Neptune region was taken into account. The filled circles show one possible initial population with 6× the current mass, the crosses the evolved population, and the stars the known distribution of Hildas in 2000. The initial population is computed as a power law with a –3 slope (incremental) and it is one case among other possibilities.

out by Marzari et al. (1995), the collisional disruption of a before escaping. Further work is needed to understand the Trojan asteroid can inject some fragments into unstable long-term effects on the size and orbital distribution of the orbits that end up in Jupiter-crossing chaotic cometary or- Trojan population. bits. A quantitative estimate of the actual flux based on the numerical simulation of Marzari et al. (1997) suggests that 4.2. Hildas the Trojan swarms could supply approximately 10% of the short-period comet and Centaurs populations. This dynami- The Hildas are objects that reside in the 3:2 mean-mo- cal connection between Trojans and short-period comets is tion resonance with Jupiter at ~4.0 AU. These asteroids are reinforced by the spectrophotometric similarity with D-type relatively isolated from physical interactions with objects asteroids, which are dominant among Trojans, and inactive of the main belt, and are weakly coupled to other outer- cometary nuclei (Hartmann et al., 1987; Shoemaker et al., belt asteroids such as Cybeles or Trojans. This is the reason 1989). their collisional activity is not very intense: The intrinsic col- The primordial Trojan population found by Marzari et lision probability for Hilda asteroids is lower than for main- al. (1997) showed only a possible lower limit to the real belt Cybele and Trojan asteroids by about a factor of 2.2 primordial population due to neglecting the depleting ef- and 3, respectively (Dahlgren, 1998; Dell’Oro et al., 2001). fect of dynamics on the population. As shown by Levison The low collisional activity is the result of two effects. et al. (1997), the stability region populated by present Tro- First, the 3:2 mean motion resonance with Jupiter has a 0.1- jans in the phase space is surrounded by isochrone curves AU-wide dynamically stable zone, surrounded by a strongly of limited lifetime. The primordial population of Trojans chaotic boundary with very short characteristic diffusion had then two independent sink mechanisms: collisions and times (Ferraz-Mello et al., 1998). Therefore, if any asteroid long-term dynamical diffusion. Collisions grind down larger is extracted from the resonance, it is quickly ejected far bodies to small fragments; dynamical diffusion ejects them away from the stable zone, no longer participating in the out of the swarm into chaotic orbits. However, these mecha- collisional evolution of the population. Then, if a fragment nisms also work in synergy: Collisions move the bodies produced by a collision reaches a relative velocity enough in the phase space, possibly refilling regions cleared up by to escape from the resonance, a depletion of the size dis- instability, and at the same time, bodies on unstable orbits tribution occurs (Gil-Hutton and Brunini, 2000). Assuming take part in the collisional evolution over a limited timespan that the relative velocity of a fragment is small with respect Davis et al.: Collisional Evolution of Small-Body Populations 555

to the orbital velocity of the parent body Vc, the semimajor belt observed populations are far from complete, but it is axis change is observationally testable: Any future survey of Hildas asteroids should not modify the currently observed size distribu- ∆V tion to a substantial degree. ∆a2a= T Vc 4.3. Interrelations with Other Populations which is a zeroth-order form of Gauss’ equation, where a is ∆ the semimajor axis and VT is the component of the rela- The existence of planetesimal populations is an integral tive velocity along the direction of the motion. Using a = part of planetary formation by accretion. Once the proto- 3.96 AU, and ∆a ~ 0.05 AU, and assuming that the relative planets grew to masses several tenths the current ones, they velocity of the fragment with respect to the parent body is started to scatter unaccreted bodies well beyond their accre- equally partitioned between the radial, tangential, and nor- tion zones. As a possible byproduct of this process, large num- mal components, the ejection velocity needed to escape bers of bodies from these populations diffused inward, reach- from the center of the resonance is ∆V > 0.163 km/s. Thus, ing the inner solar system, and a certain fraction of them small fragments produced by collisions closer to the bound- have eventually interacted with asteroids in the main belt. ary of the resonance or having large relative velocities can Collisional interactions between the asteroid belt and easily escape, depleting the Hildas’ size distribution and scattered planetesimals from the Uranus and Neptune zone producing a permanent loss of projectiles. were first considered by Brunini and Gil-Hutton (1998). The Second, the initial mass for the Hildas was not large. It possible presence of a nonnegligible amount of H and He is possible to obtain a rough value for it using the model of in the outer planetary nebula (Pollack et al., 1996) suggests planetary nebulae mass distribution proposed by Weiden- that the formation timescales of Uranus and Neptune could schilling (1977) to calculate the ratio between the total mass not have been much longer than the timescale of dissipa- of the main-belt and Hildas region. Another possibility is tion of the gaseous component of the nebula, perhaps not to assume that the number of objects currently observed is longer than a few 107 yr (Brunini and Fernández, 1999). A proportional to the initial mass, and find the ratio between significant number of scattered planetesimals reached the the number of objects larger than a certain radius that are region of the inner planets (Fernández and Ip, 1983), some observed in both populations. These methods give similar of which collided with asteroids in the main belt. Gil-Hutton results: The initial mass for the Hildas was ~22× less than and Brunini (1999) studied the collisional process between the initial mass in the main belt [between 1.5 × 10–3 and these populations and found that the planetesimal bombard- –5 3.6 × 10 M (Gil-Hutton and Brunini, 2000)]. ment was extremely efficient, removing mass of the primor- As a consequence of these effects, it is expected that dial asteroid belt in a very short time, allowing very massive there has been only a modest collisional attrition to the pri- initial populations. In spite of the fact that the collisional mordial size distribution. Nevertheless, the observed size process was very intense (a total mass infall of 15 M was distribution shows a change in the slope at a radius of scattered to the inner solar system from the Uranus-Neptune ~12 km, which was traditionally explained as an observa- zone), it lasted only 1–3 × 107 yr, so Vesta had a good chance tional bias due to the faintness of small objects in the outer of retaining its basaltic crust largely intact in this scenario. belt. However, it could be the result of a collisional pro- The interaction between the asteroid belt and the short- cess: Fernández and Ip (1983) and Brunini and Fernández period comet population was considered by Gil-Hutton (1999) found that the accretion process of Uranus and Nep- (2000). This author obtained values for the intrinsic colli- tune was extremely inefficient and a large number of bod- sion probabilities and collision velocities of 159 short-pe- ies was scattered into the inner solar system in a period not riod comets (SPC) colliding with the main belt using the longer than few times 107 yr (see section 4.3). Using this algorithm developed by Bottke et al. (1994; see Table 1). result, Gil-Hutton and Brunini (2000) found that collisions The intrinsic collision probability is about one-half that of between Hildas and scattered planetesimals from the Ura- main-belt collisions, but the mean impact speed is signifi- nus-Neptune zone could produce a large number of frag- cantly higher. However, SPC are not a serious threat for as- ments that could easily escape from the resonance. Thus, teroids due to their small population. the current size distribution of Hildas could be a result of an intense collisional process, which preferentially depleted 5. FUTURE CHALLENGES the small end of the population, leaving the Hildas without enough objects to have further collisional evolution. As a Asteroid collision studies made sizeable strides in the consequence of the fast depletion and low collisional ac- past decade, particularly with the increased volume of data tivity after the planetesimal bombardment, it is possible that on the asteroid size distribution and a better understanding the population of currently observed Hildas is complete or of fragmentation physics. However, there is still a wide dis- nearly complete at sizes >20 km in diameter (Fig. 7b), and crepancy in scaling-law prediction for the collisional energy the slope change in the size distribution could be the result needed to disrupt bodies. As numerical models become of this collisional process (Gil-Hutton and Brunini, 2000). more sophisticated and computers ever faster, new insights This result is contrary to the usual assumption that the outer are expected on this fundamental aspect of collisional studies. 556 Asteroids III

Recent results have been presented based on the mar- with that of main-belt asteroids. As in the case for main- riage of smooth particle hydrocode calculations of the frag- belt asteroids, one needs better knowledge about the size mentation process with an N-body integrator to follow the distribution of these populations down to kilometer-sized trajectories of fragments once the material interactions have bodies. Also, combining dynamical depletion mechanism ceased (Michel et al., 2001; Durda et al., 2001). Michel with collisional models will yield deeper insights into the et al. (2001) have successfully reproduced the observed size evolution of these populations. The next decade promises distribution for asteroid families with such an approach and, to show major progress in deciphering the collisional his- at the same time, have introduced a new paradigm for colli- tory of small body populations. sional outcomes. 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