Collisional Evolution of Edgeworth–Kuiper Belt Objects

ICARUS

125, 50–60 (1997) IS965595

ARTICLE NO.

Collisional Evolution of Edgeworth–Kuiper Belt Objects D. R. DAVIS Planetary Science Institute, 620 North Sixth Avenue, Tucson, Arizona 85705-8331 AND

P. FARINELLA Gruppo di Meccanica Spaziale, Dipartimento di Matematica, Universita` di Pisa, Via Buonarroti 2, 56127 Pisa, Italy Received February 20, 1996; revised July 10, 1996

theoretical evidence for a population of bodies beyond the region of the giant planets. Edgeworth (1949) and Kuiper (1951) argued for such a population based on the composition of comets and the implausibility of a sharp cutoff in the mass distribution of the proto-solar cloud at the distance of Neptune. Dynamical studies (Fernandez 1980, Duncan et al. 1988, Quinn et al. 1990, Levison and Duncan 1994) demonstrated that such a population must exist in the region beyond the outer edge of the known planetary system in order to explain the observed orbits of short period comets. This population is referred to here as Edgeworth– Kuiper Belt objects (EKOs).1 Based on the orbits and estimates of the total population, Stern, in a pioneering work (1995), showed that collisions occur frequently on the astronomical time scale in the Edgeworth–Kuiper Belt, implying that short period comets may not be primordial bodies. Such collisions could also generate dust clouds whose signatures are potentially observable in the IR (Stern 1996a). Our goal in this paper is to describe the collisional evolution of EKOs using the methodology developed for studying the collisional evolution of asteroids. The basic structure of our argument is:

The Edgeworth–Kuiper Belt contains a population of objects P103 times that of the main asteroid belt, spread over a volume P103 larger and with relative speeds P10 times lower. As for the asteroids, the size distribution of Edgeworth–Kuiper Belt objects has been modified by mutual impacts over Solar System history. We have modeled this collisional evolution process using a numerical code developed originally to study asteroid collisional evolution but modified to reflect collision rates in the Edgeworth–Kuiper Belt. Our numerical simulations show that collisional evolution is substantial in the inner part of the Edgeworth–Kuiper Belt, but its intensity decreases with increasing distance from the Sun. In the inner belt, objects with diameters D . 50–100 km are not depleted by disruptive collisions; hence they reflect the original (formative) population (many of them, however, may have been converted into ‘‘rubble piles’’). On the other hand, smaller objects are mostly multigenerational fragments, although the original population must have contained a significant number of bodies down to at least a few tens of kilometers in size in order to initiate a collisional cascade. About 10 fragments, 1–10 km in size, are produced per year in the inner Edgeworth–Kuiper Belt, with a few percent of them inserted into chaotic resonant orbits. This is in rough agreement with the required influx rate of Jupiter-family comets. Both collisions and dynamical instabilities associated with resonances are processes that can inject comets into the ‘‘escape hatches,’’ but our results indicate that most comets coming from the Edgeworth–Kuiper Belt would be fragments from larger parent bodies, rather than primitive planetesimals. However, this does not apply to Chiron-sized (D . 100 km) objects, which must be primordial and delivered to the outer Solar System by either dynamical processes or nondisruptive collisions.  1997 Academic

• Collisions occur frequently (on the astronomical time scale) among EKOs, based on the current understanding of the population and orbits of EKOs. • EKOs smaller than P100 km break up as a result of these collisions, generating a size distribution of fragments moving at speeds of tens to hundreds of meters per second relative to the original bodies. • A small fraction of the fragments are injected into

Press

1. INTRODUCTION

1

We adopt the terminology of Edgeworth-Kuiper objects in order to recognize the heretofore underappreciated contributions of K. E. Edgeworth to theoretical insights into the nature of the outer solar system. For a more complete account of Edgeworth’s career see MacFarland (1996).

The recent discovery of 1992 QB1 (Jewitt and Luu 1993) and its successors has dramatically confirmed the body of 50 0019-1035/97 $25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.

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dynamical resonances where they can be transported into the inner Solar System and become short period comets. • EKOs larger than P100 km in diameter are primordial bodies whose size distribution is the product of the formative process rather than a collisional process. Such bodies have likely been substantially cratered over Solar System history and many are probably rubble pile structures. The critical element in the above argument is the second one. We do not understand the physical structure of comets well enough to predict how they would respond during mutual collisions at hundreds of meters per second. Conceivably, such collisions would produce outcomes very different from those seen in laboratory experiments. But until such differences are demonstrated, we will assume an outcome widely observed in fragmentation events—both in laboratory experiments (Fujiwara et al. 1989) and in asteroid families (Chapman et al. 1989)—where a body breaks up into many fragments having a power-law size– frequency distribution and the fragments move with significant speed relative to the original target body. However, we will vary widely the main parameters of the collisional model (impact strength S0 , collisional energy partitioned into ejecta kinetic energy parameter fKE , and impact strength vs size scaling algorithm), reflecting our ignorance of the values appropriate for comets. The remainder of this paper is organized as follows: In Section 2 we summarize the observational constraints on our modeling work and their current uncertainties; Section 3 is devoted to a discussion of collision rates and impact outcomes in the Edgeworth–Kuiper Belt, including the similarities to and differences from the asteroid case; in Section 4 we present our main numerical results and discuss their sensitivity to the starting assumptions and parameters; and finally, in Section 5 we draw a number of conclusions relevant to the nature and history of both EKOs and short period comets and discuss some of the open issues on the topic of collisional evolution. 2. OBSERVATIONAL DATA AND THEORETICAL CONSTRAINTS

By early 1996, 32 objects have been discovered, in addition to the Pluto–Charon system, in the trans-neptunian region of the Solar System (Minor Planet Electronic Circular N03). These objects are between 100 and 350 km in diameter, assuming a geometric albedo of 0.04. The total population of EKOs is estimated at (1–3) 3 104 objects in the range p100–400 km in diameter at distances of 35–50 AU from the Sun, based on the total area searched to date and the number of discoveries (Jewitt and Luu 1995). The distribution of semimajor axes (a), eccentricities (e), and inclinations (i) is rather poorly known at this time, but for the observed objects, a, e, and i range between 35 and 48 AU, up to about 0.3 and up to about 238, respectively.

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The present size distribution of EKOs is the critical boundary condition for this study; i.e., hypothetical initial populations will be accepted or rejected based on the degree of agreement with the present population after several billion years of collisional evolution. Our estimate of the current population of EKOs as a function of size is given in Fig. 1 and is derived from three independent sources using different methodologies. The estimate of 2 3 104 bodies with diameters about 200 km is discussed above. The estimate of *108 bodies of about 20 km diameter is based on the estimate of the number of objects of this size orbiting near the 2 : 3 resonance reported by Cochran et al. (1995), based on HST searches. The total number of objects in this size range could be significantly higher; further work with HST is planned for late 1996. The number of bodies of about 2 km diameter, P5 3 109, is based on estimates of the number of short period comets together with their production and depletion rates (Holman and Wisdom 1993). Duncan et al. (1995) give an estimate of 1010 comet-sized bodies within 50 AU, consistent (within the uncertainties) with our estimate. The population of EKOs is substantially larger than the population of asteroids—there are probably nearly 10 billion objects larger than 1 km in diameter in the Edgeworth– Kuiper Belt, while there are only a few million asteroids larger than this size (Farinella and Davis 1994). Also, EKO eccentricities and inclinations, though poorly determined at this time, on average are probably lower than those for the asteroid population, since most moderately eccentric and inclined orbits are found to be unstable over the age of the Solar System (Knezˇevic´ et al. 1991, Morbidelli et al. 1995, Duncan et al. 1995). Instead of the 5.8 km/sec mean collision speed found in the asteroid belt (Farinella and Davis 1992), EKOs typically impact with speeds of several hundred meters per second. 3. COLLISION RATES AND COLLISIONAL OUTCOMES AMONG EDGEWORTH–KUIPER BELT OBJECTS

The collision rate within a population of bodies depends on two factors: (i) the orbital distribution of the bodies and (ii) their number as a function of size. We follow here the methodology for studying the collisional evolution of asteroids developed by Wetherill (1967), in which the orbital dependence is characterized by the intrinsic collision probability Pi , that is, the average collision rate per unit time and per unit cross-sectional area of the impacting bodies (apart from a factor f; see Wetherill 1967 and Greenberg 1982 for a complete analysis and some applications of this methodology). Pi is defined for any pair of non-resonant Keplerian orbits, and it vanishes when they cannot intersect each other. When an entire population of orbiting bodies is considered, the average value kPi l can be computed and used to estimate the prevailing collision

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FIG. 1. Observational constraints on the population of Kuiper belt objects at different sizes (see text). The uncertainty bars correspond to plus or minus a factor of 10 in the number of bodies and a factor of 2 in their sizes. The point for ground-based discoveries has an uncertainty bar of 6, a factor of 3 in the population, reflecting our assessment of the reduced uncertainty at this size.

rates, and the same algorithm allows one to derive the average collision speed kV l (Farinella and Davis 1992, Bottke et al. 1994). In the case of EKOs, we have derived these quantities by using four sets of 16 representative orbits each, at a 5 42, 45, 60, and 80 AU, with eccentricities and sines of inclination (sin i) of 0.01, 0.02, 0.05, and 0.12, corresponding to a mean value of 0.05. Thus for each of these orbits we derived kPi l and kV l, as shown in Table I. The average intrinsic collision probability in the inner Edgeworth–Kuiper Belt is a factor of about 2000 lower than the 2.85 3 10218 km22 yr21 value for the main asteroid belt (Farinella and Davis 1992), whereas collision velocities are typically just a factor of 10 lower. A comparison of the actual (as opposed to the intrinsic) collision rates between asteroids and those for EKOs is illustrative. The actual collision rate assuming non-resonant orbits is given by Cr 5 kPi l 3 (Rt 1 Rp)2 3 Np ,

(1)

where Cr is the collision rate for a body of radius Rt being impacted by Np projectiles of radius Rp (Wetherill 1967). Consider the impact rate of projectiles with diameter $1 km hitting a 100-km-diameter body. Using the kPi l value for a 5 42 AU, e 5 sin i 5 0.05, the collision rate is once

per 3.1 3 107 yr (assuming Np 5 1010). For comparison, in the asteroid belt a typical 100-km asteroid is hit by a projectile larger than 1 km in diameter (corresponding to Np P 3 3 106) every 4.5 3 107 yr. The substantially larger EKO population compensates for the much larger volume that they occupy, and thus, the collision rate for a given-sized body is comparable between the two populations; this was also noted by Stern (1995). However, the total number of disruptive collisions per year in the Edgeworth–Kuiper Belt is roughly 1000 times that in the asteroid belt, due to the much larger Edgeworth–Kuiper Belt population. Using the same collisional physics applied extensively in the past decade to the asteroids (see Davis et al. 1989 for a review), the outcome of a shattering collision (such that the largest fragment contains #50% of the mass of the original target body) between two EKOs is assumed to produce a size distribution of fragments whose properties (normalizing factor and characteristic exponent) depend on the specific collisional energy and the material properties of the impacting bodies. We will assume that EKOs (and comets) behave as rather weak bodies in terms of their dynamic impact strength, based on our understanding that comets are icy aggregate bodies and on the experimental data, limited though they are, relevant to such structures. It is important to realize that dynamic strength pro-

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TABLE I Average Intrinsic Collision Probabilities (in Units of 10221 km22 yr21) and Impact Velocities (in km/sec) for Hypothetical EKOs with Different Semimajor Axes (a), Eccentricities (e), and Sines of Inclination (sin i) a (AU)

e

sin i

kPi l

kV l

42 42 42 42 45 45 45 60 60 60 80 80 80 80

0.01 0.02 0.05 0.12 0.01 0.02 0.05 0.01 0.05 0.12 0.01 0.02 0.05 0.12

0.01 0.02 0.05 0.12 0.01 0.02 0.05 0.01 0.05 0.12 0.01 0.02 0.05 0.12

4.21 2.87 1.27 0.62 3.07 2.34 1.01 1.10 0.37 0.17 0.44 0.28 0.14 0.06

0.39 0.40 0.48 0.81 0.38 0.39 0.46 0.33 0.40 0.68 0.28 0.29 0.35 0.58

vides a different measure of strength than tensile strength, which is believed to be very small in comets, based on observations of their splitting and the breakup of Comet S–L 9 from very low tidal stresses. Impact experiments using weakly bound aggregate bodies (Ryan et al. 1991) found that specific energies on the order of a few times 106 erg/g are needed to shatter aggregate bodies. Impacts into ice bodies yield comparable values of the minimum shattering energy (Lange and Ahrens 1981, Kawakami et al. 1983, Cintala et al. 1985). Another assumption is that EKOs break up into a size distribution of fragments moving relative to one another. The speed of fragments is critical when the target body has a significant gravity field— fragments moving slower than the local escape speed reaccumulate to form ‘‘rubble pile’’ structures. In our model, fragment speeds are controlled by the parameter fKE , which specifies the fraction of the collisional kinetic energy that goes into fragment kinetic energy. The details of how comets break up and how fast the fragments move in response to a catastrophic collision are poorly known at present. We will vary the critical parameters of our fragmentation model to test the robustness of our conclusions to these uncertain quantities. However, the details of the fragmentation process are not critical to understanding the long-term relaxation of a population of colliding bodies. As shown by Dohnanyi (1969), Williams and Wetherill (1994), and Paolicchi (1994), a collisionally relaxed collisional population has a power-law size distribution with a fixed exponent (23.5 for the incremental form of the distribution), regardless of the details of how individual bodies in the population break up, provided (i) the fracturing process is size inde-

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pendent and (ii) there is no sharp cutoff at the small size end of the distribution (see Campo Bagatin et al. 1994 for a discussion of this point). Paolicchi has shown that even an extreme fragmentation case wherein large bodies break into only equal-sized smaller bodies leads to an equilibrium distribution consistent with the 23.5 power-law slope. Thus, as long as EKO fragmentation is consistent with the above constraints, as we expect it to be at the small size end of the distribution where gravitational effects are negligible, we are confident that the essential features of our models are correct. 4. NUMERICAL RESULTS

A numerical model developed for studying the collisional evolution of asteroids was modified for investigating the effects of collisions on the population of EKOs. This code has been described in earlier publications (Davis et al. 1989 and references therein), so only a brief overview will be given here. It uses a series of discrete diameter bins to represent populations of arbitrary size distributions and calculates the collisional interactions of each size bin with every other one during a small timestep. These interactions are summed at the end of the timestep to find the net change in the population as a function of size; the updated population is used for the next timestep. In this fashion, the time evolution of the population is found. The orbital element distribution of the population is assumed to be unaltered by the collisions—a good approximation when the orbital energy due to random motions (i.e., eccentricities and mutual inclinations) is large compared with the collisional energy needed to significantly alter the population. Or stated another way, the mean collision speed for two EKOs is larger than the typical velocities (relative to the center of mass) imparted to the fragments by the collision. This condition is met for both asteroids and EKOs. In either case, the energy available in the orbital motion is large enough to be treated as an infinite source, unchanged by withdrawals made to collisionally modify the population. The main modification to the collisional code made for the EKO problem was to use the intrinsic collision probabilities and speeds given in Table I. This code was then applied to find the collisional evolution of various hypothetical starting populations over 4.5 byr. Our results are not sensitive to the actual interval used; if it were 3.0 byr instead of 4.5 byr, reflecting the fact that the accretionary timescale was longer in the Edgeworth–Kuiper Belt region than in the terrestrial planetary zone, the results would be essentially the same, since most collisional changes occur early in the evolution. Also, different choices of collisional parameters were tested to establish how robust our conclusions were to the assumptions about collisional physics that are inherent in the model.

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TABLE II Model Parameters for the Baseline Scenario of Edgeworth–Kuiper Belt Collisional Evolution ● Initial population: Two-segment power law for the diameter (D) incremental distribution, with exponents 23.5 for D , 300 km and 220 for D $ 300 km, and with 13,000 bodies larger than 300 km ● Total initial mass: 3 M% ● Orbits: 42 , a , 46 AU, kel 5 0.05, ksin il 5 0.05 ● Density: 1.0 g cm23 ● Impact strength (for D 5 20 cm): 3.0 3 106 erg cm23 ● Strength scaling law: Energy scaling ● Fraction of impact energy partitioned into ejecta kinetic energy: 0.1 ● Exponent of the mass–velocity distribution of fragments: 29/4 ● Crater mass/projectile energy ratio for cratering impacts: 1028 g/erg

Model parameters and results for our baseline case are shown in Table II and Fig. 2a. The initial EKO population is assumed to be a power-law distribution at sizes smaller than 300 km and falls to zero for sizes between 300 and 500 km. The orbital distribution has a mean eccentricity and sin i of 0.05, consistent with results of dynamical studies showing that larger values, say 0.2 or higher, are unstable over the age of the Solar System except in special orbits (Knezˇevic´ et al. 1991, Morbidelli et al. 1995, Duncan et al. 1995). Impact speeds vary between 350 and 550 m/sec. As seen from Fig. 2a, the population at sizes larger than 100 km diameter is essentially unchanged over Solar System history, simply because at the relatively low impact speeds found in this population, even a collision between equalsized bodies cannot disrupt bodies larger than about 100– 150 km (about 5% of bodies in this size range have mutual collisions). However, at sizes smaller than 100 km, there is increasing collisional depletion with decreasing size; for sizes P20 km, the population is reduced by a factor of 10 from the initial population. The slope of the small size population is very close to 23.5, the equilibrium value for a collisionally relaxed population with size-independent collisional physics. Note that this result provides only a marginal match to the observational constraints of Fig. 1. With a 23.5 powerlaw exponent, the abundance ratio between km-sized ‘‘comets’’ and 10-km-sized HST-discovered bodies should be P300 (assuming equal logarithmic bins, as in Fig. 2) instead of P30, as suggested from the available data. However, the HST data are consistent with a power-law exponent in the range 23 to 25 (Cochran et al. 1995), which includes the equilibrium value. Actually, we do not believe the observations obtained so far provide more than just an estimate of the real population at different sizes, so we think it would be premature to draw any conclusion on the collisional physics—e.g., a possible size dependence of the collisional response parameters—from the comparison between our model runs and the data. We hope that this

will be possible when tighter observational constraints become available. Also shown in Fig. 2a is the number of bodies which survive from the original population—at sizes D P 20 km, about 50% of the current population consists of ‘‘survivors.’’ The survivor fraction increases with size, and for D . 80 km, essentially all of the bodies are survivors. Below P20 km, though, survivors are a minority of the population, and at sizes of a few kilometers, virtually all bodies are collisionally derived fragments of originally larger bodies. The same starting population and collisional parameters were rerun with the orbits pumped up to having mean e and sin i of 0.15; the results are shown in Fig. 2b. As expected, significantly more collisional evolution occurs in this case, with collisional depletion beginning at about 200 km and the transition between survivors and fragments being at about 50 km. We varied the starting population to see how sensitive the results were to the assumed initial conditions. A steeper initial population at small sizes (power-law exponent of 24.0, giving a starting population P12 times as numerous at a size around 2 km than our nominal case) again shows collisional depletion starting at about 100 km diameter, but depletion is much stronger (about a factor of 70) at sizes of D P 1 km. Again, though, the slope of the small size end of the distribution is 23.5. On the other hand, a shallow starting slope (23.0) yields less collisional depletion, only about 50% at D P 30 km, and actually gives a final population larger than the starting population by a factor of 2 at diameters of about 1 km. Also in this case the final population is collisionally relaxed with a slope of 23.5. We tested the robustness of our models to the critical fragmentation parameters (impact strength and its scaling with size and ejecta energy partitioning factor). First, we treated the case where EKOs are very weak by reducing the strength by a factor of 300 at laboratory sizes (D 5 20 cm) and assuming a strain-rate scaling law that implies a further decrease in strength with increased body size (see discussion in Davis et al. 1994). Here there is substantially more collisional depletion (and almost no survivors) at small sizes. The small size slope is 23.25, somewhat shallower than the usual collisional equilibrium value, because in this case we assumed a size-dependent impact strength (in agreement with the results of Davis et al. 1994). Next, we treated the case of very inelastic EKOs—i.e., a shattering collision produces a size distribution of fragments which move very slowly. For this case, we reduced the value of fKE from the baseline case by a factor of 10. As expected, this case shows less collisional depletion than does the baseline; significant depletion for the starting population occurs at sizes #20 km. About 50% of the bodies are survivors at D P 10 km. Finally, we combined the above

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FIG. 2. The collisionally evolved population of EKOs after 4.5 byr (asterisks), starting from a hypothetical initial population having a powerlaw size distribution with 23.5 index for diameters ,300 km (dash-dotted line). The observational constraints of Fig. 1 and the number of ‘‘survivors’’ from the original population (open circles) are also plotted. ‘‘Nominal’’ model parameters as given in Table II have been used to derive (a); in (b) we have increased the mean eccentricities and sin i to 0.15.

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FIG. 3. Rerun of the case illustrated in Fig. 2a except that the collisional parameters were changed to simulate collisionally weak, inelastic bodies. We used S0 of 104 erg/cm3, a size-dependent strain-rate scaling law (see Davis et al. 1994 for a detailed description of this algorithm), and an ejecta kinetic energy partitioning factor of 0.01.

changes to consider the case where EKOs are very weak bodies whose fragments move slowly following a disruptive collision. This case is illustrated in Fig. 3, which is similar to the results for the small fKE case alone at sizes $20 km, but has a much reduced small diameter population (it is down by a factor of 10 at sizes of 1 km), because fewer fragments escape gravitational reaccumulation. Radical changes in the starting population can result in a present population that is qualitatively different from those discussed above. For example, if the initial population had consisted only of large bodies hundreds of kilometers in size (e.g., because the formative process, either gravitational instability or accretion, was very efficient at consolidating small bodies into large ones), then there would not have been enough collisional evolution to generate the population of small bodies that supply the short period comet population. Again using a starting population close to the current one at large sizes and following a power law for D , 300 km, we can estimate that the initial smallsize slope had to be &2; otherwise, the current reservoir of short period comets would be too small. Hence at the end of the accretionary phase there must have been a sizable population of bodies down to at least a few tens of kilometers in size. However, all the smaller (kilometersized) bodies may well have been generated as fragments, as illustrated in Fig. 4. Here we show a case with all the

primordial bodies larger than about 10 km in diameter. Again, in the final population all the bodies larger than about 70 km are survivors, but the disruption of smaller bodies yields a ‘‘tail’’ of collisional fragments sufficient to supply the current comet reservoir. The amount of collisional evolution falls off significantly with distance from the Sun, as seen in Fig. 5. Figure 5a shows our nominal case but with a starting population and collision rate appropriate for heliocentric distances around 80 AU. The starting population is similar to that of Fig. 2, reflecting an assumed decrease in surface density (as a23/2; see Weidenschilling 1977), which is partially compensated for by a factor of 2 increase in the width of the heliocentric zone modeled (constant eccentricity but double the heliocentric distance). Relative speeds are also assumed to be lower by a factor of Ï2, and, as shown in Table, I the intrinsic collision probability is about one order of magnitude less than that at 40 AU (because of the larger volume available). In this case, the population is little changed over Solar System history; only at sizes below about 5 km is there a significant depletion of the original population. An intermediate case for 60 AU is shown in Fig. 5b; as expected, the results are intermediate between the 42 and 80 AU cases. As we go outward in the Edgeworth–Kuiper Belt, the size at which collisional effects manifest themselves by altering the original population

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FIG. 4. Example of a starting population close to the current one at large sizes, but containing no bodies smaller than about 10 km in diameter. Although the large bodies are never disrupted, a ‘‘tail’’ of small fragments is developed and may be sufficient to fill the comet reservoir adequately to explain the observed short period comet influx.

becomes smaller, leaving a wider size range as the signature of the accretional process. These results are in general agreement with those of Stern (1995, 1996b), who argued for a collisionally evolved zone in the inner Edgeworth– Kuiper Belt and a primordial zone further out. We would interpret this as a size-dependent transition between collisionally evolved and primordial bodies.

5. CONCLUSIONS

The main conclusions from this work can be summarized as follows: (1) The population of Edgeworth–Kuiper Belt objects larger than about 100 km diameter is not significantly altered by collisions over the age of the Solar System. The size distribution in this range must be the original (accretional) population. Many of these bodies, however, have likely been converted into rubble piles, as there is a significant energy gap between the projectile energy needed to shatter the target and that required to disrupt it, i.e., to disperse most of the target mass ‘‘to infinity’’ (Farinella et al. 1982, Davis et al. 1989). Using Eq. (1) with the kPi l values of Table I and the impact strength and initial population of Table II, it is easy to see that at 40 AU each 200-kmdiameter EKO is likely to undergo several shattering (al-

beit not disrupting) collisions over the Solar System’s lifetime. It is worth remarking that the importance of selfgravitational effects may explain why, according to Jewitt and Luu (1995), the observed size distribution for 100 , D , 400 km is relatively flat—by analogy with asteroids, in the size range where gravity becomes important a flattening of the size distribution occurs, with most objects converted by impacts into reaccumulated rubble piles. (2) Objects &50–100 km diameter currently present in the EKO population are mostly fragments undergoing a collisional cascade, with a size distribution index close to the 23.5 equilibrium value. However, their current abundance implies that there was a substantial original (accretionary) population down to a few tens of kilometers in size; otherwise, the EKO reservoir of comets would be too small to produce the present short period comet population. On the other hand, for initial values of the size distribution index &22 at diameters less than 300 km, the final population is insensitive to initial conditions, reflecting collisional relaxation to the 23.5 equilibrium value. If future observations (e.g., by HST) show that the current index has a significantly different value in the size range 1–50 km, this will probably require an interpretation in terms of size-dependent collisional response parameters. (3) In the outer Edgeworth–Kuiper Belt (around 80 AU), collisions have changed the starting population very

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FIG. 5. (a) Results of a simulation of the (limited) collisional evolution in the outer Edgeworth–Kuiper Belt (around 80 AU). This case treats a zone of width 8 AU and corresponds to a starting population consistent with a fall-off in surface mass density Ya23/2 (total mass of about 2.2 M%). The largest size bodies are arbitrarily set to be 500 km; however, a smaller upper size limit would not materially affect these results. (b) Same as (a) but for a mean distance of 60 AU, with a total mass intermediate between that of Figs. 2 and case (a).

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little, even at sizes as small as a few kilometers. Therefore, this region may be the ideal target for future observations aimed at discovering the size distribution of the planetesimal population produced by gravitational instability and/or accretion at the outer periphery of the Solar System. (4) As a by-product of the collisional process, about 10 fragments per year 1 to 10 km in size are currently produced in the Edgeworth–Kuiper Belt near 40 AU. This estimate refers to the baseline case of Fig. 2, with a factor of about 4 variability, depending on the assumed collisional response parameters. With ejection speeds of 10–100 m/ sec, similar to those inferred for asteroids, these fragments have semimajor axes about 0.1–1.0 AU different from those of their parent bodies. This is sufficient to cause at least a few percent of these fragments (say, 0.2/yr) to fall into the resonant ‘‘escape routes’’ from the Edgeworth– Kuiper Belt (Duncan et al. 1995, Morbidelli et al. 1995) and to chaotically evolve into the planetary region of the Solar System. This is roughly in agreement with the flux required to replenish the short period comets, which have an estimated population (including extinct/dormant nuclei) of 2 3 104 bodies and an average dynamical lifetime of about 3 3 105 yr (Levison and Duncan 1994). Thus, about 0.06 comets per year are needed to maintain this population. If one-third of the Edgeworth–Kuiper Belt fragments that fall into a resonant escape route become short period comets, this is adequate to maintain the population. Therefore, EKOs’ collisions are a sufficient mechanism to supply the short period comets, just as collisions in the main asteroid belt can supply near-Earth asteroids and meteorites. (5) However, disruptive collisions cannot explain how large bodies such as Chiron and other sizable members of the so-called Centaur population originate. Collisional fragments cannot be as large as 100 km or more, and indeed the largest planet-crossing asteroid, 1036 Ganymed, which is presumably a collisional fragment, is about 40 km across. A possibility is that a purely dynamical delivery mechanism, such as slow diffusion from the vicinity of the resonance zones (Holman and Wisdom 1993, Levison and Duncan 1993, Duncan et al. 1995), is at work. This does not entirely solve the puzzle of the origin of Chiron-like giant comets, however, since their dynamical lifetime is 105 to 106 yr (Hahn and Bailey 1990). Unless we are living in an unusual situation, at least some 104 Chiron-like bodies must have been supplied over Solar System history. This appears to be too many given the current EKO population (a few times 104 bodies of this size; see Section 2), unless the region where slow instability mechanisms are active is much wider than currently thought. An alternative mechanism for the insertion of big Centaurs into the delivery routes is through non-disruptive collisions: a 1022 projectile-to-target mass ratio is not enough to disrupt a 100-km-

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sized EKO, but can change its orbital velocity by several meters per second, hence its semimajor axis by P0.2% P0.1 AU. Such events may be frequent enough to provide a significant influx of Chiron-sized bodies. (6) The large changes in semimajor axis resulting from breakup of a large EKO mean that it will be very difficult to distinguish EKO dynamical families from background objects, even when a large catalogue of well-determined orbits becomes available. A small relative speed goes a long way in the outer Solar System in terms of changing orbits. On the other hand, this means that collisions far from the resonant escape hatches can inject fragments into them, so we can sample a fairly large region of the phase space for producing comets. (7) The result that most short period comets are collisional fragments (independent of the mechanism which causes their orbits to get in the planetary region) means that these bodies may have experienced some type of alteration in the interior of their parent bodies. At a minimum, there would be a modest compacting effect, due to the gravitational self-compression within a parent body’s interior. For example, at a depth of P10 km (the median mass depth in a 100-km-diameter body), the overburden pressure is about 1.5 bars (for r 5 1.0 g/cm3). Such a pressure acting over astronomical timescales could produce ‘‘cold welding’’ of the material, resulting in a nonzero tensile strength of the body. This could explain why all comets do not split when they experience thermal or tidal stresses close to the Sun or Jupiter. Interestingly, fragments from a wide range of sizes of parent bodies contribute to the population at a given size. Hence a 5km fragment may come from a 10-km parent body or a 100-km parent body and may thus exhibit very different physical properties. This might help to explain the wide diversity of behavior exhibited by comets. There is other independent evidence for collisional processing of short period comets: (i) their irregular, triaxial shapes (Jewitt and Meech 1988) resemble those of fragments produced in breakup events (see, e.g., Capaccioni et al. 1984); and (ii) the variety of colors observed among Centaur and Edgeworth–Kuiper Belt objects can be interpreted as the result of a varying degree of collisional alteration and/or resurfacing (Luu and Jewitt 1996). Thus, if short period comets come from the Edgeworth–Kuiper Belt, our collisional modeling indicates that most of them are not the primitive, ‘‘original’’ accretion products that are the current standard model for cometary structures. The next step in understanding the evolution of Edgeworth–Kuiper Belt objects would be to combine the two dominant processes acting on this population—collisions and dynamical perturbations—into a unified model. This would include time-dependent eccentricities and inclinations due to outer planet secular perturbations and population depletion due to orbital instabilities.

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DAVIS AND FARINELLA

ACKNOWLEDGMENTS We are grateful to H. Levison, A. Milani, A. Morbidelli, E. M. Shoemaker, A. Stern, and S. J. Weidenschilling for useful discussions and comments. A. Diaz provided valuable help in running the collisional evolution code and plotting the results. P.F. acknowledges financial support from the Italian Space Agency (ASI) and the Italian Ministry for University and Scientific Research (MURST). This is PSI Contribution 333.

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