Fragmentation rates of small satellites in the outer solar

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 5, NO. E7, PAGES 17,589-17,599, JULY 25, 2000 Fragmentation rates of small satellites in the outer solar system Joshua E. Colwell, Larry W. Esposito, and Danielle Bundy Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder Abstract. The narrow rings of Uranus and Neptune exist in a system of observed and hypothesized small moons. Catastrophic fragmentation of these moons by comet impact has been proposed as the mode of origin of those rings, and earlier efforts to model the process showed that small moons are destroyed by impact on short timescales, leading to rapid collisional erosion of any primordial satellite system (Colwell and Esposito, 1992). We reexamine the question of impact fragmentation of small satellites in the light of new observational data on the population of Kuiper Belt and Centaur objects that produce the impacting flux and new theoretical and computational studies of catastrophic fragmentation. We find that the impacting flux used by Colwell and Esposito (1992) is consistent with the new observations of Kuiper Belt objects and calculations of their transport into the solar system. However, new fragmentation criteria from modeling of the asteroid belt and hydrocode simulations lengthen the model collisional lifetimes of satellite systems. The observed distribution of rings, dust bands, and moons at Uranus and Neptune suggest a catastrophic disruption model with a relatively weak dependence on target radius. 1. Introduction Narrow planetary rings have short lifetimes as a result of spreading due to energy dissipation from collisions. The narrow tings of Uranus orbit within Uranus's hot exosphere, which exerts a drag force on ring particles, leading to orbital decay. This results in an upper age limit on the most massive of Uranus's rings, the œ ring, of-600 million years [Esposito and Colwell, 1989]. The other rings of Uranus reside deeper in the exosphere and are less massive, leading to even shorter lifetimes. Neptune's Adams ring contains arcs of material whose distribution in longitude suggests a single, recent formation [Esposito and Colwell, 1992]. Both ring systems contain abundant dust which is derived from a population of low optical depth moonlets or parent bodies [Smith et al., 1986, 1989; Colwell and Esposito, 1990a, b]. Catastrophic fragmentation of small moons has been proposed as a source for these short-lived ring structures [cf. Colwell, 1994]. Colwell and Esposito [ 1992, 1993] (hereinafter referred to as papers 1 and 2) studied the rate of satellite fragmentation and ring creation from external bombardment by comets. At that time much less was known about the population of cometary objects in the outer system. Using the information available, they modeled the fragmentation of satellites at Uranus and Neptune and found rapid collisional erosion of moons smaller than 0 km in radius. These papers describe the "collisional cascade" from moons to moonlets and from moonlets to rings due to catastrophic fragmentation. These models indicated that the observed small satellites of both planets are in the final % of their existence and that the primordial satellite populations must have been much more massive than the present-day populations. They suggested partial reaccretion (paper 2) and a lower impactor flux as Copyright 2000 by the American Geophysical Union. Paper number 1999JE /00/1999 JE $ ,589 possibilities for lengthening the collisional lifetime of the small moons. The discovery of objects in the Kuiper Belt (the source of the hypothesized satellite-smashing comets in papers 1 and 2) and new computer simulations of the evolution of Kuiper Belt objects into the planetary region provide a much more accurate assessment of the impacting flux at Uranus and Neptune than was possible when papers 1 and 2 were written. In this paper we reexamine the collisional evolution of satellites at Uranus and Neptune with a more realistic flux model and new models of the fragmentation process. We find that the flux of impactors adopted in papers 1 and 2 is virtually indistinguishable, within the uncertainties, from the flux derived from new observations and computer simulations. Furthermore, a detailed study of the reaccretion of disrupted satellites has shown that for the innermost small moons of Uranus and Neptune, reaccretion is entirely negligible [Canup and Esposito, 1995]. However, there is considerable variety in estimates of the collisional strength of 1-0 km radius objects. A recent strength model [Durda et al., 1998] derived from a collisional model of the asteroid belt and strength models from hydrocode simulations [Benz and Asphaug, 1999] lead to significantly longer collisional lifetimes of small moons in the outer solar system. In section 2 we describe our impacting flux and fragmentation rate results. In section 3 we provide new simulations of the satellite size distribution as a function of time, and in section 4 we discuss our results and their implications for ring and satellite origins. 2. Impacting Flux and Fragmentation Rates We are concerned here with the flux of comets large enough to disrupt moons which can result in a planetary ting: moons larger than about 1 km in radius for most of the Uranian and Neptunian rings. We derive an impact flux from the combination of observed Kuiper Belt objects and subsequent numerical integrations of the dynamical transport of those objects from the Kuiper Belt into the planetary region.

2 17,590 COLWELL ET AL.' SATELLITE FRAGMENTATION RATES lx lx -28-3O lx -32 x lx -34 lx -36 lx x -41 i Figure 1. A comparison of the model interplanetary impactor flux at Uranus and Neptune used by Colwell and Esposito [1992, 1993] (straight line) with a flux based on improved modeling and observations of Kuiper Belt objects which become planet-crossing impactors (line with slope change). Over most of the range of comet masses important for fragmentation of small planetary satellites, the fluxes agree to within a factor of a few. (g) Levison and Duncan [1997] (hereinafter referred to as LD97) performed numerical integrations of the orbits of Kuiper Belt objects and followed them into the inner region of flux of comets to get impact rates on the satellites, our results are insensitive to our precise choice of V o. Varying our choice of the eccentricity of the comets from 0.1 to 0.5, for example, the solar system, where they may be detected as short-period results in an increase of the comet flux onto the Uranian moon or Jupiter family comets. These numerical experiments are constrained in the inner solar system by the observed number Puck of only 14%. The variations are smaller for satellites closer to the planet. The flux of comets at a satellite is of comets and in the Kuiper Belt region, where the simulations F,,=F,/fp--FpRe/2as, where Fp is the flux at the planet, begin, by the statistics of the observed Kuiper Belt objects. f,=l+l/2(vo, c/voo 2 is the gravitational focusing factor for LD97 track the number of comets which impact each of the giant planets in their simulations and scale appropriately to get the rate of comet impacts on each planet. We use these impact satellites [Morrill et al., 1983; Colwell, 1994], Rp is the planet radius, a., is the satellite orbit radius, and f. and fp are >> 1. (In the expression for f,,, Vo, c refers to the escape velocity at as.) rates to calculate the flux of comets onto the small moons of This use of the results of LD97 and JH97 gives us the flux of Uranus and Neptune. A similar technique was used in papers 1 and 2, but at that time there were neither observations of Kuiper Belt objects nor extensive numerical integrations of the dynamical evolution of comets through the outer solar system. The number of comet impacts on the planets in the comets larger than 1 km in radius. The normalization to this size comet is still subject to considerable uncertainty, corresponding to an equivalent uncertainty in derived fluxes at a given impactor size of as much as a factor of (H. F. Levison, personal communication, 1999). Furthermore, simulations of LD97 is enhanced by gravitational focusing. Levison et al. [2000] recently reevaluated the LD97 planetary Because we are interested in impacts on the moons, we must impact rates and found that those values were too high by a first remove the effects of planetary gravitational focusing to factor of-2-4. We use the LD97 values below but point out arrive at an approximate unfocused interplanetary comet flux that our fragmentation and cratering rates would be decreased at Uranus and Neptune. We then apply the gravitational by this same factor of 2-4 using the Levison et al. [2000] focusing appropriate for each satellite. The planetary results. gravitational focusing factor fp depends on the approach Observation suggesthat the size distribution of comets is velocity of the cometo the planet, Voo, fp=l+(vo c/voo)2, which a two-component power law, with a break in the slope of the in turn depends on the orbital elements of the comet and the power law near km in radius [Weissman and Levison, planet. Jedicke and Herron [1997] (hereinafter referred to as 1997]. The lower and upper size cutoffs of the size JH97) give orbital element distributions of Centaurs (comets distribution are not certain. With the power-law size crossing the orbits of Saturn, Uranus, and Neptune) on the distribution of comets, the smallest comet which can destroy a basis of the integrations of LD97. We approximate the moon is also the most likely size comet to destroy a moon. gravitational focusing factor by using the mean of the distributions from JH97 and computing an average V. We used an eccentricity for the comets of ei=0.2 (LD97, JH97) and Therefore the upper cutoff will not affect our results as long as it is greater than the size of the smallest comet which can destroy a 0 km radius moon, or about km in radius. The approximated their velocities prior to gravitational lower size cutoff, however, can be important for the disruption acceleration by the planet by letting Voo=eiVp, where Vp is the planetary orbit velocity. Because we next partially refocus the of subkilometer-sized moonlets if the cutoff is larger than - m in radius (section 3). In Figure 1 we show our derived

3 COLWELL ET AL.' SATELLITE FRAGMENTATION RATES 17,591 Table 1. Estimated Cratering Rates Satellite Oberon Titania Umbriel Ariel Miranda Puck Belinda Rosalind Portia Juliet Desdemona Cressida Bianca Ophelia Cordelia Nereid Triton Proteus Larissa Despina Galatea Thalassa Naiad P(> _k2m). ' P(> km) (- 2 km yr I (- 2 km -2 Yr - ) CE92 ZDL98 Uranus Neptune Note: Cratering rates are number of craters with a diameter > km in units of - km' yr-. CE92 and ZDL98 refer to two different sets of crater scaling parameters. See text. comets roughly 1- km in radius (4x s- 8 g). Given the remaining uncertainties in the flux calculation, this agreement is partially fortuitous. Nevertheless, this suggests that a significantly lower impactor flux is not a resolution to the dilemma raised in paper 1 (namely, that satellite fragmentation proceeds at such a rapid rate that the moon systems are near the end of their observable existence). However, for the smallest comets the new flux is smaller by nearly a factor of, so a detailed study of the effects of the flux is included here. Overall, we find that the difference in flux has a negligible effect because fragmentation of the largest satellites controls the rate of the collisional cascade. These satellites are disrupted by the flux of larger comets, which is very little changed from the model of papers 1 and 2. Furthermore, the possibility that limited accretion may slow the collisional cascade can also be ruled out for the small ring moons at Uranus and Neptune. In paper 2 we used a two-body escape velocity criterion for dispersal of moon fragments, which significantly slowed the collisional cascade. However, a more detailed study of reaccretion of fragmented satellites in the Roche zone, including tidal effects, found that "accretion is not possible at the current orbital locations of Cordelia, Ophelia, Naiad, Thalassa, and Despina" and that "reaccretion alone cannot end the quandary over the existence of the current populations" of ring moons [Canup and Esposito, 1995, pp. 347, 348]. Therefore we conclude that for the moons to have a longer collisional lifetime they must be more resistanto disruption than was assumed in paper 1. Before addressing fragmentation, we estimate cratering rates with the new flux model described above and compare them to our earlier results. Crater diameter is a function of impactor mass and velocity and target properties. We use one crater-scaling law with two sets of scaling parameters. In paper 1 we used D= 12.6R'l d -l Vi 2l cm (1) [Schmidt and Holsapple, 1982], where D is crater diameter, R is target radius, d is impactor diameter, Vi is the impact speed, and the scaling constant [3 was taken to be 1/6. Unless stated otherwise, all units are cgs. In (1) and throughout we use an impactor density of 1.0 g cm -3 and a target (satellite) density of 1.5 g cm -3. A slightly different scaling with [3=0.22 and a leading constant of 22.2 was used by Zahnle et al. [1998] (hereinafter referred to as ZDL98) after Schmidt and Housen [1987] in a (more detailed) study of cratering rates on the moons of Jupiter, also using the numerical integrations of LD97. In Table 1 we present cratering rates on the moons of Uranus and Neptune with the two sets of crater-scaling constants given above. We compute the cratering rates by generating 3 comet orbits from the orbital element distributions presented in JH97 and using the technique of paper 1 from Shoemaker and Wolfe [1982] with the flux presented in Figure 1. For comparison, we use the crater scaling from paper 1 and from ZDL98. That technique involves computing the number of craters produced by a specified size comet (2.53 km diameter; cf. paper 1) and interplanetary fluxes with different assumptions about the extrapolating those to a km diameter crater using an location of the break in the size distribution and the fluxes assumed crater size distribution. Results in Table 1 are used in papers 1 and 2 for comparison. Given the uncertainty presented as the rate of production of craters larger than km inherent in our remaining calculations on crater sizes and disruption rates, the current fluxes and those from papers 1 and in diameter. (For the smallest observed moons, i.e., those with R- km, this can be regarded as a crude estimate of the 2 are virtually indistinguishable, with the newer flux actually fragmentation rate.) Using the new flux and the cratering rate slightly higher than the nominal flux used in papers 1 and 2 for calculation from paper 1, the results are virtually unchanged from those in paper 1 (compare Table 1 first column with paper 1 Table 2). The numbers in Table 1 provide the variation in the cratering rate between satellites and can be compared to results in paper 1 and crateri,ng rates on moons of other planets, but the uncertainties in the absolute rates are at least as large as the variation between the different columns. Nevertheless, the results are consistent with, or well above, those estimated in paper 1, suggesting that the small moons are heavily bombarded and that fragmentation may occur on timescales short enough to provide a source for the narrow rings of Uranus and Neptune. This result is not surprising given the small change in the flux model and also shows that other crater-scaling parameters provide even larger cratering rates than those in paper 1. In the next section we model the fragmentation process. 3. Evolution of Satellite Size Distribution To model the collisional cascade, we use the Markov chain approach described in paper 1. The Markov chain simulations follow the evolution of the fragment size distribution and transport mass from larger objects to smaller at rates determined by their fragmentation probabilities and the model

4 17,592 COLWELL ET AL.' SATELLITE FRAGMENTATION RATES x { 2.0x 8 3.0x {5 4.0x 8 Time (years) Figure 2. The average of 300 Monte Carlo simulations of the evolution of the size of the largest fragment from a series of catastrophic fragmentations by comet impact using the new flux model described in the text and presented in Figure 1. The simulations are for the Uranian moon Cordelia (Table 2); the fragmentation model is the same as was used by Colwell and Esposito [1992] (compare to their Figure 6). fragment size distribution. Paper 2 introduced dependencies satellite size distribution are the resistance of the satellites to' on the satellite fragment velocity distribution, but because catastrophic fragmentation and the size distribution of the reaccretion is not important for the innermost moons, which fragments produced in such a fragmentation. are the hypothesized precursors to planetary rings [Canup and We use the same fragment mass distribution as was used in' Esposito, 1995], we do not include any fragment velocity- paper 1, based on the results of ground-based hypervelocity dependent effects here. In addition to the flux of impactors impact experiments [Davis and Ryan, 1990]. We use the described above, the important factors in the evolution of the impactor flux as described in section 2, with a lower size o 9 8 _. ' ' Housen et al, (1991) Durda et al. (1998) Benz & Asphaug (1999) , ,0 Redius (km) Figure 3. Three models for Q*, the specific impact energy necessary for catastrophic fragmentation. The solid line is a strain-rate scaling theory from Housen and Halsapple [1990], with additional strengthening the gravity regime based on experiments by Housen et al. [1991]. The dashed line is an empirical model from Durda et al. [1998] derived to allow numerical simulations of the collisional cascade of the asteroid belt to match the observed size distribution. The dash-dot line is the result of a hydrocode model for impacts into ice at 3 km s" from Bens and Asphaug [1999].

5 _ - _ - _ COLWELL ET AL.' SATELLITE FRAGMENTATION RATES 17, g! i [,! i trc dius (krn) Plate 1. Disruption probabilities for a satellite at the location of the Uranian moon Cordelia as a function of satellite size, for different disruption models and fluxes: black, disruption model of paper 1, smallest comet radius of m; blue, same as black, but with Q* increased by a factor of 20 in the gravity regime following Housen et al. [1991]; yellow, same as black, but with a smallest comet rad(.u.s of 1 m; gray, same as blue, but with a smallest comet radius of 1 m; red, same as blue, but 5 times stronger in the strength regime; green, using Durda et al. [1998] asteroid model for Q* and smallest comet radius of m; purple, using Benz and Asphaug [1999] hydrocode model for impacts into ice at 3 km s '. At radii larger than 1 km the black and yellow curves are coincident. At radii smaller than 1 km the yellow and gray curves are coincident. E 15 t... I... ]... I r 0 1.0x g 2.0x g 3.0x g 4.0x g Time (years) Plate 2. A comparison of Monte Carlo simulations of the collisional cascade for different disruption models and fluxes. Simulations are for the Uranian moon Cordelia. Color key is the same as for Plate 1. The black and yellow curves are nearly coincident. The blue and gray curves are nearly coincident.

6 17,594 COLWELL ET AL.: SATELLITE FRAGMENTATION RATES cutoff to the impacting population, Rmi n. In papers 1 and 2 we Neptune is -15 km s 'i There is no reason a priori to suspect assumed a continuous size distribution of impactors between that these Q* models are appropriate for planetary satellites, kilometer-sized comets and micrometer-sized dust. The but they provide suitable end-members t,o test the dependence fragmentation rates of kilometer-sized moonlets depend of the ring-satellite collisional cascade on disruption models. strongly on this assumption, and the population of very small comets is not known. A Monte Carlo simulation of the evolution of the largest fragment at Cordelia, using the asteroicl model Q*, increases In paper 1 we used an expression for Q*, the specific impact the satellite lifetime by more than a factor of 5. After 4x9 energy necessary for fragmentation, based on the strain-rate years the mean largest fragment size from the Monte Carlo scaling theory of Housen and Holsapple [1990]. There is simulations is 2 km in radius. The hydrocode model is even significant variation between various expressions for Q* based stronger and results in a disruption lifetime for Cordelia which on numerical modeling, hydrocode simulations, and scaling exceeds the age of the solar system. Such a large value for Q* theory [cf. Holsapple, 1993; Durda et al., 1998; Benz and is difficult to reconcile with the creation of narrow planetary Asphaug, 1999]. In order to reproduce the observed asteroid rings from satellite disruption. This discrepancy may be size distribution with numerical simulations of the asteroid because the hydrocode model is for Qt *, the specific impact collisional cascade, Durda et al. [1998] created an empirical energy needed to fragment an object and disperse the prescription for Q*. In what follows we present results on fragments to infinity. In the Roche zone the relevant satellite fragmentation using the scaling theory of Housen and fragmentation threshold probably lies closer to Qs*, the Holsapple [1990] and Housen et al. [1991], the empirical threshold for shattering with no dispersal of fragments. expression from Durda et al. [1998], and hydrocode (Because it is based on disruption of asteroids, the asteroid simulations from Benz and Asphaug [1999]. We will refer to these as the scaling theory model, asteroid model, and hydrocode model, respectively. Because the improved impactor flux model used here is model is also a model of Qt *.) In Plate 2 we show the effect of different disruption models on the evolution in time of the largest surviving fragment from Monte Carlo simulations. Lifetimes against disruption for the similar to that used in paper 1, our new results with the scaling small moons were computed using the scaling theory model theory expression for Q* confirm our conclusions from paper [Housen and Holsapple, 1990; Housen et al., 1991] and the 1. To illustrate this, Figure 2 shows the result of a Monte asteroid model [Durda et al., 1998]. These are presented in Carlo simulation of the evolution of the largest surviving Table 2 and are the inverse of the disruption probabilities fragment from successive catastrophic fragmentations, using computed from the impacting comet flux and the resistance of the new flux presented here with Rmin = 0.01 km and the strain- the satellites to disruption parameterized by Q*. The lifetimes rate scaling for Q* adopted in paper 1. This simulation of the derived from the hydrocode model exceed the age of the solar collisional erosion of the Uranian moon Cordelia is analagous system for all moons larger than km in radius. Because the to that in Figure 6 of paper 1, and the results are virtually indjstinquishable. With this flux and disruption model the radius of the largest surviving fragment is reduced to <1 km in asteroid model for Q* increases more rapidly than the scaling theory model in the gravity regime, where all of the observed moons are, the difference in satellite lifetimes between the two 9 years. models is larger for larger moons. Both models predict The disruption model adopted by Durda et al. [1998] to match the asteroid size distribution is shown in figure 3 along with the scaling theory model of Housen et al. [1991] and a collisional lifetimes well below the age of the solar system for the small shepherd satellites of Uranus's e ring, Ophelia and Cordelia. Unseen but hypothesized smaller ring moons within model based on hydrocode simulations [Benz and Asphaug, the ring system would have even shorter collisional lifetimes, 1999]. The asteroid model is significantly stronger for moons at least down to a radius of --1 km, where the potential lower larger than a few kilometers in radius and significantly weaker for the smaller moons. The hydrocode model is nearly as size cutoff of the impacting comet flux becomes important. Smaller moonlets, which are potential sources of the dust strong as the asteroid model for large moons and stronger for bands seen between the rings at Uranus and Neptune [Smith et smaller moons. In Plate 1 we show the effects of the various al., 1986; paper 1; paper 2], could have longer collisional models for Q* on the disruption probability per year at the lifetimes owing to their smaller cross sections if the comet size orbit of Cordelia and as a function of satellite radius. The disruption rate of moons smaller than --1 km in radius is sensitive to the assumed lower size cutoff of the impacting flux. At sizes larger than -- km in radius, the various models show a varation in disruption probability of one to two orders of magnitude. The size distribution of small moons in the outer solar system is not as well constrained as the asteroid distribution rolls over near 0 m in radius (ZDL98). Next we show the results of Markov chain simulations of the satellite collisional cascade using the new flux model and the new models of Q*. In Plates 3 and 4 we show the evolution of the entire fragment size distribution from Markov chain simulations (cf. paper 1) using different disruption models. The collisional cascade size distributions in Plates 3 size distribution, so it would not be possible to arrive at a and 4 are expectation values or averages over an ensemble of unique solution for Q* from the observed populations. Instead, systems. Because they follow the evolution of fragments from we next use these new models to examine the effects of this a single precursor satellite, the distributions are predictive of a disruption model on the satellite collisional cascade. The particular size distribution only in the case where the Durda et al. model was derived by performing numerical expectation value for the precursor satellite is <<1. experiments on presumed initial asteroidal populations and Different models of Q* produce qualitatively different selecting the functional dependence of Q* on asteroid radius results for the predicted evolution of the collisional cascade of which best reproduced the present-day asteroid size moon fragments. In Plates 1-4 all disruption models are the distribution. The hydrocode model we use is based on Housen et al. [1991] strain-rate scaling model except for the simulations of impacts into ice at 3 km s -. For comparison, green curves, which are for the asteroid model, and the purple the typical impact speed of comets onto moons of Uranus and curves,,which use the hydrocode model. For the five

7 COLWELL ET AL.' SATELLITE FRAGMENTATION RATES 17, c (o) i i R dius (km) OO.OO c- 1oo.oo.OO 1.oo O. O.Ol i i I i I,, 1 O Radius (km) Plate 3. A comparison of Markov chain simulations of the collisional cascade for different disruption models and fluxes. Simulations are for (a) the Uranian moon Cordeliand (b) the Neptunian moon Naiad, after l0 9 years of bombardment by a constant flux of impactors. Values are the expectation values for the number of objects of each size, based on a starting population of one moon identical to the current satellite. See text. Color key is the same as for Plates 1 and 2. The black and yellow curves are nearly coincident. The blue and gray curves are nearly coincident.

8 17,596 COLWELL ET AL.: SATELLITE FRAGMENTATION RATES O.OO.OO (o) 1.oo o.lo O.Ol i i I 1 I I I I I I o Radius (km) O! i i i!!!! I i!!,,,! i (-- 4 (b) O Radius (km) Plate 4. A comparison of Markov chain simulations of the collisional cascade for different disruption models and fluxes. Simulations are for the Utanian moon Puck after (a) 9 and (b) 4xl 09 years of bombardment by a constant flux of impactors. Values are the expectation values for the number of objects of each size, based on a starting population of one moon identical to the current satellite. See text. Color key is the same as for Plates 1-3. The black and yellow curves are nearly coincident. The blu'e and gray curves are nearly coincident.

9 COLWELL ET AL.: SATELLITE FRAGMENTATION RATES 17,597 Table 2. Estimated Satellite Disruption Rates Radius, km Orbit Radius, 4 km Disruption Lifetime, 9 years Satellite Durda et al. [ 1998] Housen et al. [ 1991 ] Uranus Puck Belinda Rosalind Portia Juliet Desdemona Cressida Bianca Ophelia Cordelia Neptune Proteus Larissa Galatea Despina Thalassa Naiad Note: Disruption lifetimes are inverses of disruption probabilities (see Plate 1). simulations with the scaling theory model we adjust the value of Q* in the strength and gravity regimes and adjust the size of the smallest impactor. The latter makes little difference in the size ranges shown. The black and yellow curves are analogous to the results of paper 1, which used the Q* model of Housen and Holsapple [1990], with the yellow curves representing an impactor flux which continues to 1 m radius instead of cutting off at m in radius. The blue and gray curves increase Q* by a factor of 20 in the gravity regime over the Housen and Holsapple [1990] model following the experimental results of Housen et al. [1991] [cf. Holsapple, 1993], with the gray curve representing a comet cutoff radius of 1 m. The differences between the black and yellow curves and the blue and gray curves are small and appear only for moon sizes below 1 km in radius, suggesting that as long as the comet flux continues to a minimum comet radius of <0 m, the rate of satellite fragmentation is not sensitive to the exact value of the cutoff. The effects of the small comet size cutoff can be seen in the disruption probabilities in Plate 1 for objects smaller than 1 km in radius. With a m cutoff the disruption probability drops off with target size for targets smaller than 1 km, while extending the comet distribution to 1 m in radius results in a factor of larger disruption probability for moonlets 200 m in radius. In papers 1 and 2 we assumed a continuous size distribution for the impactor flux with a steeper distribution, resulting in more small impactors than used in our improved model here. This and the use of small size cutoff in the impactor distribution account for the significantly lower The observed satellite populations at Uranus and Neptune disruption probabilities in Plate 1 for subkilometer-sized are too limited to allow the same type of analysis that Durda et moonlets compared with our earlier model (Figure 2 in paper al. [1998] performed for the asteroid belt. Compositionally, 1). the satellites of Uranus and Neptune are not similar to main The red curves are for the scaling theory model (with Q* belt asteroids, but composition is not likely to play an enhanced in the gravity regime by a factor of 20) and an important role in the gravity regime (satellites larger than increased material strength corresponding to increasing Q* in the strength regime by a factor of 7. Such an increase in strength is comparable to changing the material properties from those of ice to those of basalt (cf. paper 1). This results in a lower disruption probability in the target radius range frorn 0.8 to 8 km (Plate 1). At larger sizes in Plate 1 the red curve approaches the blue and gray curves as the material properties are dominated by the effects of gravity in the gravity regime. The values in Table 2 correspond to the blue and gray curves for the scaling theory values and the green curve for the asteroid model values. 4. Discussion Our simulations using the scaling theory strength model suggest rapid erosion of the small moons of Uranus and Neptune, as was found in paper 1. The Durda et al. [1998] model, based on modeled asteroid strengths, predicts slower erosion and longer collisional lifetimes for the observed moons. All of our results with the Housen et al. [ 1991] model for Q* suggest collisional lifetimes comparable to or less than the age of the solar system for moons <0 km in radius at Uranus and Neptune. The empirical asteroid model, on the other hand, has a significantly higher value of Q* in the gravity regime, making -0 km radius moons much more resistant to disruption. With the asteroid model, moons such as Puck can survive bombardment at the current rate for the age of the solar sy?em (Plate 4). The hydrocode model [Benz and Asphaug, 1999] produces similar results to the asteroid model for Puck (Plate 4, green and purple curves); for Cordelia and Naiad the hydrocode model shows the initial satellite intact (N.-- 1). km radius) in determining response to hypervelocity impacts. The hydrocode model is for impacts at a lower velocity than occur on planetary satellites, but that is not likely to result in significantly shorter disruption lifetimes. It is more

10 17,598 COLWELL ET AL.: SATELLITE FRAGMENTATION RATES likely that these Qr>* models are inappropriately strong for small moons within their planer's Roche zone, where tidal forces help disperse rubble piles. Our original modeling of the collisional fragmentation of the small moons of Neptune and Uranus (papers 1 and 2) suggested such rapid collisional disruption of those moons that maintaining the observed populations over the age of the solar system was problematic. At that time little information about the cometary flux in the outer solar system was available, and because our results scale with impactor flux, we suggested that intermediate to that of Durda et al. [1998] and that of paper 1) a lower impactor flux could slow the collisional cascade. would allow the fragments from a -20 km radius satellite to Recent observational and computational advances have produce tings and moonlet belts over a longer period of time, advanced our understanding of the population of impactors resulting in a more steady population of rings, moonlet belts, capable of destroying small moons in the outer solar system. and small moons. The hydrocode model leads to ring creation These new results are remarkably consistent with the flux rates which are too slow to account for the continued existence model used in papers 1 and 2. Furthermore, tidal forces of short-lived narrow rings. This may simply reflect the prevent even partial reaccretion for fragments of the innermost difference between Qr>* for asteroids and for moons within the moons of Uranus and Neptune [Canup and Esposito, 1995]. In order for the observed small satellite populations at Roche zone where tidal forces can lead to dispersal and something closer to Qs* is a more appropriate Uranus and Neptune not to be on the brink of collisional criterion. The hydrocode model is stronger than the scaling destruction, the satellites must be more resistant to theory model in the gravity regime (Figure 3), but the slopes fragmentation than the model used in papers 1 and 2. The are similar, suggesting that a shift from Qr>* empirical model of Durda et al. [1998] developed to explain move the hydrocode results closer to the scaling theory curve. the collisionally evolved asteroid size distribution leads to collisional lifetimes of-1-3x9 years for the smallest observed moons of Uranus and Neptune (Table 2). 'I]fis is a factor of a few longer than our results in papers 1 and 2. However, the collisional lifetime grows much more rapidly wilh satellite size using the asteroid model, leading to lifetimes against disruption in excess of the age of the solar system for the larger, more distant satellites (Table 2). Furthermore, the probability seen in the scaling theory model than it is with the asteroid model. The latter would predict rapid ring creation from parent moonlets, so that once a larger moon is destroyed, its fragments are rapidly disrupted to produce a short-lived population of rings and moonlet belts. Such ting-moon systems would be characterized by long intervals without many tings or moonlet belts interrupted by brief (8 years) periods characterized by a large number of tings and moons. In contrast, the "strong" Housen et al. model (a model 5. Summary disruption to Qs* would Our reanalysis of the disruption rates of the moons of Uranus and Neptune includes an improved model of the interplanetary impactor flux in the outer solar system. This flux is similar to that used in papers 1 and 2 but includes a asteroid model has lower strengths in the strength regime than the scaling theory model, leading to rapid disruption of lower size cutoff at m. We derive new crateting rates on the moons of Uranus and Neptune and disruption rates with kilometer-sized moonlets. different assumptions about the resistance of the moons to A "strong" scaling theory model, with a larger strength in catastrophic fragmentation. The continued existence of a the strength regime and an increased Q* in the gravity regime population of small moons at Uranus and Neptune can be over the model of paper 1, leads to slower collisional evolution explained if they are considerably more resistant to than our earlier papers, but with a less extreme dependence on catastrophic fragmentation than implied by the strain rate target size than either the asteroid model or the hydrocode scaling model used in paper 1. An empirical model based on model. With the scaling theory model the collisional lifetime of Cordelia is 430 million years, comparable to the maximum asteroid collisional evolution [Durda et al., 1998] and a hydrocode model [Benz and Asphaug, 1999] slow the derived shepherded age of the œ ring of Uranus [Esposito and Colwell, satellite collisional cascade and make most of the observed 1989]. The weaker dependence of disruption probability on target size in this model (Plate 1) is consistent with the relatively shallow size distribution of small moons observed at Uranus and Neptune. If the tings of Uranus are the debris belts of 1- km moonlets, then the number of tings observed () provides an additional constraint on the rate of satellite disruption. moons at Uranus and Neptune have ages against disruption exceed the age of the solar system. This, coupled with the very short collisional lifetimes for ring-precursor satellites in the asteroid model for Q*, makes it difficult to maintain the observed distribution of a mix of rings, moonlet belts, and small moons at each planet. A stronger version of the strain rate scaling model used in Another 15 or so dust bands at Uranus point to unseen paper 1, with an enhanced Q* in the gravity regime and a populations of parent bodies which are also likely the debris higher material strength in the strength regime, slows the from satellite disruptions. The asteroid model for Q* predicts collisional cascade enough to explain the continued the disruption of a kilometer-sized moonlet in 8 years, while persistance of moons. This model (red curves in Plates 1-4) the strong scaling theory model (red curves) predicts lifetimes predicts shorter collisional lifetimes for the observed -0 5 times longer at that size range. Because the parent moons of km radius moons than the asteroid model but longer collisional these tings and moonlet belts are themselves the fragments lifetimes for the sub- km ring-precursor satellites. This from the less common disruption of larger moons, many are weak dependence on satellite size leads to a more steady produced in a single satellite fragmentation event. Because production of rings over time and is consistent with the mix of the Uranian ring lifetimes appear to be short (-8 years or rings, moonlet belts, and small moons observed at Uranus and less) from torques due to exospheric drag, a population of Neptune. A hydrocode model for Qr>* [Benz and Asphaug, tings at the present epoch suggests a steady production of 1999] has a similar dependence on satellite size but is too rings from precursor satellites created in rare (-9 year) strong to account for ring production by satellite disruption, fragmentation events. This is also more consistent with the suggesting that Qr>* is not the appropriate criterion for moons relatively weak dependence on satellite size of disruption within the Roche zone.

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