Icarus145, 220–229 (2000)

doi:10.1006/icar.1999.6333, available online at http://www.idealibrary.com on

Collision Rates in the Present-Day Kuiper Belt and Centaur Regions:Applications to Surface Activation and Modification on Comets,

Kuiper Belt Objects, Centaurs, and Pluto–Charon

Daniel D. Durda and S. Alan Stern

Southwest Research Institute, Space Studies Department, 1050 Walnut Street, Suite 426, Boulder, Colorado 80302E-mail: durda@boulder.swri.edu

Received March 29, 1999; revised November 29, 1999

We present results from our model of collision rates in the present-day Edgeworth–Kuiper Belt and Centaur region. We have updatedprevious results to allow for new estimates of the total disk pop-ulation in order to examine surface activation and modificationtime scales due to cratering impacts. We extend previous resultsshowing that the surfaces of Edgeworth–Kuiper Belt objects arenot primordial and have been moderately to heavily reworked bycollisions. Objects smaller than about r = 2.5 km have collisionaldisruption lifetimes less than 3.5 Gyr in the present-day collisionalenvironment and have probably been heavily damaged in their in-teriors by large collisions. In the 30- to 50-AU region, impacts of1-km-radius comets onto individual 100-km-radius objects occuron 7× 107–4× 108-year time scales, cratering the surfaces of thelarger objects with ∼8–54 craters 6 km in diameter over a 3.5-Gyrperiod. Collision time scales for impacts of 4-m-radius projectilesonto 1-km-radius comets range from 3× 107, to 5× 107 years. Thecumulative fraction of the surface area of 1- and 100-km-radius ob-jects cratered by projectiles with radii larger than 4 m ranges froma few to a few tens percent over 3.5 Gyr. The flux of Edgeworth–Kuiper Belt projectiles onto Pluto and Charon is also calculated andis found to be ∼3–5 times that of previous estimates. Our impactmodel is also applied to Centaur objects in the 5- to 30-AU region.We find that during their dynamical lifetimes within the Centaurregion, objects undergo very little collisional evolution. Therefore,the collisional/cratering histories of Centaurs are dominated by thetime spent in the Edgeworth–Kuiper Belt rather than the time spenton planet-crossing orbits. Further, we find that the predominantsurface activity of Centaur objects like Chiron is most likely notimpact-induced. c© 2000 Academic Press

Key Words: centaurs; Chiron; comets; Kuiper Belt objects; Pluto.

1. INTRODUCTION

rt

ca

and catastrophic collisions over the eons have left their mark on

an-elyandbe-col-B.by

er

ctstingal

ol-to

, andele-KB,

resse

rly,rbits

nal

a-ta,

ares of

dayhedel

2

Collisions are the dominant evolutionary process actingmost small bodies in the Solar System. In the main astebelt, for instance, cratering collisions have greatly modifiedsurfaces of individual asteroids by leaving large impact s(e.g., Greenberget al. 1994, 1996, Veverkaet al. 1997) andredistributing regolith across their surfaces (Geissleret al.1996),

2

0019-1035/00 $35.00Copyright c© 2000 by Academic PressAll rights of reproduction in any form reserved.

onoidhers

the entire population size distribution (Daviset al.1979, 1989,Durdaet al.1998).

The Edgeworth–Kuiper Belt (EKB) population representsother major population of small bodies whose evolution is largshaped by collisions (Stern 1995). Stern (1996) and DavisFarinella (1997) have further explored the rate of collisionstween comets in the region beyond 30 AU and found thatlisional evolution is a highly important process in the EKCollisional evolution in the EKB has recently been reviewedFarinellaet al. (in press).

Although intrinsic collision rates (number of collisions pkilometer2 per year) are lower by a factor of∼1000 in the EKBcompared to the main asteroid belt, the population of objethere is∼1000 times as great. As a result of these compefactors, the overall level of collisional processing of individuobjects is of similar scale to that in the main belt.

Here we seek to investigate the implications of the EKB clision rates for surface modification. In particular, we wishestimate quantities such as the surface cratering fractionsthe expected largest crater sizes. In addition to a direct rvance for understanding comets and other objects in the Ewe also seek to gain insights into what the Pluto–Kuiper Expspacecraft (Terrileet al.1997) may observe when it images thsurfaces of Pluto, Charon, and other EKB objects. Similawe seek to assess whether Centaur objects on transient oin the giant planet region undergo further significant collisioprocessing.

In what follows we first revisit previous collision rate calcultions (Stern 1995, 1996) in light of both new observational daand higher fidelity modeling. Once improved collision ratesobtained, we go on to evaluate their effect on the surfaceobjects in, and derived from, the Edgeworth–Kuiper Belt.

2. THE COLLISION RATE MODEL

Stern (1995) examined collision rates in the present-Edgeworth–Kuiper Belt beyond 30 AU, as a function of tdisk’s radial and population size structure. The numerical mo

0

G

;

e

cfi

b

r

ee

s

a

sntct

n

r

del

ar-cle-t:lcu-iesx”

93,

ly,w-

orper-

theourtingce,putethat

ere

tionon-

tedrm

ter

es-

COLLISIONS IN THE ED

for calculating collision rates is described in detail in that papso only a brief recapitulation will be presented in this sectionthe next section we will describe changes and improvementshave been made to the model to produce the results preslater in this paper.

The 1995 model is a static, multizone, multi-size-bin, particin-a-box collision rate model that calculates instantaneouslision rates. The colliding population is defined in terms ototal disk mass and a single-valued power-law size distrtion of objects in the disk, normalized by the total number∼100-km-diameter and larger objects in the 30- to 50-AU zoThis size distribution is treated as a series of monotonicallycreasing radiusr bins, with the objects in each successive1.6 times larger in size (and 4 times more massive) than thin the preceding bin.1 The model also specifies the radial distbution of heliocentric surface mass density6(r ) so that:

6(r )=6orβ, (1)

where6o is a normalization constant which in effect specifia total EKB mass in the 30- to 50-AU zone. The power-lawponentβ determines the heliocentric radial distribution of main the disk, with the two cases we consider defining a realirange of parameter space:β =−1 corresponds to a constant maper heliocentric radial bin, whileβ =−2 (more realistic, andour preferred case) corresponds to a declining mass per rbin. A disk-wide average eccentricity,〈e〉, is adopted for eachmodel run; an equilibrium condition where the disk wedge an〈i 〉= 1

2〈e〉 is assumed (see, e.g., Lissauer and Stewart 1993Once the global properties of the disk are specified, the di

binned into a series of radially concentric tori 1 AU in width, athe collision rates for objects at each semimajor axis arecalculated in a particle-in-a-box formalism. In this approathe instantaneous collision ratec (collisions/unit time) of targebodies with semimajor axisa, eccentricitye, and radiusrk beingstruck by impactors of radiusr l is

c(rk, rl ,a, e, i, R)

=a(1+〈e〉)∑

R=a(1−〈e〉)

√GM¯4π2a3

T(a, 〈e〉, R)n(rl , R) vkl(a, 〈e〉, 〈i 〉, R)

× σg(rk, rl , vkl , vesc[k+l ] ), (2)

whereT(a, 〈e〉, R) represents the time the target body speat each distanceR during its orbit.T(a, 〈e〉, R) is computed bysolving the Kepler time-of-flight equation explicitly for eve(a, 〈e〉) pair in the model’s parameter space. The number dsity of impactorsn(rl , R) in the torus centered at distanceR iscomputed from the mass of the disk, the disk’s wedge angle〈i 〉,

its population size distribution, and its heliocentric surface mdensity structure (Eq. (1)). Herevkl is the local average crossing

1 The radius of the smallest bin was 3.94 m; successive bin radii were 6.29.92 m, etc.

es-,izesy the

EWORTH–KUIPER BELT 221

er,inthatnted

le-ol-a

bu-ofne.in-inosei-

sx-sstic

ss

dial

gle).k isd

henh,

ds

yen-

ass

speed of the impactor population against the targets,vesc is theescape speed of the combined target–projectile pair, andσg isthe gravitational-focusing corrected collision cross section.

3. MODEL IMPROVEMENTS AND INPUTPARAMETER UPDATES

We have made two noteworthy improvements to the mooutlined above. These are:

• A more exact treatment of relative impact speeds. In the elier model, relative impact speeds were calculated by a “partiin-a-box” approximation of the orbital motion of the targev= (〈e〉2+〈i 〉2)1/2vk, wherevk is the average Keplerian orbitaspeed of the target. We now include in the collision rate callations the difference between the collision frequency of bodin mutual Keplerian orbits and that based on “particle-in-a-bocollisions, so thatv= ( 5

4〈e〉2+〈i 〉2)1/2vk, as well as the effect ofa Gaussian speed distribution (cf., Wetherill and Stewart 19Appendix A).• Setting realistic limits on gravitational focusing. Previous

the effects of gravitational focusing were unconstrained, alloing the collision cross sectionσg to grow unrealistically largefor the most massive targets and for very low〈e〉. In the presentmodel we now include limits on the gravitational focusing factdue to Keplerian shear, three-body effects, and velocity dission (cf., Ward 1996, Eqs. (9) and (11)).

These improvements are numerical refinements affectingfinal results at only about the 10% level as compared withprevious calculations; nonetheless, they are worth documenand make the final results more robust. Of greater importanwe have updated the input parameters necessary to comcollision rates in the EKB, based on observational advanceshave occurred since 1995. In particular, these are:• Jewitt et al. (1998) and Gladmanet al. (1998) have each

provided convincing evidence that between 30 and 50 AU thexist at least 70,000 objects withr > 50 km, and perhaps twicethat many. This is between 2 and almost 5 times the populaestimates for such bodies available in 1995. We therefore cduct new model runs with normalizations of both 7× 104 and1.4× 105 objects withr > 50 km.• Further, the population size distribution is now represen

by a more sophisticated, two-component power law of the foN(di )∝ db

i ddi , whereb=−3 for di < d0 andb=−4.5 for di>

d0, with d0= 10 km (Weissman and Levison 1997, hereafWL97).2

For reference, a WL97 size distribution, coupled with antimated population of 70,000 objects withr > 50 km, yields

2 At large sizes the WL97 size distribution is consistent with the latesttimates by other researchers (Gladmanet al. 1998, for instance). For smallercomet-size objects, simple, single power-law extrapolations from larger sappear to overestimate the number of small EKB objects needed to suppl

5 m,short-period comet flux (Duncanet al. 1995), hence the broken power law ofWL97.

N

i

t

l

ps

t

tb

.(

ij

tlr

dcts.

sionterials of

KB.

dis-the

wntlytheme-i.e.,am-dies

c-of

ted.,

dent

222 DURDA A

∼4564 objects in our model’sr = 102.4-km size bin and∼1.2×109 objects in ther = 1-km size bin. For a population of 140,00objects larger thanr = 50 km, the number of objects in all sizbins doubles accordingly. We continue to model the spatialtribution of objects in the 30- to 50-AU region as a disk, wour preferred surface mass density indexβ =−2, as describedabove.

We have compared our modeled collision rates with thcomputed from the observed distribution of EKO orbits (Bot1999, pers. commun.) and find very good agreement betwthe two independent methods. Our modeled collision rates,cussed in the following sections, are within a factor of∼2–4of those calculated based on anOpik-style collision rate mode(Bottkeet al. 1994) applied to the observed EKO orbit distribtion. Considering the fact that we have not made any attembias-correct the observed orbit distribution for this compariand the fact that our disk model has an inclination distributthat is somewhat “colder” than the observed EKO populatio3

we consider the agreement between the two calculationsquite good.4

4. NEW ESTIMATES OF COLLISION RATESIN THE KUIPER BELT

a. Collision Outcomes

We now present results of collision rate calculations for30- to 50-AU region obtained with our improved collision modand updated input parameters.

First, however, it is important to remember that givendynamical conditions of the present EKB, mutual collisionstween Edgeworth–Kuiper Belt objects (EKOs) are generally esive. That is, above some critical eccentricity,e∗, impacts occurat relative speeds high enough that most ejecta escapes thget bodies. Figure 1 shows the critical eccentricity boundbetween erosional (i.e., net mass loss) and accretional (i.emass gain) regimes for mutual collisions between EKOsStern 1996). Our contribution here, in Fig. 1, is the additionsome 128 multiopposition EKOs for which fairly reliable orbhave been determined, so that this large population of obwith moderately well established orbits can be evaluated relato the critical eccentricity boundary curves.

Notice that the critical eccentricity for mutually colliding objects in the EKB increases slightly with increasing heliocendistance due to the direct linear dependence of the typicaproach speed upon the local Keplerian orbital speed. Fafrom the Sun, higher〈e〉s are required to generate impact en

gies sufficient to guarantee erosive collisions. Thus, if〈e〉 doesnot increase with heliocentric distance, collisions will tend to

3 Relative to observed EKO eccentricities, observed inclinations are higthan the〈i 〉= 1

2〈e〉 equilibrium values assumed in our model.4 Nevertheless, we remind the reader about the large model uncerta

attributable to using simple power laws for both the orbital and the populasize distributions.

size

D STERN

0edis-th

osekeeendis-

u-t toon

ionn,o be

theel

hee-ro-

e tar-ary, netseeof

tsectstive

-ricap-therer-

FIG. 1. The critical eccentricity (e∗) boundary between erosional anaccretional outcomes for collisions between Edgeworth–Kuiper Belt objeCritical eccentricity boundaries are shown for both strong (ρ= 2 g cm−3 andSo= 3× 106 erg g−1) and weak (ρ= 0.5 g cm−3 andSo= 3× 104 erg g−1) tar-gets at two representative heliocentric distances (30 and 50 AU). The collistrengths chosen for the strong and weak cases bound a wide range of maproperties and, we believe, the likely range of collision strength parameterKuiper Belt Objects.

less erosional in nature as we move outward through the EFor 〈e〉 greater than the critical eccentricity,e∗, impacts ejectmore target mass than is retained, and the target is eitherrupted in response to a catastrophic collision or eroded incase of a cratering collision.

The plotted data points for 128 multiopposition EKOs shothat most large EKOs, like the main-belt asteroids, are curreundergoing predominantly erosive collisions, even undermost pessimistic assumption, i.e., that of strong surfacechanical properties. As to classical, kilometer-scale comets (those objects which leave the EKB to appear as the Jupiter Fily comets),e∗ values are so low as to guarantee that these bohave resided in a heavily erosional collisional environment.

b. Collision Rates and Fluxes

Figure 2 shows the typical number of collisions which ocur for 100-km- and 1-km-scale radius EKOs as a functionprojectile radius, at both 35 and 45 AU.

The calculations shown here assume the “nominal” estimapopulation5 of such objects in the 30- to 50-AU zone today, i.eN(r > 50 km)= 7× 104. Since the collision rate model usehere is static (i.e., it calculates collision rates for the presdisk and does not account for a decrease in the population

be

her

intiestion

with time as bodies are collisionally destroyed), we calculatethe total flux of impactors on a target only over the past 3.5 Gyr,

5 In what follows we will report collision time scales and impact fluxesfor the “nominal” population only. For a larger population withN(r >50 km)= 1.4× 105, reported collision time scales for individual objects willbe reduced by a factor of 2 and impact fluxes will be increased by a factor of 2.

COLLISIONS IN THE EDGEWORTH–KUIPER BELT 223

FIG. 2. The total number of impacts onto 102- and 1-km-radius EKOs over 3.5 Gyr as a function of projectile radius. The bend in the plotted curves att

cn

as

d

-dt

it

ngefiont the

hen-

ay

m-yr?era-ner-cateybys,des

dy-that

impactor radii of 10 km is a reflection of the shape of the EKO size distribu

the approximate time since the disk is expected to have rearoughly its present mass and dynamical state (e.g., WeissmaLevison 1997).6 Our interest here is in present-day collisionrates and effects. This is primarily because the small bodiethe EKB are young compared to 3.5 Gyr (Stern 1995, DavisFarinella 1997), and bodies of all sizes underwent far morenificant collisional processing in the more massive, primordKuiper disk (Stern and Colwell 1997).

Figure 2 shows that a typical, 100-km-scale radius EKO whave undergone∼8–54 cratering impacts with 1-km-radiu“comets” over the past 3.5 Gyr, depending on heliocentrictance and〈e〉. At 35 AU the collision time scales for a typical EKO are 6.5× 107 and 1.5× 108 years for〈e〉=0.0256and 0.2048, respectively. The same collision time scales1.4× 108 and 4.1× 108 years at 45 AU for the same〈e〉s. Thesevalues of〈e〉 cover the range of observede for most EKOs.For large EKOs, collision time scales are shorter for sma〈e〉s due to increased gravitational focusing effects at smaencounter speeds. Given our estimated population of 100radius EKOs in the EKB (4564 objects in the size bin with rabetween 81–129 km), there should be one such EKO–comelision somewhere in the 30- to 50-AU region every∼1.4× 104

to 9.0× 104 years. Smaller projectiles hit more frequently, wimpact time scales for 4-m-radius projectiles onto any sin100-km-scale radius target of∼1000–6000 years; across thentire EKB, such cratering impacts occur every∼80–510 days.

Figure 2 also shows the number of collisions onto a 1-kradius comet. Over 3.5 Gyr7 a 1-km-radius comet between 3

6 For this reason, we denote any object or surface unit on an object of thisas “primordial” if it is this age or older.

7 In the next subsection we show that catastrophic disruption lifetimes1-km-radius comets in the EKB range from∼1 to 10 Gyr. We calculate impactfluxes here over 3.5 Gyr in order to directly compare with results for largobjects; for shorter- or longer-lived comets, fluxes will be proportionately smaor larger, respectively.

on-

ject

enU

ion adopted in our study (Weissman and Levison 1997).

hedand

als inndig-ial

illsis-

-

are

llerllerkm-ii

col-

hglee

m-5

age

for

er

and 45 AU will experience∼90–300 cratering collisions withprojectiles larger thanr = 4 m. Collision time scales for im-pacts of 4-m-radius projectiles onto 1-km-radius targets rafrom 2.5× 107 to 4.7× 107 years. Over the entire population o∼2× 109 comets in the disk, there should be one such collisevery several days. These numbers assume, of course, thaWL97 size distribution is valid to sizes as small as 4 m. Tactual size distribution of small objects in the EKB is highly ucertain, and the cratering records of the Galilean satellites meven hint at a lack of cometary objects smaller than∼100 m(Chapman 1997, Chapmanet al.1998).

c. Catastrophic Disruption Lifetimes andDeep Interior Modification

What about larger collisions and the likelihood that a 100-kradius EKO would be catastrophically disrupted over 3.5 GThe impact scaling literature has been dominated by considtion of collisions between asteroids, so the impact-specific egies and scaling laws used here are most appropriate for silitargets. However, Ryanet al.(1999) have conducted laboratorimpact studies in which porous ice targets were impactedfractured ice projectiles to simulate collisions between EKOand found that impact specific energies and fragmentation moare similar to those for silicate targets. Smooth particle hydronamics calculations by Benz and Asphaug (1999) indicateimpact specific energies for basalt targets are only∼2–4 timesgreater than those for icy objects under the same impact cditions. Estimates of the impact specific energy,Q∗D, requiredto catastrophically disrupt and disperse a 100-km-radius obrange from∼1× 105 to 4× 105 J kg−1 (Daviset al.1989, Loveand Ahrens 1996, Melosh and Ryan 1997, Durdaet al.1998).

With 〈e〉=0.2048, effective relative impact speeds betwe100-km-radius EKOs and other objects in the 30- to 50-A

llerregion range from∼1.1 to 1.4 km s−1. The relative impact speedisvi = (U2+ v2

esc)1/2, whereU is the hyperbolic encounter speed

ero

ellnwm

r

r

i

it

itale

al-l-

es

e

tmedearwe

re-red

ex-rgeousial

a-ciesant,

d inver-esical

f the

ius

-ctile

224 DURDA AN

FIG. 3. Collisional disruption lifetime for EKOs as a function of targsize, calculated at 35 and 45 AU with〈e〉=0.2048. Critical specific energies focatastrophic disruption,Q∗D, were assumed to be in the mid-range of those frthe published scaling laws referred to in the text.

(dependent upon heliocentric distance and〈e〉8) andvesc is theescape speed of the combined target–projectile pair. At thspeeds, the smallest object capable of delivering the requspecific energy for disruption to the target is aboutr = 53–84 kmin size. The time scale for such a collision is presently 3× 1012 to8× 1012 years (using a model bin radius of 64.5 km). Still-largEKOs would have longer disruption time scales.

Our results show clearly that the vast majority of the largEKOs are not likely to have been involved in disruptive cosions over the past 3.5 Gyr. This is in agreement with collisiomodels of EKB evolution by Davis and Farinella (1997), shoing that the population at diameters larger than about 100 kessentially unchanged over Solar System history.

Figure 3 shows the catastrophic disruption lifetimes forange of EKO sizes at 35 and 45 AU with〈e〉=0.2048. Shat-tering time scales for 100-km-radius objects are shorter,∼3.9–10.5 Gyr, since projectiles as small asr ≈ 5 km may fragmentthe target without dispersing the resulting debris, resultinggravitationally reaccumulated rubble-pile EKOs.

What is the largest object size-class in which the majoof objects can be expected to have been involved in disrupcollisions in the EKB in the past 3.5 Gyr? Using our highemodeled〈e〉 of 0.2048 and the lowest estimatedQ∗D for objectsof various sizes, and searching through the many combinatof our calculated target–projectile collision time scales, we fithat objects smaller thanr ≈ 2.5 km presently have catastrophdisruption time scales at 35 AU less than 3.5 Gyr. At 45 AUdisruption time scale increases by a factor of∼2.5–3, so thatsmaller objects with radii less than about 1.6 km have lifetim

of 3.5 Gyr. At lower〈e〉 and assuming values ofQ∗D nearer themid-range of published scaling laws, objects become harde

8 The encounter speed is also dependent upon〈i 〉, but recall that we haveassumed an equilibrium condition where〈i 〉= 1

2〈e〉.

zes,of

D STERN

t

m

eseired

er

sti-al-is

a

in

itytivest

onsndche

es

r to

destroy, so thatr ≈ 2.5 km should be considered an upper limto the size object that can have a collisional disruption time scless than 3.5 Gyr anywhere in the EKB.

From the collision rates yielded by our model, we can also cculate the collisional disruption lifetime in the EKB for a typicacomet nucleus with radiusr = 1 km. We estimate the impact specific energy for disrupting such an object to be∼10–200 J kg−1

(Durdaet al.1998, Melosh and Ryan 1997). Using these valufor Q∗D, and assuming disk-wide〈e〉 of 0.0256–0.2048, we findthat at 35 AU ar = 1-km comet can be destroyed byr ≈ 0.02-to 0.25-km projectiles. The corresponding disruption lifetimranges from∼9.6× 108 to 9.6× 1010 years. At 45 AU the life-time is∼2.6× 109 to 2.6× 1011 years. Eccentricities for mosobserved EKOs tend to be near the higher of the values assuhere, so that disruption time scales for comets are likely nthe lower values reported. Across the 30- to 50-AU region,find that collision lifetimes forr = 1-km comets are then likelyof order 1–10 Gyr, assuming mid-range values ofQ∗D.

5. SURFACE MODIFICATION IN THE EKB

a. Cratering Fraction

The results of collision rate calculations presented in the pvious section demonstrate that EKOs of all sizes have suffea significant number of collisions. These collisions can bepected to have significantly affected the surfaces of both laand small objects, covering their surfaces with craters of varisizes, overturning and reworking underlying, more primordsurfaces, removing surface materials through impact sublimtion, and possibly exposing deeper and more volatile icy speand thus possibly serving as a mechanism to activate distinactive objects.

Using the encounter speeds and impactor fluxes calculatethe previous section, we can estimate the sizes and spatial coage of impact craters on typical EKOs. Holsapple (1993) givan expression for the diameter of an idealized, hemisphercrater as

D= 1.26d(Aρimpactor/ρtarget)1/3(1.61gd/v2

i

)−α/3, (3)

whered is the impactor diameter,ρ is the density,g is the sur-face gravity of the target,vi is the encounter speed, andA andαare constants dependent upon the mechanical properties otarget material. Holsapple (1993) uses values ofA andα of 0.2and 0.65, respectively, for water ice. For a 100-km-scale radEKO with an assumed density ofρ= 1.5 g cm−3, g≈ 4.3 cm s−2.The relative encounter speed,vi , as defined in the previous section, includes the escape speed of the combined target–projepair.

Figure 4 shows the fraction of ar = 102-km target’s surfacearea covered by craters produced by projectiles of various siover a 3.5-Gyr period. We find that the cumulative fraction

the surface cratered by allr > 4-m projectiles ranges from∼7 to32% for targets from 35 to 45 AU and for〈e〉=0.0256–0.2048.

COLLISIONS IN THE EDGEWORTH–KUIPER BELT 225

i

kais

t

t

h

ll

ec

a

k

a

ts

,ice

under-

en-aleand

ingersghscape

etstheringor-

leKO,d athe

ets, ande-

ionlingced

ofize

FIG. 4. Fraction of the surface area of 102- and 1-km-rad

This is a conservative estimate that does not include the ational surface area covered by crater flanks and ejecta blanwhich would cover∼4 times more area. Significant additionarea will also be covered by craters produced by projectsmaller thanr = 4 m, our lower size limit in these calculationOverall, perhaps a third of the entire surface of a typical 100-kscale EKO will have been reworked by impact cratering overpast 3.5 Gyr.

Collision rates calculated in the previous section showedcomet-sized objects (r ≈ 1 km) will also be significantly crateredduring their lifetimes, typically enduring∼90–300 impacts byobjects larger thanr = 4 m. Figure 4 shows that a comet witr = 1 km will have∼20–224% of its surface cratered by impacts withr > 4-m projectiles over 3.5 Gyr. Davis and Farine(1997) concluded that most comet-size objects are collisiofragments of larger parent objects. Our results suggest thatthe small fraction of objects that are surviving, “original” acretion products (i.e., those that are>3.5 Gyr old) will haveundergone substantial collisional processing by crateringsubcatastrophic impacts.

This result extends the notion that the surfaces of EKOsnot primordial and have been moderately to heavily reworduring their history in the EKB (Stern 1995, Luu and Jew1996, Davis and Farinella 1997, Farinellaet al. in press). Thenumerous smaller impactors, responsible for most of the spcoverage, will have comminuted and gardened the surfaceEKOs, producing a regolith of shattered, icy debris. As we wsee below, EKO surfaces are expected to display both freshprimordial (i.e.,>3.5 Gyr old) units.

Larger impactors will have penetrated deeper into theget’s surface, excavating material that is less impact-procefrom below. The most recent, large impacts may have assated bright ejecta blankets and ray systems. Brownet al.(1999)report evidence that the intensity of water bands in the sptrum of 1996 TO66 varies with rotational phase, which sugges

a “patchy” surface. Figure 2 shows that the largest impactlikely to have struck 100-km-radius EKOs over the last 3.5 G

us EKOs cratered during 3.5 Gyr, as a function of projectile radius.

ddi-ets,lles.m-he

hat

-analven-

nd

areeditt

tials ofilland

ar-sed

oci-

ec-ts

haver ≈ 5 km, resulting in craters∼26 km in diameter. Deeperless impact-processed material may contain more volatilespecies that could provide a source of surface activity arorecent impact sites. Such activity may range from small, geyslike plumes activated by impactors just large enough to petrate a processed regolith, up to the production of full-sccomae generated by the largest impactors (see FitzsimmonsFletcher [1999] in this regard).

b. Mass Loss Due to Ejecta Escape

The same impacts responsible for overturning and reworkthe surface of an EKO and covering its surface with cratwill also erode and remove a fraction of that surface throuthe escape of ejecta launched at speeds greater than the espeed of the target.

Standard impact scaling laws indicate that for strong targwith radii less than∼75–150 m, ejecta speeds should exceedescape speed of the object. In a purely strength-scaling crateregime, the volume eroded from the target is directly proptional to the volume of the impactor:Verode(D)= hD3, whereDis the projectile diameter, andh≈ 120 as suggested by Holsapp(1993). For larger targets, such as a 100-km-scale radius Elarger craters are excavated in the gravity-scaling regime anmuch smaller fraction of the ejecta is launched at greater thanescape speed. Even for targets as small as 1-km-radius comsignificant fraction of impact ejecta may be retained (Ryan aMelosh 1998). For a 100-km-scale radius icy EKO and the mchanical properties described just below Eq. (3), the transitcrater diameter between the strength-scaling and gravity-scaregimes is of order 40–50 m, about the size of craters produby the smallest objects considered in our model (r = 4 m). As inthe strength-scaling regime, the volume eroded per volumeprojectile for gravity-scaling cratering is independent of the sof the impactor,Verode(D)= h′D3, but hereh′ is much smaller

orsyrthanh, of order 3–4 for a 100-km-scale radius icy object in theEKB (see the Appendix of Geissleret al. (1996) for a more

226 DURDA AND STERN

r s.

clagw

8o

r

safn

a

n

ndve,ionn’s

n

erve

les

uck

ng

FIG. 5. Kinetic energy partitioned into impact vaporization of the su

complete discussion of mass loss from small objects due topact ejecta erosion).

With the above mass loss assumptions and the EKO poption and collision time scales calculated from our model, weestimate the magnitude of the surface loss from comets andEKOs due to escape of impact ejecta (Fig. 5). Time-averamass loss rates from a 100-km-radius EKO due to impactsr > 4-m projectiles range from∼4× 108 to 7× 109 g year−1 forR between 35 and 45 AU and〈e〉 between 0.0256 and 0.204Over 3.5 Gyr, this amounts to some 10–100 m of surface lMass loss rates from 1-km-radius comets range from 6× 105

to 2× 106 g year−1 under the end-member assumption of ctering in the strength regime, amounting to∼110–390 m ofsurface loss over 3.5 Gyr. Gravity-regime cratering would rein lower mass loss rates and less surface erosion due to the lfraction of retained ejecta. We suspect that most of the surloss effects due to impacts will be localized primarily aroularger impact sites, and the impact-undisturbed surface frac(perhaps 90% for a minimum impact radius of 4 m) will be cosiderably less damaged. In contrast, recall that fully a third ofentire surface of a typical large EKO may have been procesover the past 3.5 Gyr by mechanical overturning and regoformation.

6. COLLISION RATES ON PLUTO ANDCHARON REEVALUATED

Weissman and Stern (1994) estimated impact rates of Eand Oort Cloud comets onto both Pluto and Charon given eestimates of EKB population numbers. They showed thatouter Oort Cloud is a negligible source of impactors and thatdominant sources of impactors on both bodies are EKB and inOort Cloud comets, with the EKB population dominating ovthe inner OC by factors of a few, contributing a flux of some 24

impacts over the age of the Solar System onto Pluto’s surfaand 460 impacts onto Charon’s surface. Here we reexamine

faces of 102- and 1-km EKOs during 3.5 Gyr as a function of impactor radiu

im-

ula-anrgeedith

.ss.

a-

ultrger

aced

tionn-thesedlith

KBrly

thethenerer00

EKB flux onto Pluto and Charon using newer EKB populatioestimates described above in Section 3.

Figure 6 shows the number of EKO impacts onto Pluto aCharon over 3.5 Gyr, as a function of impactor radius. As abothe calculated flux of impactors includes impact cross-sectenhancement due to the focusing effects of Pluto’s and Charogravitational fields. Over 3.5 Gyr, the total number ofr = 1-kmcomets striking Pluto and Charon is approximately 8.9× 103

and 1.1× 103, respectively. (To compare with the Weissmaand Stern (1994) results, this amounts to some 1× 104 and1.2× 103 impacts onto Pluto and Charon, respectively, ov4.5 Gyr). Clearly, improved EKB population parameters haincreased the Weissman and Stern (1994) fluxes by∼3–5 times.Impacts of 1-km-radius comets onto Pluto occur on time scaof∼3.9× 105 years. Similar impacts on Charon occur on∼3.2×106-year time scales. The largest EKOs expected to have strPluto and Charon during the past 3.5 Gyr have radii ofr = 40and 20 km, respectively. We find from Eq. (3) that the resulti

ce,the

FIG. 6. Total number of impacts onto Pluto and Charon over 3.5 Gyr, as afunction of impactor size.

Eim

a

eses

d

f

ohtorp

y

e

or

hle

hr

r

enc

fromem,thanalodi-

lotheaursme,

inothnd

hile

ofpop-theinedvederi-ckeree

nted

hanverpling97butcketi-omnce

COLLISIONS IN THE EDG

largest crater diameters on both bodies due to present-daycollisions are roughly 123 and 75 km, respectively. Largerpact basins resulting from earlier, massive impacts may wunderly these more recent craters.

If we simply sum up the total area covered by cratersPluto’s surface over 3.5 Gyr, we find cumulative cratered surffractions of∼40, 0.4, and 0.003% forr > 4-, 40-, and 400-mprojectiles, respectively. However, the hydrodynamic escapPluto’s atmosphere (Traftonet al. 1997) implies that there habeen∼1–5 km of surface loss due to sublimation if the presescape flux has been maintained over the age of the Solar SyFor craters and basins with depths of 1 km (∼5 km diameter),Pluto’s surface may therefore be comparatively young, of or2× 108 years. On such a time scale, the cumulative cratefraction drops to∼2, 0.02, and 2× 10−4% for r > 4-, 40-, and400-m projectiles, respectively.

Unlike Pluto, Charon’s surface is not losing significant voatiles to atmospheric escape (Traftonet al.1997), and so shouldrecord the cumulative flux of projectiles encountered overlifetime. Over 3.5 Gyr, the cumulative cratered fractionsCharon are∼20, 0.2, and 0.002% forr > 4-, 40-, and 400-mprojectiles, respectively.

When the highly anticipated Pluto–Kuiper Express recnaissance mission reaches the Pluto–Charon system, one squite clearly expect Charon to display an older surface reflecthe time-integrated flux of projectiles. At the same time, Plutsurface should reflect the recent production population, theshowing the essentially instantaneous (i.e., recent-times) imflux.

Given the modeled EKB flux through the Pluto–Charon stem, we can also calculate the size of the smallest survivprimordial satellite in the system. The Pluto–Charon pair likformed as the result of a giant impact which may have left smasatellites or debris orbiting Pluto after the accretion of CharWe calculate that at 39 AU, with〈e〉=0.2048, objects smallethan r ≈ 1.5–2 km have catastrophic disruption lifetimes lethan 3.5 Gyr. Any primordial Pluto satellites smaller than tshould have been destroyed by collisions with EKB projectiAnalysis of archival HST images by Sternet al. (1994) showsthat at the 90% confidence level, no Pluto satellites larger tr ≈ 140, 46, and 42 km exist inside the Charon instability stbetween 1 and 2 arcsec from Pluto (i.e., between 1.1 andCharon’s orbital radius), and outside 2 arcsec from Pluto,spectively. If the Pluto–Charon system harbors undiscovesurviving primordial satellites, we conclude that they will molikely be in ther ≈ 2- to 46-km size range.

7. COLLISION RATES IN THE CENTAUR REGION

We have also applied our model to examine collision rain the Centaur region, between 5 and 30 AU. Slow leakagobjects from the EKB due to planetary perturbations sustaipopulation of objects in the giant planet region with dynami

lifetimes of order 5× 107 years (Duncanet al.1995, Levison andDuncan 1997). Chiron, Pholus, and the other presently kno

EWORTH–KUIPER BELT 227

KB-

ell

once

of

nttem.

erred

l-

itsor

n-ould

ing’s

ebyact

s-inglyllern.

ssiss.

anip,2.2re-ed,st

tesofs aal

FIG. 7. Same as Fig. 1, except for the population of Centaur objects.

Centaurs in this region are thus recognized as emissariesthe EKB, delivered to a less distant region of the Solar Systenabling more detailed observational studies of Centaursof EKOs. We therefore wish to understand how EKB collisionhistories recorded on the surfaces of Centaurs have been mfied in the Centaur region.

Figure 7 shows a collisional erosion/accretion boundary psimilar to Fig. 1, but for the population of known Centaurs. Thigher average orbital speeds and eccentricities of the Centconspire together to place them strongly in the erosive regiso that growth of larger objects by accretion is not possiblethat region at the present time. Such erosive collisions will bcontribute ejecta and fine dust in the 5- to 30-AU zone, acause EKOs in that region to slowly, but surely, lose mass win transit through this region.

To quantify such effects, we constructed a model diskCentaur objects based upon observational constraints on theulation by Jedicke and Herron (1997). Their determination ofdetection efficiency of the Spacewatch survey system, combwith the observed absolute magnitude distribution of obserCentaurs and a model of the orbit distribution based on numcal integrations of Levison and Duncan (1997), allowed Jediand Herron to determine that in the 5- to 30-AU region themust be fewer than∼2000 objects in the absolute magnitudrange−4< H < 10.5 (r > 26 km for pν = 0.04). Assuming thatthe Centaur absolute magnitude distribution can be represeby a power law, they find a slope parameterb= 4.05 over therelevant size range. This slope is just slightly less steep tthat favored for EKOs by Weissman and Levison (1997) othe same size range. As the Centaurs are a dynamical samof the EKB population, we therefore continue to use the WLpopulation size distribution as our favored size distribution,we will also report Centaur collision rates assuming the Jediand Herron best-fit power law. Finally, we distribute the esmated total number of Centaurs throughout a disk ranging fr5 to 30 AU with a heliocentric surface mass density depende

wnof R1.3 (Levison and Duncan 1997) and calculate collision ratesas above.

o

l

il

n

st

Oi

et

tc

eee

us

ofim-

-

few

actkm-- to

p-yr.

Ball,lli-by

nd

een

onny-par-hisdue

n’singanan

tsmee

U.tedac-

ernomem.rn

228 DURDA AN

FIG. 8. Total number of impacts onto a roughly Chiron-size Centaur otypical dynamical lifetime in the giant planet region of 50 Myr, as a functionimpactor size.

Figure 8 shows the resulting number of impacts onto a Chirsize object (r ≈ 90 km) at 14 AU, over a typical 5× 107-yeardynamical lifetime, as a function of impactor size. Collision timscales at 14 AU are∼300 times longer than for a comparabbody at 40 AU. At 14 AU, cratering collisions of 1-km-radiucomets onto Chiron-size targets occur every∼60 Gyr (every∼4 Gyr assuming the Jedicke and Herron best-fit size distrtion extends tor = 1 km). Clearly, there is very little collisionaevolution in the Centaur region. We therefore conclude thatcumulative collisional and cratering history of Centaurs is doinated by the time they spent in the Kuiper Belt and that oa negligible amount of collisional processing occurs while thare extant in the Centaur region.

Hughes (1991) has proposed that surface activity responfor outbursts and coma around Chiron might be the resulcratering impacts exposing supervolatile ices. Analysis of pdiscovery images of Chiron (Buset al.2000) indicates variabilityin surface activity on time scales of a few to several years.model calculations for Chiron yield cratering time scales w4-m-radius projectiles of∼1 Myr for a WL97 size distribution.Even assuming that the Jedicke and Herron power law sizetribution is valid to sizes as small as 4 m, the mean time betwimpacts will be of order 200 years, significantly longer thanobserved time scale of outburst activity on Chiron. We mightfortunate to witness one such cratering impact on a singleject, but given the lengthy time scales, one must conclude thapredominant surface activity on Chiron is not caused by impa

8. CONCLUSIONS AND DISCUSSION

We have updated our previous collision rate model (St1995) to include a more precise treatment of encounter spand collision cross sections, and have incorporated new

mates of the EKB population size and structure (WL97). Bason this we find:

D STERN

f aof

n-

ees

bu-

them-lyey

ibleof

re-

urth

dis-en

hebeob-thets.

rnedssti-

1. Collision time scales in the present EKB for 1-km-radicomets onto 100-km-radius objects are∼6.5× 107 to 4.1×108 years. Over 3.5 Gyr this amounts to∼8–54 such impactsonto a single 100-km target. Given the estimated populationsuch objects in the present EKB, there should be one suchpact somewhere in the 30- to 50-AU region every∼1.4× 104–9.0× 104 years. Impacts of 4-m-radius projectiles onto 1-kmradius comets occur on 2.5× 107- to 4.7× 107-year time scales,resulting in∼90–300 cratering impacts withr > 4-m projectilesonto individual comets. Over the entire population of∼2× 109

comets in the EKB, there should be one such collision everydays.

2. Assuming relative encounter speeds of∼1.1–1.4 km s−1

between objects in the 30- 50-AU region, and using impstrengths from published scaling laws, we estimate that 100-scale radius EKOs can be catastrophically disrupted by 5384-km radius projectiles, yielding disruption lifetimes of 3×1012–8× 1012 years in the present EKB. Catastrophic disrution time scales for 1-km-radius comets range from 1 to 10 G

3. Objects smaller than aboutr = 2.5 km have collisionaldisruption lifetimes less than 3.5 Gyr in the present-day EKcollisional environment. It can be expected that most smcomet-size bodies, even primordial objects not formed as cosion fragments, have been heavily damaged in their interiorslarge collisions.

4. The cumulative fraction of the surface area of 1- a100-km-radius objects cratered by projectiles withr > 4 mranges from a few to a few tens percent over 3.5 Gyr.

5. Over 3.5 Gyr, Pluto and Charon are estimated to have bimpacted by 8.9× 103 and 1.1× 103 1-km-radius comets, re-spectively. Impacts of 1-km-radius comets onto Pluto occurtime scales of∼3.9× 105 years. Similar impacts on Charooccur on∼3.2× 106-year time scales. Because of the hydrodnamic escape of Pluto’s atmosphere, its surface may be comatively young for craters with depths less than about 1 km. In tcase, fresh craters may cover less than 2% of Pluto’s surfaceto impacts by projectiles with radii greater than 4 m. Charosurface, in contrast, should appear substantially older, recorda history of impacts since its formation, and having more th20% of its surface cratered by projectiles with radii greater th4 m.

6. In the Centaur region, collisions of 1 km radius comeonto 100 km radius targets (roughly Chiron’s size) occur on tiscales as long as∼60 Gyr. Collision time scales at 14 AU ar∼300 times longer than those for comparable bodies at 40 AThe collisional and cratering histories of Centaurs are dominaby the time they spent in the EKB. The predominant surfacetivity on Chiron is not likely caused by impacts.

Our results, like those of Davis and Farinella (1997) and St(1995), show that the small bodies we call comets coming frthe EKB must be young compared to the age of the Solar SystThis will not be the case in the Oort Cloud (OC), where Ste

ed(1988) showed collisions are rare. Thus, we can predict a majordifference between OC and EKB comets: age, which should

G

ne

a,ioA

c

i

v

r

g

ip

o

n

v

r

ss

o

,

a

u

rI

ces.

.),

taur

na

ily

ls. In.

ated

belt

sed.

tory

of

a-

ialap.

isonuto–

. K.luto,

pe

ph,,97.

r07,

ns-

ss,

ron

COLLISIONS IN THE ED

manifest itself in CRE age, crater counts, regolith reworkiradiation effects on surface microstructure, and perhapsalbedo and upper crust chemistry.

ACKNOWLEDGMENTS

We thank Eileen Ryan and Stan Love for helpful reviews. We also thRobin Canup and Joel Parker for their comments on our draft manuscriptLevison for helpful discussions, and Bill Bottke for his independent calculatof collision rates as a check for our model. This work was supported by the NOrigins of Solar Systems Program.

REFERENCES

Ahrens, T. J., and S. G. Love 1996. Strength versus gravity dominance instrophic collisions.Lunar Planet. Sci.27, 1–2.

Benz, W., and E. Asphaug 1999. Catastrophic disruptions revisited.Icarus142,5–20.

Bottke, W. F., M. C. Nolan, R. Greenberg, and R. Kolvoord 1994. Velocdistributions among colliding asteroids.Icarus107, 255–268.

Brown, R. H., D. P. Cruikshank, and Y. Pendleton 1999. Water ice on KuBelt object 1996 TO66. Astrophys. J.519, L101–L104.

Bus, S. J. 1991. Detection of CN emission from (2060) Chiron.Science251,774–777.

Bus, S. J., M. F. A’Hearn, E. Bowell, and S. A. Stern 2000. 2060 Chiron: Eence for activity near aphelion.Icarus, submitted.

Chapman, C. R. 1997. Cratering on the Galilean satellites: Implications fosize distribution of cratering impacts in the solar system.Meteoritics Planet.Sci.32, A27.

Chapman, C. R., W. J. Merline, B. Bierhaus, S. Brooks, and the Galileo ImaTeam 1998. Cratering in the jovian system: Intersatellite comparisons.LunarPlanet. Sci.29. [abstract 1927]

Davis, D. R., and P. Farinella 1997. Collisional evolution of Edgeworth–Kubelt objects.Icarus125, 50–60.

Davis, D. R., C. R. Chapman, R. Greenberg, and A. W. Harris 1979. Collisievolution of asteroids: Populations, rotations, and velocities. InAsteroids(T. Gehrels, Ed.), pp. 528–557. Univ. of Arizona Press, Tucson.

Davis, D. R., E. V. Ryan, and P. Farinella 1995. On how to scale disrupcollisions.Lunar Planet. Sci.26, 319–320.

Davis, D. R., S. J. Weidenschilling, P. Farinella, P. Paolicchi, and R. P. Bi1989. Asteroid collisional history: Effects on sizes and spins. InAsteroids II(R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 805–826. UniArizona Press, Tucson.

Duncan, M. J., H. F. Levison, and S. M. Budd 1995. The dynamical structuthe Kuiper Belt.Astron. J.110, 3073–3081.

Durda, D. D., R. Greenberg, and R. Jedicke 1998. Collisional models anding laws: A new interpretation of the shape of the main-belt asteroiddistribution.Icarus135, 431–440.

Farinella, P., D. R. Davis, and S. A. Stern 2000. Formation and collisievolution of the Edgeworth–Kuiper belt. InProtostars and Planets IV(V. Mannings and S. Russel, Eds.). Cambridge Univ. Press, Cambridgein press.

Geissler, P., J.-M. Petit, D. D. Durda, R. Greenberg, W. Bottke, M. Nolan,J. Moore 1996. Erosion and ejecta reaccretion on 243 Ida and its moon.Icarus120, 140–157.

Gladman, B., J. J. Kavelaars, P. D. Nicholson, T. J. Loredo, and J. A. B1998. Pencil-beam surveys for faint trans-neptunian comets.Astron. J.116,2042–2054.

Greenberg, R., W. Bottke, M. Nolan, P. Geissler, J.-M. Petit, D. D. DuE. Asphaug, and J. Head 1996. Collisional and dynamical history ofIcarus120, 106–118.

Greenberg, R., M. Nolan, W. F. Bottke, Jr., R. A. Kolvoord, and J. Veverka 19Collisional history of Gaspra.Icarus107, 84–97.

os:

EWORTH–KUIPER BELT 229

g,ven

nkHalnsSA

ata-

ity

per

id-

the

ing

er

nal

tive

zel

. of

e of

cal-ize

nal

UK,

nd

rns

da,da.

Holsapple, K. A. 1993. The scaling of impact processes in planetary scienAnnu. Rev. Earth Planet. Sci.21, 333–374.

Hughes, D. W. 1991. Possible mechanisms for cometary outbursts. InCometsin the Post-Halley Era(R. L. Newburn, M. Neugebauer, and J. Rahe, Edspp. 825–854. Kluwer, Dordrecht.

Jedicke, R., and J. D. Herron 1997. Observational constraints on the Cenpopulation.Icarus127, 494–507.

Jewitt, D., and J. X. Luu 1993. Discovery of Kuiper belt object 1992 QB1.Nature362, 730–730.

Jewitt, D., J. X. Luu, and C. Trujillo 1998. Large Kuiper belt objects: The MauKea 8K CCD Survey.Astron. J.115, 2125–2135.

Levison, H. F., and M. J. Duncan 1997. From the Kuiper belt to Jupiter-famcomets: The spatial distribution of ecliptic comets.Icarus127, 13–32.

Lissauer, J. J., and G. R. Stewart 1993. Growth of planets from planetesimaProtostars and Planets III(E. H. Levy and J. I. Lunine, Eds.), pp. 1061–1088Univ. of Arizona Press, Tucson.

Love, S. G., and T. J. Ahrens 1996. Catastrophic impacts on gravity dominasteroids.Icarus124, 141–155.

Luu, J., and D. Jewitt 1996. Color diversity among the Centaurs and Kuiperobjects.Astron. J.112, 2310–2318.

Melosh, H. J., and E. V. Ryan 1997. Asteroids: Shattered but not disperIcarus129, 562–564.

Ryan, E. V., and H. J. Melosh 1998. Impact fragmentation: From the laborato asteroids.Icarus133, 1–24.

Ryan, E. V., D. R. Davis, and I. Giblin 1999. A laboratory impact studysimulated Edgeworth–Kuiper belt objects.Icarus142, 56–62.

Stern, S. A. 1988. Collisions in the Oort Cloud.Icarus73, 499–507.

Stern, S. A. 1995. Collisional time scales in the Kuiper disk and their implictions.Astron. J.110, 856–868.

Stern, S. A. 1996. Signatures of collisions in the Kuiper disk.Astron. Astrophys.310, 999–1010.

Stern, S. A., and J. E. Colwell 1997. Collisional erosion in the primordEdgeworth–Kuiper belt and the generation of the 30–50 AU Kuiper gAstrophys. J.490, 879–882.

Stern, S. A., J. W. Parker, M. J. Duncan, J. C. Snowdall, Jr., and H. F. Lev1994. Dynamical and observational constraints on satellites in the inner PlCharon system.Icarus108, 234–242.

Terrile, R. J., S. A. Stern, R. L. Staehle, S. C. Brewster, J. B. Carraway, PHenry, H. Price, and S. S. Weinstein 1997. Spacecraft missions to Pand Charon. InPluto and Charon(S. A. Stern and D. J. Tholen, Eds.)pp. 103–124. Univ. of Arizona Press, Tucson.

Trafton, L. M., D. M. Hunten, K. J. Zahnle, and R. L. McNutt, Jr. 1997. Escaprocesses at Pluto and Charon. InPluto and Charon(S. A. Stern and D. J.Tholen, Eds.), pp. 475–522. Univ. of Arizona Press, Tucson.

Veverka, J., P. Thomas, A. Harch, B. Clark, J. F. Bell III, B. Carcich, J. JoseC. Chapman, W. Merline, M. Robinson, M. Malin, L. A. McFaddenS. Murchie, S. E. Hawkins III, R. Farquhar, N. Izenberg, and A. Cheng 19NEAR’s flyby of 253 Mathilde: Images of a C asteroid.Science278, 2109–2114.

Ward, W. R. 1996. Planetary accretion. InCompleting the Inventory of the SolaSystem(T. W. Rettig and J. M Hahn, Eds.), ASP Conference Series, Vol. 1pp. 337–361. Astronomical Society of the Pacific, San Francisco.

Weissman, P. R., and H. F. Levison 1997. The population of the traneptunian region: The Pluto–Charon environment. InPluto and Charon(S.A. Stern and D. J. Tholen, Eds.), pp. 559–604. Univ. of Arizona PreTucson.

Weissman, P. R., and S. A. Stern 1994. The impactor flux in the Pluto–Chasystem.Icarus111, 378–386.

Wetherill, G. W., and G. R. Stewart 1993. Formation of planetary embry

94.Effects of fragmentation, low relative velocity, and independent variation of

eccentricity and inclination.Icarus106, 190–209.