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

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  • Icarus 145, 220229 (2000)doi:10.1006/icar.1999.6333, available online at on

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    We pday Edpreviouulationtime scshowinnot primcollisiondisruptenvironteriors1-km-raon 7larger operiod.onto 1-kcumulajects craa few toKuiperis foundmodel iWe findregion,the colltime spon plansurfaceimpact-

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    0019-103CopyrightAll rightsollision Rates in the Present-Day KuiApplications to Surface Activation a

    Kuiper Belt Objects, Centaur

    Daniel D. Durda and S. A

    Southwest Research Institute, Space Studies Department, 1050 WalE-mail: durda@boulder.sw

    Received March 29, 1999; revised No

    resent results from our model of collision rates in the present-geworthKuiper Belt and Centaur region. We have updateds results to allow for new estimates of the total disk pop-in order to examine surface activation and modification

    ales due to cratering impacts. We extend previous resultsg that the surfaces of EdgeworthKuiper Belt objects are

    ordial and have been moderately to heavily reworked bys. Objects smaller than about r D 2.5 km have collisional

    ion lifetimes less than 3.5 Gyr in the present-day collisionalment and have probably been heavily damaged in their in-by large collisions. In the 30- to 50-AU region, impacts ofdius comets onto individual 100-km-radius objects occur

    1074 108-year time scales, cratering the surfaces of thebjects with 854 craters 6 km in diameter over a 3.5-GyrCollision time scales for impacts of 4-m-radius projectilesm-radius comets range from 3 107, to 5 107 years. The

    tive fraction of the surface area of 1- and 100-km-radius ob-tered by projectiles with radii larger than 4 m ranges froma few tens percent over 3.5 Gyr. The flux of Edgeworth

    Belt projectiles onto Pluto and Charon is also calculated andto be 35 times that of previous estimates. Our impact

    s also applied to Centaur objects in the 5- to 30-AU region.that during their dynamical lifetimes within the Centaur

    objects undergo very little collisional evolution. Therefore,isional/cratering histories of Centaurs are dominated by theent in the EdgeworthKuiper Belt rather than the time spentet-crossing orbits. Further, we find that the predominantactivity of Centaur objects like Chiron is most likely notinduced. c 2000 Academic Pressords: centaurs; Chiron; comets; Kuiper Belt objects; Pluto.


    isions are the dominant evolutionary process acting onmall bodies in the Solar System. In the main asteroidr instance, cratering collisions have greatly modified thes of individual asteroids by leaving large impact scarsreenberg et al. 1994, 1996, Veverka et al. 1997) and

    buting regolith across their surfaces (Geissler et al. 1996),

    and catthe entDurda

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    5/00 $35.00c 2000 by Academic Press

    of reproduction in any form Belt and Centaur Regions:d Modification on Comets,, and PlutoCharonlan Sternnut Street, Suite 426, Boulder, Colorado

    vember 29, 1999

    astrophic collisions over the eons have left their mark onire population size distribution (Davis et al. 1979, 1989,et al. 1998).EdgeworthKuiper Belt (EKB) population represents an-ajor population of small bodies whose evolution is largelyby collisions (Stern 1995). Stern (1996) and Davis andla (1997) have further explored the rate of collisions be-comets in the region beyond 30 AU and found that col-l evolution is a highly important process in the EKB.onal evolution in the EKB has recently been reviewed byla et al. (in press).ough intrinsic collision rates (number of collisions perter2 per year) are lower by a factor of1000 in the EKBred to the main asteroid belt, the population of objectss 1000 times as great. As a result of these competing, the overall level of collisional processing of individualis of similar scale to that in the main belt.we seek to investigate the implications of the EKB col-

    ates for surface modification. In particular, we wish tote quantities such as the surface cratering fractions, andected largest crater sizes. In addition to a direct rele-

    for understanding comets and other objects in the EKB,seek to gain insights into what the PlutoKuiper Express

    raft (Terrile et al. 1997) may observe when it images thes of Pluto, Charon, and other EKB objects. Similarly,

    k to assess whether Centaur objects on transient orbitsiant planet region undergo further significant collisional

    sing.hat follows we first revisit previous collision rate calcula-tern 1995, 1996) in light of both new observational data,her fidelity modeling. Once improved collision rates ared, we go on to evaluate their effect on the surfaces ofin, and derived from, the EdgeworthKuiper Belt.


    n (1995) examined collision rates in the present-dayorthKuiper Belt beyond 30 AU, as a function of theadial and population size structure. The numerical model


    for calculating collision rates is described in detail in that paper,so only a brief recapitulation will be presented in this section; inthe next section we will describe changes and improvements thathave been made to the model to produce the results presentedlater in this paper.

    Thein-a-bolisiontotal dtion of100-This screasin1.6 timin thebution

    wherea totalponenin therange oper heour prbin. AmodelhiiD 12

    Oncbinnedthe cocalculathe insbodiesstruck

    c(rk; r


    whereat eachsolvin(a; heisity ofcompuits popdensity

    1 The9.92 m,

    speed of the impactor population against the targets, vesc is theescape speed of the combined targetprojectile pair, and g isthe gra














    ste1995 model is a static, multizone, multi-size-bin, particle-x collision rate model that calculates instantaneous col-

    rates. The colliding population is defined in terms of aisk mass and a single-valued power-law size distribu-objects in the disk, normalized by the total number of

    km-diameter and larger objects in the 30- to 50-AU zone.ize distribution is treated as a series of monotonically in-g radius r bins, with the objects in each successive bines larger in size (and 4 times more massive) than those

    preceding bin.1 The model also specifies the radial distri-of heliocentric surface mass density 6(r ) so that:

    6(r )D6orfl; (1)

    6o is a normalization constant which in effect specifiesEKB mass in the 30- to 50-AU zone. The power-law ex-

    t fl determines the heliocentric radial distribution of massdisk, with the two cases we consider defining a realisticf parameter space:fl D1 corresponds to a constant mass

    liocentric radial bin, while fl D2 (more realistic, andeferred case) corresponds to a declining mass per radialdisk-wide average eccentricity, hei, is adopted for eachrun; an equilibrium condition where the disk wedge anglehei is assumed (see, e.g., Lissauer and Stewart 1993).e the global properties of the disk are specified, the disk isinto a series of radially concentric tori 1 AU in width, and

    llision rates for objects at each semimajor axis are thented in a particle-in-a-box formalism. In this approach,tantaneous collision rate c (collisions/unit time) of targetwith semimajor axis a, eccentricity e, and radius rk beingby impactors of radius rl is

    l ; a; e; i; R)a(1Chei)XDa(1hei)

    rG Mfl42a3

    T (a; hei; R)n(rl ; R) vkl(a; hei; hii; R)

    g(rk; rl ; vkl ; vesc[kCl]); (2)

    T (a; hei; R) represents the time the target body spendsdistance R during its orbit. T (a; hei; R) is computed by

    g the Kepler time-of-flight equation explicitly for every) pair in the models parameter space. The number den-impactors n(rl ; R) in the torus centered at distance R isted from the mass of the disk, the disks wedge angle hii,ulation size distribution, and its heliocentric surface massstructure (Eq. (1)). Here vkl is the local average crossing

    radius of the smallest bin was 3.94 m; successive bin radii were 6.25 m,etc.

    Weoutlin A

    lier min-a-bvD (hspeedlationin mucollisa GauAppe S

    the efing thfor thmodedue tosion (

    Thfinal rprevioand mwe hacollishave J

    providexistthat mestimduct n1:4 F

    by a mN (di )d0, wWL97

    For retimate

    2 Attimatescomet-appearshort-pWL97.vitational-focusing corrected collision cross section.


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

    more exact treatment of relative impact speeds. In the ear-del, relative impact speeds were calculated by a particle-x approximation of the orbital motion of the target:i2Chii2)1=2vk , where vk is the average Keplerian orbital

    of the target. We now include in the collision rate calcu-the difference between the collision frequency of bodies

    ual Keplerian orbits and that based on particle-in-a-boxons, so that vD ( 54 hei2Chii2)1=2vk , as well as the effect ofsian speed distribution (cf., Wetherill and Stewart 1993,dix A).etting realistic limits on gravitational focusing. Previously,ects of gravitational focusing were unconstrained, allow-

    collision cross section g to grow unrealistically largemost massive targets and for very low hei. In the presentwe now include limits on the gravitational focusing factorKeplerian shear, three-body effects, and velocity disper-f., Ward 1996, Eqs. (9) and (11)).se improvements are numerical refinements affecting theesults at only about the 10% level as compared with ourus calculations; nonetheless, they are worth documentingake the final results more robust. Of greater importance,ve updated the input parameters necessary to computeon rates in the EKB, based on observational advances thatccurred since 1995. In particular, these are:witt et al. (1998) and Gladman et al. (1998) have eached convincing evidence that between 30 and 50 AU theret least 70,000 objects with r > 50 km, and perhaps twiceany. This is between 2 and almost 5 times the populationtes for such bodies available in 1995. We therefore con-ew model runs with normalizations of both 7 104 and05 objects with r > 50 km.

    urther, the population size distribution is now representedore sophisticated, two-component power law of the form/ dbi ddi , where bD3 for di < d0 and bD4:5 for di>th d0D 10 km (Weissman and Levison 1997, hereafter).2

    ference, a WL97 size distribution, coupled with an es-d population of 70,000 objects with r > 50 km, yields

    arge sizes the WL97 size distribution is consistent with the latest es-by other researchers (Gladman et al. 1998, for instance). For smaller,ize objects, simple, single power-law extrapolations from larger sizeso overestimate the number of small EKB objects needed to supply theriod comet flux (Duncan et al. 1995), hence the broken power law of


    4564 objects in our models r D 102:4-km size bin and1:2109 objects in the r D 1-km size bin. For a population of 140,000objects larger than r D 50 km, the number of objects in all sizebins doubles accordingly. We continue to model the spatial dis-tribution of objects in the 30- to 50-AU region as a disk, withour preferred surface mass density index fl D2, as describedabove.

    We have compared our modeled collision rates with thosecompu1999,the twocussedof thos(Bottktion. Cbias-coand ththat iswe con

    quite g

    a. Col

    We30- to 5and up

    Firsdynamtweensive. Tat relaget bobetweemass gStern 1some 1have bwith mto the

    Notjects indistancproachfrom tgies sunot inc

    3 Relathan the

    4 Nevattributasize dist













    )ted from the observed distribution of EKO orbits (Bottkepers. commun.) and find very good agreement between

    independent methods. Our modeled collision rates, dis-in the following sections, are within a factor of 24

    e calculated based on an Opik-style collision rate modele et al. 1994) applied to the observed EKO orbit distribu-onsidering the fact that we have not made any attempt torrect the observed orbit distribution for this comparison

    e fact that our disk model has an inclination distributionsomewhat colder than the observed EKO population,3sider the agreement between the two calculations to beood.4


    lision Outcomes

    now present results of collision rate calculations for the0-AU region obtained with our improved collision model

    dated input parameters.t, however, it is important to remember that given theical conditions of the present EKB, mutual collisions be-EdgeworthKuiper Belt objects (EKOs) are generally ero-hat is, above some critical eccentricity, e, impacts occurtive speeds high enough that most ejecta escapes the tar-dies. Figure 1 shows the critical eccentricity boundaryn erosional (i.e., net mass loss) and accretional (i.e., netain) regimes for mutual collisions between EKOs (see996). Our contribution here, in Fig. 1, is the addition of28 multiopposition EKOs for which fairly reliable orbitseen determined, so that this large population of objectsoderately well established orbits can be evaluated relativecritical eccentricity boundary that the critical eccentricity for mutually colliding ob-

    the EKB increases slightly with increasing heliocentrice due to the direct linear dependence of the typical ap-speed upon the local Keplerian orbital speed. Farther

    he Sun, higher heis are required to generate impact ener-fficient to guarantee erosive collisions. Thus, if hei doesrease with heliocentric distance, collisions will tend to be

    tive to observed EKO eccentricities, observed inclinations are higherhiiD 12 hei equilibrium values assumed in our model.ertheless, we remind the reader about the large model uncertaintiesble to using simple power laws for both the orbital and the populationributions.

    FIGaccretiCriticaSoD 3gets atstrengtproperKuiper

    less eFor hmore

    ruptedcase o

    Ththat mundermostchanithoseily cohave

    b. Co

    Figcur foprojec

    ThpopulN (r >heredisk awiththe to

    5 Infor the50 kmbe redu. 1. The critical eccentricity (e) boundary between erosional andnal outcomes for collisions between EdgeworthKuiper Belt objects.eccentricity boundaries are shown for both strong (D 2 g cm3 and106 erg g1) and weak (D 0:5 g cm3 and SoD 3 104 erg g1) tar-

    two representative heliocentric distances (30 and 50 AU). The collisions chosen for the strong and weak cases bound a wide range of material

    ies and, we believe, the likely range of collision strength parameters ofBelt Objects.

    osional in nature as we move outward through the EKB.i greater than the critical eccentricity, e, impacts ejecttarget mass than is retained, and the target is either dis-in response to a catastrophic collision or eroded in the

    f a cratering collision.plotted data points for 128 multiopposition EKOs show

    ost large EKOs, like the main-belt asteroids, are currentlyoing predominantly erosive collisions, even under the

    pessimistic assumption, i.e., that of strong surface me-al properties. As to classical, kilometer-scale comets (i.e.,

    objects which leave the EKB to appear as the Jupiter Fam-ets), e values are so low as to guarant...