The early impact histories of meteorite parent bodies

Thomas M. DAVISON ∗1,2, David P. O’BRIEN3, Fred J. CIESLA1, and Gareth S.COLLINS2

1Department of the Geophysical Sciences, University of Chicago, 5734 S Ellis Avenue, Chicago, IL 60637, USA2Department of Earth Science and Engineering, Imperial College London, London, SW7 2AZ, UK

3Planetary Science Institute, 1700 E. Ft. Lowell, Suite 106, Tucson, AZ 85719, USA

To be published in Meteoritics & Planetary Science

DOI:10.1111/maps.12193

Abstract

We have developed a statistical framework that uses collisional evolution models, shock physicsmodeling and scaling laws to determine the range of plausible collisional histories for individualmeteorite parent bodies. It is likely that those parent bodies that were not catastrophically disruptedsustained hundreds of impacts on their surfaces compacting, heating, and mixing the outer layers;it is highly unlikely that many parent bodies escaped without any impacts processing the outer fewkilometers. The first 10 – 20 Myr were the most important time for impacts, both in terms of thenumber of impacts and the increase of specific internal energy due to impacts. The model has beenapplied to evaluate the proposed impact histories of several meteorite parent bodies: Up to 10 parentbodies that were not disrupted in the first 100 Myr experienced a vaporizing collision of the typenecessary to produce the metal inclusions and chondrules on the CB chondrite parent; around 1 – 5%of bodies that were catastrophically disrupted after 12 Myr sustained impacts at times that match theheating events recorded on the IAB/winonaite parent body; more than 75% of 100 km radius parentbodies which survived past 100 Myr without being disrupted sustained an impact that excavates tothe depth required for mixing in the outer layers of the H chondrite parent body; and to protect themagnetic field on the CV chondrite parent body, the crust would have had to have been thick (∼ 20km) in order to prevent it being punctured by impacts.

Keywords: Impacts, Asteroid; Asteroids, Thermal evolution; Planetesimal; Parent body

INTRODUCTION

The early Solar System was a violent place foryoung planetesimals. Collisions with other plan-etesimals were common, and would have affectedthe evolution of the bodies that would go on to be-come the asteroids and parent bodies of the me-teorites we find on Earth today. Planetesimals,the first rocky bodies to form in the Solar Sys-tem, accreted within the solar nebula, the cloud

∗e-mail address: thomas.davison@imperial.ac.uk

of dust and gas that orbited the young Sun. Ini-tially, highly porous dust aggregates up to centime-ters in scale accreted in low-velocity, pairwise colli-sions between dust particles (Blum, 2003; Wurmet al., 2001, 2004). How these aggregates grewfrom this scale to the kilometer scale is still anopen question, although leading theories suggestthat either hydrodynamic effects and gravitationalinstabilities (Johansen et al., 2007, 2009; Cuzziet al., 2008) or further low velocity collisions ledto a population of planetesimals ∼ 1 – 100 km inscale (Weidenschilling, 2011). These early bodies

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are likely to have contained a significant fractionof pore space (Dominik and Tielens, 1997; Blandet al., 2011). Some planetesimals grew to proto-planets; the first stage of this growth is termedrunaway growth (Greenberg et al., 1978; Wether-ill and Stewart, 1989, 1993), in which the largestplanetesimals gravitationally focused smaller bod-ies, increasing the number of collisions on their sur-faces (Kokubo and Ida, 1996), eventually drawingin all bodies from their so-called feeding zone. Asthe number of small bodies was diminished dur-ing runaway growth, protoplanets grew more slowlyduring the subsequent oligarchic growth phase, asimpactors needed to be drawn from further afieldand the protoplanets began to excite the remainingplanetesimal population, reducing the number ofcollisions on their surfaces (Kokubo and Ida, 1998).

These processes are expected to have resultedin a remnant population of planetesimals and asmaller number of embryos (∼ 100 – 1000 km inradius). Collisions between and among these pop-ulations would have been common, as the numberof bodies would have been much greater than thenumber of bodies that remain in the asteroid belttoday. For example, the population in the region2 – 4 AU could have been ∼ 100 – 1000 timesmore massive than the current asteroid belt (Bottkeet al., 2005a,b). Mass was lost from the populationas bodies were scattered into the Sun or ejectedout of the Solar System by Jupiter; the majorityof this mass depletion took place in the initial 100Myr period (Petit et al., 2001; Bottke et al., 2005a),meaning that the frequency of collisions during thatperiod was much greater than it was after mass de-pletion took place. Planetary embryos could have‘stirred up’ the smaller planetesimal population, in-creasing velocities above the mutual escape velocityof the colliding pair of bodies, up to several kilome-ters per second (Kenyon and Bromley, 2001). Assome bodies fell into resonances with Jupiter andSaturn, their orbits would have been excited to higheccentricities leading to impact velocities of at least10 km/s (Weidenschilling et al., 1998, 2001; Bottkeet al., 2005a).

As meteorites provide our strongest evidence ofconditions in the early Solar System, a full under-standing of the histories of their parent bodies isvital. Collisions played a major role in these histo-ries: Indeed, impacts have been invoked to explainmany observations from the meteorite record. For

example, shock wave processes have been shown tocause deformation of minerals and localized melting(Stoffler et al., 1991) and shock blackening (Hey-mann, 1967; Britt and Pieters, 1991); Wittmannet al. (2010) show that these effects typically post-date metamorphism from radiogenic decay. Im-pacts are able to perturb the onion-shell nature ofpetrologic types; for example, Grimm (1985) andTaylor et al. (1987) proposed nearly complete mix-ing of chondrite parent bodies by impact disrup-tion and reaccumulation. More recently, it has beensuggested that impact processing may have broughtsome heated material (petrologic type 4–6) fromthe interior of the parent body into close contactwith cooler material (type 3) from the outer lay-ers of the body (Harrison and Grimm, 2010; Scottet al., 2011). Impacts have even been invoked as apossible heating mechanism leading to resetting ofthermochronometers or phase transitions in moreextreme cases, for example on the IAB/winonaiteparent body (Schulz et al., 2009; Wasson and Kalle-meyn, 2002), the CB chondrites (Campbell et al.,2002; Krot et al., 2005) and the Angrite and Eucriteparent bodies (Scott and Bottke, 2011). Recentlythe effects of individual collisions between porousbodies has been quantified (Davison et al., 2010),as well as their influence on the long timescale ther-mal evolution of a parent body (Davison et al.,2012), demonstrating that localized heating canbe significant and allow higher temperatures to bereached than previously estimated assuming non-porous bodies (Keil et al., 1997).

The extent to which impacts would have affecteda meteorite parent body will vary from body tobody, and depend on the number, sizes and timingof the collision events. To date, there has been noquantitative analysis of the types of collision his-tories that a meteorite parent body might haveexperienced during its early evolution. Analyti-cal techniques (e.g Chambers, 2006), can provideestimates of the number of mutual collisions be-tween planetesimals of a given size. However, thisapproach does not provide histories for individualparent bodies that are subject to impacts from anevolving population of smaller planetesimals. Toinvestigate the role of collisions on the thermal anddisruptional histories of meteorite parent bodies,and to find the likelihood that collisions could haveproduced the effects they have been invoked to ex-plain in the meteorite record, it is important to

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know what kinds of collisions occurred.This study aims to address this gap in our knowl-

edge of impact processes, and quantify: (1) howmany impacts a parent body is likely to sustainduring the first 100 Myr of its history; (2) howlikely a parent body is to be disrupted during thistime; (3) when collisions are most likely to occur;(4) the average and range of velocities of the im-pacts and how they change with time; (5) the timethat most disruptive impacts occur; and (6) howmuch collisional heating occurs during that time.In this paper we develop a Monte Carlo model todetermine the history of impacts on a parent bodysurface, and use this model to provide answers tothe above questions. In the next section, the detailsof the model are presented. Then, results from thismodel for a range of parent bodies and collisionalevolution scenarios are presented. Finally, the pre-dictions from the model are applied to several casestudies for which impacts have been invoked in theliterature, to test the plausibility of those scenarios.

METHODS

A large number of planetesimals would have beenpresent in the early Solar System, each with theirown collisional history determined by a series ofchance encounters with other planetesimals. There-fore the impact history of a parent body throughtime cannot be solved analytically, as each bodywould have experienced a different set of impactson its surface. A Monte Carlo approach is requiredto determine the range of possible impact historiesthat a parent body could have experienced, andwhat a typical impact history would have been.

In the Monte Carlo simulation developed for thisstudy, the collisional histories of many parent bod-ies are modeled, where the likelihood of a collisionwith a given impactor size occurring at a given timeand the velocity of each collision are calculatedfrom a probability distribution. The probabilityof a collision occurring with a given impactor sizeand the velocity-frequency distribution at a time tmust therefore be known beforehand. These dis-tributions are based on dynamical and collisionalmodels of terrestrial planet formation; their calcu-lations are outlined below, in the section Collisionalevolution modeling. For each collision, the result-ing crater size is estimated from crater scaling re-

lationships, the amount of heating is determinedfrom shock physics simulations, and whether theparent body is catastrophically disrupted can becalculated using strength estimates from simula-tions of catastrophic collisions (Benz and Asphaug,1999; Jutzi et al., 2010). These calculations areoutlined in the section ‘Collisions: Cratering anddisruption’.

Collisional and dynamical evolutionmodeling

Quantifying the impact history of a parent bodyrequires knowledge of the size-frequency distribu-tion (SFD) of planetesimals, the velocity-frequencydistribution (VFD) of collisions and the intrinsiccollisional probability, which are all known to betime-dependent. Simulations of the dynamical andcollisional evolution of the planetesimal populationin the early Solar System are able to provide es-timates of these quantities. O’Brien et al. (2006,2007) OBrien et al. (2006; 2007) performed simu-lations of terrestrial planet formation and the ex-citation and mass depletion of the early asteroidbelt using the SyMBA N-Body integrator (Dun-can et al., 1998). The gravitational interactions ofa population of Mars-mass planetary embryos andan equal mass of smaller planetesimals distributedfrom 0.3–4 AU were simulated, including the in-fluence of Jupiter and Saturn. The smaller plan-etesimal bodies are influenced by the gravity of thelarge bodies, but do not interact with each other,as is commonly done in such simulations, while thelarger bodies are influenced by the gravity of thesmall planetesimals and one another. Collisions oc-cur when an embryo runs into another embryo orplanetesimal, which leads to a perfect merger of theimpacting bodies, conserving linear momentum.

While mutual collisions between planetesimalswere ignored in the planetary accretion simulations,they are expected to have occurred during this pe-riod of planet formation. As such, the simulationswere analyzed to determine the evolution of the col-lision probability and VFD for the first 100 Myr ofdynamical evolution using the algorithm of Bottkeet al. (1994), where t = 0 is defined as the timewhen most of the gas has dissipated and gas dragno longer provides a damping effect. Bodies weredivided into two groups: Those that remained inthe asteroid belt at the end of the simulations, and

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1.0

Tot

alm

ass,

Mto

t

(nor

mal

ized

byin

itia

lm

ass)

c)

Figure 1: Evolution of the intrinsic collisional probabil-ity, the mean collision velocity and the total mass of plan-etesimals for the two different collisional evolution modelsdescribed in the text, CJS and EJS. Collisional probabilityand velocity are specifically for collisions of all planetesimalsonto the planetesimals that remain in the asteroid belt at theend of the simulations.

those that were either accreted by the terrestrialplanets or lost from the system. Collision veloc-ity distributions and intrinsic collision probabili-ties, Pi, were calculated for planetesimals withinand between these different populations through-out the time of interest (Figure 1).

The dynamical simulations just consider a sin-gle particle size for the planetesimals, so to trackthe evolution of the size distribution, we use aseparate collisional evolution model that takes thetime-dependent impact velocity, collision probabil-ities, and total mass determined from the dynami-cal simulations and evolves the size distributions ofthe planetesimal populations in time under mutualcollisions, assuming an impact strength for porousplanetesimals (Jutzi et al., 2010).

This model is modified from O’Brien and Green-

berg (2005) and OBrien (2009) to treat multipleinteracting populations and time-dependent colli-sional parameters, in an approach similar to thatof Bottke et al. (2005a). Two separate but in-teracting populations are considered: The bodiesthat eventually end up in the asteroid belt region(denoted Nrem), for the remnant population), andall other bodies (those that are eventually lost andthose that are accreted by the terrestrial planets,as described above; denoted Ndepfor the depletedpopulation). The size of the depleted populationis varied to provide a good match between the fi-nal remnant population in the model and the ob-served mass of the present-day asteroid belt, whereNdep= fNrem. For the non-asteroid belt bodies, aforced exponential decay term is applied, based onhow fast the populations deplete in the dynamicalsimulations. As time advances, the production ofcollisional fragments increases the number of smallbodies in both populations (Figure 2). Both popu-lations evolve through mutual collisions, but even-tually (after tens to ∼ 100 Myr) the non-asteroidbelt bodies are dynamically depleted and their con-tribution to the collisional evolution of the aster-oids is negligible, leaving only the asteroids. Thismethod is discussed in more detail in Bottke et al.(2005a).

These dynamical and collisional evolution sim-ulations depend upon the orbits of the gas giantsJupiter and Saturn. The initial orbital parametersof the gas giants are not well known. In this workwe consider two different end-member simulations,which were found to give reasonable agreement be-tween the final SFD of planetesimals and the cur-rent SFD of the asteroid belt. In both simulations,Jupiter and Saturn were included from the startof the simulation (Chambers, 2001; O’Brien et al.,2006). In one set of simulations, Jupiter and Sat-urn started with orbits consistent with their currentinclination and eccentricity (denoted EJS here, for‘Eccentric Jupiter and Saturn’). The embryos andplanetesimals were centered on the invariant planeof Jupiter and Saturn. In this simulation, the fi-nal mass of the planetesimals in the asteroid beltregion was a factor of 95 less than the total ini-tial mass of the planetesimals in the system (i.e.f = 95). In the other set of simulations, Jupiterand Saturn adopted initial orbits that were near-circular and co-planar (i.e. inclination and eccen-tricity were essentially zero), similar to that ex-

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Figure 2: Evolution of the two planetesimal populationsover the course of the CJS simulation with f = 100. Thenumber of small bodies rapidly increases from the initialdistribution due to collisional fragmentation. Depletion ofthe population begins at t = 0, and decays with a half-life of∼ 22 Myr (for comparison, the EJS model decays with a half-life of ∼ 11 Myr). After 4500 Myr, the depleted populationhas been completely removed from the disk, leaving only thebodies that remain in the asteroid belt.

pected based on the Nice model (Tsiganis et al.,2005; Morbidelli et al., 2005; Gomes et al., 2005,denoted CJS here, for ‘Circular Jupiter and Sat-urn’) denoted CJS here, for Circular Jupiter andSaturn). Planetesimals and embryos were centeredon the plane of Jupiters initial orbit. In these sim-ulations, the best match to the final SFD of theasteroid population was found when the final massof the planetesimals in the asteroid belt region wasa factor of 100 less than the total initial mass of theplanetesimals in the system (i.e. f = 100, which issimilar to the f value found for the EJS simula-tion and the results of Bottke et al. (2005a), whoused f values in the range ∼ 100 – 200, dependingon the formation time of Jupiter). Figure 2 shows

the evolution of the remnant and depleted popula-tions in the CJS simulation with f = 100. The fithere is in close agreement with the observed SFDof the current asteroid belt. However, the distribu-tion has fewer bodies between a few tens of km and100 km in diameter than the current asteroid belt,which may overestimate the collisional grinding inthis size-range (i.e. too many bodies of that sizehave experienced a disruptive collision), which inturn produces a larger number of small objects atthe expense of larger ones.

Figure 1 shows the evolution through time ofthe intrinsic collisional probability, Pi (Figure 1a inunits of km−2 yr−1; see Equation (1) to convert Pito a probability of a collision between a pair of plan-etesimals of radii rimp and rt, in a given time in-terval), the mean collision velocity (Figure 1b) andthe total mass of planetesimals (Figure 1c), for thetwo different simulations. For the EJS simulation,impact velocities increase sooner than for CJS, andthe collisional probability is higher in early times.However, after ∼ 1 Myr, the collisional probabilityin the EJS case decreases, while for CJS it remainsapproximately constant until about 10 Myr untildropping off. This is because in the EJS case bod-ies are rapidly excited to high inclination orbits,leading to the depletion of objects from the simula-tion, and thus fewer collisions occur after ∼ 1 Myr.In the CJS case, this excitation takes longer, andtherefore the collisional probability does not de-crease as quickly. These differences could play animportant role in the collisional evolution of a par-ent body, and the sensitivity of our results to theseassumptions is tested below.

The amount of damage done to main belt as-teroids in the first few 100 Myr is approximatelyequivalent to that done in the proceeding 4.4 Gyr,suggesting that the latter component probably can-not be ignored. However, the big difference in termsof impact heating is that (i) the parent bodies arenow cold and (ii) impact velocities are now muchlower (most impacts come from main belt projec-tiles), such that the amount of heat produced willbe much lower. Thus, for applications of impactheating, or discussions of the early impact histo-ries of parent bodies, this model is applicable. Ifwe wish to apply the results of this model to studythe complete impact history of a parent body, themodel would need to extend over the full historyof the Solar System. It should be noted that the

5

model employed here does not account for radialmigration of giant planets, such as that suggestedin the Grand Tack model (Walsh et al., 2011), andthus that scenario is not treated in this work. Theapproaches outlined here, however, are general andwe plan to investigate that particular model of So-lar System formation in a future study.

Monte Carlo Model

In this section, the model algorithm used in thiswork is described. For each parent body in ourmodel (one sampling of the Monte Carlo model), westep through time from t = 0 to t = 100 Myr. Ateach time, t, the probability, Pc, that a collision willoccur between the parent body, of radius rt, and animpactor of size rimp at this time, t, is calculated:

Pc (rimp, t) = NimpPi (rimp + rt)2

∆t (1)

The intrinsic collision probability, Pi, and num-ber of impactors in a given size bin, Nimp, at agiven time t, were estimated by interpolating be-tween data points output from the dynamical mod-els described above, where the SFD of bodies in thecollisional evolution models represented the popu-lation of potential impactors. While the collisionalevolution models provide statistics for the num-ber of planetesimals with radii, rimp, ranging from∼ 1 m to 500 km, to increase calculation speeds,and because smaller impactors are unable to dis-rupt or significantly process the parent body, onlyimpactors with rimp > 150 m are considered. Theboundaries of the impactor size bins are logarith-mic, with log10 (rmax/rmin) = 0.1, where rmax isthe upper bound of the radius of a planetesimal ina given size bin, while rmin is the lower bound. If tis between two of the output timesteps from the col-lisional evolution modeling, Nimp was determinedusing a linear interpolation on the SFD.

Next, a random number, R1, was generated,where 0 ≤ R1 ≤ 1. If R1 < Pc (rimp, t), then a col-lision was adjudged to occur between an impactorof size rimp and the parent body at this time, t. Ifa collision occurs, then the collateral effects of thatcollision are quantified: A second random number,R2, is chosen, to pick a collision velocity from theMaxwellian distribution of velocity (defined by themean velocity, v), which was also interpolated asneeded (see appendix).

Collisions: Cratering and disruption

With knowledge of the impactor size and velocity,and the parent bodys size (and hence gravity), thediameter of the transient crater formed by the im-pact, dtr, can be determined from crater scaling re-lationships determined from experiments (Schmidtand Housen, 1987; Holsapple, 1993) and numeri-cal modeling (O’Keefe and Ahrens, 1993; Pierazzoet al., 1997):

dtr = 1.61−βCDg−βd1−βimp v

2βimp

(π6

) 13

(2)

where g is the surface gravity of the parent body,dimp is diameter of the impactor (= 2rimp), andvimp is the impact velocity. The material-specificconstants for non-porous soil and sand were esti-mated by Schmidt and Housen (1987) as CD =1.6, β = 0.22 and CD = 1.54, β = 0.165, respec-tively. Based on numerical cratering simulations,Wunnemann et al. (2006) suggest that the con-stants for sand represent a good approximation formoderately porous, granular materials.

To convert the transient crater diameter to a fi-nal crater diameter, it is necessary to know whetherthe crater is a simple, bowl-shaped crater or a com-plex crater. The transition from simple to complexcraters is known to occur at ∼ 3.2 km on Earth,and ∼ 18 km on the Moon, and scales as the in-verse power of the surface gravity, g. Indeed, thisrelationship is thought to extend over a wide rangeof surface gravities, including to small bodies suchas Vesta (Melosh and Ivanov, 1999). Hence, thesimple-to-complex transition diameter, dsc, can beestimated by:

dsc =gmoonρmoondscmoon

gρt(3)

where gmoon, ρmoon and dsc are the surface grav-ity, density and simple-to-complex transition diam-eter on the moon, respectively, and ρt is the den-sity of the parent body. The final crater diame-ter can then be estimated, depending on whetherit is simple or complex, by the following relation-ships (McKinnon and Schenk, 1985; Collins andWunnemann, 2005). For a simple crater, the finaldiameter, df , is given by:

df = 1.25dtr (4)

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and the final crater diameter for a complex crateris given by:

df = 1.17d1.13tr

d1.13sc

(5)

Equations (2 – 5) assume that all craters occurin the gravity regime. Estimates of the transitionbetween strength-dominated craters and gravity-dominated craters suggest that on a 100 km ra-dius body, any crater larger than 2.3 km willbe in the gravity regime (Nolan et al., 1996). Asthe minimum size of impactors considered in thisstudy is ∼ 150 m radius, only a small proportionof craters are in the strength regime. For the simu-lations presented below, less than 0.1% of impactsare in the strength regime for 100 km and 250 kmradius parent bodies. For the simulations of 50 kmradius parent bodies, around 2 – 3% of craters arein the strength regime. As these are also the small-est craters, they have little effect on the overallstatistics of the size of craters on the parent body.If future studies are conducted to investigate thecollisional histories of smaller parent bodies, thestrength regime should be accounted for in the cal-culation of crater size.

The depth that the impact event excavates ma-terial from (i.e. the maximum depth from whichmaterial can be brought to the surface, rather thanbeing displaced downwards into the target) can alsobe determined from scaling laws and the same im-pactor parameters (impactor size and velocity, andtarget gravity). Melosh (1989) states that the ex-cavation depth, Hexc, is approximately one tenthof the transient crater diameter, i.e.:

Hexc =1

10dtr (6)

One limitation of the scaling relationships usedhere (Equations 2 – 6) is that they are defined forimpacts onto a planar surface. More studies of ex-periments and numerical simulations of impacts oncurved surfaces are required to define similar scal-ing laws appropriate for large impacts on a parentbody. Therefore, for very large craters (where thesize of the crater approaches the size of the targetbody) this scaling should be used with caution. Nu-merical cratering calculations have confirmed thisrelationship for craters of diameters up to the sizeof the planetary radius (Potter et al., 2012).

Another possible outcome of the collision eventis the catastrophic disruption of the target. Catas-trophic disruption is defined as a collision in whichthe largest fragment has a mass of less than half ofthe original target body mass. The quantity Q?Dis defined as the minimum specific impact energy(energy per unit mass of the target) required todisrupt the parent body:

Q?D =mimpv

2imp

2mt=ρimp

(r?imp

)3v2imp

2ρtr3t(7)

Hence, the disruptive impactor radius, r?i , canbe expressed in terms of the impact energy and thetarget size:

r?imp = rt

(2ρtQ

?D

ρimpv2imp

) 13

(8)

Replacing r?i with rimp in Equation 8 gives thedefinition of specific impact energy, Q:

Q =ρimpv

2imp

2ρt

(rimprt

)3

(9)

Benz and Asphaug (1999) used numerical model-ing to show that Q?D can be expressed by the func-tional form:

Q?D = Q0

( rt0.01m

)a+Bρt

( rt0.01m

)b(10)

where Q0, a, b and B are material specific con-stants. The first term on the right-hand side ofEquation (10) describes the strength regime, andis only applicable for parent bodies up to approx-imately rt = 100 m. Hence, for the parent bodiesconsidered in this work, only the second term onthe right-hand side needs to be considered. Severalrecent studies have attempted to quantify the dis-ruption threshold for a range of scenarios (e.g. Benzand Asphaug, 1999; Leinhardt and Stewart, 2009;Stewart and Leinhardt, 2009). One such study in-vestigated the effect of porosity on this importantparameter: Jutzi et al. (2010) used SPH modelingto simulate planetesimal collisions, and determinedmaterial constants for use in Equation (10). For im-pacts into a porous pumice target at 5 km/s, theyfound that B = 5.70 erg cm3/g2, and b = 1.22.The explicit porosity is not defined in that work,

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but the density is approximately half that of thenon-porous target material. While pumice is a verydifferent material to basalt or dunite, as this is theonly disruption criterion in the literature to incor-porate porosity, it is thus the best approximationfor use in our model of the early Solar System, whenporosity would have been significantly higher thantoday. For impacts into a non-porous basalt, theyfound B = 1.50 erg cm3/g2, and b = 1.29. To applythis disruption criterion to a wider range of colli-sion speeds, velocity scaling from Housen and Hol-sapple (1990) can be applied to the gravity regimeterm from Equation (10):

Q?D = Bρt

( rt0.01m

)b( vimp5000m/s

)2−b(11)

By combining Equations (8) and (11), the min-imum disruptive impactor radius can be expressedin terms of the target radius, the impact velocity,and the material specific constants of b, B, ρt andρi:

r?i =

(2B

ρ2rρimp

(0.01m)−b

(5000m/s)b−2

r3+bt v−bimp

) 13

(12)For each collision in our Monte Carlo model,

Equation (12) is used to determine if the collisionwill catastrophically disrupt the parent body. Ifdisruption does occur, the calculation on that par-ent body is stopped: No further collisions are mod-eled.

Collision heating

For each collision that occurs, the amount of heatdeposited in the target body can also be estimated.Numerical studies of collisions between planetesi-mals have quantified the amount of heating donebetween a range of planetesimal pairs, for a rangeof collision velocities up to ∼ 10 km/s (Davisonet al., 2010), and found that for an impactor of lessthan one tenth of the mass of the target, the mass ofmaterial shock heated to a given temperature wasa constant multiple of the impactor mass (wherethe constant depends on the impact velocity, targetporosity and starting temperature). For impactorsgreater than one tenth of the mass of the target,the heated mass was limited by the total mass of

Table 1: Peak shock pressures, entropies and specific inter-nal energy increases corresponding to the post-shock tem-peratures used in this worka.

Post shock Specific Peak shock Specifictemperature entropy pressure internal energy

[K] [erg/g/K] [GPa] increase [erg/g]

350 12925 1.4 5.29 × 105

400 14365 2.9 1.07 × 106

500 16810 5.8 2.16 × 106

600 18832 8.9 3.27 × 106

700 20552 11.9 4.39 × 106

800 22048 15 5.51 × 106

900 23371 18.1 6.63 × 106

1000 24557 21.3 7.76 × 106

1250 27070 28.9 1.06 × 107

1373b 28128 32.7 1.20 × 107

1500 29126 36.5 1.34 × 107

1750 30867 43.7 1.62 × 107

2000 32377 50.8 1.90 × 107

2053c 32673 52.3 1.96 × 107

aThe values in this table are derived from the ANEOSequation of state for dunite (Benz et al., 1989) and theε− α porous compaction model (Wunnemann et al., 2006),using the technique described in Davison et al. (2010).These values are appropriate for 20% porous dunite.bSoliduscLiquidus

the target. Here we extend those calculations toconsider impact velocities up to 50 km/s (greaterthan the maximum collision velocity expected fromthe collisional evolution models discussed above).

We used the iSALE-2D shock physics model(Wunnemann et al., 2006), which is based on theSALE hydrocode (Amsden et al., 1980). To sim-ulate hypervelocity impact processes in solid ma-terials SALE was modified to include an elasto-plastic constitutive model, fragmentation models,various equations of state, and multiple materials(Melosh et al., 1992; Ivanov et al., 1997). Morerecent improvements include a modified strengthmodel (Collins et al., 2004) and a porosity com-paction model (Wunnemann et al., 2006; Collinset al., 2011b). Davison et al. (2010) further adaptediSALE to allow the simulation of collisions betweentwo planetesimals. iSALE shock physics calcula-tions were performed for collisions between a 10km radius impactor into a 100 km radius targetbody (a mass ratio of 1/1000), for non-porous and20% porous planetesimals. The planetesimals weremodeled as spheres, using the ANEOS equationof state tables for dunite (Benz et al., 1989). To

8

determine the mass of material shock heated toseveral different temperatures, Lagrangian tracerparticles were placed in the computational mesh.The mass of all particles that experienced a thresh-old peak pressure required to reach the desiredfinal, post-shock temperature was determined inpost-processing (Pierazzo et al., 1997; Pierazzo andMelosh, 2000; Ivanov and Artemieva, 2002; Davisonet al., 2010). Figure 3 shows the mass of materialheated to several temperatures for 20% porosity,for the full range of velocities considered in thisstudy. To determine the heated mass in the MonteCarlo model, a linear interpolation between the twoclosest velocities is used. If the heated mass inter-polated from the iSALE models exceeds the massof the parent body, the heated mass recorded fromthis collision is limited to the parent body mass.The increase in specific internal energy for eachtemperature increase is also listed in Table 1. Foreach impact, therefore, the total amount of energythat is used for heating the parent body, Qheat, canbe approximated by:

Qheat = ∆Q2053M (> T2053) +

n−1∑j=1

∆Qj (M (> Tj)−M (> Tj+1)) (13)

where ∆Qj is the specific internal energy in-crease associated with shock heating the materialto the post-shock temperature Tj (listed in Table1), M (> Tj) is the mass heated to the given tem-perature in the collision (Figure 3). Qheat can beconverted to a specific internal energy increase forthe parent body by dividing by the parent bodymass; however it should be noted that Qheat will benon-uniformly distributed within the parent bodyenergy will be localized to the impact sites.

The techniques used here to determine the fi-nal temperature of the material are dependent onthe accuracy of the equation of state for convertingpeak shock pressures to post-shock temperatures.As the ANEOS equation of state does not accountfor latent heat, ANEOS over-estimates tempera-tures in excess of the melt temperature. To ac-count for this, the peak shock pressures and en-tropy that correspond to the post-shock tempera-tures that are used in this work are presented inTable 1. As ANEOS equations of state are im-

0 10 20 30 40 50Velocity, vimp [km/s]

0

50

100

150

200

250

300

350

M(>

T)/

Mim

p

400K

700K

1000K

1373K

2053K

Figure 3: Mass of material heated in a collision betweentwo 20% porous dunite planetesimals, where the impactorhas one tenth the radius of the target (one thousandth of themass), for a range of impact velocities. The heated mass isnormalised to the mass of the impactor. Both planetesimalshad an initial temperature of 300 K. T = 1373 K and 2053K are the inferred dunite solidus and liquidus, respectively.

proved in the future (for example by defining dif-ferent thermodynamic constants to be used for dif-ferent high-pressure phases, and by treating eachphase as a separate material in ANEOS), theseshock pressures and entropies can be used to amendthe temperatures quoted in the remainder of thiswork. However, as shock melting is minimal inmost cases presented here, the latent heat effectsintroduce only small uncertainties to our results.

Timestep

After these calculations have taken place for all im-pactor size bins, the timestep is advanced and theprocess is repeated, unless a disruptive impact hasoccurred, in which case the calculation for this par-ent body ceases, and the calculation of the colli-sional history of the next parent body starts. Thesize of the timestep is chosen so that the probabilityof a collision by an impactor in the smallest size bin,Pc (rmin, t) (which is the most likely collision to oc-cur), is less than unity. To explore the sensitivity ofthe results to the size of the timestep, simulationswere run with different timestep values, so thatPc (rmin, t) = 0.1, 0.25, 0.5 and 1. For Pc (rmin, t) =1, the total number of collisions experienced by theparent body was less than for Pc (rmin, t) 0.5, butdid not change for Pc (rmin, t) = 0.1 – 0.5. There-fore a timestep limit of Pc (rmin, t)= 0.5 was usedfor all simulations presented here. This approach

9

is an approximation of a Poisson distribution.

Assumptions and limitations

Time period

The model described above focuses on the early im-pact evolution of the Solar System. Only the first100 Myr of Solar System evolution are accountedfor, as this is the time period in which the greatestnumber of impacts occurred. By 100 Myr, the in-trinsic collision probability and the total mass ofplanetesimals are significantly lower than at thestart of the calculation. As the results below show,the frequency of collisions at t ≈ 100 Myr is muchlower than in the first ∼ 20 Myr. Collisions afterthis time are not accounted for, but in this study weare focused on the earliest effects of collisions whenimpacts were most frequent. For future studies in-terested in a later period of Solar System history,the collisional evolution models could be run for alonger period of time, and the same Monte Carlomethods applied to those results.

The parent body

To fully quantify the amount of heating possiblefrom multiple impacts on the surface of a mete-orite parent body, no other heat sources (e.g. 26Aldecay) were considered in this study. The parentbody was therefore assigned a constant initial tem-perature of 300 K, which is the reference tempera-ture of dunite from the ANEOS equation of state(Benz et al., 1989; Thompson and Lauson, 1972),used in hydrocode models to determine heatingfor specific impact scenarios (Davison et al., 2010).Such a temperature may be appropriate for a plan-etesimal inside the snow line in the early Solar Sys-tem, and therefore this assumption does not affectmany of the results of this paper. However, cratersize and impact heating calculations are dependenton the initial temperature of the material. Thus,to extend this study to a wider region of the SolarSystem, future studies investigating this parameterin more detail are required.

The conditions for solid materials in the earlySolar System are somewhat uncertain. It is likelythat the earliest solid bodies contained a significantfraction of pore space (Bland et al., 2011; Blum,2003; Dominik and Tielens, 1997; Love et al., 1993;

Wurm et al., 2001, 2004). The gentle manner inwhich dust grains must collide during planet for-mation likely leads to bodies with high porosities(Teiser and Wurm, 2009; Blum and Wurm, 2008).Even growth to planetesimal formation must oc-cur at very low velocities to ensure sticking (e.g.Weidenschilling, 1980, 1984, 1997) or to allow ag-gregates to become gravitationally bound (Cuzziet al., 2001, 2008; Johansen et al., 2007). Indeed, itis thought that the porous nature of early planetes-imals is seen in the fabric of the Allende meteorite,with initial porosities reaching 60 – 70% (Blandet al., 2011).

Once the planetesimal formed, porosity would di-minish: The rate of densification is a strong func-tion of internal pressure and temperature (Schwennand Goetze, 1978; Yomogida and Matsui, 1984;Henke et al., 2012). To simplify the calculationsin this work, it is assumed that all parent bodiesand impacting planetesimals modeled have a con-stant porosity that does not change with time. Itis possible to estimate the size of parent body forwhich porosity would be crushed out due to the in-ternal pressure at the center. Assuming the bulkdensity, ρt remains constant throughout the parentbody, then the pressure, P , at a distance r from thecenter of a body with radius rt can be expressed as(Turcotte and Schubert, 2002):

P =2

3πρ2tG

(r2t − r2

)(14)

For a 20% porous dunite parent body with a ra-dius of 250 km, the internal pressure at the centerwould be approximately 61 MPa, and for a bodywith a radius of 100 km, the pressure would beapproximately 10 MPa. Nakamura et al. (2009)performed crush experiments on a range of porousmaterials to determine the pressure at which porespace begins to collapse. For an initially 50%porous gypsum sample, crushing began at 20–30MPa, and at 50 MPa the porosity had been re-duced to 40%. An initially 40% porous sampleexperienced little reduction in pore space, evenafter 100 MPa of pressure was applied. Indeed,Menendez et al. (1996) show that static pressuresof 500 MPa to 1 GPa are required to lithify sand-stones on Earth. Therefore, the reduction of porespace in a cold parent body due to internal pres-sure alone is not likely to be significant. Densifi-cation would thus be minimal in bodies of the size

10

of interest, unless temperatures were elevated to> 700 – 800 K, at which temperatures sinteringby creep diffusion would allow for densification ontimescales of ∼ 106 years. At lower temperatures,primordial porosity would be preserved. Even withradiogenic heating, porous layers will be preservedin the outer regions of planetesimals (which aremost affected by impact) as a result of the low hy-drostatic pressures and temperatures expected inthat region (Henke et al., 2012). Studies of aster-oid densities show that most asteroids contain bothmacro- and microporosity (e.g. Britt et al., 2002;Britt and Consolmagno, 2003; Consolmagno andBritt, 2004; Consolmagno et al., 2008), suggestingthat at least some of the initial (micro) porosity ofa parent body would be maintained through time,along with the introduction of new (macro) poros-ity due to fracturing and breakup during impactevents. Each impact on a parent body could alsocrush out some pore space. This compaction wouldbe constrained to the locality of the impact site(Wunnemann et al., 2006; Davison et al., 2012).Hence, in the simulations presented here it is as-sumed that each impact occurs on a fresh, uncom-pacted target surface. Until we have better dataon the evolution of porosity through time, this as-sumption is a necessary simplification in the model.

Mass of the parent body

It is assumed that ejection velocities are suppressedduring impact into a porous target relative to thoseexpected during impact into a non porous target(Collins et al., 2011a), and therefore that all ma-terial heated during a planetesimal collision is re-tained on the surface of the parent body (unlessthe parent body is catastrophically disrupted; seebelow). As a consequence of this assumption, themass of the parent body remains the same through-out the simulation (the mass lost to ejection is ne-glected). While this is an over-simplification, it isnot expected to affect the mass of heated materialon the parent body, as it is likely that most heatedmass is not ejected during the collision (Collinset al., 2011a). In addition, the mass added fromimpactors is not accounted for in the simulation.As most impactors are small in comparison to theparent body (and those that are large are morelikely to lead to a disruption), this assumption isappropriate for a first-order approximation. In the

simulations discussed below, the average total vol-ume of impactors on a parent body ranges from 1 –20%, depending on the parent body size and colli-sional evolution model. Future work incorporatingcollisional outcome calculations (e.g. Davison et al.,2010; Leinhardt and Stewart, 2012) into the MonteCarlo simulation will address this limitation.

Impact angle

For each collision that takes place, no impact an-gle is assigned. For calculations of the disruptionthreshold, an impact angle of 45 is assumed, asit is the most probable impact angle (see above,and Jutzi et al., 2010). For calculations of craterdimensions and impact heating, the impact angleis assumed to be vertical. This limitation is be-cause much of the work defining the scaling lawswas performed using two- dimensional numericalsimulations, which employed axial symmetry. Ver-tical incidence impacts produce craters that areonly ∼ 10% larger than impacts at 45◦(Gault andWedekind, 1978; Elbeshausen et al., 2009; Davisonet al., 2011). Preliminary modeling work has shownthat the mass of material heated in an impact at45◦may be approximately 20–30% less than in ver-tical incidence impacts (Davison et al., 2012). Asthe use of three-dimensional simulations becomesmore common (as computational speeds improve),the effect of impact angle on crater size and im-pact heating can be quantified, and subsequentlyused to overcome this limitation in this work. Thisassumption only affects our estimates of the massheated in the planetesimal and does not affect thesize of the impactors or impact velocities that willbe experienced by a given body.

RESULTS

There exists a large parameter space to which thisapproach can be applied. In the following section,the results of the Monte Carlo model describedabove are presented for the collisional history ofa 100 km radius target body for the CJS modelof collisional evolution, with f = 100. A radiusof 100 km is representative of the expected size ofchondrite parent bodies (e.g. Harrison and Grimm,2010). In discussing the thermal consequences anddisruption threshold of the impacts, we assume a

11

uniform porosity of 20%, a value that is less thanthe expected initial porosity of fresh planetesimalsand close to the typical porosity seen in currentchondrites. While porosity likely evolves in com-plicated ways due to compaction from heating andimpacts as well as creation of pore space by impactfracturing and shear deformation, we ignore thisevolution for simplicity. In subsequent sections, weinvestigate the sensitivity of a parent bodys colli-sional history to the disk collisional evolution model(CJS or EJS), the size of the parent body (rt= 50,100 and 250 km) and the porosity (non-porous and20%) of the parent body.

CJS model, rt= 100 km, φ = 20%

Overview

Figure 4 shows statistics of the number of impactsexperienced by a 20% porous, 100 km radius par-ent body. On average, a parent body that survives100 Myr without being catastrophically disruptedsustains ∼ 851 ± 26 (1-σ deviations) impacts frombodies with a radius greater than 150 m (Figure4a). During this same time period, 7.6% of parentbodies experience a disruptive collision: i.e. theyexperience a collision for which Q > Q?D. Of thosethat are disrupted, the number of collisions withrimp> 150 m prior to disruption varies from 1 to∼ 900, with an approximately equal chance of re-ceiving any number of collisions within that range(Figure 4b). Figures 4c–e show the number of col-lisions sustained by the parent body for impactorslarger than 0.05 rt, 0.1 rt and 0.2 rt, respectively.On average, a parent body receives ∼ 1.04 impactswith rimp > 0.05 rt. The probability that a par-ent body will experience at least one impact of thissize or larger (defined here as η0.05rt) is 66%. Ap-proximately one in five parent bodies experiencean impact with rimp > 0.1 rt(η0.1rt = 17%) andof that number, approximately half are disrupted.The mean number of impacts for which rimp > 0.2rtis 0.09, and η0.2rt = 9%.

Of those that experience at least one impact inthis size range, around 4 out of 5 were disrupted—hence, the one in five that was not disrupted ex-perienced a collision with a large impactor at suffi-ciently low velocity to allow survival. A summaryof these statistics is presented in Table 2, along withstatistics from several other simulations.

760 780 800 820 840 860 880 900 920 940Number of impactors, rimp > 150 m (survivors)

0.00

0.02

0.04

0.06

0.08

Pro

bab

ility µ = 851.83

σ = 25.64

a)

0 250 500 750 1000Number of impactors, rimp > 150 m (disrupted)

0.00

0.01

0.02

0.03

0.04

Pro

bab

ility

b)

0 4Number of impactors, rimp > 0.05rt

0.00.10.20.30.40.5

Pro

bab

ility µ = 1.04

η0.05rt = 66.53%

c)

Survived 100Myrs Disrupted

0 1 2 3Number of impactors, rimp > 0.1rt

0.00.20.40.60.81.0

Pro

bab

ility µ = 0.18

η0.1rt = 16.88%

d)

0 1 2Number of impactors, rimp > 0.2rt

0.00.20.40.60.81.0

Pro

bab

ility µ = 0.09

η0.2rt = 9.27%

e)

Figure 4: Statistics of the number of impacts expected ona 100 km radius, 20% porous parent body, from the CJSmodel. a) The probability that a parent body that survivesto 100 Myr will receive a given number of impacts fromimpactors with radii > 150 m; µ and σ are the mean andstandard deviation. b) The probability that a parent bodythat is disrupted before 100 Myr will receive a given numberof impacts from impactors with radii > 150 m. c–e) Theprobability that a parent body will receive a given numberof impacts from impacts larger than a certain size. η is theprobability that a parent body will experience at least oneimpact of this size or larger.

12

Table 2: Statistics for disruptive events in Monte Carlo simulations for a range of parent body sizes and disk models,after 105 parent bodies had been modeled for each case.

Model f aNimp> rimp

c

rtb 150m0.05rt 0.1rt 0.2rt

Ndisruptd

[km] (survivors) %

CJS 100 50µ = 214.25 µ = 1.56 µ = 0.32 µ = 0.07

8.2σ = 13.96 η = 80.5% e η= 28.2% η = 7.3%

CJS 100 100µ = 851.83 µ = 1.04 µ = 0.18 µ = 0.09

7.6σ = 25.64 η = 66.5% η= 16.9% η = 9.3%

CJS 100 250µ = 5303.38 µ = 0.81 µ = 0.36 µ = 0.17

2.6σ = 61.84 η = 55.9% η= 30.6% η = 15.6%

EJS 95 50µ = 77.28 µ = 0.71 µ = 0.15 µ = 0.03

3.7σ = 8.55 η = 51.8% η= 14.0% η = 3.8%

EJS 95 100µ = 307.35 µ = 0.48 µ = 0.07 µ = 0.03

2.9σ = 15.86 η = 39.1% η= 6.5% η = 3.3%

EJS 95 250µ = 1912.58 µ = 0.30 µ = 0.13 µ = 0.06

1.3σ = 37.50 η = 26.0% η= 11.9% η = 6.1%

a f is the ratio of the total initial mass of the planetesimal population to the final mass ofplanetesimals in the asteroid belt.b rt is the radius of the parent body.c rimp is the impactor radius.d Ndisrupt is the percentage of parent bodies catastrophically disrupted within 100 Myr.e η is defined as the probability that a parent body will experience at least one impact greaterthan or equal to rimp. These figures are for parent bodies with 20% porosity.

Crater sizes, excavation depth and impact energyfrom individual collisions

In addition to impactor size, the energy (velocity)of the impact is critical for determining the extentto which materials are processed during a collision.For each of the impacts that occurred, we also de-termined the energy as a fraction of the disruptiveenergy (i.e. Q/Q?D). Figure 5a shows the meancumulative number of impacts on a parent bodyas a function of Q/Q?D. On average, each par-ent body will see one impact with a Q/Q?D of atleast 0.005. These impacts may be from relativelylarge impactors moving at low velocity or small im-pactors moving at very high velocities—this figuredoes not differentiate between them.

For each impact that occurs on the parent body,the depth that material is excavated from and thesize of the impact crater formed can be determinedfrom scaling laws (Equations 2 – 6). Figure 5bshows the number of impacts on a parent body asa function of the excavation depth: Each parentbody has a 50% chance of experiencing at least one

impact that excavates material from at least 10 kmdepth, while ∼ 100 impacts will excavate materialfrom a depth of at least 1 km. Figure 5c showsthe number of impacts on a parent body as a func-tion of the final crater diameter. The transitionfrom simple to complex craters on a 100 km ra-dius, 20% porous parent body occurs at Df≈ 320km: i.e. larger than the parent body itself; in otherwords, all craters that form on this parent body,and do not catastrophically disrupt it, will be sim-ple craters. On average, a parent body will experi-ence one impact that forms a crater at least 94 kmin diameter. Approximately 150 events per parentbody will form craters 10 km in diameter, or larger.On those parent bodies that are not disrupted dur-ing the initial 100 Myr, on average the total area ofall craters is ∼ 0.8 times the surface area of the par-ent body (assuming the craters are circular, with afinal diameter calculated from Equations 2 – 5).A representative example of the surface from onesurviving parent body is shown in Figure 6. Inthis figure, the craters were placed randomly on

13

−8 −6 −4 −2 0 2 4

Impact energy, log10(Q/Q?D)

10−2

100

102

104

Cu

mu

lati

ven

um

ber

ofim

pac

ts>

Q/

Q? D

a)

10−1 100 101 102 103

Excavation depth, Hexc [km]

10−2

100

102

104

Cu

mu

lati

ven

um

ber

ofim

pac

ts>

Hex

c

b)

100 101 102 103 104

Crater diameter, df [km]

10−2

100

102

104

Cu

mu

lati

ven

um

ber

ofim

pac

ts>

df

c)

Figure 5: Plots showing the average number of collisionsthat each parent body experiences as a function of the scaleof the impact. (a) Cumulative number of impacts with animpact energy greater than Q/Q?

D; (b) Cumulative numberof impacts that excavate material from greater than Hexc

depth; (c) Cumulative number of impacts that form a finalcrater larger than df . Dotted lines in (b) and (c) repre-sent events for which the scaling laws used are not suffi-cient to describe the outcome of the impact: i.e. an exca-vation depth greater than the radius of the parent body ora crater diameter larger than the parent body diameter arenot physical; these events likely represent either disruptiveor sub-catastrophic events, which may not form a traditionalimpact crater.

a square surface with an area equal to the parentbody surface area. Figure 6 shows that while somecraters will overlap one another, a large number ofthe craters will form at least partially on a fresh,uncompacted surface, and that only a small portionof the surface will be left unaffected by collisions.Thus, the assumption of the model that all cratersform on a porous surface is reasonable. This calcu-lation assumes that each crater is formed by verti-

0 100 200 300x (km)

0

100

200

300

y(k

m)

Figure 6: The craters from one 100 km radius, 20% porousparent body from the CJS simulation, randomly placed ona square with an area equal to the surface area of the parentbody, to show the coverage of craters on an average surviv-ing parent body. While some craters overlap, most form atleast partially on a fresh surface. Darker shading representsoverlapping craters.

cal impact, and thus should be viewed as an upperlimit. Experiments and three dimensional model-ing suggest that at an impact of 45◦, the crater areamay be approximately 20% smaller in area thana normal incidence crater (Burchell and Mackay,1998; Davison et al., 2011). Figure 5 shows thatwhile not all of the ∼ 850 events that are predictedto occur from the model will have a global effecton the parent body, there are likely to be hundredsof events that will influence the outer few km ofthe parent body: That is, the region most likely tobe the source region for low petrologic type chon-drites (e.g. Harrison and Grimm, 2010). Only asmall fraction of the surface area is likely to escapewithout being directly processed by impacts.

Cumulative impact energy

The total amount of impact energy deposited ona parent body can be determined by summing to-gether the impact energy, Q, for each impact. For99% of the parent bodies that survive to 100 Myrin this simulation, the total impact energy, Qtot, iswithin the range 4.5 × 107 – 6.3 × 109 erg/g, witha geometric mean Qtot of 2.3× 108 erg/g. For par-ent bodies that are disrupted, Qtot is several orders

14

of magnitude larger, mainly due to the energy ofthe final disruptive impact: 99% of disrupted bod-ies have a Qtot in the range 2.8 × 109 – 5.1 × 1013

erg/g, and the geometric mean Qtot is 6.0 × 1010

erg/g. For reference, Q?D from Jutzi et al. (2010)for an impact at 5 km/s into a porous 100 km par-ent body is ∼ 2 × 109 erg/g. Figure 7a shows ahistogram of Qtot for all parent bodies modeled. Afraction of this total impact energy is converted toheat; this is quantified below, in the section ‘Impactheating ’.

Most energetic impact

As Q?D depends on the impact velocity and the im-pactor size, the impact with the largest Q may notbe the impact closest to the disruption threshold.Indeed, in approximately 1 out of every 8 parentbodies, the most energetic impact is not the mostdisruptive. Therefore, the most energetic impactsustained on a parent body is defined here as theimpact with the largest Q/Q?D. Figure 7b showsthe distribution of the most energetic impact forall parent bodies modeled. For those parent bod-ies that are not disrupted, Q/Q?D varies between∼ 10−3 to 1, and the geometric mean is 0.01; i.e.the average impact energy of the most energeticimpact in parent bodies that are not disrupted isapproximately 1% of that needed to disrupt thebody. For reference, Q/Q?D = 0.01 is approximatelyequivalent to a 7.5 km radius projectile impactingthe 100 km radius parent body at 5 km/s. The col-lateral effects of this type of impact are discussedin more detail in Davison et al. (2012); such im-pacts can produce peak temperatures and coolingrates consistent with what is seen in type 3, 4, 5,and 6 H-chondrite meteorites. Figure 7c-d showsthe impact velocity and radius for the most ener-getic impact. A typical impact velocity in the mostenergetic impact is ∼ 7–10 km/s, but a range from∼ 1 km/s up to 25 km/s is possible. For bodiesthat are disrupted, the impactor radius is typicallyin the range 30 – 100 km. The minimum disruptiveimpactor radius in the simulations we performed is∼ 22.2 km (for which the impact velocity was 17.8km/s), and 99% of all disruptors are > 28 km inradius. For those bodies that are not disrupted inthe first 100 Myr, the average radius of the mostenergetic impactor is ∼ 7 km. For nearly 1% of sur-viving parent bodies, the most energetic impactor

7 8 9 10 11 12 13Total impact energy (all events)

log10(Qtot [erg/g])

0.00

0.04

0.08

0.12

Pro

bab

ility

b)

−4 −3 −2 −1 0 1 2 3 4Maximum impact energy in single event

log10(Q/Q?D)

0.00

0.04

0.08

0.12

0.16

Pro

bab

ility

a)

Survived 100Myr Disrupted

0 5 10 15 20 25 30Impact velocity for most energetic impact

vimp [km/s]

0.00

0.04

0.08

0.12

Pro

bab

ility

c)

0.0 0.5 1.0 1.5 2.0 2.5Impactor radius for most energetic impact

log10(rimp [km])

0.00

0.08

0.16

Pro

bab

ility

d)

−0.5 0.0 0.5 1.0 1.5 2.0Time of most energetic impact

log10(t [Myrs])

0.000.020.040.060.080.10

Pro

bab

ility

e)

Figure 7: Statistics for the amount of impact energy deliv-ered to a 100 km radius, 20% porous parent body from theCJS simulation. The ‘most energetic impact’ is defined asthe impact with the maximum Q/Q?

Don each parent body.Qtot is the total impact energy received on the parent bodyin the first 100 Myr.

was > 30 km in radius; these impactors have lowvelocities so the impact energy is below Q?D.

Time of impact

Figure 8 shows the average number of collisions ona potential meteorite parent body (impactors largerthan rimp = 150 m) during each 1 Myr period of the

15

0

5

10

15

20

25

30

35

40

Imp

acts

per

Myr

0

1

2

3

4

Sp

ecifi

cin

tern

alen

ergy

incr

ease

per

Myr

from

imp

acts

[10

7er

g/g/

Myr

]

0 20 40 60 80 100Time [Myr]

Energy increase

Number of impacts

Figure 8: The average number of impacts by impactorswith radius > 150 m that occurred in each 1 Myr period, forthe first 100 Myr (grey solid line), and the specific internalenergy increase that occurred due to those impacts (blackdashed line). These results are for a 100 km radius, 20%porous parent body from the CJS simulation. Averages weretaken after 100000 parent bodies were modeled. The mostimportant time for impacts was the first 10–20 Myr.

first 100 Myr. ∼ 19% of the collisions that a par-ent body experienced were during the first 5 Myr,∼ 39% during the first 10 Myr, and ∼ 64% duringthe first 20 Myr. 75% of impacts occur within thefirst 29 Myr. Within a factor of ∼ 2, the numberof impacts on a body in the first 100 Myr is ex-pected to be approximately equal to the number ofimpacts in the remaining 4.4 Gyr.

Figure 7e shows the time that the most energeticimpact occurs for the modeled parent bodies. Ap-proximately 21% of disruptive collisions occur inthe first 5 Myr, 43% occur in the first 10 Myr, and68% occur in the first 20 Myr. Of those parentbodies that are not disrupted in the first 20 Myr,97% will survive to the end of the 100 Myr periodsimulated by this model (for this study, ‘surviving’is defined as not being disrupted by 100 Myr).

The average time of the most energetic collisionin survivors occurs later than the disruptive colli-sions: Only 5% of these collisions occur in the first5 Myr, 16% occur in the first 10 Myr, and 39%occur in the first 20 Myr. Some of the most en-ergetic collisions experienced on the parent bodytherefore continue later than the disruptive colli-sions, although after ∼ 30–40 Myr, their frequencydecreases (only one third of the most disruptivecollisions occur after 40 Myr). The difference inthe timing of disruptive collisions and most ener-

getic collisions on surviving bodies is because: (1)if a disruptive collision occurs, the simulation isstopped, and no more collisions can occur on thatbody, which would therefore favor earlier collisions,and (2) while the average collision velocity contin-ues to increase until around 30 Myr, the numberof large impactors, capable of causing a disruption,decreases rapidly after ∼ 10 Myr.

Impact heating

Figure 9 shows the probability that a given fractionof a parent body will be heated to a given tempera-ture (assuming the impact occurs onto a fresh sur-face), for those parent bodies that survived 100 Myrwithout a disruptive collision (Figure 9(a)), andthose that were disrupted (Figures 9b–c). Thesefigures can be interpreted in two ways: First, ifone is interested in the outcome associated witha given probability, the figure can be read hori-zontally, to establish the mass fraction heated toa given temperature (this is the maximum massfraction, as oblique impacts are not considered).Second, to infer the probability of a particular out-come (e.g. 50% of the mass being heated to a giventemperature), the figure can be read vertically. Forexample, of the surviving parent bodies, there isa 20% chance that a parent body will see aroundone thirtieth of the mass heated to 400 K, and a30% chance that one hundredth of the mass willbe heated to the solidus. For those parent bod-ies that are disrupted, however, there is an 80%chance that the entire parent body will be heatedto 400 K, and a 50% chance that one fifth of themass of the parent body reaches the solidus. Com-paring Figures 9b and 9c, it is clear that the largeimpact that causes the parent body to disrupt isalso by far the most important impact in terms ofheating the target materials. Further modeling isrequired to determine how the heated material willbe distributed amongst the collisional fragments indisruptive events.

By using Equation (13), the specific internal en-ergy increase due to collisional heating Qheat canbe determined for each parent body. For those par-ent bodies that survive, the mean specific internalenergy increase due to all the collisions that occurin the first 100 Myr is 1.6× 108 ± 3.3× 108 erg/g.This represents ∼ 37.4 ± 3.6% of the predicted to-tal impact energy Qtot received by surviving bodies

16

0

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LiquidusSolidus

1000K700K400K

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10−4 10−3 10−2 10−1 100

Fraction of parent body heated,M(> T )/Mt

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Figure 9: The probability that a given fraction of a par-ent body will be heated by impacts to a range of temper-atures, for (a) those bodies that survive 100 Myr withoutbeing catastrophically disrupted, (b) those bodies that aredisrupted, immediately before the disruptive impact occurs,and (c) those bodies that are disrupted, after the disruptiveimpact.

and produces a globally averaged temperature in-crease of ∼ 15–35 K. This is consistent with earlierfindings that non-disruptive collisions are unlikelyto cause global heating on a meteorite parent body(Keil et al., 1997). For disrupted parent bodies,much more impact energy is converted to heat: Thetotal combined specific internal energy increase inthe target body is 9.6×109±6.7×109 erg/g (corre-sponding to global average temperature increases of∼ 600–1000 K). However, this represents a smallerpercentage of the total impact energy received onthose bodies (17.7 ± 11.8%), because much of that

impact energy is used mechanically to disrupt theparent body. As all the impacts used in this analy-sis are normal incidence impacts, the estimate ofthe amount of impact energy converted to heatshould be considered an upper limit: Oblique im-pacts would likely convert less impact energy intoheat.

Figure 8 also shows this increase in specific in-ternal energy as a function of time. This reiteratesthat the initial ∼ 10–20 Myr are the most impor-tant in terms of collisional processing and the de-livery of heat to the parent body, with the peakincrease of > 3× 107 erg/g/Myr occurring at ∼ 11Myr. However, this figure also shows that even af-ter 50 Myr, at which time heating from short-livedradionuclides would be negligible, collisions couldstill deliver ∼ 5 × 106 erg/g/Myr. Moreover, asmost impact energy is deposited in the outer lay-ers of the parent body (e.g. of the ∼ 850 expectedimpacts on a parent body, on average less than oneimpact will excavate material from more than 10km depth), the specific internal energy increase inthat material may be significantly higher. As afirst-order approximation, if it is assumed all thespecific internal energy is deposited in the outer 10km of the parent body, then the energies presentedabove and in Figure 8 could increase by a factor of∼ 4. This shows that while global heating by im-pacts is unlikely, heating on the local scale duringthe period of peak impact activity is possible.

Influence of parent body size

In addition to the simulation described above, simi-lar models with the CJS collisional evolution modelwere run with different parent body sizes: rt = 50km and rt = 250 km.

Number of impacts

Table 2 shows statistics for the number of collisionsand the probability of catastrophically disruptingthe parent body. A parent body of 50 km radiusexperiences a factor of ∼ 4 fewer collisions than the100 km parent body, and a 250 km parent body ex-periences ∼ 6 times more collisions than a 100 kmbody – these factors correspond to the change insurface area. Smaller parent bodies are more likelyto be disrupted, despite the lower frequency of col-lisions: 8.2% of 50 km radius bodies are disrupted,

17

7.6% of 100 km bodies, and 2.6% of 250 km bodies,due to the Q?D dependence on parent body size—the value of Q?D increases with target size as we arein the gravity regime.

The number of impactors one twentieth of the ra-dius of the parent body decreases with parent bodysize: On average, there are ∼ 1.6 impacts of thissize on a 50 km parent body, compared to ∼ 1.0on a 100 km body and ∼ 0.8 on a 250 km body.However, for larger relative impactor sizes, the re-lationship is not so simple: While one in five 100km parent bodies experiences at least one impactof at least one tenth of the parent body size, forboth 50 km and 250 km radius bodies, that numberincreases to one in three; for impactors that are atleast one fifth of the size of the parent body, 250 kmparent bodies experience approximately twice thenumber of impacts than do both the 50 km and100 km radius parent bodies. Thus, there is nota direct relationship between the number of largeimpactors and the size of the parent body. This islikely a consequence of the wavy nature of the size-frequency distribution; for smaller parent bodies,there are more large impactors because there aremore bodies in that size range in the population:For example, there are ∼ 40 times more 5 km radiusimpactors than 25 km radius impactors at 10 Myr.The transition between these two effects seems tobe somewhere between 50 km and 250 km radius,for impactors larger than one tenth of the parentbody size. Further modeling of a range of parentbody sizes could constrain this point of inflectionfurther.

Impact energy

For parent bodies that survived 100 Myr with-out disruption, the average impact energy received,Qtot, (the sum of the impact energy from all im-pacts on the parent body, normalized by the parentbody mass) is approximately the same for all sizesof parent bodies modeled here (Table 3). However,for those bodies that are disrupted, the larger par-ent bodies (rt = 250 km) receive a approximatelytwice the total specific impact energy: A larger spe-cific impact energy is required to disrupt a largerparent body (by a factor of 1.22 for porous mate-rials), which results in fewer of the larger parentbodies being disrupted. This is reflected in the av-erage impact energy from the most energetic im-

pact ((Q/Q?D)max), which decreases monotonicallywith increasing parent body size.

Sensitivity to collision evolutionmodel

Equivalent simulations were performed for thethree parent body sizes discussed above, this timeusing the second collisional evolution model param-eters (EJS model). Results for these models aresummarized in Tables 2 and 3. The total numberof impacts received on the surface of a planetesi-mal in the EJS model is a factor of ∼ 2.8 fewerthan the equivalent sized parent body in the CJSmodel. For impactors larger than a given size (0.05rt, 0.1 rt and 0.2 rt), the average number of im-pacts per parent body is a factor of 2 – 3 fewerin all cases listed in Table 2. This is in part be-cause the mass of the planetesimal population de-pletes more rapidly in the EJS simulations (Figure1c). Furthermore, the inclinations of bodies aremore rapidly excited in the EJS simulations, whichmakes collisions less likely: After 1 Myr, the intrin-sic collision probability is always lower in the EJSsimulation compared to the CJS simulation. Inthe EJS simulation, Pi = 3.4× 10−18 km−2 yr−1

at 1 Myr, and decreases monotonically to1.6× 10−18 km−2 yr−1 by 100 Myr; however, inthe CJS simulation, Pi remains steady between∼ 3.5 and 4.2× 10−18 km−2 yr−1 until 10 Myr,before decreasing to 1.6× 10−18 km−2 yr−1 at100 Myr. The fraction of parent bodies disruptedin the EJS simulation is also approximately a factorof 4 fewer than in the CJS simulation; this is as aresult of the eccentric orbits of Jupiter and Saturnexciting bodies more readily in the EJS simulation,thus leading to a more rapid reduction in the num-ber of planetesimals in the Solar System, which inturn leads to fewer collisions occurring.

Table 3 compares the amount of impact energydeposited on a parent body in the CJS and EJSsimulations. The sum of the impact energy fromall collisions that occur on surviving bodies is a fac-tor of ∼ 2 – 2.5 lower in the EJS simulations thanin the CJS simulations, again, reflecting the lowernumber of collisions. For those parent bodies thatare disrupted, the average total impact energy doesnot change significantly between the CJS and EJScases (because the impact energy required for dis-ruption is unchanged between the two models, and

18

Table 3: Average total impact energy, Qtot, sustained on a parent body, for each of the simulated scenarios.

Model f aGeometric mean Geometric mean Geometric

rtb Qtot (survivors) Qtot (disrupted) mean[km] [erg/g] [erg/g] (Q/Q?

D)maxc

CJS 100 50 2.7 × 108 7.0 × 1010 0.059CJS 100 100 2.3 × 108 6.0 × 1010 0.017CJS 100 250 2.5 × 108 1.2 × 1011 0.004

EJS 95 50 1.4 × 108 7.0 × 1010 0.022EJS 95 10 1.3 × 108 9.6 × 1010 0.007EJS 95 250 1.1 × 108 1.2 × 1011 0.001

a f is the ratio of the total initial mass of the planetesimal population to thefinal mass of planetesimals in the asteroid belt.b rt is the radius of the parent body.c (Q/Q?

D)max is the ratio of the maximum impact energy from a single impactto the impact energy required for catastrophic disruption.

if a disruptive impact occurs, that impact usuallydelivers the most energy).

As noted above, the collisional evolution mod-els that we employ in this work produce too muchcollisional grinding, leading to slightly fewer bodiesbetween 10 and 100 km in radius. This implies thatwe may be overestimating the number of disruptivecollisions in that size range, which should thus beconsidered as an upper limit.

Influence of porosity

A simulation was also performed in which the par-ent body and all impactors were assumed to haveno porosity, for the case of a 100 km radius parentbody with the CJS collisional evolution parameters.Therefore, the disruption criterion used the non-porous values for B and b (Jutzi et al., 2010), andthe calculation of heating on the parent body usedresults from iSALE simulations with non-porousmaterial. The number of collisions and the fre-quency of disruption are similar to the porous case:7.4% of non- porous parent bodies are disruptedin this simulation. The major difference betweenthe non-porous simulation and the porous simula-tion is the fraction of the impact energy receivedon the parent body that is converted to heat: Insurviving bodies in the non- porous simulation, 7.2± 2.6% of the total impact energy is converted toheat, and 2.8 ± 3.2% is converted to heat in dis-rupted bodies (in both cases, a factor of ∼ 5 less

than in the porous case). This reiterates the re-sult that the shock compaction of pore space cangreatly increase collisional heating (Davison et al.,2010).

DISCUSSION

The results in the previous section show that colli-sions on meteorite parent bodies would have beencommon events during the first 100 Myr. Manyparent bodies would have been disrupted by en-ergetic impacts during that time. Those thatsurvived without being disrupted likely sustainedmany impacts on their surfaces, processing (com-pacting, heating, and mixing) the outer layers. Theextent to which this occurred can vary by largeamounts depending on the size of the body, andthe collisional evolution of the planetesimal popu-lation, but it is unlikely that many parent bodiesescaped without a significant number of impactsprocessing their upper few kilometers.

Impact histories of specific parentbodies

In this section, the model described above is appliedto examine the plausibility of some of the invoca-tions of impacts in meteorite parent body histories.In order to examine the histories of some meteoriteparent bodies, data from meteorites (such as peak

19

temperatures, timing of impacts, etc.) can be com-pared with data from the parent bodies simulatedin the Monte Carlo model.

Timing of t = 0

There is one point of clarification that should beconsidered in this analysis: The time t = 0 in ourmodel is defined as the time that the gas dispersedfrom the disk such that it does not significantlyaffect the orbits of the planetesimals. In general,meteorite studies quote timings from the forma-tion of CAIs (Ca-Al-rich inclusions—some of theoldest known materials found in meteorites). CAIsare predicted to have formed 4568.5 ± 0.5 Myr ago(Bouvier et al., 2007). These events (gas disper-sion and CAI formation) likely did not occur atthe same time. Ciesla (2010) argued that the CAIswhich define t = 0 for the Solar System formedright when the solar nebula stopped accreting sig-nificant amounts of mass from its parent molecu-lar cloud; astronomical surveys suggest disks last∼ 2 – 5 Myr after this point (Haisch et al., 2001).Thus the timings discussed below could change byup to ∼ 2 Myr, and thereby alter the conclusionsdiscussed below. The effects of moving t = 0 by∼ 2 Myr (i.e. assuming that the gas dispersed 2Myr after CAI formation—therefore, t = 0 in theMonte Carlo model was 2 Myr after CAI forma-tion) are discussed below. An assumption in thisanalysis is that collisions between the formation ofthe CAIs and the dispersion of the gas are consid-ered too gentle to be accounted for in the model.The discussion below shows that while the choiceof t = 0 has an effect on the results of the modelwhen looking at collision effects in the very earlySolar System, its influence is diminished for eventsthat took place after ∼ 5–10 Myr. In order to fullyquantify the effects of collisions in the first severalmillion years of Solar System history, a more thor-ough understanding of the timing of gas dispersionand the onset of the dynamical excitation of theplanetesimal population is required.

CB chondrite parent body

The bencubbinite CB chondrites contain large, ex-otic metal aggregates with elemental abundancesvarying strongly with volatility; the origin of theseaggregates is unknown. One theory suggests that

an impact between a metallic planetesimal and areduced-silicate body could have produced a vaporplume that was rich in metal and not very oxidiz-ing (Campbell et al., 2002), from which the metalcondensed. In order to create an appropriate vaporcloud, Campbell et al. (2002) suggested a collisionof such bodies at velocities > 8 – 10 km/s occurredbefore 5 Myr. Chondrules are another feature of theCB parent body: The CB chondrules are relativelyyoung compared to other chondrules, and appearto have formed in a single event, leading to specu-lation that they may have an impact origin (Krotet al., 2005), and that one impact formed both thechondrules and the metal aggregates. The timingof the chondrule-forming impact has been deter-mined from 207Pb–206Pb ages, and is predicted tohave occurred between 4.3 and 7.1 Myr after CAIformation (Krot et al., 2005). It should be notedthat this chondrule formation mechanism is distinctfrom the impact mechanism recently suggested byAsphaug et al. (2011), which requires impacts withvelocities at < 100 m/s.

To test the likelihood that an impact could haveformed both the metal and the chondrules in theCB chondrite parent body, we can thus look for animpact that caused vaporization. The time of theimpact must be after 4.3 Myr (from Krot et al.,2005), but before 5 Myr (from Campbell et al.,2002). Campbell et al. (2002) do not offer a con-straint of the size of the colliding bodies. Krot et al.(2005) suggest an impact between Moon-sized bod-ies, but that constraint is simply based on requir-ing a velocity high enough to produce vapor, as thegravity of those bodies would increase the velocityat impact. As the planetesimals in our model areable to reach collisional velocities of 8 – 10 km/seven at smaller sizes (due to interactions with thegas giants and the planetary embryos), here we willconsider vaporizing impacts on each parent bodysize discussed above.

In the CJS models, on average three out of everyfive 50 km radius parent bodies will experience avaporizing collision (vimp > 8 km/s), each 100 kmradius body will experience ∼ 2.5 such collisions,and each 250 km radius body will experience over15 vaporizing collisions. The number of collisionswith vimp > 10 km/s is approximately 4 – 5 timesfewer than with vimp > 8 km/s. In the EJS mod-els, there are around 3 times more collisions withvimp > 8 km/s and around 10 times more colli-

20

sions with vimp > 10 km/s than in the CJS mod-els. While fewer collisions are expected overall inthe EJS case compared to CJS, more of these highvelocity vaporizing collisions occur, due to the higheccentricity of the planetesimals orbits.

Those predictions do not account for the size ofthe impactor, and include all vaporizing collisionswith impactors greater than 150 m in radius. If weinstead only consider impactors at least one twenti-eth of the radius of the parent body, we get a senseof the number of large-scale vaporizing collisions.In the CJS models, around one in a hundred par-ent bodies will see a collision with vimp > 8 km/swith an impactor of that size (independent of par-ent body size). In the EJS models, around one intwenty parent bodies see a large impact with vimp> 8 km/s.

If CB chondrites were formed as hypothesizedby Campbell et al. (2002) and Krot et al. (2005),at most ∼ 10 parent bodies would have undergoneevolution allowing chondrules and metallic phasesto form: To analyze whether this scenario is likelyto have occurred to the CB chondrite parent body,we can examine the number of parent bodies inthe collisional evolution model at 100 Myr of eachsize that remained in the asteroid belt. In the CJSmodel, there were 165 parent bodies of ∼ 50 kmradius (in the bins with a radius between 40 and63 km), 41 parent bodies of ∼ 100 km radius (inthe bins with a radius between 79 and 126 km),and 3 parent bodies of ∼ 250 km radius (in thebins with a radius between 199 and 315 km). Inthe EJS model, these numbers are 168, 43 and 3,respectively. So, in the CJS case, one or two 50 kmradius parent bodies that remain in the asteroidbelt would have seen a large vaporizing collision,and there is a ∼ 40% chance that a 100 km ra-dius body and a ∼ 3% chance of a 250 km radiusbody experiencing a vaporizing collision and sur-viving in the asteroid belt. In the EJS case, therewould be eight 50 km radius bodies, two 100 kmradius bodies and a 15% chance of a 250 km radiusbody surviving in the asteroid belt that match theCB chondrite criteria. By moving the definition oft = 0 by 2 Myr (see above), the number of vaporiz-ing impacts decreases by a factor of ∼ 4; this doesnot change the overall conclusion that it is possiblethat a small number of parent bodies experiencethe type of vaporizing collision needed to producethe metal and chondrules seen in the CB chondrites

and survive past 100 Myr.

CV chondrite parent body

A model for the formation and differentiation of theCV chondrite parent body to account for the uni-directional magnetic field was developed by Elkins-Tanton et al. (2011). In that model, parent bodiesof ∼ 100 – 300 km radius differentiated due to thedecay of 26Al, leaving an unmelted, chondritic crustof ∼ 6 – 20 km thickness on top of a magma ocean.A porosity of 25% is assumed for the crust, close tothe porosity used in the Monte Carlo model of thiswork. The surviving, undifferentiated crust wouldthen serve as the source of the pristine CV chon-drites. One requirement of the model is that the∼ 6 – 20 km thick crust must not be disturbed dur-ing the time that the parent body is being heatedand the magnetic field is in place (∼ 10 Myr),to prevent both impact foundering and breachesleading to magma eruptions. While Elkins-Tantonet al. (2011) advocate a parent body that begins ataround 100 km radius and continues to grow to 300km radius while heating, our model cannot accountfor changing parent body sizes. Thus, to test thismodel, here we will examine the histories of par-ent bodies with fixed radii of 100 km and 250 kmin the Monte Carlo simulation described above todetermine if a parent body could have escaped anyimpacts large enough to penetrate the crustal ma-terial. The depth of penetration, Hpen, of a craterin a gravity-dominated crater can be approximatedas Hpen = 0.28 dtr (O’Keefe and Ahrens, 1993).

On 100 km radius parent bodies for the CJS case,the sum of the area of all craters would be ∼ 0.8times the surface area of the parent body (Figures4 & 6). The average number of collisions that pene-trate to 6 km deep is ∼ 5 per parent body. Aroundone in four parent bodies will experience an impactthat penetrates to 20 km. Therefore, while lessthan 1% of 100 km radius parent bodies survivethe first 10 Myr without a collision penetrating to6 km, around 75 – 80% survive without a collisionthat penetrates to 20 km.

For 250 km parent bodies, more collisions can beexpected: ∼ 22 – 24 collisions per parent body pen-etrate to 6 km, and each parent body will experi-ence ∼ 1 collision that penetrates to 20 km depth.No 250 km radius parent bodies survives 10 Myrwithout a collision that penetrates to 6 km, and

21

∼ 40 – 45% of bodies survived without a collisionthat penetrated to 20 km.

Therefore, it seems unlikely that the CV par-ent body could have developed in the way pro-posed by Elkins-Tanton et al. (2011) if the crustwas only ≤ 6 km thick. Our Monte Carlo modeltherefore predicts that for CV chondrites to comefrom a body such as that modeled by Elkins-Tantonet al. (2011), a thicker crust (∼ 20 km thick) is re-quired: This would increase the chances that theparent body could have survived for 10 Myr with-out experiencing a collision that would disrupt thethermal structure and therefore the magnetic field.Whether these impacts would increase the densityof the chondritic crust to create a density instabilityor cause it to founder outright will be the subjectof future work.

Above (in the ‘CB Chondrite parent body ’ sec-tion), we showed that approximately 40 parentbodies of ∼ 100 km radius and approximately 3parent bodies of 250 km radius would survive to100 Myr and remain in the asteroid belt. There-fore, if a crust of 20 km was present, around 30parent bodies of 100 km radius would have sur-vived past 100 Myr and matched the criteria forthe CV chondrite parent body from Elkins-Tantonet al. (2011), and only 1 parent body of ∼ 250 kmsize would have matched these criteria.

The number of bodies that match the CV chon-drite criteria is not strongly affected by movingt = 0 by 2 Myr. The total number of collisionsthat penetrated the crust within the first 8 Myrof the model (rather than the first 10 Myr) de-creased by around a third, and the number ofbodies that escaped without a collision increasedslightly: Around 1 – 3% of 100 km radius bod-ies escaped without a collision penetrating 6 km,and around 80 – 85% survived without a collisionpenetrating 20 km. No 250 km radius parent bod-ies survived without a collision penetrating a 6 kmcrust, and approximately half of all 250 km parentbodies survived without a collision penetrating to20 km depth. Thus, changing the timing of t = 0does not change our conclusions about the Elkins-Tanton et al. (2011) model: A thick crust (∼ 20 km)would be needed to protect the CV chondrite par-ent body from a collision that could cause founder-ing and loss of the crust.

IAB/winonaite parent body

The IAB non-magmatic iron meteorites and thewinonaite meteorites are thought to share a com-mon parent body (Benedix et al., 2000). Whilethe majority of heating in these bodies likely camefrom 26Al decay (e.g. Theis et al., 2013), there aresome studies of the chronology of these meteoritesthat suggest that the parent body experienced sev-eral heating events after the decay of short-livedradionuclides. Impacts have been invoked to ex-plain these relatively late thermal events (Schulzet al., 2009, 2010, 2012) which are recorded by theHf-W chronometer as well as Sm and W isotopicabundances. An event at 2.5+2.3/−2.0 Myr causedsilicate differentiation; partial melting was causedat 5.06+0.42/−0.41 Myr; further thermal metamor-phism occurred at 10.8+2.4/−2.0 Myr with temper-atures reaching 1273 K; and finally a disruptiveevent occurred after∼ 12 Myr. The extent to whichimpacts could provide the heat required to matchthe thermal history recorded in meteorites can beevaluated using the Monte Carlo model. Figure 10shows the probability that a given fraction of a 100km radius parent body (in the CJS simulation) isheated to temperatures above those of chondriteswith petrologic type 6 (> 1273 K) or above the as-sumed (dunite) solidus (> 1373 K) in any impactsthat occurred after 10 Myr (i.e. after the decayof 26Al). In most parent bodies, only a small frac-tion is heated to those temperatures: Less than onehundredth of the parent body reaches these tem-peratures in more than half of all cases modeled.However, in some bodies, a significant fraction ofthe parent body is heated in late collisions: In 7.5%of parent bodies, more than one tenth of the parentbody is heated to the required temperatures. TheMonte Carlo model can also be used to examine theimpact history of this parent body in more detail.Of the parent bodies that were disrupted after 12Myr, each collision was examined to determine ifit heated material to the required temperature inthe given time intervals. In the simulation of 100km radius parent bodies with the CJS collisionalevolution model, 3834 parent bodies (out of 105)were disrupted after 12 Myr. Of those bodies, only3 (around 0.1% of the disrupted bodies) had en-ergetic enough collisions during each of the threegiven time periods listed above, to heat at leastone thousandth of the parent body to the required

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< 1100 Mt

1100 Mt − 1

10 Mt > 110 Mt

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Figure 10: Probability of a given fraction of a 100 kmradius parent body in the CJS model being heated above1273 K (i.e. above petrologic type 6) or 1373 K (i.e. abovethe dunite solidus) in impacts that occurred later than 10Myr after formation. In almost 90% of all parent bodies inthis simulation, less than one hundredth of the parent bodywas heated to either of those temperatures. In 7% of cases,between one hundredth and one tenth of the parent bodywas heated to those temperatures, and in around 3.5% ofcases, more than one tenth of the parent body was heatedto those temperatures.

temperature. It is unclear if the first heating event(at 2.5+2.3/−2.0 Myr) was caused by impacts or ra-dionuclide decay (Schulz et al., 2010), as it is duringthe time when 26Al decay was still active. Runningthe same analysis, but without the requirement foran impact to heat material in that event yields 54parent bodies that match the criteria (around 1.4%of the disrupted bodies). For the EJS model, 894parent bodies (out of 105) were disrupted after 12Myr; of those, 8 parent bodies (∼ 1% of the dis-rupted bodies) experienced collisions to match allthe events, and 48 parent bodies (∼ 5%) met thecriteria when the first heating event was excluded.This shows that the collisional history needed toproduce the IAB/winonaite parent body may bedifficult to achieve.

To evaluate the influence of our choice of t = 0on the analysis of the IAB/winonaite parent body,the times of the impact events were shifted by 2Myr. In the CJS model, the total number of dis-rupted parent bodies that could match the criteriadecreased from 54 to 11, and in the EJS model thatnumber decreased slightly from 42 to 37. Thus, thechoice of t = 0 does not significantly change ourconclusions about the IAB/winonaite parent body.

A caveat to the approach used here is that it is

assumed that the material starts out cold beforethe impact, and therefore requires a more energeticcollision to produce the levels of heating observedin the meteorite record; this approach does nottake into account the heating provided by short-lived radionuclides, and therefore should be takenas a lower estimate of the probability of produc-ing a parent body to match the IAB/winonaitebody. Furthermore, here we have assumed a par-ent body radius of 100 km—however, the size of theIAB/winonaite parent body is not well constrained.A different parent body size could yield different re-sults. Further work is clearly needed to fully quan-tify the early impact history of the IAB/winonaiteparent body.

H chondrite parent body

Models of the thermal evolution of the H chondriteparent body predict an onion shell structure result-ing from the heat produced by the decay of short-lived radionuclides, such as 26Al (e.g. Trieloff et al.,2003; Kleine et al., 2008; Harrison and Grimm,2010). However, Harrison and Grimm (2010), in or-der to explain some measurements of cooling rates,peak temperatures and closure times which do notfit their onion shell model, invoked an impact whichdisturbed the thermal evolution of a portion of thebody by excavating material from at least 5.6 kmdeep on a 100 km radius parent body. In the simu-lations presented above for a 100 km parent body,on average 2.3 collisions per parent body excavatedto this depth in the CJS model (see Figure 5b), andin the EJS model, 1.5 collisions per parent bodymatched this criterion. Approximately 8.4% and22.7% of parent bodies modeled escaped withoutany collisions excavating to this depth in the CJSand EJS simulations, respectively. While the re-sults depend strongly on the choice of collisionalevolution model (CJS or EJS), they show that mostparent bodies (more than 77% of all 100 km radiusparent bodies) will experience at least one impact ofthe type required by Harrison and Grimm (2010).Of the 40 or so ∼ 100 km radius parent bodies thatsurvived to 100 Myr, at least 30 would have experi-enced a collision which excavated to at least 5.6 kmdepth. Thus, the requirement of an impact disturb-ing the H chondrite parent body thermal structureis supported by this study.

Other models of the H chondrite parent body

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suggest impacts may have played a larger role inthe thermal history (Scott et al., 2011), excavat-ing and thoroughly mixing material (Krot et al.,2012). In that model, both type 4 and type 6 ma-terials are brought to the surface by impacts earlyin the evolution of the parent body. The depthto type 4 and type 6 material can be estimatedusing an onion shell model. Harrison and Grimm(2010) predict type 4 material may be as shallowas 0.9 km below the surface of a 100 km radius par-ent body. Analysis of cooling rates from one type6 meteorite, predicted type 6 material at 3.4 kmdepth. However, other analyses in that study us-ing both cooling rates and closure times predicteda depth of at least 11.2 km for type 6 meteorites.In the CJS model of a 100 km radius parent body,∼ 123 impacts per parent body excavated materialfrom 0.9 km depth, ∼ 9 impacts excavated froma depth of 3.4 km and ∼ 0.3 impacts per parentbody excavated from 11.2 km. In the EJS model,fewer impacts excavated to those depths with ∼ 56impacts excavated from 0.9 km, ∼ 5 impacts exca-vated from 3.4 km, and ∼ 0.2 impacts excavatedfrom 11.2 km. This shows that bringing type 4material to the surface is a common process, andshould occur in at least 50 events per parent bodyin the first 100 Myr. In the CJS model, the sur-face area of craters that excavated to this depthis three quarters of the area of the parent body;in the EJS model, on average around a third ofthe parent body surface would experience cratersthat excavate to this depth. If type 6 material isas shallow as 3.4 km, up to 9 events could bringthat material to the surface to cool more quickly(more than half of the surface area is affected inthis way in the CJS model; one quarter in the EJSmodel). However, if type 6 material is deeper, thenfar fewer impacts would be able to excavate it—notevery parent body is likely to experience a collisionduring the first 100 Myr of the magnitude requiredto excavate material from a depth of 11.2 km (anaverage covering of two fifths of the parent bodysurface area in the CJS model and one fifth in theEJS model).

CONCLUSIONS

In this study, we have developed a model to sim-ulate the collisional histories of meteorite parent

bodies. The model predicts that impacts were com-mon processes on parent bodies and the outer layersof those bodies would have been strongly processedby shock events. Impacts can provide a significant,secondary heat source to parent bodies, which canbe especially important after the decay of short-lived radionuclides and in the regions near the sur-face of the original parent body. The probabilitiesof different outcomes vary strongly with the dif-ferent dynamical scenarios considered for the earlySolar System. Further compilation of how shockprocesses affected meteorite parent bodies in theearly Solar System should allow us to test the dif-ferent dynamical scenarios and collisional historiesenvisioned for terrestrial planet formation.

The model has been applied to evaluate the pro-posed histories of several meteorite parent bodies:At most, around 10 parent bodies that survivedto 100 Myr underwent a collisional evolution con-sistent with the formation mechanism of the chon-drules and metallic phases of Campbell et al. (2002)and Krot et al. (2005); around 1 – 5% of bod-ies that were disrupted after 12 Myr (which com-prises ∼ 1 – 4% of the initial population of 100 kmplanetesimals) experienced impact events contem-poraneously with some late heating events on theIAB/winonoaite parent body (Schulz et al., 2009,2010); at least 30 of the ∼ 40 surviving 100 kmradius bodies would sustain an impact that exca-vated material from a depth of at least 5.6 km, aspredicted by Harrison and Grimm (2010) for theH chondrite parent body, and almost all bodies ex-perienced impacts that brought type 4 material tothe surface, as required by Scott et al. (2011); athick crust (∼ 20 km) is required on the CV parentbody to prevent foundering and disturbance of thethermal structure—a crust of this thickness wouldprovide enough shielding for 10 Myr for ∼ 30 par-ent bodies to survive for 100 Myr and fit the modelof Elkins-Tanton et al. (2011), while a thin crust(∼ 6 km) would experience too many impacts thatpuncture it and thus allow foundering.

There are some critical uncertainties in theMonte Carlo simulations which should be addressedin future iterations of the model: For example, theinitial mass of material in the planetesimal pop-ulation, the initial orbits of the gas giants (CJSor EJS), the response of the disruption thresholdto temperature and porosity changes, and the evo-lution of the porosity structure of a parent body

24

through time are all factors for which we have cho-sen the best available approximations. Nonethe-less, our approach offers a new means for evaluatingthe roles that impacts played in shaping meteoriteparent bodies. By better recognizing the signaturesand timing of impact events in meteorites, we canunderstand the collisional history of meteorite par-ent bodies and use that information to constraindynamical models for the Solar System.

Acknowledgments: We gratefully acknowledgethe major contributions of Jay Melosh, BorisIvanov, Kai Wnnemann and Dirk Elbeshausen tothe development of iSALE. We thank Ed Scott,Bill Bottke and Robert Grimm for their insightfuland constructive comments. TMD and FJC weresupported by NASA PGG grants NNX09AG13Gand NNX12AQ06. TMD and GSC were sup-ported by STFC grant ST/J001260/1. GSC wasalso supported by STFC grant ST/G002452/1.DPO was supported by grant NASA PGG grantNNX09AE36G.

APPENDIX A: VFDPARAMETERISATION

In order to use the results from the dynamicaland collisional evolution simulations, they mustbe parameterised in such a way that the MonteCarlo model can use them. In Figure 1, the meanvelocity is presented as a function of time. Inorder to allow a VFD to be produced at everytimestep in the Monte Carlo model, the mean ve-locity is interpolated, and converted into a VFDusing a Maxwellian distribution. To check that aMaxwellian distribution is appropriate, the VFDfrom the dynamical models was output at timesthroughout the simulation, and compared to theMaxwellian. The dynamical model output is wellfit by the Maxwellian distribution (with a coeffi-cient of determination, R2 > 0.96 for all timestepsin the first 100 Myr). The evolution of the VFDfor the CJS simulation is shown in Figure A1.

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0

0.2

0.4

0.6

0.8

Pro

bab

ility

P(v

=v i

mp)

a)t = 105 yrs

t = 106 yrs

t = 107 yrs

t = 108 yrs

0 5 10 15 20 25Velocity, vimp [km/s]

0.0

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0.4

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1.0

Cu

mu

lati

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rob

abili

tyP

(v<

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