The early impact histories of meteorite parent bodies · young planetesimals. Collisions with other...

The early impact histories of meteorite parent bodies

Thomas M. DAVISON1,2*, David P. O’BRIEN3, Fred J. CIESLA1, and Gareth S. COLLINS2

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

3Planetary Science Institute, 1700 E. Ft. Lowell, Suite 106, Tucson, Arizona 85719, USA*Corresponding author. E-mail: [email protected]

(Received 25 December 2012; revision accepted 23 July 2013)

Abstract–We have developed a statistical framework that uses collisional evolution models,shock physics modeling, and scaling laws to determine the range of plausible collisionalhistories for individual meteorite parent bodies. It is likely that those parent bodies thatwere not catastrophically disrupted sustained hundreds of impacts on their surfaces—compacting, heating, and mixing the outer layers; it is highly unlikely that many parentbodies escaped without any impacts processing the outer few kilometers. The first10–20 Myr were the most important time for impacts, both in terms of the number ofimpacts 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 to10 parent bodies that were not disrupted in the first 100 Myr experienced a vaporizingcollision of the type necessary to produce the metal inclusions and chondrules on the CBchondrite parent; around 1–5% of bodies that were catastrophically disrupted after 12 Myrsustained impacts at times that match the heating events recorded on the IAB/winonaiteparent body; more than 75% of 100 km radius parent bodies, which survived past 100 Myrwithout being disrupted, sustained an impact that excavates to the depth required formixing in the outer layers of the H-chondrite parent body; and to protect the magnetic fieldon the CV chondrite parent body, the crust would have had to have been thick(approximately 20 km) to prevent it being punctured by impacts.

INTRODUCTION

The early solar system was a violent place foryoung planetesimals. Collisions with other planetesimalswere common, and would have affected the evolution ofthe bodies that would go on to become the asteroids andparent bodies of the meteorites we find on Earth today.

Planetesimals, the first rocky bodies to form in thesolar system, accreted within the solar nebula, the cloudof dust and gas that orbited the young Sun. Initially,highly porous dust aggregates up to centimeters in scaleaccreted in low-velocity, pairwise collisions between dustparticles (Wurm et al. 2001, 2004; Blum 2003). Howthese aggregates grew from this scale to the kilometerscale is still an open question, although leading theoriessuggest that either hydrodynamic effects andgravitational instabilities (Johansen et al. 2007, 2009;Cuzzi et al. 2008) or further low-velocity collisions led to

a population of planetesimals approximately 1–100 kmin scale (Weidenschilling 2011). These early bodies arelikely to have contained a significant fraction of porespace (Dominik and Tielens 1997; Bland et al. 2011).Some planetesimals grew to protoplanets; the first stageof this growth is termed runaway growth (Greenberget al. 1978; Wetherill and Stewart 1989, 1993), in whichthe largest planetesimals gravitationally focused smallerbodies, increasing the number of collisions on theirsurfaces (Kokubo and Ida 1996), eventually drawing inall bodies from their so-called “feeding zone.” As thenumber of small bodies was diminished during runawaygrowth, protoplanets grew more slowly during thesubsequent oligarchic growth phase, as impactors neededto be drawn from farther afield and the protoplanetsbegan to excite the remaining planetesimal population,reducing the number of collisions on their surfaces(Kokubo and Ida 1998).

Meteoritics & Planetary Science 1–25 (2013)

doi: 10.1111/maps.12193

1 © The Meteoritical Society, 2013.

These processes are expected to have resulted in aremnant population of planetesimals and a smallernumber of embryos (approximately 100–1000 km inradius). Collisions between and among these populationswould have been common, as the number of bodieswould have been much greater than the number of bodiesthat remain in the asteroid belt today. For example, thepopulation in the region 2–4 AU could have beenapproximately 100–1000 times more massive than thecurrent asteroid belt (Bottke et al. 2005a, 2005b). Masswas lost from the population as bodies were scatteredinto the Sun or ejected out of the solar system by Jupiter;the majority of this mass depletion took place in theinitial 100 Myr period (Petit et al. 2001; Bottke et al.2005a), meaning that the frequency of collisions duringthat period was much greater than it was after massdepletion took place. Planetary embryos could have“stirred up” the smaller planetesimal population,increasing velocities above the mutual escape velocity ofthe colliding pair of bodies, up to several kilometers persecond (Kenyon and Bromley 2001). As some bodies fellinto resonances with Jupiter and Saturn, their orbitswould have been excited to high eccentricities—leading toimpact velocities of at least 10 km s�1 (Weidenschillinget al. 1998, 2001; Bottke et al. 2005a).

As meteorites provide our strongest evidence ofconditions in the early solar system, a fullunderstanding of the histories of their parent bodies isvital. Collisions played a major role in these histories:Indeed, impacts have been invoked to explain manyobservations from the meteorite record. For example,shock wave processes have been shown to causedeformation of minerals and localized melting (St€offleret al. 1991) and shock blackening (Heymann 1967; Brittand Pieters 1991); Wittmann et al. (2010) show thatthese effects typically postdate metamorphism fromradiogenic decay. Impacts are able to perturb the“onion-shell” nature of petrologic types; for example,Grimm (1985) and Taylor et al. (1987) proposed nearlycomplete mixing of chondrite parent bodies by impactdisruption and reaccumulation. More recently, it hasbeen suggested that impact processing may havebrought some heated material (petrologic type 4–6)from the interior of the parent body into close contactwith cooler material (type 3) from the outer layers ofthe body (Harrison and Grimm 2010; Scott et al. 2011).Impacts have even been invoked as a possible heatingmechanism leading to resetting of thermochronometersor phase transitions in more extreme cases, for example,on the IAB/winonaite parent body (Wasson andKallemeyn 2002; Schulz et al. 2009), the CB chondrites(Campbell et al. 2002; Krot et al. 2005), and the angriteand eucrite parent bodies (Scott and Bottke 2011).Recently, the effects of individual collisions between

porous bodies have been quantified (Davison et al.2010), as well as their influence on the long time scalethermal evolution of a parent body (Davison et al.2012b), demonstrating that localized heating can besignificant and allow higher temperatures to be reachedthan previously estimated assuming nonporous bodies(Keil et al. 1997).

The extent to which impacts would have affected ameteorite parent body will vary from body to body, andthe effects depend on the number, sizes, and timing of thecollision events. To date, there has been no quantitativeanalysis of the types of collision histories that a meteoriteparent body might have experienced during its earlyevolution. Analytical techniques (e.g., Chambers 2006)can provide estimates of the number of mutual collisionsbetween planetesimals of a given size. However, thisapproach does not provide histories for individual parentbodies that are subject to impacts from an evolvingpopulation of smaller planetesimals. To investigate therole of collisions on the thermal and disruptional historiesof meteorite parent bodies, and to find the likelihood thatcollisions could have produced the effects they have beeninvoked to explain in the meteorite record, it is importantto know what kinds of collisions occurred.

This study aims to address this gap in ourknowledge of impact processes, and quantify: (1) howmany impacts a parent body is likely to sustain duringthe first approximately 100 Myr of its history, (2) howlikely a parent body is to be disrupted during this time,(3) when collisions are most likely to occur, (4) theaverage and range of velocities of the impacts and howthey change with time, (5) the time that most disruptiveimpacts occur, and (6) how much collisional heatingoccurs 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 to theabove questions. In the next section, the details of themodel are presented. Then, results from this model fora range of parent bodies and collisional evolutionscenarios are presented. Finally, the predictions fromthe model are applied to several case studies for whichimpacts have been invoked in the literature, to test theplausibility of those scenarios.

METHODS

A large number of planetesimals would have beenpresent in the early solar system, each with its owncollisional history determined by a series of chanceencounters with other planetesimals. Therefore, theimpact history of a parent body through time cannot besolved analytically, as each body would haveexperienced a different set of impacts on its surface. A

2 T. M. Davison et al.

Monte Carlo approach is required to determine therange of possible impact histories that a parent bodycould have experienced, and what a typical impacthistory would have been.

In the Monte Carlo simulation developed for thisstudy, the collisional histories of many parent bodies aremodeled, where the likelihood of a collision with a givenimpactor size occurring at a given time and the velocityof each collision are calculated from a probabilitydistribution. The probability of a collision occurring witha given impactor size and the velocity–frequencydistribution (VFD) at a time t must therefore be knownbeforehand. These distributions are based on dynamicaland collisional models of terrestrial planet formation;their calculations are outlined below, in the CollisionalEvolution Modeling section. For each collision, theresulting crater size is estimated from crater scalingrelationships, the amount of heating is determined fromshock physics simulations, and whether the parent bodyis catastrophically disrupted can be calculated usingstrength estimates from simulations of catastrophiccollisions (Benz and Asphaug 1999; Jutzi et al. 2010).These calculations are outlined in the Collisions:Cratering and Disruption section.

Collisional and Dynamical Evolution Modeling

Quantifying the impact history of a parent bodyrequires knowledge of the size-frequency distribution(SFD) of planetesimals, the VFD of collisions, and theintrinsic collisional probability, which are all known tobe time dependent. Simulations of the dynamical andcollisional evolution of the planetesimal population inthe early solar system are able to provide estimates ofthese quantities.

O’Brien et al. (2006, 2007) performed simulations ofterrestrial planet formation and the excitation and massdepletion of the early asteroid belt using the SyMBA N-Body integrator (Duncan et al. 1998). The gravitationalinteractions of a population of Mars-mass planetaryembryos and an equal mass of smaller planetesimalsdistributed from 0.3 to 4 AU were simulated, includingthe influence of Jupiter and Saturn. The smallerplanetesimal bodies are influenced by the gravity of thelarge bodies, but do not interact with each other, as iscommonly done in such simulations, while the largerbodies are influenced by the gravity of the smallplanetesimals and one another. Collisions occur whenan embryo runs into another embryo or planetesimal,which leads to a perfect merger of the impacting bodies,conserving linear momentum.

While mutual collisions between planetesimals wereignored in the planetary accretion simulations, they areexpected to have occurred during this period of planet

formation. As such, the simulations were analyzed todetermine the evolution of the collision probability andVFD for the first 100 Myr of dynamical evolution usingthe algorithm of Bottke et al. (1994), where t = 0 isdefined as the time when most of the gas has dissipatedand gas drag no longer provides a damping effect. Bodieswere divided into two groups: those that remained in theasteroid belt at the end of the simulations, and those thatwere either accreted by the terrestrial planets or lost fromthe system. Collision velocity distributions and intrinsiccollision probabilities, Pi, were calculated forplanetesimals within and between these differentpopulations throughout the time of interest (Fig. 1).

The dynamical simulations just consider a singleparticle size for the planetesimals, so to track theevolution of the size distribution, we use a separatecollisional evolution model that takes the time-dependent

1

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Intr

insic

colli

sion

prob

abili

ty,P

i

[10−

18km

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−1]

(a)CJS EJS

02468

1012

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nve

loci

ty,v i

mp

[km

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Time [Myr]

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0.8

1.0

Tot

alm

ass,

Mto

t

(nor

mal

ized

byin

itia

lmas

s)

(b)

(c)

Fig. 1. Evolution of the intrinsic collisional probability, themean collision velocity, and the total mass of planetesimals forthe two different collisional evolution models described in thetext, CJS and EJS. Collisional probability and velocity arespecifically for collisions of all planetesimals onto theplanetesimals that remain in the asteroid belt at the end of thesimulations.

Early impact histories of meteorite parent bodies 3

impact velocity, collision probabilities, and total massdetermined from the dynamical simulations and evolvesthe size distributions of the planetesimal populations intime under mutual collisions, assuming an impactstrength for porous planetesimals (Jutzi et al. 2010).

This model is modified from O’Brien andGreenberg (2005) and O’Brien (2009) to treat multipleinteracting populations and time-dependent collisionalparameters, in an approach similar to that of Bottkeet al. (2005a). Two separate, but interacting,populations are considered: The bodies that eventuallyend up in the asteroid belt region (denoted Nrem, for theremnant population), and all other bodies (those thatare eventually lost and those that are accreted by theterrestrial planets, as described above; denoted Ndep forthe “depleted” population). The size of the depletedpopulation is varied to provide a good match betweenthe final remnant population in the model and theobserved mass of the present-day asteroid belt, whereNdep = f Nrem. For the non–asteroid belt bodies, aforced exponential decay term is applied, based on howfast the populations deplete in the dynamicalsimulations. As time advances, the production ofcollisional fragments increases the number of smallbodies in both populations (Fig. 2). Both populationsevolve through mutual collisions, but eventually (aftertens to approximately 100 Myr), the nonasteroid beltbodies are dynamically depleted and their contributionto the collisional evolution of the asteroids is negligible,leaving only the asteroids. This method is discussed inmore detail in Bottke et al. (2005a).

These dynamical and collisional evolutionsimulations depend upon the orbits of the gas giantsJupiter and Saturn. The initial orbital parameters of thegas giants are not well known. In this work, weconsider two different endmember simulations, whichwere found to give reasonable agreement between thefinal SFD of planetesimals and the current SFD of theasteroid belt. In both simulations, Jupiter and Saturnwere included from the start of the simulation(Chambers 2001; O’Brien et al. 2006). In one set ofsimulations, Jupiter and Saturn started with orbitsconsistent with their current inclination and eccentricity(denoted EJS here, for “Eccentric Jupiter and Saturn”).The embryos and planetesimals were centered on theinvariant plane of Jupiter and Saturn. In thissimulation, the final mass of the planetesimals in theasteroid belt region was a factor of 95 less than thetotal initial mass of the planetesimals in the system (i.e.,f = 95). In the other set of simulations, Jupiter andSaturn adopted initial orbits that were near-circular andcoplanar (i.e., inclination and eccentricity wereessentially zero), similar to that expected based on theNice model (Tsiganis et al. 2005; Morbidelli et al. 2005;

Gomes et al. 2005; denoted CJS here, for “CircularJupiter and Saturn”). Planetesimals and embryos werecentered on the plane of Jupiter’s initial orbit. In thesesimulations, the best match to the final SFD of theasteroid population was found when the final mass ofthe planetesimals in the asteroid belt region was a factorof 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 simulation andthe results of Bottke et al. [2005a], who used f values inthe range approximately 100–200, depending on theformation time of Jupiter). Figure 2 shows the

100

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Fig. 2. Evolution of the two planetesimal populations over thecourse of the CJS simulation with f = 100. The number ofsmall bodies rapidly increases from the initial distribution dueto collisional fragmentation. Depletion of the populationbegins at t=0, and decays with a half-life of approximately22 Myr (for comparison, the EJS model decays with a half-lifeof approximately 11 Myr). After 4500 Myr, the depletedpopulation has been completely removed from the disk,leaving only the bodies that remain in the asteroid belt.


evolution of the remnant and depleted populations inthe CJS simulation with f = 100. The fit here is in closeagreement with the observed SFD of the currentasteroid belt. However, the distribution has fewerbodies between a few tens of km and 100 km indiameter than the current asteroid belt, which mayoverestimate the collisional grinding in this size range(i.e., too many bodies of that size have experienced adisruptive collision), which in turn produces a largernumber of small objects at the expense of larger ones.

Figure 1 shows the evolution through time of theintrinsic collisional probability, Pi (Fig. 1a in units ofkm�2 yr�1; see Equation (1) to convert Pi to aprobability of a collision between a pair of planetesimalsof radii rimp and rt, in a given time interval), the meancollision velocity (Fig. 1b), and the total mass ofplanetesimals (Fig. 1c), for the two different simulations.For the EJS simulation, impact velocities increase soonerthan for CJS, and the collisional probability is higher inearly times. However, after approximately 1 Myr, thecollisional probability in the EJS case decreases, while,for CJS, it remains approximately constant until about10 Myr until dropping off. This is because in the EJScase bodies are rapidly excited to high inclination orbits,leading to the depletion of objects from the simulation,and thus fewer collisions occur after approximately1 Myr. In the CJS case, this excitation takes longer, andtherefore the collisional probability does not decrease asquickly. These differences could play an important rolein the collisional evolution of a parent body, and thesensitivity of our results to these assumptions is testedbelow.

The amount of damage done to Main Belt asteroidsin the first few 100 Myr is approximately equivalent tothat done in the proceeding 4.4 Gyr, suggesting that thelatter component probably cannot be ignored.However, the big difference in terms of impact heatingis that: (1) the parent bodies are now cold and (2)impact velocities are now much lower (most impactscome from Main Belt projectiles), such that the amountof heat produced will be much lower. Thus, forapplications of impact heating, or discussions of theearly impact histories of parent bodies, this model isapplicable. If we wish to apply the results of this modelto study the complete impact history of a parent body,the model would need to extend over the full history ofthe solar system.

It should be noted that the model employed heredoes not account for radial migration of giant planets,such as that suggested in the Grand Tack model (Walshet al. 2011), and thus, that scenario is not treated in thiswork. The approaches outlined here, however, aregeneral and we plan to investigate that particular modelof solar 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 our model(one sampling of the Monte Carlo model), we stepthrough time from t = 0 to t = 100 Myr. At each time,t, the probability, Pc, that a collision will occur betweenthe parent body, of radius rt, and an impactor of sizerimp at this time, t, is calculated:

Pcðrimp; tÞ ¼ NimpPiðrimp þ rtÞ2Dt (1)

The intrinsic collision probability, Pi, and numberof impactors in a given size bin, Nimp, at a given time t,were estimated by interpolating between data pointsoutput from the dynamical models described above,where the SFD of bodies in the collisional evolutionmodels represented the population of potentialimpactors. While the collisional evolution modelsprovide statistics for the number of planetesimals withradii, rimp, ranging from approximately 1 m to 500 km,to increase calculation speeds, and because smallerimpactors are unable to disrupt or significantly processthe parent body, only impactors with rimp > 150 m areconsidered. The boundaries of the impactor size binsare logarithmic, with log10(rmax/rmin) = 0.1, where rmax

is the upper bound of the radius of a planetesimal in agiven size bin, while rmin is the lower bound. If t isbetween two of the output timesteps from the collisionalevolution modeling, Nimp was determined using a linearinterpolation on the SFD.

Next, a random number, ℛ1, was generated, where0 ≤ ℛ1 ≤ 1. If ℛ1 < Pc(rimp,t), then a collision wasadjudged to occur between an impactor of size rimp andthe parent body at this time, t. If a collision occurs,then the collateral effects of that collision arequantified: A second random number, ℛ2, is chosen, topick a collision velocity from the Maxwelliandistribution of velocity (defined by the mean velocity, �m),which was also interpolated as needed.

Collisions: Cratering and DisruptionWith knowledge of the impactor size and velocity,

and the parent body’s size (and hence gravity), thediameter of the transient crater formed by the impact,dtr, can be determined from crater scaling relationshipsdetermined from experiments (Schmidt and Housen1987; Holsapple 1993) and numerical modeling(O’Keefe and Ahrens 1993; Pierazzo et al. 1997):

dtr ¼ 1:61�bCDg�bd1�b

imp m2bimp

p6

� �1=3(2)

where g is the surface gravity of the parent body, dimp isdiameter of the impactor (=2rimp), and vimp is the


impact velocity. The material-specific constants fornonporous soil and sand were estimated by Schmidtand Housen (1987) as CD = 1.6, b = 0.22 and CD =1.54, b = 0.165, respectively. Based on numericalcratering simulations, W€unnemann et al. (2006) suggestthat the constants for sand represent a goodapproximation for moderately porous, granularmaterials.

To convert the transient crater diameter to a finalcrater diameter, it is necessary to know whether thecrater is a simple, bowl-shaped crater or a complexcrater. The transition from simple to complex craters isknown to occur at approximately 3.2 km on Earth, andapproximately 18 km on the Moon, and scales as theinverse power of the surface gravity, g. Indeed, thisrelationship is thought to extend over a wide range ofsurface gravities, including to small bodies, such asVesta (Melosh and Ivanov 1999). Hence, the simple-to-complex transition diameter, dsc, can be estimated by:

dsc ¼ gmoonqmoondscmoon

gqt(3)

where gmoon, qmoon, and dscmoonare the surface gravity,

density, and simple-to-complex transition diameter onthe Moon, respectively, and qt is the density of theparent body. The final crater diameter can then beestimated, depending on whether it is simple orcomplex, by the following relationships (McKinnon andSchenk 1985; Collins et al. 2005). For a simple crater,the final diameter, df, is given by:

df ¼ 1:25dtr (4)

and the final crater diameter for a complex crater isgiven by:

df ¼ 1:17d1:13tr

d1:13sc

(5)

Equations (2–5) assume that all craters occur in thegravity regime. Estimates of the transition betweenstrength-dominated craters and gravity-dominatedcraters suggest that, on a 100 km radius body, anycrater larger than approximately 2.3 km will be in thegravity regime (Nolan et al. 1996). As the minimum sizeof impactors considered in this study is approximately150 m radius, only a small proportion of craters are inthe strength regime. For the simulations presentedbelow, less than 0.1% of impacts are in the strengthregime for 100 and 250 km radius parent bodies. Forthe simulations of 50 km radius parent bodies, around2–3% of craters are in the strength regime. As these arealso the smallest craters, they have little effect on the

overall statistics of the size of craters on the parentbody. If future studies are conducted to investigate thecollisional histories of smaller parent bodies, thestrength regime should be accounted for in thecalculation of crater size.

The depth that the impact event excavates materialfrom (i.e., the maximum depth from which material canbe brought to the surface, rather than being displaceddownward into the target) can also be determined fromscaling laws and the same impactor parameters (impactorsize and velocity, and target gravity). Melosh (1989)states that the excavation depth, Hexc, is approximatelyone tenth of the transient crater diameter, that is:

Hexc ¼ 1

10dtr (6)

One limitation of the scaling relationships used here(Equations 2–6) is that they are defined for impactsonto a planar surface. More studies of experiments andnumerical simulations of impacts on curved surfaces arerequired to define similar scaling laws appropriate forlarge impacts on a parent body. Therefore, for verylarge craters (where the size of the crater approaches thesize of the target body), this scaling should be used withcaution. Numerical cratering calculations haveconfirmed this relationship for craters of diameters upto the size of the planetary radius (Potter et al. 2012).

Another possible outcome of the collision event isthe catastrophic disruption of the target. Catastrophicdisruption is defined as a collision in which the largestfragment has a mass of less than half of the originaltarget body mass. The quantity Q*

D is defined as theminimum specific impact energy (energy per unit massof the target) required to disrupt the parent body:

Q�D ¼ mimpm2imp

2mt¼ qimpðr�impÞ3m2imp

2qtr3t

(7)

Hence, the disruptive impactor radius, r�i , can beexpressed in terms of the impact energy and the targetsize:

r�imp ¼ rt2qtQ

�D

qimpm2imp

!1=3

(8)

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

Q ¼ qimpm2imp

2qt

rimp

rt

� �3

(9)

Benz and Asphaug (1999) used numerical modelingto show that Q*

D can be expressed by the functionalform:


Q�D ¼ Q0

rt0:01m

� �aþ Bqt

rt0:01m

� �b(10)

where Q0, a, b, and B are material-specific constants.The first term on the right-hand side of Equation (10)describes the strength regime, and is only applicable forparent bodies up to approximately rt = 100 m. Hence,for the parent bodies considered in this work, only thesecond term on the right-hand side needs to beconsidered. Several recent studies have attempted toquantify the disruption threshold for a range ofscenarios (e.g., Benz and Asphaug 1999; Leinhardt andStewart 2009; Stewart and Leinhardt 2009). One suchstudy investigated the effect of porosity on thisimportant parameter: Jutzi et al. (2010) used SPHmodeling to simulate planetesimal collisions, anddetermined material constants for use in Equation (10).For impacts into a porous pumice target at 5 km s�1,they found that B = 5.70 erg cm3 g�2, and b = 1.22.The explicit porosity is not defined in that work, but thedensity is approximately half that of the nonporoustarget material. While pumice is a very differentmaterial to basalt or dunite, as this is the onlydisruption criterion in the literature to incorporateporosity, it is thus the best approximation for use inour model of the early solar system, when porositywould have been significantly higher than today. Forimpacts into a nonporous basalt, they found B =1.50 erg cm3 g�2, and b = 1.29. To apply this disruptioncriterion to a wider range of collision speeds,velocity scaling from Housen and Holsapple (1990)can be applied to the gravity regime term fromEquation (10):

Q�D ¼ Bqt

rt0:01m

� �b mimp

5000ms�1

� �2�b

(11)

By combining Equations (8) and (11), the minimumdisruptive impactor radius can be expressed in terms ofthe target radius, the impact velocity, and the material-specific constants of b, B, qt, and qi:

r�imp ¼ 2Bq2tqimp

ð0:01mÞ�bð5000ms�1Þb�2r3þbt m�b

imp

!1=3

(12)

For each collision in our Monte Carlo model,Equation (12) is used to determine if the collision willcatastrophically disrupt the parent body. If disruptiondoes occur, the calculation on that parent body isstopped: no further collisions are modeled.

Collisional HeatingFor each collision that occurs, the amount of heat

deposited in the target body can also be estimated.Numerical studies of collisions between planetesimalshave quantified the amount of heating done between arange of planetesimal pairs, for a range of collisionvelocities up to approximately 10 km s�1 (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 was aconstant multiple of the impactor mass (where theconstant depends on the impact velocity, targetporosity, and starting temperature). For impactorsgreater than one tenth of the mass of the target, theheated mass was limited by the total mass of the target.Here, we extend those calculations to consider impactvelocities up to 50 km s�1 (greater than the maximumcollision velocity expected from the collisional evolutionmodels discussed above).

We used the iSALE-2D shock physics model(W€unnemann et al. 2006), which is based on the SALEhydrocode (Amsden et al. 1980). To simulatehypervelocity impact processes in solid materials SALEwas modified to include an elasto-plastic constitutivemodel, fragmentation models, various equations ofstate, and multiple materials (Melosh et al. 1992;Ivanov et al. 1997). More recent improvements includea modified strength model (Collins et al. 2004) and aporosity compaction model (W€unnemann et al. 2006;Collins et al. 2011b). Davison et al. (2010) furtheradapted iSALE to allow the simulation of collisionsbetween two planetesimals. iSALE shock physicscalculations were performed for collisions between a10 km radius impactor into a 100 km radius targetbody (a mass ratio of 1/1000), for nonporous and 20%porous planetesimals. The planetesimals were modeledas spheres, using the ANEOS equation of state tablesfor dunite (Benz et al. 1989). To determine the mass ofmaterial shock heated to several different temperatures,Lagrangian tracer particles were placed in thecomputational mesh. The mass of all particles thatexperienced a threshold peak pressure required to reachthe desired final, postshock temperature wasdetermined in postprocessing (Pierazzo et al. 1997;Pierazzo and Melosh 2000; Ivanov and Artemieva 2002;Davison et al. 2010). Figure 3 shows the mass ofmaterial heated to several temperatures for 20%porosity, for the full range of velocities considered inthis study. To determine the heated mass in the MonteCarlo model, a linear interpolation between the twoclosest velocities is used. If the heated mass interpolatedfrom the iSALE models exceeds the mass of the parentbody, the heated mass recorded from this collision is


limited to the parent body mass. The increase in specificinternal energy for each temperature increase is alsolisted in Table 1. For each impact, therefore, the totalamount of energy that is used for heating the parentbody, Qheat, can be approximated by:

Qheat ¼ DQ2053Mð[T2053Þ

þXn�1

j¼1

DQj Mð[TjÞ �Mð[Tjþ1Þ� �

(13)

where DQj is the specific internal energy increaseassociated with shock heating the material to thepostshock temperature Tj (listed in Table 1), M(>Tj) isthe mass heated to the given temperature in the collision(Fig. 3). Qheat can be converted to a specific internalenergy increase for the parent body by dividing by theparent body mass; however, it should be noted thatQheat will be nonuniformly distributed within the parentbody—energy will be localized to the impact sites.

The techniques used here to determine the finaltemperature of the material are dependent on theaccuracy of the equation of state for converting peakshock pressures to postshock temperatures. As theANEOS equation of state does not account for latentheat, ANEOS overestimates temperatures in excess ofthe melt temperature. To account for this, the peakshock pressures and entropy that correspond to the post-shock temperatures that are used in this work arepresented in Table 1. As ANEOS equations of state areimproved in the future (e.g., by defining different

thermodynamic constants to be used for different high-pressure phases, and by treating each phase as a separatematerial in ANEOS), these shock pressures and entropiescan be used to amend the temperatures quoted in theremainder of this work. However, as shock melting isminimal in most cases presented here, the latent heateffects introduce only small uncertainties to our results.

TimestepAfter these calculations have taken place for all

impactor size bins, the timestep is advanced and theprocess is repeated, unless a disruptive impact hasoccurred, in which case the calculation for this parentbody ceases, and the calculation of the collisionalhistory of the next parent body starts. The size of thetimestep is chosen so that the probability of a collisionby an impactor in the smallest size bin, Pc(rmin,t) (whichis the most likely collision to occur), is less than unity.To explore the sensitivity of the results to the size of thetimestep, simulations were run with different timestepvalues, so that Pc(rmin,t) = 0.1, 0.25, 0.5, and 1. ForPc(rmin,t) = 1, the total number of collisions experiencedby the parent body was less than for Pc(rmin,t) ≤ 0.5,but did not change for Pc(rmin,t) = 0.1–0.5. Therefore, atimestep limit of Pc(rmin,t) = 0.5 was used for allsimulations presented here. This approach is anapproximation of a Poisson distribution.

Table 1. Peak shock pressures, entropies, and specificinternal energy increases corresponding to thepostshock temperatures used in this work.a

Postshocktemperature(K)

Specificentropy(erg g�1 K�1)

Peak shockpressure(GPa)

Specificinternal energyincrease (erg g�1)

350 12,925 1.4 5.29 9 105

400 14,365 2.9 1.07 9 106

500 16,810 5.8 2.16 9 106

600 18,832 8.9 3.27 9 106

700 20,552 11.9 4.39 9 106

800 22,048 15.0 5.51 9 106

900 23,371 18.1 6.63 9 106

1000 24,557 21.3 7.76 9 106

1250 27,070 28.9 1.06 9 107

1373b 28,128 32.7 1.20 9 107

1500 29,126 36.5 1.34 9 107

1750 30,867 43.7 1.62 9 107

2000 32,377 50.8 1.90 9 107

2053c 32,673 52.3 1.96 9 107

aThe values in this table are derived from the ANEOS equation of

state for dunite (Benz et al. 1989) and the e–a porous compaction

model (W€unnemann et al. 2006), using the technique described in

Davison et al. (2010). These values are appropriate for 20% porous

dunite.bSolidus.cLiquidus.

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

0

50

100

150

200

250

300

350M

(>T

)/M

imp

400K700K1000K1373K2053K

Fig. 3. Mass of material heated in a collision between two20% porous dunite planetesimals, where the impactor has onetenth the radius of the target (one thousandth of the mass),for a range of impact velocities. The heated mass isnormalized to the mass of the impactor. Both planetesimalshad an initial temperature of 300 K. T = 1373 K and 2053 Kare the inferred dunite solidus and liquidus, respectively.


Assumptions and Limitations

Time PeriodThe model described above focuses on the early

impact evolution of the solar system. Only the first100 Myr of solar system evolution are accounted for,as this is the time period in which the greatest numberof impacts occurred. By 100 Myr, the intrinsic collisionprobability and the total mass of planetesimals aresignificantly lower than at the start of the calculation.As the results below show, the frequency of collisions att � 100 Myr is much lower than in the first � 20 Myr.Collisions after this time are not accounted for, but, inthis study, we are focused on the earliest effects ofcollisions when impacts were most frequent. For futurestudies interested in a later period of solar systemhistory, the collisional evolution models could be runfor a longer period of time, and the same Monte Carlomethods applied to those results.

The Parent BodyTo fully quantify the amount of heating possible

from multiple impacts on the surface of a meteoriteparent body, no other heat sources (e.g., 26Al decay) wereconsidered in this study. The parent body was thereforeassigned a constant initial temperature of 300 K, which isthe reference temperature of dunite from the ANEOSequation of state (Thompson and Lauson 1972; Benzet al. 1989), used in hydrocode models to determineheating for specific impact scenarios (Davison et al.2010). Such a temperature may be appropriate for aplanetesimal inside the snow line in the early solarsystem, and therefore this assumption does not affectmany of the results of this paper. However, crater sizeand impact heating calculations are dependent on theinitial temperature of the material. Thus, to extend thisstudy to a wider region of the solar system, future studiesinvestigating this parameter in more detail are required.

The conditions for solid materials in the early solarsystem are somewhat uncertain. It is likely that theearliest solid bodies contained a significant fraction ofpore space (Love et al. 1993; Dominik and Tielens1997; Wurm et al. 2001, 2004; Blum 2003; Bland et al.2011). The gentle manner in which dust grains mustcollide during planet formation probably leads to bodieswith high porosities (Blum and Wurm 2008; Teiser andWurm 2009). Even growth to planetesimal formationmust occur at very low velocities to ensure sticking(e.g., Weidenschilling 1980, 1984, 1997) or to allowaggregates to become gravitationally bound (Cuzzi et al.2001, 2008; Johansen et al. 2007). Indeed, it is thoughtthat the porous nature of early planetesimals is seen inthe fabric of the Allende meteorite, with initialporosities reaching 60–70% (Bland et al. 2011).

Once the planetesimal formed, porosity woulddiminish: The rate of densification is a strong functionof internal pressure and temperature (Schwenn andGoetze 1978; Yomogida and Matsui 1984; Henke et al.2012). To simplify the calculations in this work, it isassumed that all parent bodies and impactingplanetesimals modeled have a constant porosity thatdoes not change with time. It is possible to estimate thesize of parent body for which porosity would becrushed out due to the internal pressure at the center.Assuming that the bulk density, qt, remains constantthroughout the parent body, then the pressure, P, at adistance r from the center of a body with radius rt canbe expressed as (Turcotte and Schubert 2002):

P ¼ 2

3pqtGðr2t � r2Þ (14)

For a 20% porous dunite parent body with a radiusof 250 km, the internal pressure at the center would beapproximately 61 MPa, and for a body with a radius of100 km, the pressure would be approximately 10 MPa.Nakamura et al. (2009) performed crush experiments ona range of porous materials to determine the pressure atwhich pore space begins to collapse. For an initially50% porous gypsum sample, crushing began at 20–30 MPa, and, at 50 MPa, the porosity had beenreduced to 40%. An initially 40% porous sampleexperienced little reduction in pore space, even after100 MPa of pressure was applied. Indeed, Men�endezet al. (1996) show that static pressures of 500 MPa to1 GPa are required to lithify sandstones on Earth.Therefore, the reduction of pore space in a cold parentbody due to internal pressure alone is not likely to besignificant. Densification would thus be minimal inbodies of the size of interest, unless temperatures wereelevated to > 700–800 K, at which temperaturessintering by creep diffusion would allow fordensification on time scales of approximately 106 yr. Atlower temperatures, primordial porosity would bepreserved. Even with radiogenic heating, porous layerswill be preserved in the outer regions of planetesimals(which are most affected by impact) as a result of thelow hydrostatic pressures and temperatures expected inthat region (Henke et al. 2012). Studies of asteroiddensities show that most asteroids contain both macro-and microporosity (e.g., Britt et al. 2002; Britt andConsolmagno 2003; Consolmagno and Britt 2004;Consolmagno et al. 2008), suggesting that at least someof the initial (micro) porosity of a parent body wouldbe maintained through time, along with theintroduction of new (macro) porosity due to fracturingand breakup during impact events. Each impact on aparent body could also crush out some pore space. This


compaction would be constrained to the locality of theimpact site (W€unnemann et al. 2006; Davison et al.2012b). Hence, in the simulations presented here, it isassumed that each impact occurs on a fresh,uncompacted target surface. Until we have better dataon the evolution of porosity through time, thisassumption is a necessary simplification in the model.

Mass of the Parent BodyIt is assumed that ejection velocities are suppressed

during impact into a porous target relative to thoseexpected during impact into a nonporous target (Collinset al. 2011a), and therefore that all material heatedduring a planetesimal collision is retained on the surfaceof the parent body (unless the parent body iscatastrophically disrupted; see below). As a consequenceof this assumption, the mass of the parent body remainsthe same throughout the simulation (the mass lost toejection is neglected). While this is an oversimplification,it is not expected to affect the mass of heated materialon the parent body, as it is likely that most heated massis not ejected during the collision (Collins et al. 2011a).In addition, the mass added from impactors is notaccounted for in the simulation. As most impactors aresmall in comparison to the parent body (and those thatare large are more likely to lead to a disruption), thisassumption is appropriate for a first-orderapproximation. In the simulations discussed below, theaverage total volume of impactors on a parent bodyranges from 1 to 20%, depending on the parent bodysize and collisional evolution model. Future workincorporating collisional outcome calculations (e.g.,Davison et al. 2010; Leinhardt and Stewart 2012) intothe Monte Carlo simulation will address this limitation.

Impact AngleFor each collision that takes place, no impact angle

is assigned. For calculations of the disruption threshold,an impact angle of 45° is assumed, as it is the mostprobable impact angle (see above, and Jutzi et al. 2010).For calculations of crater dimensions and impactheating, the impact angle is assumed to be vertical. Thislimitation is because much of the work defining thescaling laws was performed using two-dimensionalnumerical simulations, which employed axial symmetry.Vertical incidence impacts produce craters that are onlyapproximately 10% larger than impacts at 45° (Gaultand Wedekind 1978; Elbeshausen et al. 2009; Davisonet al. 2011). Preliminary modeling work has shown thatthe mass of material heated in an impact at 45° may beapproximately 20–30% less than in vertical incidenceimpacts (Davison et al. 2012a). As the use of three-dimensional simulations becomes more common (ascomputational speeds improve), the effect of impact

angle on crater size and impact heating can bequantified, and subsequently used to overcome thislimitation in this work. This assumption only affectsour estimates of the mass heated in the planetesimaland does not affect the size of the impactors or impactvelocities that will be experienced by a given body.

RESULTS

There exists a large parameter space to which thisapproach can be applied. In the following section, theresults of the Monte Carlo model described above arepresented for the collisional history of a 100 km radiustarget body for the CJS model of collisional evolution,with f = 100. A radius of 100 km is representative ofthe expected size of chondrite parent bodies (e.g.,Harrison and Grimm 2010). In discussing the thermalconsequences and disruption threshold of the impacts,we assume a uniform porosity of 20%, a value that isless than the expected initial porosity of freshplanetesimals and close to the typical porosity seen incurrent chondrites. While porosity probably evolves incomplicated 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 body’s collisionalhistory to the disk collisional evolution model (CJS orEJS), the size of the parent body (rt = 50, 100, and250 km), and the porosity (nonporous and 20%) of theparent body.

CJS Model, rt = 100 km, φ = 20%

OverviewFigure 4 shows statistics of the number of impacts

experienced by a 20% porous, 100 km radius parentbody. On average, a parent body that survives 100 Myrwithout being catastrophically disrupted sustainsapproximately 851 � 26 (1–r deviations) impacts frombodies with a radius greater than 150 m (Fig. 4a).During this same time period, 7.6% of parent bodiesexperience a disruptive collision; that is, they experiencea collision for which Q > Q*

D. Of those that aredisrupted, the number of collisions with rimp > 150 mprior to disruption varies from 1 to approximately 900,with an approximately equal chance of receiving anynumber of collisions within that range (Fig. 4b).Figures 4c–e show the number of collisions sustained bythe parent body for impactors larger than 0.05 rt, 0.1 rt,and 0.2 rt, respectively. On average, a parent bodyreceives approximately 1.04 impacts with rimp > 0.05 rt.The probability that a parent body will experience atleast one impact of this size or larger (defined here as


g0:05rt ) is 66%. Approximately one in five parentbodies experiences an impact with rimp > 0.1 rt(g0:1rt ¼ 17%) and, of that number, approximately half

are disrupted. The mean number of impacts for whichrimp > 0.2 rt is 0.09, and g0:2rt = 9%. Of those thatexperience at least one impact in this size range, around4 of 5 were disrupted—hence, the one in five that wasnot disrupted experienced a collision with a largeimpactor at sufficiently low velocity to allow survival. Asummary of these statistics is presented in Table 2,along with statistics from several other simulations.

Crater Sizes, Excavation Depth, and Impact Energyfrom Individual Collisions

In addition to impactor size, the energy (velocity) ofthe impact is critical for determining the extent to whichmaterials are processed during a collision. For each ofthe impacts that occurred, we also determined the energyas a fraction of the disruptive energy (i.e., Q=Q�

D).Figure 5a shows the mean cumulative number of impactson a parent body as a function of Q=Q�

D. On average,each parent body will see one impact with a Q=Q�

D of atleast 0.005. These impacts may be from relatively largeimpactors moving at low velocity or small impactorsmoving at very high velocities—this figure does notdifferentiate between them.

For each impact that occurs on the parent body,the depth that material is excavated from and the sizeof the impact crater formed can be determined fromscaling laws (Equations 2–6). Figure 5b shows thenumber of impacts on a parent body as a function ofthe excavation depth: each parent body has a 50%chance of experiencing at least one impact thatexcavates material from at least 10 km depth, whileapproximately 100 impacts will excavate material froma depth of at least 1 km. Figure 5c shows the numberof impacts on a parent body as a function of the finalcrater diameter. The transition from simple to complexcraters on a 100 km radius, 20% porous parent bodyoccurs at Df � 320 km: that is, larger than the parentbody itself; in other words, all craters that form on thisparent body, and do not catastrophically disrupt it, willbe simple craters. On average, a parent body willexperience one impact that forms a crater at least94 km in diameter. Approximately 150 events perparent body will form craters 10 km in diameter, orlarger. On those parent bodies that are not disruptedduring the initial 100 Myr, on average, the total area ofall craters is approximately 0.8 times the surface area ofthe parent body (assuming that the craters are circular,with a final diameter calculated from Equations 2–5). Arepresentative example of the surface from onesurviving parent body is shown in Fig. 6. In this figure,the craters were placed randomly on a square surfacewith an area equal to the parent body surface area.Figure 6 shows that while some craters will overlap oneanother, a large number of the craters will form at least

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

0.000.020.040.060.08

Pro

babi

lity μ = 851.83

σ = 25.64

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

0.00

0.01

0.02

0.03

0.04

Pro

babi

lity

0 4Number of impactors, rimp > 0.05rt

0.00.10.20.30.40.5

Pro

babi

lity μ = 1.04

η0.05rt = 66.53%

Survived 100Myrs Disrupted

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

0.00.20.40.60.81.0

Pro

babi

lity μ = 0.18

η0.1rt = 16.88%

0 1 2Number of impactors, rimp > 0.2rt

0.00.20.40.60.81.0

Pro

babi

lity μ = 0.09

η0.2rt = 9.27%

(a)

(b)

(c)

(d)

(e)

Fig. 4. Statistics of the number of impacts expected on a100 km radius, 20% porous parent body, from the CJS model.a) The probability that a parent body that survives to 100 Myrwill receive a given number of impacts from impactors withradii >150 m; l and r are the mean and standard deviation. b)The probability that a parent body that is disrupted before100 Myr will receive a given number of impacts fromimpactors with radii >150 m. c–e) The probability that a parentbody will receive a given number of impacts from impactslarger than a certain size. g is the probability that a parentbody will experience at least one impact of this size or larger.


partially on a fresh, uncompacted surface, and that onlya small portion of the surface will be left unaffected bycollisions. Thus, the assumption of the model that allcraters form on a porous surface is reasonable. Thiscalculation assumes that each crater is formed byvertical impact, and thus should be viewed as an upperlimit. Experiments and three-dimensional modelingsuggest that, at an impact of 45°, the crater area may beapproximately 20% smaller in area than a normalincidence crater (Burchell and Mackay 1998; Davisonet al. 2011). Figure 5 shows that, while not all of theapproximately 850 events that are predicted to occurfrom the model will have a global effect on the parentbody, there are likely to be hundreds of events that willinfluence the outer few km of the parent body: that is,the region most likely to be the source region for lowpetrologic type chondrites (e.g., Harrison and Grimm2010). Only a small fraction of the surface area is likelyto escape without being directly processed by impacts.

Cumulative Impact EnergyThe total amount of impact energy deposited on a

parent body can be determined by summing togetherthe impact energy, Q, for each impact. For 99% of theparent bodies that survive to 100 Myr in thissimulation, the total impact energy, Qtot, is within therange 4.5 9 107–6.3 9 109 erg g�1, with a geometricmean Qtot of 2.3 9 108 erg g�1. For parent bodies thatare disrupted, Qtot is several orders of magnitude larger,mainly due to the energy of the final disruptive impact:99% of disrupted bodies have a Qtot in the range

2.8 9 109–5.1 9 1013 erg g�1, and the geometric meanQtot is 6.0 9 1010 erg g�1. For reference, Q*

D fromJutzi et al. (2010) for an impact at 5 km s�1 into aporous 100 km parent body is approximately2 9 109 erg g�1. Figure 7a shows a histogram of Qtot

for all parent bodies modeled. A fraction of this totalimpact energy is converted to heat; this is quantifiedbelow, in the Impact Heating section.

Most Energetic ImpactAs Q*

D depends on the impact velocity and theimpactor size, the impact with the largest Q may not bethe impact closest to the disruption threshold. Indeed,in approximately one of every eight parent bodies, themost energetic impact is not the most disruptive.Therefore, the most energetic impact sustained on aparent body is defined here as the impact with thelargest Q=Q�

D. Figure 7b shows the distribution of themost energetic impact for all parent bodies modeled.For those parent bodies that are not disrupted, Q=Q�

D

varies between approximately 10�3 and 1, and thegeometric mean is 0.01; that is, the average impactenergy of the most energetic impact in parent bodiesthat are not disrupted is approximately 1% of thatneeded to disrupt the body. For reference, Q=Q�

D = 0.01is approximately equivalent to a 7.5 km radius projectileimpacting the 100 km radius parent body at 5 km s�1.The collateral effects of this type of impact arediscussed in more detail in Davison et al. (2012b); suchimpacts can produce peak temperatures and coolingrates consistent with what is seen in types 3, 4, 5, and 6

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

Model f a rtb (km)

Nimp ≥ rimpc

Ndisrupt,d %150 m (survivors) 0.05 rt 0.1 rt 0.2 rt

CJS 100 50 l = 214.25 l = 1.56 l = 0.32 l = 0.07 8.2

r = 13.96 g = 80.5%e g = 28.2% g = 7.3%CJS 100 100 l = 851.83 l = 1.04 l = 0.18 l = 0.09 7.6

r = 25.64 g = 66.5% g = 16.9% g = 9.3%CJS 100 250 l = 5303.38 l = 0.81 l = 0.36 l = 0.17 2.6

r = 61.84 g = 55.9% g = 30.6% g = 15.6%EJS 95 50 l = 77.28 l = 0.71 l = 0.15 l = 0.03 3.7

r = 8.55 g = 51.8% g = 14.0% g = 3.8%

EJS 95 100 l = 307.35 l = 0.48 l = 0.07 l = 0.03 2.9r = 15.86 g = 39.1% g = 6.5% g = 3.3%

EJS 95 250 l = 1912.58 l = 0.30 l = 0.13 l = 0.06 1.3

r = 37.50 g = 26.0% g = 11.9% g = 6.1%af is the ratio of the total initial mass of the planetesimal population to the final mass of planetesimals in the asteroid belt.brt is the radius of the parent body.crimp is the impactor radius.dNdisrupt is the percentage of parent bodies catastrophically disrupted within 100 Myr.eg is defined as the probability that a parent body will experience at least one impact greater than or equal to rimp. These figures are for parent

bodies with 20% porosity.


H-chondrite meteorites. Figure 7c–d shows the impactvelocity and radius for the most energetic impact. Atypical impact velocity in the most energetic impact isapproximately 7–10 km s�1, but a range fromapproximately 1 km s�1 up to 25 km s�1 is possible.For bodies that are disrupted, the impactor radius istypically in the range 30–100 km. The minimumdisruptive impactor radius in the simulations weperformed is approximately 22.2 km (for which the

impact velocity was 17.8 km s�1), and 99% of alldisruptors are >28 km in radius. For those bodies thatare not disrupted in the first 100 Myr, the averageradius of the most energetic impactor is approximately7 km. For nearly 1% of surviving parent bodies, themost energetic impactor was >30 km in radius; theseimpactors have low velocities, so the impact energy isbelow Q*

D.

Time of ImpactFigure 8 shows the average number of collisions on

a potential meteorite parent body (impactors larger thanrimp = 150 m) during each 1 Myr period of the first100 Myr. Approximately 19% of the collisions that aparent body experienced were during the first 5 Myr,approximately 39% during the first 10 Myr, andapproximately 64% during the first 20 Myr; 75% ofimpacts occur within the first 29 Myr. Within a factorof approximately 2, the number of impacts on a body inthe first 100 Myr is expected to be approximately equalto the number of impacts in the remaining 4.4 Gyr.

Figure 7e shows the time that the most energeticimpact occurs for the modeled parent bodies.Approximately 21% of disruptive collisions occur in thefirst 5 Myr, 43% occur in the first 10 Myr, and 68%occur in the first 20 Myr. Of those parent bodies that

−8 −6 −4 −2 0 2 4Impact energy, log10(Q/Q�

D)

10−2

100

102

104Cum

ulat

ive

num

ber

ofim

pact

s>

Q/Q

� D

10−1 100 101 102 103

Excavation depth, Hexc [km]

10−2

100

102

104

Cum

ulat

ive

num

ber

ofim

pact

s>

Hex

c

100 101 102 103 104

Crater diameter, df [km]

10−2

100

102

104

Cum

ulat

ive

num

ber

ofim

pact

s>

d f(a)

(b)

(c)

Fig. 5. Plots showing the average number of collisions thateach parent body experiences as a function of the scale of theimpact. a) Cumulative number of impacts with an impactenergy greater than Q/Q*D; b) cumulative number of impactsthat excavate material from greater than Hexc depth; c)cumulative number of impacts that form a final crater largerthan df. Dotted lines in (b) and (c) represent events for whichthe scaling laws used are not sufficient to describe the outcomeof the impact; that is, an excavation depth greater than theradius of the parent body or a crater diameter larger than theparent body diameter are not physical; these events probablyrepresent either disruptive or subcatastrophic events, whichmay not form a traditional impact crater.

0 100 200 300x (km)

0

100

200

300

y(k

m)

Fig. 6. The craters from one 100 km radius, 20% porousparent body from the CJS simulation, randomly placed on asquare with an area equal to the surface area of the parentbody, to show the coverage of craters on an average survivingparent body. While some craters overlap, most form at leastpartially on a fresh surface. Darker shading representsoverlapping craters.


are not disrupted in the first 20 Myr, 97% will surviveto the end of the 100 Myr period simulated by thismodel (for this study, “surviving” is defined as notbeing disrupted by 100 Myr).

The average time of the most energetic collision insurvivors occurs later than the disruptive collisions: only

5% of these collisions occur in the first 5 Myr, 16%occur in the first 10 Myr, and 39% occur in the first20 Myr. Some of the most energetic collisionsexperienced on the parent body therefore continue laterthan the disruptive collisions, although afterapproximately 30–40 Myr, their frequency decreases(only one third of the most disruptive collisions occurafter 40 Myr). The difference in the timing of disruptivecollisions and most energetic collisions on survivingbodies is because: (1) if a disruptive collision occurs, thesimulation is stopped, and no more collisions can occuron that body, which would therefore favor earliercollisions, and (2) while the average collision velocitycontinues to increase until around 30 Myr, the numberof large impactors, capable of causing a disruption,decreases rapidly after approximately 10 Myr.

Impact HeatingFigure 9 shows the probability that a given fraction

of a parent body will be heated to a given temperature(assuming that the impact occurs onto a fresh surface),for those parent bodies that survived 100 Myr without adisruptive collision (Fig. 9a), and those that weredisrupted (Figs. 9b–c). These figures can be interpretedin two ways: First, if one is interested in the outcomeassociated with a given probability, the figure can beread horizontally, to establish the mass fraction heatedto a given temperature (this is the maximum massfraction, as oblique impacts are not considered).Second, to infer the probability of a particular outcome

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

log10(Qtot [erg/g])

0.00

0.04

0.08

0.12

Pro

babi

lity

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

log10(Q/Q�D)

0.000.040.080.120.16

Pro

babi

lity

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

babi

lity

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

babi

lity

−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

babi

lity

(b)

(a)

(c)

(d)

(e)

Fig. 7. Statistics for the amount of impact energy delivered to a100 km radius, 20% porous parent body from the CJSsimulation. The “most energetic impact” is defined as the impactwith the maximum Q/Q*

D on each parent body. Qtot is the totalimpact energy received on the parent body in the first 100 Myr.

0

5

10

15

20

25

30

35

40

Impa

cts

per

Myr

0

1

2

3

4

Spec

ific

inte

rnal

ener

gyin

crea

sepe

rM

yrfrom

impa

cts

[107

erg/

g/M

yr]

0 20 40 60 80 100Time [Myr]

Energy increaseNumber of impacts

Fig. 8. The average number of impacts by impactors withradius >150 m that occurred in each 1 Myr period, for thefirst 100 Myr (gray solid line), and the specific internal energyincrease that occurred due to those impacts (black dashedline). These results are for a 100 km radius, 20% porousparent body from the CJS simulation. Averages were takenafter 100,000 parent bodies were modeled. The mostimportant time for impacts was the first 10–20 Myr.


(e.g., 50% of the mass being heated to a giventemperature), the figure can be read vertically. Forexample, of the surviving parent bodies, there is a 20%chance that a parent body will see around one thirtiethof the mass heated to 400 K, and a 30% chance thatone hundredth of the mass will be heated to the solidus.For those parent bodies that are disrupted, however,there is an 80% chance that the entire parent body willbe heated to 400 K, and a 50% chance that one fifth ofthe mass of the parent body reaches the solidus.Comparing Figs. 9b and 9c, it is clear that the large

impact that causes the parent body to disrupt is also byfar the most important impact in terms of heating thetarget materials. Further modeling is required todetermine how the heated material will be distributedamong the collisional fragments in disruptive events.

By using Equation (13), the specific internal energyincrease due to collisional heating Qheat can bedetermined for each parent body. For those parentbodies that survive, the mean specific internal energyincrease due to all the collisions that occur in the first100 Myr is 1.6 9 108 � 3.3 9 108 erg g�1. Thisrepresents approximately 37.4 � 3.6% of the predictedtotal impact energy Qtot received by surviving bodiesand produces a globally averaged temperature increaseof approximately 15–35 K. This is consistent withearlier findings that nondisruptive collisions are unlikelyto cause global heating on a meteorite parent body(Keil et al. 1997). For disrupted parent bodies, muchmore impact energy is converted to heat: the totalcombined specific internal energy increase in the targetbody is 9.69109 � 6.79109 erg g�1 (corresponding toglobal average temperature increases of approximately600–1000 K). However, this represents a smallerpercentage of the total impact energy received on thosebodies (17.7 � 11.8%), because much of that impactenergy is used mechanically to disrupt the parent body.As all the impacts used in this analysis are normalincidence impacts, the estimate of the amount of impactenergy converted to heat should be considered an upperlimit: oblique impacts would probably convert lessimpact energy into heat.

Figure 8 also shows this increase in specific internalenergy as a function of time. This reiterates that theinitial � 10–20 Myr are the most important in termsof collisional processing and the delivery of heat tothe parent body, with the peak increase of>3 9 107 erg g�1 Myr�1 occurring at approximately11 Myr. However, this figure also shows that even after50 Myr, at which time heating from short-livedradionuclides would be negligible, collisions could stilldeliver approximately 5 9 106 erg g�1 Myr�1. Moreover,as most impact energy is deposited in the outer layers ofthe parent body (e.g., of the approximately 850 expectedimpacts on a parent body, on average, less than oneimpact will excavate material from more than 10 kmdepth), the specific internal energy increase in thatmaterial may be significantly higher. As a first-orderapproximation, if it is assumed all the specific internalenergy is deposited in the outer 10 km of the parentbody, then the energies presented above and in Fig. 8could increase by a factor of approximately 4. Thisshows that while global heating by impacts is unlikely,heating on the local scale during the period of peakimpact activity is possible.

0

20

40

60

80

100

Pro

babi

lity,

%

LiquidusSolidus1000K700K400K

Survivors

0

20

40

60

80

100

Pro

babi

lity,

%

Beforedisruption

10−4 10−3 10−2 10−1 100

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

0

20

40

60

80

100

Pro

babi

lity,

%

(a)

(b)

(c)

After disruption

Fig. 9. The probability that a given fraction of a parent bodywill be heated by impacts to a range of temperatures, for a)those bodies that survive 100 Myr without beingcatastrophically disrupted; b) those bodies that are disrupted,immediately before the disruptive impact occurs; and c) thosebodies that are disrupted, after the disruptive impact.


Influence of Parent Body Size

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

Number of ImpactsTable 2 shows statistics for the number of collisions

and the probability of catastrophically disrupting theparent body. A parent body of 50 km radius experiencesa factor of approximately four fewer collisions that the100 km parent body, and a 250 km parent bodyexperiences approximately six times more collisions thana 100 km body—these factors correspond to the changein surface area. Smaller parent bodies are more likely tobe disrupted, despite the lower frequency of collisions:8.2% of 50 km radius bodies are disrupted, 7.6% of100 km bodies, and 2.6% of 250 km bodies, due to theQ*

D dependence on parent body size—the value of Q*Dincreases with target size as we are in the gravity regime.

The number of impactors one twentieth of theradius of the parent body decreases with parent bodysize: on average, there are approximately 1.6 impacts ofthis size on a 50 km parent body, compared toapproximately 1.0 on a 100 km body and approximately0.8 on a 250 km body. However, for larger relativeimpactor sizes, the relationship is not so simple: whileone in five 100 km parent bodies experiences at leastone impact of at least one tenth of the parent body size,for both 50 km and 250 km radius bodies, that numberincreases to one in three; for impactors that are at leastone fifth of the size of the parent body, 250 km parentbodies experience approximately twice the number ofimpacts than do both the 50 km and 100 km radiusparent bodies. Thus, there is not a direct relationshipbetween the number of large impactors and the size ofthe parent body. This is probably a consequence of thewavy nature of the SFD; for smaller parent bodies,there are larger impactors because there are morebodies in that size range in the population; for example,there are approximately 40 times more 5 km radiusimpactors than 25 km radius impactors at 10 Myr. Thetransition between these two effects seems to besomewhere between 50 and 250 km radius, forimpactors larger than one tenth of the parent body size.Further modeling of a range of parent body sizes couldconstrain this point of inflection further.

Impact EnergyFor parent bodies that survived 100 Myr without

disruption, the average impact energy received, Qtot (thesum of the impact energy from all impacts on the parentbody, normalized by the parent body mass), is

approximately the same for all sizes of parent bodiesmodeled here (Table 3). However, for those bodies thatare disrupted, the larger parent bodies (rt = 250 km)receive approximately twice the total specific impactenergy: A larger specific impact energy is required todisrupt a larger parent body (by a factor of 1.22 forporous materials), which results in fewer of the largerparent bodies being disrupted. This is reflected in theaverage impact energy from the most energetic impact(Q=Q�

D)max, which decreases monotonically with increasingparent body size.

Sensitivity to Collision Evolution Model

Equivalent simulations were performed for the threeparent body sizes discussed above, this time using thesecond collisional evolution model parameters (EJSmodel). Results for these models are summarized inTables 2 and 3. The total number of impacts receivedon the surface of a planetesimal in the EJS model is afactor of approximately 2.8 fewer than the equivalentsized parent body in the CJS model. For impactorslarger than a given size (0.05 rt, 0.1 rt, and 0.2 rt), theaverage number of impacts per parent body is a factorof 2–3 fewer in all cases listed in Table 2. This is, inpart, because the mass of the planetesimal populationdepletes more rapidly in the EJS simulations (Fig. 1c).Furthermore, the inclinations of bodies are morerapidly excited in EJS simulations, which makescollisions less likely: after 1 Myr, the intrinsic collisionprobability is always lower in the EJS simulationcompared with the CJS simulation. In the EJSsimulation, Pi = 3.4 9 10�18 km�2 yr�1 at 1 Myr, anddecreases monotonically to 1.6 9 10–18 km�2 yr�1 by100 Myr; however, in the CJS simulation, Pi remains

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

Model f a rtb (km)

Geometric mean

Qtot,survivors

(erg g�1)

Qtot,disrupted

(erg g�1) (Q/Q*D)max

c

CJS 100 50 2.7 9 108 7.0 9 1010 0.059CJS 100 100 2.3 9 108 6.0 9 1010 0.017CJS 100 250 2.5 9 108 1.2 9 1011 0.004

EJS 95 50 1.4 9 108 7.0 9 1010 0.022EJS 95 100 1.3 9 108 9.6 9 1010 0.007EJS 95 250 1.1 9 108 1.2 9 1011 0.001af is the ratio of the total initial mass of the planetesimal population

to the final mass of planetesimals in the asteroid belt.brt is the radius of the parent body.c(Q/Q*

D)max is the ratio of the maximum impact energy from a single

impact to the impact energy required for catastrophic disruption.


steady between approximately 3.5 and 4.2 9

10�18 km�2 yr�1 until 10 Myr, before decreasing to1.6 9 10�18 km�2 yr�1 at 100 Myr. The fraction ofparent bodies disrupted in the EJS simulation is alsoapproximately a factor of four fewer than in the CJSsimulation; this is as a result of the eccentric orbits ofJupiter and Saturn exciting bodies more readily in theEJS simulation, thus leading to a more rapid reductionin the number of planetesimals in the solar system,which, in turn, 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 from allcollisions that occur on surviving bodies is a factor ofapproximately 2–2.5 lower in the EJS simulations than inthe CJS simulations, again, reflecting the lower numberof collisions. For those parent bodies that are disrupted,the average total impact energy does not changesignificantly between the CJS and EJS cases (because theimpact energy required for disruption is unchangedbetween the two models, and, if a disruptive impactoccurs, that impact usually delivers the most energy).

As noted above, the collisional evolution modelsthat we employ in this work produce too muchcollisional grinding, leading to slightly fewer bodiesbetween 10 and 100 km in radius. This implies that wemay 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 theparent 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 nonporousvalues for B and b (Jutzi et al. 2010), and thecalculation of heating on the parent body used resultsfrom iSALE simulations with nonporous material. Thenumber of collisions and the frequency of disruptionare similar to the porous case: 7.4% of nonporousparent bodies are disrupted in this simulation. Themajor difference between the nonporous simulationand the porous simulation is the fraction of the impactenergy received on the parent body that is convertedto heat: in surviving bodies in the nonporoussimulation, 7.2 � 2.6% of the total impact energy isconverted to heat, and 2.8 � 3.2% is converted to heatin disrupted bodies (in both cases, a factor ofapproximately five less than in the porous case). Thisreiterates the result that the shock compaction of porespace can greatly increase collisional heating (Davisonet al. 2010).

DISCUSSION

The results in the previous section show thatcollisions on meteorite parent bodies would have beencommon events during the first 100 Myr. Many parentbodies would have been disrupted by energetic impactsduring that time. Those that survived without beingdisrupted probably sustained many impacts on theirsurfaces, processing (compacting, heating, and mixing)the outer layers. The extent to which this occurred canvary by large amounts depending on the size of thebody, and the collisional evolution of the planetesimalpopulation, but it is unlikely that many parent bodiesescaped without a significant number of impactsprocessing their upper few kilometers.

Impact Histories of Specific Parent Bodies

In this section, the model described above is appliedto examine the plausibility of some of the invocations ofimpacts in meteorite parent body histories. To examinethe histories of some meteorite parent bodies, data frommeteorites (such as peak temperatures, timing ofimpacts, etc.) can be compared with data from theparent bodies simulated in the Monte Carlo model.

Timing of t = 0There is one point of clarification that should be

considered in this analysis: the time t = 0 in our modelis defined as the time that the gas dispersed from thedisk such that it does not significantly affect the orbitsof the planetesimals. In general, meteorite studies quotetimings from the formation of CAIs (Ca-Al-richinclusions—some of the oldest known materials foundin meteorites). CAIs are predicted to have formed4568.5 � 0.5 Myr ago (Bouvier et al. 2007). Theseevents (gas dispersion and CAI formation) probably didnot occur at the same time. Ciesla (2010) argued thatthe CAIs which define t = 0 for the solar systemformed right when the solar nebula stopped accretingsignificant amounts of mass from its parent molecularcloud; astronomical surveys suggest that disks lastapproximately 2–5 Myr after this point (Haisch et al.2001). Thus, the timings discussed below could changeby up to approximately 2 Myr, and thereby alter theconclusions discussed below. The effects of movingt = 0 by approximately 2 Myr (i.e., assuming that thegas dispersed 2 Myr after CAI formation—therefore,t = 0 in the Monte Carlo model was 2 Myr after CAIformation) are discussed below. An assumption in thisanalysis is that collisions between the formation of theCAIs and the dispersion of the gas are considered toogentle to be accounted for in the model. The discussionbelow shows that while the choice of t = 0 has an effect


on the results of the model when looking at collisioneffects in the very early solar system, its influence isdiminished for events that took place afterapproximately 5–10 Myr. To fully quantify the effectsof collisions in the first several million years of solarsystem history, a more thorough understanding of thetiming of gas dispersion and the onset of the dynamicalexcitation of the planetesimal population is required.

CB Chondrite Parent BodyThe bencubbinite CB chondrites contain large, exotic

metal aggregates with elemental abundances varyingstrongly with volatility; the origin of these aggregates isunknown. One theory suggests that an impact between ametallic planetesimal and a reduced-silicate body couldhave produced a vapor plume that was rich in metal andnot very oxidizing (Campbell et al. 2002), from which themetal condensed. To create an appropriate vapor cloud,Campbell et al. (2002) suggested that a collision of suchbodies at velocities >8–10 km s�1 occurred before 5 Myr.Chondrules are another feature of the CB parent body:The CB chondrules are relatively young compared withother chondrules, and appear to have formed in a singleevent, leading to speculation that they may have animpact origin (Krot et al. 2005), and that one impactformed both the chondrules and the metal aggregates.The timing of the chondrule-forming impact has beendetermined 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 noted that thischondrule formation mechanism is distinct from theimpact mechanism recently suggested by Asphaug et al.(2011), which requires impacts with velocities at<100 m s�1.

To test the likelihood that an impact could haveformed both the metal and the chondrules in the CBchondrite parent body, we can thus look for an impactthat caused vaporization. The time of the impact mustbe after 4.3 Myr (from Krot et al. 2005), but before5 Myr (from Campbell et al. 2002). Campbell et al.(2002) do not offer a constraint of the size of thecolliding bodies. Krot et al. (2005) suggest an impactbetween Moon-sized bodies, but that constraint issimply based on requiring a velocity high enough toproduce vapor, as the gravity of those bodies wouldincrease the velocity at impact. As the planetesimals inour model are able to reach collisional velocities of 8–10 km s�1 even at smaller sizes (due to interactions withthe gas giants and the planetary embryos), here we willconsider vaporizing impacts on each parent body sizediscussed above.

In the CJS models, on average three of every five50 km radius parent bodies will experience a vaporizingcollision (vimp > 8 km s�1), each 100 km radius body

will experience approximately 2.5 such collisions, andeach 250 km radius body will experience over 15vaporizing collisions. The number of collisions withvimp > 10 km s�1 is approximately 4–5 times fewer thanwith vimp > 8 km s�1. In the EJS models, there arearound 3 times more collisions with vimp > 8 km s�1

and around 10 times more collisions withvimp > 10 km s�1 than in the CJS models. While fewercollisions are expected overall in the EJS case comparedwith CJS, more of these high-velocity vaporizingcollisions occur, due to the high eccentricity of theplanetesimals’ orbits.

Those predictions do not account for the size of theimpactor, and include all vaporizing collisions withimpactors greater than 150 m in radius. If we insteadonly consider impactors at least one twentieth of theradius of the parent body, we get a sense of the numberof large-scale vaporizing collisions. In the CJS models,around one in a hundred parent bodies will see acollision with vimp > 8 km s�1 with an impactor of thatsize (independent of parent body size). In the EJSmodels, around one in twenty parent bodies sees a largeimpact with vimp > 8 km s�1.

If CB chondrites were formed as hypothesized byCampbell et al. (2002) and Krot et al. (2005), at mostapproximately 10 parent bodies would have undergoneevolution, allowing chondrules and metallic phases toform: To analyze whether this scenario is likely to haveoccurred to the CB chondrite parent body, we canexamine the number of parent bodies in the collisionalevolution model at 100 Myr of each size that remainedin the asteroid belt. In the CJS model, there were 165parent bodies of approximately 50 km radius (in thebins with a radius between 40 and 63 km), 41 parentbodies of approximately 100 km radius (in the bins witha radius between 79 and 126 km), and 3 parent bodiesof approximately 250 km radius (in the bins with aradius between 199 and 315 km). In the EJS model,these numbers are 168, 43, and 3, respectively. So, inthe CJS case, one or two 50 km radius parent bodiesthat remain in the asteroid belt would have seen a largevaporizing collision, and there is an approximately 40%chance of a 100 km radius body and an approximately3% chance of a 250 km radius body experiencing avaporizing collision and surviving in the asteroid belt.In the EJS case, there would be eight 50 km radiusbodies, two 100 km radius bodies, and a 15% chance ofa 250 km radius body surviving in the asteroid belt thatmatch the CB chondrite criteria. By moving thedefinition of t = 0 by 2 Myr (see above), the number ofvaporizing impacts decreases by a factor ofapproximately four; this does not change the overallconclusion that it is possible that a small number ofparent bodies experience the type of vaporizing collision


needed to produce the metal and chondrules seen in theCB chondrites and survive past 100 Myr.

CV Chondrite Parent BodyA model for the formation and differentiation of the

CV chondrite parent body to account for theunidirectional magnetic field was developed by Elkins-Tanton et al. (2011). In that model, parent bodies ofapproximately 100–300 km radius differentiated due tothe decay of 26Al, leaving an unmelted, chondritic crustof approximately 6–20 km thickness on top of a magmaocean. A porosity of 25% is assumed for the crust, closeto the porosity used in the Monte Carlo model of thiswork. The surviving, undifferentiated crust would thenserve as the source of the pristine CV chondrites. Onerequirement of the model is that the approximately 6–20 km thick crust must not be disturbed during the timethat the parent body is being heated and the magneticfield is in place (approximately 10 Myr), to prevent bothimpact foundering and breaches leading to magmaeruptions. While Elkins-Tanton et al. (2011) advocate aparent body that begins at around 100 km radius andcontinues to grow to 300 km radius while heating, ourmodel cannot account for changing parent body sizes.Thus, to test this model, here we will examine thehistories of parent bodies with fixed radii of 100 and250 km in the Monte Carlo simulation described aboveto determine if a parent body could have escaped anyimpacts large enough to penetrate the crustal material.The depth of penetration, Hpen, of a crater in a gravitydominated crater can be approximated as 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 beapproximately 0.8 times the surface area of the parentbody (Figs. 4 and 6). The average number of collisionsthat penetrate to 6 km deep is approximately five perparent body. Around one in four parent bodies willexperience an impact that penetrates to 20 km.Therefore, while less than 1% of 100 km radius parentbodies survive the first 10 Myr without a collisionpenetrating to 6 km, around 75–80% survive without acollision that penetrates to 20 km.

For 250 km parent bodies, more collisions can beexpected: approximately 22–24 collisions per parentbody penetrate to 6 km, and each parent body willexperience approximately one collision that penetratesto 20 km depth. No 250 km radius parent bodiessurvive 10 Myr without a collision that penetrates to6 km, and approximately 40–45% of bodies survivedwithout a collision that penetrated to 20 km.

Therefore, it seems unlikely that the CV parentbody could have developed in the way proposed byElkins-Tanton et al. (2011) if the crust was only ≤6 km

thick. Our Monte Carlo model therefore predicts that,for CV chondrites to come from a body such as thatmodeled by Elkins-Tanton et al. (2011), a thicker crust(approximately 20 km thick) is required: This wouldincrease the chances that the parent body could havesurvived for 10 Myr without experiencing a collisionthat would disrupt the thermal structure and thereforethe magnetic field. Whether these impacts wouldincrease the density of the chondritic crust to create adensity instability or cause it to founder outright will bethe subject of future work.

Above (in the CB Chondrite Parent Body section),we showed that approximately 40 parent bodies ofapproximately 100 km radius and approximately 3parent bodies of 250 km radius would survive to100 Myr and remain in the asteroid belt. Therefore, if acrust of 20 km was present, around 30 parent bodies of100 km radius would have survived past 100 Myr andmatched the criteria for the CV chondrite parent bodyfrom Elkins-Tanton et al. (2011), and only one parentbody of approximately 250 km size would havematched these criteria.

The number of bodies that match the CV chondritecriteria is not strongly affected by moving t = 0 by2 Myr. The total number of collisions that penetratedthe crust within the first 8 Myr of the model (rather thanthe first 10 Myr) decreased by around a third, and thenumber of bodies that escaped without a collisionincreased slightly: around 1–3% of 100 km radius bodiesescaped without a collision penetrating 6 km, andaround 80–85% survived without a collision penetrating20 km. No 250 km radius parent bodies survivedwithout a collision penetrating a 6 km crust, andapproximately half of all 250 km parent bodies survivedwithout a collision penetrating to 20 km depth. Thus,changing the timing of t = 0 does not change ourconclusions about the Elkins-Tanton et al. (2011) model:A thick crust (approximately 20 km) would be needed toprotect the CV chondrite parent body from a collisionthat could cause foundering and loss of the crust.

IAB/Winonaite Parent BodyThe IAB nonmagmatic iron meteorites and the

winonaite meteorites are thought to share a commonparent body (Benedix et al. 2000). While the majority ofheating in these bodies probably came from 26Al decay(e.g., Theis et al. 2013), there are some studies of thechronology of these meteorites that suggest thatthe parent body experienced several heating events afterthe decay of short-lived radionuclides. Impacts havebeen invoked to explain these relatively late thermalevents (Schulz et al. 2009, 2010, 2012), which arerecorded by the Hf-W chronometer as well as Sm andW isotopic abundances. An event at 2.5+2.3/�2.0 Myr


caused silicate differentiation; partial melting wascaused at 5.06+0.42/�0.41 Myr; further thermalmetamorphism occurred at 10.8+2.4/�2.0 Myr withtemperatures reaching 1273 K; and finally a disruptiveevent occurred after approximately 12 Myr. The extentto which impacts could provide the heat required tomatch the thermal history recorded in meteorites can beevaluated using the Monte Carlo model. Figure 10shows the probability that a given fraction of a 100 kmradius parent body (in the CJS simulation) is heated totemperatures above those of chondrites with petrologictype 6 (>1273 K) or above the assumed (dunite) solidus(>1373 K) in any impacts that occurred after 10 Myr(i.e., after the decay of 26Al). In most parent bodies,only a small fraction is heated to those temperatures:Less than one hundredth of the parent body reachesthese temperatures in more than half of all casesmodeled. However, in some bodies, a significantfraction of the parent body is heated in late collisions:In 7.5% of parent bodies, more than one tenth of theparent body is heated to the required temperatures.The Monte Carlo model can also be used to examinethe impact history of this parent body in more detail.Of the parent bodies that were disrupted after 12 Myr,each collision was examined to determine if it heatedmaterial to the required temperature in the given timeintervals. In the simulation of 100 km radius parent

bodies with the CJS collisional evolution model, 3834parent bodies (of 105) were disrupted after 12 Myr. Ofthose bodies, only three (around 0.1% of the disruptedbodies) had energetic enough collisions during each ofthe three given time periods listed above, to heat atleast one thousandth of the parent body to the requiredtemperature. It is unclear if the first heating event (at2.5+2.3/�2.0 Myr) was caused by impacts or radionuclidedecay (Schulz et al. 2010), as it is during the time when26Al decay was still active. Running the same analysis,but without the requirement for an impact to heatmaterial in that event, yields 54 parent bodies thatmatch the criteria (around 1.4% of the disruptedbodies). For the EJS model, 894 parent bodies (of 105)were disrupted after 12 Myr; of those, 8 parent bodies(approximately 1% of the disrupted bodies) experiencedcollisions to match all the events, and 48 parent bodies(approximately 5%) met the criteria when the firstheating event was excluded. This shows that thecollisional history needed to produce the IAB/winonaiteparent body may be difficult to achieve.

To evaluate the influence of our choice of t = 0 onthe analysis of the IAB/winonaite parent body, thetimes of the impact events were shifted by 2 Myr. In theCJS model, the total number of disrupted parent bodiesthat could match the criteria decreased from 54 to 11,and, in the EJS model, that number decreased slightlyfrom 42 to 37. Thus, the choice of t = 0 does notsignificantly change our conclusions about the IAB/winonaite parent body.

A caveat to the approach used here is that it isassumed that the material starts out cold before theimpact, and therefore requires a more energetic collisionto produce the levels of heating observed in themeteorite record; this approach does not take intoaccount the heating provided by short-livedradionuclides, and therefore should be taken as a lowerestimate of the probability of producing a parent bodyto match the IAB/winonaite body. Furthermore, herewe have assumed a parent body radius of 100 km—however, the size of the IAB/winonaite parent body isnot well constrained. A different parent body size couldyield different results. Further work is clearly needed tofully quantify the early impact history of the IAB/winonaite parent body.

H-Chondrite Parent BodyModels of the thermal evolution of the H-chondrite

parent body predict an onion-shell structure resultingfrom the heat produced by the decay of short-livedradionuclides, such as 26Al (e.g., Trieloff et al. 2003;Kleine et al. 2008; Harrison and Grimm 2010).However, Harrison and Grimm (2010), to explain somemeasurements of cooling rates, peak temperatures, and

< 1100Mt

1100Mt − 1

10Mt > 110Mt

Fraction of parent body heated to T after 10 Myr

10−2

10−1

100

Pro

babi

lity

T = 1273 KT = 1373 K

Fig. 10. Probability of a given fraction of a 100 km radiusparent body in the CJS model being heated above 1273 K(i.e., above petrologic type 6) or 1373 K (i.e., above the dunitesolidus) in impacts that occurred later than 10 Myr afterformation. In almost 90% of all parent bodies in thissimulation, less than one hundredth of the parent body washeated to either of those temperatures. In 7% of cases,between one hundredth and one tenth of the parent body washeated to those temperatures, and, in around 3.5% of cases,more than one tenth of the parent body was heated to thosetemperatures.


closure times that do not fit their onion-shell model,invoked an impact that disturbed the thermal evolutionof a portion of the body by excavating material from atleast 5.6 km deep on a 100 km radius parent body. Inthe simulations presented above, for a 100 km parentbody, on average 2.3 collisions per parent bodyexcavated to this depth in the CJS model (see Fig. 5b),and in the EJS model, 1.5 collisions per parent bodymatched this criterion. Approximately 8.4% and 22.7%of parent bodies modeled escaped without any collisionsexcavating to this depth in the CJS and EJS simulations,respectively. While the results depend strongly on thechoice of collisional evolution model (CJS or EJS), theyshow that most parent bodies (more than 77% of all100 km radius parent bodies) will experience at least oneimpact of the type required by Harrison and Grimm(2010). Of the 40 or so approximately 100 km radiusparent bodies that survived to 100 Myr, at least 30would have experienced a collision that excavated to atleast 5.6 km depth. Thus, the requirement of an impactdisturbing the H-chondrite parent body thermalstructure is supported by this study.

Other models of the H-chondrite parent body suggestthat impacts may have played a larger role in the thermalhistory (Scott et al. 2011), excavating and thoroughlymixing material (Krot et al. 2012). In that model, bothtype 4 and type 6 materials are brought to the surface byimpacts early in the evolution of the parent body. Thedepth to type 4 and type 6 material can be estimatedusing an onion-shell model. Harrison and Grimm (2010)predict that type 4 material may be as shallow as 0.9 kmbelow the surface of a 100 km radius parent body.Analysis of cooling rates from one type 6 meteoritepredicted type 6 material at 3.4 km depth. However,other analyses in that study using both cooling rates andclosure times predicted a depth of at least 11.2 km fortype 6 meteorites. In the CJS model of a 100 km radiusparent body, approximately 123 impacts per parent bodyexcavated material from 0.9 km depth, approximately 9impacts excavated from a depth of 3.4 km, andapproximately 0.3 impacts per parent body excavated from11.2 km. In the EJS model, fewer impacts excavated tothose depths with approximately 56 impacts excavated from0.9 km, approximately 5 impacts excavated from 3.4 km,and approximately 0.2 impacts excavated from 11.2 km.This shows that bringing type 4 material to the surface isa common process, and should occur in at least 50 eventsper parent body in the first 100 Myr. In the CJS model,the surface area of craters that excavated to this depth isthree quarters of the area of the parent body; in the EJSmodel, on average, around a third of the parent bodysurface would experience craters that excavate to thisdepth. If type 6 material is as shallow as 3.4 km, up tonine events could bring that material to the surface to

cool more quickly (more than half of the surface area isaffected in this way in the CJS model; one quarter in theEJS model). However, if type 6 material is deeper, thenfar fewer impacts would be able to excavate it—not everyparent body is likely to experience a collision during thefirst 100 Myr of the magnitude required to excavatematerial from a depth of 11.2 km (an average covering oftwo fifths of the parent body surface area in the CJSmodel and one fifth in the EJS model).

CONCLUSIONS

In this study, we have developed a model tosimulate the collisional histories of meteorite parentbodies. The model predicts that impacts were commonprocesses on parent bodies and the outer layers of thosebodies would have been strongly processed by shockevents. Impacts can provide a significant, secondaryheat source to parent bodies, which can be especiallyimportant after the decay of short-lived radionuclidesand in the regions near the surface of the originalparent body. The probabilities of different outcomesvary strongly with the different dynamical scenariosconsidered for the early solar system. Furthercompilation of how shock processes affected meteoriteparent bodies in the early solar system should allow usto test the different dynamical scenarios and collisionalhistories envisioned for terrestrial planet formation.

The model has been applied to evaluate theproposed histories of several meteorite parent bodies:At most, around 10 parent bodies that survived to100 Myr underwent a collisional evolution consistentwith the formation mechanism of the chondrules andmetallic phases of Campbell et al. (2002) and Krot et al.(2005); around 1–5% of bodies that were disrupted after12 Myr (which comprises approximately 1–4% of theinitial population of 100 km planetesimals) experiencedimpact events contemporaneously with some lateheating events on the IAB/winonaite parent body(Schulz et al. 2009, 2010); at least 30 of theapproximately 40 surviving 100 km radius bodies wouldsustain an impact that excavated material from a depthof at least 5.6 km, as predicted by Harrison and Grimm(2010) for the H-chondrite parent body, and almost allbodies experienced impacts that brought type 4 materialto the surface, as required by Scott et al. (2011); a thickcrust (approximately 20 km) is required on the CVparent body to prevent foundering and disturbance ofthe thermal structure—a crust of this thickness wouldprovide enough shielding for 10 Myr for approximately30 parent bodies to survive for 100 Myr and fit themodel of Elkins-Tanton et al. (2011), while a thin crust(approximately 6 km) would experience too manyimpacts that puncture it and thus allow foundering.


There are some critical uncertainties in the MonteCarlo simulations that should be addressed in futureiterations of the model: For example, the initial mass ofmaterial in the planetesimal population, the initial orbitsof the gas giants (CJS or EJS), the response of thedisruption threshold to temperature and porositychanges, and the evolution of the porosity structure of aparent body through time are all factors for which wehave chosen the best available approximations.Nonetheless, our approach offers a new means forevaluating the roles that impacts played in shapingmeteorite parent bodies. By better recognizing thesignatures and timing of impact events in meteorites, wecan understand the collisional history of meteoriteparent bodies and use that information to constraindynamical models for the solar system.

Acknowledgments—We gratefully acknowledge themajor contributions of Jay Melosh, Boris Ivanov, KaiW€unnemann, and Dirk Elbeshausen to the developmentof iSALE. We thank Ed Scott, Bill Bottke, and RobertGrimm for their insightful and constructive comments.TMD and FJC were supported by NASA PGG grantsNNX09AG13G and NNX12AQ06. TMD and GSCwere supported by STFC grant ST/J001260/1. GSC wasalso supported by STFC grant ST/G002452/1. DPO wassupported by grant NASA PGG grant NNX09AE36G.

Editorial Handling—Dr. Edward Scott

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APPENDIX

VELOCITY–FREQUENCY DISTRIBUTION

PARAMETERIZATION

To use the results from the dynamical andcollisional evolution simulations, they must beparameterized in such a way that the Monte Carlomodel can use them. In Fig. 1, the mean velocity ispresented as a function of time. To allow a VFD to be

produced at every timestep in the Monte Carlo model,the mean velocity is interpolated, and converted into aVFD using a Maxwellian distribution. To check that aMaxwellian distribution is appropriate, the VFD fromthe dynamical models was output at times throughoutthe simulation, and compared with the Maxwellian. Thedynamical model output is well fit by the Maxwelliandistribution (with a coefficient of determination,R2 > 0.96 for all timesteps in the first 100 Myr). Theevolution of the VFD for the CJS simulation is shownin Fig. A1.

0

0.2

0.4

0.6

0.8

Pro

babi

lity

P(v

=v i

mp)

t = 105 yrs

t = 106 yrs

t = 107 yrs

t = 108 yrs

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

0.0

0.2

0.4

0.6

0.8

1.0

Cum

ulat

ive

Pro

babi

lity

P(v

<v i

mp)

(a)

(b)

Fig. A1. Evolution of the VFD for the CJS simulation, at several points in time. The VFD is fit well (R2 > 0.96) by aMaxwellian distribution.


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