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Author's personal copy

INVITED REVIEW

Similar-sized collisions and the diversity of planets

Erik Asphaug �

Earth and Planetary Sciences, University of California, 1156 High Street, Santa Cruz, CA 95064, USA

a r t i c l e i n f o

Article history:

Received 10 September 2009

Accepted 31 January 2010

Keywords:

Planets

Impact

Collisions

Accretion

Planet Formation

a b s t r a c t

It is assumed in models of terrestrial planet formation that colliding bodies simply merge. From this the

dynamical and chemical properties (and habitability) of finished planets have been computed, and our

own and other planetary systems compared to the results of these calculations. But efficient mergers

may be exceptions to the rule, for the similar-sized collisions (SSCs) that dominate terrestrial planet

formation, simply because moderately off-axis SSCs are grazing; their centers of mass overshoot. In a

‘‘hit and run’’ collision the smaller body narrowly avoids accretion and is profoundly deformed and

altered by gravitational and mechanical torques, shears, tides, and impact shocks. Consequences to the

larger body are minor in inverse proportion to its relative mass. Over the possible impact angles, hit-

and-run is the most common outcome for impact velocities vimp between � 1:2 and 2.7 times the

mutual escape velocity vesc between similar-sized planets. Slower collisions are usually accretionary,

and faster SSCs are erosive or disruptive, and thus the prevalence of hit-and-run is sensitive to the

velocity regime during epochs of accretion. Consequences of hit-and-run are diverse. If barely grazing,

the target strips much of the exterior from the impactor—any atmosphere and ocean, much of the

crust—and unloads its deep interior from hydrostatic pressure for about an hour. If closer to head-on

ð � 302451Þ a hit-and-run can cause the impactor core to plow through the target mantle, graze the

target core, and emerge as a chain of diverse new planetoids on escaping trajectories. A hypothesis is

developed for the diversity of next-largest bodies (NLBs) in an accreting planetary system—the bodies

from which asteroids and meteorites derive. Because nearly all the NLBs eventually get accreted by the

largest (Venus and Earth in our terrestrial system) or by the Sun, or otherwise lost, those we see today

have survived the attrition of merger, evolving with each close call towards denser and volatile-poor

bulk composition. This hypothesis would explain the observed density diversity of differentiated

asteroids, and of dwarf planets beyond Neptune, in terms of episodic global-scale losses of rock or ice

mantles, respectively. In an event similar to the Moon-forming giant impact, Mercury might have lost

its original crust and upper mantle when it emerged from a modest velocity hit and run collision with a

larger embryo or planet. In systems with super-Earths, profound diversity and diminished habitability

is predicted among the unaccreted Earth-mass planets, as many of these will have be stripped of their

atmospheres, oceans and crusts.

& 2010 Elsevier GmbH. All rights reserved.

1. Introduction

During the late stage of terrestrial planet formation (Wetherill,1985), giant impacts occur when similar-sized planets at or nearthe largest end of their size distribution collide at speeds rangingfrom � 1 to a few times their mutual escape velocity vesc. Thisnotion of late giant impacts emerged alongside the idea for a giantimpact origin of the Moon (Hartmann and Davis, 1975), where aMars-sized projectile is proposed to have struck the proto-Earthto liberate a new planet composed mostly of Earth-like mantle.Giant impacts can be generalized as occurring between the largestand next-largest bodies at any stage of planet formation, at

impact velocities vimp comparable to the mutual escape velocity

vesc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2G

Mþm

Rþr

rð1Þ

which is the velocity at which two spheres collide if starting out atzero velocity at infinite distance. The radii rtR and masses m andM correspond to a spherical projectile and target, andG¼ 6:673� 10�8 cm2 g�1 s�2. This generalization of giant impactis called a similar-sized collision or SSC.

Agnor and Asphaug (2004a) studied collisions between equal-sized planetary embryos (r=R) and found that merger is inefficientexcept when vimp almost equal to vesc. Impact speeds are expected tobe higher than this in the late stage of terrestrial planet formation,since the orbits must be planet-crossing. This paved the way tostudies (Asphaug et al., 2006) of geophysical and compositional

Contents lists available at ScienceDirect

journal homepage: www.elsevier.de/chemer

Chemie der Erde

0009-2819/$ - see front matter & 2010 Elsevier GmbH. All rights reserved.

doi:10.1016/j.chemer.2010.01.004

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Chemie der Erde 70 (2010) 199–219


evolution in a broader range of scenarios ðrtRÞ. They found that it iscommon in gravitationally stirred-up populations for planetaryembryos somewhat smaller than the largest to dash up against thelargest but not accrete. These hit and run collisions dismantle theimpactors (r) or catastrophically disrupt them in peculiar ways.

It is argued below that many or most of the unaccreted next-largest bodies (NLBs) surviving the late stage of planet formationbear the scars of one or more hit and run collisions. A remarkablediversity is then predicted for the final collection of NLBs, whetherthey be Mars and Mercury of the inner solar system, middle-sizedmembers of Saturn’s satellites, Vesta and Psyche in the Main Belt,Quaoar and Haumea and other oddities beyond Neptune, or Earth-mass planets in solar systems with super-Earths. Next-largest bodiesare lucky to be here, and each is lucky in its own way.

Hit-and-run can be as common as accretion, when thecharacteristic random velocity v1 of a planetesimal swarm(relative to distant circular coplanar orbits) is comparable to thecharacteristic escape velocity vesc of the largest members of thepopulation. This random velocity is added to the escape velocity,so that spherical planets collide at an impact velocity

vimp ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv21þv2

esc

qð2Þ

When random velocity v1=vesc C0 accretion is efficient, but whenv1 � vesc hit and run is the most common outcome. The uniquepetrogenetic outcomes of hit and run collisions, and the predicteddiversity of NLBs and the asteroids and meteorites that derivefrom them, may be indicative of the random velocities that prevailduring the dynamical epochs of planet formation, in the earlieststages corresponding to the evolution of chondrites and chon-drules, and in the late stages that define the characters of finishedplanets.

1.1. Accretion basics

In the classical accretion theory of Safronov (reviewed inWetherill, 1980) the random velocity is related to the escapevelocity, regulated by the gravitational stirring, according to

Y¼v2

esc

2v21

ð3Þ

The Safronov number Y is postuated to be � 325 during thecourse of planetesimal growth (Safronov and Zvjagina, 1969;Safronov, 1972) and closer to Y� 122 in the late stage ofterrestrial planet formation (Wetherill, 1976). This relation arisesfrom the assumption of planetesimal (gas-free) accretion, withrandom velocities excited gravitationally by the largest bodiesinto mutually crossing orbits. Based on N-body numericalexperiments, Agnor et al. (1999) and O’Brien et al. (2006) findthat vimp ranges from41 to a few times vesc during the late stageof giant impacts, broadly consistent with Wetherill’s result.

The ratio v1=vesc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1=2Y

pdepends on the location within the

size distribution, since vesc and random velocity both change. Theensemble gravitational drag of small planetesimals reduces thevelocity dispersion of the larger embryos, so generally v1 is lowerfor steeper mass distributions (with greater masses of smallparticles). If gravitational stirring happens to small and largebodies alike, then smaller bodies encounter one another at higherv1=vesc , so even if the largest encounters are mostly accretionary(v1=vesc � 0:3, say), colliding bodies half as large will havev1=vesc � 0:6, with outcomes that are mostly hit-and run. This iswhy hit and run is described below as an edge effect, occurring atthe margin of the population.

Under dynamically cold conditions (v15vesc) the growth ofthe largest bodies can run away (Greenberg et al., 1978;Weidenschilling, 2008) since the rate of growth dR/dt increases

with R. This is because growing bodies sweep up smallplanetesimals within an enhanced cross-sectional area that isincreased by a gravitational focusing factor

Fg ¼ 1þ2y¼ 1þv2

esc

v21

ð4Þ

accounting for slow planetesimals falling in towards the body. Theother scenario, v1bvesc is sometimes called orderly growth; sincethere is no focusing all bodies increase in radius at the same rate.But orderly growth assumes perfect sticking during a sweep-up ofplanetesimals at high random velocity. Perfect sticking can be avery poor assumption when v1bvesc , and this calls to questionwhether orderly growth is a valid concept.

Planet formation is likely to involve quiescent epochs, and alsoepochs of moderate random stirring dominated by similar-sizedcollisions. It seems incontrovertible that moderate randomvelocities are required during the late stage, since orbits mustintersect across increasing distances. Epochs of random stirringare also expected during the first few million years of solar systemformation. The severe consequences of planetary dismantling bythe mechanism of hit-and-run are likely to be vital to the finalbulk chemistry of planets. Planetary growth is after all not just theaccumulation of a feeding zone by accretionary events; it is alsothe record of a comparable or even greater number of non-accretionary hit-and-runs, each with the capacity to dismantleand segregate a next-largest body’s mantle, core, atmosphere,crust and ocean.

1.2. Collision timescale

Collisions involving bodies within a factor of 2–3 in size areextended-source phenomena, distinct in important respects fromthe point-source collisional phenomena that cause the formationof impact craters (Melosh, 1989). A key difference is that there isno physically important central point in a similar-sized colli-sion—broad regions such as the cores respond in one way, and thecolliding mantles respond in another. The outer layers (atmo-sphere, ocean, crust) respond in yet another. The impact locus as itwere is hemispheric or even global in extent. The understandingof impact cratering benefits greatly from the principle of latestage equivalence, a strong form of hydrodynamic similaritywhereby the fundamental characteristics of a collision areobtained by geometric (power law) combinations of impactorradius, density and velocity (Holsapple, 1993). In cratering thisallows meaningful extrapolations of laboratory results to largegeophysical scales. Hydrodynamical similarity applies to SSCs, butnot the cratering concept of an impact locus.

A second major distinction between similar sized collisionsand impact cratering is that the contact and compressiontimescale for an SSC equals the gravity timescale. The smallerplanet is deformed mechanically (compressed and sheared) by itsabrupt deceleration against the target, while it is deformedgravitationally. Its fate, and the outcome of the collision, maydepend upon the inter-dominance of self-gravitational instabilityand shear instability. The deceleration and deformation of theprojectile in a head-on collision occurs on a timescale

tcoll ¼ 2r=vimp ð5Þ

where vimp is the collisional speed at the time of contact. Becausevimp � 1 to a few times vesc, the collisional timescale for SSCs istcoll � r=vesc . By comparison, the self-gravitational timescale is

tgrav ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3p=Gr

pð6Þ

where the density r¼M= 43pR3 ¼m= 4

3pr3 assuming uniformbodies. This is the time for a sphere of uniform-density matterto orbit itself. Because r� R for similar-sized bodies,

E. Asphaug / Chemie der Erde 70 (2010) 199–219200


vesc �ffiffiffiffiffiffiffiffiffiffiffiffiGM=r

p� r

ffiffiffiffiffiffiffiGr

pand thus the collisional timescale

tcoll � 1=ffiffiffiffiffiffiffiGr

p� tgrav. This confluence of gravitational and impact

deformational timescales contributes to the ‘‘lava lamp’’ likequality noticed in movies of giant impact simulations, withcollisional deformation countered by gravitational restitution.

1.3. Fate of the bullet

In planetary impact cratering the discrete fate of the projectileis only of significance for the most oblique angles of incidence(Pierazzo and Melosh, 2000). For SSCs the fate of the bullet is ofprincipal consequence, even for moderate impact angles close to301 from normal. This gives rise to a third important distinctionbetween SSCs and impact cratering: for typical geometries, only afraction of the colliding matter intersects. For impact velocitiesgreater than about 1.2vesc, depending on mass ratio g¼m=M, therest goes sailing on, and the abrupt shears and unloading stressesunleash a host of planetary processes, some rather novel toconsider. Taken together, these distinctions point to the existenceof a broad range of collisions, with physical outcomes betweenthose of tidal collisions like comet Shoemaker-Levy 9 whichdisrupted near Jupiter in 1992 (Asphaug and Benz, 1994); andplanetary-scale impact cratering events (e.g. Marinova et al.,2008; Nimmo et al., 2008), with an associated diversity ofcosmochemical and planetological outcomes.

By far the best studied archetype of a similar-sized collision isthe giant impact that has been proposed for the origin of theMoon (Hartmann and Davis, 1975). In the latest studies of thisscenario, the projectile gets dismantled (literally) by gravitationaland mechanical shears and instabilities into two components, itscore which merges with the center of the Earth, and its mantlewhich interplays with Earth’s outer mantle to form a protolunardisk of several lunar masses (e.g. Stevenson, 1987; Benz et al.,1989; Cameron and Benz, 1991; Canup and Asphaug, 2001). Thisfractionation of the bulk Moon from the bulk Earth, forming a newminor planet (bound as a satellite) lacking in iron and volatiles,illustrates how SSCs can leave their permanent and formativeimprint upon unaccreted next-largest bodies.

Giant impacts such as the Moon’s formation are high-energyend members of SSCs occurring late in planet formation, whenplanets are large and vescpR is large. The phenomenon can scaledown to smaller colliding pairs, impacting at correspondinglyslower velocity, at earlier stages of planetary formation. Forinstance, two molten planetesimals colliding during the first� 1 Ma at low random velocity (vimp � vesc � 100 m=s) might looklike the Moon-forming giant impact, and take place on the sametimescale tgrav.

As for solidified planetesimals, for instance cold chondriticasteroids or solidified differentiated embryos, the outcome of ahit-and-run can involve bulk textural changes, including energeticshearing and brecciation. The shear stress in a tidal collisionexceeds the strength and internal friction of rocky or icy bodieslarger than � 100 km, or even smaller if the disruptive effect of agrazing blow is considered. Changes with scale are considered indetail below.

2. Similar sized collisions

Most of the colliding mass contributing to the formation of aplanet comes by way of the several largest impacts, bodies withinan order of magnitude in mass and a factor of a few in size thathave been stirred up gravitationally into planet-crossing orbits.During the earliest stages of terrestrial planet formation most ofthe mass is in small particulates: crystals and amorphous phasesof dust and ice grow into clusters, and these into nuggets that one

might call planetesimals, which go on to sweep up the smallerbodies in a runaway. This phase is known from chemical,dynamical and astronomical evidence (Meyer et al., 2008) toend early on, although the transition from dust to planetesimals isa strongly debated topic. Cuzzi et al. (2001, 2008) and Johansenet al. (2007) have shown that turbulence can randomly initiatethe concentration of small particles into dense clusters that thenride in pack through the gas and dust, resisting further disruptionuntil they contract gravitationally, or through dissipation andsticking, into sizable planetesimals. These mechanisms mightmake it possible for large asteroid-sized bodies to bypasshierarchical growth and accrete directly out of the planetesimalswarm. Morbidelli et al. (2009) propose on this basis thatasteroids were ‘‘born big’’ to explain the apparent factor of � 4overabundance of 100 km asteroids in the present Main Beltrelative to a power law. If so, then after the gas and dust havecleared accretion is mostly a matter of similar-sized collisions.

Even in the case of hierarchical growth, and a power lawdistribution of sizes, most of the mass that collides during planetformation interacts at the large end of the feeding chain, withMars-mass bodies colliding into Earth-mass bodies, and lunar-mass bodies into Mars-mass bodies, and so on. Consider adifferential size distribution

nðrÞpr�a ð7Þ

where n=dN/dr and N(r) is the cumulative number of planetesi-mals larger than radius r. In the case a¼ 4 there is equal mass inequal logarithmic bins, i.e. equal mass in bodies hundreds of kmdiameter as in bodies meters in diameter. For a comminuted(ground-down) population obeying size-independent fragmenta-tion physics, the theoretically derived equilibrium value is a¼ 3:5(Dohnanyi, 1969). But observed planetesimal size distributionsare shallower, with the mass in the largest bodies. In the MainBelt the four largest asteroids, within a factor of 2 in diameter,account for half the total mass. Main Belt asteroids trend witha� 223, varying with size and sub-population. Recently-disrupted comet groups and the comet population as a wholeappear to have a� 1:7 (see Weissman et al., 2004). For planet-forming planetesimals, gravitational focusing and oligarchicsweep up of feeding zones shifts the mass further into largersizes (Kokubo and Ida, 1998). Thus for characteristic minor planetand small body populations, and embryonic populations, most ofthe mass—and hence most of the colliding mass—is found in thehandful of largest and next-largest bodies, whether they are bornbig or become big.

2.1. Colliding pairs

In their studies of the evolving Main Belt asteroid sizedistribution Bottke et al. (2005) and Morbidelli et al. (2009) usea statistical code to follow the erosional and disruptive evolutionof candidate primordial Main Belts, to see which ancestral sizedistributions could have evolved to the population now observed.Their models consider impactors up to the size of a given target,but no larger—a seemingly obvious choice given that smallasteroids are usually demolished by smaller members of thepopulation; among small bodies the random velocities aretypically orders of magnitude faster than their escape velocities.Indeed, demolition by an equal sized body seems exotic, let aloneby a larger body.

But for the largest asteroids, even at modern solar system highvelocities, the impactors required to disrupt them by traditionalmeans can be as big as they are—or larger. Fig. 1 shows adifferentiated planet 500 km diameter, consisting of 30 wt% ironand 70 wt% rock, impacted by a 200 km diameter rocky asteroid atvimp ¼ 10 km=s (top four panels) and 5 km/s (bottom two panels),

E. Asphaug / Chemie der Erde 70 (2010) 199–219 201


in both cases at the most probable impact angle y¼ 451 (fromAsphaug and Agnor, 2005). The 10 km/s impact is � 223 timesthe velocity typical of modern Main Belt collisions, and � 30times the mutual escape velocity of the pair. But this barely doesthe job of leaving half the mass behind (the definition ofcatastrophic disruption) and only exposes some bits of corematerial. The 5 km/s impact, at a more typical velocity, but still� 15vesc , blasts off the outer layer but leaves the mantlereasonably intact.

Based on this and other modeling (e.g. Scott et al., 2001) itappears that once planetesimals grow large, they become difficultif not impossible to disrupt. This would present a big problem,since meteorites show evidence for the cataclysmic disruption ofnumerous asteroid parent bodies that were at least severalhundreds of km in diameter (e.g. Keil et al., 1994; Yang et al.,2007). What is required is either a very energetically excitedpopulation of Main Belt precursor bodies, with v1=vesc]30, thatis Yu10�3, yet miraculously leaving Vesta’s basaltic crust intact,or else a mechanism for disruption that operates at lower levels ofexcitation.

The process of hit-and-run is capable of causing the cata-strophic disruption of objects hundreds of km diameter at therelatively moderate random velocities that are expected. Itrequires asteroid parent bodies to run into objects larger than

themselves, a scenario consistent with the Main Belt models byChambers and Wetherill (2001), Petit et al. (2001), and O’Brienet al. (2007).

In a scenario drawn from the simulations reviewed below, adozen � 1002200 km diameter asteroids might result from thehit-and-run breakup of a single Vesta-sized asteroid that collidedat a typical impact angle and at velocity comparable to vesc, into along-gone (eventually scattered or accreted) Moon-sized world.Returning to the question of the apparent excess of 100 km bodiesin the Main Belt, it may be important to consider the hit-and-runbreakups of next-largest bodies that are torn into chains byimpacts into the largest. If the collisional physics works out thenthe mass budget is amenable, since the broad excess of 100 kmdiameter asteroids appears to be offset by a deficit of 300–400 kmdiameter asteroids.

2.2. Growing planets

A different kind of modeling approach for studying planetformation is to build up rather than break down a primordialpopulation, using accretion codes that apply advanced N-bodycomputational methods to directly integrate the orbits andinteractions of planets and asteroids around the Sun and inplanet-forming systems. Out of computational necessity thesecodes assume the simplest collisional physics—the perfectmerger—in order to facilitate precise dynamical tracking overmillions of years, with collisions = mergers leading to finishedplanets. Fragmentation can be modeled in such codes, but thedebris are impossible to integrate much further in time as discreteobjects. Planets of mass M, m and radius R, r are assumed to stickwhen they hit, forming a larger sphere of mass M+m thatconserves linear and angular momentum. Encounters are treatedas perfectly elastic gravitational encounters (no mass transfer andno dissipation of energy) if the impact parameter b4Rþr, and asperfectly inelastic mergers if brRþr, for which case accretionefficiency x¼ 1 as defined below.

Even with these simplifications, only a few hundred to athousand planetary bodies can be integrated, since the precisetreatment of close encounters slows down the calculation withthe square of the number of embryos being tracked. And so abroad size range is not allowed if one follows a physicallyreasonable size distribution. This means that every collisiontreated explicitly in these models is similar-sized. As a result,efficient merger is not generally a good assumption.

The assumption of perfect merger has been found to bestowupon planets a capacity for arbitrarily large spin angularmomentum. The impactor’s angular momentum relative to thetarget center of mass is m v

!� ~b, where ~b is the impact parameter

and m v!

the momentum, and ~v the relative velocity. Agnor et al.(1999) tracked angular momentum during terrestrial planet-forming N-body calculations and found that finished planets insuch simulations spin up to rotation periods as short as 1–2 h. Thiscan exceed dynamical stability (Chandrasekhar, 1969). Moreover,such rapid spins would have to be slowed down to present-dayrotations by some mechanism, such as loss of a large satellite, orspin–orbit coupling with another planet or its core; see Kaula(1990) in the context of Earth and Venus. The solution appears tobe that fast and/or grazing collisions do not contribute much toaccretion; this limits the accumulation of spin. How fast, and howgrazing, depends on the collisional physics and can be reliablydetermined using self-gravitating particle-based or other kinds ofhydrocode simulations.

Perfect sticking is a reliable assumption in the context of massevolution only when random velocities are slow in comparison tovesc. (The Moon formed at low v1=vesc , thus ‘‘sticking’’ may not be

Fig. 1. Differentiated planetesimals and embryos are difficult to disrupt by impact

at expected velocities. Here a Vesta-type (500 km differentiated) asteroid is struck

by a 200 km diameter rocky body at 10 km/s (top four frames) and at 5 km/s

(bottom four frames). Simulations by C. Agnor (Asphaug and Agnor, 2005). The

slower impact, at a velocity typical of contemporary Main Belt collisions, is

vimp � 15vesc and removes the top third of the mantle. The 10 km/s impact, at

� 30vesc , exposes some core iron to the surface but leaves the bottom third of the

mantle bound to the core. This suite of simulations was to study the problem of

liberating core material from a differentiated large asteroid; each case is for the

most probable impact angle y¼ 451. Snapshots for vimp ¼ 10 km=s are plotted

before and at t=120, 390, and 12,000 s after contact in the top four frames, and for

vimp ¼ 5 km=s at t=390 and 12,000 s in the bottom frames. The results deepen the

quandary of how Vesta, which retains a thick basaltic crust, did so while dozens of

other differentiated asteroids were disrupted to their core. If would seem to have

dodged quite a fusillade. However, the results lead us to contemplate impactors

larger than the target, turning the problem on its head and causing us to leave the

target-centric frame of reference behind.



quite the right word.) Dynamical friction of disk planetesimalsacting upon the embryos (O’Brien et al., 2006) are likely to reducetheir eccentricities and inclinations compared to those reportedby Agnor et al. (1999) and others. As mentioned, in the earlieststages of growth quiescent encounters were probably the norm,although turbulence and gravitational stirring by initial embryosmay excite planetesimals into relatively high random velocitiesbefore the dust has cleared. In order to have planetary encountersat all in the late stage, when orbits are increasingly wellseparated, random velocities must be rather high. If the epochsof planet formation can be discerned dynamically, then they canprobably be discerned cosmochemically owing to the dramati-cally different character of low, medium, and high velocity SSCs,as described in Section 2.7.

2.3. Scale invariance

Scale invariance applies to SSCs in the limit of incompressible,self-gravitating inviscid fluid planets. Two 100 m diameterincompressible spheres colliding at vimp ¼ 30 cm=s (a few � theirvesc) are indistinguishable from two 1000 km spheres of the samedensity colliding at vimp ¼ 30 km=s, when the velocities are scaledto vesc and distances are scaled to R. Scale-invariance thusprovides first-order physical insight rather than a fast rule. Itallows us to define this class of planetary collisions (SSCs) wherethe impactor and target are of similar size, and where the randomvelocity v1 is similar to the escape velocity vesc.

Planets are compressible, and rheologically and thermodyna-mically complex. For instance, it is quite plausible that silicatemelts are to be found deep inside most 100 km diameter or largerearly-stage terrestrial bodies. At high pressures these melts arevery soluble to water and other volatiles. In contrast, sub-kilometer bodies are unlikely to retain any appreciable melt ordissolved gas at any stage in their formation or evolution—theyare too small to retain heat and too underpressured to retain gas.They may retain ices. These smallest bodies (typical asteroids andcomets) are likely to behave as solids or granular solids duringcollisions, obeying a physics akin to landslides.

Large molten planets differentiate with iron in the core, crust-forming silicates and volatiles in the exterior, and mantleinbetween. Differentiation increases the gravitational bindingenergy of the planet making it almost 3 times as difficult tocatastrophically disrupt (in terms of impact energy) as anundifferentiated planet of the same composition, based on impactmodels similar to Fig. 1 (e.g. Benz and Asphaug, 1999). Butdifferentiation also perches the volatiles and silicates at thelowest specific binding energy, making it easier for major impactsand collisions to strip these materials from an otherwise growingplanet. Cores and deep mantles become sheltered as wasdemonstrated in Fig. 1, and so it is truly an enigma that ironshould be one of the most common representatives of ourmeteorite collections.

If we restrict ourselves to inviscid, molten, differentiatedcolliding pairs—astrophysical rather than geological ob-jects—then the most basic assumptions of scale invariance aremet. Even then there are important differences arising frompowerful shocks and high hydrostatic pressures. Large SSCs (giantimpacts) are hypervelocity because the impact velocity vimp

exceeds the sound speed of the colliding materials; they areshock-inducing collisions leading to global-scale internal heatingand Hugoniot acceleration. Assuming v1 � vesc the impactvelocity exceeds the sound speed for collisions involvingterrestrial planets that are Moon-sized and larger.

H2O and CO2 solubility in silicate melts is greatly enhanced athigh interior pressures (e.g. Dixon et al., 1988). When a large

planetary body ‘‘loses its lid’’ (Fig. 2) then its pressurized volatiles(several wt% H2O may be typical) can erupt violently, releasedover tgrav from hydrostatic pressure P. This pressure is releasedfrom a magnitude and over a timescale (kilobars, hours) that iscomparable to gas-driven kimberlite eruptions on Earth (Kelleyand Wartho, 2000; Porritt and Cas, 2009; Kamenetsky et al.,2007). As shown below in a study of purely tidal collisions, even atidal (non-impacting) impactor can have its interior pressureslowered by 50% for about an hour, with a rate of pressure release

_P � P0=tgrav � r2 ð8Þ

and � 20% permanent pressure reduction because of spin-up andmass loss. The pressure release adds VDP to the available specificenthalpy Dh¼DuþPDVþVDP, considered further below, wherevolumetric expansion PDV occurs especially in the case of bubbleformation (Gardner et al., 1999).

Most asteroids, by number and perhaps also by mass, areundifferentiated. The effects of SSCs among primitive andundifferentiated populations of smaller bodies may be moresubtle than for larger, differentiated bodies for three reasons. One,shock levels will be well below the sound speed if v1 � vesc , forobjects much smaller than the Moon. Two, the pressure unloadingeffects just described may be small. Three, surface-stripping andfragmentation may not result in a noticeable change in bulkchemistry or composition since there is no core–mantle segrega-tion, and hence no crust to remove from a mantle, or mantle fromcore. Nonetheless it is important to keep in mind that similarsized collisions do happen to primitive bodies, and that hit-and-run collisions happen when random velocity is comparableto escape velocity. These collisions would be gravity- and

Fig. 2. Impact geometry in similar-sized collisions, in side view and front view.

Only a portion of the impactor intersects any mass of the target, and commonly

the center of mass overshoots (defined as grazing). Mantles might intersect, for

instance, but not the cores. The smaller body (radius r) is by convention called the

impactor, and the larger body (R) the target, but with v1 � vesc and r� R this is

more mechanics than ballistics. Shown are two bodies a factor of 4 different in

mass (r¼ 0:6R, assuming rr ¼ rR) at the moment of collision, where y¼ 301 (left)

and y¼ 451 (right). At left the impact parameter boR; at right b4R so the center

of mass misses the target. From a mechanical point of view, the ‘‘lid’’ (� 30% of the

impactor mass at left; � 80% at right) gets sheared off as the colliding body is

stopped. The non-colliding lid is shaded grey in the plane of collision (top) and in

front view (bottom). The mechanics of hit-and-run is much more complex,

involving gravitational stresses and torques and shocks and shears. But simple

geometry explains why hit-and-run can be prevalent under typical planet-forming

conditions. Half of SSCs are more grazing than the case to the right, not counting

tidal (b4Rþr) collisions.



shear-dominated, not shock-dominated. These shears can leavetheir imprint; moreover, while pressure unloading from tens ofbars of pressure (the interior of a disrupted 100 km body) mightnot stimulate global melting or degassing as studied furtherbelow, the unbalanced pressure gradients and available enthalpycould trigger volatile migration and hydrous activity in otherwiseprimitive bodies.

2.4. Collisional geometry

In impact cratering, the projectile is rapidly buried into a semi-infinite target and effectively explodes, coupling as a point sourcewithout much downrange ballistic motion. The impact angle canbe parameterized with good results, as there is little difference tothe physics except for the shallowest collisions (Pierazzo andMelosh, 2000). Similar-sized collisions on the other hand aresensitive to impact angle over all ranges of y, simply because asimilar-sized impactor does not have semi-infinite mass to buryitself into. Fig. 2 shows how a large fraction of the mass misses thetarget for any but the most direct hits. Impacts that would not beconsidered oblique in the context of impact cratering(y� 302601) are grazing when it comes to SSCs, in the sensethat most of the mass overshoots.

The probability of impact at an angle between y and yþdy by apoint mass onto a spherical gravitating target is 1

2 sin2 ydy, afunction that peaks for impact angle y¼ 451 (Shoemaker, 1962); 451is also the median impact angle. The impact angle for undeformablecolliding spheres is the same as that of a point mass at the impactorcenter, impacting a virtual sphere of radius Rþr, thus Shoemaker’soriginal argument applies to SSCs. If one defines impact parameter b

(Fig. 2) as the offset from the impactor trajectory from the targetcenter of mass at the moment of collision, then b¼ ðrþRÞsiny andthe most likely impact parameter is b45 ¼ ðrþRÞ=

ffiffiffi2p

. (Note that b issomewhat larger than the periapse that would be computed for acollisionless encounter, at the moment of periapse when two virtualspheres are intersecting.)

For impact cratering (small r/R) the traditional definition ofgrazing is b=R, where the impactor skims tangential to the target.Abstract this notion so that the threshold for grazing is when thecenter of mass of the smaller colliding body is tangential to thelarger (Fig. 3):

yb ¼ sin�1 R

Rþr

� �ð9Þ

For equal sized planets r ¼ R and grazing requires an impact anglethat is only 301 from head-on. When r=R/2 grazing occurs fory4yb ¼ 421; that is, more than half of all collisions for bodieswithin an order of magnitude in mass. For small r, the fraction f

that is shaded grey in Fig. 2 approaches a step function,corresponding to the fact that impact cratering into a halfspaceis all-or-nothing, while for SSCs there is broad gradation over arange of impact angles. That is why their outcomes are so diverse.

If two differentiated planets collide, then their cores miss oneanother entirely for

ycore4sin�1ðrcore=rÞ ð10Þ

where the bodies have the same core fraction rcore/r=Rcore/R(Fig. 4). Among terrestrial planets rcore � r=2, so for any impactbetween 301 and 901 the cores miss one another. While this is asimplistic approach to collisions, it soundly predicts that the levelof core–core interaction during giant impacts is highly variable,with some events shredding and intermingling mantle materialsbut not cores, and others merging cores entirely. Idealizedassumptions about planetary collisions and their mixing areprobably untenable; see Nimmo and Agnor (2006) for modelingand discussion of this issue.

Fluid bodies deform during SSCs and are ellipsoidal by the timeof impact (e.g. Sridhar and Tremaine, 1992). Shoemaker’sargument for impact angle is no longer valid; impact angle isnot well defined and angular momentum goes into torques. Coresinteract with the mantles they plow through, which are abouttwice as massive, and can therefore merge even if not headedright at one another (see the Moon formation models discussedbelow). Impact trajectory is not a straight line. Mass does notcome off as a ‘‘lid’’. And lastly, concerning grazing, there is noabrupt change in the physics between an impact involving a smallfraction of the target (btrþR), and an impact that barely misses(b\rþR). The discussion of impact angle must be nuanced toaccount for tidal collisions, for which y is undefined, which maybe of great consequence to the impactor especially if the massratio g¼m=Mt0:1.

2.5. An edge effect

Hit-and-run can be thought of as an edge effect pertaining tothe planets that are similar in size to the largest, within the factorr=R� 1

3 for which grazing is common. An accreted planet that hasgrown into one of the largest, has nothing larger to run into.

0 1r/R

30

40

50

60

70

80

90

Gra

zing

Impa

ct A

ngle

0.2 0.4 0.6 0.8

Fig. 3. The threshold grazing incidence angle yb (Eq. (9)) as a function of relative

sizes of the colliding pair r/R. Grazing is defined as the center of mass of the

impactor missing the target (Fig. 1). For relative size r=R\0:4 most collisions are

grazing (ybr451). Purely tidal (non-impacting) collisions are an important class of

grazing collisions (see Fig. 13) but are not included here as y is undefined.

Fig. 4. A collision with impact angle greater than yc ¼ sin�1ðrcore=rÞmay have little

direct core–core interaction. The core in this case is half the radius, so that

ycore ¼ 301. Impacts steeper than ycore will trend towards great core interaction

(and mergers for low velocity), while shallower impacts (most of them) might

have little core interaction.



Unaccreted NLBs, though still among the ‘‘giants’’ in the context ofthe late stage, finish their evolution as part of the middle of thepack (Fig. 5). Being among the largest, they are petrologicallyinteresting, potentially thermally active, differentiated, andcomplex. If they are broken down into smaller ð � 100 kmÞobjects early on, their interiors are represented in the finalpopulation of asteroidal parent bodies and in the meteorites; ifthey are not, then their interiors remain forever sheltered, asVesta’s has been.

Collisions with largest bodies occur at v1 � vesc as stirred up bythose same bodies. Smaller bodies in the population are subject tothe same random stirring, and either accrete onto the largerbodies rather efficiently, or else catastrophically disrupt withother small bodies, considering their relatively low bindingenergy (vimpbvesc). Thus SSCs as we have defined them do notoccur throughout the size distribution, but between the largestand the next-largest for which vimp � vesc .

Hit-and-run is a third way for planets near the top of the sizedistribution to evolve and exchange material and momentumduring accretionary epochs, in addition to merger (which growslarger planets, and must at some point be dominant) anddisruption (which reverses the process of accretion and mustbecome minor for planets to form). While the associatedprocesses of impact shredding may seem exotic, hit-and-run isactually more common than merger or disruption for SSCs attypically stirred-up random velocities. It plays an important andperhaps dominant role in the physical and chemical evolution ofthe planetary bodies that grow large, but not largest, in terrestrialplanet-forming settings.

2.6. Modeling similar-sized collisions

Canup and Asphaug (2001) conducted a systematic study ofpotential Moon-forming collisions, based upon computer simula-tions using the smooth-particle hydrodynamics (SPH) methodpioneered for giant impact studies by Benz et al. (1988), Cameronand Benz (1991) and others. SPH is a Lagrangian method that usessmoothed mass elements (spherical kernel functions) to compute

the hydrodynamic and shock stresses and the pressure andgravity accelerations, and to track the trajectory and evolution ofmatter. We used a simple but appropriate nonlinear equation ofstate (Tillotson) for iron cores and rocky mantles, and set therandom velocity to zero (vimp=vesc) in order to maximize the diskmass while satisfying the constraints of final system angularmomentum. The impacts studied were otherwise characteristic ofterrestrial planet-forming collisions.

In the course of this search for the best case scenario for late-stage Moon formation around proto-Earth, we made a fewexploratory simulations with v140 and found that some ofthese impacting planets were ‘‘skipping’’ from the target Earth.Indeed, the best case scenario identified for Moon formationturned out to be an impactor which almost, but not quite, skips off(see Fig. 6). A fraction faster and there would have been no Moon,but two planets—one still rather Earth-like, and the other one lessmassive than before, missing much of its mantle—resemblingMercury, perhaps.

Fig. 7 shows frames from two simulations by Agnor andAsphaug (2004a, b) in studies of accretion efficiency and thethermodynamical aspects of SSCs (Asphaug et al., 2006). These aretypical of the two main kinds of hit-and-run collisions, one arebound and the other chain-forming. Planets before the collisionare hydrostatic, non-rotating, and start from a separation distance5Rroche. Impact velocities are vimp ¼ 1:5vesc in (a–c) and vimp ¼ 2vesc

in (d–f), corresponding to v1 ¼ 1:1vesc;1:7vesc , respectively. Theimpact angle is 301 in both cases. The first scenario results inwidespread mantle removal from the impactor; the scenario ispostulated below as a mechanism for Mercury’s mantle loss. Thesecond case results in a chain of bodies the size of the majorasteroids, all of them highly diverse in major element abundances(and most of them iron-rich), but each deriving from the sameparent body chemistry.

Research is flourishing in the area of planet-scale collisionalmodeling thanks in part to a renewed focus on the formation ofEarth-like planets around other stars (e.g. Marcus et al., 2009) andthe great strides that have been made in the development of self-gravitational hydrocodes capable of evolving millions of particles,and the computer systems to run them on. Much higherresolution giant impact simulations are now possible includingthe use of more detailed and accurate equations of state (see e.g.Benz et al., 2007). The greatest challenge may be the accuratemodeling of smaller-scale SSCs where gravity is not the only forceto be considered; complex effects such as porosity and strengthare notoriously challenging to model (Jutzi et al., 2008, 2009).Furthermore we have yet to adequately understand the long termdynamical fate of collisional ejecta—whether it reaccumulatesonto the target or onto the unaccreted impactor after many orbitsabout the Sun, or becomes background disk material, or is lost bysolar effects. Moreover, the effect of spin has not been exploredsystematically for SSCs, or in the context of accretion efficiency;Canup (2008) has made the first inroads in the context of Moonformation.

2.7. Accretion efficiency

For gravity-dominated collisions SPH is well suited, even atmoderate resolution, to computing the final total bound massesM1 and M2 where M1 is the largest collisional remnant and M2 thesecond largest. In the limit of a non-collision M1=M and M2=m.After a collision the binding energy of all particles is computedwith respect to the particle closest to the global potentialminimum, which serves as the seed for nucleating the totalbound mass of the largest aggregate M1. The binding energy ofremaining particles is computed with respect to this new position

Fig. 5. Similar sized collisions are an edge effect occurring between the largest and

the next-largest bodies in a hierarchically accreting size distribution (see text).

Here is a plot of cumulative number N versus size r, schematically a power law.

The largest and next-largest bodies have mutual collisions v1 � vesc . The ratio

v1=vesc increases with 1=r, and so is high (disruptive) towards the left of the plot.



and velocity, and the second largest aggregate M2 computed, andso on as allowed by resolution.

The accretion efficiency

x¼M1�M

mð11Þ

is determined to better than � 10% in these simulations, wherex¼ 1 for a perfect merger, M1=M+m. The determination of M1 isresolution converged using modest (� 3� 104) numbers ofparticles. That is, the answer does not change with furtherincreases in resolution. Nor has this basic result been found verysensitive to the equation of state (EOS), provided it is adequate to

Fig. 6. Hit and almost run. Moon formation in a late stage, low-velocity collision with proto-Earth, in a giant impact scenario modeled by Canup and Asphaug (2001). Color

indicates thermal energy. A proto-lunar disk of the appropriate mass, angular momentum and mantle-derived bulk composition forms after a Mars-mass impactor (0:1M� ,

coming from the right) collides at vimp ¼ vesc into a 0:9M� body. It first bounces off, as seen in the top four frames, but is now gravitationally bound, and comes back after a few

hours to be further shredded by tidal and impact shears. The impactor’s core merges with the proto-Earth’s. Molten and vaporized ejecta from the crusts and mantles of the

impactor and target shears out into a protolunar disk of about two lunar masses in this simulation. Shown are times t=0.3, 0.7, 1.4, 1.9, 3.0, 3.9, 5.0, 7.1, 11.6 h after initial contact.

Identical simulations at 30% higher impact velocity end with the impactor escaping as a novel planet, a Mercury-like body stripped of its crust and outer mantle.

Fig. 7. Hit-and-run is a common outcome when planetary bodies of similar size collide. Shown are frames from 3D SPH simulations using the Tillotson equation of state

(from Asphaug et al., 2006). In each case a Mars-mass target is struck by a planet 12 (top) and 1

10 (bottom) its mass (statistics for these and other simulations are plotted in

Fig. 8). The top is a rebounding collision at vimp ¼ 1:5vesc ; the bottom is a chain-forming collision at vimp ¼ 2vesc . Rocky mantle is labeled blue and iron red. Particles are

shown in side view before, during, and 3 h after the collision. Blue particles appearing to mix into the target core are a projection effect of particles with the same ðx; yÞ in

and out of the page; there is little if any disruption to the target core.



model shock acceleration. The Tillotson EOS and the moresophisticated ANEOS and SESAME EOS give very similar resultsfor this basic quantity of a similar-sized collision.

Modeling Moon formation using the same computationaltools is a much bolder endeavor than modeling x. Only a fewpercent of the collisional ejecta end up in a proto-lunar orbit, assatisfied only by a narrow range of ejection velocities betweenvorb and vesc, where vorb ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiGM=a

pand a is the radius of the

protolunar disk that forms outside the corotation radius. Thedynamics must then be evolved by the code for at least one orbitaltime in response to very near field gravitational torques. Thelonger timescale and higher required precision result in asensitivity to the equation of state (explored in Canup, 2004)and to radiative evolution (Stevenson, 1987) which is notincluded in these SPH models.

Accretion efficiency is plotted in Fig. 8 as a function of therandom velocity v1 (Eq. (2)) for Mars-mass planets (M¼ 0:1M�)struck by impactors m¼ 0:01M�;0:05M�;0:1M� (mass ratiog¼m=M¼ 0:1;0:5;1:0) at impact velocities vesc rvimpr3vesc .The data are derived from 144 simulations performed by Agnorand Asphaug (2004a, b) including the two shown above in Fig. 7.The results for x appear on the basis of other simulations to bescale invariant within � 10% for larger and smaller differentiatedterrestrial planetary masses (approximately Vesta sized tosuper-Earth sized) so the plot can be studied as a generalresult for terrestrial planet formation. The Moon-forminggiant impact scenario favored by Canup and Asphaug (2001)plots near the upper left red triangle. In all these simulationsthe spatial resolution is 30,000 particles, and the iron:silicatemass ratio is 30:70 as a core and mantle. Planets are hydrostaticand non-rotating prior to collision, and are placed initially at5Rroche to allow the pre-impact tidal strains and torques todevelop.

Ignoring purely gravitational (tidal) collisions (Fig. 13 below), forwhich y is undefined, the impact angles 301, 451, 601, 901each represent � 1

4 of the impact probability of collisions,dPðyÞ ¼ 1

2sin2 ydy, with 01 being head-on. The velocity range coversthe expected range of collisional velocities during late-stage planetformation. However, the simulations between 0 and 0.7vesc arerather coarsely spaced given the sensitivity of outcomes in thisrange, and need to be filled in with further simulations to betterunderstand the transition from accretion to hit-and-run behavior.Also, the range of mass ratio needs to be extended to g� 0:03 inorder to cover all similar-sized collisions; these results are for g¼ 1,0.5, 0.1. A finer resolution in impact angle would better reveal thenature of this transition as well. But the initial observation can bemade, that there are four main branches of similar-sized collisions:

1. Efficient accretion is the common occurrence for random velocitieslower than about 0.6vesc. Damped populations accrete efficiently.

2. Partial accretion is common throughout the random velocityrange � 0:722vesc . Mergers are inefficient at high velocity.Only direct hits (01, 301) are accretionary at all for randomvelocities greater than about 0.8vesc.

3. Hit and run is prevalent for the velocity range 0:7vesc

tv1t2:5vesc , i.e. 1:2vesc tvimpt2:7vesc . The clusteringaround x¼ 0 corresponds to impactors rebounding with littlenet mass contribution to the target, or little erosion.

4. Erosion and disruption occur for random velocities greater than� 2:5vesc . These tend to destroy the smaller body (most is notaccreted) and erode the larger. Catastrophic disruption,traditionally defined as M1=M/2, requires collisions at velo-cities far to the right of this plot (xcat ¼�

12M=m).

The distinction between partial accretion and hit-and-run owes toa remarkable sensitivity to impact angle; this shows in the figure

Fig. 8. Accretion efficiency x (Eq. (11)) for colliding planets as a function of random velocity, for colliding mass ratios g¼ 0:1, 0.5, 1.0 and for impact angles 01 (head-on),

301, 451 and 601. Data are from smooth-particle hydrocode impact simulations by Agnor and Asphaug (2004a, b) that assumed differentiated targets and impactors, 30 wt%

iron core and 70 wt% rocky mantle. Each angle represents � 14 the probability interval of collisions. Impact velocity vimp ranges from vesc to 3vesc; the normalized random

velocity v1=vesc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðvimp=vescÞ

2�1

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi1=2Y

pis the abscissa in this plot. The simulations reveal an abrupt transition from efficient accretion (x� 1) to hit-and-run (x� 0)

around Safronov number Y� 1 (labeled at top; Eq. (3)), corresponding to vimp � 1:2vesc and v1 � 0:7vesc . For faster impacts, half of the collisions studied (those Z451) are

effectively collisionless from the point of view of bulk mass accumulation.



as a segregation by plotted color, where red and green dominatethe hit-and-run line

xhr � 0 ð12Þ

For random velocities v1\0:7vesc half of the collisions (thoseZ451) are effectively collisionless from the point of view of bulkmass accumulation. Even for relatively normal impact angles(301) the influence of grazing is pronounced. This leads to themost striking aspect of the plot, which is the abrupt jump to thehit-and-run line with increasing impact velocity and impactangle. For a broad and significant range of impact angles andvelocities there is little net mass contribution or removal from thetarget.

2.8. Prevalence of hit-and-run

Hit-and-run is prevalent in systems that are moderatelygravitationally excited and less so in systems that are not. Asimple framework model is developed to trace the occurrences ofhit-and-run on the path to planet formation. The model is notdynamical: planetesimals are given an initial random mass(following a power law, Eq. (7)) and allowed to grow throughrandomly selected pairwise collisions. The total starting mass is aconstant. If a colliding pair is within a factor of 30 in relative mass(based on Fig. 3), and one of them is within a factor of 30 in massof the largest body in the simulation (so that vesc � v1), then it is asimilar-sized collision as defined above. Each SSC has a prob-ability of hit-and-run (x¼ 0) or merger (x¼ 1). If a merger, thenthe smaller body goes away and the largest becomes the sum ofthe masses, and carries a mass-averaged tally of the number ofhit-and-run occurrences which starts off at 0 for all bodies. Hitand run does not change either body in the model, but incrementsthe hit-and-run tally of the smaller by 1. Large colliding pairs withmass ratio go 1

30 are presumed to be efficiently accreted. Smallcolliding pairs, however, in which both masses are smaller than 1

30

the largest in the simulation, have vesc 5v1 and so the bodies areregarded as disrupted and removed from the population. Pairwisecollisions proceed, under the assumption of constant Safronovnumber, until the final number of bodies has been reducedto Nfinal.

Results are plotted in Fig. 9 for a moderately excitedpopulation (v1 � vesc) for which the probability of hit-and-run isabout 50%. Hit-and-run happens commonly to most of the next-largest bodies in a population in that case. If one mass-averagesthe history of hit-and-run in the assembly of the largest finalbodies, then it is important to the largest as well. This sampling-with-replacement result is not surprising to anyone familiar withstatistics, but it may lead to a revised thinking as to how weinterpret planets and their acquisition and retention of mantles.The scenario beginning with Ninit ¼ 10;000 ends up with apopulation of largest and next-largest objects (black diamonds)having had 4 or 5 hit and runs each; the largest are ratherhomogeneous in terms of mass and hit-and-run tally. The ‘‘latestage’’ scenario beginning with Ninit=100 allows two bodies toavoid hit and run by chance. The two largest are again similar insize and tally, and accreted from bodies that had 1 hit and run onaverage. The NLBs in this case are a very diverse group, somehaving no hit and runs and others having 2 or 3.

Much of the material stripped off by hit and run early on mayend up back in the disk, to be accreted later. It is a challengingstudy, how to model disk replenishment during N-body integra-tions. Nevertheless, a more physical study than the above ispossible, regarding the hit-and-run characteristics of accretion.N-body integrations could import the outcomes from SPH codes,allowing the dynamics (including spin) of growing/eroding/disrupting planets to change with every impact. With the advent

of commodity GPU computing it is not far-fetched to imaginerunning a fairly quick SPH simulation at every detected collision.Adding mass back to the disk, providing drag, could perhaps bedone by tracking millions of collisionless planetesimals. But theproblem is extremely complex. The size-frequency distribution ofthe disk planetesimals changes the random velocities of theembryos, through stirring damped by dynamical friction, whilethe random velocities of the embryos changes the size distribu-tion of planetesimals by changing the collisional physics.

2.9. Accreted and unaccreted

Two bodies that come into close proximity once in their orbitsabout the Sun, at the moderate random velocities considered here,are likely to do so again unless their orbits are externallyperturbed. Thus, bodies involved in a hit and run collision areexpected to come back again to try once more. Given a few tries,accretion is likely. Accretion must be overall efficient for it isknown from isotopic age dating (e.g. Yin et al., 2002) thataccretion won out after a few � 10 Ma, and terrestrial planetformation effectively ended in our solar system, in concordancewith N-body simulations that assume perfect merger. Planetarybodies on colliding orbits, for the most part, eventually accrete.

But the surviving next-largest bodies that did not accrete ontothe largest bodies are not ‘‘the most part’’. There is a severeselection among the surviving NLBs. As with an epic militarycampaign in which nearly all of the soldiers have fallen, thesurvivors have tales of miraculous good fortune to recount. Thoseplanets and planetesimals that participated in accretion, but by arelatively small chance of dynamics neither merged nor werescattered, thus remain behind as a rather exotic population ofNLBs. Planet growth proceeds apace for the largest, in the sensethat merger happens when it happens, and non-merger does notgrossly affect their mass (x� 0) except at the highest relativevelocities. Even if it takes several tries, one can to first orderignore a few unsuccessful attempts at accretion. There is someshock and mechanical processing and impact erosion of the target,but not much loss or exchange of matter compared to thedevastation that happens to the bounced-off impactors.

1000 10000 100000Final Planet Mass (normalized)

1

10

Hit

and

Run

Occ

urre

nces

(mas

s av

erag

ed)

10,000 -> 10100 -> 10

Hit and Run Occurrences for v∞∼vesc

Fig. 9. Accretion including the occurrence of hit-and-run is modeled schematically

as a collection of Ninit planets growing by random pairwise collisions (see text) into

the 10 final planets that are plotted. Hit and run, accretion and disruption occur in

approximate concordance with Fig. 8. Here the assumption is a moderately stirred

up population (v1 ¼ vesc) so the probability of hit and run is � 50%. The y-axis

shows the number of hit and run events each final body has experienced (mass

averaging the hit-and-run tally of accreting bodies) by the end of accretion. The

runs start with either Ninit ¼ 10;000 or 100 bodies, of the same total mass.



The evolutionary tendency is towards a compositional changethat discriminates, over time, the largest from the next-largestbodies. With each non-accretionary event the NLBs get stripped oftheir outer layers, becoming increasingly iron-rich and volatile-poor. If it takes an NLB several collisions to accrete, by the time itdoes merge it will tend to increase the iron fraction in the largest.Its stripped outer silicates and volatiles go back into the disk; anydiscrete parcels of this stripped material have similar dynamicalparameters and future encounters are likely with either of thecolliding pair, perhaps favoring the larger on account of its greatergravitational cross-section. If so then the largest bodies can rob,over time, the exterior materials from the NLBs, leading to acompositional dichotomy between the accreted and the unac-creted.

The late stage of planet formation ends with planets achievingstable orbits. Remnants of these final collisions are trapped andherded into relatively stable dynamical regions (Morbidelli andMoons, 1993), providing a snapshot in our solar system of howterrestrial planet formation all ended some 4.6 billion years ago,punctuated by dynamical hiccups (e.g. Chambers, 2007; Gomeset al., 2005). By the very nature of the late stage, most of thecolliding matter is in the largest sizes; nearly all of the debris thatis produced is the result of similar sized collisions, from mega-cratering events (Marinova et al., 2008) to the unaccreted strandsand clumps of hit-and-run debris. The Main Belt should thus bereplete with mantle- and crustal-derived rocks from collisionsbetween differentiated late-stage embryos (the spray of debriscolored blue in the simulations of Fig. 7). However, stonyachondrite meteorites are uncommon compared to the greatdiversity of irons, which are presumably the core relics, andrelatively few are unequivocally mantle-derived. As for spectro-scopically characterized asteroids, the vast majority are undiffer-entiated; if there is much mantle rock in the Main Belt it is hiding.Burbine et al. (1996) considered this puzzle and hypothesized thatbecause mantle rock is friable, once liberated from a parent bodyit rapidly degrades into sub-millimeter sizes and is swept away byPoynting–Robertson drag or the solar wind, or solar radiationpressure. Iron, being more resilient, survives to produce meteor-ites. Asphaug et al. (2006) argue that the debris may begin asmantle rock but undergo petrogenic transformation duringrelease from hydrostatic pressure, or if solid, be fragmentedin situ by the unloading stresses.

3. Departures from scale invariance

Most impact simulations are based on fully compressible fluiddynamics computations including the calculation of shocks andthe reasonably accurate calculation of the equation of staterelations Pðr;uÞ where P is pressure and r;u are density andinternal energy of the represented material. Simulations using thesame SPH code and Tillotson EOS indicate approximate scaleinvariance for equivalent impacts involving planets ranging fromVesta-sized to Earth-sized, when the velocities are scaled to vesc.Using a different SPH code and EOS model (and minor differencesin setup) Marcus et al. (2009) reproduce part of Fig. 8 for super-Earth-sized planets, and obtain very similar results at a muchlarger scale.

But similarity in computational results is not proof of scaleinvariance in nature. We consider these invariances by approx-imating some of the geophysical complexities, consideringstrength at small scales, enthalpy at large scales, and viscosityinbetween. In addition one must acknowledge that impactphysics itself changes fundamentally as one transitions fromhypersonic collisions (much faster than the sound speed in therock, the largest events) to subsonic small-scale events, the

transition velocity being a few km/s for rocky bodies but muchslower for uncompacted bodies. Along these lines one can identifyfour scenarios to explore for departure from scale invariance asone transitions from large, differentiated, fluid planets (the easiestto model) to SSCs involving smaller colliding pairs.

Rheology: Brittle mechanical strength is size and rate variant(Grady and Kipp, 1985; Melosh et al., 1992), and is not wellunderstood for the tens of m/s velocities that may be commonamong colliding planetesimals. For deformation on the timescaletgrav of a similar sized collision, large bodies are likely to bedominated by viscous rather than brittle deformation, firstbecause they are massive—their interior pressures far exceedthe stresses associated with the failure of elastic solids—andbecause they retain heat and are more likely to remain ductile.

Differentiation: Closely related to the thermal state, or pastthermal state, is the transition with increasing size towardsdifferentiated internal structures. Colliding bodies whose iron hassegregated to the center, and whose atmosphere and volatilecondensates (oceans) have migrated to the exterior, behavedifferently than undifferentiated colliding spheres. Impactspreferentially remove the lower-density outer layers whichreceive the brunt of the impact energy. Impedance mismatch atthe core–mantle boundary (Asphaug, 1997) may also enhance thevelocity of ejected materials from the outer layers. These effectschange the physics with scale, and work to segregate planetarymaterials.

Shocks: Because impact velocity scales approximately with thesize of the colliding bodies (vesc in m/s is equal to R in km, for asphere of uniform density 1.9 g cm�3) there is a tendency for SSCsinvolving \1000 km bodies to produce shocks, at impactvelocities exceeding the sound speed of their material. For giantimpacts at tens of km/s shocks can induce global melting, whilefor primary accretion the collisions may be only a few m/s.Subsonic impacts may only result in damage (if solid) or incompaction or shear bulking (if granular; see e.g. Schafer et al.,2007), or splashing if liquid. Shocks are less important for verygrazing SSCs in which the bulk of the matter does not intersect.

Unloading: The lithostatic overburden pressure of a non-rotating incompressible planet of radius r is PðaÞ ¼ 2

3pGr2ðr2�a2Þ

where a is the distance from the center. The characteristicpressure P0 ¼ Gr2r2 is released to a much lower value duringthe collision timescale tcoll. The effect of pressure release can besubtle for small bodies—it may have been expressed in thevigorous dust production of comet Shoemaker-Levy 9 after itstidal disruption by Jupiter (Hahn and Rettig, 2000)—and it can behugely important for large bodies. If one thinks of planetarydisruption as an enthalpy-conserving event, and ignores shocks,then the change in specific enthalpy goes as r2. Moon-sized andlarger bodies have potential for widespread eruptive degassingand hydrothermal action at global scales, even in response topurely gravitational (tidal) collisions. Based on Earth analogues,pressure release melting and plinian-type magmatic responsesmight occur at global scales.

3.1. Rheology

Rheology pertains to how geologic materials deform and flow.The simplest planetary rheology invokes a strength (that is, acohesion or yield stress, which could be zero) beyond whichmaterial deforms, for instance as a viscous fluid. Strength and floware quite complicated, but a simple approach is suitable here.

3.1.1. Strength

Strength is scale variant primarily because large objects havebigger flaws. If one thinks of an object’s volume as sampling a



probability distribution of possible flaws (Weibull, 1939) thenstatic tensile strength S expressed as the weakest flaw in avolume, decreases approximately with size to a power S� r�3=m

where m� 629 for typical rocks (Grady and Kipp, 1985). Thisleads to large asteroids requiring much less specific energy tocatastrophically fragment than small ones, whether by tidal or bycollisional stress. In addition strength is rate variant for the sameunderlying reasons (see Melosh et al., 1992), materials beingstronger in proportion to the strain rate. It is worth noting as anaside that ejection velocity vej from a rocky target scalesapproximately with the square root of tensile strength, Y � v2

ej,because strength is a specific energy. Thus, the strength of anasteroid’s rock type probably biases which meteorites we look at,by sending the strong ones on the quickest journeys to Earth,where in turn they are most likely to survive the voyage throughspace, and atmospheric entry and terrestrial residence prior todiscovery and curation.

Catastrophic disruption requires acceleration of the fragmen-ted materials to escape velocity. As Fig. 1 showed, large bodies aretightly bound and difficult to disrupt even if they have nostrength. The corollary is that rubble piles are ubiquitous forasteroids larger than some size, believed to be about 300 mdiameter (Benz and Asphaug, 1999; Holsapple et al., 2002) sinceglobal fragmentation energy is lower than binding energy.Beyond some further size, perhaps a few 100 km diameter, self-gravity and deformation begins to compactify the rubbleinto a coherent body (perhaps with the assistance of impactcratering) as the central pressure exceeds the compactionstrength of rubble, and as thermal deformation and meltingbecome important.

Cohesion, as the threshold for shear deformation, has beenstudied in the context of planetary tidal disruption. Jeffreys(1947) analyzed the gravitational disruption of planetesimalspassing inside the Roche limit of Earth, treating them not as fluidbodies but as elastic solids subject to a deformational stress.Because the tidal stress increases with r2, bodies smaller than acertain size do not fragment. Jeffreys found that a monolithicrocky asteroid smaller than a few 100 km diameter survives agrazing encounter with Earth. This assumes that tensile strengthis not size-varying; a Weibull distribution of flaws pushes thetransition to smaller sizes, between fragmenting and non-fragmenting interlopers, and more abruptly.

As with most pioneering research, Jeffreys’ specific result waseventually rendered somewhat moot, in this case by therecognition over the past decade that asteroids of that size arelikely to be rubble piles. Resistance to deformation for rubble pileobjects is not measured by tensile strength, but is a complicatedgranular rheology that depends on the overburden pressure andthe total stress condition. Small monolithic bodies might notcome apart by tides, but rubble piles might (Richardson et al.,1998). Consider the imprint of tidally disrupted comets uponJupiter’s satellites—or rather, the unexpected absence of suchimprints from disrupted parent bodies smaller than about 800 m(Schenk et al., 1996; see Fig. 10). There are more than a dozenlarger records of tidally disrupted comets, and none smaller. Itmay be that Jupiter-family comets are structurally competent atsome small value but that once their cohesion is exceeded theydisaggregate and behave rather like a fluid (Asphaug and Benz,1996). The abrupt transition from solid to fluidized behavior is acommon aspect of granular materials, and may be pronouncedunder microgravity conditions. The structural competency ofJupiter-family comets (JFCs) overall appears to be comparable tothe jovian tidal stress, which across such a small body is not muchgreater than that of a dry snowball. In modern times cometShoemaker-Levy 9 disrupted at Jupiter, an event matched by a1.6 km diameter progenitor of bulk density 0.6 g cm�3 (Asphaug

and Benz, 1994) and strength � 100 dyn cm�2 disrupting as afluid body during a few hours inside the Roche limit.

If small planetesimals behave somewhat as self-gravitatinggranular fluids during randomly stirred planetary encounters,then perhaps SSCs are dynamically similar from scales rangingfrom giant impacts to planetesimals the size of small hills.Detailed numerical models (Schafer et al., 2007) and laboratoryexperiments (Wurm and Blum, 2006; Dominik et al., 2007) ofcolliding aggregate bodies give various results, not alwaysconsistent, showing how crushable or fractal solids can behavein unexpected ways. Polyhedral rubble pile models by Korycanskyand Asphaug (2006) illustrate the transition from individual toensemble behavior and the phenomenon of granular collapse in aself-gravitating system. There is much to be learned about this‘‘mesoscale’’ of accretion, where there is no well-understoodcharacteristic stress or characteristic size.

3.1.2. Viscosity

If colliding planets are large enough, or molten, then strengthand cohesion are effectively zero and the bodies are compact. Theabove complexities go away and what remain are the equation ofstate (EOS) and the deformation rheology. Concerning the latter,the easiest approach is to consider a linear Newtonian viscosity; ifthe planet resists deformational shears on the collision timescalethen it will not disrupt.

Newtonian viscosity is the ratio of stress to the strain rate

Z¼ s=_e ð13Þ

A similar sized collision takes place over the gravitationaltimescale tgrav ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3p=Gr

p(Eq. (6)), a couple of hours for density

corresponding to terrestrial planet-forming materials. The strainrate of deformation is then

_e ¼ e=tgrav ð14Þ

for some shear strain e. If the stress is primarily gravitational(tidal or shear disruption against self-gravity) then the character-istic stress is

s� P0 � Gr2r2 ð15Þ

where r is the radius of the disrupted body.

Fig. 10. Tidal disruption remnant imprinted as a chain of craters on Jupiter’s

satellite Ganymede, imaged in 1997 by the Galileo orbiter (see Schenk et al., 1996).

North is up; sun is from left. This is Enki catena, about 160 km long, produced

when a comet broke apart in a near-parabolic tidal collision with Jupiter and hit

Ganymede on the way out. Its dozen equant fragments are indicative of

gravitational breakup of a fluidized body (Asphaug and Benz, 1996). Galileo SSI/

NASA.



The smallest impactor rmin that can come apart in an SSC in thisviscous limit is found to be

r2min ¼ Ze=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3pGr3

qð16Þ

by requiring the deformation rate _e ¼ e=tgrav to be accommodatedby the viscosity Z at a stress � P0. This expression (Asphaug et al.,2006) is plotted for various values of global strain e in Fig. 11, forterrestrial bodies of bulk density r� 4 g cm�3. Strains of 1, 10, and100 are plotted to bracket catastrophic disruption. Strain e\10 isallowed when viscosity is less than

ZoZmax � 1013ðr=1000 kmÞ2 ð17Þ

where units are poise (g cm�1s�1). For comparison, mantleviscosity is thought to be � 109 poise in convective models ofearly Earth (Walzer et al., 2004) and Z� 109

21013 poise in Ioasthenosphere models (Tackley, 2001). Partially molten Moon-sized planets (ZC1014 poise) can be approximated as fluidbodies, as can impactors as small as 10 km that are molten(Z5108 poise). Because Zmax increases pr2, while Z decreasessensitively with size due to heat retention, the transition fromviscous-limited to inviscid behavior is likely to be abrupt acrosssome size threshold rmin.

A number of effects make the response to similar-sizedcollisions fluid for bodies smaller than are plotted in Fig. 11.Rocks, partial melts and magmas are nonlinear fluids, whoseeffective linear viscosity decreases with a power of the stress. Thismeans that viscosity Z� s=_e is much lower for larger-scalecollisions because overburden stress P0 is larger by r2. Also,pressure release melting acts to lower the viscosity during tidalpressure unloading, of great significance to partially or nearlymolten planets. The exolution of volatiles during pressureunloading can initially act to increase the viscosity of a magmaby stiffening it with bubbles, but at high enough deformation anddegassing rates the low viscosity of the gas wins out, and bulkviscosity plummets (see e.g. Gardner et al., 1999; Alidibirov andDingwell, 1996). Interesting textures are expected. And lastly, any

shocks or damaging stress waves accompanying the collisionfurther act to fluidize a small body (Asphaug and Melosh, 1993).All in all, it appears that similar sized collisions larger than a few100 km in scale, and perhaps as small as � 10 km if activelyheated internally, can be modeled using an inviscid approach.

3.2. Differentiation

Viscosity’s exponential dependence on 1/T leads to a con-sideration for the heat sources available in planets before, during,and after a similar-sized collision, and to the behavior ofdifferentiated (previously or presently melted) bodies versusundifferentiated bodies.

During the primary phases of planetesimal formation, thermalenergy is available from many sources, including impact shocks(from turbulence or infall), nebular shocks, solar heating, and thedecay of short-lived radionuclides (see for instance Wasserburgand Papanastassiou, 1982). Burts of intense heating from an earlysun would lead to some silicate melting in the innermost disk, butthis heat source is not believed to be volumetrically important forterrestrial planet formation. Nebular heating by shocks can beintense, and shocks are proposed by Desch and Connolly (2002) tobe responsible for the melting of silicate chondrules. The range ofchondrule ages then requires that the solar nebula was present forseveral million years and that substantial gas and dust werepresent. Nebular shock heating would be prevalent and thendiminish rapidly in pace with the accretion of planetesimals, dueto the clearing out of the gas and dust which carries the shocks. Asfor impact shock heating, this is certainly a significant heat sourceduring collisions in the late stage of giant impacts, but not duringprimary accretion when velocities are slow, following the initialstage of infall onto the disk. Compaction heating of porouscohesive aggregates during infall, or during the earliest growthwithin the disk, may be a relevant precursor to primary accretion.But hypervelocity collisions cannot contribute to the bulk meltingof low-gravity bodies, simply because any melt products areshock-accelerated and escape.

It is now generally acknowledged that the most important heatsource for thermal processing during primary accretion was, inour solar system, the decay of 26Al - 26Mg, a radionuclide withhalf-life t1=2 ¼ 7:2� 105 yr (Bizzarro et al., 2005). While the originof 26Al is debated, its original abundance in our solar system ismeasurable in meteorites. For chondrites the initial 26Al/27AlC5� 10�6, whose decay over t1=2 releases several times moreheat (in erg g�1) than required to bring cold, dry dust to themelting point. Other short-lived radionuclides, notably 60Fe, weretrapped in early-forming rocks in our solar system, also long-spent but evidenced by their daughter products; their heatproduction is not believed to be as significant as 26Al in ourown solar system.

The prevalence of 26Al in other solar systems is unknown; thisis a critical piece of missing knowledge since its presence orabsence has the potential to dramatically alter the mechanism ofprimary accretion. A solar system with a small fraction of ourmeasured 26Al might not produce enough heat for its early smallbodies to melt. If small planetesimals remain unmelted, even asthey grow larger than � 100 km, then the character of theirimpact coagulation might change, conceivably even shifting thegrowth of planets away from the terrestrial planet-formingregion, or biasing the favored sizes of finished planets.

The rate of heat production dq=dt from radionuclide decay isproportional to the planet’s mass � 4

3pr3r. Thermal energydissipates by conducting through the solid and radiating fromthe planet’s surface area � 4pr2 at a temperature T, per unit areawith a blackbody flux sT4. The heat produced, divided by heat

Fig. 11. Bodies of viscosity Z that are larger than rmin can deform to a strain eduring a tidal collision. Smaller or more viscous bodies cannot accumulate the

level of tidal viscous strain e on the timescale tgrav (Eq. (16)). The onset of

significant prolate deformation is marked by e¼ 1, while e¼ 100 is catastrophi-

cally disruptive. The bulk density r¼ 4 g cm�3 is assumed in the calculation.

Viscosity is non-Newtonian and is reduced by stress-dependent effects, pressure

release, and by impact shocks in an actual collision; what is plotted is therefore an

upper limit to rmin.



radiated, thus increases with r, and increases greatly withtemperature. Large planets thus attain radiative equilibrium athigher temperature and get hot enough to melt. According to 1Dthermal modeling by Merk et al. (2002), a 10 km diameterhomogeneous chondritic body melts if it accretes much fasterthan t1=2 (see Ghosh et al., 2006).

Timing is everything, and so is location: planetesimals formingcloser to the Sun acquire a greater fraction of Al-bearing silicatesthan those forming where ices dominate; they also accrete muchmore rapidly. They are thus much more prone to melting. Thetransition from fluid to solid behavior, plotted in Fig. 11 in termsof planetary radius, might be thought of as a transition in timefrom fluid to solid behavior, for the half-life of 26Al is—interest-ingly enough—comparable to the timescale of primary accretion.

3.3. Energy of collision

Catastrophic disruption is defined as leaving a target body withno more than half of its original mass intact or gravitationallybound. In the target-centric view of things, the characteristicthreshold of catastrophic disruption is traditionally expressed interms of the specific impact kinetic energy 1

2 mv2imp per unit target

mass M:

Q ¼1

2mv2

imp=M ð18Þ

The disruption threshold Q* is the value of Q at which the finallargest remnant M1=M/2. For SSCs where m�M the specificenergy of impact must be defined as Q ¼ 1

2 mv2imp=ðMþmÞ. One

might then by analogy want to define Q* as forming a largestremnant with half the combined mass, M1 ¼ ðMþmÞ=2. This isinadequate for SSCs, since two just-grazing, equal-mass planetswould each have half the total mass for any impact velocity. Wemust think of disruption in terms of the fates of both bodies.

The impact kinetic energy per unit mass Q � v2imp � v2

esc

� GM=R� R2 is partitioned between the impactor and the target.The smaller of the colliding pair always suffers the greatest harm,and this is what makes hit-and-run collisions so transformativefor the next-largest bodies in an accreting terrestrial planetarysystem. The tidal stress on the smaller by the larger, comparedwith the tidal stress on the larger by the smaller, is of greatermagnitude in the smaller body in inverse proportion to its mass.In the case of direct collisions, the contact stress wave or shockwave is generated symmetrically about the contact front of acolliding pair, so that energy is partitioned equally into bothbodies; energy density is also inversely proportion to mass. As forthe impact differential stress, this can be thought of as thedifferential deceleration across the diameter of each collidingbody, for instance in an off-axis SSC where � half of the collidingmass is abruptly decelerated and half is not. If the contact forcesare symmetric, then the smaller body decelerates more abruptlythan the larger in inverse proportion to its mass.

And so, as a rule of thumb, when planets of sizes rtR collide,then the specific tidal, gravitational, shear and shock stresses feltby each body scale inversely to the mass ratio g¼m=M. In thecase of the Moon-forming simulation of Fig. 6, the impactor, beingan order of magnitude less massive than the target, suffered anorder of magnitude more damage, expressed as the gravitational,mechanical and shock energy of collision per unit mass. Whethershocks, tides or shears dominate an impactor’s disruption is afunction of geometry, from just-grazing (where tides dominate) tohead-on (where impact stresses and shocks dominate), and ofscale. In an isolated planetesimal swarm with no bodies largerthan � 1000 km, random speeds are generally subsonic. But onceMoon-sized planets exist, stirring the swarm to km/s velocities,the shock effects of impacts can become significant.

3.4. Enthalpy of unloading

The gravitational binding energy of a uniform planet of mass M

and radius R

UB ¼3

5GM2=R ð19Þ

is the energy required to disassemble a planet gently to infinity,and thus represents the lower limit of the kinetic energy� 1

2 v2escdm of all masses dm that contributed to the formation of

the planet from v1 ¼ 0, with vesc increasing as the planet grows.The total kinetic energy of impacts is several times the binding

energy for typically stirred up populations, going as 1þ1=2Y. Thisenergy dissipates as heat, through shock and friction. For a planetthe size of Mars, the gravitational binding energy is� 8� 1010 erg g�1. Assuming Safronov number Y� 1 and divid-ing by the heat capacity of rock (cp � 8� 106 erg g�1 K�1) gives anestimate for the temperature increase due to impacts, of order20,000 K (in which case constant heat capacity is not the rightassumption). For a Vesta-sized planet, the same calculation givesan impact heating of only � 100 K. Accretional heading of smallerasteroids is insignificant.

A Mars-sized planet never gets this hot; it radiates to space.But some of the accretional energy is stored as internal energy u,and some is stored as enthalpy of compression, solution, andphase change. The specific enthalpy of a planetary interior is

h¼ uþP=r ð20Þ

where P=r is the pressure times the specific volume V ¼ 1=r.Changes in enthalpy drive reactions:

dh¼ duþP dVþV dP ð21Þ

It is useful to think of SSCs in terms of enthalpy for the samereasons that enthalpy is the guiding state variable for themodeling of rising magmatic conduits (e.g. Wilson and Keil,1991; Gardner et al., 1999). During large-scale similar sizedcollisions, the drop in pressure V dP as the planet comes apartleads to an abrupt change dh over the timescale tgrav, and this candrive various reactions forward.

Hydrostatic pressure P increases with more than the square ofa planet’s radius, since rocks are compressible. Holding thetimescale of collisions tcoll � tgrav as invariant in SSCs, both thepressure release DP and the rate of pressure release dP/dt scalegreater than R2. The high magnitude and rate of unloading duringdisruption can drive cavitation, the dissolution of volatiles, bubblenucleation and coalescence, magmatic forcing, and pressurerelease melting.

This causes us to look at the volcanic eruption modeling ofGardner et al. (1999) and others. Pressure unloading and itstimescale for Vesta- to Mars-sized SSCs is comparable tostrombolian-type eruptions, which are driven by gas bubblesrising faster than the surrounding melt. Measurements atStromboli volcano in Italy (Burton et al., 2009) show gas slugsoriginating at � 3 km depth. These same pressures are attained inthe middle of a 500 km diameter planetary embryo. One mightreason that strombolian eruption physics might be directlyrelevant to the degassing of large molten planetesimals in theaftermath of hit-and-run collisions, and their magmatic fragmen-tation (Alidibirov and Dingwell, 1996).

As pressure is released, uncompensated pressure gradientsresult in accelerations that can change the dynamics of disruption,possibly contributing to the shedding of material or theemplacement of a debris disk. Pressure gradients played a rolein the formation of the protolunar disk (Stevenson, 1987), and thedisk formed not only from material released from impact shockpressure, but also from material that released from a high



pre-impact hydrostatic state P0. Most of the disk-forming materialin the Moon-forming simulations (e.g. Fig. 6) emerges in thecourse of a few hours from several tens of kilobars of pressure, soeven apart from the shock, the thermodynamic path is of criticalimportance.

3.4.1. Tidal collisions

The best way to appreciate the physics of unloading is toisolate it in a purely tidal collision b\rþR, so that the smallerplanet does not hit the larger, but still comes deep inside theRoche limit

Rroche ¼ 2:423RðrR=rrÞ1=3

ð22Þ

Rroche is the threshold distance from a planet of radius R, densityrR where an incompressible small fluid spheroid r, rr on circularorbit will be disrupted (Roche, 1850). The Roche limit is notstrictly applicable to a parabolic or hyperbolic encounter, forwhich tidal disruption requires a somewhat closer periapse(Sridhar and Tremaine, 1992). For small strengthless incompres-sible bodies encountering massive planets at v1 ¼ 0, Asphaug andBenz (1996) found that periapse inside � 0:69Rroche is thethreshold for the shedding of matter, in agreement with theanalytical result of Sridhar and Tremaine (1992); they found thatpassage inside of � 0:55Rroche along a parabolic (v1 ¼ 0) trajectoryresults in the stripping of half the mass from the outer layers.

The analysis breaks down for similar-sized tidal collisionsbecause r� R. The impactor is extensive, and part of it is oncollision course when b� rþR. But ignoring this, if rR ¼ rr then0:55RrocheC1:3R, so that to first order we expect catastrophic tidaldisruption when a planet rt0:3R is on grazing parabolicincidence. Fig. 12 summarizes purely tidal encounters from thecontext of small spherical bodies of density 0.6 g cm�3 (‘‘comets’’)and 3 g cm�3 (‘‘asteroids’’) encountering planets of variousdensity. It is seen that in the limit of r5R, a catastrophically

disruptive tidal collision is about half as likely between anasteroid and Earth, as a physical collision. For comets tidaldisruption is 50% more likely than a collision. For equal densitybodies on parabolic encounters with r5R, tidal catastrophicdisruption (M2=m/2) is � 1

3 as likely as collision.For studying compressible, differentiated planets undergoing

tidal collisions, a self-gravitating hydrocode such as SPH remainsthe appropriate tool. Fig. 13 shows the result of a simulation(Asphaug et al., 2006) in which a Moon-size (0:01M�)differentiated terrestrial impactor, with composition 70 wt%rocky mantle and 30 wt% iron core as in the previoussimulations, and the same Tillotson equation of state,encounters a Mars-size (0:1M�) impactor. The impactor isinitially a non-rotating, isostatically equilibrated sphere. There isno physical contact, only gravitation and pressure unloading, sothe target planet has been represented in the simulation as a pointmass. The closest-approach velocity is 1:05vesc and the closestapproach distance is b=1.05(R+r), and the bodies are of equaldensity. The two planets are represented in the center of massframe, so that the larger (not shown) is displaced towards the topof the figure in each time step, while the smaller swings from theright towards the bottom and is severely deformed and stripped.

The result of this slightly hyperbolic gravitational encounter ismass loss, spin-up, and global pressure unloading. The pressuredrop is plotted in Fig. 14, which shows pressure inside theimpactor normalized to the initial central pressure Pc ¼

23pGr2

r r2,measured at the center of the planet (black) and at the core–mantle boundary (grey). Pressure unloading begins about half anhour before periapse, as the larger planet’s tidal field competeswith the smaller’s self-gravity, and reaches a maximum ofDP=P0 � 40250% about half an hour after the event. Theunloading at the center takes slightly longer to complete, as thesignal from surface mass removal propagates inward, and goes onfor an hour after periapse. Permanent unloading by � 20% results

Fig. 12. Tidal disruption of small incompressible spheres of density rc ¼ 0:6 g cm�3 (‘‘comets’’, left) and ra ¼ 3 g cm�3 (‘‘asteroids’’, right) during parabolic (v1 ¼ 0)

encounters with planets in our solar system: r¼ 5:52 g cm�3 (Earth), 1.66 (Neptune), 1.31 (Jupiter), 1.19 (Uranus), and 0.69 (Saturn), fitted to suites of simulations

(Asphaug and Benz, 1996). The probability of a tidal event with a given level of disruption (M2=m between 0 and 1, in the previous discussion), is plotted normalized to the

probability of a physical collision. For example, the disruption of comet Shoemaker-Levy 9 was supercatastrophic (M2=m� 0:2) with periapse 1.31R. A tidal encounter this

close or closer, but not impacting, is 0.2 times as likely as impact with Jupiter. Asteroids are catastrophically disrupted (M2=m¼ 0:5) by tides near terrestrial planets � 12 as

often as they collide; comets are catastrophically disrupted by tides near terrestrial planets � 32 as often as they collide.



from spin-up (the final body is rotating with a period � 6 h) andmass loss. Faster hyperbolic tidal collisions involve unloading ofshorter duration.

In a hit-and-run collision with borþR, pressure unloa-ding is greater, as the periapse is closer, but shock andcollisional shearing play an increasing significant role asthe fraction of intersected mass f increases. For disruptivehit-and-run collisions such as Fig. 7 d–f, the pressure unloadingin the disrupted fragments is nearly 100%, because the material isnow found in bodies with much smaller radii than before. In thecase of Fig. 7d–f, the pressure drop experienced by material in the

largest remnants is � 90%, although the prelude to this pressuredrop is a devastating shock-inducing collision. The encounterresembles two core bodies interacting gravitationally in a tidalcollision, with mantle coming along for the ride; howeverthe physics once fully explored is unlikely to be that simple.Fluid instabilities along the accelerating density boundarybetween the core and mantle may turn out to be as importantas self-gravitational instability, in disrupting impactor cores;clues to the process are sought in the petrology of iron–silicatemeteorites.

The fate of undifferentiated bodies in response to disruptivetidal collisions is also complex. Shear localization is expected togenerate frictional heating in a gravitational collision such asFig. 13. The larger the planet the more energetic this frictionalheating, producing melting (pseudotachylites) along shear planes.Planetesimals that are partially molten could melt entirely, orlocally, as they cross the phase boundary during unloading fromhydrostatic pressure; this could trigger core differentiation andinitiate degassing.

4. Discussion

Earth finished accretion by mopping up dozens of Ceres- toMars-sized bodies, becoming an amalgam of numerous smallerfeeding zones (e.g. Chambers and Wetherill, 1998). One of thesegiant impacts formed the Moon, which is the archetypalunaccreted body. That is to say, the Moon is the unaccreted(though gravitationally bound) remnant of a highly selectiveprocess we call, as a whole, pairwise accretion, but which in fact isan ensemble of processes that can be seen in Fig. 8 to fall into fourbroad categories: (1) efficient accretion, (2) partial accretion, (3)erosion and disruption, and (4) hit-and-run.

Fig. 13. Tidal collision with closest approach velocity vimp ¼ 1:05vesc and closest approach distance b¼ 1:05ðRþrÞ, just beyond grazing; see Fig. 14. The impactor is Moon-

size (0:01M�). The target is Mars-size (0:1M�), represented as an undeformable sphere which accretes any intersecting impactor particles. The simulation begins with a

hydrostatic, non-rotating impactor at 5Rroche from the target, coming from the upper right. The snapshots are one per hour for 5 h around the periapse. The left plot shows

the material (blue=rock, red=iron) in a slice through the symmetry plane. To the right are the corresponding pressures, in log code units, where orange � 1011 dyn cm�2

and blue t107 dyn cm�2. Tidal stresses and the shedding of matter and rotational stresses result in greatly reduced pressure, and greatly increased pressure gradients,

making enthalpy available for melting and degassing in the disrupted arms of material.

Fig. 14. Pressure unloading corresponding to the encounter in Fig. 13. Vertical axis

is the pressure, divided by the central pressure of the planet. Pressures are

averaged over particles near the planet’s center (black) and over the core–mantle

boundary region (grey). Time is measured in hours before and after periapse.

Global pressure unloading DP=P0 � 40250% begins about 12 h before periapse, due

to tides, and continues for more than an hour after. Global pressure rises back to a

base level that is � 20% lower in the aftermath, due to spin-up and mass loss.

From Asphaug et al. (2006).



Each has a unique outcome. Efficient accretion buries theevidence, resulting in isotopic and compositional homogeneity ifthe colliding planets are molten, or become molten as an outcomeof the collision. It can also evidently lead to the formation ofmassive debris disks and major satellites like the Moon. Efficientaccretion at the smallest scales might preserve the impactingbodies, for instance the layered-pile model for comets proposedby Belton et al. (2007). Partial accretion accretes higher densitymaterials and loses lower-density materials, so that a planet thatgrows in this manner grows denser (iron rich and silicate andvolatile poor) over time. Erosion is a continuum of the partialaccretion curve in Fig. 8, in that a very inefficient partial accretionevent is identical in process to an erosional collision, preferen-tially removing the outer materials which receive the brunt of theblow, and merging the inner materials; one is a net gain and theother a net loss. Disruption is in continuum with erosion.

Catastrophic disruption requires impact energies far to theright of the plotted simulations. Except, that is, when one focuseson impactors that are disrupted by the targets that they strike,vice-versa. These hit-and-run planetary collisions are relativelynewly explored planetary phenomena (Agnor and Asphaug,2004a; Asphaug et al., 2006; Yang et al., 2007; Asphaug, 2009)and the degree to which they are relevant depends on the level ofexcitation of the accretionary system. In the regime whererandom velocities are several times faster than escape velocity,hit-and-run can largely be ignored since disruption and erosionare more likely to dominate (the case in the present asteroid belt,for instance) and cannot lead to the growth of planets. In theregime where random velocities are close to zero, hit-and-run isan exotic occurrence and perfect merger is the norm. But forvelocity regimes inbetween, they are prevalent, so I conclude byventuring some specific contexts for hit-and-run.

4.1. Embedded embryos

Wetherill (1994), in a series of groundbreaking papersestablishing late stage planetary collisions, conducted MonteCarlo integrations of planet accumulation and found among otherthings that planets the size of the Moon and Mars may well haveroamed the Main Belt for a few tens of Ma, until being scatteredby mutual gravitational interactions and by resonant interactionswith the giant planets. This opened up a new way of thinkingabout terrestrial planet evolution, and the origin and evolution ofasteroids and the context of meteorites. Chambers (2007)postulated on the basis of N-body integrations that a planetmight have survived between Mars and the inner Main Belt forhundreds of millions of years, until it became destabilized bychaotic interactions, as a candidate dynamical mechanism for thelate heavy bombardment (LHB), the greatly enhanced impact fluxrecorded in lunar samples dated � 3:924:0 Ga before present(reviewed in Chapman et al., 2007) and which may have beensolar-system-wide. A planet near the Main Belt, once destabilized,would stir up the asteroids and greatly enhance the flux ofimpactors striking the terrestrial bodies. Petit et al. (2001) studiedthe primordial excitation and clearing of the Main Belt and foundthat Moon- to Mars-sized bodies would persist there for at leastthe first � 10 Ma or longer. O’Brien et al. (2007) conducted similarN-body integrations and found that embryos of roughly a lunarmass could remain among the asteroids for up to the time of theLHB, to be scattered away along with the majority of asteroids,thus contributing to the excitation and loss of asteroidal massuntil the larger embryos themselves were lost.

This is all to say that the asteroids we see today in the MainBelt, and the somewhat larger progenitors from which theyderived, are widely believed to have evolved early on, perhaps

even for the first � 600 Ma until the LHB, in the presence ofMoon-sized or larger embryos. The dynamical environment wasexcited by these embedded embryos, and thus was likely to haveinvolved moderate random velocities v1 � vesc . For large asteroidprogenitors encountering these embedded embryos, hit-and-runwould have been prevalent according to Fig. 8. If the scenario ofembedded embryos is correct, then guided by Fig. 9 we canconclude that the Main Belt should be replete with hit and runsurvivors.

In the outer solar system, where chaos during giant planetmigration may have scattered as many as 10 Earth-masses of icybodies beyond Neptune, of which perhaps 0:1M� remains today(Levison and Morbidelli, 2003), the scenario is characteristicallysimilar—almost total mass depletion that would include the lossof an ancestral population of somewhat larger bodies. The densityratio of rock:ice is similar to that of iron:rock, and thus, in lieu ofsuites of simulations for self-gravitating planetary bodies of ice–rock composition, the regimes of similar-sized collisions for icybodies might be comparable to those shown in Fig. 8. If so, thenamong the differentiated bodies of the outer solar system, anumber are expected to be stripped by hit-and-run collisions inthe manner just described for the Main Belt. The dwarf planetsbeyond Neptune appear to have highly varying density, rangingfrom ice-like to rock-like (most recently, Fraser and Brown, 2009),and the relative velocities are too slow to allow for cataclysmicmantle stripping (e.g. Fig. 1). This diversity of ice:rock ratio isconsistent with the idea that they, like their cousins in the MainBelt, underwent a late stage of growth involving hit-and-runcollisions.

4.2. Recycled planetoids

Hit-and-run events can lead to the formation of chains ofrecycled planetoids, the largest of them a few times smaller thanr, as depicted in Fig. 7 d–f. The remnants in this particularsimulation include about a dozen major bodies which form whena differentiated Moon-sized body strikes a Mars-sized body, at 301from normal, at a velocity vimp ¼ 2vesc . The results are scaleinvariant given the caveats above. If we close our eyes to the rockymantles of the colliding planets and consider only the cores, thenthe appearance is that of a planetary tidal disruption event suchas comet Shoemaker-Levy 9 and its post-perijove ‘‘string ofpearls’’ (Asphaug and Benz, 1996)—a linear structure with ratherequant size and spacing. The intensely varying gravitationalstresses, acting over tgrav, appear to play a dominant role inpulling the projectile core into pieces. As the core is deformed,instabilities at the rapidly shearing core–mantle interface mayalso play a major role in establishing fragment size and spacing.Because � 2

3 of the colliding mass is in mantle silicates, whereasthe iron cores appear to have their own intense tidal interaction,the event is very complex gravitationally, mechanically (coreblobs plowing through the mantle rock), and in terms of theintense shock evolution, as the mantles respond to the direct hit.

A layer of mantle silicates ranging from a dominant fraction to athin veneer remain gravitationally bound to the blobs of core materialthat emerge from the collision. These new small planets are recycledfrom selected volumes of their parent planet materials, and have ironand silicate components that would not make sense in the context ofstandard models of planet formation and evolution. The majorfragments are significantly iron-enriched, and the smaller debris aremostly of surface or mantle composition. At these relatively lownumerical resolutions (tens of thousands of particles) more nuancedaspects of the collisions are unresolved. Higher fidelity simulationsmade possible by parallel SPH codes running on commoditysupercomputers shall give a far better understanding of the majorcompositions, thermal and shock histories, spin states, and other



aspects of the recycled planetoids, and to the long term fate of theescaping material. It is a complex phenomenon involving impactphysics, tidal physics, solar system dynamics, meteoritics, and forinternally molten planets, volcanology—the task is not just improvingmodel resolution but also the model physics. But for now, a fewobservations can be made:

The final bodies in a chain-forming hit and run collision arecomposed of deep interior materials that find themselves, in thecourse of hours, at greatly reduced hydrostatic pressure P0. Thiscreates a potential for volumetric degassing and the fluxing ofwater and other previously dissolved volatiles in response tointense, abrupt pressure disequilibrium. Shock intensity is highlyvariable, since they form from materials at a considerable distancefrom the collisional contact surface. So in general one expects awide variety of metamorphic and igneous evolution to occur inthis small stretch of time, leading to diverse pathways formeteorite petrogenesis.

Regarding iron and iron–silicate meteorites, the surface area ofiron to silicate is greatly increased during a chain-forming hit andrun event, since the original impactor core is parceled into adozen or so new planetoids, and is sheared against the silicateinterface as it happens. When a core body is sheared intomultiple components, shear localization is expected to occuralong the core–mantle interface, causing detachment and intensefriction. Thus, the prevalence of mesosiderites, pallasites, andother iron–silicate mechanical mixtures, and their diversity, isconsistent with this type of parent body mechanical-drivenevolution.

Shear localization does not require a core–mantle interface;evidence for pseudotachylitic clasts and textures in ordinarychondrite meteorites (van der Bogert et al., 2003) may indicatethe mechanical shearing of undifferentiated planetary bodies aswell, perhaps during smaller-scale early-epoch SSCs at lowervelocity. Indeed while most of this review has focused on theimplications of SSCs for differentiated bodies, for which thebefore-and-after change is the most spectacular, the processapplied to undifferentiated bodies can lead to various kinds ofevolution ranging from hydrothermal action, to pseudotachyliticmodification, to brecciation.

Direct evidence for a chain-forming hit-and-run collision isperhaps recorded in the IVA iron meteorites, one of the 14 majormeteorite groups. Each iron meteorite group corresponds to anoriginal reservoir (planetesimal core) that is a unique alloy of ironand nickel, plus trace metals such as gold. Thus the 14 major ironmeteorite groups represent 14 parent bodies, while there arehundreds of unclassified iron meteorites, bringing the totalnumber of disrupted parent bodies represented by iron meteor-ites up to 50–100. One immediately must ask: How is it that somany large differentiated asteroids underwent catastrophicbreakup—especially when other asteorids, especially 4 Vesta(Davis et al., 1985), did not?

According to Ni–Fe measurements by Rasmussen et al. (1995),the IVA irons have metallographic cooling rates spanning almost 2orders of magnitude, from 19–3400 K/Ma. Yang et al. (2007)report revised cooling rates for this group spanning 100–6000K/Ma, and show a trend of faster cooling rate for lower bulk Ni. Tosatisfy the most rapid of these cooling rates, and to conform to thebulk Ni data, Yang et al. develop a model where the IVAmeteorites cooled within a � 300 km diameter metallic bodythat was stripped bare of its mantle. Mantle removal is requiredby the most rapid cooling rates, simply because an insulatingmantle would cause the iron core to cool slowly, and under nearlyisothermal conditions. The wide span in cooling rates within asingle stripped core body remains problematic in this scenario,but it is consistent with the cosmic ray exposure ages thatindicate cooling within a single final body.

Stripping a Vesta-sized mantle bare is also possible in a chain-forming hit-and-run collision; also recall that repeat hit-and-runsmay have occurred (Fig. 9). Diverse cooling rates are an expectedresult when a core is pulled into a chain of new planets, each withits own core, its own mantle (or lack thereof), its own diameterand cooling rate. The scenario of Fig. 7 d–f would lead to theformation of a handful of new bodies each with the same majorelemental and initial isotopic core composition; each wouldsubsequently follow its own evolution as a minor planet. Ironmeteorites derived from these bodies would come from the sameparent body, compositionally, but from different bodies in termsof their solidification, cooling, and post-formation physical andchemical history, and cosmic-ray exposure ages of their resultantmeteorites. As for the silicate portions of these strung-out newplanets, there would be much greater variation in compositionowing to the initial shock state, varying provenance within theinitial projectile’s mantle, mixing with the target mantle, andvolatile dissolution.

As for that broad and most controversial topic of early solarsystem origins, the formation of chondrules, it is certainly not far-fetched to consider that hot, possibly molten bodies tens tohundreds of km diameter would have undergone collisionalevolution at random speeds comparable to their vesc, in the firstfew Ma when 26Al was active. If so, then a similar-sized collisionscenario for chondrule formation is worth considering, in whichmolten asteroids are torn apart once or twice for every efficientaccretion, as part of the inefficient process of accretion duringrandom stirring. If hit and run collisions happen to a given parcelof matter many times over as Fig. 9 suggests, and if those parcelsare molten and gas-rich, then pressure unloading would result inthe dispersal of material over tgrav � hours, under phreaticconditions. It can be thought of as an evolution of the chondruleformation hypothesis of Sanders and Taylor (2005), but withpressure unloading and the associated droplet formation physicsand timescale taking the place of impact splashing. It isdynamically and chemically plausible, and recommends furtherresearch into the earliest thermophysical processing of terrestrialplanet-forming materials during similar-sized collisions.

4.3. Mercury and Mars

Mercury is anomalously iron-rich, about 70% by mass. Benzet al. (1988, 2007) showed that a giant impact with randomvelocity � 6vesc by an projectile r� R=2 (depending on impactangle) is capable of shock-accelerating half of Mercury’s mantle toescape velocity, in an intensely energetic collision bearing over 30times the specific energy of the giant impact proposed to haveformed the Moon. One of the challenges to the hypothesis iscompositional, for there is geophysical and spectroscopic evi-dence (for instance Kerber et al., 2009; Sprague et al., 1995) forvolatiles in the crust of Mercury in much greater abundance thanon the Moon. This could be challenging to reconcile with thehypothesis of the planet finishing its evolution by having itsmantle shocked and dispersed into space, and part-reaccumu-lated. Benz et al. (2007) show that the ejected material, whichgoes to occupy a torus around the Sun centered on Mercury’sorbit, is fragmented into sizes small enough for most to beremoved by Poynting–Robertson drag; the rest reaccretes to formthe new upper mantle. However, Gladman and Coffey (2009) findthat the opacity of this debris cloud will severely limit the rate ofmass removal (and incidentally, will put the other planets intototal shadow). We knew from the start (Fig. 1) that it is not easy toblast off a planetary mantle. If the event requires a collision that isanomalously fast and energetic, it seems odd that the lowestpossible impact velocity (parabolic encounter) is invoked to



explain the Moon’s formation, whereas an anomalously hyper-bolic impact (vimpbvesc) is invoked to explain Mercury, bothduring the same late stage of solar system history.

By directly scaling from Figs. 8 and 9, an alternative giantimpact scenario is proposed that would remove Mercury’s mantleat a characteristic impact velocity v1 � vesc , assuming Mercurywas the impactor rather than the target. What did it run into?Wetherill (1992) showed that Mercury could have originatedbeyond Mars, leading to the possibility that it may haveencountered proto-Earth or proto-Venus. Perhaps more dynami-cally probable is for Mercury, under quieter circumstances, tohave become sufficiently eccentric to encounter Venus or proto-Venus in the course of later chaos (perhaps swapping its volatilesover time onto the larger planet as described earlier).

The basic hypothesis is that Mercury started out rather Mars-like and collided with a larger planetary embryo along the way,emerging from the hit-and-run as a planet of roughly Mercury’smass and composition, in a process akin to Fig. 7a–c. In thissimulation the final impactor lost 35% of its original mass(M2=0.65 m) which equals half its mantle. Largest remnantmasses (normalized to impactor mass) for the simulations plottedin Fig. 8 were computed by Agnor and Asphaug (2004a, b) for thesimulations described above and are plotted in Fig. 15. Mercurymay thus have formed more closely akin to how the Moonformed: a Mars-mass planet running into a larger planet,but in Mercury’s case a fraction faster. Still, we must keep inmind that Mercury’s bulk composition may not be definedby any single giant impact. For instance, if somewhat higherrandom velocities v1=vesc existed near Mercury throughout thelate stage, this might account for a systematic increase indisruptive and hit-and-run collisions (e.g. the black diamonds inFig. 9) leading to a hierarchical evolution towards iron-richcomposition.

5. Conclusions

Numerical simulations have shown that impacting bodies ofsimilar size can experience hit and run collisions for the randomvelocities typical of late stage planet formation. This causes us toconsider giant impacts from the surviving impactor’s perspective.Hit-and-run is an efficient mechanism for dismantling the smallerof a colliding pair, removing its outer layers and causing globaltransformations, while leaving the target body comparativelyintact.

If Main Belt bodies Moon-sized or larger once existed, as isexpected on the basis of dynamical studies, then Main Beltasteroids ought to be replete with hit and run collisional relics.Vesta, whose crust appears to be largely intact, may have beenlucky to avoid a major hit-and-run, but suffered a typicalbombardment by smaller impactors. This scenario requires onlya moderate anomaly for Vesta, compared to the alternative, thatPsyche, Kleopatra, and the parent bodies of the dozens of familiesof iron meteorites were beaten down to their cores by erosive anddisruptive impacts, with Vesta somehow dodging a cosmicfusillade. If Vesta avoided a hit and run collision until the largerbodies left the Main Belt, while Psyche was dismantled to its core,perhaps hit-and-run evolution happens to about half the NLBs inlate solar system history. For the small terrestrial planets, wemight in this context look at Mars as a body which has alwaysbeen among the very largest of its collisional population,efficiently sweeping up smaller planetesimals and embryos andbeing little bothered by hit and run collisions. We might then lookat Mercury as a typical next-largest body, a hit-and-run remnanthaving lost its mantle.

Meteorites show evidence for hit and run collisions. A hit-and-run collision can account for the wide range in cooling ratesexhibited within one or two families of iron meteorites (Yanget al., 2007). Chondrite meteorites show the kinds of frictional oreven pseudotachylitic textures that are expected during similarsized collisions, although hit-and-run is difficult to discern amongundifferentiated planets where there is no core–mantle segrega-tion. These frictional and breccia textures, and evidence foraltered and metamorphosed silicate bodies (Keil, 2000), and theoverall stunning variety and relative abundance of iron and iron–silicate meteorites, are not direct evidence for hit and runcollisions per se, but are certainly indicative that similar sizedcollisional processes have been at work, along with the associatedprocesses described above.

Chondrites, the most abundant meteorites, derive from the mostcommon parent bodies in the Main Belt. They have not beensubjected to temperatures close to melting since the time ofchondrule formation, a few million years after solar system formation.Chondrules, which might account for over half the mass of the MainBelt (Scott, 2007), are solidified silicate droplets from an epoch oftransient heating episodes. In the first 105–106 yr of planetesimalgrowth, if molten 10–100 km diameter precursor bodies were keptheated by 26Al decay, but undifferentiated because of their very lowgravity, then it is conceivable that the abrupt (strombolian) pressureunloading experienced by molten but undifferentiated small pre-cursor bodies could result in ubiquitous small melt droplets amongthe remnants that did not accrete. Chondrules forming by such amechanism would either disperse into space, and either be removedor reaccreted, or collapse en masse gravitationally on a timescale tgrav,which for a dense plume of droplets would be a few hours.

If a planet grows to become one of the largest, then there isnothing larger to collide into and it becomes an amalgam of itsfeeding zone (Chambers and Wetherill, 1998). Impacts by smallerbodies batter their outer layers for the remainder of theirevolution, and can remove atmospheres and oceans (Genda andAbe, 2005) and even hemispheres of crust (Nimmo et al., 2008;

0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

M2/

m

60

45

30

v-innity / v-escape

Fig. 15. Mass of the largest impactor remnant (M2 rm) after a hit-and-run

collision, for the g¼ 0:1 (m¼ 0:1M) subset of the collisions in Fig. 8. Calculations

by Agnor and Asphaug (2004b). The curves are for different impact angles, labeled.

The missing points to the left of the graphs for 301 and 451 are events where the

impactor and target accrete, so that there is no sizable M2; the transition from

accretion to hit-and-run is abrupt with changes in y and v1 . To the lower right of

the 301 graph the transition is gradual towards M2-0; here the impactor is

unaccreted but the escaping matter is disrupted with increasing energy into

smaller sizes. Inbetween are outcomes that leave behind a substantial portion (or

portions) of impactor mass m. The chain-forming collision of Fig. 7d–f plots here

with a largest fragment mass M2=0.2m; the outcome is actually several new

bodies of which the largest is plotted. For the most typical impact angle of 451, for

moderately excited random velocities, the impactor loses between 13 and 2

3 its mass

in a hit-and-run, a conjecture to be studied further for the origin of Mercury.



Marinova et al., 2008), but the removal of their mantles requirescollisions of unusually high energy. The cores of accreted planetsare not disrupted for expected impact velocities (Scott et al.,2001). A very massive and wayward impactor is required toremove the fraction of Mercury’s mantle that appears to bemissing, assuming it began with Earth-like bulk composition(Benz et al., 2007). This scenario is dynamically opposite theMoon-forming scenario, for which the impactor and target mustbe almost identical dynamically and chemically, falling in atv1 � 0. But in Moon-forming giant impact scenarios, models(Canup and Asphaug, 2001) show that the Mars-sized impactorwould escape if going a fraction faster. This motivates analternative scenario in which Mercury sheds its mantle in a hitand run collision with a larger protoplanet early on, at v1 � vesc

(e.g. Fig. 7a–c) or conceivably during later chaos encounteringVenus.

Hit and run collisions occur in extrasolar planet-formingsystems. Exoplanets are being discovered through transit ob-servations to include ‘‘super-Earths’’ that are several times moremassive than our own (e.g. Ribas et al., 2008); the discovery of thefirst Earth-mass planet is imminent. But how Earth-like can aplanet be, in a solar system that has accreted one or more super-Earths? Where Earths are the unaccreted NLBs, most will have losttheir crusts, oceans and atmospheres at one time or another, orbeen dismantled like Mercury, on the perilous path of planetarygrowth. It may be that in order to have water-bearing Earth-massterrestrial planets, they need to accrete as the largest bodies in thepopulation.

Acknowledgments

This research was sponsored by NASA’s Planetary Geology andGeophysics Program (‘‘Small Bodies and Planetary Collisions’’) andOrigins of Solar Systems Program (‘‘Meteorite and DynamicalConstraints on Planetary Accretion’’) under Research Opportu-nities in Space and Earth Sciences. The SPH computations wereperformed on the NSF-funded supercomputer upsand at UCSC-IGPP. My research into planet-scale collisions began as a thesisproject with Willy Benz, whose subsequent collaboration on tidaldisruption led me to try understanding hit and run collisions. Thisresearch evolved through collaborations with Robin Canup onMoon formation, and with Craig Agnor, a patient explainer ofplanetary dynamics. I am grateful to John Chambers and BillBottke for their careful reviews and unique insights. I thank EdScott, Quentin Williams, Francis Nimmo, Jeff Cuzzi and NaorMovshovitz for creative critical discussions, and Klaus Keil for hisoriginal thinking on igneous and evolved asteroids and forinviting me to write this review.

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