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Chemie der Erde 70 (2010) 199–219
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Chemie der Erde
journal homepage: www.elsevier.de/chemer
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 in fo
abstract
Article history:
Received 10 September 2009
Accepted 31 January 2010
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, hitand-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.
Keywords:
Planets
Impact
Collisions
Accretion
Planet Formation
1. Introduction
impact velocities vimp comparable to the mutual escape velocity
During the late stage of terrestrial planet formation (Wetherill,
1985), giant impacts occur when similar-sized planets at or near
the largest end of their size distribution collide at speeds ranging
from 1 to a few times their mutual escape velocity vesc. This
notion of late giant impacts emerged alongside the idea for a giant
impact origin of the Moon (Hartmann and Davis, 1975), where a
Mars-sized projectile is proposed to have struck the proto-Earth
to liberate a new planet composed mostly of Earth-like mantle.
Giant impacts can be generalized as occurring between the largest
and next-largest bodies at any stage of planet formation, at
vesc ¼
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0009-2819/$ - see front matter & 2010 Elsevier GmbH. All rights reserved.
doi:10.1016/j.chemer.2010.01.004
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Mþm
2G
Rþr
ð1Þ
which is the velocity at which two spheres collide if starting out at
zero velocity at infinite distance. The radii r t R and masses m and
M correspond to a spherical projectile and target, and
G ¼ 6:673 108 cm2 g1 s2 . This generalization of giant impact
is called a similar-sized collision or SSC.
Agnor and Asphaug (2004a) studied collisions between equalsized planetary embryos (r=R) and found that merger is inefficient
except when vimp almost equal to vesc. Impact speeds are expected to
be higher than this in the late stage of terrestrial planet formation,
since the orbits must be planet-crossing. This paved the way to
studies (Asphaug et al., 2006) of geophysical and compositional
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E. Asphaug / Chemie der Erde 70 (2010) 199–219
evolution in a broader range of scenarios ðr t RÞ. They found that it is
common in gravitationally stirred-up populations for planetary
embryos somewhat smaller than the largest to dash up against the
largest but not accrete. These hit and run collisions dismantle the
impactors (r) or catastrophically disrupt them in peculiar ways.
It is argued below that many or most of the unaccreted nextlargest bodies (NLBs) surviving the late stage of planet formation
bear the scars of one or more hit and run collisions. A remarkable
diversity is then predicted for the final collection of NLBs, whether
they be Mars and Mercury of the inner solar system, middle-sized
members of Saturn’s satellites, Vesta and Psyche in the Main Belt,
Quaoar and Haumea and other oddities beyond Neptune, or Earthmass planets in solar systems with super-Earths. Next-largest bodies
are lucky to be here, and each is lucky in its own way.
Hit-and-run can be as common as accretion, when the
characteristic random velocity v1 of a planetesimal swarm
(relative to distant circular coplanar orbits) is comparable to the
characteristic escape velocity vesc of the largest members of the
population. This random velocity is added to the escape velocity,
so that spherical planets collide at an impact velocity
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2Þ
vimp ¼ v21 þv2esc
When random velocity v1 =vesc C0 accretion is efficient, but when
v1 vesc hit and run is the most common outcome. The unique
petrogenetic outcomes of hit and run collisions, and the predicted
diversity of NLBs and the asteroids and meteorites that derive
from them, may be indicative of the random velocities that prevail
during the dynamical epochs of planet formation, in the earliest
stages corresponding to the evolution of chondrites and chondrules, and in the late stages that define the characters of finished
planets.
1.1. Accretion basics
In the classical accretion theory of Safronov (reviewed in
Wetherill, 1980) the random velocity is related to the escape
velocity, regulated by the gravitational stirring, according to
Y¼
v2esc
2v21
ð3Þ
The Safronov number Y is postuated to be 325 during the
course of planetesimal growth (Safronov and Zvjagina, 1969;
Safronov, 1972) and closer to Y 122 in the late stage of
terrestrial planet formation (Wetherill, 1976). This relation arises
from the assumption of planetesimal (gas-free) accretion, with
random velocities excited gravitationally by the largest bodies
into mutually crossing orbits. Based on N-body numerical
experiments, Agnor et al. (1999) and O’Brien et al. (2006) find
that vimp ranges from41 to a few times vesc during the late stage
of giant impacts, broadly
consistent
with Wetherill’s result.
ffi
pffiffiffiffiffiffiffiffiffiffiffiffi
The ratio v1 =vesc ¼ 1=2Y depends on the location within the
size distribution, since vesc and random velocity both change. The
ensemble gravitational drag of small planetesimals reduces the
velocity dispersion of the larger embryos, so generally v1 is lower
for steeper mass distributions (with greater masses of small
particles). If gravitational stirring happens to small and large
bodies alike, then smaller bodies encounter one another at higher
v1 =vesc , so even if the largest encounters are mostly accretionary
(v1 =vesc 0:3, say), colliding bodies half as large will have
v1 =vesc 0:6, with outcomes that are mostly hit-and run. This is
why hit and run is described below as an edge effect, occurring at
the margin of the population.
Under dynamically cold conditions (v1 5vesc ) the growth of
the 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 small
planetesimals within an enhanced cross-sectional area that is
increased by a gravitational focusing factor
Fg ¼ 1 þ2y ¼ 1 þ
v2esc
v21
ð4Þ
accounting for slow planetesimals falling in towards the body. The
other scenario, v1 b vesc is sometimes called orderly growth; since
there is no focusing all bodies increase in radius at the same rate.
But orderly growth assumes perfect sticking during a sweep-up of
planetesimals at high random velocity. Perfect sticking can be a
very poor assumption when v1 b vesc , and this calls to question
whether orderly growth is a valid concept.
Planet formation is likely to involve quiescent epochs, and also
epochs of moderate random stirring dominated by similar-sized
collisions. It seems incontrovertible that moderate random
velocities are required during the late stage, since orbits must
intersect across increasing distances. Epochs of random stirring
are also expected during the first few million years of solar system
formation. The severe consequences of planetary dismantling by
the mechanism of hit-and-run are likely to be vital to the final
bulk chemistry of planets. Planetary growth is after all not just the
accumulation of a feeding zone by accretionary events; it is also
the record of a comparable or even greater number of nonaccretionary hit-and-runs, each with the capacity to dismantle
and 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 are
extended-source phenomena, distinct in important respects from
the point-source collisional phenomena that cause the formation
of impact craters (Melosh, 1989). A key difference is that there is
no physically important central point in a similar-sized collision—broad regions such as the cores respond in one way, and the
colliding mantles respond in another. The outer layers (atmosphere, ocean, crust) respond in yet another. The impact locus as it
were is hemispheric or even global in extent. The understanding
of impact cratering benefits greatly from the principle of late
stage equivalence, a strong form of hydrodynamic similarity
whereby the fundamental characteristics of a collision are
obtained by geometric (power law) combinations of impactor
radius, density and velocity (Holsapple, 1993). In cratering this
allows meaningful extrapolations of laboratory results to large
geophysical scales. Hydrodynamical similarity applies to SSCs, but
not the cratering concept of an impact locus.
A second major distinction between similar sized collisions
and impact cratering is that the contact and compression
timescale for an SSC equals the gravity timescale. The smaller
planet is deformed mechanically (compressed and sheared) by its
abrupt deceleration against the target, while it is deformed
gravitationally. Its fate, and the outcome of the collision, may
depend upon the inter-dominance of self-gravitational instability
and shear instability. The deceleration and deformation of the
projectile 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. Because
vimp 1 to a few times vesc, the collisional timescale for SSCs is
tcoll r=vesc . By comparison, the self-gravitational timescale is
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tgrav ¼ 3p=Gr
ð6Þ
where the density r ¼ M= 43 pR3 ¼ m= 43 pr 3 assuming uniform
bodies. This is the time for a sphere of uniform-density matter
to orbit itself. Because r R for similar-sized bodies,
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E. Asphaug / Chemie der Erde 70 (2010) 199–219
pffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi
vesc GM=r
pffiffiffiffiffiffiffi r Gr and thus the collisional timescale
tcoll 1= Gr tgrav . This confluence of gravitational and impact
deformational timescales contributes to the ‘‘lava lamp’’ like
quality noticed in movies of giant impact simulations, with
collisional deformation countered by gravitational restitution.
1.3. Fate of the bullet
In planetary impact cratering the discrete fate of the projectile
is only of significance for the most oblique angles of incidence
(Pierazzo and Melosh, 2000). For SSCs the fate of the bullet is of
principal consequence, even for moderate impact angles close to
301 from normal. This gives rise to a third important distinction
between SSCs and impact cratering: for typical geometries, only a
fraction of the colliding matter intersects. For impact velocities
greater than about 1.2vesc, depending on mass ratio g ¼ m=M, the
rest goes sailing on, and the abrupt shears and unloading stresses
unleash a host of planetary processes, some rather novel to
consider. Taken together, these distinctions point to the existence
of a broad range of collisions, with physical outcomes between
those of tidal collisions like comet Shoemaker-Levy 9 which
disrupted near Jupiter in 1992 (Asphaug and Benz, 1994); and
planetary-scale impact cratering events (e.g. Marinova et al.,
2008; Nimmo et al., 2008), with an associated diversity of
cosmochemical and planetological outcomes.
By far the best studied archetype of a similar-sized collision is
the giant impact that has been proposed for the origin of the
Moon (Hartmann and Davis, 1975). In the latest studies of this
scenario, the projectile gets dismantled (literally) by gravitational
and mechanical shears and instabilities into two components, its
core which merges with the center of the Earth, and its mantle
which interplays with Earth’s outer mantle to form a protolunar
disk of several lunar masses (e.g. Stevenson, 1987; Benz et al.,
1989; Cameron and Benz, 1991; Canup and Asphaug, 2001). This
fractionation of the bulk Moon from the bulk Earth, forming a new
minor planet (bound as a satellite) lacking in iron and volatiles,
illustrates how SSCs can leave their permanent and formative
imprint upon unaccreted next-largest bodies.
Giant impacts such as the Moon’s formation are high-energy
end members of SSCs occurring late in planet formation, when
planets are large and vesc pR is large. The phenomenon can scale
down to smaller colliding pairs, impacting at correspondingly
slower velocity, at earlier stages of planetary formation. For
instance, two molten planetesimals colliding during the first
1 Ma at low random velocity (vimp vesc 100 m=s) might look
like the Moon-forming giant impact, and take place on the same
timescale tgrav .
As for solidified planetesimals, for instance cold chondritic
asteroids or solidified differentiated embryos, the outcome of a
hit-and-run can involve bulk textural changes, including energetic
shearing and brecciation. The shear stress in a tidal collision
exceeds the strength and internal friction of rocky or icy bodies
larger than 100 km, or even smaller if the disruptive effect of a
grazing blow is considered. Changes with scale are considered in
detail below.
2. Similar sized collisions
Most of the colliding mass contributing to the formation of a
planet comes by way of the several largest impacts, bodies within
an order of magnitude in mass and a factor of a few in size that
have been stirred up gravitationally into planet-crossing orbits.
During the earliest stages of terrestrial planet formation most of
the mass is in small particulates: crystals and amorphous phases
of dust and ice grow into clusters, and these into nuggets that one
201
might call planetesimals, which go on to sweep up the smaller
bodies in a runaway. This phase is known from chemical,
dynamical and astronomical evidence (Meyer et al., 2008) to
end early on, although the transition from dust to planetesimals is
a strongly debated topic. Cuzzi et al. (2001, 2008) and Johansen
et al. (2007) have shown that turbulence can randomly initiate
the concentration of small particles into dense clusters that then
ride in pack through the gas and dust, resisting further disruption
until they contract gravitationally, or through dissipation and
sticking, into sizable planetesimals. These mechanisms might
make it possible for large asteroid-sized bodies to bypass
hierarchical growth and accrete directly out of the planetesimal
swarm. Morbidelli et al. (2009) propose on this basis that
asteroids were ‘‘born big’’ to explain the apparent factor of 4
overabundance of 100 km asteroids in the present Main Belt
relative to a power law. If so, then after the gas and dust have
cleared accretion is mostly a matter of similar-sized collisions.
Even in the case of hierarchical growth, and a power law
distribution of sizes, most of the mass that collides during planet
formation interacts at the large end of the feeding chain, with
Mars-mass bodies colliding into Earth-mass bodies, and lunarmass bodies into Mars-mass bodies, and so on. Consider a
differential size distribution
nðrÞpr a
ð7Þ
where n =dN/dr and N(r) is the cumulative number of planetesimals larger than radius r. In the case a ¼ 4 there is equal mass in
equal logarithmic bins, i.e. equal mass in bodies hundreds of km
diameter as in bodies meters in diameter. For a comminuted
(ground-down) population obeying size-independent fragmentation physics, the theoretically derived equilibrium value is a ¼ 3:5
(Dohnanyi, 1969). But observed planetesimal size distributions
are shallower, with the mass in the largest bodies. In the Main
Belt the four largest asteroids, within a factor of 2 in diameter,
account for half the total mass. Main Belt asteroids trend with
a 223, varying with size and sub-population. Recentlydisrupted comet groups and the comet population as a whole
appear to have a 1:7 (see Weissman et al., 2004). For planetforming planetesimals, gravitational focusing and oligarchic
sweep up of feeding zones shifts the mass further into larger
sizes (Kokubo and Ida, 1998). Thus for characteristic minor planet
and small body populations, and embryonic populations, most of
the mass—and hence most of the colliding mass—is found in the
handful of largest and next-largest bodies, whether they are born
big or become big.
2.1. Colliding pairs
In their studies of the evolving Main Belt asteroid size
distribution Bottke et al. (2005) and Morbidelli et al. (2009) use
a statistical code to follow the erosional and disruptive evolution
of candidate primordial Main Belts, to see which ancestral size
distributions 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 small
asteroids are usually demolished by smaller members of the
population; among small bodies the random velocities are
typically orders of magnitude faster than their escape velocities.
Indeed, demolition by an equal sized body seems exotic, let alone
by a larger body.
But for the largest asteroids, even at modern solar system high
velocities, the impactors required to disrupt them by traditional
means can be as big as they are—or larger. Fig. 1 shows a
differentiated planet 500 km diameter, consisting of 30 wt% iron
and 70 wt% rock, impacted by a 200 km diameter rocky asteroid at
vimp ¼ 10 km=s (top four panels) and 5 km/s (bottom two panels),
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E. Asphaug / Chemie der Erde 70 (2010) 199–219
themselves, a scenario consistent with the Main Belt models by
Chambers and Wetherill (2001), Petit et al. (2001), and O’Brien
et al. (2007).
In a scenario drawn from the simulations reviewed below, a
dozen 1002200 km diameter asteroids might result from the
hit-and-run breakup of a single Vesta-sized asteroid that collided
at a typical impact angle and at velocity comparable to vesc, into a
long-gone (eventually scattered or accreted) Moon-sized world.
Returning to the question of the apparent excess of 100 km bodies
in the Main Belt, it may be important to consider the hit-and-run
breakups of next-largest bodies that are torn into chains by
impacts into the largest. If the collisional physics works out then
the mass budget is amenable, since the broad excess of 100 km
diameter asteroids appears to be offset by a deficit of 300–400 km
diameter asteroids.
2.2. Growing planets
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 y ¼ 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.
in both cases at the most probable impact angle y ¼ 451 (from
Asphaug and Agnor, 2005). The 10 km/s impact is 223 times
the velocity typical of modern Main Belt collisions, and 30
times the mutual escape velocity of the pair. But this barely does
the job of leaving half the mass behind (the definition of
catastrophic disruption) and only exposes some bits of core
material. The 5 km/s impact, at a more typical velocity, but still
15vesc , blasts off the outer layer but leaves the mantle
reasonably intact.
Based on this and other modeling (e.g. Scott et al., 2001) it
appears that once planetesimals grow large, they become difficult
if not impossible to disrupt. This would present a big problem,
since meteorites show evidence for the cataclysmic disruption of
numerous asteroid parent bodies that were at least several
hundreds of km in diameter (e.g. Keil et al., 1994; Yang et al.,
2007). What is required is either a very energetically excited
population of Main Belt precursor bodies, with v1 =vesc ]30, that
is Yu103 , yet miraculously leaving Vesta’s basaltic crust intact,
or else a mechanism for disruption that operates at lower levels of
excitation.
The process of hit-and-run is capable of causing the catastrophic disruption of objects hundreds of km diameter at the
relatively moderate random velocities that are expected. It
requires asteroid parent bodies to run into objects larger than
A different kind of modeling approach for studying planet
formation is to build up rather than break down a primordial
population, using accretion codes that apply advanced N-body
computational methods to directly integrate the orbits and
interactions of planets and asteroids around the Sun and in
planet-forming systems. Out of computational necessity these
codes assume the simplest collisional physics—the perfect
merger—in order to facilitate precise dynamical tracking over
millions of years, with collisions = mergers leading to finished
planets. Fragmentation can be modeled in such codes, but the
debris are impossible to integrate much further in time as discrete
objects. Planets of mass M, m and radius R, r are assumed to stick
when they hit, forming a larger sphere of mass M+ m that
conserves linear and angular momentum. Encounters are treated
as perfectly elastic gravitational encounters (no mass transfer and
no dissipation of energy) if the impact parameter b 4 R þr, and as
perfectly inelastic mergers if br R þr, for which case accretion
efficiency x ¼ 1 as defined below.
Even with these simplifications, only a few hundred to a
thousand planetary bodies can be integrated, since the precise
treatment of close encounters slows down the calculation with
the square of the number of embryos being tracked. And so a
broad size range is not allowed if one follows a physically
reasonable size distribution. This means that every collision
treated 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 bestow
upon planets a capacity for arbitrarily large spin angular
momentum. The impactor’s angular momentum relative to the
!
target 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 planetforming N-body calculations and found that finished planets in
such simulations spin up to rotation periods as short as 1–2 h. This
can exceed dynamical stability (Chandrasekhar, 1969). Moreover,
such rapid spins would have to be slowed down to present-day
rotations by some mechanism, such as loss of a large satellite, or
spin–orbit coupling with another planet or its core; see Kaula
(1990) in the context of Earth and Venus. The solution appears to
be that fast and/or grazing collisions do not contribute much to
accretion; this limits the accumulation of spin. How fast, and how
grazing, depends on the collisional physics and can be reliably
determined using self-gravitating particle-based or other kinds of
hydrocode simulations.
Perfect sticking is a reliable assumption in the context of mass
evolution only when random velocities are slow in comparison to
vesc. (The Moon formed at low v1 =vesc , thus ‘‘sticking’’ may not be
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E. Asphaug / Chemie der Erde 70 (2010) 199–219
203
quite the right word.) Dynamical friction of disk planetesimals
acting upon the embryos (O’Brien et al., 2006) are likely to reduce
their eccentricities and inclinations compared to those reported
by Agnor et al. (1999) and others. As mentioned, in the earliest
stages of growth quiescent encounters were probably the norm,
although turbulence and gravitational stirring by initial embryos
may excite planetesimals into relatively high random velocities
before the dust has cleared. In order to have planetary encounters
at all in the late stage, when orbits are increasingly well
separated, random velocities must be rather high. If the epochs
of planet formation can be discerned dynamically, then they can
probably be discerned cosmochemically owing to the dramatically 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 diameter
incompressible spheres colliding at vimp ¼ 30 cm=s (a few their
vesc) are indistinguishable from two 1000 km spheres of the same
density colliding at vimp ¼ 30 km=s, when the velocities are scaled
to vesc and distances are scaled to R. Scale-invariance thus
provides first-order physical insight rather than a fast rule. It
allows us to define this class of planetary collisions (SSCs) where
the impactor and target are of similar size, and where the random
velocity v1 is similar to the escape velocity vesc.
Planets are compressible, and rheologically and thermodynamically complex. For instance, it is quite plausible that silicate
melts are to be found deep inside most 100 km diameter or larger
early-stage terrestrial bodies. At high pressures these melts are
very soluble to water and other volatiles. In contrast, subkilometer bodies are unlikely to retain any appreciable melt or
dissolved gas at any stage in their formation or evolution—they
are too small to retain heat and too underpressured to retain gas.
They may retain ices. These smallest bodies (typical asteroids and
comets) are likely to behave as solids or granular solids during
collisions, obeying a physics akin to landslides.
Large molten planets differentiate with iron in the core, crustforming silicates and volatiles in the exterior, and mantle
inbetween. Differentiation increases the gravitational binding
energy of the planet making it almost 3 times as difficult to
catastrophically disrupt (in terms of impact energy) as an
undifferentiated planet of the same composition, based on impact
models similar to Fig. 1 (e.g. Benz and Asphaug, 1999). But
differentiation also perches the volatiles and silicates at the
lowest specific binding energy, making it easier for major impacts
and collisions to strip these materials from an otherwise growing
planet. Cores and deep mantles become sheltered as was
demonstrated in Fig. 1, and so it is truly an enigma that iron
should be one of the most common representatives of our
meteorite collections.
If we restrict ourselves to inviscid, molten, differentiated
colliding pairs—astrophysical rather than geological objects—then the most basic assumptions of scale invariance are
met. Even then there are important differences arising from
powerful shocks and high hydrostatic pressures. Large SSCs (giant
impacts) are hypervelocity because the impact velocity vimp
exceeds the sound speed of the colliding materials; they are
shock-inducing collisions leading to global-scale internal heating
and Hugoniot acceleration. Assuming v1 vesc the impact
velocity exceeds the sound speed for collisions involving
terrestrial planets that are Moon-sized and larger.
H2O and CO2 solubility in silicate melts is greatly enhanced at
high interior pressures (e.g. Dixon et al., 1988). When a large
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 (r ¼ 0:6R, assuming rr ¼ rR ) at the moment of collision, where y ¼ 301 (left)
and y ¼ 451 (right). At left the impact parameter b o R; at right b 4R 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 (b 4 R þ r) collisions.
planetary body ‘‘loses its lid’’ (Fig. 2) then its pressurized volatiles
(several wt% H2O may be typical) can erupt violently, released
over tgrav from hydrostatic pressure P. This pressure is released
from a magnitude and over a timescale (kilobars, hours) that is
comparable to gas-driven kimberlite eruptions on Earth (Kelley
and Wartho, 2000; Porritt and Cas, 2009; Kamenetsky et al.,
2007). As shown below in a study of purely tidal collisions, even a
tidal (non-impacting) impactor can have its interior pressures
lowered by 50% for about an hour, with a rate of pressure release
P_ P0 =tgrav r 2
ð8Þ
and 20% permanent pressure reduction because of spin-up and
mass loss. The pressure release adds V DP to the available specific
enthalpy Dh ¼ Du þ PDV þV DP, considered further below, where
volumetric expansion PDV occurs especially in the case of bubble
formation (Gardner et al., 1999).
Most asteroids, by number and perhaps also by mass, are
undifferentiated. The effects of SSCs among primitive and
undifferentiated populations of smaller bodies may be more
subtle than for larger, differentiated bodies for three reasons. One,
shock levels will be well below the sound speed if v1 vesc , for
objects much smaller than the Moon. Two, the pressure unloading
effects just described may be small. Three, surface-stripping and
fragmentation may not result in a noticeable change in bulk
chemistry or composition since there is no core–mantle segregation, and hence no crust to remove from a mantle, or mantle from
core. Nonetheless it is important to keep in mind that similar
sized collisions do happen to primitive bodies, and that hit-andrun collisions happen when random velocity is comparable
to escape velocity. These collisions would be gravity- and
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E. Asphaug / Chemie der Erde 70 (2010) 199–219
shear-dominated, not shock-dominated. These shears can leave
their imprint; moreover, while pressure unloading from tens of
bars of pressure (the interior of a disrupted 100 km body) might
not stimulate global melting or degassing as studied further
below, the unbalanced pressure gradients and available enthalpy
could trigger volatile migration and hydrous activity in otherwise
primitive bodies.
2.4. Collisional geometry
In impact cratering, the projectile is rapidly buried into a semiinfinite target and effectively explodes, coupling as a point source
without much downrange ballistic motion. The impact angle can
be parameterized with good results, as there is little difference to
the physics except for the shallowest collisions (Pierazzo and
Melosh, 2000). Similar-sized collisions on the other hand are
sensitive to impact angle over all ranges of y, simply because a
similar-sized impactor does not have semi-infinite mass to bury
itself into. Fig. 2 shows how a large fraction of the mass misses the
target for any but the most direct hits. Impacts that would not be
considered oblique in the context of impact cratering
(y 302601) are grazing when it comes to SSCs, in the sense
that most of the mass overshoots.
The probability of impact at an angle between y and y þ dy by a
point mass onto a spherical gravitating target is 12 sin2 y dy, a
function that peaks for impact angle y ¼ 451 (Shoemaker, 1962); 451
is also the median impact angle. The impact angle for undeformable
colliding spheres is the same as that of a point mass at the impactor
center, impacting a virtual sphere of radius R þr, thus Shoemaker’s
original argument applies to SSCs. If one defines impact parameter b
(Fig. 2) as the offset from the impactor trajectory from the target
center of mass at the moment of collision, then p
b¼
ffiffiffi ðr þ RÞsiny and
the most likely impact parameter is b45 ¼ ðr þRÞ= 2. (Note that b is
somewhat larger than the periapse that would be computed for a
collisionless encounter, at the moment of periapse when two virtual
spheres are intersecting.)
For impact cratering (small r/R) the traditional definition of
grazing is b= R, where the impactor skims tangential to the target.
Abstract this notion so that the threshold for grazing is when the
center of mass of the smaller colliding body is tangential to the
larger (Fig. 3):
R
ð9Þ
yb ¼ sin1
Rþr
For equal sized planets r ¼ R and grazing requires an impact angle
that is only 301 from head-on. When r = R/2 grazing occurs for
y 4 yb ¼ 421; that is, more than half of all collisions for bodies
within 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 halfspace
is all-or-nothing, while for SSCs there is broad gradation over a
range of impact angles. That is why their outcomes are so diverse.
If two differentiated planets collide, then their cores miss one
another entirely for
ycore 4sin1 ðrcore =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 impact
between 301 and 901 the cores miss one another. While this is a
simplistic approach to collisions, it soundly predicts that the level
of core–core interaction during giant impacts is highly variable,
with some events shredding and intermingling mantle materials
but not cores, and others merging cores entirely. Idealized
assumptions about planetary collisions and their mixing are
probably untenable; see Nimmo and Agnor (2006) for modeling
and discussion of this issue.
Fluid bodies deform during SSCs and are ellipsoidal by the time
of impact (e.g. Sridhar and Tremaine, 1992). Shoemaker’s
argument for impact angle is no longer valid; impact angle is
not well defined and angular momentum goes into torques. Cores
interact with the mantles they plow through, which are about
twice as massive, and can therefore merge even if not headed
right at one another (see the Moon formation models discussed
below). Impact trajectory is not a straight line. Mass does not
come off as a ‘‘lid’’. And lastly, concerning grazing, there is no
abrupt change in the physics between an impact involving a small
fraction of the target (b t r þR), and an impact that barely misses
(b \ r þR). The discussion of impact angle must be nuanced to
account for tidal collisions, for which y is undefined, which may
be of great consequence to the impactor especially if the mass
ratio g ¼ m=M t 0:1.
2.5. An edge effect
Hit-and-run can be thought of as an edge effect pertaining to
the planets that are similar in size to the largest, within the factor
r=R 13 for which grazing is common. An accreted planet that has
grown into one of the largest, has nothing larger to run into.
90
Grazing Impact Angle
80
70
60
50
40
30
0
0.2
0.6
0.4
0.8
1
r/R
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 (yb r 451). 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 sin1 ðrcore =rÞ may 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.
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E. Asphaug / Chemie der Erde 70 (2010) 199–219
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.
Unaccreted NLBs, though still among the ‘‘giants’’ in the context of
the late stage, finish their evolution as part of the middle of the
pack (Fig. 5). Being among the largest, they are petrologically
interesting, potentially thermally active, differentiated, and
complex. If they are broken down into smaller ð 100 kmÞ
objects early on, their interiors are represented in the final
population of asteroidal parent bodies and in the meteorites; if
they are not, then their interiors remain forever sheltered, as
Vesta’s has been.
Collisions with largest bodies occur at v1 vesc as stirred up by
those same bodies. Smaller bodies in the population are subject to
the same random stirring, and either accrete onto the larger
bodies rather efficiently, or else catastrophically disrupt with
other small bodies, considering their relatively low binding
energy (vimp bvesc ). Thus SSCs as we have defined them do not
occur throughout the size distribution, but between the largest
and the next-largest for which vimp vesc .
Hit-and-run is a third way for planets near the top of the size
distribution to evolve and exchange material and momentum
during accretionary epochs, in addition to merger (which grows
larger planets, and must at some point be dominant) and
disruption (which reverses the process of accretion and must
become minor for planets to form). While the associated
processes of impact shredding may seem exotic, hit-and-run is
actually more common than merger or disruption for SSCs at
typically stirred-up random velocities. It plays an important and
perhaps dominant role in the physical and chemical evolution of
the planetary bodies that grow large, but not largest, in terrestrial
planet-forming settings.
2.6. Modeling similar-sized collisions
Canup and Asphaug (2001) conducted a systematic study of
potential Moon-forming collisions, based upon computer simulations using the smooth-particle hydrodynamics (SPH) method
pioneered for giant impact studies by Benz et al. (1988), Cameron
and Benz (1991) and others. SPH is a Lagrangian method that uses
smoothed mass elements (spherical kernel functions) to compute
205
the hydrodynamic and shock stresses and the pressure and
gravity accelerations, and to track the trajectory and evolution of
matter. We used a simple but appropriate nonlinear equation of
state (Tillotson) for iron cores and rocky mantles, and set the
random velocity to zero (vimp = vesc) in order to maximize the disk
mass while satisfying the constraints of final system angular
momentum. The impacts studied were otherwise characteristic of
terrestrial planet-forming collisions.
In the course of this search for the best case scenario for latestage Moon formation around proto-Earth, we made a few
exploratory simulations with v1 40 and found that some of
these impacting planets were ‘‘skipping’’ from the target Earth.
Indeed, the best case scenario identified for Moon formation
turned 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 less
massive than before, missing much of its mantle—resembling
Mercury, perhaps.
Fig. 7 shows frames from two simulations by Agnor and
Asphaug (2004a, b) in studies of accretion efficiency and the
thermodynamical aspects of SSCs (Asphaug et al., 2006). These are
typical of the two main kinds of hit-and-run collisions, one a
rebound and the other chain-forming. Planets before the collision
are hydrostatic, non-rotating, and start from a separation distance
5Rroche . 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. The
impact angle is 301 in both cases. The first scenario results in
widespread mantle removal from the impactor; the scenario is
postulated below as a mechanism for Mercury’s mantle loss. The
second case results in a chain of bodies the size of the major
asteroids, all of them highly diverse in major element abundances
(and most of them iron-rich), but each deriving from the same
parent body chemistry.
Research is flourishing in the area of planet-scale collisional
modeling thanks in part to a renewed focus on the formation of
Earth-like planets around other stars (e.g. Marcus et al., 2009) and
the great strides that have been made in the development of selfgravitational hydrocodes capable of evolving millions of particles,
and the computer systems to run them on. Much higher
resolution giant impact simulations are now possible including
the use of more detailed and accurate equations of state (see e.g.
Benz et al., 2007). The greatest challenge may be the accurate
modeling of smaller-scale SSCs where gravity is not the only force
to be considered; complex effects such as porosity and strength
are notoriously challenging to model (Jutzi et al., 2008, 2009).
Furthermore we have yet to adequately understand the long term
dynamical fate of collisional ejecta—whether it reaccumulates
onto the target or onto the unaccreted impactor after many orbits
about the Sun, or becomes background disk material, or is lost by
solar effects. Moreover, the effect of spin has not been explored
systematically for SSCs, or in the context of accretion efficiency;
Canup (2008) has made the first inroads in the context of Moon
formation.
2.7. Accretion efficiency
For gravity-dominated collisions SPH is well suited, even at
moderate resolution, to computing the final total bound masses
M1 and M2 where M1 is the largest collisional remnant and M2 the
second largest. In the limit of a non-collision M1 = M and M2 = m.
After a collision the binding energy of all particles is computed
with respect to the particle closest to the global potential
minimum, which serves as the seed for nucleating the total
bound mass of the largest aggregate M1. The binding energy of
remaining particles is computed with respect to this new position
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E. Asphaug / Chemie der Erde 70 (2010) 199–219
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
1
(from Asphaug et al., 2006). In each case a Mars-mass target is struck by a planet 12 (top) and 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.
and velocity, and the second largest aggregate M2 computed, and
so on as allowed by resolution.
The accretion efficiency
x¼
M1 M
m
ð11Þ
is determined to better than 10% in these simulations, where
x ¼ 1 for a perfect merger, M1 =M +m. The determination of M1 is
resolution converged using modest ( 3 104 ) numbers of
particles. That is, the answer does not change with further
increases in resolution. Nor has this basic result been found very
sensitive to the equation of state (EOS), provided it is adequate to
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E. Asphaug / Chemie der Erde 70 (2010) 199–219
model shock acceleration. The Tillotson EOS and the more
sophisticated ANEOS and SESAME EOS give very similar results
for this basic quantity of a similar-sized collision.
Modeling Moon formation using the same computational
tools is a much bolder endeavor than modeling x. Only a few
percent of the collisional ejecta end up in a proto-lunar orbit, as
satisfied only by a narrow range
of ejection velocities between
pffiffiffiffiffiffiffiffiffiffiffiffiffi
vorb and vesc, where vorb ¼ GM=a and a is the radius of the
protolunar disk that forms outside the corotation radius. The
dynamics must then be evolved by the code for at least one orbital
time in response to very near field gravitational torques. The
longer timescale and higher required precision result in a
sensitivity to the equation of state (explored in Canup, 2004)
and to radiative evolution (Stevenson, 1987) which is not
included in these SPH models.
Accretion efficiency is plotted in Fig. 8 as a function of the
random velocity v1 (Eq. (2)) for Mars-mass planets (M ¼ 0:1M )
struck by impactors m ¼ 0:01M ; 0:05M ; 0:1M (mass ratio
g ¼ m=M ¼ 0:1; 0:5; 1:0) at impact velocities vesc rvimp r 3vesc .
The data are derived from 144 simulations performed by Agnor
and Asphaug (2004a, b) including the two shown above in Fig. 7.
The results for x appear on the basis of other simulations to be
scale invariant within 10% for larger and smaller differentiated
terrestrial planetary masses (approximately Vesta sized to
super-Earth sized) so the plot can be studied as a general
result for terrestrial planet formation. The Moon-forming
giant impact scenario favored by Canup and Asphaug (2001)
plots near the upper left red triangle. In all these simulations
the spatial resolution is 30,000 particles, and the iron:silicate
mass ratio is 30:70 as a core and mantle. Planets are hydrostatic
and non-rotating prior to collision, and are placed initially at
5Rroche to allow the pre-impact tidal strains and torques to
develop.
207
Ignoring purely gravitational (tidal) collisions (Fig. 13 below), for
which y is undefined, the impact angles 301, 451, 601, 901
each represent 14 of the impact probability of collisions,
dPðyÞ ¼ 12sin2 y dy, with 01 being head-on. The velocity range covers
the expected range of collisional velocities during late-stage planet
formation. However, the simulations between 0 and 0.7vesc are
rather coarsely spaced given the sensitivity of outcomes in this
range, and need to be filled in with further simulations to better
understand the transition from accretion to hit-and-run behavior.
Also, the range of mass ratio needs to be extended to g 0:03 in
order to cover all similar-sized collisions; these results are for g ¼ 1,
0.5, 0.1. A finer resolution in impact angle would better reveal the
nature of this transition as well. But the initial observation can be
made, that there are four main branches of similar-sized collisions:
1. Efficient accretion is the common occurrence for random velocities
lower than about 0.6vesc. Damped populations accrete efficiently.
2. Partial accretion is common throughout the random velocity
range 0:722vesc . Mergers are inefficient at high velocity.
Only direct hits (01, 301) are accretionary at all for random
velocities greater than about 0.8vesc.
3. Hit and run is prevalent for the velocity range 0:7vesc
t v1 t2:5vesc , i.e. 1:2vesc tvimp t 2:7vesc . The clustering
around x ¼ 0 corresponds to impactors rebounding with little
net 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 not
accreted) and erode the larger. Catastrophic disruption,
traditionally defined as M1 = M/2, requires collisions at velocities far to the right of this plot (xcat ¼ 12M=m).
The distinction between partial accretion and hit-and-run owes to
a 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 g ¼ 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
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi
velocity v1 =vesc ¼ ðvimp =vesc Þ2 1 ¼ 1=2Y is 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 Z 451) are
effectively collisionless from the point of view of bulk mass accumulation.
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E. Asphaug / Chemie der Erde 70 (2010) 199–219
as a segregation by plotted color, where red and green dominate
the hit-and-run line
xhr 0
ð12Þ
For random velocities v1 \ 0:7vesc half of the collisions (those
Z451) are effectively collisionless from the point of view of bulk
mass accumulation. Even for relatively normal impact angles
(301) the influence of grazing is pronounced. This leads to the
most striking aspect of the plot, which is the abrupt jump to the
hit-and-run line with increasing impact velocity and impact
angle. For a broad and significant range of impact angles and
velocities there is little net mass contribution or removal from the
target.
2.8. Prevalence of hit-and-run
Hit-and-run is prevalent in systems that are moderately
gravitationally excited and less so in systems that are not. A
simple framework model is developed to trace the occurrences of
hit-and-run on the path to planet formation. The model is not
dynamical: planetesimals are given an initial random mass
(following a power law, Eq. (7)) and allowed to grow through
randomly selected pairwise collisions. The total starting mass is a
constant. 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 mass
of the largest body in the simulation (so that vesc v1 ), then it is a
similar-sized collision as defined above. Each SSC has a probability of hit-and-run (x ¼ 0) or merger (x ¼ 1). If a merger, then
the smaller body goes away and the largest becomes the sum of
the masses, and carries a mass-averaged tally of the number of
hit-and-run occurrences which starts off at 0 for all bodies. Hit
and run does not change either body in the model, but increments
the hit-and-run tally of the smaller by 1. Large colliding pairs with
1
mass ratio g o 30
are presumed to be efficiently accreted. Small
1
colliding pairs, however, in which both masses are smaller than 30
the largest in the simulation, have vesc 5v1 and so the bodies are
regarded as disrupted and removed from the population. Pairwise
collisions proceed, under the assumption of constant Safronov
number, until the final number of bodies has been reduced
to Nfinal.
Results are plotted in Fig. 9 for a moderately excited
population (v1 vesc ) for which the probability of hit-and-run is
about 50%. Hit-and-run happens commonly to most of the nextlargest bodies in a population in that case. If one mass-averages
the history of hit-and-run in the assembly of the largest final
bodies, then it is important to the largest as well. This samplingwith-replacement result is not surprising to anyone familiar with
statistics, but it may lead to a revised thinking as to how we
interpret planets and their acquisition and retention of mantles.
The scenario beginning with Ninit ¼ 10; 000 ends up with a
population of largest and next-largest objects (black diamonds)
having had 4 or 5 hit and runs each; the largest are rather
homogeneous in terms of mass and hit-and-run tally. The ‘‘late
stage’’ scenario beginning with Ninit =100 allows two bodies to
avoid hit and run by chance. The two largest are again similar in
size and tally, and accreted from bodies that had 1 hit and run on
average. The NLBs in this case are a very diverse group, some
having no hit and runs and others having 2 or 3.
Much of the material stripped off by hit and run early on may
end up back in the disk, to be accreted later. It is a challenging
study, how to model disk replenishment during N-body integrations. Nevertheless, a more physical study than the above is
possible, 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
Hit and Run Occurrences for v∞∼vesc
Hit and Run Occurrences (mass averaged)
208
10
10,000 -> 10
100 -> 10
1
1000
10000
100000
Final Planet Mass (normalized)
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.
of commodity GPU computing it is not far-fetched to imagine
running a fairly quick SPH simulation at every detected collision.
Adding mass back to the disk, providing drag, could perhaps be
done by tracking millions of collisionless planetesimals. But the
problem is extremely complex. The size-frequency distribution of
the disk planetesimals changes the random velocities of the
embryos, through stirring damped by dynamical friction, while
the random velocities of the embryos changes the size distribution of planetesimals by changing the collisional physics.
2.9. Accreted and unaccreted
Two bodies that come into close proximity once in their orbits
about the Sun, at the moderate random velocities considered here,
are likely to do so again unless their orbits are externally
perturbed. Thus, bodies involved in a hit and run collision are
expected to come back again to try once more. Given a few tries,
accretion is likely. Accretion must be overall efficient for it is
known from isotopic age dating (e.g. Yin et al., 2002) that
accretion won out after a few 10 Ma, and terrestrial planet
formation effectively ended in our solar system, in concordance
with N-body simulations that assume perfect merger. Planetary
bodies on colliding orbits, for the most part, eventually accrete.
But the surviving next-largest bodies that did not accrete onto
the largest bodies are not ‘‘the most part’’. There is a severe
selection among the surviving NLBs. As with an epic military
campaign in which nearly all of the soldiers have fallen, the
survivors have tales of miraculous good fortune to recount. Those
planets and planetesimals that participated in accretion, but by a
relatively small chance of dynamics neither merged nor were
scattered, thus remain behind as a rather exotic population of
NLBs. Planet growth proceeds apace for the largest, in the sense
that merger happens when it happens, and non-merger does not
grossly affect their mass (x 0) except at the highest relative
velocities. Even if it takes several tries, one can to first order
ignore a few unsuccessful attempts at accretion. There is some
shock and mechanical processing and impact erosion of the target,
but not much loss or exchange of matter compared to the
devastation that happens to the bounced-off impactors.
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E. Asphaug / Chemie der Erde 70 (2010) 199–219
The evolutionary tendency is towards a compositional change
that discriminates, over time, the largest from the next-largest
bodies. With each non-accretionary event the NLBs get stripped of
their outer layers, becoming increasingly iron-rich and volatilepoor. If it takes an NLB several collisions to accrete, by the time it
does merge it will tend to increase the iron fraction in the largest.
Its stripped outer silicates and volatiles go back into the disk; any
discrete parcels of this stripped material have similar dynamical
parameters and future encounters are likely with either of the
colliding pair, perhaps favoring the larger on account of its greater
gravitational cross-section. If so then the largest bodies can rob,
over time, the exterior materials from the NLBs, leading to a
compositional dichotomy between the accreted and the unaccreted.
The late stage of planet formation ends with planets achieving
stable orbits. Remnants of these final collisions are trapped and
herded into relatively stable dynamical regions (Morbidelli and
Moons, 1993), providing a snapshot in our solar system of how
terrestrial planet formation all ended some 4.6 billion years ago,
punctuated by dynamical hiccups (e.g. Chambers, 2007; Gomes
et al., 2005). By the very nature of the late stage, most of the
colliding matter is in the largest sizes; nearly all of the debris that
is produced is the result of similar sized collisions, from megacratering events (Marinova et al., 2008) to the unaccreted strands
and clumps of hit-and-run debris. The Main Belt should thus be
replete with mantle- and crustal-derived rocks from collisions
between differentiated late-stage embryos (the spray of debris
colored blue in the simulations of Fig. 7). However, stony
achondrite meteorites are uncommon compared to the great
diversity of irons, which are presumably the core relics, and
relatively few are unequivocally mantle-derived. As for spectroscopically characterized asteroids, the vast majority are undifferentiated; if there is much mantle rock in the Main Belt it is hiding.
Burbine et al. (1996) considered this puzzle and hypothesized that
because mantle rock is friable, once liberated from a parent body
it rapidly degrades into sub-millimeter sizes and is swept away by
Poynting–Robertson drag or the solar wind, or solar radiation
pressure. Iron, being more resilient, survives to produce meteorites. Asphaug et al. (2006) argue that the debris may begin as
mantle rock but undergo petrogenic transformation during
release from hydrostatic pressure, or if solid, be fragmented
in situ by the unloading stresses.
3. Departures from scale invariance
Most impact simulations are based on fully compressible fluid
dynamics computations including the calculation of shocks and
the reasonably accurate calculation of the equation of state
relations Pðr; uÞ where P is pressure and r; u are density and
internal energy of the represented material. Simulations using the
same SPH code and Tillotson EOS indicate approximate scale
invariance for equivalent impacts involving planets ranging from
Vesta-sized to Earth-sized, when the velocities are scaled to vesc.
Using a different SPH code and EOS model (and minor differences
in setup) Marcus et al. (2009) reproduce part of Fig. 8 for superEarth-sized planets, and obtain very similar results at a much
larger scale.
But similarity in computational results is not proof of scale
invariance in nature. We consider these invariances by approximating some of the geophysical complexities, considering
strength at small scales, enthalpy at large scales, and viscosity
inbetween. In addition one must acknowledge that impact
physics itself changes fundamentally as one transitions from
hypersonic collisions (much faster than the sound speed in the
rock, the largest events) to subsonic small-scale events, the
209
transition velocity being a few km/s for rocky bodies but much
slower for uncompacted bodies. Along these lines one can identify
four scenarios to explore for departure from scale invariance as
one transitions from large, differentiated, fluid planets (the easiest
to 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 well
understood for the tens of m/s velocities that may be common
among colliding planetesimals. For deformation on the timescale
tgrav of a similar sized collision, large bodies are likely to be
dominated by viscous rather than brittle deformation, first
because they are massive—their interior pressures far exceed
the stresses associated with the failure of elastic solids—and
because they retain heat and are more likely to remain ductile.
Differentiation: Closely related to the thermal state, or past
thermal state, is the transition with increasing size towards
differentiated internal structures. Colliding bodies whose iron has
segregated to the center, and whose atmosphere and volatile
condensates (oceans) have migrated to the exterior, behave
differently than undifferentiated colliding spheres. Impacts
preferentially remove the lower-density outer layers which
receive the brunt of the impact energy. Impedance mismatch at
the core–mantle boundary (Asphaug, 1997) may also enhance the
velocity of ejected materials from the outer layers. These effects
change the physics with scale, and work to segregate planetary
materials.
Shocks: Because impact velocity scales approximately with the
size of the colliding bodies (vesc in m/s is equal to R in km, for a
sphere of uniform density 1.9 g cm 3) there is a tendency for SSCs
involving \1000 km bodies to produce shocks, at impact
velocities exceeding the sound speed of their material. For giant
impacts at tens of km/s shocks can induce global melting, while
for primary accretion the collisions may be only a few m/s.
Subsonic impacts may only result in damage (if solid) or in
compaction or shear bulking (if granular; see e.g. Schäfer et al.,
2007), or splashing if liquid. Shocks are less important for very
grazing SSCs in which the bulk of the matter does not intersect.
Unloading: The lithostatic overburden pressure of a nonrotating incompressible planet of radius r is PðaÞ ¼ 23 pGr2 ðr 2 a2 Þ
where a is the distance from the center. The characteristic
pressure P0 ¼ Gr2 r 2 is released to a much lower value during
the collision timescale tcoll . The effect of pressure release can be
subtle for small bodies—it may have been expressed in the
vigorous dust production of comet Shoemaker-Levy 9 after its
tidal disruption by Jupiter (Hahn and Rettig, 2000)—and it can be
hugely important for large bodies. If one thinks of planetary
disruption as an enthalpy-conserving event, and ignores shocks,
then the change in specific enthalpy goes as r2. Moon-sized and
larger bodies have potential for widespread eruptive degassing
and hydrothermal action at global scales, even in response to
purely gravitational (tidal) collisions. Based on Earth analogues,
pressure release melting and plinian-type magmatic responses
might 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, a
cohesion or yield stress, which could be zero) beyond which
material deforms, for instance as a viscous fluid. Strength and flow
are quite complicated, but a simple approach is suitable here.
3.1.1. Strength
Strength is scale variant primarily because large objects have
bigger flaws. If one thinks of an object’s volume as sampling a
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probability distribution of possible flaws (Weibull, 1939) then
static tensile strength S expressed as the weakest flaw in a
volume, decreases approximately with size to a power S r 3=m
where m 629 for typical rocks (Grady and Kipp, 1985). This
leads to large asteroids requiring much less specific energy to
catastrophically fragment than small ones, whether by tidal or by
collisional stress. In addition strength is rate variant for the same
underlying reasons (see Melosh et al., 1992), materials being
stronger in proportion to the strain rate. It is worth noting as an
aside that ejection velocity vej from a rocky target scales
approximately with the square root of tensile strength, Y v2ej ,
because strength is a specific energy. Thus, the strength of an
asteroid’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 through
space, and atmospheric entry and terrestrial residence prior to
discovery and curation.
Catastrophic disruption requires acceleration of the fragmented materials to escape velocity. As Fig. 1 showed, large bodies are
tightly bound and difficult to disrupt even if they have no
strength. The corollary is that rubble piles are ubiquitous for
asteroids larger than some size, believed to be about 300 m
diameter (Benz and Asphaug, 1999; Holsapple et al., 2002) since
global fragmentation energy is lower than binding energy.
Beyond some further size, perhaps a few 100 km diameter, selfgravity and deformation begins to compactify the rubble
into a coherent body (perhaps with the assistance of impact
cratering) as the central pressure exceeds the compaction
strength of rubble, and as thermal deformation and melting
become important.
Cohesion, as the threshold for shear deformation, has been
studied in the context of planetary tidal disruption. Jeffreys
(1947) analyzed the gravitational disruption of planetesimals
passing inside the Roche limit of Earth, treating them not as fluid
bodies but as elastic solids subject to a deformational stress.
Because the tidal stress increases with r2, bodies smaller than a
certain size do not fragment. Jeffreys found that a monolithic
rocky asteroid smaller than a few 100 km diameter survives a
grazing encounter with Earth. This assumes that tensile strength
is not size-varying; a Weibull distribution of flaws pushes the
transition to smaller sizes, between fragmenting and nonfragmenting interlopers, and more abruptly.
As with most pioneering research, Jeffreys’ specific result was
eventually rendered somewhat moot, in this case by the
recognition over the past decade that asteroids of that size are
likely to be rubble piles. Resistance to deformation for rubble pile
objects is not measured by tensile strength, but is a complicated
granular rheology that depends on the overburden pressure and
the total stress condition. Small monolithic bodies might not
come apart by tides, but rubble piles might (Richardson et al.,
1998). Consider the imprint of tidally disrupted comets upon
Jupiter’s satellites—or rather, the unexpected absence of such
imprints from disrupted parent bodies smaller than about 800 m
(Schenk et al., 1996; see Fig. 10). There are more than a dozen
larger records of tidally disrupted comets, and none smaller. It
may be that Jupiter-family comets are structurally competent at
some small value but that once their cohesion is exceeded they
disaggregate and behave rather like a fluid (Asphaug and Benz,
1996). The abrupt transition from solid to fluidized behavior is a
common aspect of granular materials, and may be pronounced
under microgravity conditions. The structural competency of
Jupiter-family comets (JFCs) overall appears to be comparable to
the jovian tidal stress, which across such a small body is not much
greater than that of a dry snowball. In modern times comet
Shoemaker-Levy 9 disrupted at Jupiter, an event matched by a
1.6 km diameter progenitor of bulk density 0.6 g cm 3 (Asphaug
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.
and Benz, 1994) and strength 100 dyn cm2 disrupting as a
fluid body during a few hours inside the Roche limit.
If small planetesimals behave somewhat as self-gravitating
granular fluids during randomly stirred planetary encounters,
then perhaps SSCs are dynamically similar from scales ranging
from giant impacts to planetesimals the size of small hills.
Detailed numerical models (Schäfer et al., 2007) and laboratory
experiments (Wurm and Blum, 2006; Dominik et al., 2007) of
colliding aggregate bodies give various results, not always
consistent, showing how crushable or fractal solids can behave
in unexpected ways. Polyhedral rubble pile models by Korycansky
and Asphaug (2006) illustrate the transition from individual to
ensemble behavior and the phenomenon of granular collapse in a
self-gravitating system. There is much to be learned about this
‘‘mesoscale’’ of accretion, where there is no well-understood
characteristic stress or characteristic size.
3.1.2. Viscosity
If colliding planets are large enough, or molten, then strength
and cohesion are effectively zero and the bodies are compact. The
above complexities go away and what remain are the equation of
state (EOS) and the deformation rheology. Concerning the latter,
the easiest approach is to consider a linear Newtonian viscosity; if
the planet resists deformational shears on the collision timescale
then it will not disrupt.
Newtonian viscosity is the ratio of stress to the strain rate
Z ¼ s=e_
ð13Þ
A similar sized pcollision
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi takes place over the gravitational
timescale tgrav ¼ 3p=Gr (Eq. (6)), a couple of hours for density
corresponding to terrestrial planet-forming materials. The strain
rate 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 characteristic stress is
s P0 Gr2 r2
where r is the radius of the disrupted body.
ð15Þ
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E. Asphaug / Chemie der Erde 70 (2010) 199–219
211
shocks or damaging stress waves accompanying the collision
further act to fluidize a small body (Asphaug and Melosh, 1993).
All in all, it appears that similar sized collisions larger than a few
100 km in scale, and perhaps as small as 10 km if actively
heated internally, can be modeled using an inviscid approach.
3.2. Differentiation
Fig. 11. Bodies of viscosity Z that are larger than rmin can deform to a strain e
during 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 e ¼ 1, while e ¼ 100 is catastrophically disruptive. The bulk density r ¼ 4 g cm3 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.
The smallest impactor rmin that can come apart in an SSC in this
viscous limit is found to be
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
¼ Ze= 3pGr3
rmin
ð16Þ
by requiring the deformation rate e_ ¼ e=tgrav to be accommodated
by 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, for
terrestrial bodies of bulk density r 4 g cm3 . Strains of 1, 10, and
100 are plotted to bracket catastrophic disruption. Strain e \ 10 is
allowed when viscosity is less than
Z o Zmax 1013 ðr=1000 kmÞ2
ð17Þ
where units are poise (g cm 1s 1). For comparison, mantle
viscosity is thought to be 109 poise in convective models of
early Earth (Walzer et al., 2004) and Z 109 21013 poise in Io
asthenosphere models (Tackley, 2001). Partially molten Moonsized planets (Z C 1014 poise) can be approximated as fluid
bodies, as can impactors as small as 10 km that are molten
(Z 5108 poise). Because Zmax increases pr 2 , while Z decreases
sensitively with size due to heat retention, the transition from
viscous-limited to inviscid behavior is likely to be abrupt across
some size threshold rmin.
A number of effects make the response to similar-sized
collisions fluid for bodies smaller than are plotted in Fig. 11.
Rocks, partial melts and magmas are nonlinear fluids, whose
effective linear viscosity decreases with a power of the stress. This
means that viscosity Z s=e_ is much lower for larger-scale
collisions because overburden stress P0 is larger by r2. Also,
pressure release melting acts to lower the viscosity during tidal
pressure unloading, of great significance to partially or nearly
molten planets. The exolution of volatiles during pressure
unloading can initially act to increase the viscosity of a magma
by stiffening it with bubbles, but at high enough deformation and
degassing rates the low viscosity of the gas wins out, and bulk
viscosity plummets (see e.g. Gardner et al., 1999; Alidibirov and
Dingwell, 1996). Interesting textures are expected. And lastly, any
Viscosity’s exponential dependence on 1/T leads to a consideration for the heat sources available in planets before, during,
and after a similar-sized collision, and to the behavior of
differentiated (previously or presently melted) bodies versus
undifferentiated bodies.
During the primary phases of planetesimal formation, thermal
energy is available from many sources, including impact shocks
(from turbulence or infall), nebular shocks, solar heating, and the
decay of short-lived radionuclides (see for instance Wasserburg
and Papanastassiou, 1982). Burts of intense heating from an early
sun would lead to some silicate melting in the innermost disk, but
this heat source is not believed to be volumetrically important for
terrestrial planet formation. Nebular heating by shocks can be
intense, and shocks are proposed by Desch and Connolly (2002) to
be responsible for the melting of silicate chondrules. The range of
chondrule ages then requires that the solar nebula was present for
several million years and that substantial gas and dust were
present. Nebular shock heating would be prevalent and then
diminish rapidly in pace with the accretion of planetesimals, due
to the clearing out of the gas and dust which carries the shocks. As
for impact shock heating, this is certainly a significant heat source
during collisions in the late stage of giant impacts, but not during
primary accretion when velocities are slow, following the initial
stage of infall onto the disk. Compaction heating of porous
cohesive aggregates during infall, or during the earliest growth
within the disk, may be a relevant precursor to primary accretion.
But hypervelocity collisions cannot contribute to the bulk melting
of low-gravity bodies, simply because any melt products are
shock-accelerated and escape.
It is now generally acknowledged that the most important heat
source for thermal processing during primary accretion was, in
our solar system, the decay of 26Al - 26Mg, a radionuclide with
half-life t1=2 ¼ 7:2 105 yr (Bizzarro et al., 2005). While the origin
of 26Al is debated, its original abundance in our solar system is
measurable in meteorites. For chondrites the initial 26Al/27Al
C 5 106 , whose decay over t1=2 releases several times more
heat (in erg g 1) than required to bring cold, dry dust to the
melting point. Other short-lived radionuclides, notably 60Fe, were
trapped in early-forming rocks in our solar system, also longspent but evidenced by their daughter products; their heat
production is not believed to be as significant as 26Al in our
own solar system.
The prevalence of 26Al in other solar systems is unknown; this
is a critical piece of missing knowledge since its presence or
absence has the potential to dramatically alter the mechanism of
primary accretion. A solar system with a small fraction of our
measured 26Al might not produce enough heat for its early small
bodies to melt. If small planetesimals remain unmelted, even as
they grow larger than 100 km, then the character of their
impact coagulation might change, conceivably even shifting the
growth of planets away from the terrestrial planet-forming
region, or biasing the favored sizes of finished planets.
The rate of heat production dq=dt from radionuclide decay is
proportional to the planet’s mass 43pr 3 r. Thermal energy
dissipates by conducting through the solid and radiating from
the planet’s surface area 4pr 2 at a temperature T, per unit area
with a blackbody flux sT 4 . The heat produced, divided by heat
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radiated, thus increases with r, and increases greatly with
temperature. Large planets thus attain radiative equilibrium at
higher temperature and get hot enough to melt. According to 1D
thermal modeling by Merk et al. (2002), a 10 km diameter
homogeneous chondritic body melts if it accretes much faster
than t1=2 (see Ghosh et al., 2006).
Timing is everything, and so is location: planetesimals forming
closer to the Sun acquire a greater fraction of Al-bearing silicates
than those forming where ices dominate; they also accrete much
more rapidly. They are thus much more prone to melting. The
transition from fluid to solid behavior, plotted in Fig. 11 in terms
of planetary radius, might be thought of as a transition in time
from fluid to solid behavior, for the half-life of 26Al is—interestingly enough—comparable to the timescale of primary accretion.
3.3. Energy of collision
Catastrophic disruption is defined as leaving a target body with
no more than half of its original mass intact or gravitationally
bound. In the target-centric view of things, the characteristic
threshold of catastrophic disruption is traditionally expressed in
terms of the specific impact kinetic energy 12 mv2imp per unit target
mass M:
1
Q ¼ mv2imp =M
2
ð18Þ
The disruption threshold Q* is the value of Q at which the final
largest remnant M1 =M/2. For SSCs where m M the specific
energy of impact must be defined as Q ¼ 12 mv2imp =ðM þ mÞ. One
might then by analogy want to define Q* as forming a largest
remnant with half the combined mass, M1 ¼ ðM þmÞ=2. This is
inadequate for SSCs, since two just-grazing, equal-mass planets
would each have half the total mass for any impact velocity. We
must think of disruption in terms of the fates of both bodies.
The impact kinetic energy per unit mass Q v2imp v2esc
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 transformative
for the next-largest bodies in an accreting terrestrial planetary
system. The tidal stress on the smaller by the larger, compared
with the tidal stress on the larger by the smaller, is of greater
magnitude in the smaller body in inverse proportion to its mass.
In the case of direct collisions, the contact stress wave or shock
wave is generated symmetrically about the contact front of a
colliding pair, so that energy is partitioned equally into both
bodies; energy density is also inversely proportion to mass. As for
the impact differential stress, this can be thought of as the
differential deceleration across the diameter of each colliding
body, for instance in an off-axis SSC where half of the colliding
mass is abruptly decelerated and half is not. If the contact forces
are symmetric, then the smaller body decelerates more abruptly
than the larger in inverse proportion to its mass.
And so, as a rule of thumb, when planets of sizes r t R collide,
then the specific tidal, gravitational, shear and shock stresses felt
by each body scale inversely to the mass ratio g ¼ m=M. In the
case of the Moon-forming simulation of Fig. 6, the impactor, being
an order of magnitude less massive than the target, suffered an
order of magnitude more damage, expressed as the gravitational,
mechanical and shock energy of collision per unit mass. Whether
shocks, tides or shears dominate an impactor’s disruption is a
function of geometry, from just-grazing (where tides dominate) to
head-on (where impact stresses and shocks dominate), and of
scale. In an isolated planetesimal swarm with no bodies larger
than 1000 km, random speeds are generally subsonic. But once
Moon-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
3
UB ¼ GM 2 =R
5
ð19Þ
is the energy required to disassemble a planet gently to infinity,
and thus represents the lower limit of the kinetic energy
12 v2esc dm 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 1 þ 1=2Y. This
energy dissipates as heat, through shock and friction. For a planet
the size of Mars, the gravitational binding energy is
8 1010 erg g1 . Assuming Safronov number Y 1 and dividing by the heat capacity of rock (cp 8 106 erg g1 K1 ) gives an
estimate for the temperature increase due to impacts, of order
20,000 K (in which case constant heat capacity is not the right
assumption). For a Vesta-sized planet, the same calculation gives
an impact heating of only 100 K. Accretional heading of smaller
asteroids 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, and
phase change. The specific enthalpy of a planetary interior is
h ¼ u þP=r
ð20Þ
where P=r is the pressure times the specific volume V ¼ 1=r.
Changes in enthalpy drive reactions:
dh ¼ du þP dV þ V dP
ð21Þ
It is useful to think of SSCs in terms of enthalpy for the same
reasons that enthalpy is the guiding state variable for the
modeling of rising magmatic conduits (e.g. Wilson and Keil,
1991; Gardner et al., 1999). During large-scale similar sized
collisions, the drop in pressure V dP as the planet comes apart
leads to an abrupt change dh over the timescale tgrav , and this can
drive various reactions forward.
Hydrostatic pressure P increases with more than the square of
a planet’s radius, since rocks are compressible. Holding the
timescale of collisions tcoll tgrav as invariant in SSCs, both the
pressure release DP and the rate of pressure release dP/dt scale
greater than R2. The high magnitude and rate of unloading during
disruption can drive cavitation, the dissolution of volatiles, bubble
nucleation and coalescence, magmatic forcing, and pressure
release melting.
This causes us to look at the volcanic eruption modeling of
Gardner et al. (1999) and others. Pressure unloading and its
timescale for Vesta- to Mars-sized SSCs is comparable to
strombolian-type eruptions, which are driven by gas bubbles
rising faster than the surrounding melt. Measurements at
Stromboli volcano in Italy (Burton et al., 2009) show gas slugs
originating at 3 km depth. These same pressures are attained in
the middle of a 500 km diameter planetary embryo. One might
reason that strombolian eruption physics might be directly
relevant to the degassing of large molten planetesimals in the
aftermath of hit-and-run collisions, and their magmatic fragmentation (Alidibirov and Dingwell, 1996).
As pressure is released, uncompensated pressure gradients
result in accelerations that can change the dynamics of disruption,
possibly contributing to the shedding of material or the
emplacement of a debris disk. Pressure gradients played a role
in the formation of the protolunar disk (Stevenson, 1987), and the
disk formed not only from material released from impact shock
pressure, but also from material that released from a high
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pre-impact hydrostatic state P0. Most of the disk-forming material
in the Moon-forming simulations (e.g. Fig. 6) emerges in the
course of a few hours from several tens of kilobars of pressure, so
even apart from the shock, the thermodynamic path is of critical
importance.
3.4.1. Tidal collisions
The best way to appreciate the physics of unloading is to
isolate it in a purely tidal collision b \r þ R, so that the smaller
planet does not hit the larger, but still comes deep inside the
Roche limit
Rroche ¼ 2:423RðrR =rr Þ1=3
ð22Þ
Rroche is the threshold distance from a planet of radius R, density
rR where an incompressible small fluid spheroid r, rr on circular
orbit will be disrupted (Roche, 1850). The Roche limit is not
strictly applicable to a parabolic or hyperbolic encounter, for
which tidal disruption requires a somewhat closer periapse
(Sridhar and Tremaine, 1992). For small strengthless incompressible bodies encountering massive planets at v1 ¼ 0, Asphaug and
Benz (1996) found that periapse inside 0:69Rroche is the
threshold for the shedding of matter, in agreement with the
analytical result of Sridhar and Tremaine (1992); they found that
passage inside of 0:55Rroche along a parabolic (v1 ¼ 0) trajectory
results in the stripping of half the mass from the outer layers.
The analysis breaks down for similar-sized tidal collisions
because r R. The impactor is extensive, and part of it is on
collision course when b r þ R. But ignoring this, if rR ¼ rr then
0:55Rroche C 1:3R, so that to first order we expect catastrophic tidal
disruption when a planet r t 0:3R is on grazing parabolic
incidence. Fig. 12 summarizes purely tidal encounters from the
context of small spherical bodies of density 0.6 g cm 3 (‘‘comets’’)
and 3 g cm 3 (‘‘asteroids’’) encountering planets of various
density. It is seen that in the limit of r 5R, a catastrophically
213
disruptive tidal collision is about half as likely between an
asteroid and Earth, as a physical collision. For comets tidal
disruption is 50% more likely than a collision. For equal density
bodies on parabolic encounters with r 5 R, tidal catastrophic
disruption (M2 = m/2) is 13 as likely as collision.
For studying compressible, differentiated planets undergoing
tidal collisions, a self-gravitating hydrocode such as SPH remains
the 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 previous
simulations, and the same Tillotson equation of state,
encounters a Mars-size (0:1M ) impactor. The impactor is
initially a non-rotating, isostatically equilibrated sphere. There is
no physical contact, only gravitation and pressure unloading, so
the target planet has been represented in the simulation as a point
mass. The closest-approach velocity is 1:05vesc and the closest
approach distance is b= 1.05(R+ r), and the bodies are of equal
density. The two planets are represented in the center of mass
frame, so that the larger (not shown) is displaced towards the top
of the figure in each time step, while the smaller swings from the
right towards the bottom and is severely deformed and stripped.
The result of this slightly hyperbolic gravitational encounter is
mass loss, spin-up, and global pressure unloading. The pressure
drop is plotted in Fig. 14, which shows pressure inside the
impactor normalized to the initial central pressure Pc ¼ 23 pGr2r r 2 ,
measured at the center of the planet (black) and at the core–
mantle boundary (grey). Pressure unloading begins about half an
hour before periapse, as the larger planet’s tidal field competes
with the smaller’s self-gravity, and reaches a maximum of
DP=P0 40250% about half an hour after the event. The
unloading at the center takes slightly longer to complete, as the
signal from surface mass removal propagates inward, and goes on
for an hour after periapse. Permanent unloading by 20% results
Fig. 12. Tidal disruption of small incompressible spheres of density rc ¼ 0:6 g cm3 (‘‘comets’’, left) and ra ¼ 3 g cm3 (‘‘asteroids’’, right) during parabolic (v1 ¼ 0)
encounters with planets in our solar system: r ¼ 5:52 g cm3 (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 =m ¼ 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.
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Fig. 13. Tidal collision with closest approach velocity vimp ¼ 1:05vesc and closest approach distance b ¼ 1:05ðRþ rÞ, just beyond grazing; see Fig. 14. The impactor is Moonsize (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 cm2
and blue t107 dyn cm2 . 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).
largest remnants is 90%, although the prelude to this pressure
drop is a devastating shock-inducing collision. The encounter
resembles two core bodies interacting gravitationally in a tidal
collision, with mantle coming along for the ride; however
the physics once fully explored is unlikely to be that simple.
Fluid instabilities along the accelerating density boundary
between the core and mantle may turn out to be as important
as self-gravitational instability, in disrupting impactor cores;
clues to the process are sought in the petrology of iron–silicate
meteorites.
The fate of undifferentiated bodies in response to disruptive
tidal collisions is also complex. Shear localization is expected to
generate frictional heating in a gravitational collision such as
Fig. 13. The larger the planet the more energetic this frictional
heating, producing melting (pseudotachylites) along shear planes.
Planetesimals that are partially molten could melt entirely, or
locally, as they cross the phase boundary during unloading from
hydrostatic pressure; this could trigger core differentiation and
initiate degassing.
4. Discussion
from spin-up (the final body is rotating with a period 6 h) and
mass loss. Faster hyperbolic tidal collisions involve unloading of
shorter duration.
In a hit-and-run collision with b or þ R, pressure unloading is greater, as the periapse is closer, but shock and
collisional shearing play an increasing significant role as
the fraction of intersected mass f increases. For disruptive
hit-and-run collisions such as Fig. 7 d–f, the pressure unloading
in the disrupted fragments is nearly 100%, because the material is
now found in bodies with much smaller radii than before. In the
case of Fig. 7d–f, the pressure drop experienced by material in the
Earth finished accretion by mopping up dozens of Ceres- to
Mars-sized bodies, becoming an amalgam of numerous smaller
feeding zones (e.g. Chambers and Wetherill, 1998). One of these
giant impacts formed the Moon, which is the archetypal
unaccreted body. That is to say, the Moon is the unaccreted
(though gravitationally bound) remnant of a highly selective
process we call, as a whole, pairwise accretion, but which in fact is
an ensemble of processes that can be seen in Fig. 8 to fall into four
broad categories: (1) efficient accretion, (2) partial accretion, (3)
erosion and disruption, and (4) hit-and-run.
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Each has a unique outcome. Efficient accretion buries the
evidence, resulting in isotopic and compositional homogeneity if
the colliding planets are molten, or become molten as an outcome
of the collision. It can also evidently lead to the formation of
massive debris disks and major satellites like the Moon. Efficient
accretion at the smallest scales might preserve the impacting
bodies, for instance the layered-pile model for comets proposed
by Belton et al. (2007). Partial accretion accretes higher density
materials and loses lower-density materials, so that a planet that
grows in this manner grows denser (iron rich and silicate and
volatile poor) over time. Erosion is a continuum of the partial
accretion curve in Fig. 8, in that a very inefficient partial accretion
event is identical in process to an erosional collision, preferentially removing the outer materials which receive the brunt of the
blow, and merging the inner materials; one is a net gain and the
other a net loss. Disruption is in continuum with erosion.
Catastrophic disruption requires impact energies far to the
right of the plotted simulations. Except, that is, when one focuses
on impactors that are disrupted by the targets that they strike,
vice-versa. These hit-and-run planetary collisions are relatively
newly 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 of
excitation of the accretionary system. In the regime where
random velocities are several times faster than escape velocity,
hit-and-run can largely be ignored since disruption and erosion
are more likely to dominate (the case in the present asteroid belt,
for instance) and cannot lead to the growth of planets. In the
regime where random velocities are close to zero, hit-and-run is
an exotic occurrence and perfect merger is the norm. But for
velocity regimes inbetween, they are prevalent, so I conclude by
venturing some specific contexts for hit-and-run.
4.1. Embedded embryos
Wetherill (1994), in a series of groundbreaking papers
establishing late stage planetary collisions, conducted Monte
Carlo integrations of planet accumulation and found among other
things that planets the size of the Moon and Mars may well have
roamed the Main Belt for a few tens of Ma, until being scattered
by mutual gravitational interactions and by resonant interactions
with the giant planets. This opened up a new way of thinking
about terrestrial planet evolution, and the origin and evolution of
asteroids and the context of meteorites. Chambers (2007)
postulated on the basis of N-body integrations that a planet
might have survived between Mars and the inner Main Belt for
hundreds of millions of years, until it became destabilized by
chaotic interactions, as a candidate dynamical mechanism for the
late heavy bombardment (LHB), the greatly enhanced impact flux
recorded in lunar samples dated 3:924:0 Ga before present
(reviewed in Chapman et al., 2007) and which may have been
solar-system-wide. A planet near the Main Belt, once destabilized,
would stir up the asteroids and greatly enhance the flux of
impactors striking the terrestrial bodies. Petit et al. (2001) studied
the primordial excitation and clearing of the Main Belt and found
that Moon- to Mars-sized bodies would persist there for at least
the first 10 Ma or longer. O’Brien et al. (2007) conducted similar
N-body integrations and found that embryos of roughly a lunar
mass could remain among the asteroids for up to the time of the
LHB, to be scattered away along with the majority of asteroids,
thus contributing to the excitation and loss of asteroidal mass
until the larger embryos themselves were lost.
This is all to say that the asteroids we see today in the Main
Belt, and the somewhat larger progenitors from which they
derived, are widely believed to have evolved early on, perhaps
215
even for the first 600 Ma until the LHB, in the presence of
Moon-sized or larger embryos. The dynamical environment was
excited by these embedded embryos, and thus was likely to have
involved moderate random velocities v1 vesc . For large asteroid
progenitors encountering these embedded embryos, hit-and-run
would have been prevalent according to Fig. 8. If the scenario of
embedded embryos is correct, then guided by Fig. 9 we can
conclude that the Main Belt should be replete with hit and run
survivors.
In the outer solar system, where chaos during giant planet
migration may have scattered as many as 10 Earth-masses of icy
bodies beyond Neptune, of which perhaps 0:1M remains today
(Levison and Morbidelli, 2003), the scenario is characteristically
similar—almost total mass depletion that would include the loss
of an ancestral population of somewhat larger bodies. The density
ratio of rock:ice is similar to that of iron:rock, and thus, in lieu of
suites of simulations for self-gravitating planetary bodies of ice–
rock composition, the regimes of similar-sized collisions for icy
bodies might be comparable to those shown in Fig. 8. If so, then
among the differentiated bodies of the outer solar system, a
number are expected to be stripped by hit-and-run collisions in
the manner just described for the Main Belt. The dwarf planets
beyond Neptune appear to have highly varying density, ranging
from ice-like to rock-like (most recently, Fraser and Brown, 2009),
and the relative velocities are too slow to allow for cataclysmic
mantle stripping (e.g. Fig. 1). This diversity of ice:rock ratio is
consistent with the idea that they, like their cousins in the Main
Belt, underwent a late stage of growth involving hit-and-run
collisions.
4.2. Recycled planetoids
Hit-and-run events can lead to the formation of chains of
recycled planetoids, the largest of them a few times smaller than
r, as depicted in Fig. 7 d–f. The remnants in this particular
simulation include about a dozen major bodies which form when
a differentiated Moon-sized body strikes a Mars-sized body, at 301
from normal, at a velocity vimp ¼ 2vesc . The results are scale
invariant given the caveats above. If we close our eyes to the rocky
mantles of the colliding planets and consider only the cores, then
the appearance is that of a planetary tidal disruption event such
as comet Shoemaker-Levy 9 and its post-perijove ‘‘string of
pearls’’ (Asphaug and Benz, 1996)—a linear structure with rather
equant size and spacing. The intensely varying gravitational
stresses, acting over tgrav , appear to play a dominant role in
pulling the projectile core into pieces. As the core is deformed,
instabilities at the rapidly shearing core–mantle interface may
also play a major role in establishing fragment size and spacing.
Because 23 of the colliding mass is in mantle silicates, whereas
the iron cores appear to have their own intense tidal interaction,
the event is very complex gravitationally, mechanically (core
blobs plowing through the mantle rock), and in terms of the
intense shock evolution, as the mantles respond to the direct hit.
A layer of mantle silicates ranging from a dominant fraction to a
thin veneer remain gravitationally bound to the blobs of core material
that emerge from the collision. These new small planets are recycled
from selected volumes of their parent planet materials, and have iron
and silicate components that would not make sense in the context of
standard models of planet formation and evolution. The major
fragments are significantly iron-enriched, and the smaller debris are
mostly of surface or mantle composition. At these relatively low
numerical resolutions (tens of thousands of particles) more nuanced
aspects of the collisions are unresolved. Higher fidelity simulations
made possible by parallel SPH codes running on commodity
supercomputers shall give a far better understanding of the major
compositions, thermal and shock histories, spin states, and other
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aspects of the recycled planetoids, and to the long term fate of the
escaping material. It is a complex phenomenon involving impact
physics, tidal physics, solar system dynamics, meteoritics, and for
internally molten planets, volcanology—the task is not just improving
model resolution but also the model physics. But for now, a few
observations can be made:
The final bodies in a chain-forming hit and run collision are
composed of deep interior materials that find themselves, in the
course of hours, at greatly reduced hydrostatic pressure P0. This
creates a potential for volumetric degassing and the fluxing of
water and other previously dissolved volatiles in response to
intense, abrupt pressure disequilibrium. Shock intensity is highly
variable, since they form from materials at a considerable distance
from the collisional contact surface. So in general one expects a
wide variety of metamorphic and igneous evolution to occur in
this small stretch of time, leading to diverse pathways for
meteorite petrogenesis.
Regarding iron and iron–silicate meteorites, the surface area of
iron to silicate is greatly increased during a chain-forming hit and
run event, since the original impactor core is parceled into a
dozen or so new planetoids, and is sheared against the silicate
interface as it happens. When a core body is sheared into
multiple components, shear localization is expected to occur
along the core–mantle interface, causing detachment and intense
friction. Thus, the prevalence of mesosiderites, pallasites, and
other iron–silicate mechanical mixtures, and their diversity, is
consistent with this type of parent body mechanical-driven
evolution.
Shear localization does not require a core–mantle interface;
evidence for pseudotachylitic clasts and textures in ordinary
chondrite meteorites (van der Bogert et al., 2003) may indicate
the mechanical shearing of undifferentiated planetary bodies as
well, perhaps during smaller-scale early-epoch SSCs at lower
velocity. Indeed while most of this review has focused on the
implications of SSCs for differentiated bodies, for which the
before-and-after change is the most spectacular, the process
applied to undifferentiated bodies can lead to various kinds of
evolution ranging from hydrothermal action, to pseudotachylitic
modification, to brecciation.
Direct evidence for a chain-forming hit-and-run collision is
perhaps recorded in the IVA iron meteorites, one of the 14 major
meteorite groups. Each iron meteorite group corresponds to an
original reservoir (planetesimal core) that is a unique alloy of iron
and nickel, plus trace metals such as gold. Thus the 14 major iron
meteorite groups represent 14 parent bodies, while there are
hundreds of unclassified iron meteorites, bringing the total
number of disrupted parent bodies represented by iron meteorites up to 50–100. One immediately must ask: How is it that so
many large differentiated asteroids underwent catastrophic
breakup—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 2
orders of magnitude, from 19–3400 K/Ma. Yang et al. (2007)
report revised cooling rates for this group spanning 100–6000
K/Ma, and show a trend of faster cooling rate for lower bulk Ni. To
satisfy the most rapid of these cooling rates, and to conform to the
bulk Ni data, Yang et al. develop a model where the IVA
meteorites cooled within a 300 km diameter metallic body
that was stripped bare of its mantle. Mantle removal is required
by the most rapid cooling rates, simply because an insulating
mantle would cause the iron core to cool slowly, and under nearly
isothermal conditions. The wide span in cooling rates within a
single stripped core body remains problematic in this scenario,
but it is consistent with the cosmic ray exposure ages that
indicate cooling within a single final body.
Stripping a Vesta-sized mantle bare is also possible in a chainforming hit-and-run collision; also recall that repeat hit-and-runs
may have occurred (Fig. 9). Diverse cooling rates are an expected
result when a core is pulled into a chain of new planets, each with
its own core, its own mantle (or lack thereof), its own diameter
and cooling rate. The scenario of Fig. 7 d–f would lead to the
formation of a handful of new bodies each with the same major
elemental and initial isotopic core composition; each would
subsequently follow its own evolution as a minor planet. Iron
meteorites derived from these bodies would come from the same
parent body, compositionally, but from different bodies in terms
of their solidification, cooling, and post-formation physical and
chemical history, and cosmic-ray exposure ages of their resultant
meteorites. As for the silicate portions of these strung-out new
planets, there would be much greater variation in composition
owing to the initial shock state, varying provenance within the
initial projectile’s mantle, mixing with the target mantle, and
volatile dissolution.
As for that broad and most controversial topic of early solar
system origins, the formation of chondrules, it is certainly not farfetched to consider that hot, possibly molten bodies tens to
hundreds of km diameter would have undergone collisional
evolution at random speeds comparable to their vesc, in the first
few Ma when 26Al was active. If so, then a similar-sized collision
scenario for chondrule formation is worth considering, in which
molten asteroids are torn apart once or twice for every efficient
accretion, as part of the inefficient process of accretion during
random stirring. If hit and run collisions happen to a given parcel
of matter many times over as Fig. 9 suggests, and if those parcels
are molten and gas-rich, then pressure unloading would result in
the dispersal of material over tgrav hours, under phreatic
conditions. It can be thought of as an evolution of the chondrule
formation hypothesis of Sanders and Taylor (2005), but with
pressure unloading and the associated droplet formation physics
and timescale taking the place of impact splashing. It is
dynamically and chemically plausible, and recommends further
research into the earliest thermophysical processing of terrestrial
planet-forming materials during similar-sized collisions.
4.3. Mercury and Mars
Mercury is anomalously iron-rich, about 70% by mass. Benz
et al. (1988, 2007) showed that a giant impact with random
velocity 6vesc by an projectile r R=2 (depending on impact
angle) is capable of shock-accelerating half of Mercury’s mantle to
escape velocity, in an intensely energetic collision bearing over 30
times the specific energy of the giant impact proposed to have
formed the Moon. One of the challenges to the hypothesis is
compositional, for there is geophysical and spectroscopic evidence (for instance Kerber et al., 2009; Sprague et al., 1995) for
volatiles in the crust of Mercury in much greater abundance than
on the Moon. This could be challenging to reconcile with the
hypothesis of the planet finishing its evolution by having its
mantle shocked and dispersed into space, and part-reaccumulated. Benz et al. (2007) show that the ejected material, which
goes to occupy a torus around the Sun centered on Mercury’s
orbit, is fragmented into sizes small enough for most to be
removed by Poynting–Robertson drag; the rest reaccretes to form
the new upper mantle. However, Gladman and Coffey (2009) find
that the opacity of this debris cloud will severely limit the rate of
mass removal (and incidentally, will put the other planets into
total shadow). We knew from the start (Fig. 1) that it is not easy to
blast off a planetary mantle. If the event requires a collision that is
anomalously fast and energetic, it seems odd that the lowest
possible impact velocity (parabolic encounter) is invoked to
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5. Conclusions
1
0.8
60
M2/m
0.6
45
0.4
0.2
30
0
0.5
1
1.5
2
v-innity / v-escape
2.5
3
Fig. 15. Mass of the largest impactor remnant (M2 r m) after a hit-and-run
collision, for the g ¼ 0:1 (m ¼ 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 23 its mass
in a hit-and-run, a conjecture to be studied further for the origin of Mercury.
explain the Moon’s formation, whereas an anomalously hyperbolic impact (vimp b vesc ) is invoked to explain Mercury, both
during the same late stage of solar system history.
By directly scaling from Figs. 8 and 9, an alternative giant
impact scenario is proposed that would remove Mercury’s mantle
at a characteristic impact velocity v1 vesc , assuming Mercury
was the impactor rather than the target. What did it run into?
Wetherill (1992) showed that Mercury could have originated
beyond Mars, leading to the possibility that it may have
encountered proto-Earth or proto-Venus. Perhaps more dynamically probable is for Mercury, under quieter circumstances, to
have become sufficiently eccentric to encounter Venus or protoVenus in the course of later chaos (perhaps swapping its volatiles
over time onto the larger planet as described earlier).
The basic hypothesis is that Mercury started out rather Marslike and collided with a larger planetary embryo along the way,
emerging from the hit-and-run as a planet of roughly Mercury’s
mass and composition, in a process akin to Fig. 7a–c. In this
simulation the final impactor lost 35% of its original mass
(M2 = 0.65 m) which equals half its mantle. Largest remnant
masses (normalized to impactor mass) for the simulations plotted
in Fig. 8 were computed by Agnor and Asphaug (2004a, b) for the
simulations described above and are plotted in Fig. 15. Mercury
may thus have formed more closely akin to how the Moon
formed: a Mars-mass planet running into a larger planet,
but in Mercury’s case a fraction faster. Still, we must keep in
mind that Mercury’s bulk composition may not be defined
by any single giant impact. For instance, if somewhat higher
random velocities v1 =vesc existed near Mercury throughout the
late stage, this might account for a systematic increase in
disruptive and hit-and-run collisions (e.g. the black diamonds in
Fig. 9) leading to a hierarchical evolution towards iron-rich
composition.
Numerical simulations have shown that impacting bodies of
similar size can experience hit and run collisions for the random
velocities typical of late stage planet formation. This causes us to
consider giant impacts from the surviving impactor’s perspective.
Hit-and-run is an efficient mechanism for dismantling the smaller
of a colliding pair, removing its outer layers and causing global
transformations, while leaving the target body comparatively
intact.
If Main Belt bodies Moon-sized or larger once existed, as is
expected on the basis of dynamical studies, then Main Belt
asteroids ought to be replete with hit and run collisional relics.
Vesta, whose crust appears to be largely intact, may have been
lucky to avoid a major hit-and-run, but suffered a typical
bombardment by smaller impactors. This scenario requires only
a moderate anomaly for Vesta, compared to the alternative, that
Psyche, Kleopatra, and the parent bodies of the dozens of families
of iron meteorites were beaten down to their cores by erosive and
disruptive impacts, with Vesta somehow dodging a cosmic
fusillade. If Vesta avoided a hit and run collision until the larger
bodies left the Main Belt, while Psyche was dismantled to its core,
perhaps hit-and-run evolution happens to about half the NLBs in
late solar system history. For the small terrestrial planets, we
might in this context look at Mars as a body which has always
been among the very largest of its collisional population,
efficiently sweeping up smaller planetesimals and embryos and
being little bothered by hit and run collisions. We might then look
at Mercury as a typical next-largest body, a hit-and-run remnant
having lost its mantle.
Meteorites show evidence for hit and run collisions. A hit-andrun collision can account for the wide range in cooling rates
exhibited within one or two families of iron meteorites (Yang
et al., 2007). Chondrite meteorites show the kinds of frictional or
even pseudotachylitic textures that are expected during similar
sized collisions, although hit-and-run is difficult to discern among
undifferentiated planets where there is no core–mantle segregation. These frictional and breccia textures, and evidence for
altered and metamorphosed silicate bodies (Keil, 2000), and the
overall stunning variety and relative abundance of iron and iron–
silicate meteorites, are not direct evidence for hit and run
collisions per se, but are certainly indicative that similar sized
collisional processes have been at work, along with the associated
processes described above.
Chondrites, the most abundant meteorites, derive from the most
common parent bodies in the Main Belt. They have not been
subjected to temperatures close to melting since the time of
chondrule formation, a few million years after solar system formation.
Chondrules, which might account for over half the mass of the Main
Belt (Scott, 2007), are solidified silicate droplets from an epoch of
transient heating episodes. In the first 105–106 yr of planetesimal
growth, if molten 10–100 km diameter precursor bodies were kept
heated by 26Al decay, but undifferentiated because of their very low
gravity, then it is conceivable that the abrupt (strombolian) pressure
unloading experienced by molten but undifferentiated small precursor bodies could result in ubiquitous small melt droplets among
the remnants that did not accrete. Chondrules forming by such a
mechanism would either disperse into space, and either be removed
or 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 is
nothing larger to collide into and it becomes an amalgam of its
feeding zone (Chambers and Wetherill, 1998). Impacts by smaller
bodies batter their outer layers for the remainder of their
evolution, and can remove atmospheres and oceans (Genda and
Abe, 2005) and even hemispheres of crust (Nimmo et al., 2008;
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Marinova et al., 2008), but the removal of their mantles requires
collisions of unusually high energy. The cores of accreted planets
are not disrupted for expected impact velocities (Scott et al.,
2001). A very massive and wayward impactor is required to
remove the fraction of Mercury’s mantle that appears to be
missing, assuming it began with Earth-like bulk composition
(Benz et al., 2007). This scenario is dynamically opposite the
Moon-forming scenario, for which the impactor and target must
be almost identical dynamically and chemically, falling in at
v1 0. But in Moon-forming giant impact scenarios, models
(Canup and Asphaug, 2001) show that the Mars-sized impactor
would escape if going a fraction faster. This motivates an
alternative scenario in which Mercury sheds its mantle in a hit
and run collision with a larger protoplanet early on, at v1 vesc
(e.g. Fig. 7a–c) or conceivably during later chaos encountering
Venus.
Hit and run collisions occur in extrasolar planet-forming
systems. Exoplanets are being discovered through transit observations to include ‘‘super-Earths’’ that are several times more
massive than our own (e.g. Ribas et al., 2008); the discovery of the
first Earth-mass planet is imminent. But how Earth-like can a
planet be, in a solar system that has accreted one or more superEarths? Where Earths are the unaccreted NLBs, most will have lost
their crusts, oceans and atmospheres at one time or another, or
been dismantled like Mercury, on the perilous path of planetary
growth. It may be that in order to have water-bearing Earth-mass
terrestrial planets, they need to accrete as the largest bodies in the
population.
Acknowledgments
This research was sponsored by NASA’s Planetary Geology and
Geophysics Program (‘‘Small Bodies and Planetary Collisions’’) and
Origins of Solar Systems Program (‘‘Meteorite and Dynamical
Constraints on Planetary Accretion’’) under Research Opportunities in Space and Earth Sciences. The SPH computations were
performed on the NSF-funded supercomputer upsand at UCSCIGPP. My research into planet-scale collisions began as a thesis
project with Willy Benz, whose subsequent collaboration on tidal
disruption led me to try understanding hit and run collisions. This
research evolved through collaborations with Robin Canup on
Moon formation, and with Craig Agnor, a patient explainer of
planetary dynamics. I am grateful to John Chambers and Bill
Bottke for their careful reviews and unique insights. I thank Ed
Scott, Quentin Williams, Francis Nimmo, Jeff Cuzzi and Naor
Movshovitz for creative critical discussions, and Klaus Keil for his
original thinking on igneous and evolved asteroids and for
inviting me to write this review.
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