Download Accretion of the Terrestrial Planets and the Earth-Moon

Canup and Agnor: Accretion of the Terrestrial Planets
113
Accretion of the Terrestrial Planets and the Earth-Moon System
Robin M. Canup
Southwest Research Institute
Craig B. Agnor
University of Colorado
Current models for the formation of the terrestrial planets suggest that the final stage of
planetary accretion is characterized by collisions between tens to hundreds of lunar to Marssized planetary embryos. In this view, large impacts are an inevitable outcome as a system of
embryos destabilizes to yield the final few planets. One such impact is believed to be responsible for the origin of the Moon. Improvements in numerical methods have recently allowed
for the first direct orbit integrations of the final stage of accretion, which is believed to persist
for ~108 yr. The planetary systems produced by these simulations bear a general resemblence
to the terrestrial planets, but on average differ from our system in the final number of planets
(fewer), their orbital spacings (wider) and their eccentricities and inclinations (larger). The discrepancy between these predictions and the nearly circular orbits of both Earth and Venus is
significant, and is likely a result of the approximations inherent to the late-stage accretion simulations performed to date. Results from these works further highlight the important role of stochastic impact events in determining final planetary characteristics. In particular, impacts capable
of supplying the angular momentum of the Earth-Moon system are predicted to be common.
1.
INTRODUCTION
In the planetesimal hypothesis, the growth of terrestrial
planets is the result of the process of collisional accumulation from initially small particles in the protoplanetary disk.
The accretion process is typically described in terms of three
stages of growth, which are distinguished by our basic understanding of the relevant physical processes involved in
forming solid bodies in a particular size range. The first
stage involves the formation of kilometer-sized planetesimals from an initial protoplanetary disk of gas and dust. By
the end of this stage of growth (discussed in chapter by
Ward, 2000), planetesimals have reached sizes large enough
so that their dynamical evolution is determined primarily
by gravitational interactions with the central star and with
other planetesimals, rather than by surface, electromagnetic,
or sticking forces. The middle stage consists of the accumulation of a swarm of kilometer-sized planetesimals into
lunar- to Mars-sized planetary embryos (see chapter by
Kortenkamp et al., 2000). Numerous works have demonstrated that in this stage, dynamical friction acts to reduce
encounter velocities with the largest bodies, facilitating the
“runaway” growth of ~1025–1027 g (M = 5.98 × 1027 g)
planetary embryos in as little as 105 yr (e.g., Greenberg et al.,
1978; Wetherill and Stewart, 1993). The last stage then consists of the formation of the final few planets via the collision and merger of tens to hundreds of planetary embryos.
Evolution during this period is thought to be driven by distant interactions between the embryos, and requires a few
times ~108 yr (e.g., Wetherill, 1992).
In one of the first modern works to examine terrestrial
planet formation, Safronov (1969) proposed that planets
accreted in radially confined feeding zones, in a relatively
quiescent manner through the accumulation of small bodies. Developments in the past two decades suggest that the
generally localized, runaway stage persists only until bodies grow to the size of the Moon or Mars, leaving many
planetary embryos throughout the terrestrial region. Such a
system of embryos is dynamically unstable on timescales
that are short (~106 yr) compared to the age of the solar
system, and highly energetic collisions between embryos
then occur to yield the four terrestrial planets. In this scenario, the characteristics of the final planets are determined
mainly by the specifics of the last few large impacts that
each planet experiences. The stochastic nature of the final
stage thus yields an inherent degree of uncertainty for the
outcome of accretion in any given system. Indeed, a wide
range of possible planetary architectures is found to arise
from even nearly identical initial conditions, suggestive of
the great variety of terrestrial planet systems that might exist
in extrasolar systems.
The physical and dynamical environment in which the
accretion of the terrestrial planets took place is directly relevant to models of lunar formation. In the giant impact scenario, the Moon forms as a result of a single impact with
Earth late in its formation history. While works to date have
generally considered the various stages of planet accretion
and the formation of the Moon separately, a more holistic
approach may be required. In particular, models of the proposed lunar-forming impact and the accretion of the Moon
113
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Origin of the Earth and Moon
have yet to identify a single impact that can yield the final
masses of the Earth and Moon, together with the current
system angular momentum (Cameron and Canup, 1998;
Cameron, 2000). However, recent terrestrial accretion studies suggest that impacts subsequent to the lunar-forming
event may have contributed significantly to the final mass
and/or angular momentum of the Earth-Moon system, offering a possible resolution to this dilemma (Agnor et al., 1999).
In this chapter, we review recent simulations of late-stage
accretion and discuss the successes and weaknesses of these
models (section 2). We then address issues especially relevant to the formation of the Moon via giant impact, including late-stage impact statistics and the potential role of
multiple impacts in affecting planetary spin angular momenta (section 3). A brief discussion of open issues is included in section 4.
2. ACCRETION OF THE
TERRESTRIAL PLANETS
The environment in which the final accretion of terrestrial-type planets takes place is dependent upon the outcome
of the preceeding runaway growth stage. Midstage accretion models utilizing both statistical treatments that model
the entire terrestrial region, and N-body simulations of runaway growth within a local radial zone of the disk, yield
qualitatively similar results: embryos with masses ~0.01–
0.1M , occupying nearly circular, low-inclination orbits
after 105–106 yr. The embryos have typical orbital spacings
of ∆a ~ 10 RHill, where RHill is the mutual Hill radius given
by
RHill ≡
a1 + a 2 m1 + m 2
2
3M
1/3
(1)
where a1, a2, m1, and m2 are the semimajor axes and masses
of adjacent embryos, and M is the mass of the Sun. A system of two bodies on initially circular orbits will be stable
against mutual collision so long as ∆a > 3.5 RHill (e.g.,
Gladman, 1993). However, multiplanet systems (or planets
with nonzero initial eccentricities) require larger separations
for stability. Chambers et al. (1996) performed numerical
integrations of like-sized planets intially on circular orbits
and found a separation of ∆a ≈ 8–10 RHill provided stability for ~106–107 yr, in fair agreement with the predicted
spacings from the midstage accretion simulations (e.g.,
Weidenschilling et al., 1997). Ito and Tanikawa (1999) conducted numerical stability analyses of planetary embryos,
including initial eccentricities and inclinations, a range of
embryo masses, and Jupiter and Saturn. They found much
shorter stability times of ~104–105 yr for a system of N =
14 embryos with ∆a ≈ 8–10 RHill separations and initial
values of 〈e〉1/2 = 2〈i〉1/2 = 0.005. The latter timescales are
close to those found by the late-stage N-body simulations
(described below) that begin with similar initial eccentricities. For comparison, the current terrestrial planets have
mutual separations ranging from about 26 to 40 RHill.
The boundary between the middle and late stages is generally believed to be representative of a dynamical transition: from a stage in which growth is dominated by collisions with local material, to one in which distant interactions among the embryos lead to collisions on much longer
timescales. Recent simulations that model embryo formation in the full terrestrial zone (0.5–1.5 AU) find that 90%
of the system mass is contained in a few tens of embryos
after about a million years, with the remaining ~10% of the
mass contained in a swarm of much smaller planetesimals
(Weidenschilling et al., 1997). These simulations have included effects due to gas drag and a parameterization of
distant perturbations between embryos, but to date have not
included collisional fragmentation. At present, it is not clear
to what extent the planetesimal swarm persists or is regenerated via collisional erosion throughout the late stage. It
is often assumed that all the small material in the disk would
be rapidly swept up by the embryos, since the timescale for
embryo formation (105–106 yr) is much shorter than the
timescale for the accumulation of the final planets (~2 ×
108 yr). In this case, the dynamics of the final stage are
governed solely by gravitational interactions among and
collisions between the large embryos. Interactions among
embryos initially on nearly circular orbits lead to eccentricity growth, and then to orbit crossing; once orbital isolation is overcome, the embryos collide and merge. The accumulation of embryos into larger bodies then proceeds until
the secular orbital oscillations (primarily the eccentricities)
of the remaining bodies are insufficient to allow bodies to
encounter each other, and a few planets remain on wellseparated orbits. Until recently, simulations of this final stage
were limited to statistical treatments due to the large number of orbital times involved. However, numerical techniques now exist that allow for direct integration of systems
of N ~ 10–100 embryos for ~108 yr. Results from simulations using both methods are reviewed in the next section.
2.1. Late-Stage Simulations
Late-stage terrestrial accretion has been modeled using
two basic (and complementary) techniques. Monte Carlo
simulations follow the orbital evolution of embryos in a
statistical manner based on two-body scattering events (e.g.,
Wetherill, 1985), while N-body orbital integrations directly
track the trajectories of each embryo at all times (Chambers and Wetherill, 1998, hereafter CW98; Agnor et al.,
1999, hereafter ACL99). Under comparable sets of assumptions, both methods produce similar configurations of final
planets.
Wetherill was the first to model the dynamical evolution
of systems of planetary embryos throughout the terrestrial
region. His Monte Carlo scheme, with its approximate treatment of dynamical interactions, is computationally fast and
has been used to generate large numbers of planetary systems for a range of starting conditions and assumptions (e.g.,
Wetherill, 1985, 1992, 1994, 1996). In this method, the
probability of each body experiencing a close encounter
Canup and Agnor: Accretion of the Terrestrial Planets
with any other body is determined using an adaptation of
the Opik (1951) formalism; the outcome of the encounter
is then a function of a randomly selected distance of closest approach. Wetherill (1992) presented results of 434
Monte Carlo simulations of the evolution of a few hundred
embryos throughout the terrestrial and asteroid belt region.
Fragmentation was assumed to occur when the impact energy exceeded a chosen threshold, at which point the total
mass involved in a collision was divided into a small number (i.e., 4) of equal-sized pieces. Bodies with masses
smaller than 8 × 1025 g (or about the mass of the Moon)
were removed from the simulation, in order to limit the total
number of objects to a computationally manageable level.
Effects due to perturbations from and resonances with Jupiter and Saturn were included in the form of simple parameterizations. The planetary systems resulting from these
simulations broadly resembled the current planets in number and size, with some discrepancies. In particular, Earthlike planets with large eccentricities were common outcomes; of the final planets with masses greater than 3.5 ×
1027 g (0.58 M ), ~40% had eccentricities greater than 0.05
and ~75% had eccentricities larger that of Earth or Venus.
Direct integration methods explicitly track the gravitational interactions among all bodies in a simulation. However, this increase in dynamical accuracy comes at the cost
of computational speed, which limits the number of simulations and bodies in a simulation that can be considered.
Direct integrations of the late stage have recently become
possible due to developments in symplectic integration techniques, which afford system energy and angular momentum conservation for the requisite ~108 orbits (e.g., Wisdom
and Holman, 1991; Duncan et al., 1998; Chambers, 1998b).
CW98 performed 27 late-stage integrations that each
began with a system of roughly 25–50 embryos on circular
orbits distributed throughout the terrestrial region (0.5–
1.8 AU). All collisions were assumed to result in complete
and inelastic merger. Figure 1 shows the evolution of a system of planetary embryos in mass and semimajor axis space
for 100 m.y. (CW98, their Fig. 1). The first encounter between embryos often occurs in the inner part of the disk,
and results in a scattering of the two bodies involved such
that their neighbors then become subject to close encounters. This leads to what is described as a “wave” of close
encounters that travels outward through the terrestrial region. Interactions among the embryos, including both resonances and scattering events, drive the accretion process
until the final planets are so well-separated and few in number that their mutual perturbations are insufficient to allow
for further collisions. Note that in the last frame of Fig. 1
there are still crossing orbits, and so accretion is this case
is likely not yet complete.
CW98 also investigated the effect of Jupiter and Saturn
on embryo accretion in the 0.55–4.0-AU region by adding
the giant planets (with their current masses and orbits) to a
subset of their simulations after 107 yr (see Fig. 2). The
extent to which the giant planets were present and at their
current locations during the late stage is uncertain. Requir-
115
ing that Jupiter and Saturn formed before the nebula dispersed implies a formation time of less than ~107 yr, suggesting that they were most likely present during most of
the final accretion period. However, these planets may have
migrated significantly during and subsequent to their formation. In general, perturbations from outer giant planets
act to decrease the stability of a system of protoplanets (e.g.,
Ito and Tanikawa, 1999), as both mean motion and secular
resonances drive large-amplitude eccentricity oscillations.
Of particular importance are the ν5 and the ν6 secular resonances, which occur where the apsidal precession rate of
orbiting material is near to that of Jupiter or Saturn respectively (or at about 0.6 AU for the ν5 and 2.1 AU for the ν6,
assuming current planetary orbital elements and no nebular
gas disk).
The CW98 simulations indicate that planetary embryos
with orbital radii larger than ~1.2 AU are typically scattered
into the ν6 or other, mean-motion resonances where they
become dynamically coupled to the outer planets, leading
to either collisions with embryos in the terrestrial region or
ejection from the solar system following close encounters
with Jupiter. In their current positions, Jupiter and Saturn
thus appear to prevent the accumulation of embryos into
terrestrial-sized planets for a > 1.2 AU. Given that the mass
of Mars is only ≈0.1 M , it is possible that Mars itself is
a leftover planetary embryo formed via runaway growth
(Wetherill, 1992; CW98). In the inner disk, the ν5 resonance
can lead to embryo removal via collisions with the Sun. In
simulations that include Jupiter and Saturn, at least 15% of
the total initial embryo mass is typically lost through collisions with the Sun and hyperbolic ejection events (CW98).
ACL99 performed direct integrations of the late stage that
began with either 50 (m ~ 0.04 M ) or 22 (m ~ 0.10 M )
planetary embryos distributed between 0.5 and 1.5 AU with
small eccentricities and inclinations (e ~ 0.01 and i ~ 0.05°).
In the latter case, the initial embryos were those produced
by the Weidenschilling et al. (1997) full terrestrial zone simulation of the midstage (see chapter by Kortenkamp et al.,
2000). Giant planets were not included. Figure 3 shows the
evolution of the mass distribution of embryos from one of
the ACL99 simulations using the Weidenschilling et al. (1997)
initial conditions. The general pattern of the dynamics is
similar in the simulations of both CW98 and ACL99, despite their use of different initial distributions of embryos
and different numerical integration methods.
Figure 4 shows all the final planets from the 37 simulations performed in both CW98 and ACL99. Note that the
ACL99 simulations that began with the Weidenschilling et
al. initial conditions considered a smaller total planetary
embryo mass (~1.8 M ) than the CW98 runs, and thus yield
somewhat smaller final planets. The most massive planets
tend to form near or interior to 1 AU, where the surface
density and the collision frequency are the highest. All simulations of the late stage, whether direct integrations or Monte
Carlo simulations, appear to produce planets with distributions similar to Fig. 4 when considering a total mass ~2 M
in the terrestrial region (see, e.g., Wetherill, 1992).
116
Origin of the Earth and Moon
START
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Semimajor Axis (AU)
Fig. 1. Masses and semimajor axes of the surviving objects at six different times from a simulation by Chambers and Wetherill (1998)
that began with embryos in the terrestrial region and did not include Jupiter and Saturn (their model A). The horizontal line through
each symbol connects the perihelion and aphelion of the embryo’s orbit. Mutual interactions between embryos perturb them into crossing
orbits and drive the accretion process until the final bodies’ orbits are well separated.
Canup and Agnor: Accretion of the Terrestrial Planets
1 e 5 yr
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Fig. 2. Same as Fig. 1, except this simulation included embryos in the present-day asteroid belt and Jupiter and Saturn (from Chambers and Wetherill, 1998, an example of their model C). In this case, the asteroid belt is cleared of embryos and the terrestrial region
has a smaller number of final planets than in Fig. 1.
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Origin of the Earth and Moon
Mass (m/M )
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Fig. 3. Masses and semimajor axes of the surviving objects at six different times from one of the simulations of Agnor et al. (1999,
their Fig. 3). This simulation used as its starting condition the 22 largest embryos from the multizone embryo formation calculation of
Weidenschilling et al. (1997). Despite the different integration techniques and initial conditions, the evolution of the system is broadly
similar to those shown in Figs. 1 and 2.
Canup and Agnor: Accretion of the Terrestrial Planets
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3
ACL99
CW98 Model A
CW98 Model B
CW98 Model C
Terrestrial Planets
Mass (m/M )
2
1
0
0
1
2
3
Semimajor Axis (AU)
Fig. 4. Masses and semimajor axes of all of the final planets produced by the 27 simulations of Chambers and Wetherill (1998) and
the 10 simulations of Agnor et al. (1999). Initial conditions for models A and C are shown in Figs. 1a and 2a. Note that model B of
Chambers and Wetherill (1998) had an identical distribution of initial embryos to that in model A, except that in model B Jupiter and
Saturn were included in the simulation. The values for the terrestrial planets are included for comparison.
Figure 5 shows the time-averaged eccentricities and inclinations of the final planets with masses greater than
0.5 M from CW98 and ACL99. The average final eccentricity of the largest body from each simulation done by
ACL99 (who did not include Jupiter and Saturn) was 0.08,
with time-averaged eccentricities all greater than 0.05.
CW98 report comparable eccentricities for simulations performed with similar initial conditions. Even larger eccentricities resulted for runs that included Jupiter and Saturn;
in this case the average eccentricity of the most massive final
planet in each system was 0.18.
The large eccentricities of the planets produced by simulations also yields larger angular momentum deficits for the
systems formed. The angular momentum deficit, or AMD,
is given by
N
AMD = ∑ m k n k a 2k 1 − 1 − e 2k cos i k
(2)
k =1
where N is the total number of planets, nk is the planet’s
mean-motion about the Sun, and mk is the planet’s mass.
In the 10 simulations of ACL99, the average value of the
AMD of the final planets formed was between 4.5 and 17
times larger than that of the terrestrial planets, despite an
initial system angular momentum that exceeded the terres-
trial system by no more than 5%. Larger AMD values require final planets more widely spaced (and therefore typically fewer in number) than their terrestrial counterparts for
system stability. In the ACL99 simulations that yielded two
adjacent planets with masses greater than 0.5 M (80% of
their runs), the two massive planets had an average spacing
of 〈∆a〉 = 44 RHill (with individual values ranging from 33
to 55 RHill), in comparison to the ∆a = 26.25 RHill separation between Earth and Venus.
While the results obtained with various models of the late
stage are thus in fairly close agreement, they continue to
yield systems that are different in character than the current planets. An important question is why such discrepancies persist, even as modeling methods of embryo interactions have improved. The answer is likely that computational and model limitations continue to restrict simulations
to simplified scenarios that neglect potentially influential
processes. For example, current simulations are not capable
of handling the large numbers of smaller planetesimals
present during the postrunaway evolution of planetary embryos, and perhaps during the final stage as well. Dynamical interactions with such a small body population might
yield lower planetary eccentricities, although this has yet to
be convincingly demonstrated. This issue is discussed in
more detail in section 4. However, assuming the basic con-
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Origin of the Earth and Moon
20
ACL99
CW98 Model A
CW98 Model B
CW98 Model C
Earth and Venus
Inclination (degrees)
15
10
5
0
0.00
0.10
0.20
0.30
0.40
Eccentricity
Fig. 5. The time-averaged inclinations and eccentricities of the final planets larger than 0.50 M are shown. Note the general absence of planets with eccentricities of 0.03 or less. The current values for Earth and Venus are shown for comparison (squares). Timeaveraged values are e ~ 0.03, and I ~ 2° for Earth and Venus.
clusion that runaway growth yields embryos significantly
smaller than either Earth or Venus is correct, the final accretion stage must have been characterized by large impacts
between embryos. In section 3, we review the implications
of such impacts for the final stages of growth of a terrestrial planet.
2.2. Radial Mixing During Late-Stage Accretion
The Earth, Moon, Mars, and meteorites originating in the
asteroid belt display isotopic commonalities and differences
that were generally thought to reflect their formation from
distinct radial reservoirs of material in the protoplanetary
disk. For example, the O-isotopic signatures of terrestrial
and lunar material fall on the same fractionation line, which
is distinct from that of meteorites believed to have originated in the asteroid belt or Mars. The standard explanation is that the Earth and Moon formed from material
contained in the same radial zone. This conceptually agreed
well with earlier views of planet formation, which employed
the concept of planetary growth from a local “feeding zone”
(e.g., Safronov, 1969; Lewis, 1972). Recent studies suggest
that the concept of feeding zones may be relevant to the
midstages of planet formation, when eccentricities and inclinations are low.
However, the evolution of a swarm of embryos into a few
planets yields a large degree of mixing throughout the terrestrial zone. Wetherill (1994) studied the initial location of
the embryos that comprised the final planets that formed in
his Monte Carlo simulations. In general, all the embryos
initially in the region between 0.5 and 2.5 AU became radially mixed, and the final planets were comprised of material that had its origins throughout this region. However,
the average provenance of a planet was found to be somewhat correlated to its semimajor axis, i.e., planets in the
outer terrestrial region accreted more material from larger
heliocentric distances than the inner planets. This general
result is also evident in the recent N-body simulations, as
shown in Fig. 6. CW98 find that final planets with a < 1 AU
are comprised of material that originated primarily in the
0.5–1.5-AU region (~75% of their final mass), but also contain material from further out in the protoplanetary disk
(~25% from material originating between 2.5 and 3.5 AU). It
is thus consistent with the current accretion models that terrestrial planets may “remember” the initial compositional zoning of the protoplanetary disk to a limited degree. Whether
Canup and Agnor: Accretion of the Terrestrial Planets
Fraction of Final Mass
1.0
0.8
0.6
0–1 AU
0.4
1–2 AU
0.2
2–3 AU
0.0
0
1
2
3
4
Initial Semimajor Axis (AU)
Fig. 6. The fraction of the planet mass obtained from different
regions of the protoplanetary disk are shown for the final planets
of the simulations presented in Chambers and Wetherill (1998,
their Fig. 21). The three curves are for planets with final semimajor axes of 0 < a < 1 AU, 1 < a < 2 AU, and 2 < a < 3 AU. While
planets form from some material near their end orbital radius,
radial mixing of material causes planets to acquire a significant
fraction of their mass from more distant regions.
such predictions can be reconciled with geochemical constraints that appear to favor compositional zoning is an
important and open question.
3. LATE-STAGE IMPACT EVENTS
It is during the final stage that most of the end planetary
characteristics — e.g., mass, spacing, orbital elements, spin
angular momentum, presence of impact-generated satellites,
etc. — are determined. For example, growing planets acquire their spins and obliquities as they accrete material,
with the relative motions of the bodies at impact determining the contributions of each accreted body to the spin angular momentum of the final planet. As a protoplanet grows,
both the average impact velocity and average moment arm
of colliding bodies increase, so that the rotation state of a
planet is determined primarily during its final accretion (see
chapter by Lissauer et al., 2000).
Large impact events could be either accretionary or erosive. The fate of debris generated during a planet-scale impact event would depend upon the specifics of the impact,
e.g., impact angle, velocity relative to escape velocity, and
mass ratio of impactor to target. The two-body escape velocity is
vesc ≡ 2G (m1 + m 2 ) / (R1 + R2)
(3)
where R is radius. Collisions with vimp >> vesc would likely
be primarily erosive or even disruptive; such an impact has
been invoked to account for the loss of portions of Mercury’s early mantle, resulting in the planet’s current anom-
121
alously high density (e.g., Benz et al., 1988). Oblique impacts can result in the ejection of material into bound orbit
around the target protoplanet, which can then rapidly accrete
to form satellites. Satellite formation via this mechanism has
been proposed for the origin of the Earth-Moon system
(Hartmann and Davis, 1975; Cameron and Ward, 1976), the
Pluto-Charon binary (e.g., Stern et al., 1997, and references
therein), and for asteroid-satellite systems (e.g., Durda,
1996).
Determination of impact outcome as a function of specific impact energy, material strength, and target size has
been the focus of a great body of experimental and theoretical research. General scaling laws that predict quantities
such as crater size and total ejected mass exist for impacts
that span the size range from centimeter-sized particles up
to target radii of about ~200 km (e.g., Housen and Holsapple, 1990; Ryan and Melosh, 1998; Benz and Asphaug,
1999). To date, numerical simulations of planet-scale impacts have focused primarily on the particular impact believed responsible for the Earth-Moon system (Benz et al.,
1986, 1987, 1989; Cameron and Benz, 1991). In general,
an approximately Mars-sized body with an impact angular
momentum of at least the current angular momentum of
the Earth-Moon system appears to be required to yield a
lunar-sized moon. However, general scaling relationships for
planet-scale impacts are just starting to be developed (e.g.,
Canup et al., 2000, and discussion in section 4). Given the
lack of such relationships, late-stage accretion simulations
have modeled collisional outcomes through the use of
simple extrapolations from scaling laws derived for much
smaller impacts (in the case of Wetherill’s Monte Carlo
simulations), or by simply assuming that every impact results in complete accretion (in the case of the N-body simulations).
3.1. Occurrence and Timing of Large Impacts
The Monte Carlo simulations of Wetherill (e.g., 1985,
1986, 1992) were among the first to examine the predicted
timing and size of the largest impactors to collide with Earth
during its accretion (see also Hartmann and Vail, 1986).
Wetherill (1985) performed three-dimensional simulations
that each began with 500 bodies whose masses ranged from
about 6 × 1024 to 1 × 1027 g, and with orbital distances between 0.7 and 1.1 AU. In every case giant impacts (i.e.,
impacts by objects with at least the mass of Mars) occurred,
typically ~20 m.y. after the start of a simulation. In 1992,
Wetherill performed similar calculations that began with an
initial distribution of embryos that spanned the entire terrestrial region and the asteroid belt (0.4–3.8 AU). Similar
numbers of giant impacts occurred, but at somewhat later
times of 5 × 107–108 yr. In both studies, about one impact
onto a large planet by a body at least as massive as Mars
occurred for each simulation that contained a final planet
with mass similar to that of Venus or Earth.
ACL99 characterized the collisions in their N-body simulations in terms of impactor mass, velocity, and impact an-
Origin of the Earth and Moon
Impactor Mass (m/M )
122
0.40
0.30
(a)
Simulations 1–8
Simulations 9–10
0.20
0.10
0.00
105
106
107
108
Time (yr)
8
(b)
Vimp /Vesc
6
4
2
0
105
106
107
108
Time (yr)
4
(c)
L imp/L
–M
3
2
1
0
105
106
107
108
Time (yr)
Fig. 7. The impactor mass, impact velocity, and the angular momentum of the collisional encounter are shown as a function of the
time at which each impact occurred. All collisions from the 10 simulations of Agnor et al. (1999, their Fig. 8) are shown. The angular
momentum of the Earth-Moon system (L –M) is shown by the dashed line. Simulations 1–8 utilized the Weidenschilling et al. (1997)
embryos as their starting conditions.
gular momentum as a function of time (see Fig. 7). The impactor was defined to be the less massive member of a colliding pair. Figure 8 gives a breakdown of the impact
velocity and collision angular momentum as a function of
impactor mass from the same simulations. The initial em-
bryos considered in ACL99 had masses of 0.04–0.15 M .
As a simulation proceeds, embryos collide and merge, and
a spectrum of embryo masses develops. A weak degree of
dynamical friction (see discussion in chapter by Kortenkamp
et al., 2000) causes the smaller bodies in the disk to obtain
Canup and Agnor: Accretion of the Terrestrial Planets
123
8
(a)
Simulations 1–8
Simulations 9–10
Vimp /Vesc
6
4
2
0
0.00
0.10
0.20
Impactor Mass (m/M )
0.30
0.40
0.10
0.20
Impactor Mass (m/M )
0.30
0.40
4
(b)
L imp /L
–M
3
2
1
0
0.00
Fig. 8. The impact velocity and angular momentum of the collisional encounter are shown as a function of the impactor mass for the
same collisions displayed in Fig. 7 (Agnor et al., 1999, their Fig. 9). Note that some small, high-velocity impactors are capable of
delivering more than 1 L –M to the system.
larger eccentricities and inclinations (and thus higher impact velocities) than their more massive counterparts. After
a few million years (at which time the embryo masses span
about a factor of five), some impacts occur at several times
the escape velocity (see Fig. 7b).
These results generally resemble those of Wetherill (1992)
in terms of the timing of the largest impacts. The ACL99
final planets with m > 0.5 M experienced their largest
impact at an average time of 29 m.y. (with individual values ranging from 1.4 to 95 m.y.); the last impact onto these
planets occured at an average time of 46 m.y. (with individual values ranging from 3.1 to 108 m.y.). These timings
are in relatively good agreement with isotopic constraints
on the age of the Earth and Moon. The Hf-W chronometer
places the time of core formation in the Earth and Moon at
~50 m.y. (e.g., Lee et al., 1997; see chapter by Halliday et
al., 2000), while the formation interval for Earth based on the
terrestrial excess of the heaviest isotopes of Xe is ~100 m.y.
(see chapter by Podosek and Ozima, 2000).
In an attempt to identify impacts that could result in the
formation of a lunar-sized satellite, ACL99 described those
collisions with an encounter angular momentum equal to
or exceeding the current angular momentum of the EarthMoon system (L –M) as “potential moon-forming impacts.”
This classification reflects the very rudimentary knowledge
of the impact dynamics required to form an impact-generated satellite, and is somewhat different than the “giant
impacts” of Wetherill (1985, 1992), which were classified
by the impactor mass. In the 10 ACL99 accretion simulations, there were a total of 20 potential moon-forming impacts (see Figs. 7b and 8a). In addition, 25% of the final
planets larger than 0.50 M experienced more than one such
impact.
In the ACL99 simulations that followed the evolution of
the 22 embryos from Weidenschilling et al. (1997), the
number of moon-forming collisions that occurred in each
simulation ranged from 0 to 3, with an average of about 1.6
per simulation. In the simulations that began with 50 initial
1.0
0.8
0.6
0.4
0.2
0.0
0
1.0 × 107
2.0 × 107
3.0 × 107
4.0 × 107
5.0 × 107
4.0 × 107
5.0 × 107
4.0 × 107
5.0 × 107
4.0 × 107
5.0 × 107
Time (yr)
3
–M
embryos (and a slightly larger total embryo mass of 2 M ),
there were an average of 3.5 moon-forming collisions per
simulation. The impact velocities of the moon-forming collisions were typically somewhat larger than the the two-body
escape velocity. With the exception of one (0.04 M ) impactor that had an impact velocity of 3.44 vesc, the Limp >
L –M collisions occurred with velocities in the range 1.00–
1.63 vesc, with an average of 1.20 vesc (see Fig. 8a). Highresolution simulations of the moon-forming impact performed to date (e.g., Cameron, 1997; Cameron, 2000) assume vimp = vesc, and so have considered only the lower limit
of possible specific impact energies.
Impactor Mass (m/M )
Origin of the Earth and Moon
L spin /L
124
2
1
0
0
1.0 × 107
2.0 × 107
3.0 × 107
The spin angular momentum of an embryo will evolve
due to each of the large impacts it experiences. As a basic
model for this process, ACL99 assumed that for each collision, the spin angular momentum of the merged body was
just the sum of the spin angular momenta of the two colliding bodies and the orbital angular momentum of the two
bodies about their center of mass. The rotational periods of
the growing planets were then calculated by assuming that
all bodies were spheres of uniform density. In general, the
accretion of material and angular momentum was likely less
than 100% efficient during collisions between planetary
embryos, and the assumption of inelastic mergers to model
collisions will tend to overestimate both of these quantities.
In addition, no account was made for the precession of spin
axes that will result for oblate planets, or for the evolution
of planetary obliquities due to spin-orbit coupling (see chapter by Williams and Pollard, 2000).
Figure 9 shows the evolution of the mass, the magnitude
of the spin angular momentum (in units of L –M), obliquity, and rotation period of a typical Earth-like planet (defined to be a planet with m ≥ 0.50 M ). In this particular
case, each collision results in a net increase in the magnitude of the spin angular momentum of the planet. Impacts
reorient the direction of the spin angular momentum vector, and in general, the spins and obliquities of the growing
planets during the late stage are quite large (see chapter by
Lissauer et al., 2000). For collisions with vimp = vesc, the
angular momentum of the impact, Limp, scaled by the critical angular momentum for rotational stability, Lcrit, is just
L imp
L crit
=
2
1/3
ci (1 − ci) c1/3
sin ξ
i + (1 − ci )
K
(4)
with
Lcrit ≡ MT5/3 K G 3 /(4πρ)
1/6
(5)
where MT and RT is the total colliding mass and the radius
of the combined body, ρ and K are the density and gyration constants of the colliding bodies, ci is the fraction of
the total mass contained in the smaller of the colliding bodies, K = I/MTRT2 where I is the moment of inertia, and ξ is
the impact angle. The angular momentum imparted from a
single collision can approach or exceed Lcrit for large ci. For
0
50
100
150
0
1.0 × 107
2.0 × 107
3.0 × 107
Time (yr)
Rotation Period (hr)
3.2. Following the Growth of a Planet
Obliquity (degrees)
Time (yr)
4
3
2
1
Rotationally unstable
0
0
1.0 × 107
2.0 × 107
3.0 × 107
Time (yr)
Fig. 9. The mass, magnitude of the spin angular momentum in
units of the Earth-Moon system, obliquity, and rotation period of
a typical planet (the 0.73 M final planet shown in Fig. 3) are plotted as a function of time (Agnor et al., 1999, their Fig. 10). The
dotted line of the rotation period vs. time graph indicates the rotational stability limit. The small oscillations in the obliquity are
due to the motion of the body’s orbit; the spin axis of the body was
assumed to remain fixed in the inertial frame between collisions.
example, with like-sized bodies (and assuming a gyration
constant equal to that of Earth, or K = 0.335), Limp/Lcrit =
1.33 sin ξ. Indeed, the ACL99 simulations find planetary
rotation rates that often exceed the rotational stability limit.
Clearly the assumption of merger in such cases is invalid,
as the excess angular momentum would instead be carried
away in ejected debris or in debris placed into circumplanetary orbit.
Even small embryo collisions (mimp ≈ 0.04 M ) can make
angular momentum contributions to a planet that are significant in comparison to L –M. In Fig. 10 the mass and the
magnitude of the spin angular momentum of a 0.80 M
planet from one of the ACL99 simulations are shown as a
function of time. This planet experiences two collisions with
initial embryos (at t ~ 32 × 106 and ~50 × 106 yr), which
both act to reduce the magnitude of the planet’s spin angular momentum, slowing its spin rate. The second of these
collisions occurred with vimp = 1.4 vesc, and had a collisional
angular momentum of 0.55 L –M. The largest impact to
strike this planet was also the last impact the planet experienced, which was the case for about one-half the Earth-like
Canup and Agnor: Accretion of the Terrestrial Planets
Mass (m/M )
1.0
0.8
0.6
0.4
0.2
0.0
0
2 × 107
4 × 107
6 × 107
8 × 107
1 × 108
8 × 107
1 × 108
Time (yr)
2.5
L spin /L
–M
2.0
1.5
1.0
0.5
0.0
0
2 × 107
4 × 107
6 × 107
Time (yr)
Fig. 10. The mass and magnitude of the spin angular momentum in units of the Earth-Moon system are shown as a function
of time for a planet from a simulation that began with smaller but
more closely spaced initial embryos (Agnor et al., 1999, their
Fig. 13). Small impacts (~0.04 M ), such as those occurring at
3.2 × 106 yr and 5.0 × 107 yr, can make significant contributions
to the angular momentum accretion of the planets.
Mass (m/M )
1.0
0.8
0.6
0.4
0.2
0.0
0
1.0 × 107
2.0 × 107
3.0 × 107
4.0 × 107
5.0 × 107
4.0 × 107
5.0 × 107
Time (yr)
L spin /L
–M
4
3
2
1
0
0
1.0 × 107
2.0 × 107
3.0 × 107
Time (yr)
Fig. 11. The planet mass and spin angular momentum as a function of time for a planet that experienced a net reduction in its
spin angular momentum due to cancellation between contributions
from multiple impacts (Agnor et al., 1999, their Fig. 11).
planets in the ACL99 simulations. During the growth of the
planet shown in Fig. 11, the largest impact (at t = 13 ×
106 yr) was followed by two smaller impacts, the last of
which resulted in a net decrease of 0.85 L –M in the magnitude of the spin angular momentum. Approximately 40%
of the Earth-like planets produced by the ACL99 simulations experienced collisions that decreased the magnitude
of the spin angular momentum of the planet by 0.10 L –M
or more.
For the Earth-like planets produced in ACL99, the largest and second largest impactor contributed an average of 30%
125
and 19% of the final planet mass and contributed an average of 1.44 L –M and 0.67 L –M respectively. For nearly
50% of these planets, the last impactor was also the largest. However, for an equal number of final planets, the last
impactor was less massive than both the largest and secondlargest impactor. In these cases, the last impact contributed
an average of only 8.3% of the final planet mass, but contributed an average angular momentum of 0.76 L –M with
values ranging from 0.10 to 1.83 L –M. Furthermore, these
planets accreted an average of 31% (range 7–62%) of their
final planet mass after experiencing their first Limp ≥ L –M
impact. These statistics were derived from a limited number of simulations; however, they suggest that the spin angular momentum of a terrestrial planet is likely the result
of more than one large impact, and that moon-forming collisions can occur before planetary growth is completed.
3.3. Implications for Lunar Origin
Two fundamental constraints on models of the formation
of the Earth-Moon system are the masses of the bodies and
the current system angular momentum. Models of lunar
formation have generally made the assumption that the
Moon-forming impact was the last impact Earth experienced. However, recently this constraint has been relaxed
in an order to yield a lunar-mass Moon as described below.
SPH (“smoothed-particle hydrodynamics”) simulations
of the lunar-forming impact have typically considered collisions with impactor to target mass ratios of 3:7 and 2:8,
no initial spin angular momentum of the impactor or target,
and vimp = vesc (Cameron, 1997; see also chapter by Cameron, 2000). These studies have investigated a variety of
impact angular momenta and system masses (0.5–1.0 M ).
Recent N-body simulations of the accumulation of the Moon
from an impact-generated disk indicate that a disk mass of
at least two lunar masses is required to form the Moon (e.g.,
Ida et al., 1997; see chapter by Kokubo et al., 2000). The
only impacts that SPH simulations have shown to be capable
of producing a circumterrestrial disk massive enough to
form the Moon are those with a total mass of 1.0 M and
an impact angular momentum greater than 2.0 L –M, or
those with a total mass of ~0.6 M and 1.0 L – M (Cameron and Canup, 1998; Canup et al., 2000). Each of these
scenarios is unable to simultaneously account for both the
mass and angular momentum of the Earth-Moon system
with a single impact. The first case, with Limp ~ 2 L –M,
requires that the Earth-Moon system somehow rid itself of
an angular momentum excess ~1 L –M after the moon-forming event. As solar tides could remove only a small fraction of this, this scenario would seem to require that Earth
experienced another impact to reset Earth’s spin. In the
second case, Earth acquires another ~0.3–0.4 M of material after the Moon has formed. This scenario also requires
that Earth experienced later impacts.
The results of the recent late-stage simulations show that
several large impacts typically affect a planet’s end state,
implying that Moon-forming impacts with total masses and
angular momenta different from that of the Earth-Moon
126
Origin of the Earth and Moon
system may be plausible. The ACL99 simulations find that
planets occasionally experienced glancing collisions at several times the escape velocity with smaller (~0.05 M ) impactors that were capable of delivering up to 1.1 L –M. At
present it isn’t known whether or not this type of collision
would be efficient at producing a circumplanetary disk. If
this type of collision occurred after the Moon had already
formed, it could conceivably reset the spin angular momentum of Earth, possibly offering a resolution to the high
angular momentum impact scenario described above. The
late-stage simulations also predict that embryos have rapid
spin rates throughout the final accumulation stage. In particular, the final planets of the ACL99 simulations that experienced Limp ≥ L –M impacts had an average angular
momentum of ~0.91 L –M (an average rotational period of
2.4 hr) immediately prior to these collisions. While the future refinements of planet formation models (e.g., inclusion
of the effects of small bodies during the final accretion
stage) may modify this estimate, it is significant enough to
suggest that the spin of Earth prior to the Moon-forming
event was likely not negligible, and that Moon-forming
collisions in which the impactor and/or the target are spinning should be studied. Perhaps, e.g., an Earth with a prograde spin prior to the lunar-forming impact would result
in higher yields of bound orbiting debris for a given impact angular momentum. Finally, the late-stage simulations
indicate that large impacts occur with vimp > vesc, and so such
collisions should also be explored in the context of the lu2 )
nar-forming event. Higher specific impact energies (∝ vimp
might act to increase the yield of ejected material for a given
impact angular momentum (∝ vimp), and potentially help to
mitigate the difficulty to date in forming a massive protolunar disk with an Limp ~ L –M impact.
3.4. Implications for the Frequency of
Impact-triggered Satellites
To some extent, the early perception that large impact
events were rare coincided well with the fact that among the
current terrestrial planets, only Earth has a sizeable, impactgenerated moon. However, the accretion simulations performed over the past decade depict a terrestrial planet formation process whose final stages are dominated by stochastic
impact events. In fact, it seems likely that terrestrial planets may have experienced multiple giant impacts in their
histories, and that many such impacts may have led to the
formation of satellites (e.g., Canup et al., 2000).
If accretion models now suggest that most terrestrial
planets undergo such impact events, is it not a contradiction that, e.g., Venus does not currently possess a satellite?
Not necessarily. First, while giant impacts appear to have
been commonplace in the inner solar system, there are still
only certain impact orientations that have been shown to
yield significant amounts of debris in circumplanetary orbit. So while oblique, relatively low-velocity collisions
might generate a circumplanetary disk, head-on collisions,
near misses, or collisions with vimp >> vesc might generate
debris that reimpacts the planet or escapes entirely to heliocentric orbit. Large impacts may be common, but satellite-generating impacts will be less so.
Second, it is also plausible that other terrestrial planets
once had impact-generated satellites that simply did not
persist against continuing bombardment during the late stage
or against orbital evolution due to tidal interaction. A simple
criterion for satellite stability is that the orbit initially expands rather than contracts as it tidally evolves, or a > aco,
where a is the initial semimajor axis of the satellite, and aco
is the co-rotation radius
a co ≡
GMp
1/3
(6)
ω2
where Mp and ω are the mass and angular velocity of the
planet. Since an impact-generated satellite cannot conceivably accrete within the Roche radius (see chapter by Kokubo
et al., 2000), a ≥ aRoche, with
a Roche ≈ 2.5
ρp
1/3
ρs
Rp
(7)
where ρs is the satellite density, and ρp and Rp are the density and radius of the planet. The requirement that aRoche >
aco then yields a constraint on the initial spin rate of the
planet
ωo ≥
4/3 πG
ρ
(2.5)3 s
1/2
(8)
For a satellite density equal to that of the Moon, a planet’s
day must be shorter than about 7 hr for co-rotation to fall
within the Roche limit. Slower rotators would quickly lose
their impact-generated moons to inward tidal decay on very
short timescales (e.g., approximately years for an Earthsized planet and a lunar-mass satellite with a = 3 R ). The
long-term survival of a satellite that initially forms outside
the co-rotation radius is not guaranteed, and will depend on
the interplay of solar and satellite tides, which both act to
slow the planet’s rotation and cause aco to evolve outward
(Burns, 1973; Ward and Reid, 1973). If aco overtakes a, the
eventual fate of the satellite will again be a collision with
the planet (although in this case the timescale for the orbital decay can be much longer).
4.
OPEN ISSUES
Models of the final stage of terrestrial accretion are generally successful, producing a few planets from a system of
planetary embryos in a timeframe consistent with isotopic
constraints for the age of the Earth and Moon. Some disparities between current predictions and the terrestrial planets persist, in particular with regard to accounting for the
Canup and Agnor: Accretion of the Terrestrial Planets
very low eccentricities of Earth and Venus. However, such
differences are likely the result of simplified assumptions
employed by the late-stage models, which to date have all
ignored the production and dynamical influence of small
material. From studies of the runaway growth phase, it is
well known that the general role of a background population of small bodies is to decrease the relative velocities (and
therefore eccentricities and inclinations) of the largest bodies via dynamical friction. A key open question is then how
the presence of a planetesimal swarm will alter the dynamical environment in which embryos accrete to form planets.
Whether the influence of small bodies in the late stage could
be sufficient to yield Earth-like planets with nearly circular
orbits remains uncertain, and addressing this question will
require an improved understanding of both the generation
of debris during planet-scale collisions, and the persistence
of a background population as it co-evolves with a system
of embryos.
The outcomes of planet-scale collisions are still poorly
understood, and the type of hydrodynamic simulations utilized to model the lunar-forming impact could be extended
to a broader range of collisional parameter space in order
to better generalize impact outcomes. Accounting for the
amount of ejected debris, its velocity, and its angular momentum will affect estimates of both the potential background population of small material and planetary spin rates.
Current scaling relationships derived for meter- to 100-kilometer-sized targets relate QD*, the critical specific energy
required to disperse one-half the target mass, to target size
(e.g., Benz and Asphaug, 1999). Given these relationships
for QD*, and assuming a linear relationship between the specific impact energy, Q, and the amount of material ejected
(as implied in Love and Ahrens, 1996), ACL99 estimated
the fraction of material that would be dispersed for each of
the collisions in their simulations. This fraction ranged from
~1% to 20% of the total mass involved in a collision (their
Fig. 15). SPH simulations of the lunar-forming impact (Cameron, 2000; Canup et al., 2000) suggest that the amount of
ejected mass will also depend upon target-to-impactor mass
ratio and impact angular momentum, in addition to the specific impact energy.
Given that some nonneglible fraction of material is
ejected during collisions between planetary embryos, the
question is then how long this material will persist as a
background population during the final stage. Much of the
debris ejected in an impact might rapidly reaccumulate onto
its parent body during subsequent orbital passes, unless it
is dynamically perturbed by other embryos or the giant planets. If an impact was in fact responsible for the loss of much
of Mercury’s mantle, we have at least one example case in
which the great majority of the ejected material must not
have reaccreted onto the source planet. Assuming immediate reaccretion is avoided, the timescales for sweep-up of
the remaining debris could be comparable to (or longer than)
the time between embryo collisions. A simple fragmentation model, which assumed that a few percent of the mass
127
involved in each collision was ejected as fragments, was
included in two-dimensional N-body simulations of the
accretion of embryos in the terrestrial region by Alexander
and Agnor (1998). Interestingly, that work found that the
ejected fragments quickly contained up to 50% of the total
system mass. Another possible contribution to a late-stage
background population could be leftover remants from the
earlier runaway stage. The Weidenschilling et al. (1997)
simulation did not include fragmentation, and its prediction
that 10% of the system mass remained in small bodies after a million years should therefore be viewed as a lower
limit. Recent orbit integrations of test particles placed
throughout the current terrestrial region (see Fig. 5 in chapter by Hartmann et al., 2000) find that near 50% of the
particles remain in the 0.7–1.3-AU region after 100 m.y.
Modeling the possible interactions between a system of
embryos and a background population of small material for
timescales ~108 yr presents many computational and algorithmic challenges for the next generation of planet formation simulations. An important step in this regard will involve an improved understanding of the transition from the
runaway growth period to the late stage. While the embryos
in the Weidenschilling et al. (1997) simulations (which contained about 90% of the system mass) remained in loweccentricity, stable orbits for times longer than 105 yr, the
evolution of the same embryos using direct integrations in
ACL99 (but ignoring the 10% of mass contained in smaller
material) destabilized on times ~104 yr. Determining the
source of these differences will be closely related to efforts
to determine the role of small material in late-stage accretion.
While important processes relevant to the final accumulation of the terrestrial planets are thus not yet completely
understood, the basic predictions of existing models are in
fairly good agreement with the current terrestrial planets.
Additional attention to issues neglected in simulations performed to date may help to resolve the outstanding question of forming low-eccentricity planets. But in general, a
stochastic phase dominated by large impacts appears an
inevitable consequence of the evolution of tens to hundreds
of planetary embryos into a final few planets. Barring a
major change in our understanding of the outcome of the
preceeding runaway growth stage, this conclusion appears
robust, and will likely persist even as late-stage accretion
models are further refined.
Acknowledgments. The authors wish to acknowledge helpful reviews by G. Wetherill and J. Chambers, and support from
NASA’s Origins of Solar Systems and Graduate Student Researchers programs.
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