Origin of Terrestrial Planets and the Earth-Moon Systemsappho.eps.mcgill.ca/~olivia/666/2009/Origin of the Terrestrial... · gravitational surface sticking forces (such as van der

According to current theories, the overall architecture ofour solar system was established more than 4 billion

years ago through an era of planet formation lasting from10 million to several hundred millions of years. Before webegan learning about other planetary systems, it was nat-ural to assume that our own was quite representative. Butin the past decade, the discovery of about 100 planetarysystems around other stars has challenged this view; thesesystems display a broad variation in structure and mostdo not closely resemble our own (see the article by TristanGuillot on page 63).1

Observational capabilities currently are limited to thedetection of giant, Jupiter-sized planets around otherstars, so that we are uncertain of the existence and distri-bution of smaller Earth-like planets in such systems. Thusplanetary scientists rely on our own solar system as thecase study for understanding the formation of terrestrial,solid planets and their satellites—such as Earth and theMoon. But our solar system as a whole may not be partic-ularly characteristic, and recent observations suggest thatthe process of planet formation is one from which manypotential outcomes may emerge. Theoretical models andcomputer simulations that strive to recreate the planetaryformation process must therefore be able to account forboth the primary characteristics of our system and the ap-parent diversity of extrasolar systems.

The planetesimal hypothesisUnderstanding of stellar formation processes and obser-vations of other young stars suggest that the early SolarSystem consisted of the newly formed Sun and an orbitingdisk of gas and dust (see figure 1). If one assumes it had aroughly solar composition, such a disk—also referred to asthe solar nebula—would contain a mass in hydrogen gasabout 100 times that contained in solid particles. Isotopicdating of the oldest known meteorites indicates thatmacroscopic solids began to form within the gas-rich solarnebula about 4.56 billion years ago. Observations of otherstellar systems suggest that the Sun’s hydrogen-rich neb-ula would have been lost—possibly due to a strong solar

wind or photoevaporation—afterabout 1–10 million years. At that time,the protoplanetary disk transitionedfrom one whose mass was predomi-nantly gas to one composed solely ofsolid objects orbiting the Sun.

In this context, how did our solarsystem’s large inner objects—Mercury,Venus, Earth, the Moon, and Mars—originate? The so-called planetesimal

hypothesis, which in its modern form has been developedover the past 40 years, proposes that solid planets growfrom initially small particles in the protoplanetary diskthrough collisional accumulation, or accretion. As solid ob-jects orbit the Sun, mutual gravitational interactions andinteractions with the gaseous nebula cause their ellipticalorbits to cross, which leads to collisions. The outcome of agiven collision depends primarily on the ratio of the im-pact speed, vimp, to the gravitational escape velocity of thecolliding objects, vesc, where vesc ⊂ √2G(m1 + m2)/(R1 + R2),G is Newton’s gravitational constant, and mi and Ri are theobject masses and radii. The impact speed is a function ofboth vesc and the relative velocity vrel between the objects atlarge separation with v2

imp ⊂ v2esc + v2

rel, so that vimp is al-ways greater than or equal to vesc.

For vimp � vesc, collisions result in rebound, erosion, oreven fragmentation, while for lower-impact velocities,with vimp ~ vesc, energetically dissipative collisions lead tothe formation of gravitationally bound aggregates. Re-peated collisions with low impact velocities thus allow forthe accretional growth of larger and larger objects.

Terrestrial planet accretion in our solar system is typ-ically described in three stages: growth of approximatelykilometer-sized “planetesimals” from dust and small par-ticles; accretion of planetesimals into planetary embryoscontaining around 1–10% of the Earth’s mass M−; and thecollision of tens to hundreds of planetary embryos to yieldthe final four terrestrial planets.

The processes that control growth during the firststage are the least well understood. In general, the inter-action of small, subkilometer-sized particles with the neb-ular gas causes their mutual impact velocities to greatlyexceed their gravitational escape velocities, so that growthduring two-body collisions must instead rely on either non-gravitational surface sticking forces (such as van derWaals or electrostatic forces) between the colliding parti-cles or the collective gravitational influence of neighboringparticles. Such collective effects could become important ifdynamical mechanisms exist that can concentrate solidsin a region of the disk; in that case, an enhanced local spa-tial density of particles can allow groups of particles to col-lapse under their self-gravity to form larger objects.Whether kilometer-sized planetesimals were formed bysticking and binary collisions or by gravitational instabil-ity—or both—is a subject of active research.2

56 April 2004 Physics Today © 2004 American Institute of Physics, S-0031-9228-0404-030-X

Robin Canup ([email protected]) is assistant director ofthe department of space studies in the instrumentation andspace research division at the Southwest Research Institute.She is based in Boulder, Colorado.

Increasingly sophisticated computer simulations showhow the four solid planets could have emerged throughcollisions and accretion. One late, giant collision with Earth is the likely origin of the Moon.

Robin M. Canup

Origin of Terrestrial Planetsand the Earth–Moon System

Planetesimals to planetary embryosOnce planetesimals become large enough (approximatelykilometer-sized) for their dynamics and collisional growthto be controlled by gravitational interactions, a much bet-ter understanding exists of how growth proceeds. That un-derstanding is due in large part to improvements in com-putational modeling techniques.

The rate of accretion is primarily controlled by therate of collisions among orbiting planetesimals. Consideran annulus in the protoplanetary disk centered at some or-bital radius a with volume V ⊂ Ah, where A is the mid-plane area and h is the disk thickness. If the annulus con-tains N small planetesimals with some characteristicrandom velocity vran relative to that of a circular orbit atradius a, then a larger embedded object of radius R willaccumulate the small particles at an approximate rate

(1)

where Fg ⊂ 1 + (vesc/vran)2 is the gravitational enhancementto the collisional cross section due to two-body scattering,n ⊂ N/A is the number of particles per surface area in thedisk, and h � vran/W, with W the Keplerian orbital angularvelocity. Equation 1 has its roots in kinetic theory and isknown as the “particle-in-a-box” collision rate. In the caseof an orbiting “box” of planetesimals, the random particlevelocities arise from the planetesimals’ orbital eccentrici-ties and inclinations and are analogous to the random ther-mal velocities of gas molecules confined to some volume. Asthis simple expression shows, the rate of collisions, andtherefore of accretional growth, depends on the local Kep-lerian orbital velocity (which increases with decreasing dis-tance from the Sun, so that regions closer to the Sun gen-erally accrete more rapidly), the number density ofplanetesimals, and their sizes and relative velocities.

The mass and velocity distributions of a swarm ofplanetesimals are themselves dynamically coupled. Gravi-

tational scattering amongparticles tends to increasevran, while energy dissipa-tion during collisions anddrag exerted on particlesby the gaseous nebula bothact to decrease vran. For adistribution of objects, anequilibrium between theseprocesses yields vran on theorder of the escape velocityof the object class that con-tains the majority of thetotal mass.

Interactions amongorbiting particles of differ-ent masses tend to drive

the system toward a state of equipartition of kinetic en-ergy, with smaller particles having typically higher vranand the largest objects having the lowest. This effect isknown as dynamical friction: A swarm of background smallparticles acts to damp the velocities of the largest objects.From equation 1, if the largest objects in a given region ofthe disk also have the lowest velocities, then their colli-sional cross sections will be significantly enhanced, com-pared to those of smaller objects, due to the gravitationalfocusing factor, Fg. For large objects, vesc will be large com-pared to vran, which will be controlled by the smaller plan-etesimals that still contain most of the system’s mass. Thelargest objects thus grow the fastest, and a single objecttypically ends up “running away” with the great majorityof the total available mass in its annular region in the disk.Through this so-called runaway growth, approximatelylunar-sized objects, containing roughly 1% of M−, grow inthe inner Solar System in as little as 105 years.

Once a large object has consumed most of the mass inits annulus, growth slows, primarily due to the reducedamount of locally available material and planetesimal ve-locities that have increased to around vesc of the largest em-bryo. Figure 2 shows the predicted distribution of plan-etesimal masses in a region extending between the currentorbits of Mercury and Mars from a 106-year accretion sim-ulation performed by Stuart Weidenschilling (of the Plan-etary Science Institute) and colleagues.3 After a millionyears, 22 large planetary embryos have formed in theinner Solar System and contain 90% of the total mass. Theembryos are radially well separated on nearly circular,coplanar orbits, with each containing a mass of at least1026 g (for comparison, M− � 6 × 1027 g).

Late-stage terrestrial accretionGiven our four terrestrial planets, the state shown in fig-ure 2 with tens of “miniplanets” must have been a transi-tory one. The gaseous component of the solar nebula is ex-pected to have dispersed after 106–107 years, and with itwent an important source of velocity damping for small

Collision rate ran g esc

ran

…v N R F

Vn R

vv

pp

22

2

21( )

≈ ( ) +

W ,

http://www.physicstoday.org April 2004 Physics Today 57

Figure 1. The early solarnebula, as painted byWilliam Hartmann of thePlanetary Science Institute.Dust and small particlesorbiting the young Sun ac-crete to form kilometer-sized “planetesimals,” thefirst step in the formationof the terrestrial planets inthe inner Solar System.

~

objects. As the random velocities of small objects increased,their ability to damp the velocities of the larger embryosthrough dynamical friction would decrease. Mutual grav-itational interactions among the embryos would then be-come more potent and lead to the excitation of their orbitaleccentricities, mutual orbit crossings, and finally em-bryo–embryo collisions and mergers. As the embryos col-lided and accreted, the number of planets would decreaseand the dynamical stability of the system would increase,until finally a few planets on stable orbits remained. Thefinal configuration of planets would thus be established ina stochastic “process of elimination,” and a planet’s dy-namical characteristics—its mass, orbital radius, rotationrate, and rotational axis, for example—would be greatlyinfluenced by its last few large collisions.

Models of the accretion of planetary embryos into ter-restrial planets were pioneered in the 1980s by GeorgeWetherill (Carnegie Institution of Washington’s Depart-ment of Terrestrial Magnetism), who utilized a MonteCarlo approach for tracking embryo orbits. Such statisti-cal models use analytic approximations to estimate thelikelihood of collisions and to describe the effects of mu-tual gravitational perturbations among the planetary em-bryos. Those techniques, however, could not incorporatesome important dynamical effects, including the potentialfor successive and correlated close encounters between agiven pair of embryos.

In the past decade, late-stage accretion models havebeen revolutionized by advances in methods for directly in-tegrating the equations of motion of objects that orbit amore massive central primary. The key breakthrough wasmade in 1991 by Jack Wisdom (MIT) and Matthew Hol-man (now at the Harvard–Smithsonian Center for Astro-physics). The Wisdom–Holman mapping (WHM) methodallows for accurate integrations with relatively long inte-gration time steps.4 Whereas classic orbit integrators re-quire 500–600 time steps per orbit, WHM saves time byanalytically tracking the Keplerian motion and integrat-ing only the small perturbations that arise from themasses of the orbiting objects, so that only a few tens oftime steps per orbit are needed to ensure accuracy. TheWHM method is also symplectic: Although it does not ex-

actly conserve energy, the predicted energy oscillatesabout a fixed value so that energy errors do not accumu-late with time. Modern techniques5 based on the WHMmethod can track the dynamical evolution of systems ofseveral hundred accreting planetary embryos for morethan 108 years.6,7 Such simulations follow not only the ac-tual orbit of each embryo but also the dynamical encoun-ters between embryos, including collisions or close passes.

Figure 3 shows the final planetary systems producedin eight late-stage accretion simulations recently per-formed using such direct integration techniques by JohnChambers of NASA’s Ames Research Center.7 The simu-lated systems display a wide variety of architectures butare generally similar to our solar system’s terrestrial plan-ets in terms of the number of final planets and theirmasses. A notable difference is that the planets in the sim-ulated systems have eccentricities and inclinations muchhigher than those of Earth and Venus, whose orbits arevery close to circular and coplanar. This difference is likelya result of simplifications made in most of the models todate, namely ignoring the influence of potential coexistingsmall objects or a remnant of the gas nebula in the latestage. Both would generally act to decrease eccentricitiesand inclinations. While it is conceptually simplest to con-sider a sharp division between the middle and late stagesso that in the late stage such effects can be ignored, Na-ture may not be so accommodating. Recent models that in-clude more initial objects or a small portion of the nebulargas have found systems with orbits closer to those in ourSolar System, although accounting for the nearly circularorbits of Earth and Venus remains an open issue.

A seemingly inherent feature of the late stage is giantimpacts, in which lunar- to Mars-sized objects mutually col-lide to yield the final few terrestrial planets. Figure 4 showsthe mass of impactors as a function of time for collisions thatoccurred in 10 late-stage simulations performed by CraigAgnor, Harold Levison, and me at the Southwest ResearchInstitute.6 The “Earths” produced in those simulations re-quire an average of 10–50 million years to accrete, with thelargest late-stage impacts occurring predominantly in the107- to 108-year time interval. The collisions display a ran-dom distribution of impact orientation, so that a final planet

58 April 2004 Physics Today http://www.physicstoday.org

106

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Figure 2. Planetesimal ac-cretion into planetary em-bryos is thought to be anintermediate stage of ter-

restrial planet formation. Inthese simulation results byStuart Weidenschilling and

colleagues, initial plan-etesimals 15 kilometers in

radius (~1019 g) accreteinto objects containing

about 10% of Earth’s mass(about 6 × 1027 g) after

about 106 years. The plotshows the number N of

solar-orbiting objects as afunction of their mass and

their orbital semimajoraxis, given in units of

Earth’s semimajor axis (de-fined to be 1 astronomicalunit, or 1 AU). The largest

22 planetary embryos con-tain 90% of the total mass.

(Adapted from ref. 3.)

is as likely to end up rotating in the prograde sense (that is,in the same sense as its orbit about the Sun, as is the casefor Mercury, Earth, and Mars) as in a retrograde sense (asis the case for Venus).

Origin of the Earth–Moon systemAccording to current thinking, the growing Earth experi-enced one such late-stage impact that ejected into orbit thematerial from which our Moon later formed. The giant im-pact theory for lunar origin (see the box on page 60) is fa-vored for its ability to account for the high angular mo-mentum of the Earth–Moon system and for the iron-poorcomposition of the Moon. Reconciling the type of impact re-quired with the actual Earth–Moon system thus providesan important benchmark for terrestrial-accretion models.

The impact theory proposes that the collision that cre-ated the Moon was also the primary source of the angularmomentum L−–M of the Earth–Moon system. The angularmomentum delivered by an impactor of mass gMT is

(2)

where b ⊂ sin j is the impact parameter normalized to thesum of the impactor and target radii, j is the angle betweenthe surface normal and the impact trajectory (so that agrazing impact has b ⊂ 1 and j ⊂ 90°), MT is the total com-bined mass of the impactor and target, and g is the frac-tion of the total mass contained in the impactor. From thisequation, a minimum impactor mass of about 0.08 M− isrequired to yield L−–M in the limit of a b ⊂1 grazing colli-sion for MT � M− and vimp � vesc. Imparting the Earth–Moon angular momentum by an oblique impact with Earththus implies an impactor roughly the size of Mars—thatis, about 0.1 M−.

For impact-ejected material to achieve Earth-boundorbit, some modification to standard ballistic trajectoriesmust occur, otherwise ejecta launched from the planet’ssurface either re-impacts or escapes. Two nonballistic ef-fects are gravitational torques due to mutual interactionsamong ejected material or to interaction with a nonspher-ical distortion of the target planet, and pressure gradientsassociated with vaporization. These effects become impor-tant for large impacts: the first when the impactor is a sig-nificant fraction of the target’s mass, and the second whenthe specific impact energy (that is, the impact energy perunit mass) exceeds the latent heat of vaporization for rock,about 1011 erg/g , which occurs for vimp � 5 km/s.

For a lunar-forming impact, the expected impact ve-locity is around 10 km/s, and both torques and vaporiza-tion could be important. Modeling potential lunar-formingimpacts thus requires a full hydrodynamic approach thatincludes both an explicit treatment of self-gravity and anequation of state appropriate to describe the thermody-namic response of material subjected to very high impactenergies and pressure.

Models of lunar-forming giant impacts have primarilyused smooth particle hydrodynamics. SPH, developed overthe past 25 years,8 represents a significant advance in themodeling of deforming and spatially dispersing hydrody-namic systems, including giant impacts. SPH is a La-grangian method, which is advantageous because its nu-

merical resolution tracks the spatial distribution of theevolving material, and compositional histories can be easilyfollowed. In SPH, a planetary object is represented by agreat number of spherically symmetric overlapping “parti-cles,” each containing a quantity of mass of a given compo-sition, whose three-dimensional spatial distribution is spec-ified by a density-weighting function, the kernel, and by thecharacteristic width of the particle, the smoothing length.

For impacts between planet-scale objects, each parti-cle’s kinematic variables (position and velocity) and statevariables (internal energy and density) evolve due to grav-ity, compressional heating and expansional cooling, andshock dissipation. The forces between particles thus in-clude attraction due to gravity, which acts inversely withthe squared distance between particles, and a repulsivepressure for adjacent particles closer than approximatelythe sum of their smoothing lengths. The equation of staterelates a particle’s specific internal energy and density topressure as a function of input material constants.

The use of SPH in modeling lunar-forming impactswas pioneered by Willy Benz (now at the University ofBern), Alastair Cameron (now at the University of Ari-zona), and colleagues in the 1980s.9 The general approachin performing such numerical impact experiments hasbeen to vary the four impact variables—b, MT, g, and vimp—of equation 2 in a series of simulations to determine whatsets of impact conditions yield the most favorable results.The challenge is that the possible parameter space is largeand individual impact simulations are computationally in-tensive. Recent works10,11 have successfully identified im-pacts capable of simultaneously accounting for the massesof Earth and the Moon, the Earth–Moon system angularmomentum, and the lunar iron depletion.

Figure 5 shows a time series of a lunar-forming impactsimulation11 using high-resolution SPH and a sophisticatedequation of state first developed at Sandia National Labo-ratories and recently improved by Jay Melosh of the Uni-versity of Arizona’s Lunar and Planetary Laboratory12 to in-clude a treatment of both molecular and monatomic vaporspecies. The simulation offers the most realistic treatmentof vaporization of any SPH simulations performed to date,and each impact simulation requires several days of com-putational time on a high-speed workstation.

In the impact simulation, the colliding objects are de-scribed by a total of 60 000 SPH particles. The normalizedimpact parameter is b ~ 0.7 (that is, a 45° impact angle);

L L bMM

vMimp

T imp

esc

≈

⊕−

⊕

1 30 1

5 3

..

,/

gv

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Figure 3. Final planetary systems generated by numericalsimulations, compared to the actual inner planets in our

solar system (middle panel). (Courtesy of John Chambers,NASA’s Ames Research Center).

g

the impactor contains 1.2 times the mass of Mars; the im-pact velocity is 9 km/s; and the impact angular momentumLimp = 1.25 L−–M. Before the impact, both objects are dif-ferentiated into iron cores and silicate mantles—a rea-sonable assumption given the amount of heating thatwould have been induced as the objects accreted to sizesthis large. Both are 30% iron by mass.

After the initial oblique impact (figure 5a, after about20 simulated minutes), a portion of the impactor is shearedoff and continues forward ahead of the impact site. A dis-torted arm of impactor material extends to a distance ofseveral Earth radii, and the proto-Earth surface and theinner portions of this arm rotate ahead of the more distantmaterial (figure 5b at 80 minutes). This configuration pro-vides a positive torque to the outer portions of material,helping them to gain sufficient angular momentum toachieve bound orbit. In the 3- to 5-hour time frame, theinner portions of the orbiting material (composed prima-rily of the impactor’s iron core) gravitationally contractinto a semicoherent object (figure 5c) that collides again

with the planet after about 6 hours(figure 5d). Thus at this point, mostof the impactor’s iron has been re-moved from orbit. The outer clumpof impactor material (figure 5e)—composed entirely of mantle mate-rial—passes close to Earth and issheared apart by planetary tidalforces that leave a circumplanetarydisk after about a day (figure 5f at27 hours).

At the end of this impact, the re-sulting planet and disk are a closeanalog to that needed to produce the

Earth–Moon system. The planet contains about an Earthmass and its rotational day is about 4.6 hours, and the or-biting disk contains about 1.6 lunar masses. Of the orbit-ing material, approximately a lunar mass has sufficientangular momentum to orbit beyond a distance known asthe Roche limit, located about 3 Earth radii from the cen-ter of the Earth for lunar-density material. Inside theRoche limit, planetary tidal forces inhibit accretion; it iswithin this distance, for example, that planetary ring sys-tems are found around the outer planets. Accretion willoccur for material orbiting beyond the Roche limit, so theorbiting disk produced by this impact would be expectedto yield a lunar-sized moon at an initial orbital distance ofabout 3–5 Earth radii. The short 4.6-hour day of postim-pact Earth causes the distance at which the orbital periodis equal to the terrestrial day—the so-called synchronousradius—to fall within the Roche limit, at about 2.2 Earthradii. Because the Moon forms beyond this distance, tidalinteraction with Earth will lead to the expansion of theMoon’s orbit while Earth’s rotation slows.

The impact must also account for the Moon’s iron de-pletion. Whereas the colliding objects in figure 5 both con-tained 30% iron by mass, the orbiting material is derivedoverwhelmingly from the outer mantle portions of the im-pactor. The protolunar disk contains only a few percent ironby mass, with the iron originating from the impactor’s core.The lunar-forming impact dramatically raises Earth’s tem-perature, with about 30% of the planet’s mass heated to tem-

60 April 2004 Physics Today http://www.physicstoday.org

Figure 4. Giant impacts in the late stages of planetary accre-tion are thought to play a critical role in determining the ul-timate properties of the emerging planets. Shown here arethe collisions produced in 10 simulations of the accretion ofterrestrial planets in our solar system. The mass of the im-pactor in units of Earth’s mass is shown as a function oftime. (Adapted from ref. 6.)

105 106 107 108

TIME (years)

0.4

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0.2

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IMPA

CT

OR

MA

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(/

)m

M�

Although the Moon is by far the most familiar satellite, it isa rather unusual planetary object. Whereas solid objects

in the inner Solar System typically contain about 30% iron bymass, the Moon’s low density implies that it is severely iron-depleted, with an iron core that likely is only 1–3% of itsmass. The Moon is further distinguished by its large size rela-tive to its parent planet: It contains about 1% of Earth’s mass.Mercury, Venus, and Mars, in contrast, lack large moons. Theangular momentum of the Earth–Moon system is also unusu-ally high. If it were contained solely in Earth’s rotation, itwould yield an approximately 4-hour day—much shorterthan those of the other inner planets. And, due to tidal inter-actions with Earth, the Moon’s orbital radius has expandedmore significantly over its history than any other planetarysatellite, so the Moon in its early stages was about 15 timescloser to Earth than it is currently.

Prior to the Apollo era, three lunar origin hypotheses pre-dominated: capture, fission, and coformation.17 However,each of those models failed to account for one or more of themajor characteristics of the Earth–Moon system. Capturing anindependently formed Moon into an Earth-bound orbit doesnot offer a natural explanation for the lunar iron depletion,and that scenario appears dynamically unlikely. In fission, arapidly spinning Earth becomes rotationally unstable, causinglunar material to be flung out from the equator. That hypoth-

esis requires the Earth–Moon angular momentum to be sev-eral times higher than its actual value. Coformation supposesthat the Moon grew in Earth orbit from the sweeping up ofsmaller material from the solar nebula. Although coformationmodels were successful in producing satellites, they didn’treadily explain both the lunar iron deficiency and theEarth–Moon angular momentum, since growth via manysmall impacts typically delivers little net angular momentumand produces slow planetary rotation.

In 1975–76, two independent groups proposed an alterna-tive model. William Hartmann and Donald Davis (both at thePlanetary Science Institute) suggested that the impact of alunar-sized object with the early Earth had ejected into Earth-bound orbit material from which the Moon then formed. Ifsuch material were derived primarily from the outer mantlesof the colliding objects, then an iron-depleted moon might re-sult. Alastair Cameron (now at the University of Arizona) andWilliam Ward (now at the Southwest Research Institute) fur-ther recognized that if the collision had been a grazing one bya much larger, planet-sized impactor—one roughly the size ofMars, containing � 10% of Earth’s mass—the angular mo-mentum delivered by the impact could account for Earth’srapid initial rotation. The concepts described in those re-searchers’ works contain the basic elements of the now fa-vored giant impact theory of lunar origin.18

Theories of the Moon’s Origin

peratures in excess of 7000 K. Thus postimpact Earth wouldhave been engulfed in a silicate vapor atmosphere, with themajority of the planet likely in a molten state.

Since large collisions appear typical in the late stagesof terrestrial planet formation, how often do such impactsproduce satellites? Results of impact simulations suggestthat low-velocity, oblique collisions (those with b > 0.5) be-tween planetary embryos generate some amount of orbit-ing material around the larger of the colliding objects. Forrandom impact orientation, the most likely value for b is0.7 (which is what has been found to be optimal for theMoon-forming impact), and 75% of all collisions will haveb > 0.5. Thus the inner Solar System may have initiallycontained many impact-generated satellites, with the ma-jority lost as they were destroyed or dislodged by subse-quent impacts or as their orbits contracted due to tidal in-teractions with a slow- or retrograde-rotating planet.

Isotopic timing constraintsThe general agreement between the type of impact appar-ently required to yield Earth’s Moon and those predictedby accretion simulations provides an important corrobo-ration of current models of solid-planet formation. Otherimportant pieces of independent evidence are the forma-

tion times implied by the isotopic compo-sitions of Earth and the Moon.

A key development in the pastdecade has been the use of thehafnium–tungsten chronometer for dat-ing planetary core formation and giantimpacts.13 Radioactive 182Hf decays to182W with a halflife t1/2 of 9 million years.A critical distinction between hafniumand tungsten is that hafnium islithophilic (“silicate-liking,” tending to beconcentrated in oxygen-containing com-pounds such as silicates), whereas tung-sten is siderophilic (“iron-liking,” ortending to enter metallic phases). Duringcore formation in a planetary object,whatever tungsten is present in theplanet’s mantle—radiogenic 182W as wellas nonradiogenic W-isotopes such as 183Wand 184W—will be largely removed fromthe mantle and incorporated into theiron core, while hafnium will remain inthe mantle. Thus the mantle of a differ-entiated planetary object will have aHf/W ratio larger than that of bulk Solar-

System composition. The bulk Solar-System compositioncan be inferred from the composition of primitive mete-orites, called chondrites.

The Hf/W ratio and W-isotope compositions of a dif-ferentiated object such as Earth provide timing con-straints on the formation of its core and potentially on thetiming of its last large collision. The tungsten compositionof chondrites includes both nonradiogenic isotopes and182W produced by the decay of primordial 182Hf, and thechondritic W-isotope composition provides a referencevalue believed indicative of early Solar System abun-dances. If a planet’s core formed on a timescale shorterthan about 5t1/2, its mantle, compared to chondrites, wouldcontain excess 182W (relative to the abundance of other Wisotopes) produced by decay of 182Hf after core formation.If the core formed later when 182Hf was essentially extinct,all isotopes of W would have been equally depleted by in-corporation into the iron core, leaving the mantle with achondritic W-isotopic composition.

Earth’s mantle contains an excess of 182W compared tothe most recent assessments of chondritic W-isotope compo-sition,14,15 which implies that Earth’s accretion and core for-mation were largely completed in 10–30 million years (seePHYSICS TODAY, January 2003, page 16). The Moon has a

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KFigure 5. A single giant impact can ac-count for the masses and angular momen-tum of the current Earth and Moon. In thisstate-of-the-art simulation, an object 1.2times the mass of Mars collides with earlyEarth. Part of the impactor’s mantle mate-rial is ejected into an orbiting disk, fromwhich the Moon will accrete. Panels (a)through (f) look down onto the plane ofthe impact at times of 0.3, 1.4, 4.9, 5.9,13.5, and 27.0 hours after the collision.Color scales with temperature. Red indi-cates temperatures above 6440 K, whichcorresponds, in the case of silicate parti-cles, to a completely vaporized state fororbits in the disk where the density andpressure are low. Distances are in units of1000 km. (Adapted from ref. 11.)

similar Hf-W–derived formation time of about 25–30 millionyears.14 These specific timings can be affected by model as-sumptions, such as the degree to which accreting materialisotopically equilibrates with Earth’s mantle.16 However, ingeneral, the Hf–W timings and the late-stage dynamicalmodels both yield estimates in the 10- to 50-million-year timeinterval for planetary accretion, giant impacts, and the finalepisodes of terrestrial-planet core formation. Broadly simi-lar formation times of 107 to 108 years also result from otherisotopic systems such as uranium–lead, iodine–xenon, andplutonium–xenon.16 If the terrestrial planets had grown totheir final sizes through runaway growth, their formationtimes would have been much shorter, on the order of 106

years or less. The agreement between the dynamically andgeochemically derived timings thus provides significant sup-port to the existence of a protracted phase of late accretiondominated by large impacts.

Whereas early models proposed that Earth-like plan-ets form through the orderly accretion of nearby small ma-terial in the protoplanetary disk, current work insteadsuggests that solid planets are sculpted by a violent, sto-chastic final phase of giant impacts. The implication isthat our terrestrial planets—and Moon—may only repre-sent one possible outcome in a wide array of potentialsolar-system architectures. With future space missions(such as NASA’s Terrestrial Planet Finder) devoted to de-tecting Earth-like planets around other stars, we maysomeday be able to directly test such concepts.

The author gratefully acknowledges support from NSF andNASA.

References1. G. W. Marcy, P. R. Butler, Annu. Rev. Astron. Astrophys. 36,

57 (1998); M. Mayor, Annu. Rev. Astron. Astrophys (in press).2. S. J. Weidenschilling, J. N. Cuzzi, in Protostars and Planets

III, E. H. Levy, J. I. Lunine, eds., U. of Ariz. Press, Tucson(1993), p. 1031; W. R. Ward, in Origin of the Earth and Moon,R. M. Canup, K. Righter, eds., U. of Ariz. Press, Tucson(2000), p. 75; A. N. Youdin, E. I. Chiang, Astrophys. J. 601,1109 (2004).

3. S. J. Weidenschilling, D. Spaute, D. R. Davis, F. Marzari, K.Ohtsuki, Icarus 128, 429 (1997).

4. J. Wisdom, M. Holman, Astron. J. 102, 1528 (1991).5. M. J. Duncan, H. F. Levison, M. H. Lee, Astron. J. 116, 2067

(1998); J. E. Chambers, Monthly Not. Royal Astron. Soc. 304,793 (1999).

6. C. B. Agnor, R. M. Canup, H. F. Levison, Icarus 142, 219(1999).

7. J. E. Chambers, Icarus 152, 205 (2001).8. J. J. Monaghan, Annu. Rev. Astron. Astrophys. 30, 543

(1992).9. See, for example, the review by A. G. W. Cameron, in Origin

of the Earth and Moon, R. M. Canup, K. Righter, eds., U. ofAriz. Press, Tucson (2000), p. 133.

10. R. M. Canup, E. Asphaug, Nature 412, 708 (2001); R. M.Canup, Annu. Rev. Astron. Astrophys. (in press).

11. R. M. Canup, Icarus 168, 433 (2004).12. H. J. Melosh, Lunar Planetary Sci. Conf. 31, 1903 (2000).13. A. N. Halliday, D. C. Lee, S. B. Jacobsen, in Origin of the

Earth and Moon, R. M. Canup, K. Righter, eds., U. of Ariz.Press, Tucson (2000), p. 45.

14. Q. Yin, S. B. Jacobsen, K. Yamashita, J. Blichert-Toft, P.Telouk, F. Albarede, Nature 418, 949 (2002).

15. T. Kleine, C. Münker, K. Mezger, H. Palme, Nature 418, 952(2002).

16. A. N. Halliday, Nature 427, 505 (2004).17. W. K. Hartmann, R. J. Phillips, G. J. Taylor, eds., Origin of

the Moon, Lunar and Planetary Institute, Houston, TX(1986).

18. R. M. Canup, K. Righter, eds., Origin of the Earth and Moon,U. of Ariz. Press, Tucson (2000). �

62 April 2004 Physics Today

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