DOI: 10.1126/science.1225542, 1047 (2012);338 Science

Matija Cuk and Sarah T. StewartFollowed by Resonant DespinningMaking the Moon from a Fast-Spinning Earth: A Giant Impact

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Making the Moon from a Fast-SpinningEarth: A Giant Impact Followedby Resonant DespinningMatija Cuk´ *† and Sarah T. Stewart

A common origin for the Moon and Earth is required by their identical isotopic composition.However, simulations of the current giant impact hypothesis for Moon formation find that mostlunar material originated from the impactor, which should have had a different isotopic signature.Previous Moon-formation studies assumed that the angular momentum after the impact wassimilar to that of the present day; however, Earth-mass planets are expected to have higher spinrates at the end of accretion. Here, we show that typical last giant impacts onto a fast-spinningproto-Earth can produce a Moon-forming disk derived primarily from Earth’s mantle. Furthermore,we find that a faster-spinning early Earth-Moon system can lose angular momentum and reachthe present state through an orbital resonance between the Sun and Moon.

The origin of the Moon by a giant impact(1, 2) is the leading theory to explainmultiple features of the Earth-Moon sys-

tem (3), including the current angular momen-tum, the Moon’s small core compared with thoseof rocky planets, and the compositional similar-ity between the Moon and Earth. In the canon-ical scenario (4), a ~0.1 Earth-mass (ME) bodyobliquely strikes the proto-Earth near the escapevelocity to generate a circumterrestrial debris diskfrom which the Moon accretes. Since the for-mulation of the giant impact hypothesis, ever-improving analytical techniques have revealedthat the Moon and Earth are identical in theiroxygen, tungsten, chromium, and titanium iso-topes (5–8). These isotope systems show con-siderable variations between planetary bodiesand most meteorite groups; thus, the simplestexplanation for the isotopic similarity is that theMoon was formed from Earth’s mantle (9). Incontrast, giant impact simulations find that thelunar disk is predominantly [>60 weight percent(wt %)] composed of material originating fromthe impactor (4, 10, 11), which is expected tohave a different isotopic signature than Earth.

To reconcile the impact simulations with theobservations, post-impact isotopic equilibrationby mixing material between the lunar disk andEarth has been suggested as a means to miti-gate an initial compositional difference (12, 13).However, based on recent isotopic data from thedeep mantle, the whole mantle was not com-pletely mixed at the end of accretion (14, 15).Second, recent simulations find that increasingthe impactor mass and velocity combined with asteeper impact angle could reduce the impac-

tor mass fraction in the lunar disk to ~40 wt %at the expense of a small excess in the final an-gular momentum (16), but the isotopic similarityrequires more efficient mixing of impactor andtarget material (9). Third, one could invoke thespecial case of an impactor with identical iso-topes as Earth, but such a body is unlikely to alsosatisfy other geochemical constraints such as therelative abundances of moderately siderophileelements [for example, see (17)]. To date, noneof the proposed variations on the giant impactmodel satisfy all of the geochemical observations.

All previous giant impact scenarios wereconstrained by the present angular momentumof the Earth-Moon system. The Moon accretedfrom the disk just beyond Earth’s Roche radius[RRoche ~ 2.9 Earth radii (RE)] (18), the distanceat which tidal forces no longer disrupt a satellite.Subsequent tidal interactions between the twobodies (19–21) expanded the lunar orbit to itscurrent 60RE. During this process, angular mo-mentum was transferred from Earth to the Moon,but the total angular momentum of the systemdid not change. Tides raised by the Sun have aminor effect on the Earth-Moon system, changingthe angular momentum by, at most, ~1% (10).Thus, the present spin of Earth and the orbitof the Moon imply that the post-impact Earthcould not have spun faster than once every4 hours. However, simulations of the accretionof Earth-mass planets produce final spin periodsmuch faster than 4 hours due to multiple giantimpacts (22–24). Starting with a fast-spinningproto-Earth, simulations of giant impacts that re-duced Earth’s angular momentum to the presentvalue did not produce disks massive enough toform the Moon (11).

Here, we present a different model for theorigin of the Earth-Moon system. An erosivegiant impact onto a fast-spinning proto-Earthproduced a disk that was massive enough toform the Moon and was composed primarily

of material from Earth, but the system had moreangular momentum than is the case today. Sub-sequently, the excess angular momentum waslost during tidal evolution of the Moon via a res-onance between Earth’s orbital period and theperiod of precession of the Moon’s perigee.

Impacts onto a fast-spinning proto-Earth. Wedefine successful Moon-forming impact scenar-ios by the following observational constraints:(i) the isotopic similarity between the Moon andEarth, (ii) the mass of the Moon (MM = 0.012ME),and (iii) the mass of the lunar core. First, theisotopic data limit the difference in the projectilemass fraction between the silicate Earth and thesilicate portion of the lunar disk, but the max-imum difference depends on the projectile com-position. If the impactor had the same isotopiccomposition as Mars, the difference in projectilemass fraction (∆proj) is limited to only a few toseveral weight percent (6, 8, 9). Because theprojectile may have been more similar to Earththan Mars, impact scenarios in which ∆proj ≤ 15wt % are considered successful, although a widerrange may be permitted. Second, the mass ofthe satellite that accretes from the disk must begreater than or equal to one lunar mass (MM =0.012ME). We did not model the accretion of theMoon from the disk. Instead, we used the resultsfrom previous simulations of lunar accretion(18, 25), which found that the satellite mass isapproximated by

MS ≃ 1:9Ldisk=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGMERRoche

p−

1:1Mdisk − 1:9Mesc ð1Þ

where Ldisk and Mdisk are the angular momen-tum and mass of the disk, and G is the gravita-tional constant. As in (11), we neglected the massthat escapes during disk evolution,Mesc. We alsoestimated the satellite mass by angular momen-tum conservation (MS,AM) and found values with-in ~10% of Eq. 1 (26). Third, the mass of thelunar core was estimated to be only a few weightpercent (27, 28). Following (10, 11), we required≤10 wt % of the disk be composed of materialoriginating from the iron cores of the impactorand target.

We used a smooth particle hydrodynamics(SPH) code (29, 30) to model high-velocity col-lisions between differentiated planets [2/3 sili-cate mantle and 1/3 iron core (26)]. We assumedthat Earth was nearly fully formed at the time ofthe Moon-forming impact, or subsequent accre-tion would increase any compositional differ-ence between the planet and satellite. Hence,we modeled impacts onto a ~1ME target. At theend of accretion, the average angular momen-tum of Earth-mass planets is estimated to be ~2.7times the present value (LEM), with possible spinperiods up to the instability limit of ~2 hours (24).The minimum stable spin period achieved forthe SPH model Earth-mass planets was ~2.3 hours.Our model proto-Earths began with initial spin

RESEARCHARTICLE

Department of Earth and Planetary Sciences, Harvard Uni-versity, 20 Oxford Street, Cambridge, MA 02138, USA.

*To whom correspondence should be addressed. E-mail:cuk@eps.harvard.edu†Present address: Carl Sagan Center, SETI Institute, 189 NorthBernardo Avenue, Mountain View, CA 94043, USA.

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periods from 2.3 to 2.7 hours, corresponding toangular momenta from 1.9 to 3.1LEM. The char-acteristics of the projectiles were constrained byterrestrial planet-formation simulations, where thetypical last giant impactor onto Earth-mass bodieshad a mass ≤ 0.10ME and an impact velocity ofone to three times the mutual escape velocity(Vesc) (31). We calculated the properties of acircumterrestrial disk 1 to 2 days after impact(Table 1).

An example of a successful impact scenariois shown in Fig. 1. The post-impact planet has ahot, massive atmosphere that grades into a ro-tationally supported vapor-dominated disk. Thedisk is defined by SPH particles that have suf-ficient angular momentum such that the equiv-alent circular Keplerian orbital radius is outsidethe equatorial radius of the planet. The disk iscompact with 85% of its mass within the Rocheradius. The planet’s post-impact equatorial andpolar radii are estimated by a density contour of1 g cm−3. The post-impact silicate atmosphere,approximated by lower-density material lackingthe angular momentum to remain in orbit, has amass of several weight percent of the planet(Table 1). In this example, the iron core materialin the disk is <1 wt %, and the predicted satellitemass is 1.0MM. The mass fraction of projectilein the disk (dprojdisk) is only 8 wt %, and the projectilemass fraction in the silicate Earth is 2 wt %.Hence, the compositional difference betweenthe silicate portions of the disk and Earth is only6 wt % and is within the range allowed by theisotopic data.

A wide range of probable terminal giant im-pacts onto an Earth-mass planet with a 2.3-hourrotation period produces potential Moon-formingdisks that are composed primarily of materialderived from Earth (Fig. 2 and Table 1). We findthat these giant impacts typically result in partialaccretion of the impactor and net erosion fromthe proto-Earth (a small final mass deficit isneglected in the Moon-formation criteria, as alarger initial planet mass can compensate for thedifference). Head-on and slightly retrograde im-pacts with impact velocities of ~1.5 to ~2.5Vescgenerated the most successful Moon-formingdisks. In these cases, the impactor mantle is dis-tributed between Earth and disk, and less ma-terial escapes compared with prograde impacts,which tend to deposit more impactor mantle inthe disk and put more Earth mantle material onescaping trajectories. A wide range of impact an-gles and velocities produced potential Moon-forming disks with properties very close to thedesired traits (Table 1, also bold numbers in Fig.2). For the impact velocities and projectile massesconsidered here, oblique impacts at angles of45° and greater were hit-and-run events (32) thatdid not create disks massive enough to form theMoon. Head-on impacts with velocities above3Vesc begin to substantially decrease the finalmass of the planet (32).

Giant impacts onto planets with spin pe-riods of 2.5 and 2.7 hours produced smaller disk

masses compared with the 2.3-hour cases. In ad-dition, prograde impacts onto the slower-spinningplanets have larger iron core mass fractions in thedisk (table S1). The results imply a more narrowrange for potential Moon-formation events forimpact scenarios with less angular momentum.Increasing the total angular momentum by add-ing spin to the impactors generated successfuldisks from the slower-spinning planets. Becauseangular momentum is carried away with debrisfrom these erosive giant impacts, the spin periodof the planet decreases. Thus, the spin state ofEarth is not required to be near fission before orafter the Moon-forming impact in our scenario(for example, last entry in Table 1). However,

our simulations suggest that the impact-drivenformation of a sufficiently massive disk derivedprimarily from Earth’s mantle is easiest whenthe total angular momentum of the event (fromthe spin of each body and the impact geometry)is near the stability limit.

Our candidate Moon-forming events havemore than double the kinetic energy of previousscenarios, and the impact velocities were suf-ficient to substantially vaporize silicates (33). Asa result, the silicate atmosphere and vapor-richdisk are more massive and hotter than foundin previous work (34). At the resolution of thesimulations, the projectile-to-target mass ratio isuniform from the atmosphere to the Roche radius.

Fig. 1. Formation of thelunar disk from Earth’smantle. Example impactof a 0.05ME impactor at20 km s−1 and b = −0.3onto a 1.05ME Earth spin-ning with a period of 2.3hours (‡ in Table 1). Graycircles denote the Rocheradius. (A to F) View ofSPH particles in the lowerhemisphere looking downthe counterclockwise spinaxis, where colors denotethe silicate mantles andiron cores of the Earthand the impactor. The diskis dominated by materialoriginating from Earth’smantle near the impactsite (fig. S1 and movie S1).(G) Lower hemisphere viewwith particle colors de-noting the planet (blue),atmosphere (yellow), anddisk (green). (H) Densityin the equatorial plane ofthe disk and planet, whichis stably stratified.

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The disk contains both volatile and refrac-tory components from the mantles of the collidingbodies, and the observed depletion of volatileelements in the Moon is a result of the separa-tion of volatile and refractory material duringlunar accretion from the disk (35, 36). Detailedcomparisons between our Moon-formation sce-nario with the isotopic data require modelinglunar accretion coupled to the chemical evolutionof a disk with our calculated initial conditions.

Tidal evolution and angular momentum loss.The successful Moon-forming impacts onto afast-spinning proto-Earth leave the Earth-Moonsystem with excess angular momentum (Table 1).To test if our lunar origin scenario can be rec-onciled with the present angular momentum,we simulated the early tidal evolution of the Earth-Moon system using a custom-made orbital in-tegrator based on a symplectic mapping methodcommonly used in solar system dynamics (21, 37),which includes mutual precession, both Earthand Moon tides, and solar perturbations (26).

We find that the evection resonance betweenthe Moon and the Sun (38, 39), which occurswhen the period of precession of lunar perigeeequals the period of Earth’s orbit, can substan-tially reduce the angular momentum of the Earth-Moon system. After capture into this resonance,

the long axis of lunar orbit librates around 90°from the Earth-Sun line, and the lunar perigeeprecession period is fixed at 1 year. The evectionresonance is encountered soon after lunar for-mation, and the efficiency of capture is a strongfunction of the lunar semimajor axis at whichthe resonance happens, which increases with in-creased flattening of Earth. At larger distancesfrom Earth, solar gravitational perturbations arestronger, and more importantly, lunar tidal reces-sion is slower, enabling more efficient capture. Ithas been found that capture is possible even foran Earth spinning once every 5 hours (39), aslong as tidal evolution is very slow (implying alarge tidal quality factor Q of ~104 for Earth; Qis an inverse measure of the dissipation of tidalenergy as heat within a body).

In our simulations (Fig. 3), robust captureinto the evection resonance happened at ~7RE,for a standard tidal quality factor of Q ≃ 100,2.5-hour rotation and a simplified flattening mod-el for Earth. After the Moon was captured inthe resonance, the lunar orbit continued to evolveoutward while keeping a constant precession pe-riod, which led to a rapid increase of eccentricity(39). The eccentricity increased until a balancebetween Earth and lunar tides was reached, butthe exact eccentricity at which this happened is

model-dependent because the mechanical proper-ties of both Earth and the Moon are uncertain.We used a standard satellite tidal parameter forthe Moon in Fig. 3, but other values for Q giveus similar outcomes (Fig. 4).

There was always a substantial period ofbalance between Earth and Moon tides, wherethe Moon stayed in the evection resonance witha roughly constant eccentricity. During this pe-riod, Earth tides were transferring angular mo-mentum to the Moon, and Earth’s rotation wasslowing down (movies S2 and S3). Satellitetides cannot remove angular momentum fromlunar orbit, but the Sun can absorb angular mo-mentum through the evection resonance. Be-cause the resonance couples lunar perigee andeccentricity with Earth’s orbital period, angularmomentum of the lunar orbit can be transferredto the angular momentum of Earth’s orbit aroundthe Sun. Earth tides pass angular momentum tolunar orbit, and the resonance-locked lunar orbittransfers angular momentum to the heliocentricorbit of Earth. Our integrator keeps Earth’s orbitstationary, but in reality, this process makes theEarth-Sun distance slightly larger.

As Earth lost its spin, Earth’s flattening de-creased, and the position of the evection reso-nance for the equilibrium eccentricity slowly

Table 1. PotentialMoon-forminggiant impacts onto a fast-spinning proto-Earth.Mproj, projectile mass; Vi, impact velocity in kilometers per second; b, impactparameter (negative value indicates retrograde impact; all impacts in theequatorial plane); Mplanet, mass of post-impact planet including atmosphere; Req,

estimated equatorial radius of post-impact planet in kilometers; f, flattening ofpost-impact planet (Req − Rpole)/Req (where Rpole is the polar radius); T, spin periodof post-impact planet in hours; LZ, z-component of angular momentum of planetand disk;Matm,mass of the atmosphere;Miron/Mdisk, iron coremass fraction in disk.

MprojME

Vi bMplanetME

Req f T LZLEM

MatmMM

MdiskMM

MSMM

MS,AMMM

ddiskproj

∆proj MironMdisk

Result

0.99ME target with 2.3 hours spin and 3.0LEM0.026 20.0 –0.30 0.96 7700 0.39 2.6 2.43 4.0 1.54 0.82 0.85 0.064 0.052 0.003 *0.026 25.0 –0.30 0.94 7500 0.37 2.6 2.32 5.6 2.07 0.87 0.96 0.049 0.039 0.010 *0.026 30.0 –0.30 0.91 7300 0.36 2.7 2.22 6.8 2.33 0.90 1.01 0.046 0.038 0.049 *0.026 20.0 0.30 0.96 8100 0.45 2.4 2.76 2.9 1.59 0.88 0.79 0.167 0.157 0.037 *0.050 15.0 –0.30 0.98 7800 0.42 2.5 2.56 4.5 1.81 0.91 0.93 0.113 0.087 0.002 *0.050 20.0 –0.30 0.94 7500 0.38 2.6 2.40 6.6 2.45 0.96 1.09 0.086 0.064 0.008 *0.050 25.0 –0.30 0.90 7300 0.38 2.7 2.22 8.0 2.54 0.91 1.06 0.077 0.059 0.034 *0.050 15.0 –0.15 0.98 7900 0.43 2.5 2.67 4.5 1.84 0.93 0.90 0.122 0.093 0.007 *0.050 20.0 –0.15 0.94 7600 0.40 2.6 2.52 6.4 2.62 1.14 1.22 0.090 0.067 0.024 †0.050 15.0 0.00 0.98 8100 0.46 2.5 2.84 4.2 1.97 1.02 0.95 0.176 0.147 0.015 †0.050 20.0 0.00 0.94 7800 0.42 2.5 2.68 6.0 2.85 1.33 1.36 0.115 0.091 0.054 †0.100 15.0 –0.30 0.98 7400 0.38 2.8 2.30 8.1 2.16 0.89 0.90 0.121 0.069 0.021 *0.100 20.0 –0.30 0.93 7100 0.35 2.9 2.07 10.4 2.22 0.81 0.86 0.103 0.056 0.078 *0.100 10.0 0.00 1.04 8100 0.43 2.5 2.90 4.5 1.60 0.91 0.74 0.255 0.187 0.014 *0.100 15.0 0.00 0.99 7800 0.43 2.6 2.79 7.4 2.81 1.32 1.23 0.168 0.111 0.083 †

1.05ME target with 2.3 hours spin and 3.1LEM0.050 20.0 –0.30 1.01 7500 0.37 2.7 2.47 7.0 2.20 1.02 1.02 0.081 0.059 0.005 †‡0.050 22.5 –0.30 0.99 7400 0.36 2.8 2.40 7.9 2.32 0.90 0.97 0.070 0.050 0.007 *0.050 25.0 –0.30 0.97 7300 0.35 2.8 2.31 8.7 2.37 0.93 0.95 0.066 0.048 0.019 *

1.05ME target with 2.5 hours spin and 2.5LEM0.050 25.0 0.00 0.98 7200 0.30 3.1 2.15 8.7 2.14 1.29 1.09 0.084 0.061 0.092 †0.050 20.0 0.15 1.03 7500 0.35 2.8 2.42 6.3 1.67 1.01 0.81 0.187 0.161 0.076 *

1.05ME target with 2.7 hours spin and 2.1LEM0.050 20.0 0.30 1.03 7400 0.35 2.8 2.25 5.1 1.26 0.91 0.70 0.196 0.174 0.098 *0.05§ 25.0 0.00 1.00 7100 0.29 3.4 1.94 9.9 1.77 1.24 0.82 0.101 0.076 0.084 †

*Potential Moon-forming simulations with more relaxed criteria of 0.8 ≤ MS/MM ≤ 1.5, ∆proj ≤ 0.20, and Miron/Mdisk ≤ 0.1. †Successful Moon-forming simulations with MS ≥ 1.0MM, ∆proj ≤ 0.15,and Miron/Mdisk ≤ 0.1. ‡Example in Fig. 1. §Projectile with a 2.9-hour prograde spin; other projectiles have no spin. Targets have 1 to 2 × 105 SPH particles [additional simulation results andmethods in (26)].

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shifted inward. Eventually, the lunar semimajoraxis evolved within 5RE, whereas the Moon main-tained substantial eccentricity, and Earth’s spinslowed down to ~6 hours. This was typicallythe point at which the resonance broke in oursimulations. The reason for the end of resonanceis simple: Tidal acceleration of the Moon at peri-gee weakened once the rates of Earth’s rotationand the Moon’s orbital motion became compara-ble. In other words, the Moon at perigee startedcatching up with the bulge it raised on Earth,reducing the efficiency of Earth tides. Then, lunartides dominated and (unlike Earth tides) pushedthe Moon away from the center of the resonance,leading to larger and larger amplitude of resonantlibration. Once librations exceeded the width ofthe resonance, the lunar orbit exited the resonancein the direction of lower eccentricities. After break-ing the resonance, lunar tides damped the eccen-tricity, whereas Earth tides restarted the outwardtidal evolution. As the Moon moved away fromsynchronous orbit, its eccentricity stabilized, andthe standard tidal evolution continued.

For a range of initial spin periods and tidalevolution paths (Fig. 4), the final angular mo-mentum is close to the observed value. Toumaand Wisdom (39) started the evolution of the

Earth-Moon system with its current momentumand found that capture in the evection resonanceis possible, but in their case, the resonance wasbroken soon after capture with no long high-eccentricity phase and no large angular momen-tum loss. The observed state is actually closeto the lowest angular momentum reachable byresonance, and this result only weakly dependson the model of tides used when close to syn-chronous rotation. An analytic calculation (26)shows how the parameters of the system nat-urally lead to evection resonance breaking whenthe Moon has a semimajor axis of ~5RE andEarth has a spin period of ~6 hours (assumingthat the resonance persists close to the synchro-nous orbit). Therefore, Earth could have had arange of fast spin periods before capture intoevection, and the present angular momentumof the system does not carry information aboutEarth’s primordial spin.

Our model predictions of capture into theevection resonance and exiting near the presentangular momentum depend on a number of pa-rameters, some of which are poorly constrained.Awide range of tidal evolution rates could havedelivered the system to its present state, as longas the ratio of tidal dissipation rates within Earth

and the Moon is within ~50% of the value op-timal for their balance (26). This balance of tidesrequires similar dissipation factors Q for the twobodies (assuming modern-day response to de-formation) or Earth being about an order of mag-nitude more dissipative than the Moon (assumingfluid bodies).

Discussion. Our tidal evolution simulationsare consistent with the two prevailing modelsfor generating the Moon’s high inclination and,similarly, require a low post-impact obliquity forEarth [<10°, (26)]. Interaction with the evectionresonance does not excite lunar inclination, andany primordial lunar inclination would decreasesomewhat during evolution through the reso-nance. As the Moon does not interact with theevection resonance until 7RE or more, our modelis compatible with the disk-interaction hypothesisfor the origin of lunar inclination (40). Becausethe evection resonance in our model breaks atabout the same configuration as in Touma andWisdom (39), our model is also consistent withsubsequent generation of lunar inclination throughtemporary inward migration and capture into amixed resonance (39).

A high spin rate during the giant impactphase of planet formation would affect all major

Fig. 2. Summary of the range of outcomes for expected terminal giantimpacts onto the proto-Earth: Mproj ≤ 0.1ME and 1 to 3Vesc (Vesc ~ 10 km s−1).The target was a 0.99ME body with a 2.3-hour spin. Projectiles had no spinand masses of 0.026, 0.05, or 0.10ME. The radius of each filled colored circleis proportional to the satellite mass; the black circle indicates MS = 1.0MM.Color indicates the difference in projectile composition between the silicate

disk and silicate Earth. Within a colored circle, a gray dot denotes too muchiron core mass fraction in the disk. The number above each symbol gives thefinal mass of the planet; bold numbers indicate cases that satisfy the relaxedMoon-formation criteria in Table 1. Collisions in the middle region of thefigure, head-on and slightly retrograde impacts from 10 to 30 km s−1, are thebest fit to the observational constraints for Moon-forming impacts.

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processes on the growing Earth, including man-tle convection patterns and overturn rates. With-in 100 million years of solar system formation,major chemical reservoirs were established inEarth’s lower mantle that were not destroyed bya Moon-forming impact (14, 15). The isotopicconstraints require the Moon’s formation to oc-cur at the end of Earth’s accretion, but the exacttiming remains uncertain (41). Although the SPHtechnique generally underestimates the mixingof materials, our simulations show that the rela-tively cooler and denser material from the lowermantle in the hemisphere opposite the impact is

not well mixed with material from the impactedhemisphere and upper mantle during gravita-tional reequilibration (fig. S1). The post-impactplanet is stably stratified with the entropy of theupper mantle higher than the entropy of the lowermantle, which would inhibit deep convective mix-ing. Hence, our Moon-formation scenario neednot destroy preexisting chemical differentiationwithin the proto-Earth.

Our model for the origin of the Moon blendsaspects of the original impact hypothesis, inwhich material was ejected from Earth by a latelarge impact (1), and the fission hypothesis firstproposed by Darwin (19), in which Earth lostmaterial via spin instability. We show that an ero-sive giant impact onto a fast-spinning proto-Earthfollowed by despinning during passage throughthe evection resonance can reproduce the isotopichomogeneity and present angular momentum ofthe Earth-Moon system.

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(1879).20. P. Goldreich, Rev. Geophys. 4, 411 (1966).21. J. Touma, J. Wisdom, Astron. J. 108, 1943 (1994).22. C. B. Agnor, R. M. Canup, H. F. Levison, Icarus 142, 219

(1999).23. E. Kokubo, S. Ida, Astrophys. J. 671, 2082 (2007).24. E. Kokubo, H. Genda, Astrophys. Lett. 714, L21

(2010).

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Fig. 3. Tidal evolution of the Moon through theevection resonance, starting with an Earth spinperiod of 2.5 hours. The Moon is captured into theresonance at ~9 thousand years (kyr) [at a semi-major axis of 6.8RE in (A)] and stays in the reso-nance until ~68 thousand years, when the Moonalmost reaches an orbit that is geosynchronous atperigee (gray line). During this time, the long axisof lunar orbit is locked to 90° from the Earth-Sunline. At first, the Moon keeps evolving outward (A)in the resonance while the eccentricity (B) increases,until the eccentricity stabilizes and a slower inwardmigration ensues, ending at ~5RE. During the res-onance lock, Earth’s rotation slows down dramati-cally (C), with the spin period increasing from justover 2.5 hours to almost 6 hours. During resonancecapture, resonant argument Y = 2lSun – 2ϖMoon(lSun, the Sun’s mean longitude; ϖMoon, longitudeof perigee) librates around 180° (26) (D). Alsosee movie S2.

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Fig. 4. Change in total angular momentum of the Earth-Moon system during tidal evolution of theMoon for different simulation parameters. (A) Simulations starting with Earth’s spin period of 2.5 hourswith different tidal quality factors for Earth (QE = 96, where not noted otherwise) and the Moon (QM).(B) Simulations starting with 2-, 2.25-, 2.5-, and 3-hour spin periods for Earth (QE = 96 and QM = 97,where not noted otherwise). The current angular momentum of the Earth-Moon system is 0.35 inour units [aM

ffiffiffiffiffiffiffiffiffiGMR

p; a, dimensionless moment of inertia; M, mass; R, radius (26)], where a spherical

Earth spinning at the break-up rate would have an angular momentum of 1.

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25. E. Kokubo, S. Ida, J. Makino, Icarus 148, 419 (2000).26. Supplementary materials are available on Science Online.27. R. C. Weber, P.-Y. Lin, E. J. Garnero, Q. Williams,

P. Lognonné, Science 331, 309 (2011).28. M. A. Wieczorek et al., Rev. Mineral. Geochem. 60, 221

(2006).29. V. Springel, Mon. Not. R. Astron. Soc. 364, 1105 (2005).30. R. A. Marcus, S. T. Stewart, D. Sasselov, L. Hernquist,

Astrophys. Lett. 700, L118 (2009).31. S. T. Stewart, Z. M. Leinhardt, Astrophys. J. 751, 32

(2012).32. Z. M. Leinhardt, S. T. Stewart, Astrophys. J. 745, 79

(2012).33. R. G. Kraus et al., J. Geophys. Res. 117, E09009 (2012).34. It is difficult to accurately calculate the mass of the

atmosphere and the vapor fraction of the disk, as currentnumerical methods lack equations of state for multiplesilicate phases in the mantle and multiphase flow ofpartially vaporized material (26). Although early work on

the accretion of the Moon focused on a gas-poor disk(18, 25), recent studies indicate that the Moon may alsoaccrete from a more vapor-rich atmosphere-disk structure(35, 36).

35. R. Machida, Y. Abe, Astrophys. J. 617, 633 (2004).36. W. R. Ward, Astrophys. J. 744, 140 (2012).37. J. Wisdom, M. Holman, Astron. J. 102, 1528 (1991).38. W. M. Kaula, C. F. Yoder, Proc. Lunar Planet. Sci. Conf. 7,

440 (1976).39. J. Touma, J. Wisdom, Astron. J. 115, 1653 (1998).40. W. R. Ward, R. M. Canup, Nature 403, 741 (2000).41. G. Yu, S. B. Jacobsen, Proc. Natl. Acad. Sci. U.S.A. 108,

17604 (2011).

Acknowledgments: This work, which was supported byNASA’s Origins of Solar Systems program and Harvard’sDaly and Smithosonian’s Clay Fellowships, was improved byhelpful discussions with S. Jacobsen, J. Melosh, D. Minton,S. Mukhopadhyay, D. Stevenson, J. Wisdom, and comments

from the anonymous reviewers. The impact simulations wererun on the Odyssey cluster supported by the Harvard Facultyof Arts and Sciences Research Computing Group. S.T.S.conducted the impact simulations, M.Ć. calculated the tidalevolution, and both wrote the paper. Numerical codes areavailable as supplementary materials.

Supplementary Materialswww.sciencemag.org/cgi/content/full/science.1225542/DC1Supplementary TextFigs. S1 to S7Table S1References (42–52)Movies S1 to S3Computer Codes

4 June 2012; accepted 26 September 2012Published online 17 October 2012;10.1126/science.1225542

REPORTS

Forming a Moon with an Earth-likeComposition via a Giant ImpactRobin M. Canup*

In the giant impact theory, the Moon formed from debris ejected into an Earth-orbiting diskby the collision of a large planet with the early Earth. Prior impact simulations predict thatmuch of the disk material originates from the colliding planet. However, Earth and the Moonhave essentially identical oxygen isotope compositions. This has been a challenge for theimpact theory, because the impactor’s composition would have likely differed from that ofEarth. We simulated impacts involving larger impactors than previously considered. We showthat these can produce a disk with the same composition as the planet’s mantle, consistentwith Earth-Moon compositional similarities. Such impacts require subsequent removal ofangular momentum from the Earth-Moon system through a resonance with the Sun asrecently proposed.

The oblique, low-velocity impact of a rough-ly Mars-mass planet with Earth can producean iron-depleted disk with sufficient mass

and angular momentum to later produce our iron-poor Moon while also leaving the Earth-Moonsystem with roughly its current angular momen-tum (1–3). A common result of simulations ofsuch impacts is that the disk forms primarily frommaterial originating from the impactor’s mantle.The silicate Earth and the Moon share compo-sitional similarities, including in the isotopes ofoxygen (4), chromium (5), and titanium (6). Thesewould be consistent with prior simulations if thecomposition of the impactor’s mantle was com-parable with that of Earth’s mantle. It had beensuggested that this similarity would be expectedfor a low-velocity impactor with an orbit similarto that of Earth (4, 7, 8). However, recent work(9) finds that this is improbable given the degree

of radial mixing expected during the final stagesof terrestrial planet formation (10). Explainingthe Earth-Moon compositional similarities wouldthen require post-impact mixing between the va-porized components of Earth and the disk beforethe Moon forms (9), which is a potentially re-strictive requirement (11).

A recent development is the work of Ćukand Stewart (12, 13), who find that the angularmomentum of the Earth-Moon system could havebeen decreased by about a factor of 2 after theMoon-forming impact because of the evectionresonance with the Sun. This would allow fora broader range of Moon-forming impacts thanpreviously considered, including those involvinglarger impactors.

Prior works (1–3, 14) focus primarily on im-pactors that contain substantially less mass thanthat of the target, with impactor masses Mimp ~0.1 to 0.2MT, where MT ≈ M⊕ is the total col-liding mass and M⊕ is Earth’s mass. If thetarget and impactor have different isotopic com-positions, creating a final disk and planet withsimilar compositions then requires that the disk

be formed overwhelmingly from material derivedfrom the target’s mantle. However, gravitationaltorques that produce massive disks tend to placesubstantial quantities of impactor material intoorbit (2, 3).

We considered a larger impactor that is com-parable in mass with that of the target itself. Afinal disk and planet with the same composi-tion are then produced if the impactor contrib-utes equally to both, which for large impactorsis possible even if the disk contains substantialimpactor-derived material because the impactoralso adds substantial mass to the planet. For ex-ample, in the limiting case of an impactor whosemass equals that of the target and in the absenceof any pre-impact rotation, the collision is com-pletely symmetric, and the final planet and anydisk that is produced will be composed of equalparts impactor and target-derived material andcan thus have the same silicate compositions evenif the original impactor and target did not.

We describe the impactor and target as dif-ferentiated objects with iron cores and overlyingsilicate mantles (15). We simulated impacts usingsmooth particle hydrodynamics (SPH) (Fig. 1)as in (1–3, 15, 16), representing the impactorand target with 300,000 SPH particles. Eachparticle was assigned a composition (either ironfor core particles or dunite for mantle particles)and a corresponding equation of state (17, 18),and its evolution was tracked with time as itevolved owing to gravity, pressure forces, andshock dissipation.

We simulated a given impact for approx-imately 1 day of simulated time. We used an it-erative procedure (1–3, 15) to determine whethereach particle at the end of the simulation is inthe planet, in bound orbit around the planet (inthe disk), or escaping. Given the calculated diskmass MD and angular momentum LD, we esti-mated the mass of the moon that would laterform from the disk,MM, using a conservation ofmass and angular momentum argument (19, 20).Assuming that the disk would later accumulate

Planetary Science Directorate, Southwest Research Institute,1050 Walnut Street, Suite 300, Boulder, CO 80302, USA.

*To whom correspondence should be addressed. E-mail:robin@boulder.swri.edu

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