Moon formation sasaki

The Origin of the Moon

佐々木貴教

月・人類が降り立った唯一の地球外天体・地球からの距離：38万4400km ・半径：1737km（地球の0.27倍）・質量：7.3×1022kg（地球の0.012倍）・地球に常に同じ面を向けている・地球と比べてコア（Fe）が小さい

表裏

•  ikd T

T (Fe) hT

•  i t T hT

•  Ws d Wht Su Su

0

月の起源の研究史• 黎明期（紀元前～1960年代）

• 開化期（1970年代～1980年代）

• 円熟期（1990年代～2000年代前期）

• 混乱期（2000年代後期）

• 革命期（2010年代）

• 混迷を極める現代

黎明期

• 古事記イザナギノミコトが右目を洗い月読命が誕生（左目が天照大神）

• 旧約聖書創造主が4日目に天の太陽と月と星を誕生させた

• エジプト創造主ラーの左目が月となった（右目が太陽）

• 中国創造神盤古の右目が月となった（左目が太陽）

神話における月の起源

[ 447 ]

XIII. On the Precession of a Viscous Spheroid, and on the remote History of the Earth.

By G. H. Darwin, M.A., Fellow of Trinity College, Cambridge.

Communicated by J. W. L. Glaisher, M.A., FM.S.

Received July 22,—Read December 19, 1878,

Plate 36.

The following paper contains the investigation of the mass-motion of viscous and

imperfectly elastic spheroids, as modified by a relative motion of their parts, produced

in them by the attraction of external disturbing bodies ; it must be regarded as

the continuation of my previous paper/" where the theory of the bodily tides of such

spheroids was given.

The problem is one of theoretical dynamics, but the subject is so large and complex,

that I thought it best, in the first instance, to guide the direction of the speculation

by considerations of applicability to the case of the earth, as disturbed by the sun

and moon.

In order to avoid an incessant use of the conditional mood, I speak simply of the

earth, sun, and moon ; the first being taken as the type of the rotating body, and the

two latter as types of the disturbing or tide-raising bodies. This course will be justi-

fied, if these ideas should lead (as I believe they will) to important conclusions with

respect to the history of the evolution of the solar system. This plan was the more

necessary, because it seemed to me impossible to attain a full comprehension of the

physical meaning of the long and complex formulas which occur, without having

recourse to numerical values ; moreover, the differential equations to be integrated were

so complex, that a laborious treatment, partly by analysis and partly by numerical

quadratures, was the only method that I was able to devise. Accordingly, the earth,

sun, and moon form the system from which the requisite numerical data are taken.

It will of course be understood that I do not conceive the earth to be really a

homogeneous viscous or elastico-viscous spheroid, but it does seem probable that the

earth still possesses some plasticity, and if at one time it was a molten mass (which is

highly probable), then it seems certain that some changes in the configuration of the

three bodies must have taken place, closely analogous to those hereafter determined.

And even if the earth has always been quite rigid, the greater part of the same effects

would result from oceanic tidal friction, although probably they would have taken

place with less rapidity.

# " On the Bodily Tides of Viscous and Semi-elastic Spheroids," &c, Phil. Trans. 1879, Part I,

MPCCCLXXIX. 3 M

“Fission Theory” による月形成シナリオ

月の起源説

捕獲説

分裂説

双子説

原始地球が高速回転によりふくらみ, その一部がちぎれて月が誕生

地球軌道付近での微惑星の集積により, 地球とは独立に月が形成

地球とは別の場所で作られた月が, 地球の近くを通ったときに捕らえられた

開化期

Lunar Rock by Apollo 11

Constraints of Moon Formation(1) 地球ー月系の角運動量（Ltotal が保存）(2) 地球より低密度（コアが小さい）[Hood & Zuber, 2000]

(3) 揮発性元素が強く枯渇 [Jones & Palme, 2000]

(4) 表面が大規模溶融を経験 [Warren, 1985]

(5) 酸素同位体比が地球とほぼ一致 [Wiechert et al., 2001]

both yield !18O " 5.5‰ and therefore are, onaverage, more enriched in 18O than Mars[!18O " 4.3‰ (12)] or Vesta [!18O " 3.3‰(5)].

Computer simulations of the Moon-form-ing Giant Impact indicate that the Moon in-herited a significantly higher proportion ofmaterial from Theia than the proto-Earth[e.g., see (1, 2)]. The resolution of smoothparticle hydrodynamic modeling of potentialMoon-forming Giant Impacts has been im-proved recently. These high-resolution stud-ies show that a potential impactor must havebeen very close in size to Mars. A signifi-cantly larger or smaller impactor would pro-duce a higher angular momentum than ob-served for the Earth-Moon system or wouldthrow too much iron into orbit, which wouldhave created a more iron-rich Moon (2). Onthe basis of these high-resolution models, ithas been estimated that 70 to 90% of theMoon is derived from Theia. An identicaloxygen mass fractionation line for the Moonand Earth, therefore, cannot be explained byassuming that similar proportions of materialcame from the silicate portions of the proto-Earth and Theia. Only if the proto-Earth andTheia #17O values were identical to within0.03‰ would it be possible that the average#17O value of the Moon plots within 0.005‰on the terrestrial fractionation line.

Some computer models assume a largersize for the impactor, i.e., a mass ratio of 7:3between the proto-Earth and Theia (2, 13).All these models assume that the Earth hadonly achieved about two-thirds of its finalmass after the Giant Impact, because a largerproto-Earth would produce greater angularmomentum for the Earth-Moon system thanthat observed. Models assuming that theproto-Earth had reached just 66% of its massafter the Giant Impact (2, 3) and identical#17O of the Moon and Earth require that lateincoming material came from the same res-ervoir as the material that made up Theia and

the proto-Earth. If this was another planetes-imal, it must have formed from an identicalmix of components as the proto-Earth andTheia. Because this is unlikely, the oxygenisotope data are easier to reconcile with aGiant Impact model involving a Mars-sizedimpactor.

The three isotopes of oxygen are hetero-geneously distributed in the solar system(14). The largest mass-independent oxygenisotope variations of more than 40‰ are mea-sured on minerals from calcium-aluminum–rich inclusions (CAIs) of the Allende mete-orite (15, 16). The different mass-dependentfractionation lines for asteroid 4 Vesta, Mars,and the Earth-Moon system (Fig. 2) provideevidence that the average provenance of theraw material of these objects is significantlydifferent (5). There is, however, no obviousrelation between oxygen isotopes and cur-rent heliocentric distance from the sun asfound for 53Cr/52Cr (17 ). This might indi-cate that oxygen isotope compositions werenot monotonically zoned through the solarsystem or that oxygen isotope alterationcontinued on icy planetesimals (18). How-ever, computer simulations of the colli-sional growth stage of the inner solar sys-tem (19) demonstrate that terrestrial planetswere fed from a zone with a heliocentricdistance of 0.5 to 2.5 astronomical unitsand beyond. Regardless of how heteroge-neous the early inner solar system was atthe beginning, it developed toward a homo-geneous composition by collisional growth.This is endorsed by the small #17O differ-ences of about 0.6‰ observed for theEarth-Moon system, Mars, and Vesta com-pared with more than 10‰ differencesamong chondrites. Collisional growth willsmooth out pre-existing heterogeneities butis unlikely to result in identical oxygenisotopic compositions for all planets be-cause a correlation between final heliocen-tric distance and average provenance of aplanet is predicted (19). The differences in#17O among large planetary embryos and

planets depend on final heliocentric dis-tance because oxygen isotopes were heter-ogeneously distributed in the early solarsystem. Therefore, the progenitors of theMoon and Earth formed at a similar helio-centric distance. A similar orbit of Theiacompared with the proto-Earth would resultin a relatively small encounter velocity.This is consistent with the assumptions ofmost Giant Impact simulations (20).

If the proto-Earth and Theia grew from asimilar mix of components at a similar dis-tance from the Sun, then not only the oxygenisotopes but also the chemical compositionand other isotope ratios should be similar.Therefore, differences between the Moon andEarth, such as the depletion of volatiles or thehigh FeO content of the lunar mantle com-pared with the terrestrial mantle, may be ofsecondary origin. Such differences might beproduced during accretion and differentiationof the proto-Earth, Theia, the Moon, and thepresent Earth (21, 22).

Significant amounts of meteoritic material

Fig. 1. Comparison between conventional andnew laser 16O, 17O, and 18O measurements oflunar samples. #17O gives displacement fromthe terrestrial fractionation line experimentallydefined as #17O $ !17O – !18O % 0.5245.Squares, Apollo Missions (4); diamonds, lunarmeteorites (5); circles, this study.

Fig. 2. Three oxygen isotope plots of lunar rocks.Composition of Martian meteorites (12) andHED meteorites (5), supposed to be fragmentsof asteroid 4 Vesta, are given for comparison.

Fig. 3. The #17O values for lunar samples plotwithin standard deviation (2&i) error of '0.016‰ (long-dashed lines) on the TFL. If theimpactor had formed from the same raw ma-terial as Mars or the HED parent body, then alllunar samples must have obtained, within 2%,the same portion from the impactor and proto-Earth as obtained by Earth using the triplestandard error of the mean (3&mean) as signif-icant, shown by short-dashed lines. On average,the H-chondrites plot 0.7‰ above the TFL,allowing a maximum of 3% chondritic materialmixed into any of the studied lunar samples,2& confidence level. Other chondrite groupslike L, LL, or carbonaceous chondrites show aneven larger deviation from the TFL and, there-fore, even less of these primitive materials canbe mixed into the lunar samples. For the mixinglines in this figure, identical oxygen abundanceshave been assumed for all objects.

R E P O R T S

www.sciencemag.org SCIENCE VOL 294 12 OCTOBER 2001 347

[Wiechert et al., Science, 2001]

月の起源説

捕獲説

分裂説

双子説

原始地球が高速回転によりふくらみ, その一部がちぎれて月が誕生

地球軌道付近での微惑星の集積により, 地球とは独立に月が形成

地球とは別の場所で作られた月が, 地球の近くを通ったときに捕らえられた

高速回転が難しい & 角運動量が大きすぎる

月の内部構造が説明できない & 月を残せない

捕獲確率が低い & 化学的制約を満たせない

[Schreiber & Anderson, Science, 1970]

月はチーズでできている？

ジャイアントインパクト説

[Hartman & Davis, Icarus, 1975][Cameron & Ward, LPI Conference, 1976]

520 BENZ, SLATTERY, AND CAMERON

T: 7.76217 [ -2.0. 2.0. -2.0. 2.01 T: 8.83375 ( -2.0. 2.0. -2,0. 2.0]

,,:iii:i~iiiiii!iii!i,li!~iliiiiiii~!!!iiii:~iii~:i ,:

T: 9 , q0986 ( - 2 . 0 . 2 . 0 . 2 . 0 , 2.01 T: 11.03161 I - ~ . 0 . ~,.0, - q , 0 . ~ .0]

/ , , l - - ¢ / t ~

FIG. 2. Snapshots of run 1. (u~ = 0 km/sec; rmi, = 0 . 7 7 R e a r t h ; Eint = 107 erg/g). Velocity vectors are plotted at particle locations. The velocity has been normalized to its maximum value in each frame. Time and coordinates of the four corners of the plotted field are given in the upper line (in units defined in Section 3). For particles in the vapor phase a "O" is plotted.

b e f o r e the t ime at w h i c h the pa r t i c l e s s p r e a d ou t in space . S ince this h a p p e n s af- te r the t ime o f c l o se s t a p p r o a c h , the t ra jec- t o r i e s o f the va r i ous c l u m p s fo rming a f te r co l l i s ion a re c a l c u l a t e d a c c u r a t e l y .

T h e to ta l " v i s c o u s " fo rce t he r e fo re be- c o m e s

F visc= F/bulk -I- F~ rag

and this c o m p l e t e s the d e s c r i p t i o n o f the e q u a t i o n o f mo t ion .

4.2. Energy Conservation Equation T h e v a r i a t i o n o f the in te rna l e n e r g y is

g iven b y t h e r m o d y n a m i c s and is wr i t t en

du dV dQ d--i = - P --d? + d--7'


T: 7.76217 [ -2.0. 2.0. -2.0. 2.01 T: 8.83375 ( -2.0. 2.0. -2,0. 2.0]


T: 9 , q0986 ( - 2 . 0 . 2 . 0 . 2 . 0 , 2.01 T: 11.03161 I - ~ . 0 . ~,.0, - q , 0 . ~ .0]

/ , , l - - ¢ / t ~


b e f o r e the t ime at w h i c h the pa r t i c l e s s p r e a d ou t in space . S ince this h a p p e n s af- te r the t ime o f c l o se s t a p p r o a c h , the t ra jec- t o r i e s o f the va r i ou s c l u m p s fo rming a f te r co l l i s ion a re c a l c u l a t e d a c c u r a t e l y .






du dV dQ d--i = - P --d? + d--7'


T: 7.76217 [ -2.0. 2.0. -2.0. 2.01 T: 8.83375 ( -2.0. 2.0. -2,0. 2.0]


T: 9 , q0986 ( - 2 . 0 . 2 . 0 . 2 . 0 , 2.01 T: 11.03161 I - ~ . 0 . ~,.0, - q , 0 . ~ .0]

/ , , l - - ¢ / t ~


b e f o r e the t ime at w h i c h the pa r t i c l e s s p r e a d ou t in space . S ince this h a p p e n s af- te r the t ime o f c l o s e s t a p p r o a c h , the t ra jec- t o r i e s o f the va r i ou s c l u m p s fo rming a f te r co l l i s ion a re c a l c u l a t e d a c c u r a t e l y .






du dV dQ d--i = - P --d? + d--7'


T: 7.76217 [ -2.0. 2.0. -2.0. 2.01 T: 8.83375 ( -2.0. 2.0. -2,0. 2.0]


T: 9 , q0986 ( - 2 . 0 . 2 . 0 . 2 . 0 , 2.01 T: 11.03161 I - ~ . 0 . ~,.0, - q , 0 . ~ .0]

/ , , l - - ¢ / t ~


b e f o r e the t ime at w h i c h the pa r t i c l e s s p r e a d ou t in space . S ince this h a p p e n s af- te r the t ime o f c l o se s t a p p r o a c h , the t ra jec- t o r i e s o f the va r i ous c l u m p s fo rming a f te r co l l i s ion a re c a l c u l a t e d a c c u r a t e l y .






du dV dQ d--i = - P --d? + d--7'

T: 1%68976

C O L L I S I O N A L ORIGIN OF T H E M O O N

-6 .0 . 6 .0 . -6 .0 . 6 .0} T: 22.19287 -6 .0 , 6 .0 , -6 .0 . 6.01

521

", S s

3. " j'q~k','" '. 0 ~,'e~4 "

FIG. 2--Continued.

i %

where dQ is the amount of energy absorbed by the system from its surroundings. By writing this equation for each particle one finds that the change of internal energy of the ith particle is given by the sum of the following two terms:

_ (p dV Pj Pi) v i j V i W ( r i j ' h)

N ( d O ) : 0 . 5 £ "i jv i j~iW(ri j , h) -37, j = l

N + 2 viF~ rag"

j - I

These two expressions allow us to compute the change in specific internal energy and therefore assure conservat ion of total energy.

Up to this point we have written all the equations without specifying any special form for the kernel W(r, h). To complete the above description and to allow numerical computat ions it is necessary to define this kernel. It can be shown mathematically that there are only a small number of constraints on W(r, h) such as that fW(r, h) = m (m being the mass of one particle), and that W should be continuous, together with

its first derivative, to assure the continuity of the forces. In practice, however , it is useful to choose a function for which all derivatives as well as the integral can be obtained analytically. One natural choice is the exponential function which we adopted in these simulations. Gingold and Monaghan adopted a Gaussian function (this type of kernel leads to a erf-type function when integrated and an analytical ex- pression is unobtainable). The detailed effects of choosing a different kernel have not been studied thoroughly; however , theoret- ically there should not be a significant difference as long as the basic requirements for W are met. In practice, the only way is to test the code against known analytical solutions or against results produced by other independent codes. This last procedure is the one that we have chosen and it is described in the next section.

The exponential kernel we chose was proposed by Wood (1981) and has the analytical form

m ( r ) W(r, h) = ~ exp -

with m the mass of one particle.

T: 1%68976

C O L L I S I O N A L ORIGIN OF T H E M O O N

-6 .0 . 6 .0 . -6 .0 . 6 .0} T: 22.19287 -6 .0 , 6 .0 , -6 .0 . 6.01

521

", S s

3. " j'q~k','" '. 0 ~,'e~4 "

FIG. 2--Continued.

i %

where dQ is the amount of energy absorbed by the system from its surroundings. By writing this equation for each particle one finds that the change of internal energy of the ith particle is given by the sum of the following two terms:

_ (p dV Pj Pi) v i j V i W ( r i j ' h)

N ( d O ) : 0 . 5 £ "i jv i j~iW(ri j , h) -37, j = l

N + 2 viF~ rag"

j - I

These two expressions allow us to compute the change in specific internal energy and therefore assure conservat ion of total energy.

Up to this point we have written all the equations without specifying any special form for the kernel W(r, h). To complete the above description and to allow numerical computat ions it is necessary to define this kernel. It can be shown mathematically that there are only a small number of constraints on W(r, h) such as that fW(r, h) = m (m being the mass of one particle), and that W should be continuous, together with

its first derivative, to assure the continuity of the forces. In practice, however , it is useful to choose a function for which all derivatives as well as the integral can be obtained analytically. One natural choice is the exponential function which we adopted in these simulations. Gingold and Monaghan adopted a Gaussian function (this type of kernel leads to a erf-type function when integrated and an analytical ex- pression is unobtainable). The detailed effects of choosing a different kernel have not been studied thoroughly; however , theoret- ically there should not be a significant difference as long as the basic requirements for W are met. In practice, the only way is to test the code against known analytical solutions or against results produced by other independent codes. This last procedure is the one that we have chosen and it is described in the next section.

The exponential kernel we chose was proposed by Wood (1981) and has the analytical form

m ( r ) W(r, h) = ~ exp -

with m the mass of one particle.

Giant Impact by SPH（初）

[Benz et al., Icarus, 1986]





Giant Impact によって形成される周惑星円盤の~80% が Impactor 起源のマントル物質である

円熟期

太陽系形成標準理論（林モデル）

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©Newton Press [Hayashi et al., 1985]

184 KOKUBO AND IDA

FIG. 1. Time evolution of a planetesimal system on the (a) a–e and (b) a–i planes. The circles represent planetesimals and their radii areproportional to the radii of planetesimals. The system initially consists of 3000 equal-mass (1023 g) bodies. We used the radii of planetesimals fivetimes larger than the realistic ones, f 5 5. The numbers of planetesimals are 1533 (t 5 5000 years), 1294 (t 5 10,000 years), and 1059 (t 5 20,000 years).

The increase of radii artificially strengthens collisional f 5 1. Thus, the increased collisional damping with f 5 5hardly changes the dynamics of largest bodies and hencedamping of random velocities. However, the effect of in-

creased collisional damping with f 5 5 on dynamics of does not change the growth mode in the stage we areconcerned with (it may affect the growth mode in the lastlarge bodies is small. In a real system with f 5 1, as for

large bodies, the strength of dynamical friction is always stage of accretion where dynamical friction is weak becauseof relatively small total mass of small bodies).larger than that of collisional damping in the stage we are

concerned with: the stage where the total mass of small In summary, the acceleration of accretion with the in-crease factor f 5 5 does not change the growth mode andbodies is much larger than the masses of largest bodies.

In this stage, collisional damping for largest bodies is the dynamical processes, although time scale is reducedby some factor.mainly due to merging with small bodies. In this case, the

ratio of damping due to collisional damping to that dueto dynamical friction is given by Scollision/Sclose ln L (e.g., 4. RESULTSWetherill and Stewart 1989). As shown above, Scollision/Sclose ln L , 1 when f 5 5, which means that not collisional We present the result of a simulation starting from an

equal-mass (1023 g) planetesimal system with the numberdamping but dynamical friction dominates energy dampingin our simulations in the same way of a real system with of bodies N 5 3000 and using the increase factor of radii

[Kokubo & Ida, Icarus, 1996]

OLIGARCHIC GROWTH OF PROTOPLANETS 177

runaway stage, while most planetesimals remain small. Thetypical orbital separation of protoplanets kept while grow-ing is about 10rH. This value depends only weakly on themass of protoplanets, the surface density of the solid mate-rial, and the semimajor axis. This self-organized structureis a general property of self-gravitating accreting bodiesin a disk when gravitational focusing and dynamical frictionare effective.

If we assume that the oligarchic growth continues tillthe final stage of planetary accretion, the mass of proto-planets is estimated by M 5 2fabS. In the solar nebulamodel that is 50% more massive than the minimum massmodel, the surface mass density of the solar nebula isgiven by

S 5 510 S a1 AUD23/2

[g cm22] a , 2.7 AU

4 S a5 AUD23/2

[g cm22] a . 2.7 AU.

(12)

Adopting this S and b 5 10rH, we have M Q 0.2M% andb Q 0.07 AU at 1 AU (S 5 10 g cm22), M Q 7M% andb Q 2 AU at 7 AU (S 5 2.4 g cm22), and M Q 17M% andb Q 8 AU at 25 AU (S 5 0.36 g cm22), where M% is theEarth mass. In the terrestrial planet region, the estimatedmass and the orbital separation of protoplanets are stillsmaller than the present planets. This may suggest thatoligarchic growth does not continue till the final stage ofplanetary accretion in the terrestrial planet region. TheFIG. 4. The same as Fig. 1 but for the system initially consists of

4000 equal-mass planetesimals (m 5 3 3 1023 g). The radius increase orbital separation may get larger in the terrestrial planetfactor is 6. In the final frame, the filled circles represent protoplanets region, if the radial excursion of planetesimals ea that isand lines from the center of the protoplanets to both sides have the proportional to the random velocity gets larger than 10rHlength of 5rH. The protoplanets are selected if their masses are larger than

due to, for example, the clearance of solar nebula gas in1/5 of the maximum mass of the system. The numbers of planetesimals arethe late stage of planetary accretion. The absence of gas1977 (t 5 5000 years), 1514 (t 5 10,000 years), and 1116

(t 5 20,000 years). drag leads to the higher velocity dispersion and thus widerradial excursion.

In the jovian planet region, however, the oligarchicgrowth may be consistent with the formation of the presentare formed, while most planetesimals remain small. Theplanets. As for Jupiter and Saturn, which have massive gasfive protoplanets have the 34% of the total mass of theenvelopes, the estimated mass of protoplanets is as largesystem. The lines with the length of 5rH are drawn fromas the critical mass to onset the gas accretion onto thethe center of the protoplanets to both sides in the finalprotoplanets. As for Uranus and Neptune, which consistframe. This Hill radius is slightly modified to include onlymainly of solid material, the estimated mass of proto-the mass of a protoplanet. The separations are roughlyplanets and the orbital separation are consistent with theirconstant with the typical value of 5–10rH, which agreespresent values. These results suggest that jovian planetswell with the result of the two-protoplanet system and themay have been formed along the line of oligarchic growth.analytical estimation.However, we should be careful when we apply oligarchicgrowth to the jovian planet region. Oligarchic growth is4. CONCLUSION AND DISCUSSIONobtained from the local area simulation where the semima-jor axis is much larger than the width of the simulationWe have shown the oligarchic growth of protoplanets in

the post-runaway stage. Protoplanets with the same order region. It is uncertain that oligarchic growth takes placein the wide jovian planet region in the same way as the localmasses with the orbital separation larger than about 5rH

is the inevitable outcome of planetary accretion in the post- area simulation. Further work on this issue is required.

[Kokubo & Ida, Icarus, 1998]

暴走成長＆寡占成長

ORIGIN OF MOON AND SINGLE IMPACT HYPOTHESIS 129

FIG. 2. Series of snapshots (panels 6a–6i) showing the progression of a Giant Impact between a Protoearth of 0.8M% and an Impactor of 0.2M%

(case 6). The bottom row shows views of the final panels from Figs. 1–3 rotated 908; these are designated 2r, 6r, and 8r.

[Cameron, Icarus, 1997]

Giant Impact by SPH

Nature © Macmillan Publishers Ltd 1997

the final moon mass, we do not need to know the details of initialmass, size and velocity distributions of the disk particles. Thepredicted moon mass from the disks obtained by the previousimpact simulations might be as large as the present lunar mass insome cases. However, we cannot make a definitive conclusion atpresent, as the previous impact simulations did not provide enoughdata about the disk angular momenta. Improved simulations areneeded to provide total mass and angular momentum of the disk.The combination of more refined N-body and impact simulationswould clarify whether a giant impact could indeed have producedthe Moon or not.

Model description

We simulated the formation of a moon from a (three-dimensional)circumterrestrial debris disk initially consisting of 1,000–2,700particles with mass m < 10 2 5 to 10−2 ML (see Fig. 1), assumingthat solid particles had condensed and attained such sizes throughaccretion. (In the inner part of the disk, the particles might remainvery small, as accretion becomes increasingly inhibited inside theRoche limit. Furthermore, the disk material might remain liquidowing to the longer cooling time in the inner part. We will commenton these effects later.)

We calculated disks with as many different initial conditions aspossible, as we do not have enough knowledge about disk con-ditions after the vapour/liquid phase and initial collisionalevolution. The parameters we examined are summarized inTable 1, where we show 19 runs of the 27 simulations for whichwe retained detailed output data. As shown below, the final outcomeof accretion has only a weak dependence on the details of conditionsof a starting disk. We scale the orbital radii by the Roche radiusdefined by aR ¼ 2:456ðr!=rÞ1=3R! where (r!/r) is the ratio of theinternal density of the Earth to that of the disk particles. For disk

particles with r ¼ 3:34 g cm 2 3 (the bulk density of the Moon), aR islocated at about 2.9R!. Using aR, the physical radii of disk particleswith mass m are given by R ¼ ð1=2:456Þðm=M!Þ1=3aR, independentof (r!/r).

Near the Roche radius, tidal forces of the proto-Earth affectwhether colliding particles rebound or accrete. Within ,0.8aR,tidal forces preclude accretion, whereas in the transitional zone,0.8–1.35aR, limited accretional growth can occur11. Exterior tothis zone, accretion is largely unaffected by tidal forces. Thistransitional zone will be referred to as the Roche zone. We adopthere the accretional criteria of Canup and Esposito11, which includethis transition in addition to the impact velocity condition that,for accretion, the calculated rebound velocity must be smallerthan some critical value corresponding to the (mutual) surfaceescape velocity11. If the colliding bodies in our simulation satisfythe criteria, we produce a merged body, conserving momentum.If not, the bodies rebound with given restitution coefficients(Table 1).

Characteristics of moon accretion

Below we present the results from several of the 27 disk simulationsthat were calculated. In most of the simulations, a single large bodyis formed near the Roche radius. In Figs 2 and 3, we show snapshotsof the results for the disks with initial mass Mdisk ¼ 0:03M!

(¼ 2:44ML). The unit of time is the kerplerian rotation time ataR, which is ,7 h; t ¼ 100 realistically corresponds to 1 month.Figure 2 shows a centrally confined disk case (run 4 in Table 1) inwhich the semimajor axes of all the particles are initially within theRoche radius, whereas Fig. 3 is a rather extended disk case (run 9).The extension of a disk is indicated by Jdisk/Mdisk, where Jdisk is thetotal angular momentum of the starting disk. For the disks in Figs 2and 3, Jdisk/Mdisk are 0:692 GM!aR and 0:813 GM!aR, respectively.

articles

354 NATURE | VOL 389 | 25 SEPTEMBER 1997

Figure 2 Snapshots of disk particles plotted in geocentric cylindrical coordinates

(r; z). (Particles at negative z are plotted at z jj .) The units of length and time are

the Roche limit radius, aR, and Kepler time, TKep at aR (,7h). The solid and dotted

circles are disk particles and the Earth, respectively. The sizes of the circles

indicate physical sizes. The snapshots here are the result of run 4 in Table 1. The

mean specific angular momentum, Jdisk/Mdisk, is initially 0:692 GM!aR. At

t ¼ 1,500 the moon has mass 0.40ML, semimajor axis 1.20aR, eccentricity 0.09

and inclination (radian) 0.02. The second body’s mass is only 0.025ML. The

masses ejected from the system (M`) and that hit the Earth are 0.026ML and

1.95ML, respectively.

Figure 3 The same snapshots as in Fig. 2 but for run 9 of a more extended disk

(Jdisk=Mdisk ¼ 0:813 GM!aR). At t ¼ 1,000 the largest moon mass is 0.71ML.

[Ida et al., Nature, 1997]

FORMATION OF A SINGLE MOON 425

FIG. 2. Snapshots of the circumterrestrial disk projected on the R–z plane at t = 0, 10, 30, 100, 1000TK for runs (a) 29a, (b) 13a, and (c) 15a. The semi-circlecentered at the coordinate origin stands for Earth. Circles represent disk particles and their size is proportional to the physical size of disk particles.

[Kokubo et al., Icarus, 2000]

Moon Formation by N-body

Moon Formation by N-body

N = 1,000~3hours@MacPro

数ヶ月～数年で、ひとつの月ができる

Giant Impact by SPH

[Canup & Asphaug, Nature, 2001]

1. Hf-W Chronometry　Hf → 　W（半減期～ 9My）182 182

・Hf：親石性　W：親鉄性・Hf, Wともに難揮発性元素

metal/silicate分離年代を示す時計！

t =τでコア形成による平衡化が起きたとすると

ε(t) = f (τ )

※初期比は始源的なコンドライトから決定

ε(t) =

182W /

184W( )

182W /

184W( )

CHUR

−1

×104

ε(t) が観測可能量なのでコア形成年代τが求まる

Giant Impact による平衡化プロセス

３. Equilibration by G.I.

Giant Impact !

全球的なマグマオーシャン(target + impactor)のマントルと impactor のコアの mixture

現在の地球

マントルと impactor のメタルとの間で平衡化が起こり、コア形成の際に過剰　Ｗがコアに持ち去られる182

impactorのメタル粒によってマントルが平衡化

AcknowledgementsWe thank the Smithsonian Institution, Harvard Mineralogical Museum, U. Marvin andH. Palme for providing the samples, and A. N. Halliday and D.-C. Lee for comments onthis paper. This work was supported by NASA’s Cosmochemistry and Origin of SolarSystem programmes and the National Science Foundation.

Competing interests statementThe authors declare that they have no competing financial interests.

Correspondence and requests for materials should be addressed to Q.Z.Y.(e-mail: [email protected]).

..............................................................

Rapid accretion and early coreformation on asteroids and theterrestrial planets from Hf–WchronometryT. Kleine*, C. Munker*, K. Mezger* & H. Palme†

* Institut fur Mineralogie, Universitat Munster, Corrensstrasse 24,D-48149 Munster, Germany† Institut fur Mineralogie und Geochemie, Universitat zu Koln,Zulpicherstrasse 49b, D-50674 Koln, Germany.............................................................................................................................................................................

The timescales and mechanisms for the formation and chemicaldifferentiation of the planets can be quantified using the radio-active decay of short-lived isotopes1–10. Of these, the 182Hf-to-182W decay is ideally suited for dating core formation inplanetary bodies1–5. In an earlier study, the W isotope compo-sition1 of the Earth’s mantle was used to infer that core formationwas late1 ($60 million years after the beginning of the SolarSystem) and that accretion was a protracted process11,12. Thecorrect interpretation of Hf–Wdata depends, however, on accu-rate knowledge of the initial abundance of 182Hf in the SolarSystem and the W isotope composition of chondritic meteorites.Here we report Hf–W data for carbonaceous and H chondritemeteorites that lead to timescales of accretion and core formationsignificantly different from those calculated previously1,3,5,11,12.The revised ages for Vesta, Mars and Earth indicate rapidaccretion, and show that the timescale for core formationdecreases with decreasing size of the planet. We conclude thatcore formation in the terrestrial planets and the formation of theMoon must have occurred during the first ,30 million years ofthe life of the Solar System.Chemically and mineralogically, carbonaceous chondrites rep-

resent some of the most primitive material in the Solar System. Thechemical composition of type 1 (CI) carbonaceous chondrites is,except for extremely volatile elements (H, C, N, O, rare gases),identical to that of the bulk Solar System. Other types of carbon-aceous chondrites are fractionated relative to CI chondrites, butrefractory elements occur in chondritic (solar) proportions in alltypes. Hence, all carbonaceous chondrites should have the sameratio of the two refractory elements Hf and W, and therefore auniform W isotope composition. Variations in the Hf/W ratioamong different groups of carbonaceous chondrites of up to,30% would result in W isotope variations indistinguishable atthe currently obtained analytical resolution. We analysed the Wisotope compositions of seven carbonaceous chondrites—Orgueil(CI), Allende (CV), Murray (CM), Murchison (CM), Cold Bokke-veld (CM), Nogoya (CM) and Karoonda (CK)—using multicollec-tor inductively coupled plasma mass spectrometry (MC-ICP-MS),

and found them to be identical within analytical error. The weightedaverage of the 182W/184W ratios for these samples differs by21.9 ^ 0.2 1 units from the terrestrial standard (Table 1, Fig. 1),but agrees with Hf–W data for enstatite chondrites13.

The most reliable method for determining the initial abundanceof 182Hf is by measuring internal Hf–W isochrons of samples withindependently known age. The H chondrites Ste Marguerite andForest Vale were selected because their phosphates have been datedpreviously by the high-precision Pb–Pb method. The preservationof U–Pb phosphate ages of 4.5627 ^ 0.0006Gyr for Ste Marguer-ite14 and 4.5609 ^ 0.0007 Gyr for Forest Vale14 precludes latedisturbance or alteration of these meteorites. A four-point metal-silicate isochron for Ste Marguerite (Fig. 2a) defines an initial182Hf/180Hf ratio of ð0:85^ 0:05Þ£ 1024 at the time of closure ofthe Hf–W system. Likewise, a four-point isochron for Forest Valegives an initial 182Hf/180Hf ratio of ð1:0^ 0:5Þ£ 1024 (Fig. 2b).These two isochrons pass through the newly defined chondritic 1wat the chondritic Hf/W ratio of ,1.1.

Back calculation of these initial values to the Pb–Pb age of Ca-Al-rich inclusions (CAI; 4.566 ^ 0.002Gyr, ref. 15), which is com-monly used as reference for condensation of the first solid matter inthe Solar System, yields a 182Hf/180Hf ratio of ð1:09^ 0:09Þ£ 1024

for Ste Marguerite and ð1:5^ 0:7Þ£ 1024 for Forest Vale. Thesevalues define the initial abundance of 182Hf in the Solar System,provided that the 182Hf–182W system in metal and silicate closed atapproximately the same time as the U–Pb system in phosphates.This is likely, as these H chondrites do not show evidence for laterthermal overprint. The more precise value of ð1:09^ 0:09Þ£ 1024

for Ste Marguerite is our preferred approximation for the initial182Hf/180Hf ratio of the Solar System. This precisely defined value isin agreement with previous estimates obtained from internalchondrite isochrons16, the comparison of W isotopes in ironmeteorites and chondrites16,30, and the W isotope compositions of

Figure 1 1w values of carbonaceous chondrites compared with those of the Toluca ironmeteorite and terrestrial samples analysed in this study. The values for Toluca, Allende,

G1-RF and IGDL-GD are the weighted averages of four or more independent analyses.

Also included are data from ref. 16 (indicated by a), ref. 30 (b), and ref. 2 (c). For the

definition of 1w see Table 1. The vertical shaded bar refers to the uncertainty in the W

isotope composition of chondrites. Terrestrial samples include IGDL-GD (greywacke), G1-

RF (granite) and BB and BE-N (basalts).

letters to nature

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[Kleine et al., Nature, 2002]

inconsistent with the 3-Myr timescale implied by our present Hf–Wdata, as well as with evidence for now extinct 53Mn (half-life t1/2¼ 3.7Myr, ref. 1) and 26Al (t1/2 ¼ 0.7Myr, refs 2, 3) in eucrites.From our present Hf–W data and the available 53Mn and 26Alchronologies, we conclude that the eucrites formed ,3Myr afterAllende’s CAIs. The eucrite isochron in Fig. 1 exhibits a high initial1w of about21 compared to the solar initial 1w of about23.5. Themantle of the eucrite parent body (EPB; probably the asteroid Vesta)must have evolved as a high Hf/W reservoir in order to develop

radiogenic 182W signatures. Core formation in the EPB resulted in af La=W < f Hf=W < 15 for the EPB mantle25. Thus, using the eucriteisochron6, the EPB mantle should have a present-day 1w ofþ17. As90–95% ofW in the EPB is in its core, it follows bymass balance thatthe W isotope composition of the core is in the range 0.9 to 1.7 1units below the present CHUR value of 22. This is (within error)the same as the difference between present and initial CHURof 1.5 1units as obtained in our work, and also consistent with W isotopedata for most iron meteorites and metals in chondrites8,15–19. Thisrequires that the time of eruption of these basalts on the surface ofthe EPB post-dates core formation in the EPB. Thus, the difficultiesencountered in interpretingHf–W systematics between eucrites andiron meteorites6 is resolved by using the chondrite data determinedin the present study.

Using our present estimate of the CHUR values for the182Hf–182W system, we can explore the implications for the timingof accretion and core formation of the Earth. The difference between1w in the BSE and CHUR ðD1w ¼ 1wðBSEÞ2 1wðCHURÞÞ togetherwith the fHf/W of ,12 for the BSE15 provide the basis for such acalculation. The D1w value of the BSE is þ2, and a plot of D1wversus the mean time of core formation is shown in Fig. 2. A two-stage model age for the BSE of 29Myr since the formation of the

Figure 1 Hf–W systematics for the early Solar System. Shown is a plot of 1w versus180Hf/183W represented as f Hf/W (see Table 1 for definitions of 1w and f

Hf/W). a, Data formetal and silicate fractions from ordinary chondrites Dalgety Downs (L4) and Dhurmsala

(LL6), and from carbonaceous chondrites Allende and Murchison, define a good fossil

isochron, identical within error of the individual isochrons for the two ordinary chondrites.

Least-squares fitting of the data include the Allende and Murchison whole-rock data, but

exclude the Allende CAI. Including or excluding the Murchison and Allende whole-rock

data or the CAI data does not significantly change the slope or the intercept. Our Juvinas

eucrite datum plots on the eucrite isochron6. The Moon, with a residual 1w ¼ 1.3 ^ 0.4

from 182Hf decay27 and f Hf/W ¼ 18 defined by the lunar La/W ratio28, falls within error on

the extension of the tie-line between the bulk chondrite (CHUR) and bulk silicate Earth

(BSE) points. b, Magnified area for bulk chondrite data. Dotted curves show the 2j error

band. Our results are consistent with E-chondrite data19, the zircon data for the Simmern

(H5) chondrite30, Ste Marguerite (H4) and Richardton (H5) ordinary chondrites18, and

inconsistent with published Allende and Murchison data7,8 and the published initial182Hf/180Hf value of Forest Vale (H4)18. The position of our isochron relative to the Forest

Vale isochron cannot be explained by late metamorphism. In that case the two isochrons

would intersect at the bulk chondrite point (f Hf/W ¼ 0), which is not observed. Plausible

reasons for discrepancy are: (1) uncorrected or improperly corrected interference; (2)

contamination of the meteorite with terrestrial W either during or before chemical

separation; and (3) incomplete dissolution, yielding a non-representative isotopic

composition.

Figure 2 Models for timing of core formation in the Earth. Shown is the expected

radiogenic 182W/183W value in the Earth relative to chondrites ðD1w ¼ ½1wðBSEÞ21wðCHURÞ& for a range of mean times of core formation (given by T 0 2 kTlcf; where T 0 is

the age of the Solar System and kT lcf is the mean age of core formation; see ref. 14) in theEarth for two different models of core segregation: a two-stage model, and a magma

ocean model. For the D1 w value of þ1.9 ^ 0.20 reported in this work, we obtain as

shown a two-stage model age of 29.5 ^ 1.5Myr and a mean time of core formation of

11 ^ 1.0Myr (63% of core formed) for the continuous model (calculated as described in

ref. 15). The vertical hatched bars represent the age uncertainties. We believe that the

magma-ocean segregation model yields the most realistic estimate (11 ^ 1.0Myr) for

the mean time of core formation. For comparison, we show the previous results8 with a

D1 w value of þ0.17 ^ 0.29. The timescale with the previous data8 is about a factor of

220.3þ153 longer (two-stage model age ¼ 62214

þ4504 Myr) than the one obtained in this work.

With the old data of ref. 8 the time of core formation is highly uncertain (62214þ4504 Myr), as

the upper error limit corresponds to no radiogenic 182W compared to chondrites in the

silicate Earth.

letters to nature

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[Yin et al., Nature, 2002]

Age of the Moon Formation

CAI 形成から約3,000万年後に last giant impact = 月形成

“CAI 形成から約3,000万年後原始地球に火星サイズの原始惑星が斜め衝突し飛び散ったマントル物質が周惑星円盤を形成し

それらが約数年かけて集積し月が誕生した”

Copyright Mitsumura Tosho Publishing Co.,Ltd. All rights reserved.

シーズン・インタビュー光村図書

　研究者によっては，ぶつかってきた原始惑星の大きな塊が，地球の中心部にある核のところに残っているのではと考える人もいます。それが見つかれば，巨大衝突説の証拠となるのではと――。でも，混ざってしまっていたら，痕跡を見つけるのは難しいかもしれませんね。　このように道は遠いけど，まだだれも知らないことを自分が最初に知ることの喜びやわくわくする感じがあって，きっとそれが好きだから，研究を続けているのだと思います。

　お話をいただいたときは，理科ではなく国語の教科書ということにまずびっくりしました。「えっ，僕で大丈夫？」と（笑）。　中学生にわかるような説明の文章にするために，いろいろと苦労しました。例えば，この文章には，まとまりごとに小見出しが付いているでしょう。これは論文の形式です。その章に書かれている内容が端的に把握できるよう，論文では章題を付けるのが普通です。でも，これまで教科書ではそういう文章をあまり扱っていなかったと聞き，それが意外でしたね。読みやすく，わかりやすい文章であるために，小見出しは絶対に必要だと思いました。　それから，この文章では図も重要です。編集部と何度もやり取りしながら，そのような点を検討し，教科書に掲載されている形に仕上げていきました。だから，これまでにないような種類の文章になっているかもしれませんね。

◆ 編集部から，教科書への書きおろしの依頼があったときには，どう思われましたか。

“国語” の教科書に載る

混乱期





Giant Impact によって形成される周惑星円盤の~80% が Impactor 起源のマントル物質である

of the Earth will freeze without having the opportunity to differentiate (because the crystals are advected vigorously by the turbulent convec-tive motions that accompany the cooling).

The Moon probably did not form immediately after the giant impact, even though orbital times for material placed about Earth are less than a day. Instead, it seems to be necessary to wait for hundreds to thou-sands of years, the timescale of disk cooling, as it is thought likely that the Moon did form completely molten. For reasons not fully under-stood, the need to cool the disk is of greater importance than the shorter timescales of dynamical evolution. Perhaps lunar formation should not be thought of as disconnected from the provenance and evolution of the deep Earth. The reason is that, after the giant impact, some exchange of material may have taken place between Earth and the disk, aided by the vigorous convection of both the liquid and vapour parts of each and the presence of a common silicate atmosphere. This picture of rapid exchange makes the disk more Earth-like, rather than like the projec-tile that was responsible for its formation. The picture was originally motivated by a desire to understand the remarkable similarity of Earth and Moon oxygen isotopes8 but also finds support in tungsten9 and possibly silicon10 isotopic evidence. However, we do not yet have a fully integrated model of lunar formation that is dynamically satisfactory as well as chemically acceptable.

Core formationThe core-formation events (one event per giant impact) are particularly important because core formation is the biggest differentiation process of Earth: it involves one-third of Earth’s mass and a large energy release, because the iron is about twice as dense as the silicates. To a substantial extent, it also defines the composition of Earth’s mantle. In the imme-diate aftermath of a giant impact, we expect a substantial part of the core of the projectile to be emulsified with the molten mantle of the pre-impact proto-Earth. The core and mantle materials are thought to be immiscible (like water and oil) despite the very high temperatures, perhaps as high as 10,000 K for some of the material. If the material is mixed down to a small scale (perhaps even to the point where there are centimetre-sized droplets of iron immersed in the liquid silicate) then the iron and silicate can chemically and thermally equilibrate at high temperature and pressure (Fig. 3a). The composition of the core and the iron content of the mantle were presumably set during these equili-bration episodes. The silicon and hydrogen contents of the mantle may also be affected by this equilibration, as both are soluble in iron at high pressure and temperature. These elements are particularly significant: silicon content affects the mineralogy of Earth’s mantle, and the fate of hydrogen may have much to say about the total water inventory of Earth at this early epoch and the flow of mantle rocks. However, much of Earth’s water may have been delivered later.

It is likely that some of the projectile iron is not mixed down to the smallest scales but instead finds its way to the core just hours after the impact (Fig. 3b). This iron will not equilibrate, either thermally or chemically, and it thus carries a memory of previous core-forming events at earlier times in smaller bodies (the embryos discussed earlier). The emerging picture is a complex one in which we should not expect the core or mantle of Earth to have a simple chemical relationship that involves the last equilibration at a particular pressure and temperature, but rather to have been formed under a range of thermodynamic condi-tions involving a number of significant events at different times2,11.

Earth’s atmosphere at the time of a giant impact might have been mostly steam and carbon dioxide (CO2) — probably both were impor-tant. It is possible, but not certain, that a large part of the atmosphere was blown away immediately after the giant impact. Water vapour is, however, much more soluble than CO2 in magma, so that even if the atmosphere were ejected into space, outgassing from the underlying magma ocean would replenish much of it. An important feature of water vapour is that it has a strong greenhouse effect, and that may have allowed the reten-tion of an underlying magma ocean, even for the long periods between giant impacts. However, this type of atmosphere can rain out if there is insufficient energy supplied to its base (sunlight alone is insufficient) and,

as a consequence, any steam atmosphere may collapse on a geologically short timescale, leading to an Earth surface that is actually cool (able to have liquid water) even while the interior is very hot.

Mantle differentiationThe mantle of the post-giant impact Earth will cool very fast at first12, limited only by the black-body radiation that can escape from the top of the transient (initially silicate vapour) atmosphere. The thermal structure of the mantle is expected to be close to isentropic because that is the state of neutral buoyancy and therefore the state preferred by convection, pro-vided that viscosity is low. The nature of the freezing within this convect-ing state is of great importance and is thermodynamically determined. Many materials have the property that if they are squeezed isentropically, they undergo freezing even as they get hotter. Equivalently, they melt if they are decompressed isentropically from a frozen but hot, high-pressure state. The former correctly describes the freezing of Earth’s solid inner core (the hottest place in Earth, yet frozen) whereas the latter correctly describes the melting responsible for the generation of basaltic magma, the dominant volcanism on Earth and most voluminously expressed at the low mantle pressures immediately beneath mid-ocean ridges. Recent work13,14 suggests that this picture may not apply for the deeper part of Earth’s mantle, so that freezing may begin at mid-depths.

Even so, there will eventually come a point (perhaps as soon as a few thousand years) after a giant impact when the bottom part of the mantle

a

b

c

Lunar-forming giant impact

Core

Core

Magma disk

Silicate vapouratmosphere

Radiative cooling

Blobs of iron settlingto core

Partlysolidified mantle

Rest of disk fallsback on Earth

Newly formed Moon, mostly or

partly molten

Figure 2 | The effect on Earth of the giant impact that formed the Moon. a, A giant planetary embryo collides with the nearly complete Earth. b, A magma disk is in orbit about Earth, while blobs of iron from the planetary embryo settle down through the mantle to join the existing core. c, The outermost part of the magma disk coalesces to form the Moon as the result of radioactive cooling, while the rest falls back to Earth. Inside Earth, the mantle nearest the core has partly solidified, and the mantle might acquire a layered structure.

263

NATURE|Vol 451|17 January 2008 YEAR OF PLANET EARTH FEATURE

Mixing in the Magma Disk

[Stevenson, Nature, 2008]

[Pahlevan & Stevenson, EPSL, 2007]

原始地球と原始月円盤の間で数100年間 mixing すればよい

The Astrophysical Journal, 760:83 (18pp), 2012 November 20 Salmon & Canup

Table 3Hybrid Simulation Parameters

Run Ld/Md Ld Md Min Mout q amax(√

GM⊕aR) (LEM) (M!) (M!) (M!) (R⊕)

1 0.843 0.304 2.00 2.00 0.00 N/A 2.92 0.843 0.365 2.50 2.50 0.00 N/A 2.9

3 0.955 0.345 2.00 1.00 1.00 5 44 0.960 0.347 2.00 1.00 1.00 3 45 0.965 0.348 2.00 1.00 1.00 1 46 0.955 0.414 2.40 1.20 1.20 5 47 0.960 0.416 2.40 1.20 1.20 3 48 0.965 0.418 2.40 1.20 1.20 1 49 0.899 0.325 2.00 1.50 0.50 5 410 0.901 0.326 2.00 1.50 0.50 3 411 0.904 0.326 2.00 1.50 0.50 1 412 0.899 0.390 2.40 1.80 0.60 5 413 0.901 0.391 2.40 1.80 0.60 3 414 0.904 0.392 2.40 1.80 0.60 1 415 0.888 0.401 2.50 2.00 0.50 5 416 0.890 0.402 2.50 2.00 0.50 3 417 0.892 0.403 2.50 2.00 0.50 1 418 0.880 0.477 3.00 2.50 0.50 5 419 0.882 0.478 3.00 2.50 0.50 3 420 0.884 0.479 3.00 2.50 0.50 1 4

21 0.986 0.356 2.00 1.00 1.00 5 622 1.009 0.365 2.00 1.00 1.00 3 623 1.036 0.374 2.00 1.00 1.00 1 624 0.986 0.427 2.40 1.20 1.20 5 625 1.009 0.437 2.40 1.20 1.20 3 626 1.036 0.449 2.40 1.20 1.20 1 627 0.914 0.330 2.00 1.50 0.50 5 628 0.926 0.335 2.00 1.50 0.50 3 629 0.940 0.339 2.00 1.50 0.50 1 630 0.914 0.396 2.40 1.80 0.60 5 631 0.926 0.401 2.40 1.80 0.60 3 632 0.940 0.407 2.40 1.80 0.60 1 633 0.900 0.406 2.50 2.00 0.50 5 634 0.909 0.411 2.50 2.00 0.50 3 635 0.920 0.416 2.50 2.00 0.50 1 636 0.890 0.482 3.00 2.50 0.50 5 637 0.898 0.487 3.00 2.50 0.50 3 638 0.907 0.492 3.00 2.50 0.50 1 6

39 1.068 0.386 2.00 1.00 1.00 1 740 1.068 0.463 2.00 1.20 1.20 1 741 0.998 0.361 2.00 1.00 1.00 5 842 1.043 0.377 2.00 1.00 1.00 3 843 1.099 0.397 2.00 1.00 1.00 1 844 0.998 0.433 2.40 1.20 1.20 5 845 1.043 0.452 2.40 1.20 1.20 3 846 1.098 0.476 2.40 1.20 1.20 1 8

Notes. Simulation parameters with a Roche-interior fluid disk and Roche-exterior individual particles. Md , Ld , and amax are the disk’s total initial mass,angular momentum, and outer edge. Min and Mout are the masses of thefluid disk, and of the solid bodies, respectively. Ld/Md is the disk’s totalspecific angular momentum (in units of

√GM⊕aR). q is the exponent for

the initial surface density distribution (σ (a) ∝ a−q ) of the Roche-exterior disk.Units of mass, distance, and angular momentum are the present lunar massM!, Earth radius R⊕, and angular momentum of the Earth–Moon system(LEM = 3.5 × 1041 g cm2 s−1). The normal and tangential coefficients ofrestitution ϵn and ϵt are set to 0.01 and 1, respectively. The particle-sizedistribution index p is set to 1.5, and the number of orbiting particles N isset to 1500. Runs 1 and 2 start with only a Roche-interior fluid disk.

the Moon to continue its accretion, or get ejected or scatteredclose to the planet where they are absorbed by the inner disk.When the Moon accretes spawned moonlets, its semi-major axis

Figure 2. Snapshots of the protolunar disk, projected on the R − z plane, att = 0, 0.03, 1, 30, 200, and 1000 years, for Run 34 using the hybrid model witha fluid inner disk. The size of circles is proportional to the physical size of thecorresponding particle. The horizontal thick line is the Roche-interior disk. Thevertical dashed line is the Roche limit at 2.9 R⊕.

tends to decrease slightly, since the specific angular momentumof the spawned moonlets is typically smaller than that of theMoon. However interactions between the Moon and moonletsthat are scattered into the inner disk cause the Moon’s semi-major axis to increase, as the Moon generally gains angularmomentum from the inner scattered bodies (Figures 2(e) and(f), see also next section). The latter effect dominates the end ofthe system’s evolution.

Contrary to accretion timescales of less than a year foundwith pure N-body simulations, here the initial confinement ofthe inner disk by outer bodies and the slow spreading of theRoche-interior disk back out to the Roche limit delay the finalaccretion of the Moon by several hundreds of years. We can

7

[Salmon & Canup, ApJ, 2012]

粘性モデルRoche 半径以内は流体的に振る舞う・固体への凝縮・物質の輸送

N 体計算Roche 半径以遠は固体的に振る舞う・衝突合体成長

月の形成時間~1,000年





LETTERSPUBLISHED ONLINE: 25 MARCH 2012 | DOI: 10.1038/NGEO1429

The proto-Earth as a significant source oflunar materialJunjun Zhang1*, Nicolas Dauphas1, AndrewM. Davis1, Ingo Leya2 and Alexei Fedkin1

A giant impact between the proto-Earth and a Mars-sizedimpactor named Theia is the favoured scenario for theformation of the Moon1–3. Oxygen isotopic compositionshave been found to be identical between terrestrial andlunar samples4, which is inconsistent with numerical modelsestimating that more than 40% of the Moon-forming diskmaterial was derived from Theia2,3. However, it remainsuncertain whether more refractory elements, such as titanium,show the same degree of isotope homogeneity as oxygen in theEarth–Moon system. Here we present 50Ti/47Ti ratios in lunarsamples measured by mass spectrometry. After correctingfor secondary effects associated with cosmic-ray exposureat the lunar surface using samarium and gadolinium isotopesystematics, we find that the 50Ti/47Ti ratio of the Moon isidentical to that of the Earth within about four parts permillion, which is only 1/150 of the isotopic range documented inmeteorites. The isotopic homogeneity of this highly refractoryelement suggests that lunar material was derived from theproto-Earth mantle, an origin that could be explained byefficient impact ejection, by an exchange of material betweenthe Earth’s magma ocean and the protolunar disk, or by fissionfrom a rapidly rotating post-impact Earth.

Apart from the effects of radioactive decay, the isotopiccompositions of most terrestrial rocks are related by the laws ofmass-dependent fractionation. Meteorites show departures fromthis rule that can be ascribed to unusual chemical processes,inheritance of nucleosynthetic anomalies, or nuclear transmu-tations (cosmogenic effects and radioactive decay). In the zooof elements that show well-documented isotopic anomalies ata bulk planetary scale5–8, highly refractory titanium, with largenucleosynthetic anomalies on 50Ti, is the most promising toassess the degree of homogeneity in the Earth–Moon system9.Taking advantage of our new chemical procedure for titaniumseparation and developments in multicollector inductively cou-pled plasma mass spectrometry (MC-ICPMS; see Methods andSupplementary Information for details; ref. 10), we measuredthe titanium isotopic compositions of 5 terrestrial samples, 37bulk chondrites, and 24 lunar samples (8 whole rocks, 6 ilmeniteseparates, 1 pyroxene separate, and 9 soil samples) with sub-stantially higher precision (0.04–0.11 "-unit for "50Ti, 2SE, where"50Ti = [(50Ti/47Ti)sample/(50Ti/47Ti)rutile � 1]⇥ 104) than previousstudies8,9,11,12 (Table 1 and Fig. 1). In agreement with earlier work8,we found that terrestrial rocks have constant titanium isotopiccomposition with an "50Ti value of +0.01±0.01 (average weightedby uncertainties, n= 19), whereas bulk meteorites show a spread in"50Ti values of ⇠6 "-units.

1Origins Laboratory, Department of the Geophysical Sciences, Enrico Fermi Institute, and Chicago Center for Cosmochemistry, The University of Chicago,5734 South Ellis Avenue, Chicago, Illinois 60637, USA, 2Physical Institute, Space Sciences and Planetology, University of Bern, Sidlerstrasse 5, Bern 3012,Switzerland. *e-mail: [email protected].

¬2 ¬1 0 1 2 3 4 5 650Ti

Pre-exposure lunar value ( 50Ti = ¬0.03±0.04)

ε

ε

Ordinary chondrites

Enstatite chondrites

Moon

Earth

Carbonaceous chondrites

Achondrites

CI

CM

CRCO

CVCK

EH

ELHL

LL

HEDs

Angrites

Aubrites

UngroupedAcapulcoite

Figure 1 | Titanium nucleosynthetic heterogeneity,"50Ti= [(50Ti/47Ti)sample/(50Ti/47Ti)rutile�1]⇥104, for carbonaceous,enstatite, ordinary chondrites, and achondrites. The filled and opensymbols are from this study and ref. 8, respectively, after internalnormalization to 49Ti/47Ti = 0.749766 (refs 8–10). The uncertainties are95% confidence intervals (2SE; see Supplementary Information for details).Grey areas cover the ranges of "50Ti values for each meteorite group in thisstudy. The red area indicates the pre-exposure lunar "50Ti value of�0.03±0.04 from extrapolation of the linear correlation between "50Tiand 150Sm/152Sm (see Fig. 2a for details).

The majority of lunar samples have titanium isotopiccompositions identical to terrestrial samples within the level of

NATURE GEOSCIENCE | VOL 5 | APRIL 2012 | www.nature.com/naturegeoscience 251

[Zhang et al., Nature Geo., 2012]

LETTERS

Late formation and prolonged differentiation of theMoon inferred from W isotopes in lunar metalsM. Touboul1, T. Kleine1, B. Bourdon1, H. Palme2 & R. Wieler1

The Moon is thought to have formed from debris ejected by a giantimpact with the early ‘proto’-Earth1 and, as a result of the highenergies involved, the Moon would have melted to form a magmaocean. The timescales for formation and solidification of theMoon can be quantified by using 182Hf–182W and 146Sm–142Ndchronometry2–4, but these methods have yielded contradictingresults. In earlier studies3,5–7, 182W anomalies in lunar rocks wereattributed to decay of 182Hf within the lunar mantle and were usedto infer that the Moon solidified within the first ,60 million yearsof the Solar System. However, the dominant 182W componentin most lunar rocks reflects cosmogenic production mainly byneutron capture of 181Ta during cosmic-ray exposure of the lunarsurface3,7, compromising a reliable interpretation in terms of182Hf–182W chronometry. Here we present tungsten isotope datafor lunar metals that do not contain any measurable Ta-derived182W. All metals have identical 182W/184W ratios, indicating thatthe lunar magma ocean did not crystallize within the first ,60 Myrof the Solar System, which is no longer inconsistent with Sm–Ndchronometry8–11. Our new data reveal that the lunar and terrestrialmantles have identical 182W/184W. This, in conjunction with147Sm–143Nd ages for the oldest lunar rocks8–11, constrains theage of the Moon and Earth to 62z90

{10 Myr after formation ofthe Solar System. The identical 182W/184W ratios of the lunarand terrestrial mantles require either that the Moon is derivedmainly from terrestrial material or that tungsten isotopes in theMoon and Earth’s mantle equilibrated in the aftermath of the giantimpact, as has been proposed to account for identical oxygen iso-tope compositions of the Earth and Moon12.

We obtained tungsten isotope data for metals from two KREEP-rich samples (KREEP stands for enrichment in potassium (K), rareearth elements (REE) and phosphorus (P)), four low-Ti and fivehigh-Ti mare basalts (Fig. 1 and Table 1). We processed fourfold tofivefold more material than an earlier study3 and monitored thepurity of our metal separates by determining their Hf/W ratios.These indicate that for the analyses reported here any possible con-tamination from silicate and oxide grains has no measurable effect on182W/184W. Most of the samples investigated here had relatively shortexposure times and required corrections of ,0.1 e units (e 5 0.01%)for burnout of tungsten isotopes13,14; only for samples 15556 and70017 (exposure ages ,220 and ,500 Myr) were corrections larger(,0.4 and ,0.2 e units). Details of the corrections are givenin Supplementary Information. All samples analysed here haveidentical 182W/184W ratios within 60.32 e units (2 s) and agree withpreviously reported data for metals from KREEP-rich samples3.Combined, these data average at e182W 5 0.09 6 0.10 (2 s.e.m.),n 5 15; e182W is defined in Table 1).

In contrast to earlier studies3,5,6, we do not find 182W/184W varia-tions within the lunar mantle. Lee et al.5 reported e182W < 1.4 for

low-Ti mare basalt 15555; however, the exposure age of this sample,combined with its Sm isotopic composition and Ta/W ratio, indi-cates that this anomaly might be due entirely to cosmogenic 182W.Kleine et al.3 reported elevated e182W < 2 for a magnetic separatefrom high-Ti mare basalt 79155 but we determined Hf/W 5 7.5 foran aliquot from the same magnetic separate, most probably indi-cating the presence of some ilmenite and hence cosmogenic 182Win this separate. The calculated cosmogenic 182W component is ,1.7

1Institute for Isotope Geochemistry and Mineral Resources, Department of Earth Sciences, Eidgenossische Technische Hochschule Zurich, Clausiusstrasse 25, 8092 Zurich,Switzerland. 2Institut fur Mineralogie und Geochemie, Universitat zu Koln, Zulpicherstrasse 49b, 50674 Koln, Germany.

–2 –1 0 1 2 3 4 5

–2 –1 0 1 2 3 4 5

e182W

Ref. 3

This study

Ref. 5

Corrected in this study

KREEP-rich samples

Low-Ti mare basalts

High-Ti mare basalts

14310154456223565015

6811568815

7215579155

750757751670035

70017

70035

15475

15555 (WR)

1549915556

15058

15555

75035

7425574275

12004

Figure 1 | e182W of lunar metals analysed in this study compared with datafrom refs 3 and 5. Some of the previous data (shown with black dots insidethe symbols) were corrected for cosmogenic 182W (see the text for details).Error bars indicate 2s external reproducibilities. The hatched area indicatesthe average e182W 5 0.09 6 0.10 (2 s.e.m., n 5 15) of lunar metals from thisstudy combined with previously reported data for metals from KREEP-richsamples5. The dashed lines indicate a 2s of 60.32 e182W of these data.

Vol 450 | 20/27 December 2007 | doi:10.1038/nature06428

1206Nature ©2007 Publishing Group

[Touboul et al., Nature, 2007]

難揮発性元素の同位体も一致





(6) 難揮発性元素の同位体比が地球とほぼ一致[Touboul et al., 2007]

すぐに凝結してしまう元素は十分に mixing できない

6. Results

6.1. Chemical fractionation

Fractionation calculations in the silicate vapor atmosphere ofthe Earth require two steps: a closed-system step, described earlier(Section 4), whereby the liquid and vapor equilibrate at a newpressure, and an open-system step, whereby the newly equilibratedliquid partially rains out, shifting the properties of the parcel (entropy,composition) toward that of the vapor. At each altitude, we assumethat a fraction, fL, of the liquid is removed via rainout. We treat fL as afree parameter, and explore the consequences of varying it for theresulting lunar composition inherited from the silicate Earth. Whenthe liquid is partially removed, the vapor fraction in the remainingparcel increases, and is calculated according to mass balance:

fV‐new =1

1 + 1= fV‐old−1ð Þ × 1−fLð Þ : ð14Þ

Using the new value of the vapor fraction and Eqs. (9) and (10), wecalculate the entropy and composition of the parcel after rainout.Subsequent episodes of droplet-vapor equilibration at a lowerpressure – representing adiabatic ascent – and partial liquid removalmake it possible to calculate the compositional structure of the silicatevapor atmosphere undergoing varying degrees of rainout. Thisprocedure is exactly analogous to the calculation of pseudoadiabatsin atmospheric science (Holton, 1992).

Plotted in Figure 2 is the result of a sample calculation with 40%liquid removal via rainout (fL=0.4) every three-fold decrease inpressure (roughly every scale height), starting at a pressure of 1 bar.This particular parameter was chosen because this level of rainout overtwo orders-of-magnitude of atmospheric pressure can generate – at thetop of the atmosphere – thewidely postulated ~2× enhancement of thelunar FeO/MgO ratio.

In addition to themajor elements, we are interested in following thebehavior of chemical tracers during rainout. However, for this purpose,we must know the partitioning behavior of elements between liquid

and vapor, which requires a solution model that yields reliable activitycoefficients for trace elements at the temperatures of relevance(T=2500 K–3500 K). At present, no such solution model exists.However, we can still calculate the behavior of elements in certainlimiting cases. For elements that quantitatively partition into the liquid,we have:

XRnew = XR

old ×1−fLð Þ

fV‐old + 1−fLð Þ × 1−fV‐oldð Þ ð15Þ

where XR represents the mole fraction of the refractory tracer in theparcel before and after rainout. We use this approach to follow the fateofmajor-element refractory oxides (i.e. CaO, Al2O3) in the context of therainout scenario just described. The range of lunar bulk compositionsthat can be evolved in this way from the Earth mantle is plotted inFigure 3.We note that compositions derived from the terrestrial mantleenhanced in FeO/MgO through liquid rainout are depleted in refractory

Fig. 1. Chemical fractionation on an unstratified Earth. A single convective columncharacterizes the Earth from the deep magma ocean, where only one phase is present,through the top of the two-phase atmosphere. Rainout of Mg-rich droplets in ascendingparcels shifts the composition of the upper atmosphere towards an Fe-rich vaporcomposition. The degree of rainout in these models is a free parameter.

-2

-1

00 0.2 0.4 0.6

log

P (b

ars)

Fe/Fe+Mg

liquid

parcel

vapor

Fig. 2. Chemical structure of the silicate vapor atmosphere of the Earth undergoingrainout. The parcel represents the composition of the atmosphere (forsteritic dropletssuspended in a fayalitic vapor) and shifts with altitude toward that of the Fe-rich vaporas the droplets separate via rainout. The lower atmosphere is directly sourced byconvection from the underlying magma ocean and is therefore of the samecomposition. This calculation assumes that 40% of the liquid is removed via rainoutevery three-fold decrease in pressure (fL=0.4). This parameter is chosen such that – atthe top of the atmosphere – a two-fold enhancement in the FeO/MgO ratio evolves. Thisenhancement is comparable to a widely postulated silicate Earth–Moon difference, andhas observable consequences (see text).

2

3

4

5

6

0 0.1 0.2 0.3Fe/Fe+Mg

Al 2

O3

(wt %

)

Higher degrees of liquid rainout

TerrestrialMagmaOcean

fL= 0.4

fL= 0.3

fL= 0.2

fL= 0.1

Fig. 3. Co-evolution of the Fe/Fe+Mg ratio and refractory element abundances –

represented by alumina – of atmospheric columns undergoing rainout. The aluminaabundance (=4.7 wt.%) and Fe/Fe+Mg (=0.1) of the modern mantle is assumed forthe magma ocean. Plotted are upper atmospheric compositions evolved by rainout atlower levels and inherited by the proto-lunar disk. The process of phase separation inthe vapor atmosphere shifts the expected lunar composition towards the compositionof the vapor with an enrichment in Fe/Fe+Mg and depletion in refractory elements,but small degrees of phase separation can yield lunar compositions nearly isochemicalwith the terrestrial mantle.

438 K. Pahlevan et al. / Earth and Planetary Science Letters 301 (2011) 433–443

[Pahlevan et al., EPSL, 2011]

“Unstratified” Magma Disk

難揮発性元素についても mixing の可能性を提案






All fractionations measured were mass dependent, indeedmass dependence was used as a data quality check to ensurecomplete resolution of the Si mass spectrum from poly-atomic interfering species. Mass dependent mass bias waschecked by constructing a three isotope plot of all the dSidata (Fig. 2), which was regressed to give a slope of0.520 ± 0.015. This best fit slope is within error of calcu-lated equilibrium (0.5178) and kinetic (0.5092) fraction-ation slopes based on the exact masses of 28Si, 29Si and30Si (Audi and Wapstra, 1993) following Young et al.(2002).

3. RESULTS

The Si isotopic data for the 20 bulk lunar rocks and 4lunar glasses are given in Table 1 and Fig. 3. The averageof all the bulk lunar samples is d30Si = !0.29 ± 0.08(±2rSD) which is identical to the recent value of Savageet al. (2010) for bulk silicate Earth of d30Si =!0.29 ± 0.08 (±2rSD). The average of the four lunar bas-alts (d30Si = !0.31 ± 0.07, 2rSD) from Georg et al.’s(2007) and Fitoussi et al.’s (2010) mean lunar compositionof d30Si = !0.30 ± 0.05& (2rSD) are consistent with thenarrow observed range of Si isotope compositions acrossthe variety of samples observed in this study (Fig. 4). Thelunar lithologies analysed in this study are identicalwithin error (2rSD): d30SiLow-Ti basalt = !0.29 ± 0.06;d30SiHigh-Ti basalt = !0.32 ± 0.09; d30Silunar glass = !0.29 ±0.05; d30SiHighland rocks = !0.27 ± 0.10. The similarity of

Fig. 2. d29Si versus d30Si plot. The error bars represent ±2rSEM forthe samples. The calculated slopes for mass dependent equilibriumfractionation (0.5178) and mass dependent kinetic fractionation(0.5092) are also plotted.

Fig. 3. d30Si isotope compositions of bulk lunar samples. The errorbars represent ±2rSEM. (II) signifies a separate fusion of anotheraliquot of sample powder. The solid line is the mean of the lunarsamples, while the dashed lines are ±2rSD.

Fig. 4. Histograms of d30Si values lunar samples from this studyand bulk silicate Earth samples (BSE) from Savage et al. (2010).The lunar breccia from Chakrabarti and Jacobsen (2010) has alsobeen plotted. The solid black line is the calculated normaldistribution of the Georg et al. (2007) lunar basalt data while thedashed black is the calculated normal distribution of mare basaltsof Fitoussi et al. (2010).

d Data for the BHVO-2 mean comes from Zambardi and Poitrasson (2010).e When there are duplicate fusions the average of their values is used when calculating the mean.f The value for the bulk silicate Earth average (±2rSD) comes from Savage et al. (2010).g The meteorite average is from Armytage et al. (2011).h All these samples have been analysed on the MC-ICPMS using the USGS basalt standard BHVO-2 as a bracketing standard rather than

NBS-28, giving a direct measurement of the offset between terrestrial and lunar basalts.i For all the BHVO-2 bracketed data (n = 1).

508 R.M.G. Armytage et al. / Geochimica et Cosmochimica Acta 77 (2012) 504–514

[Armytage et al., GCA, 2012]

Si 同位体比も一致






Si 同位体分配には圧力（＝サイズ）依存性がある

(7) Si 同位体比が地球とほぼ一致 [Armytage et al., 2012]

Hf-W Chronometry の弱点

[Sasaki & Abe, PPV, 2005]

どう考えても Hf-W 系の完全平衡化は実現できない

T. SASAKI AND Y. ABE: IMPERFECT EQUILIBRATION OF HF-W SYSTEM 1041

Fig. 6. The age of the last giant impact as a function of the resetting ratioof each giant impact, fitting to the observational data (ϵ = 2) from Earthsamples. The number of giant impacts is assumed to be five. The initialstate is ϵ = 10 at t = 10. The formation age of the Earth for perfectresetting (resetting ratio = 1) is about 30 Myr, in agreement with aprevious study (Yin et al., 2002).

metal-silicate equilibration. This would not be a realisticassumption. However, these calculations give us the higherlimit of equilibration at each giant impact event. Therefore,from the viewpoint of obtaining the observed isotopic ratio,we can obtain the lower limit of the resetting ratio requiredfor each giant impact. That is, we use assumptions that leadto the “higher limit of equilibration” on calculating isotopicevolution to obtain “the lower limit of required resetting ra-tio” to meet the observed epsilon value. The number ofgiant impacts, n, was varied from 2 to 10. Figure 6 showsthe result for n = 5, for example. The estimated age ofthe last giant impact depends on the resetting ratio of eachgiant impact, which must be greater than 0.3 to yield theobserved ϵ value. The effect of the number of giant impactsis shown in Fig. 7. It shows that the resetting ratio of eachgiant impact and the number of giant impacts both affectthe estimation of the age of the last giant impact. The re-sults indicate that the average resetting ratio of each giantimpact must be greater than 0.2 to yield a good fit with theobservations, even if giant impacts occurred ten times.

Although we use the f -value of 12 in Eq. (4) to solveEq. (6) and Eq. (7), as mentioned earlier, the f -value inEq. (4) is considerably uncertain (from 10 to 40). Therefore,we check how does this uncertainty affect our conclusionshere. Figure 8 shows the age of last giant impact assumedto have occurred ten times to form the Earth for f = 10,12, 20, 30, 40, which was calculated in a manner similar tothe case of f = 12. It shows that the lower limit of averageresetting ratio of each giant impact is still about 0.2 in therange from f = 10 to f = 40, which would not alter ourconclusions.

Our calculations tend to give an overestimation of theequilibration rate of the Hf-W system, as all of the im-pactor’s core and mantle is assumed to be equilibrated bya giant impact. In practice, because some fraction of theimpactor’s core (mantle) may be added to the target’s core(mantle) without equilibrating, the required resetting ratiomay be larger than that was estimated in this section. There-fore, our value of 0.2 should be regarded as a lower limit ofthe required resetting ratio of each giant impact, for a total

Fig. 7. The age of the last giant impact as a function of the resetting ratioof each giant impact, fitting to the observational data (ϵ = 2) from Earthsamples. The number of giant impacts is 2 to 10 from left to right. Theinitial state is ϵ = 10 at t = 10.

Fig. 8. The age of the tenth giant impact as a function of the resettingratio of each giant impact, fitting to the observational data (ϵ = 2) fromEarth samples. The f -value is 10, 12, 20, 30, 40 from bottom to top.Each initial ϵ-value was calculated using Eq. (6) and Eq. (7) for eachf -value.

number of giant impacts less than or equal to ten.Nimmo and Agnor (2006) considered two extreme sce-

narios after giant impacts: complete metal-silicate equili-bration and core merging without any significant equilibra-tion. They claimed that the observed isotopic data requirere-equilibration of impacting bodies with the target mantleand ruled out direct core merging even for the largest im-pacting bodies. Although the direct core merging is onetype of imperfect equilibration, it is not the only possibleway of imperfectness. The Rayleigh-Taylor instability dis-cussed in Section 2 is another type of imperfect equilibra-tion, which leaves some fraction of silicate without equili-bration with iron, while direct core merging leaves somefraction of iron without equilibration with silicate. Herewe showed that the observational data does not rule out in-complete equilibration due to the Rayleigh-Taylor instabil-ity. Our results indicate that the collision conditions and thenumber of giant impacts are essential parameters to estimatethe age of the core formation event.

4. DiscussionWe have shown that complete metal-silicate equilibration

by a giant impact cannot be expected even if the impact

[Sasaki & Abe, EPS, 2007]

[Wood & Halliday, Nature, 2005]

Age of the Moon Formation?

Hf-W chronometry では月形成の年代は決まらない

2.3. Initial Conditions

We followCanup&Asphaug (2001) andCanup (2004) for theorbital parameters of the impactor for which the most massivesatellite is expected. The masses of the proto-Earth and the im-pactor are assumed to be 1.0 and 0:2 M!, whereM! is the Earthmass. The radii of the proto-Earth and protoplanet are rE ¼ 1:0and 0:64rE, respectively. Note that no significant differences inthe results for smaller impactors (e.g., 0:1 M!) were found in oursimulations. The initial orbits of the impactor are assumed tobe parabolic, and the angular momentum is 0.86Lgraz, where Lgrazis the angular momentum for a grazing collision (Canup &Asphaug 2001). Initially, the impactor is located at 4:0rE fromthe proto-Earth.

3. RESULTS

3.1. Disk Evolution and the Predicted Lunar Mass

Figure 1 shows a typical time evolution of the giant impactwith EOS-1 (model A). This model corresponds to the ‘‘late’’impact model in Canup&Asphaug (2001). After the first impact(t ’ 1 hr), the disrupted impactor is reaccumulated to form aclump at t ’ 3 hr, which finally collides with the proto-Earth att ’ 6 hr. During the second impact, the impactor is destroyed,and a dense part of the remnant spirals onto the proto-Earth(t ’ 10 hr), and a circumterrestrial debris disk is formed aroundt ’ 18 hr. Note that many strong spiral shocks are generated inthis process as seen in the density map (Fig. 2) and azimuthaldensity profile (Fig. 3).

Fig. 1.—Giant impact simulation with EOS-1, which represents a state in which most of the impactor mass is vaporized. Left, face-on views of the system; right, edge-on views. The numbers in the upper right corners of the panels show the time in units of hours. The color represents log-scaled density (the units are !0 ¼ 12:6 g cm#3).

Fig. 2.—Snapshot of the density field of model A at t ¼ 12:3 hr. Strongspiral shocks in the debris are resolved.

WADA, KOKUBO, & MAKINO1182 Vol. 638

[Wada et al., ApJ, 2005]

高解像度格子法による G.I. 計算

蒸発した原始月円盤内に衝撃波が立ちまくって円盤が角運動量を失い数日で全て地球に落下！？

革命期

Ćuk & Stewart, Science (2012)

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.

References and Notes1. W. K. Hartmann, D. R. Davis, Icarus 24, 504

(1975).2. A. G. W. Cameron, W. R. Ward, Proc. Lunar Planet.

Sci. Conf. 7, 120 (1976).3. D. J. Stevenson, Annu. Rev. Earth Planet. Sci. 15, 271

(1987).

4. R. M. Canup, E. Asphaug, Nature 412, 708 (2001).5. G. W. Lugmair, A. Shukolyukov, Geochim. Cosmochim. Acta

62, 2863 (1998).6. U. Wiechert et al., Science 294, 345 (2001).7. M. Touboul, T. Kleine, B. Bourdon, H. Palme, R. Wieler,

Nature 450, 1206 (2007).8. J. Zhang, N. Dauphas, A. M. Davis, I. Leya, A. Fedkin,

Nat. Geosci. 5, 251 (2012).9. M. M. M. Meier, Nat. Geosci. 5, 240 (2012).

10. R. M. Canup, Icarus 168, 433 (2004).11. R. M. Canup, Icarus 196, 518 (2008).12. K. Pahlevan, D. J. Stevenson, Earth Planet. Sci. Lett. 262,

438 (2007).13. K. Pahlevan, D. J. Stevenson, J. M. Eiler, Earth Planet.

Sci. Lett. 301, 433 (2011).14. M. Touboul, I. S. Puchtel, R. J. Walker, Science 335, 1065

(2012).15. S. Mukhopadhyay, Nature 486, 101 (2012).16. A. Reufer, M. M. M. Meier, W. Benz, R. Wieler, Icarus

221, 296 (2012).17. A. E. Ringwood, Earth Planet. Sci. Lett. 95, 208

(1989).18. S. Ida, R. M. Canup, G. R. Stewart, Nature 389, 353

(1997).19. G. H. Darwin, Philos. Trans. R. Soc. London 170, 447

(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).

-200-150-100

-50 0

50 100 150 200

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entr

icity

B

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i-maj

or a

xis

(RE) A

Synchronous at perigee

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.

0.3

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QE=48 QM=48QE=96 QM=97

QM=117QM=73QM=57

A

B

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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地球-月-太陽の間の永年共鳴で系の角運動量が減少






(7) Si 同位体比が地球とほぼ一致 [Armytage et al., 2012]

Collision Scenariosw/o angular momentum constraints


PERSPECTIVES

formed from a magma ocean ( 5), implying an

intensely energetic fi ery start at a time when

heat-producing short-lived nuclides (26Al and 60Fe) were extinct. Third, the oxygen isoto-

pic composition of the Moon is identical to

that of Earth ( 6) to within 5 parts per million,

whereas that of nearly all asteroidal and plan-

etary objects are different.

The Giant Impact Theory is the resul-

tant widely accepted current explanation for

the Moon’s late, molten start as a low-den-

sity object that now contributes most of the

angular momentum in the Earth-Moon sys-

tem. Following an early similar model ( 7),

dynamic models using smooth particle hydro-

dynamic code have been deployed ( 8, 9) to

track the temperature and fate of silicate and

dense iron metal resulting from the oblique

collision of two planets. In most models, this

has been an Earth when it was about 85 to

90% formed and another planet, sometimes

called Theia, that was about 10 to 15% of the

mass of Earth or roughly the size of Mars (see

the fi gure, panel A). Simulations of this cata-

strophic glancing blow show that most of the

material from Theia was added to Earth, with

a small fraction of silicate-rich material left as

a disk from which the Moon accreted.

In nearly all “successful” Giant Impact

simulations, the material that ends up in

the Moon is mainly derived from Theia. To

account for the identical isotopic composi-

tion, it was proposed that Theia and Earth

must have formed at similar heliocentric

distances—but then it is diffi cult to explain

why the Giant Impact was so late. Perhaps

at the high temperatures achieved during the

Giant Impact, Earth’s mantle and lunar accre-

tion disk would have remained in vapor and

liquid form long enough to achieve isotopic

equilibration by mixing and diffusion ( 10).

This would work for elements as volatile as

oxygen. However, refractory elements such

as tungsten and titanium are also isotopically

identical ( 3, 11). Further mixing of refractory

elements might have occurred during rainout

of condensates ( 12). However, the identical

silicon isotopic composition of Earth and the

Moon ( 13) is not readily explained; the rain-

out process is expected to generate a silicon

isotopic difference, so the problem persists.

The papers by Ćuk and Stewart and by

Canup remove the major constraint that the

initial angular momentum was generated by

the Giant Impact. Ćuk and Stewart propose

instead that after the Moon formed the sys-

tem was rotating far faster, that Earth could

also have been doing so beforehand, and that

it has been slowed subsequently as a conse-

quence of a resonance in tidal forces with the

Sun. This opens up the possibility of different

impact scenarios. Earth itself could have been

left spinning rapidly after its prior accretion

history, such that a relatively small proportion

of the angular momentum of the Earth-Moon

system today is the result of the Giant Impact.

Without this constraint, it is then possible to

investigate a broader array of impact scenar-

ios, and this is where the two papers diverge.

Ćuk and Stewart investigate solutions

with a relatively small Theia (less than 10%

of Earth’s mass) and a pre-impact angular

momentum two to three times that of today.

In one example with a mass of Theia of only

2% of Earth’s fi nal mass (see the fi gure, panel

B), they achieve formation of a lunar mass of

material in orbit with only 8% being derived

from Theia, which compares with a final

Earth with as little as 2%. This small pro-

portion of impactor material in both objects

limits the possibilities for there being Earth-

Moon isotopic differences.

Canup goes to the opposite extreme with

models exploring the possibility that Theia

was 30 to 45% of the current Earth (see the

fi gure, panel C). As Theia gets bigger, the

proportions of the proto-Earth:Theia mix

become closer in the two objects. The result

is tested using assumptions of isotopic diver-

sity, the strongest constraint for which comes

from oxygen that a Theia of more than 40%

of the total mass appears to satisfy.

Another class of models has been pro-

posed simultaneously ( 14), arguing for very

energetic hit-and-run collisions between a

Theia with the more conventional 10% Earth

mass and removal of angular momentum by

loss of material from the system (see the fi g-

ure, panel A). As with the Ćuk and Stewart

model, most of the material in the Moon is

derived from the proto-Earth.

Distinguishing among these three mod-

els is going to involve further simulation and

debate. Geochemical constraints may again

prove decisive in three ways.

First, tungsten isotopes are sensitive to

equilibration between incoming metal from

the impactor’s core and tungsten in the sili-

cate Earth ( 15). Such equilibration will vary

with impact angle ( 16) and should lead to dif-

ferent isotopic compositions between silicate

Earth and Moon after further post-impact

equilibration and core formation. This will be

less of an issue with a small impactor.

Similarly, the silicon isotopic compo-

sition of Earth and the Moon are identical

( 13), and this is a signature of high-pressure

core formation that has been transferred to

the Moon. If there is a major increase in

the size of Earth, as in the Canup model, it

might be expected to further fractionate sili-

con isotopes relative to the Moon.

Finally, for the particularly energetic mod-

els ( 2, 14), this would be expected to lead to

widespread melting and mixing. Yet recent

results ( 17) provide evidence of deep reser-

voirs with noble gas isotopic heterogeneities

that have been preserved since about the time

of the Giant Impact. It is not known whether

these are localized or widespread, but their

presentation is intriguing in a planet that sup-

posedly was built by repeated highly ener-

getic accretion.

References

1. M. Ćuk, S. T. Stewart, Science 338, 1047 (2012); 10.1126/science.1225542.

2. R. M. Canup, Science 338, 1052 (2012); 10.1126/science.1226073.

3. M. Touboul, T. Kleine, B. Bourdon, H. Palme, R. Wieler, Nature 450, 1206 (2007).

4. J. E. Chambers, Earth Planet. Sci. Lett. 223, 241 (2004). 5. G. J. Taylor, Elements 5, 17 (2009). 6. U. Wiechert et al., Science 294, 345 (2001). 7. W. K. Hartmann, D. R. Davis, Icarus 24, 505 (1975). 8. A. G. W. Cameron, W. Benz, Icarus 92, 204 (1991). 9. R. M. Canup, E. Asphaug, Nature 412, 708 (2001). 10. K. Pahlevan, D. J. Stevenson, Earth Planet. Sci. Lett. 262,

438 (2007). 11. J. Zhang, N. Dauphas, A. M. Davis, I. Leya, A. Fedkin, Nat.

Geosci. 5, 251 (2012). 12. K. Pahlevan, D. J. Stevenson, J. M. Eiler, Earth Planet. Sci.

Lett. 301, 433 (2011). 13. R. M. G. Armytage, R. B. Georg, H. M. Williams, A. N.

Halliday, Geochim. Cosmochim. Acta 77, 504 (2012). 14. A. Reufer, M. M. M. Meier, W. Benz, R. Wieler, Icarus 221,

296 (2012). 15. A. N. Halliday, Nature 427, 505 (2004). 16. T. W. Dahl, D. J. Stevenson, Earth Planet. Sci. Lett. 295,

177 (2010). 17. S. Mukhopadhyay, Nature 486, 101 (2012).

A

Standard impactor Small impactor Large impactor

B C

Collision scenarios. Examples of the three new models of the Moon-forming Giant Impact, each of which allows more angular momentum to be lost and thereby achieves oxygen isotopic compositions that cannot be resolved between Earth and the Moon. (A) “Standard” impactor, 10% of Earth’s fi nal mass, works with “hit and run” collision ( 14). (B) “Small” impactor, 2.5% of Earth’s fi nal mass ( 1). (C) “Large” impactor, 45% of Earth’s fi nal mass ( 2).

10.1126/science.1229954

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(A) 質量比 10:1 で “Hit-and-Run” collision [Reufer et al., 2012]

(B) 質量比 40:1 で “Fission-like” collision [Ćuk & Stewart, 2012]

(C) 質量比 1:1 で “Twins” collision [Canup, 2012]

[Halliday, 2012]

ratio of 9:1 and a total mass of 1.05 ME (Canup, 2004). Both the impactor and thetarget are assumed to be differentiated bodies with a 30 wt% iron core and a70 wt% silicate mantle. In these low-velocity collisions, the impactor loses kineticenergy during its grazing collision with the target, before it is dispersed into a diskaround the target. The resulting proto-lunar disk is therefore mainly composed ofimpactor material. We will call this the ‘‘canonical scenario’’.

When the assumption that no mass is lost is dropped however, the collisionalangular momentum is no longer tightly constrained, as lost mass also carries awayangular momentum. The total collisional angular momentum can therefore be con-siderably higher than the final angular momentum in the Earth–Moon system. Withthis additional degree of freedom, new regions in the collision parameter space be-come feasible.

Apart from the disk mass, another interesting quantity is the origin of the mate-rial which ends up in the proto-lunar disk, especially for the silicate part.

We call the fraction of target silicate to total silicate material in the disk

fT ¼ ðMsilctarg=Msilc

tot Þdisk ð1Þ

where Msilctarg and Msilc

tot denote the mass of the silicate fraction of the disk derived fromthe target, and the total disk mass, respectively. If we define a similar target-derivedsilicate fraction for the post-impact Earth, we can deduce a deviation factor

dfT ¼ Msilctarg

! .Msilc

tot

"

disk

.Msilc

targ

! .Msilc

tot

"

post-impact Earth$ 1 ð2Þ

which directly reflects the compositional similarity between the silicate part of theproto-lunar disk and the silicate part of the post-impact Earth.

Isotopic measurements show (Wiechert et al., 2001; Zhang et al., 2012) a strongisotopic similarity between the silicate fractions of today’s Moon and Earth. Assum-ing isotopic heterogeneity of the pre-impact bodies, this suggests that either thematerial of the bodies mixed during the collision or re-equilibrated their isotopicsignatures after the collision. Either scenario is represented by a dfT % 0 between to-day’s Earth and the Moon. The value of dfT right after the impact thus serves as astarting point, from which a re-equilibration mechanism leads to todays value ofdfT % 0.

In a typical simulation of the canonical scenario, only about 30% of the diskmaterial and 90% of the material of the post-impact Earth is derived from the target(the proto-Earth) respectively (Canup, 2004), yielding a dfT of $67%.

4. Results

The new class of collisions presented here falls into the broad regime of slowhit-and-run collisions (Asphaug et al., 2006) with impact velocities between 1.20and 1.40 vesc. Hit-and-run occurs up to half the time for collisions with impactvelocities in this range. Because of the higher impact velocities in this type of col-lisions, substantial mass and angular momentum can be lost in the process. There-fore, the initial angular momentum is less constrained and can be considerablyhigher than the post-impact 1.0–1.1 LE–M angular momentum of the Earth–Moon-system. The higher impact velocities used in these simulations are also encouragedby more recent models of terrestrial planet formation (O’Brien et al., 2006). In hit-and-run collisions, a significant part of the impactor escapes, so that the disk frac-tion is enriched in target-derived materials compared to the canonical case. Fig. 1ashows four consecutive snapshots of such a hit-and-run collision. While the overallcharacteristics of the collision resemble the canonical scenario, here a considerablepart of the impactor is ejected.

In the new class of giant impact simulations presented in the following para-graphs, for the first time a significantly higher fraction of the material constitutingthe disk is derived from the Earth’s mantle. Table 1 shows a selection of around 60simulations performed by us. A canonical reference run (cA08) uses initial param-eters and conditions similar to those used in the canonical scenario (Canup,2004), successfully reproducing an iron-depleted proto-lunar disk massive enoughto form a Moon. For each of our runs, a final Moon mass is calculated using the diskmass and the specific angular momentum (Kokubo et al., 2000). We employed threedifferent impactor types with different compositions: chondrictic impactors with70 wt% silicates and 30 wt% Fe (vesc = 9.2 km/s, c-runs), iron-rich impactoswith 30 wt% silicates and 70 wt% Fe (vesc = 9.3 km/s, f-runs) and icy impactors with50 wt% water ice, 35 wt% silicates and 15 wt% Fe (vesc = 8.9 km/s, i-runs). Note thatthe scaled impact velocity vimp/vesc determines the type of collision, as similar-sizedcollisions in the gravity regime are self-similar (Asphaug, 2010; Leinhardt andStewart, 2012). Initial temperatures of the iron cores are between 4000 and5000 K, for the silicate layers between 1600 K and 2200 K and around 300 K forwater ice layers. As mentioned before the layers are isentropic and the temperaturetherefore varies with depth.

5. Discussion

The ratio of target- vs. impactor-derived material that ends up in the proto-lu-nar disk is mainly defined by the geometry of the collision during the very earlyphase when the impactor is accelerating target material. This can be seen inFig. 1b, where the particles which later end up in the disk are highlighted in bright

colors. In the canonical scenario, the impactor grazes around the target’s mantleand is deformed. Due to the low impact velocity, material supposed to end up in or-bit around the Earth must not be decelerated too strongly in order to retain enoughvelocity to stay in orbit. This is only achieved for the parts of the impactor mantlemost distant to the point of impact, and some minor part of the target’s mantle. Butif impact velocity is increased from 1.00 (cA08) to 1.30 vesc (cC01), parts from dee-per within the target mantle receive the right amount of energy for orbit insertion,while the outer regions of the target mantle, retain too much velocity and leave thesystem, thereby removing mass and angular momentum. Both processes work to-wards increasing the target material fraction in the proto-lunar disk. While in runcB04 only %10% of the initial angular momentum is removed, %45% are removedin run cC06.

We have found that collisions with an impact angle of 30–40! and impact veloc-ities of 1.2–1.3 vesc are successful in putting significant amounts of target-derivedmaterial into orbit, when using differentiated impactors with a chondritic iron/sil-icates ratio (30 wt% Fe, 70 wt% silicates) and masses between 0.15 and 0.20 ME.Some runs in this regime show an iron excess of >5 wt% in the proto-lunar diskand are rejected, as in previous work (Canup, 2004). While none of the runs doneso far provide a ‘‘perfect match’’ in terms of the constraints from the actualEarth–Moon-system, several simulations come close to that. The best runs comingclose to matching the constraints (cC03 and cC06) are obtained using impact anglesof 32.5! and 35! and velocities of 1.25 and 1.20 vesc, resulting in 54% and 56% of thesilicate material deriving from the target, and dfT thus increasing to $35% and $37%compared to $66% in the reference run of the canonical case. While the satellitemasses match well (1.01 and 1.24 ML), the angular momentum of the runs is some-what too high (1.28 LE–M). This should, however, be contrasted with other runs, e.g.cB03, where the reduction of the impactor mass to 0.15 ME results in a similar disk-composition (dfT = $33%), but also a lower Moon mass (0.53 ML) and a smallerangular momentum of 1.06 LE–M. As collision geometry predominantly determinesthe fraction of target material in the proto-lunar disk, altering the size of the impac-tor by density changes should also change the target material fraction in the disk. Adenser impactor with the same mass delivers the same momentum, while reducing‘‘spill-over’’ of impactor material into the disk, as it can be seen in Fig. 1b. To includesuch a high density, iron-rich impactor in this study is also motivated by the workof Asphaug (2010), where the population of second-largest bodies in a planet-forming disk becomes slowly enriched in iron through composition-changing hit-and-run collisions. We investigated impactors with 50 wt% and 70 wt% iron corefractions, respectively. With a 0.2 ME impactor at 30! impact angle and 1.30 vesc

impact velocity, the target material fraction fT increases from 57% (dfT = $34%) inthe ‘‘chondritic’’ run (cC01), to 64% (dfT = $28%) in the 50 wt% iron core run(fA01), and up to 75% (dfT = $19%) in the 70 wt% iron core run (fB06). But at thesame time, the iron content of the disk increases to values incompatible with lunardata, and also the bound angular momentum increases to unrealistic values. Appar-ently, reducing the ‘‘spill-over’’ also reduces the lost mass and therefore the lostangular momentum. We also looked into less dense, but still fully differentiatedimpactors with a composition of 50 wt% ice, 35 wt% silicate and 15 wt% iron, typicalfor small bodies accreted in regions of the Solar System beyond the snow-line. In

Fig. 1a. Five snapshots from the 30! impact angle and 1.30 vesc impact velocity case(cC06) showing cuts through the impact plane. Color coded is the type and origin ofthe material. Dark and light blue indicate target and impactor iron; Red and orangeshow corresponding silicate material. The far right shows the situation at the timeof impact. At 0.52 h, it can be seen how the impactor ploughs deep through thetargets mantle and pushes considerable amount of target material into orbit. Aspiral arm of material forms and gravitationally collapses into fragments. The outerportions of the arm mainly consist of impactor silicates and escapes due to havingretained a velocity well above escape velocity. The silicate fragments further inwardare stronger decelerated and enter eccentric orbits around the target. Theimpactor’s iron core also looses much of its angular momentum to the outer partsof the spiral arm and re-impacts the proto-Earth. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

A. Reufer et al. / Icarus 221 (2012) 296–299 297“Hit-and-Run” Collision

[Reufer et al., Icarus, 2012]

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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“Fission-like” Collision

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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into a single moon at an orbital distance of about3.8R⊕, where R⊕ is Earth’s radius (19, 20),

MM

MD≈ 1:9

LDMD

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2:9GM⊕R⊕

p" #

−

1:1 − 1:9Mesc

MD

" #ð1Þ

where Mesc is the mass that escapes from thedisk as the Moon accretes. To estimate MM, weused Eq. 1 and made the favorable assumptionthat Mesc = 0.

We tracked the origin (impactor versus tar-get) of the particles in the final planet and thedisk. To quantify the compositional difference be-tween the silicate portions of the disk and planet,we define a deviation percentage

dfT ≡ [FD,tar/FP,tar − 1] × 100 (2)

where FD,tar and FP,tar are the mass fractions ofthe silicate portions of the disk and of the planetderived from the target’s mantle, respectively (21).Identical disk-planet compositions have dfT = 0,whereas a disk that contains fractionally moreimpactor-derived silicate than the final planet hasdfT < 0, and a disk that contains fractionally lessimpactor-derived silicate than the final planet hasdfT > 0.

Prior impact simulations (1–3, 14, 15) thatconsider g ≡Mimp/MT ≈ 0.1 to 0.2 produce diskswith −90% ≤ dfT ≤ −35% for cases with MM >ML, where ML is the Moon’s mass. Results withlarger impactors having g = 0.3, 0.4, and 0.45are shown in Figs. 1 and 2 and Table 1. As therelative size of the impactor (g) is increased, thereis generally a closer compositional match be-tween the final disk and the planet. For g ≥ 0.4,some disks have both sufficient mass and an-gular momentum to yield the Moon and nearlyidentical silicate compositions to that of the final

Fig. 1. An SPH simulationof a moderately oblique,low-velocity (v∞ = 4 kms–1) collision between animpactor and target withsimilar masses (Table 1,run 31). Color scales withparticle temperature inkelvin, per color bar, withred indicating tempera-tures >6440 K. All particlesin the three-dimensionalsimulation are overplotted.Time is shown in hours,and distances are shownin units of 103 km. Afterthe initial impact, the plan-ets recollided, merged,and spun rapidly. Theiriron cores migrated to thecenter, while the mergedstructure developed a bar-type mode and spiral arms(24). The arms wrappedup and finally dispersedto form a disk containing~3 lunar masses, whosesilicate composition dif-fered from that of thefinal planet by less than1%. Because of the nearsymmetry of the colli-sion, impactor and targetmaterial are distributedapproximately proportion-ately throughout the finaldisk, so that the disk’s dfTvalue does not vary ap-preciably with distancefrom the planet.

Fig. 2. Compositional differ-ence between the disk and finalplanet (dfT) (Eq. 2) produced bysimulations with (A) g = 0.3and (B) g = 0.4 (triangles) and0.45 (squares) versus the pre-dicted mass of the moon thatwould accrete from each disk(MM) (Eq. 1) scaled to the finalplanet’s mass (MP). There is achange in y axis scales betweenthe two plots. Gray, purple, darkblue, light blue, green, yellow,orange, and red points corre-spond to vimp/vesc = 1.0, 1.1,1.2, 1.3, 1.4, 1.6, 1.8, and 2.0,respectively. The open squareis run 60* from Table 1, whichincludes pre-impact rotation.Forming an appropriate-massMoon mass requires MM/MP > 0.012, the region to the right of the vertical solid line. Constraints on dfT needed to satisfy Earth-Moon compositional similaritiesare shown by horizontal lines for oxygen (solid), titanium (dotted), and chromium (dot-dashed), assuming a Mars-composition impactor.


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standard

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[Simulations by Miki Nakajima]

混迷を極める現代






tot Þdisk ð1Þ



dfT ¼ Msilctarg

! .Msilc

tot

"

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.Msilc

targ

! .Msilc

tot

"





4. Results



5. Discussion






A. Reufer et al. / Icarus 221 (2012) 296–299 297

“Hit-and-Run” Collision?

[Reufer et al., Icarus, 2012]

ever, Canup (2012) suggests that this issue can be resolved if Theiahad a mass comparable to that of the proto-Earth. In this case, boththe Earth and Moon-forming disk are a roughly even mixture of theproto-Earth and Theia. (This scenario relies on the angular momen-tum of the Earth–Moon system later decreasing via an evectionresonance with the Sun (Cuk and Stewart, 2012).)

With our large number of terrestrial planet formation simula-tions, we can estimate the statistical likelihood that Theia’s masswas comparable to the proto-Earth. To do this, we simply look atthe distribution of mass ratios for Earth analogs struck by Theiaanalogs in our simulations. This distribution is shown in Fig. 17,where we plot the parameter c, which is the ratio of Theia’s massto the combined mass of Theia and the proto-Earth at the time ofimpact. To mix the Earth and Moon evenly enough, Canup (2012)finds that Theia must have had cJ 0:4. In Fig. 17, we see that suchcollisions are not found in any of our simulations. Out of the 104Earth analogs generated in our collisions, the largest recorded cis 0.325, and only 8.7% of our Earth analogs experienced impactswith c > 0:3. Late impacts with cJ 0:4 must be exceedingly rare,implying that a comparably massed Theia and proto-Earth is a veryunlikely event. This result agrees with Jacobson and Morbidelli(2014), who also found that major mergers between protoplanetswith similar masses are rare.

Instead of comparable masses for Theia and the proto-Earth,both Reufer et al. (2012) and Cuk and Stewart (2012) invoke ahigher impact velocity between Theia and the proto-Earth to pro-duce an Earth-like Moon. Unfortunately, our CJS and EJS simula-tions did not record the collisional velocities between embryos,so the impact velocity distribution in these simulation sets is notknown. However, we do have the collisional velocity data fromthe ANN simulations, which had 35 impacts between Earth andTheia analogs. The cumulative distribution of impact velocitiesbetween Theia analogs and Earth analogs in ANN simulations isshown in Fig. 18. In this plot we see that the median impact veloc-ity is just a few percent greater than the mutual escape velocity ofthe proto-Earth and Theia analogs with m > 0:1 M!. Moreover, thelargest impact velocity seen in our simulations is 126% of themutual escape velocity. For comparison, most of the successful col-lisions in Cuk and Stewart (2012) require an impact velocity atleast 150% of the mutual escape speed. Thus, our ANN simulationssuggest that such high velocity impacts are rare for Theia analogswith m > 0:1 M!.

Cuk and Stewart (2012) also explore impact scenarios involvingTheia masses of 0:05 M! and 0:025 M! and find that such collisionscan successfully produce an Earth-heavy composition for the Moon

if the proto-Earth was spinning very rapidly before impact. Becauseof this finding, we also look at our collision statistics for last majormergers on Earth analogs that involve impacting bodies withmasses below 0:1 M!. These are also shown in Fig. 18. We see thatsmaller impactors do collide with the Earth at higher velocities, butthe large majority are still below 150% of escape velocity, the min-imum velocity preferred in Cuk and Stewart (2012). Of the Theiaanalogs with 0:025 M! < m < 0:05 M!, 17% strike the proto-Earthwith a velocity that is more than 150% of the mutual escape veloc-ity. For Theia analogs with 0:05 M! < m < 0:1 M!, the percentageof high-velocity collisions falls to 7.9%. Because the CJS and EJSsimulations involve collisions of bodies spanning a larger rangeof semimajor axes than the ANN set, we may expect larger frac-tions of energetic collisions in these two simulation sets. Unfortu-nately, these fractions are not known. However, in order for theCuk and Stewart (2012) mechanism to succeed, the Earth’s spinrate also has to be exceptionally high (a period of "2.3 h). Such ahigh spin rate is likely achieved with a very large merger eventbefore the Moon-forming impact, and we have already shown inFig. 17 that massive impactors are rare. Thus, production of ourMoon with a high velocity impact may also be a relatively lowprobability event.

3.5. Comparison to venus analogs

With four different families of initial D17O distributions set bythe accretion histories of Earth and Mars analogs, we can now pre-dict the D17O values that these would yield for Venus by examiningthe accretion histories of Venus analogs in our simulations.Because we need Earth and Mars analogs to define our initialD17O distributions, we only use simulations that form analogs forVenus, Earth, and Mars. (This set of 70 simulations is slightly differ-ent from the last section since we drop the requirement for Theiaanalogs and replace it with Venus analogs.) As in the previous sec-tion we examine the outcomes of four different initial D17O distri-butions for our embryos and planetesimals: a linear distribution, astep-function distribution, a random distribution, and two differ-ent values for planetesimals and embryos.

In Fig. 19, we show the distributions of jD17Oj found for ourVenus analogs if we assume the initial D17O among embryos andplanetesimals varies linearly with their heliocentric distance. Asbefore, we force the linear distributions to yield jD17Oj ’ 0:32‰for the Mars analog in each simulation. When this is done, we findthat the Venus analogs are unlikely to have similar oxygen isotopecompositions to the Earth. In fact for each of our three simulation

Fig. 17. The distribution of c values seen among Earth analogs in our simulations (cbeing the ratio of Theia’s mass to the combined mass of Theia and the proto-Earth atthe time of impact). Results from the CJS, EJS, and ANN simulations are shown withthe black solid, dashed, and dotted CDFs, respectively. The combined CDF for allEarth analogs is marked with the thicker gray line.

Fig. 18. The cumulative distribution of impact velocities between Earth and Theiaanalogs in the ANN simulations. Theia analogs are split into three different massbins: m = 0.025–0.05 M! (solid line), m = 0.05–0.1 M! (dashed line), andm > 0:1 M! (dotted line). Impact velocity is calculated in terms of the mutualescape velocity of the Earth and Theia analogs.

N.A. Kaib, N.B. Cowan / Icarus 252 (2015) 161–174 171

[Kaib & Cowan, Icarus, 2012]

N 体計算で Giant Impacts の過程を追ったところ衝突速度が必要な大きさに達しないことが判明

L24 KOKUBO & GENDA Vol. 714

Figure 2. Average semimajor axes and masses of the largest (filled symbols) andsecond-largest (open symbols) planets for realistic (circle) and perfect (square)accretion models. The error bars indicate 1σ .(A color version of this figure is available in the online journal.)

while the second-largest planets with M ≃ 0.7 M⊕ are widelyscattered in the initial protoplanet region. We also find nodifference in their eccentricities and inclinations, and those are≃ 0.1. These eccentricities and inclinations are an order ofmagnitude larger than the proper eccentricities and inclinationsof the present terrestrial planets. Some damping mechanismsuch as gravitational drag (dynamical friction) from a dissipatinggas disk (Kominami & Ida 2002) or a residual planetesimal disk(Agnor et al. 1999) is necessary after the giant impact stage.

3.3. Statistics of Spin

In 50 runs of the realistic and perfect accretion models,we have 128 and 124 planets that experience at least oneaccretionary collision, respectively. The average values of each

component of the spin angular velocity ω of the planets and itsdispersion σ are summarized in Table 2 together with the rmsspin angular velocity ⟨ω2⟩1/2 and the spin anisotropy parameterβ = ⟨ω2

z⟩/⟨ω2⟩.The spin angular velocity averaged in mass bins against

mass is shown in Figure 3 (left). We show clearly that theaverage angular velocity is almost independent of mass forboth accretion models. For the perfect accretion model, theaverage values are as high as the critical angular velocity ωcr.These are natural outcomes for the giant impact stage under theassumption of perfect accretion (Kokubo & Ida 2007). The rmsspin angular velocity for the realistic accretion model is about30% smaller than that for the perfect accretion model. Thisis because in the realistic accretion model, grazing and high-velocity collisions that have high angular momentum result in ahit-and-run, while nearly head-on or low-velocity collisions thathave small angular momentum lead to accretion. In other words,small angular momentum collisions are selective in accretion.Thus, the accretion inefficiency leads to slower planetary spin,compared with the perfect accretion model. Indeed, the rmsspin angular velocity is almost consistent with the estimationby Equation (4) in Section 3.1. The rms spin angular velocityslightly larger than the estimation based on a single dominantaccretionary collision is due to the contribution of other non-dominant accretionary collisions.

We confirm that each component of ω follows a Gaussiandistribution. We perform a Kolmogorov–Smirnov (K–S) test toconfirm the agreement with a Gaussian distribution. For thedistribution of each component, we obtain sufficiently highvalues of the K–S probability QK–S > 0.5 in both accretionmodels.

In Figure 3 (right), we show the obliquity distribution withan isotropic distribution, ndε = (1/2) sin εdε. We find thatthe obliquity ranges from 0◦ to 180◦ and follows an isotropicdistribution which is consistent with Agnor et al. (1999)and Kokubo & Ida (2007). By a K–S test, we obtain highK–S probabilities of 0.3 and 0.9 for the realistic and perfectaccretion models, respectively. This is also confirmed by thespin anisotropy parameter β ≃ 1/3 in Table 2. The isotropic

Figure 3. Left: average spin angular velocity of all planets formed in the 50 runs of the realistic (circle) and perfect (square) accretion models is plotted against theirmass M with mass bin of 0.1 M⊕. The error bars indicate 1σ and the dotted line shows ωcr. Right: normalized cumulative distributions of ε for the realistic (solidcurve) and perfect (dashed curve) accretion models with an isotropic distribution (dotted curve).(A color version of this figure is available in the online journal.)

[Kokubo & Genda, ApJ, 2010]

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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衝突破壊の効果も考慮すると原始地球を高速回転できない[Ćuk & Stewart, Science, 2012]


MM

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signal, with the MORB mantle more similar to the meteoritic Ne-B20

composition and the deep mantle similar to a solar Ne composition20,21.Taken together, the differences in 20Ne/22Ne, He/Ne and Ne/Ar

ratios between MORBs and the Iceland plume require that heteroge-neities from the early Earth still exist in the present day mantle, and thenew Xe measurements provide conclusive evidence for this interpreta-tion. Previous studies have observed lower 129Xe/130Xe ratios in OIBscompared to MORBs3,17,18. The lower measured 129Xe/130Xe ratios inOIBs could reflect syn- to post-eruptive atmospheric contamination ofthe lavas, mixing between subducted atmospheric Xe and MORB-type Xe3,4,18, or different I/Xe ratios for the plume and MORB sources1.Ar–Xe mixing systematics (Fig. 3) constrain the Iceland mantle source129Xe/130Xe ratio to be 6.98 6 0.07, significantly lower than the MORBsource ratio of 7.9 6 0.14 (ref. 3). Therefore, syn- to post-eruptive

20Ne/22Ne

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Icelandmantlesource

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Figure 1 | Differences in neon and argon isotopic composition betweenMORB and the Iceland plume. a, Neon three-isotope plot showing the newanalyses of the DICE 10 sample (filled circles) from Iceland in comparison topreviously published data for this sample (open circles; ref. 18) and the gas-rich‘popping rock’ (2PD43) from the north Mid-Atlantic Ridge (open triangles; ref.17). Error bars are 1s, and for clarity, two previous analyses18 with large error barshave not been shown. Step-crushing of a mantle-derived basalt produces a lineartrend that reflects variable amounts of post-eruptive air contamination in vesiclescontaining mantle Ne. The slope of the line is a function of the ratio of nucleogenic21Ne to primordial 22Ne, with steeper slopes indicating a higher proportion ofprimordial 22Ne and, thus, a less degassed mantle source. The slope of the Icelandline based on the new analyses is consistent with that obtained previously18.Importantly, 20Ne/22Ne ratios of 12.88 6 0.06 are distinctly higher than theMORB source 20Ne/22Ne of #12.5 as constrained from continental well gases20.b, Ne–Ar compositions of individual step crushes of the DICE 10 sample. 40Ar isgenerated by radioactive decay of 40K, and low 40Ar/36Ar ratios are indicative of aless degassed mantle. The data reflect mixing between a mantle component andpost-eruptive atmospheric contamination. A least-squares hyperbolic fit throughthe data yields a 40Ar/36Ar ratio of 10,745 6 3,080, corresponding to a mantlesolar 20Ne/22Ne ratio of 13.8. This Ar isotopic ratio is used as the mantle sourcevalue for Iceland in Figs 2 and 3. Symbols as in a; error bars are 1s.

Kinetic fractionation

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Iceland; this studyMORB (2ΠD43)

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Figure 2 | Differences in elemental abundances and isotope ratios betweenMORB and the Iceland plume. Error bars are 1s. a, 3He/22Ne versus 20Ne/22Ne;b, 3He/36Ar versus 40Ar/36Ar; and c, 22Ne/36Ar versus 40Ar/36Ar. The mantlesource composition for 2PD43 (filled grey square in all panels) is based on the40Ar/36Ar and 20Ne/22Ne ratios as defined in ref. 30, and the mantle sourcecomposition for Iceland (filled black square in all panels) is based on Fig. 1. Thegrey and black arrows at the top of the figure indicate how elemental ratios evolveas a result of kinetic fractionation and solubility controlled degassing, respectively.Good linear relationships are observed between isotope ratios and elementalratios, which reflect mixing between mantle-derived noble gases and post-eruptive atmospheric contamination. Lines are robust linear regressions throughthe data with the atmospheric contaminant near the origin and the mantle sourceat the other end. Arrow in c indicates the minimum 22Ne/36Ar ratio of the Icelandmantle source given the measured 40Ar/36Ar ratio of 7,047 (Supplementary Table6). Because of systematic differences in noble gas solubilities and diffusivities, thedifferences in elemental abundances are not likely to be generated throughancient fractionation associated with diffusion or magmatic outgassing. Forexample, kinetic fractionation should lead to higher 3He/22Ne and higher3He/36Ar–22Ne/36Ar ratios. However, the Iceland source has a lower 3He/22Neand higher 3He/36Ar–22Ne/36Ar. Likewise, adding recycled atmospheric gases tothe MORB source cannot produce the plume noble gas compositions. Finally,c shows that preferential recirculation of atmospheric Ar into the plume sourcedoes not explain the higher 22Ne/36Ar of the plume source and because of thedifference in MORB and OIB 22Ne/36Ar ratios, adding radiogenic 40Ar to theplume composition is not likely to generate the 40Ar/36Ar ratio in MORBs.

RESEARCH LETTER

1 0 2 | N A T U R E | V O L 4 8 6 | 7 J U N E 2 0 1 2

Macmillan Publishers Limited. All rights reserved©2012

signal, with the MORB mantle more similar to the meteoritic Ne-B20

composition and the deep mantle similar to a solar Ne composition20,21.Taken together, the differences in 20Ne/22Ne, He/Ne and Ne/Ar

ratios between MORBs and the Iceland plume require that heteroge-neities from the early Earth still exist in the present day mantle, and thenew Xe measurements provide conclusive evidence for this interpreta-tion. Previous studies have observed lower 129Xe/130Xe ratios in OIBscompared to MORBs3,17,18. The lower measured 129Xe/130Xe ratios inOIBs could reflect syn- to post-eruptive atmospheric contamination ofthe lavas, mixing between subducted atmospheric Xe and MORB-type Xe3,4,18, or different I/Xe ratios for the plume and MORB sources1.Ar–Xe mixing systematics (Fig. 3) constrain the Iceland mantle source129Xe/130Xe ratio to be 6.98 6 0.07, significantly lower than the MORBsource ratio of 7.9 6 0.14 (ref. 3). Therefore, syn- to post-eruptive

20Ne/22Ne

10 11 12 13

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MORB (2ΠD43)Iceland; ref. 18

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b

0.040.03 0.05 0.06

Figure 1 | Differences in neon and argon isotopic composition betweenMORB and the Iceland plume. a, Neon three-isotope plot showing the newanalyses of the DICE 10 sample (filled circles) from Iceland in comparison topreviously published data for this sample (open circles; ref. 18) and the gas-rich‘popping rock’ (2PD43) from the north Mid-Atlantic Ridge (open triangles; ref.17). Error bars are 1s, and for clarity, two previous analyses18 with large error barshave not been shown. Step-crushing of a mantle-derived basalt produces a lineartrend that reflects variable amounts of post-eruptive air contamination in vesiclescontaining mantle Ne. The slope of the line is a function of the ratio of nucleogenic21Ne to primordial 22Ne, with steeper slopes indicating a higher proportion ofprimordial 22Ne and, thus, a less degassed mantle source. The slope of the Icelandline based on the new analyses is consistent with that obtained previously18.Importantly, 20Ne/22Ne ratios of 12.88 6 0.06 are distinctly higher than theMORB source 20Ne/22Ne of #12.5 as constrained from continental well gases20.b, Ne–Ar compositions of individual step crushes of the DICE 10 sample. 40Ar isgenerated by radioactive decay of 40K, and low 40Ar/36Ar ratios are indicative of aless degassed mantle. The data reflect mixing between a mantle component andpost-eruptive atmospheric contamination. A least-squares hyperbolic fit throughthe data yields a 40Ar/36Ar ratio of 10,745 6 3,080, corresponding to a mantlesolar 20Ne/22Ne ratio of 13.8. This Ar isotopic ratio is used as the mantle sourcevalue for Iceland in Figs 2 and 3. Symbols as in a; error bars are 1s.

Kinetic fractionation

10

13

Iceland; this studyMORB (2ΠD43)

a

Air

20N

e/22

Ne

3He/22Ne

12

11

3He/36Ar

40A

r/36

Ar

5,000

10,000

15,000

20,000

25,000

30,000b

Air

Iceland mantlesource

MORB(2ΠD43)mantlesource

0.0 0.2 0.4 0.6 0.8

22Ne/36Ar

40A

r/36

Ar

5,000

10,000

15,000

20,000

25,000

30,000Air

Sea water

c

0.0 0.1 0.2 0.3 0.4

Degassing

0 1 2 3 4 5 6

Figure 2 | Differences in elemental abundances and isotope ratios betweenMORB and the Iceland plume. Error bars are 1s. a, 3He/22Ne versus 20Ne/22Ne;b, 3He/36Ar versus 40Ar/36Ar; and c, 22Ne/36Ar versus 40Ar/36Ar. The mantlesource composition for 2PD43 (filled grey square in all panels) is based on the40Ar/36Ar and 20Ne/22Ne ratios as defined in ref. 30, and the mantle sourcecomposition for Iceland (filled black square in all panels) is based on Fig. 1. Thegrey and black arrows at the top of the figure indicate how elemental ratios evolveas a result of kinetic fractionation and solubility controlled degassing, respectively.Good linear relationships are observed between isotope ratios and elementalratios, which reflect mixing between mantle-derived noble gases and post-eruptive atmospheric contamination. Lines are robust linear regressions throughthe data with the atmospheric contaminant near the origin and the mantle sourceat the other end. Arrow in c indicates the minimum 22Ne/36Ar ratio of the Icelandmantle source given the measured 40Ar/36Ar ratio of 7,047 (Supplementary Table6). Because of systematic differences in noble gas solubilities and diffusivities, thedifferences in elemental abundances are not likely to be generated throughancient fractionation associated with diffusion or magmatic outgassing. Forexample, kinetic fractionation should lead to higher 3He/22Ne and higher3He/36Ar–22Ne/36Ar ratios. However, the Iceland source has a lower 3He/22Neand higher 3He/36Ar–22Ne/36Ar. Likewise, adding recycled atmospheric gases tothe MORB source cannot produce the plume noble gas compositions. Finally,c shows that preferential recirculation of atmospheric Ar into the plume sourcedoes not explain the higher 22Ne/36Ar of the plume source and because of thedifference in MORB and OIB 22Ne/36Ar ratios, adding radiogenic 40Ar to theplume composition is not likely to generate the 40Ar/36Ar ratio in MORBs.

RESEARCH LETTER

1 0 2 | N A T U R E | V O L 4 8 6 | 7 J U N E 2 0 1 2


contamination processes are ruled out as the reason for the lower129Xe/130Xe ratios at Iceland.

The data in Fig. 3a demonstrate that the Iceland and MORB sourcemantles evolved with different I/Xe ratios, requiring the two mantlesources to have separated by 4.45 Gyr ago with limited subsequent mix-ing between the two. As atmosphere is located near the origin in this plot(Fig. 3a), and mixing in this space is linear, adding subducted atmo-spheric Xe to the MORB source clearly cannot produce the Icelandmantle source composition. Similarly, the Iceland source cannot supplyXe to the MORB source unless it is augmented by radiogenic 129Xeproduced from decay of 129I. However, 129I became extinct at,4.45 Gyr ago. Therefore, the two sources must have been separatedbefore 4.45 Gyr ago and subsequently the sources were not homogenized,as otherwise the differences in 129Xe/130Xe would not have been pre-served in the present day mantle. Plumes, therefore, cannot have suppliedXe and all of the primordial volatiles to the MORB source (Figs 1–3),contradicting predictions of the steady-state mantle models8,9.

The new Xe isotopic measurements also indicate a difference in129Xe/136Xe ratios between MORBs and the Iceland plume that cannotbe related solely to subduction of atmospheric Xe, or to addition of136Xe to MORB Xe (Fig. 4). 136Xe, along with 131Xe, 132Xe and 134Xe, isproduced by fission from extinct 244Pu (half-life 80 Myr) and extant238U. However, Pu and U produce the four fission Xe isotopes indifferent proportions, and so measurements of the fissiogenic isotopescan be used to deconvolve Pu- from U-derived Xe (ref. 23). A reservoirthat has remained closed to volatile loss over Earth history should have,97% of the fission Xe isotopes produced from 244Pu. Progressivedegassing of a reservoir, particularly after 244Pu becomes extinct, leadsto increasing proportions of U-derived fission Xe in the reservoir.

Compared to the MORB source, the Iceland plume source has asubstantially higher proportion of Pu- to U-derived fission Xe. Thus,the Iceland plume source is less degassed than the MORB source.Depending on whether the initial Xe isotopic composition of the mantlewas solar or chondritic24, the MORB source has a few per cent to43 6 16% of fission 136Xe derived from 244Pu. The corresponding valuesfor Iceland are 66 6 19% to 99z1

{3% (Supplementary Table 4); the lattervalue, based on an initial chondritic mantle Xe composition, isidentical to values expected for closed system evolution. Hence,irrespective of the initial Xe isotopic composition of the mantle, theIceland plume sample has a higher proportion of Pu- to U-derived fissionXe compared to MORB samples when both sets of data are processed inthe same manner during the deconvolution (Methods, SupplementaryTable 4). The requirement for a lower degree of degassing for the Iceland

source, based on its higher proportion of Pu- to U-derived fission Xe, is aconclusion that is independent of the absolute concentrations of noblegases and the relative partition coefficients of the noble gases with respectto their radiogenic parents.

The combined I–Pu–Xe system has been used to constrain theclosure time for volatile loss of a mantle reservoir through the129*Xe/136*XePu ratio1,2,6,25, where 129*Xe is the decay product of 129Idecay and 136*XePu is 136Xe produced from 244Pu fission. 129I has ashorter half-life than 244Pu, and as a result higher 129*Xe/136*XePu ratiosare indicative of earlier closure to volatile loss1,2,6,25. Depending on theinitial mantle Xe composition, the 129*Xe/136*XePu ratio varies between2:9z0:1

{0:1 and 5:8z1:1{0:9 for the Iceland mantle and the corresponding

values for MORBs are between 7:9z3:3{2:9 and 64:9z132

{31:2 (Methods;Supplementary Table 4). If the mantle had a homogenous I/Pu ratio,the lower 129*Xe/136*XePu ratio in the plume source would paradox-ically imply that the deep mantle became closed to volatile loss after theshallow mantle. A simpler explanation is that the lower 129*Xe/136*XePuratio reflects a lower I/Pu ratio for the plume source compared to theMORB source. These differences would indicate that the initial phase ofEarth’s accretion was volatile-poor compared to the later stages ofaccretion, a conclusion consistent with a recent Pd–Ag study26.

6.6

6.8

7.0

7.2

7.4

129Xe/ 130Xe

40Ar/36Ar

2,000 4,000 6,000 8,000 10,000

Iceland mantle129Xe/130Xe

Air

b

3He/130Xe

0 200 400 600 800 1,000Air

129 X

e/13

0 Xe

6.6

6.8

7.0

7.2

7.4

7.6

7.8

MORB(2ΠD43)source

Icelandmantlesource

a

Figure 3 | Differences in Xe isotopic composition between MORB and theIceland plume. a, Correlation between 129Xe and 3He in the ‘popping rock’MORB (2PD43)17 and Iceland (DICE 10). Error bars are 1s. Data points areindividual step crushes that reflect different degrees of post-eruptive atmosphericcontamination in the vesicles. Air lies near the origin and the mantlecompositions at the other end of the linear arrays. The straight lines are robustregressions through the data. Because mixing in this space is linear, the lines alsorepresent the trajectories along which the mantle sources will evolve when mixed

with subducted air. The new observations from Iceland demonstrate that theIceland plume 129Xe/130Xe ratio cannot be generated solely through addingrecycled atmospheric Xe to the MORB source, and vice versa. Thus, two mantlereservoirs with distinct I/Xe ratios are required. The mantle 129Xe/130Xe ratio of6.98 6 0.07 for Iceland was derived from a hyperbolic least-squares fit throughthe Ar-Xe data (b) corresponding to a mantle 40Ar/36Ar ratio of 10,745. Note thatgiven the curvature in Ar–Xe space, the 129Xe/130Xe in the Iceland mantle sourceis not particularly sensitive to the exact choice of the mantle 40Ar/36Ar ratio.

130Xe/136Xe

129 X

e/13

6 Xe

2.97

2.98

2.99

3.00MORB; ref. 25Iceland; this study

Air

136Xe from Pu and U fission

129Xe from 129I

Air subduction

Air subduction

0.43 0.44 0.45 0.46

Figure 4 | Difference in the measured 129Xe/136Xe ratio between MORB andthe Iceland plume. Unlike the traditional 129Xe/130Xe–136Xe/130Xe plot, the xand y errors are de-correlated. The arrows illustrate how the MORB and OIBsource compositions evolve as subducted air is added. Error bars are 1s. Thefigure demonstrates a small Xe isotopic difference between the Iceland plumeand MORBs that cannot be related solely through recycling atmospheric Xe orby adding fissiogenic 136Xe to MORB Xe. The data points represent theweighted means of the different step crushes for MORBs (n 5 38) and Iceland(n 5 51; this study). The MORB 129Xe/136Xe ratio was calculated from theweighted means of the 129Xe/130Xe and 136Xe/130Xe ratios25.

LETTER RESEARCH

7 J U N E 2 0 1 2 | V O L 4 8 6 | N A T U R E | 1 0 3


[Mukhopadhyay, Nature, 2012]

地球深部の希ガス同位体不均一地球深部まで melting していない

discussed here, as shown in the Methods. The comparison is then doneas follows. First we compare the feeding zones of the planet and impac-tor, as shown in the examples in Fig. 1 (the cumulative plots for theseand all other cases can be found in the Methods). We calculate theprobability P that the feeding zones of the impactor and the planet aredrawn from the same distribution, using a two-group Kolmogorov–Smirnov test (probabilities shown in the plots and in Table 1). In 3 outof 20 cases the feeding zones contributing to the Moon and those con-tributing to the planet are consistent with being drawn from the same

parent distribution. In other words, the Moon’s feeding zones, if derivedsolely from the impactor, are consistent with the Earth’s in 15% of theimpacts. The consistency further improves if we assume that a fractionof the proto-Earth was mixed into the Moon (as suggested by detailedcollision simulations showing a 10%–40% contribution from the proto-Earth14). For the typical 20% mix of proto-Earth material with theimpactor material forming the Moon (as found in simulations), 35%of cases are consistent with their feeding zones being drawn from thesame parent distribution, and the success rate increases further for a

Table 1 | The modelled planet–impactor systems and observations of Solar System bodiesModel Number MP (M›) MI (M›) MI/MP NP NI tcoll

(Myr)Ccal 3D17O (p.p.m.) Kolmogorov–Smirnov probability

0% 20% 40% 0% 20% 40%

cjs15 1 0.94 0.43 0.46 123 97 50.7 13 6 14 10 6 13 8 6 12 0.0039 0.13 0.67cjs15 2 0.78 0.27 0.35 209 78 80.9 (21.05 6 0.26)

3 102284 6 23 263 6 22 1.3 3 1027 3.1 3 1024 0.079

cjs1 3 1.25 0.42 0.34 219 73 149.7 64 6 13 52 6 11 39.9 6 9.6 5.3 3 10211 8.1 3 1027 0.0092cjs1 4 1.05 0.39 0.37 128 78 186.2 (21.97 6 0.20)

3 102(21.57 6 0.20)3 102

(21.18 6 0.20)3 102

1.1 3 10229 5.1 3 10218 0.052

cjs1 5* 1.21 0.38 0.31 219 75 123.5 224 6 17 220 6 15 215 6 13 0.0023 0.056 0.19cjsecc 6 0.94 0.36 0.38 117 79 80.4 251 6 34 240 6 30 231 6 27 3.4 3 1024 0.025 0.33cjsecc 7 1.01 0.32 0.32 144 68 75.5 212 6 16 210 6 13 27 6 12 0.038 0.050 0.20cjsecc 8 1.02 0.42 0.41 148 89 36.9 13 6 79 11 6 71 8 6 66 0.054 0.21 0.53eejs15 9 0.70 0.19 0.27 111 52 77.1 9.1 6 7.4 7.3 6 6.2 5.5 6 5.2 0.071 0.18 0.32eejs15 10 0.55 0.13 0.24 263 65 24.6 98 6 29 78 6 24 59 6 22 1.6 3 10212 2.3 3 1028 0.0025eejs15 11 0.78 0.22 0.29 256 69 102.3 (21.08 6 0.35)

3 102287 6 31 265 6 27 2.9 3 10213 7.9 3 1027 0.0069

eejs15 12 0.73 0.26 0.36 298 87 105.6 26 6 18 21 6 16 16 6 14 1.3 3 10210 5.5 3 1026 0.034eejs15 13* 1.30 0.33 0.25 525 126 199.8 93 6 1.5 3 102 74 6 1.3

3 10255 6 1.1 3 102 7.1 3 1028 5.4 3 1026 0.0043

eejs15 14 0.50 0.18 0.36 170 53 33.1 226 6 75 221 6 65 216 6 56 0.14 0.13 0.24eejs15 15 0.50 0.13 0.26 234 61 32.0 273 6 56 258 6 48 244 6 43 2.7 3 1028 1.8 3 1027 0.062eejs15 16 0.67 0.33 0.49 213 120 145.0 (1.37 6 0.51)

3 102(1.10 6 0.45)3 102

83 6 42 2.4 3 1025 0.013 0.43

eejs15 17* 1.15 0.41 0.36 177 69 168.3 (7.9 6 1.3)3 102

(6.3 6 1.1)3 102

(4.75 6 0.93)3 102

3.5 3 10219 1.2 3 10210 0.0028

ejs15 18 0.81 0.36 0.44 63 55 76.3 281 6 48 265 6 42 249 6 37 1.5 3 1025 0.0039 0.29jsres 19 1.04 0.32 0.31 166 89 79.5 55 6 22 44 6 19 33 6 17 3.0 3 10210 1.5 3 1026 0.0078jsres 20 1.27 0.63 0.50 134 89 176.8 284 6 29 267 6 25 250 6 23 5.6 3 1027 0.011 0.50Measured D17O (p.p.m.)Earth 1 0 6 3Moon 0.012 12 6 3Mars 0.07 321 6 134 Vesta 4.33 3 1025 2250 6 80

MP, NP and MI, NI are the mass and number of particles in the planet and the impactor, respectively; tcoll is the collision time in the simulations; Ccal is the calibration pre-factor (see main text). TheD17O compositiondifference and the Kolmogorov–Smirnov probability (of the distribution of the feeding zones of the planet and impactor being sampled from the same parent distribution) are shown for both the case of nocontribution of planetary material to the newly formed Moon, and the cases of 20% and 40% contribution of material from the planet.*Three-planet systems; calibration was done on the second and third planets.

50

100

50

100

50

100

N

50

100

0.5 1 1.5 2 2.5 3 3.5 40

50

100

a (AU)

aNP = 123, NI = 97

P = 0.0039

10%, P = 0.023

20%, P = 0.13

30%, P = 0.32

40%, P = 0.67

cjs15 number 1

0204060

02040

02040

N0

2040

1 2 3 40

2040

a (AU)

cjs1 number 4

10%, P = 6.7 × 10−27

NP = 128, NI = 78, P = 1.1 × 10−29

20%, P = 5.1 × 10−18

30%, P = 2.7 × 10−10

40%, P = 0.052

b

Figure 1 | The distribution of planetesimals composing the planet and theimpactor. a, A case where the origins of the planetesimals composing theplanet (red) and the impactor (blue) are consistent with being sampled from thesame parent distribution for the expected typical 20% contribution of planetarymaterial in moon-forming impacts (Kolmogorov–Smirnov test probability.0.05). b, A case where the planet and impactor compositions are inconsistent

(P , 0.05), but become consistent once a large (40%) contribution of materialfrom the planet is considered. The lower plots in each panel show the resultswhen different contributions from the planet are assumed (four cases areshown 10%; 20%; 30% and 40%). The cumulative distribution for these cases aswell as all other planet–impactor pairs in Table 1 can be found in the Methods.

9 A P R I L 2 0 1 5 | V O L 5 2 0 | N A T U R E | 2 1 3

LETTER RESEARCH

G2015 Macmillan Publishers Limited. All rights reserved

ProtoEarth ≒ Theia?

[Mastrobuono-Battisti et al., Nature, 2015]

2003). This outward migration occurred because of the back-reac-tion from planetesimal scattering, which causes the orbits of Sat-urn, Uranus and Neptune to expand and the orbit of Jupiter tocontract (Fernandez and Ip, 1984). In addition, the ‘‘Nice model”of giant planet evolution, which explains several observed charac-teristics of the Solar System, proposes that Jupiter and Saturnformed interior to their mutual 2:1 mean motion resonance, per-haps in fact in the 3:2 resonance and migrated apart (Tsiganiset al., 2005; Gomes et al., 2005; Morbidelli et al., 2005, 2007). Thus,Jupiter and Saturn may very well have been in a more compactconfiguration at early times.

We tested a range of configurations for Jupiter and Saturn,although we did not perform an exhaustive search given the largecomputational expense of each simulation. However, to accountfor stochastic variations in outcome we performed four simula-tions for each giant planet configuration. The configurations wetested were:

! CJS (‘‘Circular Jupiter and Saturn”). These are the initial condi-tions for the Nice model, as in Tsiganis et al. (2005) and alsoused in O’Brien et al. (2006). Jupiter and Saturn were placedon circular orbits with semimajor axes of 5.45 and 8.18 AUand a mutual inclination of 0.5!. We note that even though Jupi-ter and Saturn begin with zero eccentricities, they induce small,non-zero eccentricities in each others’ orbits.

! CJSECC (‘‘CJS with ECCentric orbits”). Jupiter and Saturn wereplaced at their CJS semimajor axes of 5.45 and 8.18 AU witheJ ¼ 0:02 and eS ¼ 0:03 and a mutual inclination of 0.5!.

! EJS (‘‘Eccentric Jupiter and Saturn”). Jupiter and Saturn wereplaced on approximately their current orbits: aJ ¼5:25 AU; eJ ¼ 0:05; aS ¼ 9:54 AU, and eS ¼ 0:06, with a mutualinclination of 1.5!.

! EEJS (‘‘Extra Eccentric Jupiter and Saturn”). Jupiter and Saturnwere placed at their current semimajor axes but with higherorbital eccentricities: aJ ¼ 5:25 AU; aS ¼ 9:54 AU, andeJ ¼ eS ¼ 0:1, with a mutual inclination of 1.5!. These casesproved to be interesting, so we ran eight cases in addition tothe original four. The next four cases (referred to as EEJS 5-8)had the same configuration of Jupiter and Saturn but 2000planetesimals rather than 1000. The final four cases (EEJS 9-12) also had 2000 planetesimals but had eJ ¼ 0:07 and eS ¼ 0:08.

! JSRES (‘‘Jupiter and Saturn in RESonance”). Jupiter and Saturnwere placed in their mutual 3:2 mean motion resonance, follow-ing directly from simulations of their evolution in the gaseousSolar Nebula (Morbidelli et al., 2007): aJ ¼ 5:43 AU; aS ¼7:30 AU; eJ ¼ 0:005, and eS ¼ 0:01, with a mutual inclinationof 0.2!.

! JSRESECC (‘‘Jupiter and Saturn in RESonance on ECCentricorbits”). As for JSRES but with eJ ¼ eS ¼ 0:03.

The EJS and EEJS simulations assume that Jupiter and Saturndid not undergo any migration. The EEJS simulations are moreself-consistent than the EJS simulations, because scattering ofremnant planetesimals and embryos tends to decrease the eccen-tricities and semimajor axes of Jupiter and Saturn (e.g., Chambers,2001). Thus, to end up on their current orbits, Jupiter and Saturnwould have had to form on more eccentric and slightly more dis-tant orbits. The CJS, JSRES and JSRESECC simulations all followfrom the Nice model and assume that Jupiter and Saturn’s orbitschanged significantly after their formation, with Saturn migratingoutward and Jupiter inward (Tsiganis et al., 2005). If migration ofthe giant planets is really associated with the late heavy bom-bardment (Gomes et al., 2005; Strom et al., 2005), then at leastmost of the migration of Jupiter and Saturn must have occurredlate, well after the completion of the terrestrial planet formationprocess.

3.2. Properties of the protoplanetary disk

For all of our simulations, the disk of solids extended from 0.5 to4.5 AU and contained populations of planetary embryos and plane-tesimals. For most cases, we assumed that the disk’s surface den-sity in solids R followed a simple radial power-law distribution:

RðrÞ ¼ R1r

1AU

! "%x: ð3Þ

For the minimum-mass Solar Nebula (MMSN) model,R1 & 6—7 g cm%2 and x ¼ 3=2 (Weidenschilling, 1977b; Hayashi,1981). For most of our simulations we assumed x ¼ 3=2 but we alsoperformed some cases with x ¼ 1 for the CJS and EJS giant planetconfiguration. Cases with x ¼ 1 are labeled by the x value; for exam-ple, the EJS15 simulations have x ¼ 3=2 and the EJS1 simulationshave x ¼ 1 (see Table 2). For each case, we calibrated our disks tocontain a total of 5 M' in solids between 0.5 and 4.5 AU, dividedequally between the planetesimal and embryo components.

Fig. 2 shows a sample set of initial conditions. We assumed thatembryos are spaced by D ¼ 3—6 mutual Hill radii RH , whereRH ¼ 0:5ðr1 þ r2Þ ½ðM1 þM2Þ=3M*+1=3, where a1 and M1 are the ra-dial distance and mass of embryo 1. The embryo mass thereforescales with orbital distance as M , r3=2 ð2%xÞD3=2 (Kokubo and Ida,2002; Raymond et al., 2005). The disks contained 85–90 embryoswith masses between 0.005 and 0:1 M'. In Mars’ vicinity the typi-cal embryo mass was roughly 1/6–1/3 of a Mars mass. Planetesi-mals were laid out as Np , rxþ1 to follow the annular mass, andhad masses of 0:0025 M'. Embryos and planetesimals were givenrandomly-chosen starting eccentricities of less than 0.02 and incli-nations of less than 0.5!. In a few EEJS cases we performed addi-tional simulations with 2000 planetesimals, which followed thesame distribution but had correspondingly smaller masses.

We assume that there existed a radial compositional gradientfor rocky bodies in the Solar Nebula. This gradient was presumablyimprinted on planetesimals by the local temperature during theirformation (e.g., Boss, 1998), although heating by short-lived radio-nuclides such as 26Al may have played a role (Grimm and McS-ween, 1993). We assume the same water distribution as inRaymond et al. (2004, 2006), using data for primitive meteoritesfrom Abe et al. (2000). The ‘‘water mass fraction”, WMF, i.e. thewater content by mass, varies with radial distance r as

WMF ¼10%5; r < 2AU10%3; 2AU < r < 2:5AU5%; r > 2:5AU

8><

>:ð4Þ

This water distribution is imprinted on planetesimals and em-bryos at the start of each simulation. During accretion the water

Fig. 2. Sample initial conditions for a disk with R , r%3=2 containing 97 planetaryembryos and 1000 planetesimals. Embryos are shown in gray with their sizesproportional to their mass(1/3) (but not to scale on the x axis).

S.N. Raymond et al. / Icarus 203 (2009) 644–662 647

[Raymond et al., Icarus, 2009]

planets is hnM i ’ 2:0 ! 0:6, which means that the typical result-ing system consists of two Earth-sized planets and a smallerplanet. In thismodel, we obtain hnai ’ 1:8 ! 0:7. In other words,one or two planets tend to form outside the initial distribution ofprotoplanets. In most runs, these planets are smaller scatteredplanets. Thus we obtain a high efficiency of h fai ¼ 0:79 ! 0:15.The accretion timescale is hTacci ¼ 1:05 ! 0:58ð Þ ; 108 yr. Theseresults are consistent with Agnor et al. (1999), whose initial con-ditions are the same as the standard model except for !1 ¼ 8.

The left and right panels of Figure 3 show the final planets onthe a-M andM–e, i planes for 20 runs. The largest planets tend to

cluster around a ¼ 0:8 AU, while the second-largest avoid thesame semimajor axis as the largest, shown as the gap around a ¼ha1i. Most of these are more massive thanM%/2. The mass of thelargest planet is hM1i ’ 1:27 ! 0:25M%, and its orbital elementsare ha1i ’ 0:75 ! 0:20 AU, he1i ’ 0:11 ! 0:07, and hi1i ’0:06 ! 0:04. On the other hand, the second-largest planet hashM2i’ 0:66 ! 0:23M%, ha2i ’ 1:12 ! 0:53AU, he2i ’ 0:12 !0:05, and hi2i ’ 0:10 ! 0:08. The dispersion of a2 is large, sincein some runs, the second-largest planet forms inside the largestone, while in others it forms outside the largest. In this model, wefind a1 > a2 in seven runs.

Fig. 2.—Snapshots of the system on the a-e (left) and a-i (right) planes at t ¼ 0, 106, 107, 108, and 2 ; 108 yr for the same run as in Fig. 1. The sizes of the circlesare proportional to the physical sizes of the planets.

Fig. 3.—All planets on the a-M (left) and M–e, i (right) planes formed in the 20 runs of the standard model (model 1). The symbols indicate the planets first(circles), second (hexagons), third (squares), and fourth (triangles) highest in mass. The filled symbols are the final planets, and the open circles are the initialprotoplanets in the left panel. The filled and open symbols mean e and i in the right panel, respectively. [See the electronic edition of the Journal for a color versionof this figure.]

KOKUBO, KOMINAMI, & IDA1134 Vol. 642

[Kokubo et al., ApJ, 2006]

そもそも原始地球と衝突天体は同じ材料で形成初期条件が極めて恣意的（標準シナリオではない）結果は初期条件を反映した自然な帰結にすぎない

meteorite with the same W concentration and with m182W 5 2330,comparable to IVA irons17, would have lowered the W isotopic com-positions of the impact-melt rocks by only ,1 p.p.m., relative to theindigenous lunar signature (Extended Data Table 1).

As further evidence that the W isotopic compositions of the metalseparates are nearly completely derived from the lunar target rocks, wenote that the metals from the two pieces of 68815 are characterized byconsiderably different absolute and relative abundances of HSE, as wellas 187Os/188Os, yet their W isotopic compositions are identical withinuncertainties. The differences in HSE are likely to reflect the incorp-oration of different proportions of HSE from two impactors into thetwo pieces. Incorporation of HSE from more than one impactor iscommon in lunar impact-melt rocks18. If our assumptions about themass balance of W among target rocks and impactors are grosslyincorrect, and significant but variable proportions of the W presentin the metal separates were derived from different impactors, it is verylikely they would also have different W isotopic compositions. This isnot observed. We conclude that modifications to the indigenous lunarW isotopic composition by contamination from basin-forming impac-tors were minor, and that the average m182W value of 120.6 6 5.1(2 s.d.) for the three metal separates provides the current best estimateof the W isotopic composition of their parental mantle KREEPdomain.

The observed isotopic offset between the Moon and the silicateEarth might also reflect in situ decay of 182Hf in a high-Hf/W domain,formed as a consequence of the characteristics of the materials fromwhich the Moon coalesced, fractionation of the two elements duringcore–mantle segregation of the Moon, or crystallization of the lunarmagma ocean (LMO). Although the Hf/W ratio of the bulk lunarmantle probably increased slightly as a result of lunar core formationand extraction of an unknown proportion of the siderophile W into thecore, recent studies have concluded that the silicate portions of theEarth and Moon had nearly identical Hf/W (see, for example, ref. 19).Consequently, if the Moon formed while 182Hf was still extant, and thesilicate portions of the Earth and Moon had identical W isotopiccompositions at the time of formation, the isotopic compositions ofW would not have evolved to the different compositions observed.

In contrast, fractional crystallization of the LMO almost certainlyled to the creation of mantle domains with both higher and lowerHf/W ratios, compared to the bulk lunar mantle. This is due to themore incompatible nature of W in silicate systems, compared with Hf(ref. 20). Crystal–liquid fractionation would, therefore, have led to thecreation of 182W-enriched and 182W-depleted domains in the mantle,compared to the 182W of the bulk lunar mantle, if LMO crystallization

was rapid while 182Hf was extant. The comparatively large amount ofW needed to make sufficiently high-precision measurements, coupledwith sample mass limitations for the Apollo samples, prevented usfrom making isotopic measurements on rocks derived from lunarmantle domains with different Hf/W from the KREEP source. Twoobservations, however, suggest that radiogenic ingrowth inside theMoon was not the cause of the 182W-enriched nature of the metalsexamined here. First, the coupled 146,147Sm–142,143Nd systematics ofcrustal rocks derived from the lunar mantle indicate that late stages ofLMO crystallization occurred more than 100 Myr after Solar Systemformation21, well after 182Hf was extinct. Second, regardless of thetiming of LMO crystallization, the mantle source of KREEP was likelyto have been a low-Hf/W reservoir, given the W-enriched nature ofKREEP. Thus, if the KREEP mantle source rapidly formed during thelifetime of 182Hf, it would have developed a 182W deficit relative to theEarth–Moon system, rather than the observed enrichment (ExtendedData Fig. 1), assuming that the mantles of both bodies were in isotopicequilibrium at the time of the Moon’s formation. We conclude that

µ182W

–10 0 10 20 30 40 50

68815,396

68115,114

Average 68115,114

68815,394

Figure 1 | Values of m182W of lunar metals separated from KREEP-richimpact melts analysed by negative thermal ionization mass spectrometry inthis study. The data for 68115,114, 68815,394, and 68815,396 are shown ascircles, diamond, and square respectively; error bars for our analysis showinternal precision of one single measurement, for which the 2 standarddeviations (s.d.) external reproducibility is ,4.5 ppm, as demonstrated byreplicated standard measurements over the two year period. The white-dottedcircle corresponds to the average of the three replicated analyses of 68115,114metal; error bars show 2 s.d. of these data. The dark grey area and blackdashed line indicates the average m182W 5 120.6 6 5.1 (2 s.d., n 5 3) of thethree metal separates from Apollo 16 impact melt rocks analysed here. The lightgrey dashed line corresponds to the W isotope composition of the modernterrestrial mantle, and the light grey area at m182W 5 0 corresponds to the2 standard errors (s.e.) uncertainty for repeated analyses of the Alfa Aesar Wstandard.

Table 2 | Highly siderophile element contents and Os isotopic compositions of lunar metalsSamples Re

(p.p.b.)Os

(p.p.b.)Ir

(p.p.b.)Ru

(p.p.b.)Pt

(p.p.b.)Pd

(p.p.b.)

187Os/188Os 62smean

68115,113 metal 0.13825 0.00007Replicate 0.13837 0.00004

68815,394 metal 6.750 57.59 61.99 144.5 167.2 163.6 0.13720 0.0001168815,396 metal 128.6 1256 1188 2479 3497 2347 0.13480 0.00006

2smean corresponds to 2 standard errors of an individual 187Os/188Os measurement.

Table 1 | Tungsten isotopic compositions and W and Hf abundancesof lunar metalsSamples W

(p.p.m.)Hf

(p.p.m.)m182W

68115,114 metal 32.7 6 0.3 2.23 6 0.05 125.3 6 4.6121.5 6 2.6123.0 6 1.7

68115,114 metal average (62 s.d.) 123.3 6 3.8

68815,394 metal 22.8 6 0.3 1.41 6 0.02 118.1 6 2.5

68815,396 metal 36.3 6 0.5 0.27 6 0.01 120.4 6 2.9

Bulk lunar mantle (n 5 3, 62 s.d.) 120.6 6 5.1

LETTER RESEARCH


2 3 A P R I L 2 0 1 5 | V O L 5 2 0 | N A T U R E | 5 3 1

[Touboul et al., Nature, 2015]

of 10.27 6 0.04 is significantly higher than the previously obtainedmean value of 0.09 6 0.10 for lunar metal samples (ref. 10), but fornon-irradiated samples (68115, 68815) there is good agreementbetween our data and previous data (Fig. 2). For more strongly irra-diated samples, however, the e182W of the metals tends to be slightlylower10, resulting in an overall decrease of the mean e182W inferredfrom the lunar metals. Therefore, the higher pre-exposure e182W of10.27 6 0.04 determined here reflects not only the better precision ofour measurements, but also that the previous study10 did not fullyquantify neutron capture effects in the metals.

The well-resolved 182W excess of the Moon compared to the pre-sent-day BSE (Fig. 2) places important constraints on the occurrence,mass and timing of the late veneer as well as on the origin of the Moon.Below we first evaluate the magnitude of any e182W difference betweenthe BSE and the Moon induced by the late veneer, and then we assesswhether there is a resolvable 182W anomaly in the Moon resulting fromthe mixing of impactor and proto-Earth material during the giantimpact. The mass and composition of the late veneer is constrainedthrough absolute and relative HSE abundances and ratios of S, Se andTe in Earth’s primitive mantle2,19,20. On this basis, the late veneerprobably had a carbonaceous-chondrite-like composition with aminor fraction of iron-meteorite-like material16, corresponding to

,0.35% of Earth’s mass. This composition can explain several geo-chemical signatures of the Earth’s mantle, including its chondritic Os/Ir, Pt/Ir and Rh/Ir but suprachondritic Ru/Ir and Pd/Ir, as well as its187Os/188Os value2 and Se–Te systematics19. Mass balance considera-

Table 1 | Tungsten and Hf isotope data for KREEP-rich samples analysed by MC-ICPMSSample tCRE 1 (Ma) N e182/184W (6/4)meas.* e182/184W (6/4)corr.{ Wmet{ (%)c e180Hf

(62s) (62s) (695% confidence interval)

Weakly irradiated samples14321, 1827 2 0.2960.10 0.2960.10 0.1 20.0260.0814321, 1856 6 0.2760.05 0.2760.05 0.1 20.0260.0814321 (weighted mean) 23.8 0.2760.04 0.2760.04 0.168115, 295 4 0.2460.06 0.2860.06 1.4 0.0260.1368115, 112 3 0.2760.10 0.3160.10 1.4 0.0260.1368115 (weighted mean) 2.08 0.2560.05 0.2960.05 1.468815, 400 2.04 2 0.1860.10 0.2160.10 1.0 20.0360.1614321, 68115, 68815 (weighted mean) 0.2560.03 0.2760.03

Strongly irradiated samples14163, 921 NA 5 2.3560.04 ND ND 23.7960.0812034, 120 NA 3 1.2660.10 ND ND 21.7960.0914310, 676 259 3 1.8760.10 ND ND 22.9460.0862235, 122 153 2 1.6360.10 ND ND 22.4660.10

All Hf isotope data are from ref. 13, except for samples 14321 and 14163 which were newly analysed (see Methods). N, number of measurements of each sample; NA, not available; ND, not determined.*Measured e182W internally normalized to 186W/184W 5 0.92767, denoted by (6/4).{ e182W corrected for meteoritic contamination on the lunar surface using measured HSE and W abundances (see Methods).{Percentage of W in sample that derives from meteoritic impactor component added at lunar surface.1 Cosmic-ray exposure ages (tCRE) of lunar samples (ref. 14 and references therein).

–4–3–2–100

1

2

ε180Hf

ε182 W

68115

12034

14310

14321

62235

14163KREEP-rich samples

ε182Wpre-exposure

68815

Figure 1 | Plot of e182W versus e180Hf determined for KREEP-rich samples.e182W has been internally normalized to 186W/184W 5 0.92767: elsewhere thisis referred to as e182W (6/4) (see Methods and Table 1). Solid line is a best-fitlinear regression through the data (slope 5 20.549 6 0.019; MSWD 5 0.36)with the intersection at e180Hf 5 0 (arrowed) defining the pre-exposure e182W(5 10.27 6 0.04, 695% confidence interval). Error bars, external uncertainties(95% confidence interval or 2 s.d.; Extended Data Table 2).

–0.2 0 0.2 0.4 0.6

14321, 1827(n = 2)

14321, 1856(n = 6)

68115, 295(n = 4)

68115, 112(n = 3)

68115, 112(n = 4), ref. 10

BCR-2 (n = 22)

AGV-2 (n = 12)

BHVO-2 (n = 3)

68815, 400(n = 2)

68815, 400(n = 4), ref. 10

KREEP-rich samples

Terrestrial rock standards

ε182W

Figure 2 | e182W data of KREEP-rich samples and terrestrial rock standards.Top panel, data from this study (filled symbols) and for metal samples from ref.10 (open symbols). Data points of 68115 and 68815 (this study) were correctedfor a minor contribution from meteoritic contamination at the lunar surface(Table 1). Error bars indicate external uncertainties derived from the 2 s.d.obtained for terrestrial rock standards analysed in this study (if N , 4) or 95%confidence interval of multiple solution replicates of a sample (if N $ 4)(Extended Data Table 1). Bottom panel, data from terrestrial rock standards.Top panel, weighted mean (n 5 5) e182W 5 10.27 6 0.03 (95% confidenceinterval, blue shaded area); bottom panel, mean e182W (n 5 37) 5 0.00 6 0.10(2 s.d.) 5 0.00 6 0.02 (95% confidence interval, green shaded area). e182W hasbeen internally normalized to 186W/184W 5 0.92767, and dashed grey lineshows e182W 5 0.

LETTER RESEARCH


2 3 A P R I L 2 0 1 5 | V O L 5 2 0 | N A T U R E | 5 3 5

[Kruijer et al., Nature, 2015]

わずかな W 同位体の差

G.I. 直後は完全に identical・その後の天体衝突で変化

Earth more than the materials from the deformedimpactor and hence contribute more to the Moon.

The composition of a magma ocean can beestimated from the phase diagram (element parti-tioning) of peridotite and the principles of theisotopic fractionation. I use the experimental dataon Fe/Mg partitioning between olivine and ultra-mafic melt14) to estimate the Mg# of magma oceanfor a various degree of melting. Inferred value of Mg#of the Moon (983 ’ 5) can be explained by a broadrange of degree of partial melting (920–80%) orfractional crystallization. The degree of chemicalsegregation upon melting depends on the processof melting. Melting by a giant impact and partialmelting will have somewhat different consequence onchemical segregation. However, the experimentallyobserved Fe/Mg partitioning between the ultramaficmelts and olivine implies that melt will always bemore FeO rich than co-existing solids under theseconditions. In contrast to the major element compo-sition, isotopic fractionation elements is much lessaffected by partial melting (e.g., ref. 13). However,the chemical composition of the Moon can besomewhat different from the composition of thematerials ejected by the impact because the processesof condensation from the gas-liquid mixture can

modify the composition (e.g., ref. 24). The presentmodel could also explain the presence of a smallFe-rich core,25) if the influence of reduction at hightemperatures is included.

3. Summary and discussions

The current model of planetary formationsuggests that a large degree of heating occurs in thelate stage of planetary formation. The efficientheating in this stage is a result of quick and henceefficient conversion of large gravitational energy toheat due to the collisions of relatively large objects.Consequently, high degree of heating will occurleading to melting (magma ocean formation) in alarge planet (larger than Mars, e.g., ref. 26), but not

proto-Earth proto-Earth

impactorimpactor

magma ocean

vapor je

t

(a) (b)

Fig. 4. Schematic drawing of processes of ejection of materialsupon a giant impact. (a) A case where the proto-Earth does nothave a magma ocean. (b) A case where the proto-Earth has amagma ocean.

R

hA

BC

Fig. 3. A schematic diagram showing possible paths of materialsejected at a certain height. Only a fraction of materials goes tothe orbit (shaded region) from which the Moon was formed. Thefate of ejected materials depends on the ratio h/R and materialswith only for modest value of h/R and velocity will become thesource of the Moon.

0

5000

10000

15000

20000

25000

0 2 4 6 8 10 12

tem

pera

ture

(K

)

collision velocity (km/s)liq

uid

q= -2

-1.5

-1

0

0 1

solid

melting temperature

o= V

Vo

q

Fig. 2. Collision velocity versus temperature. The temperatureincrease by a collision is calculated using the relation [1] inthe text. This temperature corresponds to temperature at thehighest shock pressure. Parameters used are summarized inTable 1. .: the Grüneisen parameter, V: volume, q: a non-dimensional parameter in relation !

!o¼ ð"o" Þ

q (q 9 1 for solids,q < 0 for liquids).

Asymmetric shock heating and the terrestrial magma ocean origin of the MoonNo. 3] 101

[Karato, Proc. Jpn. Acad., 2014]

Giant Impact on Magma Ocean

Magma Ocean 状態の原始地球への Giant Impact

アイデアの提案だけで数値計算等は行われていない

Dynamical models alone do not indicate whether the Moon-formingimpact occurred early (about 30 Myr after condensation) or late (about50–100 Myr after condensation), because the result depends on theinitial disk conditions. However, we find a clear statistical correlationbetween the time of the Moon-forming impact and the mass subse-quently accreted, known as the late-accreted mass. This era of LateAccretion includes no giant impacts by definition and so all of the late-accreted mass comes from the planetesimal population. As shown inFig. 1, this correlation exists across all simulations of both types: clas-sical and Grand Tack. We interpret this correlation by considering thatthe planetesimal population decays over a characteristic time, so that ifthe last giant impact occurs earlier, then the remaining planetesimalpopulation is larger. A larger remaining planetesimal population deli-vers a larger late-accreted mass. Strengthening the correlation, a largerinitial planetesimal population leads to a shorter timescale for giantimpacts owing to enhanced dynamical friction. For any given last-giant-impact time, Earth-like planets in the classical simulationsacquire larger late-accreted masses than those in the Grand Tack simu-lations (see Fig. 1), because the planetesimal population is more dis-persed in the classical scenario and therefore decays more slowly.

The correlation displayed in Fig. 1 can be used as a clock that isindependent of radiometric dating systems. The late-accreted mass isinput into this clock and the time of the last giant impact is read out. Atraditional estimate for the late-accreted mass can be obtained from thehighly siderophile element (HSE) abundances in Earth’s mantle rela-tive to the HSE abundances in chondritic meteorites11,12. HSEs par-tition strongly into iron, and so are transported from the mantle to thecore during core formation. In this process, the element ratios arestrongly fractionated relative to chondritic proportions19. The HSEsin Earth’s mantle are significantly depleted relative to chondriticbodies—a clear consequence of core formation—and yet the remainingHSEs are in chondritic or near-chondritic proportions relative to eachother13,14. This is commonly interpreted11,12,19,20 as evidence that all or alarge portion of the HSEs currently in the mantle were delivered bychondritic bodies after the closure of Earth’s core, an accretion phaseknown as the Late Veneer20. To account for the observed mantle budgetof HSEs, we estimate that a chondritic mass of 4.8 6 1.6 3 1023M› isnecessary, where M› represents an Earth mass. This mass does includecontributions from the era known as the Late Heavy Bombardment.Current mass estimates for this very late (approximately 500 Myr aftercondensation) accretion are21 1023M›, which we added to the late-accreted masses of our synthetic Earth-like planets, but it only accountsfor about 2% of the chondritic mass and therefore does not play animportant part in our analysis of the correlation.

The chondritic mass can only be identical to the late-accreted massor to the Late Veneer mass if the Moon-forming event stripped all ofthe HSEs from Earth’s mantle or was the last episode of growth forEarth’s core, respectively (as is traditionally assumed). However, theseconditions are not necessarily true. Consider that some projectilescolliding with Earth after the Moon-forming event might have beendifferentiated, so that their HSEs were contained in their cores. If partof these cores had merged with Earth’s core22, then the late-accretedmass would clearly be larger than the chondritic mass, because therewould be no HSE record of this fraction of the projectile cores inEarth’s mantle. Additionally, in this case, given that iron (and thereforeHSEs) would have been simultaneously added to Earth’s mantle and itscore, the chondritic mass would be larger than the Late Veneer mass,which is geochemically defined as the mass accreted to Earth after thecore has stopped growing.

In fact, as explained in detail in the Methods and in Extended DataFigs 3 and 4, it is unlikely that more than 50% of a projectile’s core directlyreaches Earth’s core, otherwise geochemical models cannot reproducethe tungsten isotope composition of Earth’s mantle23. Moreover, a largelate-accreted mass, delivered in only a few objects so as to explain therelative HSE abundances of Earth and Moon12, would have left a detect-able isotopic signature on Earth relative to the Moon24,25. Thus, evenwhen considering these more complex possibilities, geochemical evid-ence constrains the late-accreted mass probably not to exceed 0.01M›(see Methods).

For these reasons, we first make the usual assumption that the late-accreted mass and the HSE-derived chondritic mass are identical. Inthis case, not a single simulated Earth-like planet with a last giantimpact earlier than 48 Myr since condensation has a late-accreted massin agreement with the value estimated from HSEs (see Fig. 1). Of thoseforming in less than 48 Myr, only one planet is near the upper 1sbounds of the chondritic mass. Only after 67 Myr since condensationare there Earth-like planets with late-accreted masses consistentlywithin the 1s uncertainty bounds for the chondritic mass. After126 Myr since condensation, the late-accreted masses of Earth-likeplanets are often significantly below the lower limit set by the HSEmeasurements.

We calculate the log-normal mean and standard deviation of thelate-accreted masses of all Earth-like planets with last giant impactswithin a range around a chosen time (see Fig. 1). We interpret thesedistributions as a model of the likelihood of a specific late-accretedmass given a last giant impact time. Given this likelihood model, we

Running geometric mean ofall Earth-like planetsRunning geometric mean of only Earth-likeplanets from Grand Tack simulations

Earth-like planets fromclassical simulationsEarth-like planets fromGrand Tack simulations

10 1005020 3015 1507010–4

0.001

0.01

0.1

1

Rel

ativ

e la

te a

ccre

ted

mas

s

Time of last giant impact (Myr)

Figure 1 | The late-accreted mass relative to each synthetic Earth-likeplanet’s final mass as a function of the time of the last giant impact.Triangles represent Earth-like planets from the first category: classicalsimulations with Jupiter and Saturn near their contemporary orbits7,8. Circlesrepresent Earth-like planets from the second category: Grand Tack simulationswith a truncated protoplanetary disk9,10. The black line resembling a staircase isthe moving geometric mean of the late-accreted masses in the Grand Tacksimulations evaluated at logarithmic time intervals with a spacing parameter of0.025 and a width parameter twice that. The blue region encloses the 1svariance of the late-accreted mass, computed assuming that the latter isdistributed log-normally about the geometric mean. Always predicting largerlate-accreted masses for each last giant impact time, the dotted staircase is thegeometric mean obtained by also considering the classical simulations,although those simulations do not fit Solar System constraints as well as theGrand Tack simulations do. The horizontal dashed line and enclosing darkenedregion are the best estimate and 1s uncertainty of the late-accreted massinferred from the HSE abundances in the mantle (chondritic mass):4.8 6 1.6 3 1023M›. The best estimate for the intersection of the correlationand the chondritic mass is 95 6 32 Myr after condensation. The dark and lightred regions highlight Moon-formation times that are ruled out with 99.9%(40 Myr) and 85% (63 Myr) confidence or greater, respectively.

LETTER RESEARCH

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[Jacobson et al., Nature, 2014]


地球マントルの HSE 量を説明するためには、最後の G.I. は CAI 形成後 ~100My であるべき

地球マントルの HSE 過剰　→ 地球形成後の late veneer

※ Grand Tack Model を仮定した場合の年代である

~1% come back to strike the Moon within 400million years (My) (Fig. 1) (8). Because the Moononly has ~25 ancient (Pre-Nectarian) lunar basins(16), probably made by the impact of diameterD > 20 km projectiles >4.1 Ga (13, 17), an impactprobability of ~1% implies the GI ejecta popula-tion could—at best—only contain a few thou-sand D > 20 km bodies (the order of 25/0.01).Mass balance therefore requires the majority ofGI ejecta to be in a steep size frequency distribu-tion dominated by D < 20 km bodies (8). Thisleads us to predict that ~1010-km-sized projec-tiles were thrown out of the Earth-Moon system(fig. S8) (8).Although GI simulations lack the resolution to

confirm the nature of this steep size frequencydistribution, insights gleaned from numericalimpact experiments on D = 100 km bodies showthat such steep slopes are common outcomeswhen the targets are largely left intact (6). Ananalog in nature for this may be the formation ofthe ~500-km Rheasilvia basin on the D = 530 kmasteroid Vesta; the largest body in Vesta’s family offragments is D ~ 8 km, a factor of 70 smaller thanVesta itself, whereas the exponents of its cumu-lative power law size distribution are extremelysteep, with –3.7 and –8 observed for D > 3 kmand > 5 km bodies, respectively (fig. S6) (7, 8, 18).The shape of this size distribution implies thatmuch of the mass of GI ejecta was initially inthe form of 0.1 < D < 20 km fragments ratherthan of dust and small debris (8).A consequence of a steep GI ejecta size fre-

quency distribution is that the fragments shouldundergo vigorous collisional evolution withthemselves. Tests using collision evolution codes(13, 19) indicate that D < 1 km bodies rapidlydemolished themselves, enough so to reduce thepopulation by several orders of magnitude inmass within 0.1 to 1 My of the GI (fig. S8) (8). Thiswould lead to a huge dust spike, with smallparticles either thrown out of the solar systemvia radiation pressure or lost to the Sun viaPoynting-Robertson drag (14, 20). The survivingfragments were depleted enough that they set-tled into a quasi-collisional steady state, withsubsequent mass loss dominated by dynamicalprocesses. The net effect is that ~105 to 107 D >1 km bodies were left in the GI ejecta populationfor many tens of millions of years (fig. S8).A substantial fraction of GI ejecta reached as-

teroid belt–crossing orbits after the GI, either bybeing launched onto such orbits or by dynami-cally evolving there via planetary perturbationsand resonances (Fig. 1 and fig. S1). We exploredtheir impact consequences for large main-beltasteroids by calculating how they affected a rep-resentative main belt target, Vesta. Vesta waschosen because it is a likely source of the eucritemeteorites (21), Vesta’s fragments have access tothe gravitational resonances thought to providemost meteorites to Earth (22), and its eccentric-ity and inclination are close to average main-beltvalues (23). By calculating collision probabilityand impact velocity distributions over time be-tween GI ejecta and Vesta (fig. S2) (8, 23), wefound that many fragments should have hit

322 17 APRIL 2015 • VOL 348 ISSUE 6232 sciencemag.org SCIENCE

Fig. 2. Compilations of impact agesfound within chondritic meteorites.(A) A representation of 40Ar-39Arshock degassing ages for 34 ordinaryand enstatite chondrites whose meanages are between ~4.32 billion and4.567 billion years (9–11). All sampleswere heavily shocked, shock-melted,or otherwise had some evidence forhaving been part of a large collision. Tocreate this age-probability distribution,we separated the sample ages byparent body (EL, EH, E-melt/Aubrites,L, LL, and H chondrites) and computedthe sum probability of ages withineach class by adding Gaussian profiles,with centers and widths correspondingto the most probable age and 1s errorsof each dated sample (8). The profileswere then normalized before theywere summed in order to prevent anysingle class from dominating thedistribution (fig. S9A). We caution thatsystematic errors in measured Ar decayrates could make these ages slightlyolder (8). (B) The age-probabilitydistribution of U-Pb ages for 24 L, LL,and H chondrites (table S1) created byusing the same method (fig. S9B). U-Pbages >60 My after CAIs are interpretedto be from impact heating alone, whereas those <60 My after CAIs are an unknown mixture of formation,metamorphic, and impact ages (26). Both distributions show a feature ~80 to 120 My after CAIs (~4.45to 4.49 Ga).

Fig. 3. A sample comparisonbetween our model and ran-domly derived 40Ar-39Ar shockdegassing ages for asteroidalmeteorites. (A) The combined40Ar-39Ar age distribution, in blue,was created by assuming thatleftover planetesimals and giantimpact ejecta struck main beltasteroids such as Vesta early insolar system history (8). Bothmodel contributions have thesame shape, with the former(red) ~1.8 times as large as thelatter (green), respectively.(B) A single representation of ourmodel results, shown as a blueline, compared with 34 40Ar-39Arshock reset ages randomlydrawn from Fig. 2A. In thisexample, the giant impact takesplace at 112 My after CAIs. Theplotted results are close to ourderived age for the giant impact,105 T 25 My after CAIs (equiva-lently 4.46 T 0.03 Ga), and ourderived contribution ratio of1.9 T 0.9 between leftoverplanetesimals and giant impactejecta.

RESEARCH | REPORTS

[Bottke et al., Science, 2015]

17 APRIL 2015 • VOL 348 ISSUE 6232 27 1SCIENCE sciencemag.org

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By Eric Hand

It was the biggest cataclysm the solar

system has ever seen. About 100 million

years after the planets began to take

shape, a Mars-sized body crashed into

the proto-Earth, creating a halo of hot

debris that coalesced into the moon.

There was collateral damage, it turns

out. Scientists now suspect that fragments

of the giant impact were flung all the way to

the fledgling asteroid belt. When this plan-

etary shrapnel crashed into bodies there, it

shock heated them, leaving an imprint that

can still be detected billions of years later

in meteorites. On page 321, planetary scien-

tists show that these shock-heating signa-

tures provide a new way to date the moon’s

formation, pegging it at 105 million years

after the beginning of the solar system

4.6 billion years ago.

The result could help settle debates

about the age of the moon and suggests that

meteorites, which are mostly fragments of

asteroids, could harbor other evidence of

tumult in the inner solar system. “The as-

teroid belt is almost primordial,” says Bill

Bottke, a planetary scientist at Southwest

Research Institute in Boulder, Colorado,

and lead author of the new study. “A lot of

objects there have been witness to activity

in the inner solar system. We now have a

way to probe that.”

Scientists have long tried to pin down the

age of the moon by analyzing lunar samples

returned from the Apollo missions. But be-

cause of disagreements about the isotope

systems used for dating, the calculated ages

vary from about 30 million years after the

start of the solar system

to 100 million or even

200 million years younger.

A more precise age would

help scientists work out

when the bumper-car pro-

cess of planet formation

began winding down. The

moon-forming impact is

thought to have come late

in the process, because

the composition of Earth’s

mantle reflects only a

short period of impacts

by smaller bodies after the

mammoth collision.

Researchers who study

the giant impact have

typically ignored the bits

that didn’t end up in the moon. But Bottke

realized that an event so large would have

created fragments moving fast enough to

escape the collective gravity of the Earth-

moon system. “You create this huge swarm

of material,” he says. His models suggest

that 10 billion kilometer-sized bodies would

have been flung out into the solar system—

where many of them could strike asteroids.

Asteroids constantly collide with each

other, but at relatively slow speeds. Some

high-speed projectiles from the giant im-

pact, in contrast, would have struck at

speeds upward of 10 kilometers a second,

melting and transforming asteroid miner-

als into darker, glassy materials. The shock

heating would also have altered a standard

radio active “clock” used for dating, in which

a radioactive isotope of potassium decays

into argon that remains trapped in the crys-

tal structure of the rock. “If you heat it up

enough, argon moves through the crystal

structures, and you can reset [the clock],”

says study co-author Tim Swindle, director

of the Lunar and Planetary Laboratory at

the University of Arizona in Tucson.

Searching through the literature for me-

teorites that had already been dated, the

team found 34 samples that fit their profile:

those with shock-heating alteration and

ancient argon ages. A significant fraction

of these 34 samples have ages that cluster

around 105 million years after the solar sys-

tem began; that, the team believes, is the

age of the moon-forming impact.

Other scientists are excited about the

method but worried about the small sam-

ple size. The authors used their own judg-

ment to identify meteorites with the right

type of shock heating, and their 34 meteor-

ite samples could hail from as few as five

or six parent asteroid bodies. “Is that really

representative of everything the asteroid

belt saw?” asks Sarah Stewart, a planetary

scientist at the University of California,

Davis. “It’s not a robust

conclusion, but it’s a

robust method.”

Swindle says the new

moon age estimate—a

signal “strong enough

to look like more than a

curiosity”—will improve

as his lab and others calcu-

late dates for more shock-

heated meteorites. And

Bottke hopes the method

will be used for more than

just dates. He says mete-

oriticists should return

to these 34 samples and

inspect them carefully.

Perhaps amid the veins

of glassy materials are

fragments of the giant impactor, or the

proto-Earth itself. “There may still be

traces of the primordial Earth in the aster-

oid belt, and they may be in our meteor-

ite collections today,” Bottke says. “To me

that’s fun.” ■

Moon-forming impact left

scars in distant asteroids

Planetary collision dated through analysis of meteorites

PLANETARY SCIENCE

The giant impact that

formed the moon may

have flung copious debris

into the solar system.

Dark impact melt (top) in a meteorite

that fell near Chelyabinsk, Russia, may

have been created by projectiles from

the moon-forming impact.

Published by AAAS

on

April

16,

201

5w

ww

.sci

ence

mag

.org

Dow

nloa

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(c) Science


Giant Impact Ejecta が高速で小惑星に衝突・年代をリセットした証拠が隕石に刻まれているはず※ サンプルが少なすぎる＆モデルがシンプルすぎる

結局どうしたらいいの？

• 各 Giant Impact モデルの妥当性がわからない→ 多様な Giant Impact のより詳細な計算を行う

• 地球と月の化学組成が一致しすぎている？→ 新たな月の石を取得・サンプルバイアスを除く

• 全ての化学データを満たす解が存在しない→ いっそ Giant Impact 説から離れて考えてみる

• 月形成年代をどうやって決める？→ 月の酸素同位体比を持つ欠片＠隕石を探す

topically with the disk in so short a timescale, and we have no evidence yet thatthe deep interior is substantially differ-ent from Earth’s near surface in oxygenisotopes.

Formation directly from Earth.To get a circumplanetary disk that is de-rived mostly from Earth, it seems nec-essary to have an impact that violatesthe angular-momentum constraint.Planetary scientists have suggestedtwo possibilities. One is to hit an Earththat is already close to fission with afast-moving projectile. That could bethought of as impact-triggered fission.Another possibility is for the collisionto happen between two ”sub-Earths,”two bodies each about half an Earthmass.

Figure 2 presents hydrodynamicsnapshots of three kinds of giant im-pacts. Only the first—the standardgiant impact of a smaller body collid-ing with Earth—satisfies the angular-momentum constraint.9 But the otherimpact scenarios show how the mate-rial used to make the Moon can comemainly from Earth; for them to be can-didates, one must find a way of gettingrid of excess angular momentum. Onemethod for doing so, proposed byMatija Ćuk and Sarah Stewart twoyears ago,10 is an evection resonance,in which the precession rate of theMoon’s orbit matches Earth’s meanmotion about the Sun (see box 3). Al-though the resonance is well known,its application to account for a loss ofangular momentum is new.

At present, no one knows the an-swer to the formation puzzle. It couldbe a combination of all three possibil-ities or something else entirely. In thecase of the evection resonance idea, which re-searchers are still analyzing, at issue is not the exis-tence of the resonance but rather the need to have itin place for an extended period of time during whichthe eccentricity of the lunar orbit is large. That ap-pears to require a particular and possibly narrowrange of parameters for the tidal behaviors of Earthand the Moon.

What about Venus?In 1672 Giovanni Cassini “discovered“ the moon ofVenus. Astronomers of the time thought it was ob-vious that Venus should have a moon because Earthwas so endowed. The moon was named Neith, afteran Egyptian goddess. Cassini’s observation was “con-firmed” by many others, and the orbit of Neith wasconfirmed by Joseph Lagrange in 1761. That sameyear, French mathematician Jean le Rond d’Alembertlamented to Voltaire in a letter, “I do not know whathas happened with the lackey of Venus. I am afraidit cannot be a hired lackey which has ceased to staywith her for a long time, but rather that the said

lackey has declined to follow his mistress during herpassage over the Sun.” It gradually became apparentthat Neith was a false discovery.11

Planetary scientists no longer think that a moonof Venus should be obvious. Nor do they seek to un-derstand Earth and the Moon in isolation but as partof a broader picture. Although Venus is closer thanEarth to the Sun, an Earth–Moon system placed inthe orbit of Venus would be stable for the age of thesolar system. A stable moon for Mercury is far moredifficult to imagine because of both the planet’s lowmass and greater proximity to the Sun.

Notwithstanding Mercury, the absence of amoon for Venus is puzzling considering that moonsare thought to be readily made by giant impact. Onepossibility is simply to appeal to chance. Not all giantimpacts lead to moons, and perhaps Venus lackedsuch an event. The alternative is to suppose thatVenus once had a moon but lost it. The giant impactthat made Earth’s moon was very late, the last majorevent in the evolution of the inner solar system, ac-cording to isotopic evidence.

www.physicstoday.org November 2014 Physics Today 37

a

b

c

Standard

Fast-spinningEarth

Sub-Earth

TIME

Figure 2. Hydrodynamic snapshots of giant impacts that might have been. Ineach of three cases, a projectile, whose mantle and core are shown in orange andwhite, respectively, obliquely hits Earth, whose mantle and core are shown in greenand gray. Earth’s North Pole points out of the page. The aftermath of each collision,projected onto the equatorial plane, is pictured from left to right, with several hourselapsing between each snapshot. (a) In the standard scenario,9 the angular momentumof the impact equals that of the current Earth–Moon system, but the material thatends up in orbit is mainly projectile orange, a result at odds with the nearly identicalisotopic ratios of oxygen, silicon, tungsten, and titanium observed in the real Earthand Moon. In the two other cases, (b) a small projectile smashes into a rapidly rotatingplanet,10 and (c) two bodies collide, each with half of Earth’s mass.13 In all three cases,very little metallic iron ends up in orbit, a result borne out by observation. But only inpanels b and c does primarily Earth’s mantle (green) end up in orbit. For videos of thesimulations, see the online version of this article. (Figure prepared by Miki Nakajimaand adapted from results in ref. 14.)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:173.250.176.23 On: Wed, 19 Nov 2014 00:39:58

[Nakajima & Stevenson, 2014]

Earth more than the materials from the deformedimpactor and hence contribute more to the Moon.

The composition of a magma ocean can beestimated from the phase diagram (element parti-tioning) of peridotite and the principles of theisotopic fractionation. I use the experimental dataon Fe/Mg partitioning between olivine and ultra-mafic melt14) to estimate the Mg# of magma oceanfor a various degree of melting. Inferred value of Mg#of the Moon (983 ’ 5) can be explained by a broadrange of degree of partial melting (920–80%) orfractional crystallization. The degree of chemicalsegregation upon melting depends on the processof melting. Melting by a giant impact and partialmelting will have somewhat different consequence onchemical segregation. However, the experimentallyobserved Fe/Mg partitioning between the ultramaficmelts and olivine implies that melt will always bemore FeO rich than co-existing solids under theseconditions. In contrast to the major element compo-sition, isotopic fractionation elements is much lessaffected by partial melting (e.g., ref. 13). However,the chemical composition of the Moon can besomewhat different from the composition of thematerials ejected by the impact because the processesof condensation from the gas-liquid mixture can

modify the composition (e.g., ref. 24). The presentmodel could also explain the presence of a smallFe-rich core,25) if the influence of reduction at hightemperatures is included.

3. Summary and discussions

The current model of planetary formationsuggests that a large degree of heating occurs in thelate stage of planetary formation. The efficientheating in this stage is a result of quick and henceefficient conversion of large gravitational energy toheat due to the collisions of relatively large objects.Consequently, high degree of heating will occurleading to melting (magma ocean formation) in alarge planet (larger than Mars, e.g., ref. 26), but not

proto-Earth proto-Earth

impactorimpactor

magma ocean

vapor je

t

(a) (b)

Fig. 4. Schematic drawing of processes of ejection of materialsupon a giant impact. (a) A case where the proto-Earth does nothave a magma ocean. (b) A case where the proto-Earth has amagma ocean.

R

hA

BC

Fig. 3. A schematic diagram showing possible paths of materialsejected at a certain height. Only a fraction of materials goes tothe orbit (shaded region) from which the Moon was formed. Thefate of ejected materials depends on the ratio h/R and materialswith only for modest value of h/R and velocity will become thesource of the Moon.

0

5000

10000

15000

20000

25000

0 2 4 6 8 10 12

tem

pera

ture

(K

)

collision velocity (km/s)

liqui

d

q= -2

-1.5

-1

0

0 1

solid

melting temperature

o= V

Vo

q

Fig. 2. Collision velocity versus temperature. The temperatureincrease by a collision is calculated using the relation [1] inthe text. This temperature corresponds to temperature at thehighest shock pressure. Parameters used are summarized inTable 1. .: the Grüneisen parameter, V: volume, q: a non-dimensional parameter in relation !

!o¼ ð"o" Þ

q (q 9 1 for solids,q < 0 for liquids).

Asymmetric shock heating and the terrestrial magma ocean origin of the MoonNo. 3] 101

[Karato, 2014]

2.3. Initial Conditions

We followCanup&Asphaug (2001) andCanup (2004) for theorbital parameters of the impactor for which the most massivesatellite is expected. The masses of the proto-Earth and the im-pactor are assumed to be 1.0 and 0:2 M!, whereM! is the Earthmass. The radii of the proto-Earth and protoplanet are rE ¼ 1:0and 0:64rE, respectively. Note that no significant differences inthe results for smaller impactors (e.g., 0:1 M!) were found in oursimulations. The initial orbits of the impactor are assumed tobe parabolic, and the angular momentum is 0.86Lgraz, where Lgrazis the angular momentum for a grazing collision (Canup &Asphaug 2001). Initially, the impactor is located at 4:0rE fromthe proto-Earth.

3. RESULTS

3.1. Disk Evolution and the Predicted Lunar Mass

Figure 1 shows a typical time evolution of the giant impactwith EOS-1 (model A). This model corresponds to the ‘‘late’’impact model in Canup&Asphaug (2001). After the first impact(t ’ 1 hr), the disrupted impactor is reaccumulated to form aclump at t ’ 3 hr, which finally collides with the proto-Earth att ’ 6 hr. During the second impact, the impactor is destroyed,and a dense part of the remnant spirals onto the proto-Earth(t ’ 10 hr), and a circumterrestrial debris disk is formed aroundt ’ 18 hr. Note that many strong spiral shocks are generated inthis process as seen in the density map (Fig. 2) and azimuthaldensity profile (Fig. 3).

Fig. 1.—Giant impact simulation with EOS-1, which represents a state in which most of the impactor mass is vaporized. Left, face-on views of the system; right, edge-on views. The numbers in the upper right corners of the panels show the time in units of hours. The color represents log-scaled density (the units are !0 ¼ 12:6 g cm#3).

Fig. 2.—Snapshot of the density field of model A at t ¼ 12:3 hr. Strongspiral shocks in the debris are resolved.

WADA, KOKUBO, & MAKINO1182 Vol. 638

[Wada et al., 2005]

[Pahlevan & Stevenson, 2007]






tot Þdisk ð1Þ



dfT ¼ Msilctarg

! .Msilc

tot

"

disk

.Msilc

targ

! .Msilc

tot

"





4. Results



5. Discussion






A. Reufer et al. / Icarus 221 (2012) 296–299 297

[Reufer et al., 2012]

T. SASAKI AND Y. ABE: IMPERFECT EQUILIBRATION OF HF-W SYSTEM 1041

Fig. 6. The age of the last giant impact as a function of the resetting ratioof each giant impact, fitting to the observational data (ϵ = 2) from Earthsamples. The number of giant impacts is assumed to be five. The initialstate is ϵ = 10 at t = 10. The formation age of the Earth for perfectresetting (resetting ratio = 1) is about 30 Myr, in agreement with aprevious study (Yin et al., 2002).

metal-silicate equilibration. This would not be a realisticassumption. However, these calculations give us the higherlimit of equilibration at each giant impact event. Therefore,from the viewpoint of obtaining the observed isotopic ratio,we can obtain the lower limit of the resetting ratio requiredfor each giant impact. That is, we use assumptions that leadto the “higher limit of equilibration” on calculating isotopicevolution to obtain “the lower limit of required resetting ra-tio” to meet the observed epsilon value. The number ofgiant impacts, n, was varied from 2 to 10. Figure 6 showsthe result for n = 5, for example. The estimated age ofthe last giant impact depends on the resetting ratio of eachgiant impact, which must be greater than 0.3 to yield theobserved ϵ value. The effect of the number of giant impactsis shown in Fig. 7. It shows that the resetting ratio of eachgiant impact and the number of giant impacts both affectthe estimation of the age of the last giant impact. The re-sults indicate that the average resetting ratio of each giantimpact must be greater than 0.2 to yield a good fit with theobservations, even if giant impacts occurred ten times.

Although we use the f -value of 12 in Eq. (4) to solveEq. (6) and Eq. (7), as mentioned earlier, the f -value inEq. (4) is considerably uncertain (from 10 to 40). Therefore,we check how does this uncertainty affect our conclusionshere. Figure 8 shows the age of last giant impact assumedto have occurred ten times to form the Earth for f = 10,12, 20, 30, 40, which was calculated in a manner similar tothe case of f = 12. It shows that the lower limit of averageresetting ratio of each giant impact is still about 0.2 in therange from f = 10 to f = 40, which would not alter ourconclusions.

Our calculations tend to give an overestimation of theequilibration rate of the Hf-W system, as all of the im-pactor’s core and mantle is assumed to be equilibrated bya giant impact. In practice, because some fraction of theimpactor’s core (mantle) may be added to the target’s core(mantle) without equilibrating, the required resetting ratiomay be larger than that was estimated in this section. There-fore, our value of 0.2 should be regarded as a lower limit ofthe required resetting ratio of each giant impact, for a total

Fig. 7. The age of the last giant impact as a function of the resetting ratioof each giant impact, fitting to the observational data (ϵ = 2) from Earthsamples. The number of giant impacts is 2 to 10 from left to right. Theinitial state is ϵ = 10 at t = 10.

Fig. 8. The age of the tenth giant impact as a function of the resettingratio of each giant impact, fitting to the observational data (ϵ = 2) fromEarth samples. The f -value is 10, 12, 20, 30, 40 from bottom to top.Each initial ϵ-value was calculated using Eq. (6) and Eq. (7) for eachf -value.

number of giant impacts less than or equal to ten.Nimmo and Agnor (2006) considered two extreme sce-

narios after giant impacts: complete metal-silicate equili-bration and core merging without any significant equilibra-tion. They claimed that the observed isotopic data requirere-equilibration of impacting bodies with the target mantleand ruled out direct core merging even for the largest im-pacting bodies. Although the direct core merging is onetype of imperfect equilibration, it is not the only possibleway of imperfectness. The Rayleigh-Taylor instability dis-cussed in Section 2 is another type of imperfect equilibra-tion, which leaves some fraction of silicate without equili-bration with iron, while direct core merging leaves somefraction of iron without equilibration with silicate. Herewe showed that the observational data does not rule out in-complete equilibration due to the Rayleigh-Taylor instabil-ity. Our results indicate that the collision conditions and thenumber of giant impacts are essential parameters to estimatethe age of the core formation event.

4. DiscussionWe have shown that complete metal-silicate equilibration

by a giant impact cannot be expected even if the impact

[Sasaki & Abe, 2007]

o

i Tk ab

W WPHi!!

(c) Tetsuya Kawase

New Perspectives by WPH

• 各 Giant Impact モデルの妥当性がわからない→ 多様な Giant Impact のより詳細な計算を行う

• 地球と月の化学組成が一致しすぎている？→ 新たな月の石を取得・サンプルバイアスを除く

• 全ての化学データを満たす解が存在しない→ いっそ Giant Impact 説から離れて考えてみる

• 月形成年代をどうやって決める？→ 月の酸素同位体比を持つ欠片＠隕石を探す

Education

Moon formation sasaki