月の謎を解く

10/19/19 1

Shun-ichiroKarato唐戸　俊一郎YaleUniversity

DepartmentofGeology&GeophysicsNewHaven,CT,USA

IncollaboraIonwith

N.Hosono,J.Makino,T.Saitoh

•  InalaterstageofEarthformaIon,arelaIvelylargebodycollidedobliquely.àlargeangularmomentumàmantlematerialsareejectedàtheMoonismadeofrockymaterials(nobigcore).

•  Classiccalcula)on(1986):theMoonismostlyfromtheimpactor•  ThisisOKifweareconcernedonlywiththebulkcomposi)on.•  Recenthigh-resolu)ongeochemicalmeasurements

à(i)closesimilarityofisotopesbetweentheMoonandEarth.à(ii)“wet”Moon

(i)IftheMoonismademainlyoftheimpactor,whyaretheisotopicraIosoftheMoonsosimilartothoseofEarth?à isotopiccrisis(howtoexplaintheisotopicsimilaritytogetherwiththelarge

angularmomentum?)

(ii)“wet”Moon:Shouldn’twaterbelostduringagiantimpact?

10/19/19 2

ThegiantimpactmodelHartmann-Davis(1975),Cameron-Ward(1976)

😅

•  Conven)onalapproaches:–  Fateofagiantimpactiscontrolledbythe“mechanics”ofanimpactnotbythephysicalproper)esofmaFer(convenIonalmodelingapproach).

–  Vola)lecontentisdeterminedbythecondensa)onfromagastosolids(convenIonalcosmochemistry).

àPhysicsandchemistryofmaFersmaFer!•  Theroleofliquidsinvola)lereten)on(“wetMoon”)•  Theroleofliquidsingiantimpact(àvaporiza)onàdiskforma)on:isotopicsimilarity,FeOcontent)

àCansolvemostofpuzzles(?)10/19/19 3

Limita)onsofpreviousapproachesàanewapproach

“wet”Moon?

measurements, and Cl contents of <50 ppm thatwere limited by electron microprobe detectionlimits. Reheated Apollo 12 melt inclusions, con-taining medium-Ti magmas (5 to 6 wt % TiO2),show sulfur contents that are 20% higher thanour data on average (23). In our data set, weobserve a correlation of all the volatiles witheach other (Fig. 3), pointing toward the degassedcompositions of the matrix glass rinds and vol-canic glass beads (4).

Themost important aspect of our volatile dataon lunar melt inclusions is their similarity to meltinclusions from primitive samples of terrestrialmid-ocean ridge basalts (MORBs), like thoserecovered from spreading centers located withintransform faults (24); the melt inclusions from74220 are markedly similar to melt inclusionsfrom the Siqueiros Fracture Zone on the EastPacific Rise, some of the most primitive mid-ocean ridge magmas that have been measured(Fig. 3). These similarities suggest that the vol-atile signature of the lunar mantle source of thehigh-Ti melt inclusions is very similar to that ofthe upper mantle source of MORB.

It is important that we have made thesemeasurements on inclusions from olivine crys-tals contained within primitive lunar volcanicglasses. These inclusions were quenched withinminutes after their eruption (4), providing min-imal opportunity for posteruptive hydrogen dif-fusion out of the inclusions and affording adirect H2O measurement on primary lunar mag-ma samples that have not experienced poster-uptive degassing and associated loss of volatiles.The water concentrations that we measured are20 to 100 times as high as previous direct mea-surements of the lunar glass beads from thissame sample, which was estimated to have suf-fered 95 to 98% loss of H2O via degassing (4),and they are higher than estimates derived fromlunar apatite measurements, which require a 95to 99% correction for fractional crystallizationto estimate primary magma volatile contents(5, 6). Our results are direct measurements onprimary lunar magma compositions that requireno such extrapolations.

Our melt inclusion data allow us to placesome constraints on the volatile content of thelunar mantle source that generated the high-Tipicritic magmas. Using the most water-rich meltinclusion composition after correction for post-entrapment crystallization, and an estimation thatthe high-Ti magmas originated from 5 to 30%batch partial melting with partitioning similar tothat of terrestrial mantle-derived melts (17), weestimate lunar mantle volatile concentrations of79 to 409 ppm H2O, 7 to 26 ppm F, 193 to 352ppm S, and 0.14 to 0.83 ppm Cl. These estimatesoverlap most estimates for the volatile content ofthe terrestrial MORB mantle (24–27) and aremuch higher than previous estimates for the lunarmantle based on the volatile content of lunarapatite (5, 6) and the variation of Cl isotopes inlunar rocks (28), including the sample 74220 thatwe have studied here. The melt inclusions indi-

cate definitively that some reservoirs within theinteriors of Earth and the Moon not only havesimilar water contents, but also similar contentsof fluorine, sulfur, and chlorine associated withthis water, a volatile abundance signature sharedby both bodies.

These results show that the Moon is the onlyplanetary object in our solar system currently

identified to have an internal reservoir with avolatile content similar to that of Earth’s uppermantle, and that previous estimates of the lunarinventory for highly volatile elements are biasedto low concentrations owing to the degassed na-ture of lunar samples thus far studied. The Moonhas erupted a wide variety of magmas during itshistory, and it remains to be seen whether other

AB

FED

C16OH/30Si

Fig. 2. (A to F) NanoSIMS scanning isotope images of olivines A1, A2, N3, N6, N8, and N9 fromApollo 17sample 74220, showing the distribution of water within melt inclusions from the olivine grains shown inFig. 1. The images show the distribution of the isotope ratio 16OH/30Si indicated by the color scale shownin (A), which ranges from dark regions corresponding to low 16OH/30Si ratios (e.g., olivine surroundingmelt inclusions), to red regions within melt inclusions with 16OH/30Si ratios approaching 0.25(corresponding to ~1400 ppm H2O). The color scale is the same in all images, and all images show a scalebar of 1 mm. Rectangular areas are regions of interest within which each isotope ratio is calculated andconverted to a concentration.

A C

E

B

D F

Fig. 1. (A to F) Optical photographs of olivines A1, A2, N3, N6, N8, and N9 from Apollo 17 sample74220. Inclusions within circles indicate the inclusions that were imaged in Fig. 2. Scale bars are 10 mmin all photos.

8 JULY 2011 VOL 333 SCIENCE www.sciencemag.org214

REPORTS

on J

uly

21, 2011

ww

w.s

cie

ncem

ag.o

rgD

ow

nlo

aded fro

m

10/19/19

lunar mantle sources are as volatile rich as thesource ofApollo 17 high-Timagmas.Nevertheless,the hydrated nature of at least part of the Moon’sinterior is a result that is not consistent with thenotion that the Moon lost its entire volatile in-ventory to the vacuum of space during degassingafter a high-energy giant impact, which wouldbe expected to leave a highly desiccated lunarinterior.

If the bulk of the lunar interior has a volatilecontent similar to our estimate for the high-Timantle source, then our results present difficultiesfor late-accretion models that require volatiledelivery to Earth and the Moon after their for-mation, because these two bodies have very dif-ferent accretion cross sections that would predictdifferent internal volatile contents. An Earth-Moonsimilarity in volatiles could indicate that chemicalexchange of even the most volatile elements be-tween the molten Earth and the proto-lunar discmight have been pervasive and extensive, result-ing in homogenization at the very high temper-atures expected after a giant impact; this couldhave been aided by the presence of a high-temperature convective atmospheric envelope sur-rounding Earth and the proto-lunar disc as theMoon solidified (29). Alternatively, it is con-ceivable that a portion of the lunar interior

escaped the widespread melting expected in theaftermath of a giant impact and simply inheritedthe inventory of water and other volatiles that ischaracteristic of Earth’s upper mantle. Any modelfor the formation of Earth-Moon system mustmeet the constraints imposed by the presence ofH2O in the lunar interior, with an abundance sim-ilar to that of Earth’s upper mantle and with acomplement of fluorine, sulfur, and chlorine alsopresent at terrestrial levels. To the extent that lu-nar formation models predict very different vol-atile contents of Earth and the Moon, our resultson the volatile content of lunar melt inclusionssuggest that we lack understanding on some crit-ical aspects of the physics of planetary moon for-mation by collisional impact.

Our findings also have implications for theorigin of water ice in shadowed lunar craters,which has been attributed to cometary andmeteoritic impacts (30). It is conceivable thatsome of this water could have originated frommagmatic degassing during emplacement anderuption of lunar magmas (31). These results alsounderscore the importance of pyroclastic volcan-ic samples in unraveling the history and compo-sition of theMoon’s interior; indeed, such depositshave been identified and mapped on the surfacesof all the terrestrial planets and many satellites.

References and Notes1. R. M. Canup, Annu. Rev. Astron. Astrophys. 42, 441

(2004).2. S. R. Taylor, in Origin of the Moon, W. K. Hartmann,

R. J. Phillips, G. J. Taylor, Eds. (Lunar Planetary Institute,Houston, TX, 1986), pp. 125–143.

3. F. Albarède, Nature 461, 1227 (2009).4. A. E. Saal et al., Nature 454, 192 (2008).5. F. M. McCubbin et al., Proc. Natl. Acad. Sci. U.S.A. 107,

11223 (2010).6. J. W. Boyce et al., Nature 466, 466 (2010).7. J. P. Greenwood et al., Nat. Geosci. 4, 79 (2011).8. J. W. Delano, J. Geophys. Res. 91, D201 (1986).9. K. A. Kelley et al., J. Geophys. Res. 111, B09208

(2006).10. A. M. Shaw, E. H. Hauri, T. P. Fischer, D. R. Hilton,

K. A. Kelley, Earth Planet. Sci. Lett. 275, 138 (2008).11. E. H. Hauri, Chem. Geol. 183, 115 (2002).12. J. C. Lassiter, E. H. Hauri, I. K. Nikogosian,

H. G. Barsczus, Earth Planet. Sci. Lett. 202, 525(2002).

13. N. Shimizu, Phys. Earth Planet. Inter. 107, 183(1998).

14. A. M. Shaw, M. D. Behn, S. E. Humphris, R. A. Sohn,P. M. Gregg, Earth Planet. Sci. Lett. 289, 311(2010).

15. H. Y. McSween Jr., R. P. Harvey, Science 259, 1890(1993).

16. L. L. Watson, I. D. Hutcheon, S. Epstein, E. M. Stolper,Science 265, 86 (1994).

17. Detailed information on melt inclusion identificationand analytical methods used in this study, as well asall data, are contained in supporting online materialsavailable at Science Online.

18. E. Roedder, P. W. Weiblen, Proc. Lunar Sci. Conf. 1, 801(1970).

19. E. Roedder, P. W. Weiblen, Proc. Lunar Sci. Conf. 2, 507(1971).

20. E. Roedder, P. W. Weiblen, Earth Planet. Sci. Lett. 13,272 (1972).

21. N. Klein, M. J. Rutherford, Lunar Planet. Sci. 29, 1448(1998).

22. C. M. Weitz, M. J. Rutherford, J. W. Head III, D. S. McKay,Meteorit. Planet. Sci. 34, 527 (1999).

23. D. J. Bombardieri, M. D. Norman, V. S. Kamenetsky,L. V. Danyushevsky, Meteorit. Planet. Sci. 40, 679 (2005).

24. A. E. Saal, E. H. Hauri, C. H. Langmuir, M. R. Perfit,Nature 419, 451 (2002).

25. A. Jambon, J. L. Zimmermann, Earth Planet. Sci. Lett.101, 323 (1990).

26. P. Michael, Earth Planet. Sci. Lett. 131, 301 (1995).27. J. E. Dixon, L. Leist, C. Langmuir, J. G. Schilling,

Nature 420, 385 (2002).28. Z. D. Sharp, C. K. Shearer, K. D. McKeegan, J. D. Barnes,

Y. Q. Wang, Science 329, 1050 (2010).29. K. Pahlevan, D. J. Stevenson, Earth Planet. Sci. Lett. 262,

438 (2007).30. W. C. Feldman et al., J. Geophys. Res. 106, 23231

(2001).31. A. P. S. Crotts, Astrophys. J. 687, 692 (2008).Acknowledgments: This work was supported by the

Carnegie Institution of Washington, the NASA LASERand Cosmochemistry programs, the NASA LunarScience Institute, and the NASA Astrobiology Institute.We thank J. Wang for NanoSIMS assistance, andL. Nittler and Z. Peeters for help with imageprocessing software.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/science.1204626/DC1Materials and MethodsFigs. S1 and S2Tables S1 and S2References (32–37)

21 February 2011; accepted 10 May 2011Published online 26 May 2011;10.1126/science.1204626

Fig. 3. (A to C) Volatile abun-dances for lunar melt inclusions(orange circle with black rims) andmatrix glasses (orange circles) fromApollo 17 sample 74220. Melt in-clusions show the highest concen-trations (>600 ppm H2O) whereasmatrix glasses show the lowest con-centrations due to degassing (≤30ppm H2O). The black curves showlunar magma degassing trends,scaled from the volatile-volatilecorrelations observed in core-rimNanoSIMS data on a lunar glassbead reported by Saal et al. (4);the core-rim data were scaled bymultiplying the originally reporteddata for each element, by the ratioof the highest melt inclusion com-position to that of the core com-position reported in table 2 of (4).The gray field surrounds data formelt inclusions from the SiqueirosFracture Zone on the East PacificRise, as an example of depletedMORB (24).

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 500 1000 1500

H2O ppmC

l ppm

0

200

400

600

800

1000

S p

pm

0

20

40

60

80

100

120

140

F p

pm

A

B

C

Siqueiros MORB

Siqueiros MORB

Siqueiros MORB

www.sciencemag.org SCIENCE VOL 333 8 JULY 2011 215

REPORTS

Saaletal.(2008,2013)(olivine)Haurietal.(2011)(olivine)[Greenwoodetal.(2011)(apa)te)]

InclusionsinolivineinsomelunarrocksshowvolaIlecontentsimilartoEarth.à LunarinteriorisaswetasEarth’suppermantle(depletedbutnot-so-dry

(~100ppmwtwater)).à Butarethesesamplesrepresenta)veoftheMoon?

4

Haurietal.(2011)

The“dryMoon”paradigmischallengedbyhigh-resoluIonchemicalanalyses.

(sample:74220(Apollo17))MORB(Earth)

lunarsample(Moon)

H2O

F

S

Cl

Howaboutgeophysicalobserva)ons?

•  Geophysicalobserva)ons=global(indirect)•  WhichobservaIons?

– SeismicwavevelociIes– ElectricalconducIvity– TidalQ(viscosity)

10/19/19 5

3 7 6 P. GOLDREICH AND S. SOTER

B ~ , m

FIG. 1. The force of attraction between the satellite m and the nearer tidal bulge A exceeds that between m and B; a component of the net torque retards the rotation of the planet M and accelerates the satellite in its orbit.

The asymmetrical position of lhe tidal bulges with respect to the line of centers, Mm, introduces a net torque between the planet and satellite. Because the satellite is a t t racted more strongly by the near side bulge which is leading it in longitude, the torque acts to transfer angular momentum and energy from the planet's rotation into the satellite's orbital revolution. The excess planetary spin energy is dissii.~ated as heat in the planet's interior. As a familiar exam- ple, the Earth 's rotation is gradually slowing down while the lunar semimajor axis is expanding. In addition to affecting the satellite's semimajor axis, the frictionally retarded tides on the planet also produce secular changes in eccentricity, inclination, and obliquity. As we are particularly inter- ested in the changes of eccentricity, we shall briefly describe the mechanism by which they are produced.

The tidal torque on a satellite which moves in an eccentric orbit is larger at pericenter than at apocenter. For this rea- son, we may approximate the total addition of angular momentum to the satellite orbit by one impulse at pericenter and by another, somewhat smaller impulse at apocenter. Due to the periodic nature of bound orbits in an inverse-square-law force field, it is evident that an impulse at pericenter increases the apocenter distance without altering the distance to pericenter. Similarly, an impulse at apocenter increases the pcricenter dis- tance but doesn't affect, the distance Co apocenter. Because the larger impulse occurs at pericenter, the net effect of the ~idal

torque is to increase the eccentricity, as well as the semimajor axis, of the sateIlite's orbit. The tangential component of force on the satellite, which is responsible for the tidal torque, is not the only component which affects the eccentricity. The radial component also plays a role in this process. Consider the situation where the satellite's orbital period just equals the planet's rotation period. High tide on a perfectly elastic planet would occur when the satellite was at pericenter. In reality, the maximum occurs some time after pericenter due to dissipation in these radial tides. Now con- sider the more usual case of relative rotation between the planet and satellite: the tides still retain a periodic radial component, pro- vided e ~ 0. Although this component in- volves no net torques that transfer angular momentum between the planet and satellite, it nonetheless dissipates mechanical energy of the system. Because they decrease the orbital energy without changing the orbital angular momentum, the radial tides must diminish the eccentricity of the relative orbit. The net change in eccentricity, due to both the tidal torque and the radial forces, may be shown to be positive if the planet rotates much faster than the satellite revolves in its orbit. For constant Q, the planet's spin rate must be at least 50% faster than the satellite's mean motion in order that the eccentricity be increasing.

Up to now we have discussed the tides raised by a satellite on its planet. Tides raised on a satellite by its planet work to retard the satellite's spin (e.g., the Moon's

Electro-magneIcinducIon(electricalconducIvity)

TidaldissipaIon(viscosity)

Constrainingwatercontentandtemperatureusingbothconduc)vityand)dalQ

10/19/19 6

à LunarmantleiscoolerthanEarth’smantle,butitswatercontentissimilartotheEarth’sasthenosphere(orslightlyless).

Earth’sasthenosphere

Karato(2013)

10/19/19 7

Whywasn’twaterlostduringtheMoonforma)onfromahigh-Tgas?

Karato(2013)

gasàsolid:largewaterlossgasàliquid(melt):notmuchwaterlossWhatcontrolsthecondensaIontosolidorliquid?

Thepressureofagasdetermineseitherliquidorsolidcondensates(thephasediagram).

10/19/19 8

3424 S. Yoneda and L. Grossman

Temperature (K) = 1600 1800 2000 2200 2400

2

7 6.5 6 5.5 5 4.5 4

. 1/T(K) x 10 4

FtG. 5. The ranges of total pressure and temperature over which CMAS condensate liquids are stable in a system of solar composition. Liquid in and liquid out refer to the appearance and disappearance of liquid, respectively, upon isobaric cooling. Liq, liquid; V, vapor; Xls, crystalline phases. Other abbreviations as used previously.

1.5 atm except for a progressive shift of the equilibria to lower temperatures with decreasing total pressure, temperature gaps where liquids do not exist which widen with decreasing P " ' , and small changes in the relative temperatures of the inflection points as the sequence of appearance of solid phases with falling temperature changes slightly with decreasing P"'. Ex- amples of the latter at 1 atm are the break in the rate of fall of the A1203 content where melilite precipitates (b) at 1821 K, the crystallization of spinel at the temperature at which the liquid first disappears, and the complete melting of melilite (d) at the temperature at which the liquid reappears. At P " = 0.3 atm, the liquid first disappears when melilite, rather than spinel, crystallizes from it, with spinel forming in the absence of liquid at a temperature above that at which the liquid reappears. As p,o, falls, truncation of the increase in MgO content of the liquid with falling temperature by con- densation of forsterite occurs at progressively lower MgO concentrations, causing the low-temperature parts of the MgO and CaO curves to come closer together until, at 0.1 atm, the CaO contents of the liquids are higher than the MgO contents.

These liquids are highly non-ideal. Using the liquids which form at 1 atm as examples, the activity coefficients for CaO, MgO, A1203, and SiO2 are 3.3 x 10 ~, 0.21, 0.35, and 4.5 × 10 2, respectively, relative to pure liquid oxides at 1991 K, the temperature of initial condensation of liquid. When the liquid disappears due to precipitation of spinel and melilite at 1781 K, the activity coefficients are 7.1 × 10 4, 0.17, 0.27, and 0.11, respectively. They are 3.5 × 10 4, 5.1 × 10 2,0.19, and 0.25, respectively, when forsterite condenses at 1717 K and 1.2 × 10 -s, 9.6 × 10 3, 4.0 x 10 2, and 1.01, respec- tively, when the liquid disappears due to precipitation of fas- saite and plagioclase at 1526 K.

The variation of XAk in melilite with temperature is shown at different total pressures in Fig. 7. At all pressures, XAk is lowest in the highest-temperature melilite to form, rises

steadily with falling temperature, and reaches a maximum at a temperature just above the temperature of melilite disap- pearance. In all cases, the maximum XAk reached increases with the width of the temperature interval for melilite stability. With increasing total pressure, the latter increases up to O.l atm and then decreases, the reversal being due to the fact that melilite disappears by reaction with the gas to form either fassaite or rankinite at P'"' - 1 x 10 2 atm but by a different mechanism, dissolution in condensate liquids, at higher pres- sures. As a result, the maximum XAk increases from 0.13 at 1 atm to 0.57 at 0.1 atm and then decreases to 0.16 at 1 × 10 *' atm. The fiat top on the curve for 1 x 10 2 atm is due to the coincidental coexistence of both spinel and forsterite with melilite, making the XAk/Xc,,. ratio of the latter almost constant over a small temperature range at this ptot. The unusually large gap between the curves for 1 X 10- 2 and 0.1 atm is due to our use of the thermodynamic data of Berman (1983) for 5kermanite at P~"' ----- 0.1 atm, where liquids are stable, and of the data of Charlu et al. ( 1981 ) at lower pressures. The added stability of ~kermanite in Berman's data shifts all the high- pressure curves higher by about 15 K.

The variation of the composition of high-temperature, A1- rich spinel with temperature is shown at different total pres- sures in Fig. 8b and d. The form of the variation of the molar Fe/Fe + Mg ratio with temperature is different at different total pressures, due entirely to the pressure-dependent varia- tion of the relative condensation temperatures of the metal alloy, spinel, and forsterite. At P~'" -> 0.1 atm, the ratio is a maximum at the initial formation temperature of spinel, falls rapidly with decreasing temperature as co-condensing metal- lic iron removes Fe from the gas, then increases gradually with decreasing temperature after forsterite condensation be- gins to remove significant Mg from the gas. With increasing P'" ' , the fraction of the Fe already condensed as metal prior to spinel formation increases steadily, causing the initial mo- lar Fe/Fe + Mg ratio in the spinel to fall from 1.7 x 10 3 at 0.1 atm to 1.1 x 10 3 at 1.5 atm. In this pressure range, the minimum and final molar Fe/Fe + Mg ratios are 3 .7-4.5 x 10 4 and 5 .6-6 .3 x 10 4, respectively. At 1 x 10 2 and 1 x 10 3 atm, Al-rich spinel condenses at a higher tempera- ture than metallic iron. The resulting curves are similar to those at higher pressures except for the addition of high-tem- perature segments in which the molar Fe/Fe + Mg ratios fall gradually with decreasing temperature due to formation of MgAI204 as a by-product of Mg enrichment of melilite. In the pressure range of 1 x 10 ~ to 1 x 10 6 atm, the final Fe/Fe + Mg ratio is higher than the initial one. At high temperature, the ratio falls with decreasing temperature due to the melilite reaction, then rises after forsterite condensation begins to con- sume Mg. At 1 x 10 4 atm, forsterite and metallic iron con- dense at almost the same temperature, but iron condenses after forsterite at lower pressure. At 1 x 10 4 and 1 x 10 5 atm, the relative rate of condensation of Mg in forsterite is faster than that of Fe in the alloy, allowing the Fe/Fe + Mg ratio of coexisting spinel to continue to rise with falling temperature after metal alloy condensation. At 1 × 10 6 atm, spinel dis- appears prior to metallic iron condensation. The initial, min- imum, and final molar Fe/Fe + Mg ratios vary from 6.0 × 10 4, 4.9 × 10 4, and 7.1 × 10 4 respectively, at 1

Yoneda-Grossman(1995)

Pdisk ≈ π2 Gσ

2 ≈ 12π G

M2

R4

vaporonly

Temperatureà

LogP(atm

)

solids+vapor

liquid+vaporMoon-formingdisk

Solarnebula

gasàsolid:Solarnebula(planetformaIon)(lowP)gasàliquid:Moon-formingdisk(highP)

(becauseofthesmallspaceduetothegravityofEarth)

Supportforhigh-Pcondensa)onduringtheMoonforma)on

10/19/19 9

IsotopecomposiIonofKofthelunarrocksàcondensaIoninthehigh-Penvironment(~1MPa(~10bar))

Wang-Jacobsen(2016)

But,liquidshouldfinallysolidify.Thenallwaterwillbegone.CantheMoonbeformedbeforethecompletesolidifica)on?

Nature © Macmillan Publishers Ltd 1997NATURE | VOL 389 | 25 SEPTEMBER 1997 353

articles

Lunar accretion from an

impact-generated disk

Shigeru Ida*†, Robin M. Canup

†& Glen R. Stewart

†

* Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Meguro-ku, Tokyo 152, Japan† LASP, University of Colorado, Campus Box 392, Boulder, Colorado 80309-0392, USA

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

Although the mechanism by which the Moon was formed is currently unknown, several lines of evidence point to its

accretion from a circumterrestrial disk of debris generated by a giant impact on the Earth. Theoretical simulations

show that a single large moon can be produced from such a disk in less than a year, and establish a direct relationship

between the size of the accreted moon and the initial configuration of the debris disk.

Many models have been proposed for formation of the Moon1, butno one has succeeded in showing the formation satisfactorily. Thepopular ‘‘giant impact’’2,3 model states that a Mars-sized proto-planet hit the proto-Earth and generated a circumterrestrial debrisdisk from which the Moon accreted. This model has been favouredas it may well account for the dynamical and geochemical char-acteristics of the Moon (large angular momentum of Earth–Moonsystem, depletion of volatiles and iron). Many hydrodynamicsimulations (a smoothed particle method) have modelled theimpact process4–7. They calculated the impact between two largeprotoplanets with iron cores and silicate mantles and followed theorbital evolution of the debris after the impact for short timescales(on the order of a few orbital periods). It is found that an impact by aMars-sized body usually results in formation of a circumterrestrialdisk rather than direct formation of a clump. (This trend is mostclear in recent simulations5–7.) The disk mass is usually smaller than2.5ML, where ML is the present lunar mass (0.0123M!; M! is theEarth mass). Most of the disk material is distributed near or interiorto the radius aR of the Roche limit (,2.9R!, where R! is the radiusof the Earth) if the orbital angular momentum of the impact is 1–2JEM, where JEM is the angular momentum of the present Earth/Moonsystem. Within and near the Roche limit, the tidal force of the Earthinhibits accretional growth.

In contrast, little has been done to simulate the accretionalprocess of the Moon. The only published accretion calculation isthat of Canup and Esposito8 with a gas dynamic approach. Theyapproximated disk particles as particles in a box and tracked theevolution of the mass distribution function at individual regions ofthe disk, modelling velocity evolution, accretion and rebounding of

the disk particles. They showed that, in general, many small moon-lets are formed initially rather than a single large moon andconcluded that the simplest way to form the present-sized moonis to begin with at least a lunar mass of material outside the Rochelimit. However, in gas-dynamic calculations it is difficult toinclude non-local effects such as radial migration of the diskmaterial and global interaction between formed moons and thedisk. The importance of the radial diffusion out from the Rochelimit has been pointed out through analytical argument9.

Here we perform directly N-body simulations, which automati-cally include non-local effects, to investigate global lunar accretionprocesses. The sequence of accretion of the moon from an impact-generated disk might be as follows8,10 (see Fig. 1). Initially, the diskwould probably be a hot, silicate-vapour atmosphere/torus6,7. Solidparticles condense owing to cooling of the disk, possibly after someradial migration10. Subsequent collisions and fragmentation of theparticles would damp initially large orbital eccentricities andinclinations of the particles to moderate values in a few orbitalperiods. Our simulations start from this stage and follow thecollisional evolution to a moon(s). On a longer timescale, one ormore formed moons gradually migrate outwards by tidal inter-action with the Earth8,10, sweeping remnants. We do not pursue suchlong-term evolution here.

We present the results of 27 simulations with different initial diskconditions. We found that a single large moon, rather than multiplemoons, is usually formed at similar distance from the proto-Earth in100–1,000 orbital periods (about a month to a year). We also foundthat the final moon mass is mostly determined by a simple functionof initial total mass and angular momentum of the disk. To estimate

Figure 1 Schematic illustrationsof the formation of the Moon by a giant impact: a,

a Mars-sized body’s impact on the proto-Earth; b, a hot, silicate vapour

atmosphere/torus; c, a solid particle disk from which one or more moons

accrete; d, outward migration of the formed moon(s) by tidal interaction with the

Earth. We adopted stage c as the initial conditions for N-body simulations. The

particle disk is modelled as follows. The disk consists of solid particles with a

power-law size distribution as nðmÞdm ~ m2 pdm, where m is mass of the

particles. The surface density of the disk is given by SðaÞ ~ a2 q for

0:35aR , a , amax, where a is the semimajor axis. The orbits of the disk particles

are integrated bya fourth-order hermitian integrator16 with a hierarchical individual

time step17, calculating all gravitational interactions between the particles, in

geocentric cartesian coordinates. We take out particles from the system if they

collide with the Earth or are scattered into hyperbolic orbits. We adopt the

accretion criteria of Canup and Esposito11 (see text).

10/19/19

τ cooling

τ accretion

10

Condensedmaterials(liquidsdominate?)

Lunar-forming impacts 441

Fig. 2. Continued.

Figures 3c–3e show a mapping of final particle state(escape, orbiting, or in planet) onto the original objects(Figs. 3c–3d), and the objects just after the initial impact(Fig. 3e); here yellow–green particles are those that com-prise the final disk, red particles escape, and blue particlesare accreted by the planet. Most of the material that endsup in orbit originates from the leading face of the impactorthat was just exterior (e.g., at greater radial distance fromthe center of the target) to the primary impact interface.A region of escaping particles on the impactor just belowthis region is associated with the front edge of the initialimpact site, which from Figs. 3a–3b is shown to be highly-heated/vaporized material; this is likely a result of “jetting”(e.g., Vickery and Melosh, 1987) from the initial oblique im-

pact. Figure 3f shows the instantaneous particle temperaturesat the time step shown in Fig. 3e; only particles within a4000-km slice centered on the z = 0 plane are plotted, to-gether with vectors whose length is proportional to particlevelocity. From the velocity vectors in Fig. 3f, it can be seenthat the leading material in Fig. 3e that eventually escapeshas been significantly accelerated as a result of the initialimpact (e.g., the highest magnitude velocity at this time is∼ 14 km/sec, vs. an impact velocity ∼ 9 km/sec).Comparison of Fig. 3d with the temperature map in

Fig. 3b shows that the impactor material that eventuallycomprises the orbiting disk is primarily the least thermallyheated of all of the material originally in the impactor, hav-ing for the most part avoided direct impact with the pro-

Moon-forming disk High P (high mass density) à condensation to liquids and ( 100 y, 1-100 y) à a large fraction of materials accrete as liquids à little depletion in volatiles Proto-solar nebula Low P (low mass density) à condensation to solids [and ] à high degree of depletion in volatiles

😅

Earth

Moon

(Zhangetal.,2012)

The“isotopiccrisis”Verysimilarisotopiccomposi)ons(e.g.,Ti,Si,O)

10/19/19 11

ε 50Ti = 50Ti / 47Ti( )sample / 50Ti / 47Ti( )standard −1⎡⎣ ⎤⎦ ×104

-2 -1 10 2 3 4 5 6

meteo

rites

IsotopiccomposiIonsaredifferentamongdifferentmeteorites.But,isotopiccomposiIonoftheMoonisverysimilartothatofEarth.

carbonaceouschondrite

enstaItechondrite

ordinarychondrite

achondrite

10/19/19 12

Moon

Earth

Khanetal.(2006)

DifferentFeOcontent

Moon

IsotopeàverysimilartoEarth[FeO]àhigherthanthatof(average)EarthHowcanweexplainboth?[FeOcontentissueisusuallyignored]

Earth

ßHighFeOcontent Melosh(2014)

Whatdoweneedtohaveaderacollision?MassbalanceandtheisotopicraIouponagiantimpact

10/19/19 13

εM =xy( )Mxy( )E

−1 =ξ α1

α2− β1β2( ) ε1−ε2( )

β1β2

xy( )2xy( )E

⎛

⎝⎜⎜

⎞

⎠⎟⎟+ξ

⎡

⎣⎢⎢

⎤

⎦⎥⎥

α1α2

xy( )1xy( )E

⎛

⎝⎜⎜

⎞

⎠⎟⎟+ξ

⎡

⎣⎢⎢

⎤

⎦⎥⎥

≈ fE − fM( ) ε1 − ε2( )

fE,M : target fraction

for E(Earth),M (Moon)

ε1,2 =xy( )1,2xy( )E

−1 e.g.,50Ti47 Ti( )

1,250Ti47 Ti( )

E

−1⎡

⎣⎢⎢

⎤

⎦⎥⎥

isotopicratioof“1”:target “2”:impactorrelativetothatofEarth

WeneedtohaveasmallfE-fM

toexplainsmallforlarge

εMε1 − ε2( )

10/19/19 14

Howtoexplaintheisotopicsimilari)es?theisotopiccrisis

“classicmodel”:obliquecollisionHartmann-Davis(1975),Cameron-Ward(1976)Benzetal.(1986),Canup(2004)à  largedistorIonofanimpactorà  theMoonmainlyfromtheimpactor

Ćuk-Stewart(2012)

Canup(2012)

SeeksoluIonsbychangingthe“mechanics”ofcollisionà DifficulIesinexplainingthelargeangularmomentum(andFeOcontent)à They(needto)invokerareevents(howprobable??)

“newmodels”:head-oncollision

Ćuk-Stewart(2012)

ProblemswiththeĆuk-Stewartmodel1.  Onlyinasmallparameterspace,canonehavethecomposiIonsimilartoEarth(bychance?).2.  PredictsFeOcontentinconsistentwiththeobservaIon.3.  Angularmomentum?

10/19/19 15

10/19/19 16

Canup(2012)

ProblemswiththeCanup(2012)model1.  OnlyinasmallparameterspaceonecanhavecomposiIonsimilartoEarth(bychance?).2.  PredictsFeOcontentinconsistentwiththeobservaIon.3.  Difficulttoexplainthelargeangularmomentum

Whatcontrolsthecomposi)onoftheMoonformedbyagiantimpact?

Fateofejectedmaterialsaderanimpact

10/19/19 17

A:escapeB:orbiIngEarthàMoonC:re-impact

CollisionejectsmaterialsàmaterialsejectedtotherelaIvelyhighlevel(andvelocity)willbecometheMoon

impactor

target(proto-Earth)

Melosh-Soner(1986)

Togetmoreproto-Earthmaterialstotheorbit,oneneedstohaveamechanismtoheattheproto-Earthmorethanthetarget

x = hR⊕

Stevenson(1987)

ShockheaIngàgasàexpandàlargeàmorechancetogetintotheproto-EarthsurroundingorbittobecometheMoonà Isthereaphysicalmechanismtoheatthetarget()theproto-

Earth)morethantheimpactor?àthermodynamicsofmaFer10/19/19 18

x = hR⊕

x = hR⊕

collisionàhea)ng:materialdependent

•  Inallpreviousstudies,thesamematerialsproperIes(equa)onofstate)wasusedfortheproto-Earthandtheimpactor(Theia).

•  Proto-Earthislikelycoveredbyamagmaocean(liquids),butanimpactorislikelysolids.

•  SolidsandliquidshaveverydifferentequaIonofstate(nextslide;Jing-Karato,2011).àdifferentdegreeofheaIng

10/19/19 19

10/19/19 20

Uniquecompressionalproper)esofliquidsà Whenaliquid(magmaocean)collidesasolidbody

à  theliquidwillbeheatedmorethansolid.

Bulkmoduliofcomplexliquids(silicate,oxidemelts)havelirlecorrelaIonwiththoseofcorrespondingsolidsàlirleroleofchemicalbonding(internalenergy)

Grüneisenparameterdecreaseswithcompressioninsolids,butitincreaseswithcompressioninliquidsàintenseheaInguponcompression

(modifiedfromJing-Karato(2011)) (Mosenfelderetal.(2007))

bridgmanite

γ = ∂logT∂logρ( )ad

10/19/19 21

Uniquecompressionalproper)esofliquidscont.RIVERS AND CARMICHAEL' ULTRASONIC STUDIES OF SILICATE MELTS 9267

2.50

2.25

2.00

o 1.75

1.50

1.25

1.00.

ld 10 -4 10 -3 10 -: 10 -1 10 ¸ 101 10: 10 3

COTs Fig. 10. Ratio of the sound speed to the low-frequency (relaxed) sound speed as a function of roz s. The line segments

join data for the same composition at different frequencies and temperatures. The smooth line is the response of a fluid with a single relaxation time and R = 3.84.

mum value of sit. Differentiating (29) above with respect to for

and equating to zero yields

(røZ)m"x = (R + 1) •/2 (32) Substitution yields

r•R

(0•'•)max = 2(R + 1) •/2 (33) The data in Figure 11 indicate that (•;t)m,x = 1.4. Solving (33)

yields R = 1.4 also.

Individually, Figures 10 and 11 would seem to indicate that

our dispersion and absorption data for silicate melts can be

modeled with a single relaxation time equal to O.010sfls. The

width of the "band" around the theoretical curve is equivalent

to a factor of 2-3 in the shear viscosity, and this is equal to the

estimated uncertainty in 0s, particularly for the very high vis- cosity melts which most influence the relaxation model.

However, the values of R which one determines from the

two types of data are very different, indicating failure of the

single relaxation time mechanism. The value R = 3.8 from the

dispersion data would predict (•;t)m,, = 2.7, nearly twice the

observed value. Similarly, the value R = 1.4 from the absorp-

tion data predicts c•/c o = 1.5, much less than the observed value.

Either the assumption that z v = r s and/or the assumption

that either is really a single relaxation time must be incorrect.

In their detailed study of B20 3 melt, Macedo and Litovitz

[1965] and Tauke et al. [1968] showed that the data below

1073 K required a distribution of relaxation times which was

Gaussian in logarithmic frequency. The effect of such a spec-

trum of relaxation times is to broaden both the dispersion and

absorption curves. It will also lower the value of (0d0m• be-

cause at any given value of for not all of the relaxation

"modes" are equally active. The value of c•/c o is not affected,

since at very high frequencies all of the modes will be "frozen OUt."

Figures 10 and 11 are quite useful for predicting the acous-

tic behavior of silicate melts as a function composition, tem-

perature, and frequency, For instance, if one is interested in 10-Hz seismic waves, then one will obtain "relaxed" sound

velocities when for = 2r•lOflsO.010 s < 0.1 (Figure 10). For a

melt with ils=7 x 10 -• Pa -•, then 0s must be less than 2 x 10 9 Pa s, or 2 x 108 P. Only a very dry obsidian melt

would display dispersive behavior at these frequencies. The

absorption per wavelength of such a wave would be very close to zero for melts less viscous than this, and there will be very little attenuation of seismic P waves in silicate melts.

CONCLUSIONS

We have shown that the relaxed ultrasonic sound speeds of silicate melts can be modeled as a linear function of the

volume fraction of the component oxides with an error of

about 4%. The sound speed coefficients decrease with in- creased cation radius for the alkali and alkaline earth oxide

components.

While there is substantial variation in the derived iso-

thermal compressibilities with composition for the range of

liquids we have studied, the variation in natural compositions

is much more restricted. For instance, we predict (using the

values in Table 8) that a picritic melt with a density of 2.71

Rivers and Carmichael (1987)

frequency-dependentsoundvelocityàcompressioninvolvessomeviscous(Ime-dependent)processes

10/19/19 22

Compressionofsolidschangesmostlytheirinternalenergy.Resistanceforcompressionofcomplexliquidshasnocorrelationwiththatofcorrespondingsolids.à Compressionofliquidschangesentropy

P = − ∂F

∂V( )T = − ∂U∂V( )T +T ∂S

∂V( )T

à uponcompression,muchofthefreeenergychangeischangeinentropyà highdegreeofheatingà liquidsexpandandgototheorbit(toformtheMoon)

liquid

solid

solids

liquids

Asimpleanaly)calmodel(a“flatEarth”model)

10/19/19 23

Karato(2014)

heaIngofliquids>>heaIngofsolidsà morematerialsgototheorbitfromthemagmaoceanà  theMoonismainlyfromthemagmaoceanoftheproto-Earth?

dT = − TγV + 1

2CυP − Po( ) + Vo −V( ) dPdV⎡⎣ ⎤⎦{ }dV

γ = ∂logT∂logρ( )ad

10/19/19 24

(a)  convenIonalmodel:impactorisdistortedbyanobliquecollisionàmostmaterialsfortheMoonarefromtheimpactor

(b)  withamagmaoceanonthetarget(proto-Earth)àmostofejectedmaterialsarefromthemagmaoceanàexplaintheisotopicsimilarity,FeO-enrichment(togetherwithlargeangularmomentum)?

Karato(2014)

BeyondtheflatEarth3-DnumericalmodelingusingamodifiedSPH*

(K(京)-computer,RIKEN,Kobe,Japan)

10/19/19 25

*:AstandardSPH(SmoothedParIcleHydrodynamics)codecannottreatadensitydisconInuityproperly.à“DISPH”(density-independentSPH)

Hosonoetal.(2019)

P-TcondiIonsaxeragiantimpact

10/19/19 26

ConsequenceofacollisiondependsstronglyonEoS(equa)onofstate)

10/19/19 27

Solid+gasEoS(TillotsonEoS)

Mie-GrüneisenEoSformagmaocean

Hard-SphereEoSformagmaocean

DISPH

StandardSPH

(Hosonoetal.2019)

10/19/19 28

Target(proto-Earth)materialsformalargefracIonoftheMoon.

(Hosonoetal.2019)

Angularmomentum/(M-Eangularmomentum)

Impactvelocity

/escapevelocity

10/19/19 29

AsmallfE-fMisneededtoexplainthesimilarisotopiccomposiIonbetweentheMoonandEarth.InaconvenIonalmodelwithanobliquecollision(thatexplainslargeangularmomentum),fE-fMistoolargeà  Needstoinvoke“unusual”collisioncondiIons.Withamagmaocean+improvedSPH(DISPH),fE-fMcanbereducedsubstanIally.àthecomposiIonandtheangularmomentumoftheMooncanbeexplainedmorenaturally.

Composi)onoftheMoon(andEarth)axeragiantimpactthatproducesrequiredangularmomentum

(Hosonoetal.2019)

Whattypeofimpactors?Acomparisontoisotopicobserva)ons

10/19/19 30

l  ForaconvenIonalmodel,impactorhastobeverysimilartoEarth.l  Forthepresentmagmaoceanmodel,abroaderrangeofmaterials

areacceptableasanimpactor.àTheMoonasobservedcanbeexplainedmorenaturally.

(Hosonoetal.2019)

10/19/19 31

FeOgoesmoretothemelt(butlirlechangeinisotopiccomposiIon)à magmaoceanwillhavehigherFeOthanthebulkofEarthà Iftheejectedmaterialsaremostlyfromthemagmaocean,thisexplainsthehighFeOoftheMoon

ThemagmaoceanmodelalsoexplainshighFeOoftheMoon.

Conclusions•  Many“puzzles”ofthelunarcomposiIoncanbeunderstoodasa

naturalconsequenceoftheMoonforma)on(giantimpactmodel)iftheimportanceofliquidsisincludedinthemodel.

•  IsotopiccomposiIonsandFeOcontent:Agiantimpactàmagmaoceanmaterialsontheproto-Earthmaterialsareheatedmorethantheimpactoràmajorityofthedisk(tobecometheMoon)wasthemagmaoceanmaterialsàisotopicsimilarity,FeOcontentdifference.

•  “wetMoon”:Whenthehotdiskgascooled,condensa)onoccurredtoliquids(nottosolids)duetohighPofthedisk(~1MPa)àonlysmalllossofvola)les.

10/19/19 32

What’snext?–  HowodenistheMoon-typesatelliteformed?

•  IstheEarth-Moonsystemso“rare”thatEarth-likeplanetwillbeveryunique(“RareEarth”hypothesis)?

•  Some)mes,mantleofthetargetplanetisstrippedoff(Mercury).–  Whenisasatelliteformed?Whendoesmantlematerialsescape?

–  OurmodelexplainisotopicsimilariIes(Ti,Si,O---).Butsomeisotopesshowdifferences(e.g.,Zn,K).

•  WhatdotheytellusabouttheMoonformaIon?

–  Explainthemajorelementchemistry(FeO,Al2O3,CaO)–  Morecompleteequa)onofstate(limitofthehard-spheremodel):thehard-sphereequaIonofstatewillnotworkatveryhighdegreeofcompression.

10/19/19 33

10/19/19 34

10/19/19 35

geology(petrology)[parIalmelIng] Solid

(doen’tlikevolaIles)

gas

Liquid(lovesvolaIles)

MoonformaIon(cosmochemistry) (mostof)

cosmochemistry

Earth(terrestrialplanets)lostmostofvola)lesduringforma)on,buttheMoondidnotlosemuchwater:why?

gassolid

liquid

pressure

temperature

10/19/19 36

NotmuchwaterlossduetothecondensaIontoliquid(majorwaterlossduetothecondensaIontosolid)

Karato(2013)

total disk(Moon) target(Earth) escapingparIcles

impactor(rock)impactor(Fecore)targetmagmaocean

EvoluIonofcomposiIonofthediskduringanimpact

(Hosonoetal.2019)

•  Explainsthelargeangularmomentum+chemistryoftheMoon(lackofaFe-richcore)

•  Giantimpactmodelisques)onedbytheresultsofmoderngeochemicalmeasurements.– Unexpectedobserva)ons:

(1)“wet”Moon?(2)verycloseagreementin(most)isotopicraIos (higherFeOcontent)

10/19/19 38

Agiantimpactmodel

BackgroundPlanetaryforma)onandforma)onoftheMoon

10/19/19 39

16 V.S. S A F R O N O V A N D E . L . R U S K O L

.~,~'~". . .... :,-~.. - :: .. ,.,. ," .: :z -. ~.. . . ' ~-: '~.~~

.-:, i:~~:.~'__:;~; ," .':-.~-.-~ ;:«~, -:icl._ ù , . . - .: : ~ ~ S : ~ » " - : ~ ~ : " ~ : ": :¢~.;~:.~',~.

~ . ' - " ~ i . . . . . . '''4":~'r~~:~»''':''¢~';':'«'z'72:'~'~~l~~'~:'¢~::'5~~"

~ ~ ~ ~ . . . , ~ ~ ¢ ~ - f ~ - a ~ « . . . . . . . . . . . . . . ,%

F i g . 1. S e e t e x t .

like that described above, only conditions for gravitational instability in the subdisk were more severe. Dispersion equation (1) can be applied to the dust subdisk if we take into account the interaction of particles with the gas. There were four types of such interactions: (a) due to turbulent stirring of the subdisk, particles

k2v 2 acquired random velocities vp, (therefore term k2c 2 should be changed to pz,, (b) the term taking into account the resistance of gas to a perturbation motion of particles n/te should be added where te = v/(dv/dt) is the response time to the gas drag force, which depends on sizes of bodies and on their velocities relative to gas, (c) initially gas perturbs together with particles, but its perturbation does not develop and it returns soon to equilibrium state. It could be carried by the dust only at an implausibly high dust density Pp/Pa > 106" Therefore, without appreciable error perturbations in the gas can be neglected, (d) gas lag from the Keplerian rotation caused a radial rnotion of particles toward the sun, which depended on their

StarformaIon:gravitaIonalcollapseofamolecularcloudà  heaIngà  formaIonofahotnebulargasdiskà  cooling(byradiaIon)à  condensa)onàdustformaIonà  gravitaIonalinstabilityà “planetesimals”à planetesimalsizeincreasesà bigonesgethot(amagmaocean)à smallonesremaincold()à collisionofbigbodies(giantimpacts)à hotgasàcoolingàcondensa)onà theMoon

Mc ≈ 2MMars

Hayashi (林忠四郎) and Safronov

KeyfeaturesoftheMoon

•  TheMoonisarelaIvelylargeplanetarybody(~1/4oftheEarthsize,~1/100oftheEarthmass)yetcomposiIonisnearly

homogeneous(verysmallcore),andmademostlyofrocks.

•  Mostofotherplanetarybodiesofthissizearedifferen)ated:mantle-corestructure(e.g.,parentbodiesofironmeteorites).

•  FormedinthelaterstageofplanetaryformaIon(~50-60Ma).•  TheEarth-Moonsystemhaslargeangularmomentum.àHowcanweexplainrockycomposi)onoftheMoon(andthelargeangularmomentum)?[needtounderstandhowcomposiIoniscontrolledduringthecomplexprocessesofMoonformaIon]

10/19/19

40

ModelsoftheMoonforma)on

10/19/19 41

Fissionmodel Capturemodel Agiantimpactmodel

Darwin.jpeg (JPEG Image, 268 × 326 pixels) http://www-history.mcs.st-and.ac.uk/BigPictures/Darwin.jpeg

1 of 1 9/19/12 11:55 AM

GeorgeDarwin(1845-1912)

Hartmann-Davis(1975)Cameron-Ward(1976)

DynamicallynotfeasibleChemicallynotfeasible