Icarus 211 (2011) 97–100

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Icarus

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Lunar shape does not record a past eccentric orbit

Matija Cuk ⇑Department of Earth and Planetary Sciences, Harvard University, 20 Oxford Street, Cambridge, Massachusetts 02138, United States

a r t i c l e i n f o

Article history:Received 11 May 2009Revised 23 August 2010Accepted 27 August 2010Available online 15 September 2010

Keywords:Rotational dynamicsMoonSatellites, FormationSatellites, DynamicsSatellites, Shapes

0019-1035/$ - see front matter � 2010 Elsevier Inc. Adoi:10.1016/j.icarus.2010.08.027

⇑ Fax: +1 617 495 7093.E-mail address: cuk@eps.harvard.edu

a b s t r a c t

The Moon has long been known to have an overall shape not consistent with expected past tidal forces. Ithas recently been suggested (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652–655)that the present lunar moments of inertia indicate a past high-eccentricity orbit and, possibly, a pastnon-synchronous spin–orbit resonance. Here I show that the match between the lunar shape and the pro-posed orbital and spin states is much less conclusive than initially proposed. Garrick-Bethell et al. (Gar-rick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652–655) spin and shape evolution scenariosalso completely ignore the physics of the capture into such resonances, which require prior permanentdeformation, as well as tidal despinning to the relevant resonance. If the early lunar orbit was eccentric,the Moon would have been rotating at an equilibrium non-synchronous rate determined by it eccentric-ity. This equilibrium supersynchronous rotation would be much too fast to allow a synchronous spin–orbit lock at e = 0.49, while the capture into the 3:2 resonance is possible only for a very constrained lunareccentricity history and assuming some early permanent lunar tri-axiality. Here I show that large impactsin the early history of the Moon would have frequently disrupted this putative resonant rotation, makingthe rotation and eccentricity solutions of Garrick-Bethell et al. (Garrick-Bethell, I., Wisdom, J., Zuber, M.T.[2006]. Science 313, 652–655) unstable. I conclude that the present lunar shape cannot be used to sup-port the hypothesis of an early eccentric lunar orbit.

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1. Introduction

The Moon has a pronounced tri-axial shape, the origin of whichhas been long debated. The most recent and prominent modelexplaining the lunar shape has been that of Garrick-Bethell et al.(2006, hereafter GWZ). However, their model suffers from serioustheoretical faults and internal inconsistencies, which will be de-tailed in this paper. I will argue that the putative high lunar eccen-tricity is therefore irrelevant to explaining lunar shape, andtherefore we cannot make inferences about past lunar eccentricityon the basis of its shape.

The origin of the lunar shape is an unresolved question. Thereare two basic possibilities for the origin of lunar shape: a frozen-in tidal deformation and stochastic irregularity due to non-uniformmass distribution (see Lambeck and Pullan (1980) for review ofprior literature). It is well known that non-uniform mass distribu-tion affects the higher order harmonics of lunar shape (Lambeckand Pullan, 1980; Konopliv et al., 1998), and extrapolation fromthese higher orders in the lunar gravity field can give us an esti-mate of the expected irregularities in the leading quadrupole termsof the gravity field. While some authors think that these irregular-ities are sufficient to explain the Moon’s overall shape (Goldreich

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and Toomre, 1969; Lefftz and Legros, 1993; Bills and Lemoine,1995), others believe that past tidal deformation has contributedto the present lunar tri-axiality (Lambeck and Pullan, 1980). HereI will not address the question whether the data suggest a tidalcontribution to lunar shape, but will argue that the model ofGWZ advocating a fossil tidal bulge is physically inconsistent.

Unlike most small moons, which have significantly irregularshapes supported by material strength, large satellites are oftenconsidered to be in hydrostatic equilibrium with the gravitationaland inertial forces. Tri-axiality of a body is measured through prin-cipal moments of inertia A–C (where C > B > A). More sensitivedimensionless parameters b = (C � A)/B and c = (B � A)/C are oftenused to describe the deformation of a near-spherical body likethe Moon (Soler, 1984; Bills, 1995). A strengthless satellite on a cir-cular orbit and in synchronous rotation will have the ratio c/b = 0.75, or equivalently1 (2C � A � B)/(B � A) = 5/3 (Hubbard andAnderson, 1978). Such a satellite should be somewhat prolate, withthe difference between the two larger principal moments of inertiabeing about a quarter of that between the largest and the smallest.Note that in a stable synchronous rotation, axes associated withthe three principal moments A–C are parallel to the radial, tangentialand normal directions, respectively. The amplitude of b and c isdetermined by orbital separation, component masses and secondary

1 These expressions are equivalent if C � B� B, C.

98 M. Cuk / Icarus 211 (2011) 97–100

size. The order of magnitude of b and c is roughly consistent with ti-dal forces at the Earth–Moon distance of 22–30 Earth radii (the pres-ent distance is about 60 Earth radii). Since the Moon is known tohave migrated away from Earth due to tidal dissipation, it has beenproposed that the tidal deformation might have been frozen-in whenthe Moon was at this distance (Lambeck and Pullan, 1980).

However, the Moon has c/b = 0.36, which means that it is cur-rently more oblate than we would expect from tidal deformation.In other words, a ‘‘frozen” tidal equilibrium figure at any pastEarth–Moon distance cannot match the observed values of b andc, assuming that the Moon had synchronous rotation and a circularorbit. Lambeck and Pullan (1980) suggested that the lunar shapecould have originated as a combination of a ‘‘frozen in” past tidaldeformation and stochastic irregularities. Since about 2/3 of thedeformation can be explained by tidal deformation, Lambeck andPullan (1980) argued that the remaining degree-2 departures fromequilibrium are more consistent with the ‘‘noise spectrum” foundfor higher order harmonics. This hypothesis, however, suffers fromphysical inconsistencies, as pointed out by Stevenson (2001). Lam-beck and Pullan (1980) require that the largest moment C (i.e., theone around the polar axis) is larger than the value dictated by tides,while the intermediate moment B (around the shortest equatorialdiameter) should be somewhat smaller, due to mass concentrationirregularities. For simplicity, we assume that the third moment A isunchanged. But this would require the freezing-in of the tidalshape while the Moon is in a synchronous state with the axis defin-ing the intermediate (B), rather than with the axis defining thelowest moment of inertia (A) pointing toward Earth. Stevenson(2001) pointed out that such a state would be unstable, as a 90-de-gree rotation would settle the Moon into a configuration with low-er potential energy. Therefore, mass distribution inhomogeneitiespredating a freeze-in cannot be used to explain why the principalmoments A and B differ more than expected for a body that solid-ified while in tidal equilibrium.

Recently, Garrick-Bethell et al. (2006) have suggested a novelsolution to the lunar shape problem, relaxing the constraint of a cir-cular lunar orbit. An early eccentric lunar orbit was previously sug-gested by Touma and Wisdom (1998) as a part of their detailedmodel of the Moon’s orbital evolution. GWZ claim that, if lunareccentricity is significant, its effect on tides is equivalent to Earth fol-lowing an epicycle in a reference frame rotating with the Moon.Since Earth’s tides are now acting from more than one direction (rel-ative to the body of the Moon), these epicyclic librations will de-crease the difference between the two smaller moments of inertia(A and B), assuming the lunar shape was in equilibrium with theaverage tidal forces.

GWZ also suggest that the Moon might have been in a spin–or-bit resonance other than synchronous (with a Mercury-like 2:3resonance being their favored solution). Higher-order spin–orbitresonances can produce an average tidal equilibrium shape thatwill differ from the one expected for synchronous rotation at thesame eccentricity. However, GWZ do not furnish a satisfactory res-olution of the puzzle of lunar shape. In following sections, I willshow that their model is both internally inconsistent and is a lessconvincing fit with the data than they initially claimed.

2. De Facto linear dependence of the lunar moment differences

In GWZ, the authors state explicitly that the parameters b, c andC20 = (2C � B � A)/(2MR2) (C20 quantifies oblateness; M and R arethe Moon’s mass and radius; for the Moon C20 = 2.034 � 10�4)can be considered independent of each other. It is easy to showthat C20 is a function of b, c and principal moment of inertia C:

C20 ¼Cð2b� cÞ

2MR2 � Cb

MR2

b� cbþ 1

� �ð1Þ

The second term is quadratic in small parameters (b = 6.315 � 10�4,c = 2.279 � 10�4), so it can be ignored when comparing quantitiesknown with relative precision of 10�4. It is clear that C20 can be con-sidered independent only if C is independent. Formally, C is mathe-matically independent of b and c as they are functions of all threeprincipal moments of inertia, A–C. However, relative variation ofthe scaled moment C/(MR2) is much smaller than that of the mo-ment differences b and c when considering shapes of realistic lu-nar-sized bodies dominated by gravity. While b and c can differby orders of magnitude depending on the degree of a body’s tri-axi-ality, C/(MR2) typically varies between 0.3 and 0.4, depending moreon the size of the body’s core, rather than its shape. C/(MR2) is closeto 0.4 for bodies with small or no cores (i.e. with a nearly-uniformdensity). GWZ explicitly assume that the Moon is a homogeneousbody, so C/(MR2) = 0.4 + O(10�3) is implied.

GWZ calculate the semi-major axis a and eccentricity e thatproduce the measured C20, b and c for various postulated rota-tional states. When they present their results (GWZ, Figs. 1 and2), they plot three lines, each representing combinations of aand e for which a calculated value for b, c or C20 matches the ob-served value. But as we have just seen, their theory implies C20

given by Eq. (1) using C/(MR2) = 0.4. As the real value of C/(MR2)is 0.393 (due to the small core of the Moon; Williams et al.,2001), C20 predicted by this theory will always be within 2% ofthe correct value whenever b and c are matched to their observedvalues (Eq. (1) gives C20 = 2.07 � 10�4 using C/(MR2) = 0.4). In or-der to show that the third solution would not always intersectthe other two, GWZ plot a curve for C20 that is 80% of the presentone, and this line does not intersect the other two curves. How-ever, for this value of C20 to be consistent with those of b andc, the Moon would have to have C/(MR2) = 0.32, which implies alarge core – explicitly contravening the assumption of uniformdensity made by GWZ.

While the lack of independence of the third parameter does notinvalidate the solutions obtained by GWZ, it makes the match ofthe lunar shape to their proposed tidal fields appear less extraordi-nary. I conclude that the current shape of the Moon has no partic-ular affinity to the solutions proposed by GWZ, and that suchsolutions could be found for any value of c and b that satisfy0 < c/b < 0.75.

3. Capture into a spin–orbit resonance

GWZ find three past rotation states that produce a lunar equi-librium shape matching the observed one. One is a high-eccentric-ity orbit with synchronous rotation. The other two are higher order(3:2 and 1:2) spin–orbit resonances. GWZ express a preference forthe one of the tidal distortion solutions within the 3:2 resonance(similar to that of Mercury) as it requires lower eccentricity thanall other solutions (including the synchronous; GWZ, Fig. 1). How-ever, Mercury’s spin–orbit resonance is possible only because of itspermanent quadrupole moment (i.e. tri-axiality). GWZ use the ti-dal forces experienced by the Moon in a spin–orbit resonance tocreate the tri-axiality, but tri-axiality is required to maintain theresonance. This poses a ‘‘chicken and egg” problem. For the sakeof argument, I will assume that the Moon could have had somepermanent deformation before the final freeze-in of its shape. I willalso assume that this original deformation was two orders of mag-nitude smaller than the present one, and therefore contributednegligibly to the Moon’s shape. Both of these assumptions areneeded in order for the results of GWZ to be applicable to theMoon.

Lunar eccentricity could have been generated very soon afterthe Moon’s formation, with one possible source being the solarevection resonance (Touma and Wisdom, 1998). As this resonance

M. Cuk / Icarus 211 (2011) 97–100 99

was likely encountered within millennia of lunar formation, no sig-nificant solidification could have taken place before this putativeeccentricity-exciting event. I will also assume that the lunar eccen-tricity was damped since the breaking of the evection resonance(cf. Touma and Wisdom, 1998). In order for the Moon to be cap-tured in the evection resonance, Touma and Wisdom (1998) as-sume Earth’s tidal quality factor Q to be 104, which is likely to beunrealistic (see Section 5).

A satellite that is strengthless has no memory of its past defor-mation, so unless it is on a strictly circular orbit, once tidally de-spun it will be rotating with a frequency that is notcommensurable with its mean motion (Goldreich and Peale,1966; Greenberg and Weidenschilling, 1984):

X ¼ n 1þ 192

e2� �

ð2Þ

where X is the rotation rate and n is the mean motion. A 3:2 reso-nance can be reached only if e = 0.229, unless the Moon had a per-manent quadrupole moment. The same is true for synchronousrotation for any e > 0.

Alternatively, if the Moon possessed a permanent quadrupolemoment sufficient for resonance capture, it could have enteredthe resonance and experienced tidal freeze-in described byGWZ. This scenario requires that the lunar shape must be a com-bination of stochastic and tidal contributions, as in the suggestionof Lambeck and Pullan (1980), if not necessarily in the same pro-portions. The size of the initial permanent deformation (quanti-fied by c) needed for the Moon’s capture into these spin–orbitresonances is about c ’ 10�8 � 10�6 (Murray and Dermott,1999), much smaller than the present value. This would requireonly a minor component of the Moon’s present shape to originatefrom mass inhomogeneities. While this scenario still suffers fromsomewhat questionable assumption that one can have permanentdeformation before the freeze-in (cf. Stevenson, 2001), we will as-sume that this state of affairs was possible at some time in lunarhistory.

This two-component shape scenario requires that the equilib-rium spin given by Eq. (2) be within the spin–orbit resonance inquestion, or the capture could not have occurred. If the orbitaleccentricity (and therefore the equilibrium spin rate given byEq. (2)) changed over time, capture into resonance requires thatthe equilibrium spin at least passed through the resonance atsome point. Additionally, the eccentricity required to producethis spin rate should ultimately be consistent with the value ob-tained by GWZ from the shape assuming tidal freeze-in. How-ever, this scenario would not work for most of GWZ solutions.The synchronous solution implies X = n at e = 0.5, and the high-eccentricity 3:2 solution requires X = 1.5n at e = 0.6, but theequilibrium spin rates (Eq. (2)) for those eccentricities are signif-icantly faster than the resonances in question. Therefore thesecombinations of X and e could not have been reached by captureinto spin–orbit resonance at optimal eccentricity and subsequente-damping. The lower-e 3:2 solution (X = 1.5n at e = 0.17) mighthave been achievable if the resonance capture happened at high-er e ’ 0.23 which was subsequently damped to e = 0.17 beforethe Moon’s figure solidified (while the spin–orbit resonancewas maintained).

Note that Eq. (2) assumes MacDonald (1964) tides and fre-quency-independent tidal lag (i.e. constant tidal quality factor Q).Using Darwin tides with a constant time lag, one obtains a similarexpression with a e2 coefficient of 6 instead of 9.5 (Hut, 1981; Fer-raz-Mello et al., 2008). As lunar Q appears to be only very weaklydependent on frequency (Williams et al., 2001) I prefer using aconstant-Q solution, although my conclusions here do not changeif a constant time lag is used.

4. Stability of resonant rotation against impacts

Another major shortcoming of GWZ argument is that it does nottake into account the effect of large impacts on the Moon, that canaffect its rotation. Since GWZ assume that the freezing-in of the lu-nar shape took about 100 Myr, it is important to ask if the spin–or-bit resonances in question could have been maintained againstimpacts. Note that, once out of resonance, the lunar spin wouldevolve toward the optimal non-synchronous rotation rate deter-mined by the eccentricity, as discussed in the previous section.Therefore, the ‘‘precarious” 2:3 spin orbit resonance that hasevolved to eccentricities much below the optimal one could onlylast as long as no large impacts are able to break the resonancelock.

Lissauer (1985) calculated the size of impactors needed to breakthe synchronous lock for a range of Solar System satellites. He con-cludes that only the formation of a large basin (diameter ’900 km)can break the synchronous rotation of the Moon if present shapeand orbit are assumed. The mass of the required impactor in-creases linearly with higher past rotation rates, so formation ofan even larger basin would be required to break the synchronousrotation at the distance of 25–30 Earth radii (an impact requiredto break another spin–orbit resonance would be of the same orderor magnitude). Lissauer’s calculation assumes the current amountof deformation. However, as per GWZ model, the deformationbuild-up gradually in a spin–orbit resonance. Consequently, intheir scenario an impactor producing a basin smaller than900 km could have disrupted lunar rotation before the final solid-ification. If we assume that the initial spin-resonance capture oc-curred due to a small initial tri-axiality (say, c ’ 10�6, this lockcould be broken by an impact of a projectile that is an order ofmagnitude less massive than the Imbrium impactor (Lissauer,1985).

The width of the 3:2 spin–orbit resonance at a = 25RE ande = 0.17 (GWZ preferred solution) is about 60% of the width ofthe synchronous state for the same shape (Murray and Dermott,1999). Assuming original c = 10�6, the impact breaking the 3:2 res-onant state at e = 0.17 should also form at least a 500-km basin. AsGWZ suggest that the Moon spend the interval between about 100and 200 Myr after formation in this state, it is very likely that mul-tiple impacts of that magnitude would break the resonance lockover the same period. Bottke et al. (2007) calculate that aD > 900 km basin would have formed about every 10 Myr on theMoon at this time, assuming a post-accretion inner-Solar Systemleftover population of 0.05 Earth masses at ‘‘t = 0”.

If the collisionally-perturbed Moon finds itself rotating slowerthan the 3:2 resonance at e = 0.17, tidal despinning would placeit into an equilibrium non-resonant rotation with a rate of about1.27n. Presumably its rotation would become synchronous oncethe eccentricity has decreased below about 0.052 (for the presentshape) or even lower (for a more fluid body). While under certainspecial circumstances described in Section 3 the Moon might be-come captured into the 3:2 spin–orbit resonance as GWZ propose,this lock cannot be long-term stable at the required eccentricity inthe presence of heavy bombardment.

Therefore, we must conclude that the present lunar shape can-not be taken as evidence of a past high-eccentricity orbit, as all ofthe solutions proposed by GWZ are either completely unreachableby tidal despinning or unstable against basin-forming impactevents.

5. Conclusions

Here I have shown that the past lunar spin–orbit states pro-posed by Garrick-Bethell et al. (2006) as explanations for lunar

100 M. Cuk / Icarus 211 (2011) 97–100

shape could not have been stable in the long term. Very eccentricsynchronous and 3:2 states are far from the end-points of tidaldespinning for suggested eccentricities. Capture into the lesseccentric of the two proposed 3:2 states could in principle hap-pen for some special scenarios of eccentricity evolution, but thesesolutions would still be unstable against spin-up by large impacts.If the after-impact spin is slower than the resonant one, than tidaldespinning would evolve rotation into stable non-resident statesdictated by the solution’s eccentricity. I also show that the solu-tions of Garrick-Bethell et al. (2006) do not match the shape ofthe Moon as dramatically as suggested by their Figs. 1 and 2,due to the mutual dependence of the three parameters that arebeing fitted.

Once the evidence from lunar shape is ruled out, the case forpast high eccentricity lunar orbit becomes very weak. Touma andWisdom (1998) proposed the high eccentricity post-evection orbitas a route to exciting the lunar inclination through another solarresonance. However, lunar inclination could have been also pro-duced through resonant interactions of the Moon with the circum-terrestrial debris ring (Ward and Canup, 2000).

The Moon likely formed on a low-e orbit (Canup, 2004), and forvalues of the Earth’s tidal quality factor Q < 1000, Touma and Wis-dom (1998) find that the Moon does not become captured in theevection resonance and does not acquire a high eccentricity. Mosttheoretical estimates of past terrestrial Q (Bills and Ray, 1999) areconsistent with the damping of lunar e during the first part of itsorbital evolution and more recent increase (that has been ob-served; Williams et al., 2001). As most of the current lunar eccen-tricity can be generated by planetary resonances encounteredrelatively late in the system’s history (Cuk, 2007), there is currentlyno compelling evidence for a significantly higher past lunareccentricity.

Acknowledgments

I wish to thank Professors Sylvio Ferraz-Mello, Richard O’Con-nell and Jack Wisdom for insightful discussions, as well as KarthikaSasikumar for her help with the manuscript. The author is a DalyPostdoctoral Fellow at Harvard University.

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