why do we see the man in the moon?

3
Note Why do we see the man in the Moon? Oded Aharonson a,, Peter Goldreich a , Re’em Sari b a California Institute of Technology, MC 150-21, Pasadena, CA 91125, United States b Racah Institute of Physics, The Hebrew University, 91904 Jerusalem, Israel article info Article history: Received 12 July 2011 Revised 8 February 2012 Accepted 17 February 2012 Available online 27 February 2012 Keywords: Moon Moon, Interior Satellites, Dynamics abstract Numerical simulations and analysis show that the Moon locks into resonance with a statistical preference of facing either the current near-side or far-side toward Earth. The near-side is largely covered by dense, topographically low, dark mare basalts, the pattern of which to some, resembles the image of a man’s face. Although the Moon is locked in this configuration at present, the opposite one, with the current far-side facing Earth, is of lower potential energy and hence might be naively expected. Instead, we find that the probability of selecting each configuration depends upon the ratio of the asymmetry of the potential energy maxima, dominated by the octupole moment of the Moon, to the energy dissipated per tidal cycle within the Moon. If this ratio is small, the two configurations are equally likely. Otherwise, interesting dynamical behavior ensues. In the Moon’s present orbit, with the best-estimated geophysical parameters and dissipation parameter Q = 35, trapping into the current higher-energy configuration is preferred. With Q = 100 in analogy with the solid Earth, the current configuration is nearly certain. The ratio of energies and corre- sponding probabilities were different in the past. Relative crater counts on the leading and trailing faces indicate an impact may have unlocked the Moon before it settled into the present configuration. Our analysis constrains the geo- physical parameters at the time of the last such event. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction The face of the Moon that faces Earth is largely covered by dense, topographi- cally low, dark mare basalts. In some cultures, the pattern of these compositionally distinct plains is reported to resemble the image of a man’s face, which continu- ously points towards Earth in the current spin–orbit resonance. The origin of the hemispheric dichotomy itself is not investigated here; it may be the consequence of processes internal (e.g., Garrick-Bethell et al., 2010; Loper and Werner, 2002; Wasson and Warren, 1980), or external (e.g., Zuber et al., 1994; Wood, 1973; Jutzi and Asphaug, 2011) to the Moon. Internal dissipation causes the Moon to spin about its axis of maximum mo- ment of inertia and point its axis of minimum inertia towards the Earth. The axis of intermediate moment points along the orbit. This leaves the two options of the Moon facing either its current near-side or far-side towards the Earth. Both config- urations are local potential energy minima, because the energy is dominated by the second moment of the Moon’s mass distribution, while higher moments break this symmetry. Using the measured gravity coefficients (Konopliv et al., 2001), we com- pute the potential energy of the configuration as a function of the alignment angle / evaluated at the equatorial plane, U ¼ GM a þ GM a P 1 l¼2 P l m¼0 R a l P lm ð0ÞðC lm cos m/ þ S lm sin m/Þ; ð1Þ where G is the universal gravitational constant, M the mass of the Moon, R its radius, and a the orbital distance to Earth. P lm are the Legendre polynomials, C lm and S lm are the measured gravity coefficients. The result is shown in Fig. 1. The potential energy minima differ by dU 2.4 10 17 J, and the two potential energy maxima between them differ by DU 7.2 10 16 J. If the Moon were to re- verse its orientation such that the current far-side would face the Earth, the total potential energy of the system would be slightly lower than at present. The Moon is currently ‘‘stranded’’ in an energy state higher than the global minimum by an amount that is roughly R/a 1/240 times the energy barrier separating these min- ima. However, as shown below, the choice of minima depends not on the difference in minima, dU, but rather on the difference in maxima DU. 2. Dynamical evolution In order to understand how the Moon arrived at its current state, we analyze a simplified system that exhibits the relevant dynamics. The system consists of damped motion of a particle in an approximate potential, neglecting the Moon’s small obliquity, orbital eccentricity and inclination. An analogy is made with the motion of a particle in a periodic potential composed of degree two and three com- ponents, slowed by a constant frictional force. The angle / = h nt is the angle of the Earth as measured from the Moon, h is the orientation angle of the Moon in iner- tial space, n is the orbital angular frequency, and t is time. In a frame rotating at a constant rate n, the kinetic energy T ¼ 1 2 C _ / 2 and potential U ¼ 1 2 u cosð2/Þ 1 3 u cosð3/ þ / 0 Þ govern the conservative dynamics. Including the dissipative tidal torque, the equation of motion in this potential conveniently reads (Goldreich and Peale, 1966): C / þ u sinð2/Þþ u sinð3/ þ / 0 Þ¼ssignð _ /Þ; ð2Þ where the amplitudes of the degree 2 and 3 components are u ¼ 3 2 n 2 ðB AÞ and u respectively, the phase difference between them is / 0 . For small motions, the reso- nant frequency of the approximate harmonic oscillator is ffiffiffiffiffiffiffiffiffiffiffi 2u=C p . The dissipation is due to a constant torque s acting in a direction opposite to that of _ /. A, B, and C, and the two equatorial and polar moments of inertia, respectively. An arbitrary mass configuration would have an additional sin/ term arising from the degree 3 order 1 moment, but it is not needed to capture the essential dynamical behavior. The par- ticle’s potential energy oscillates while its total energy decays at a rate obtainable from the expressions above, _ T þ _ U ¼sj _ /j: ð3Þ For 1, the two minima and two maxima of the degree 2 potential are perturbed such that the energy difference between the minima is dU ¼ 2 3 u cos / 0 and between the maxima DU ¼ 2 3 u sin / 0 . The Moon’s spin undergoes an analogous motion to that of this particle. 0019-1035/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2012.02.019 Corresponding author. Present address: Weizmann Institute of Science, Rehovot 76100, Israel. E-mail address: [email protected] (O. Aharonson). Icarus 219 (2012) 241–243 Contents lists available at SciVerse ScienceDirect Icarus journal homepage: www.elsevier.com/locate/icarus

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Page 1: Why do we see the man in the Moon?

Icarus 219 (2012) 241–243

Contents lists available at SciVerse ScienceDirect

Icarus

journal homepage: www.elsevier .com/locate / icarus

Note

Why do we see the man in the Moon?

Oded Aharonson a,⇑, Peter Goldreich a, Re’em Sari b

a California Institute of Technology, MC 150-21, Pasadena, CA 91125, United Statesb Racah Institute of Physics, The Hebrew University, 91904 Jerusalem, Israel

a r t i c l e i n f o

Article history:Received 12 July 2011Revised 8 February 2012Accepted 17 February 2012Available online 27 February 2012

Keywords:MoonMoon, InteriorSatellites, Dynamics

0019-1035/$ - see front matter � 2012 Elsevier Inc. Adoi:10.1016/j.icarus.2012.02.019

⇑ Corresponding author. Present address: Weizmann76100, Israel.

E-mail address: [email protected] (O. Aharonson).

a b s t r a c t

Numerical simulations and analysis show that the Moon locks into resonance with a statistical preference of facingeither the current near-side or far-side toward Earth. The near-side is largely covered by dense, topographicallylow, dark mare basalts, the pattern of which to some, resembles the image of a man’s face. Although the Moon is lockedin this configuration at present, the opposite one, with the current far-side facing Earth, is of lower potential energyand hence might be naively expected. Instead, we find that the probability of selecting each configuration dependsupon the ratio of the asymmetry of the potential energy maxima, dominated by the octupole moment of the Moon,to the energy dissipated per tidal cycle within the Moon. If this ratio is small, the two configurations are equally likely.Otherwise, interesting dynamical behavior ensues. In the Moon’s present orbit, with the best-estimated geophysicalparameters and dissipation parameter Q = 35, trapping into the current higher-energy configuration is preferred. WithQ = 100 in analogy with the solid Earth, the current configuration is nearly certain. The ratio of energies and corre-sponding probabilities were different in the past. Relative crater counts on the leading and trailing faces indicate animpact may have unlocked the Moon before it settled into the present configuration. Our analysis constrains the geo-physical parameters at the time of the last such event.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction amount that is roughly R/a � 1/240 times the energy barrier separating these min-

The face of the Moon that faces Earth is largely covered by dense, topographi-cally low, dark mare basalts. In some cultures, the pattern of these compositionallydistinct plains is reported to resemble the image of a man’s face, which continu-ously points towards Earth in the current spin–orbit resonance. The origin of thehemispheric dichotomy itself is not investigated here; it may be the consequenceof processes internal (e.g., Garrick-Bethell et al., 2010; Loper and Werner, 2002;Wasson and Warren, 1980), or external (e.g., Zuber et al., 1994; Wood, 1973; Jutziand Asphaug, 2011) to the Moon.

Internal dissipation causes the Moon to spin about its axis of maximum mo-ment of inertia and point its axis of minimum inertia towards the Earth. The axisof intermediate moment points along the orbit. This leaves the two options of theMoon facing either its current near-side or far-side towards the Earth. Both config-urations are local potential energy minima, because the energy is dominated by thesecond moment of the Moon’s mass distribution, while higher moments break thissymmetry. Using the measured gravity coefficients (Konopliv et al., 2001), we com-pute the potential energy of the configuration as a function of the alignment angle /evaluated at the equatorial plane,

U ¼ GMaþ GM

aP1l¼2

Pl

m¼0

Ra

� �l

Plmð0ÞðClm cos m/þ Slm sin m/Þ; ð1Þ

where G is the universal gravitational constant, M the mass of the Moon, R its radius,and a the orbital distance to Earth. Plm are the Legendre polynomials, Clm and Slm arethe measured gravity coefficients. The result is shown in Fig. 1.

The potential energy minima differ by dU � 2.4 � 1017 J, and the two potentialenergy maxima between them differ by DU � 7.2 � 1016 J. If the Moon were to re-verse its orientation such that the current far-side would face the Earth, the totalpotential energy of the system would be slightly lower than at present. The Moonis currently ‘‘stranded’’ in an energy state higher than the global minimum by an

ll rights reserved.

Institute of Science, Rehovot

ima. However, as shown below, the choice of minima depends not on the differencein minima, dU, but rather on the difference in maxima DU.

2. Dynamical evolution

In order to understand how the Moon arrived at its current state, we analyze asimplified system that exhibits the relevant dynamics. The system consists ofdamped motion of a particle in an approximate potential, neglecting the Moon’ssmall obliquity, orbital eccentricity and inclination. An analogy is made with themotion of a particle in a periodic potential composed of degree two and three com-ponents, slowed by a constant frictional force. The angle / = h � nt is the angle ofthe Earth as measured from the Moon, h is the orientation angle of the Moon in iner-tial space, n is the orbital angular frequency, and t is time. In a frame rotating at aconstant rate n, the kinetic energy T ¼ 1

2 C _/2 and potentialU ¼ � 1

2 u cosð2/Þ � 13 �u cosð3/þ /0Þ govern the conservative dynamics. Including

the dissipative tidal torque, the equation of motion in this potential convenientlyreads (Goldreich and Peale, 1966):

C €/þ u sinð2/Þ þ �u sinð3/þ /0Þ ¼ �ssignð _/Þ; ð2Þ

where the amplitudes of the degree 2 and 3 components are u ¼ 32 n2ðB� AÞ and �u

respectively, the phase difference between them is /0. For small motions, the reso-nant frequency of the approximate harmonic oscillator is

ffiffiffiffiffiffiffiffiffiffiffi2u=C

p. The dissipation is

due to a constant torque s acting in a direction opposite to that of _/. A, B, and C, andthe two equatorial and polar moments of inertia, respectively. An arbitrary massconfiguration would have an additional sin/ term arising from the degree 3 order1 moment, but it is not needed to capture the essential dynamical behavior. The par-ticle’s potential energy oscillates while its total energy decays at a rate obtainablefrom the expressions above,

_T þ _U ¼ �sj _/j: ð3Þ

For �� 1, the two minima and two maxima of the degree 2 potential are perturbedsuch that the energy difference between the minima is dU ¼ 2

3 �u cos /0 and betweenthe maxima DU ¼ 2

3 �u sin /0. The Moon’s spin undergoes an analogous motion tothat of this particle.

Page 2: Why do we see the man in the Moon?

−300

−250

−200

−150

−100

−50

0

Pote

ntia

l Ene

rgy

(1018

J)

(a)

−180 −90 0 90 180−0.5

−0.25

0

0.25

0.5

φ (degrees)

Pote

ntia

l Ene

rgy

(10

18J)

E

xclu

ding

Qua

drup

ole

ΔU δU

(b)

Fig. 1. Potential energy of the Moon (Konopliv et al., 2001) as a function oforientation angle /. (a) The variations (excluding the monopole) are dominated bythe degree 2 terms. (b) Contributions from higher than quadrupole (l > 2) terms aredue mostly to degree 3. State E which precedes the higher maximum at / = 90� hasa higher energy than state W. The energy difference between the potential minimais dU, and between the maxima is DU.

242 Note / Icarus 219 (2012) 241–243

An example solution to Eq. (1) obtained by numerical integration is shown inFig. 2. The initial energy is reduced by 2ps per cycle, obtained by integrating thedissipation rate over one cycle. When the total energy drops below the maximumpotential energy the direction of motion reverses and the particle continues tooscillate and lose an energy of dE � ps between stopping points separated byapproximately p radians. Eventually the energy drops below the lower of the twopotential maxima, and the particle falls into one of the two potential minima.The choice between the two minima is governed by the energy difference betweenthe maxima, DU, and the energy dissipated per cycle between stops, dE.

We follow this simple dynamical system by direct integration of the trajecto-ries, choosing parameters which set the difference in the potential energy maximaDU. We vary the dissipation parameter s, thus varying the energy dissipated per cy-cle dE. For each resulting ratio of DU/dE we run 1000 simulations, randomizing the

−180 −90 0 90 180

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

φ (degrees)

Ene

rgy

(u)

Fig. 2. Representation of the total energy time evolution (blue), and the configuration porelevant energies for locking are indicated. (For interpretation of the references to color

initial angle and kinetic energy, always above the energy required for complete cir-culation. We call the minimum preceding (East of) the higher potential maximum E

and the minimum West of the maximum W. The resulting probability of captureinto the East minimum, PðEÞ, is plotted in Fig. 3. Several features are noteworthy.When the energy dissipation, d E, and potential maxima asymmetry, DU, are equal,the probability PðEÞ ¼ 1. This may be understood as follows. The particle passes thehighest potential maximum for the last time before stopping with an energy suffi-cient to carry out less than one, but more than one-half a cycle. It therefore capturesinto the adjacent potential minimum (E), which with our choice of potential, is thehigher of the two minima. The additional peaks in PðEÞ similarly correspond to ra-tios of DU/dE that correspond to arriving at the potential maximum with kinetic en-ergy sufficient for increasing integral number of oscillations before the total energydrops below the lower maximum. In the limit of high dissipation, stopping occurs inless than half a revolution after the last passage above one of the maxima, andhence the probability of capture into each of the minima approaches 1/2. At theother extreme, of low dissipation relative to the asymmetry energy, the probabilityfunction, PðEÞ, oscillates between 0 and 1, with an average of 1/2. As this discussionillustrates, the relevant energies are that of dissipation per cycle, and the differencein the potential maxima, not the difference in the potential minima.

An approximate analytical solution is possible. Denote the energy above thehigher of the potential maxima, at the last cycle before stopping by E0 (seeFig. 2). After stopping, / oscillates until the energy falls below the lower of the po-tential maxima, trapping the particle into one of the two minima. The energy at thelast stopping point before trapping relative to the lower potential maximum is ob-tained by subtracting the energy dissipated during an integer number of oscillationsbetween the potential barriers. For uniform random initial phase and energy, E0 isuniformly distributed. If the barriers are approximated as knife-edges, the probabil-ity of trapping into E is

PðEÞ ¼ 12

1� DUdEþ 4

DU þ dE4dE

�������

�������; ð4Þ

where sxt is the greatest integer 6 x. In order to match the results when the dissi-pation is small compared with the asymmetry energy, we correct for the reduceddissipation due to the cycles in / actually occurring between finite width potentialbarriers. In this case the cycles are shorter than p radians assumed earlier. The cor-rection may be obtained by considering the integral of the potential curve betweenthe maxima, that is the region missed by the oscillations (indicated as gray in Fig. 2).To first order in DU/u, the correction is Ec ¼ 2

3DUp

ffiffiffiffiffiffiffi2DU

u

q, a quantity which may be

added to the initial energy when computing PðEÞ to adjust the phase at the momentof locking. The correction should only be applied when the dissipation is small, i.e.when the number of oscillations between stopping and trapping is large, and hencethe accumulated phase error by the approximation substantial. Because the analyt-ical approximation depends on the difference in the potential maxima, arbitrary har-monics of a potential that contribute to this difference may be included, although thecomplete shape of the resulting potential may not. The uncorrected and correctedanalytical solutions are shown in Fig. 3.

3. Discussion

The torque, s, which despins the satellite, may be obtained (Kaula, 1964;Goldreich and Peale, 1966) by considering the Moon’s response to the degree twotidal potential parametrized by the Love number k2 and quality factor Q:

−180 −90 0 90 1800.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

φ (degrees)

ΔU

E0

δE El

Total Energy U+TPotential Energy UPhase Correction

tential (black), for a case where DU/dE = 1.54. In the magnified plot on the right, thein this figure legend, the reader is referred to the web version of this article.)

Page 3: Why do we see the man in the Moon?

0.1 1 2 3 4 5 10 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

↓ Q=35

↓ Q=100

↓Q=60

Fig. 3. The probability of the E configuration obtained from sets of 1000 Monte-Carlo simulations with numerical integration of the synchronization process (pointswith error bars), a simple analytical solution for knife-edge potential (green line),and a correction to that solution appropriate for small dissipation (purple line). Forconcreteness we choose values describing the potential of u/C = 1, � = 0.1, and/0 = p/4. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

Note / Icarus 219 (2012) 241–243 243

s ¼ �32

k2

Qn4

GR5signð _/Þ: ð5Þ

Gravity measurements indicate (Konopliv et al., 2001) k2 = 0.026 ± 0.003, whereas Qis constrained by lunar laser ranging (Williams and Boggs, 2008) to be �35. Assum-ing these values, the energy dissipated per cycle due to the integrated torque isdE � 4.2 � 1016 J, similar in magnitude to the asymmetry energy DU � 7.2 � 1016 J.The two energies are comparable, resulting in interesting locking dynamics; theMoon despins into the current configuration with a mild preference, at a probabilityof �63%. If, however, Q = 100 in analogy with Earth’s solid body, the preference ismuch stronger, with a probability for the current near-side of �97%. Using Bayes’theorem and an assumption of apriori uniform distribution of Q, it is possible to solvefor the probability distribution of Q given the current configuration, as a function ofthe probability of the configuration given Q. With these assumptions the conditionalprobabilities are simply proportional to each other. We find that the current E con-figuration and geophysical parameters are consistent with values of Q of 20 ± 17, or101 ± 17 (as well as other higher values). A value of Q � 60 is incompatible withlocking in the current state.

The early Moon was hotter, closer to Earth, and its mass distribution could havebeen different from today’s. It formed at a distance corresponding to an orbital per-iod of a few days. Much closer orbits are within the Roche limit, preventing accre-tion, and impact simulations show it is difficult to deliver sufficient material muchfurther from Earth (Canup and Asphaug, 2001). With a residue of accretional heatand a higher rate of tidal dissipation at a closer distance, the Moon’s mantle convec-tion would have been more vigorous and could have dynamically supported differ-ent geoid anomalies. The relative timing of maria emplacement and locking intoresonance is not clear. The mass asymmetry may have been enhanced after lockinginto resonance by internal/external processes (e.g. Loper and Werner, 2002; Jutziand Asphaug, 2011). Dissipation in the past depends upon the orbital frequencyand Q, which are difficult to constrain. The dissipation energy dE scales as� k2

QR5

a6 GM2E where ME is the Earth’s mass. The energy difference between the poten-

tial minima and maxima arises dominantly due to the octupole gravity moment of

the Moon. This quantity scales with C3mMMER3/a4, where C3m denotes the degree 3gravity coefficients. Hence the ratio of energies scales as

DUdE� C3m

Qk2

MME

aR

� �2: ð6Þ

In a smaller orbit, the dissipative energy term was larger and may have been com-parable to, or even exceeded the asymmetry energy.

The minimum radius of an impactor capable of knocking the Moon out of syn-chronous rotation is several 10s of kilometers, and multiple impact craters createdby bodies of this size are recorded in the lunar crust, suggesting the Moon waskicked out of resonance and relocked after its formation (Melosh, 1975; Wieczorekand Le Feuvre, 2009). It is difficult to identify or date the youngest such event,although it appears the majority occurred more than 3.8 byr ago (Stöffler and Ryder,2001; Wieczorek and Le Feuvre, 2009). A statistical estimate of the clustering of an-cient large craters indicates that the current trailing hemisphere might have beenthe leading one during the time the oldest subset of these large basins formed(Wieczorek and Le Feuvre, 2009). These basins occur early enough in lunar historythat using present day parameters for the capture dynamics is not justified. How-ever, in the event that the last such reorienting impact occurred sufficiently latein lunar history such that the Moon’s gravity, orbital period, and Love number weresimilar to today’s, then the estimates of Q given above are relevant.

We show that for the current Moon, the asymmetry and dissipative energies areclose to each other, resulting in the interesting capture dynamics. This, rather thanthe semblance of a man’s face on the near-side, may be the surprising coincidenceof our Moon.

Acknowledgments

O.A. thanks Bruce Bills for introducing him to this question, Mark Wieczorek forhelpful discussions, and two anonymous reviewers for their constructive reviews.We are thankful for the support of the Lunar Reconnaissance Orbiter Project (Grant# NNX08AZ54G).

References

Canup, R.M., Asphaug, E., 2001. Origin of the Moon in a giant impact near the end ofthe Earth’s formation. Nature 412, 708–712.

Garrick-Bethell, I., Nimmo, F., Wieczorek, M., 2010. Structure and formation of thelunar farside highlands. Science 330, 949–951.

Goldreich, P., Peale, S., 1966. Spin–orbit coupling in the Solar System. Astron. J. 71,425–438.

Jutzi, M., Asphaug, E., 2011. Forming the lunar farside highlands by accretion of acompanion Moon. Nature 476, 69–72.

Kaula, W.M., 1964. Tidal dissipation by solid friction and the resulting orbitalevolution. Rev. Geophys. Space Phys. 2, 661–685.

Konopliv, A.S., Asmar, S.W., Carranza, E., Sjogren, W.L., Yuan, D.N., 2001. Recentgravity models as a result of the lunar prospector mission. Icarus 150, 1–18.

Loper, D.E., Werner, C.L., 2002. On lunar asymmetries 1: Tilted convection andcrustal asymmetry. J. Geophys. Res. Planet 107, 131–137.

Melosh, H.J., 1975. Large impact craters and the Moon’s orientation. Earth Planet.Sci. Lett. 26, 353–360.

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Wieczorek, M.A., Le Feuvre, M., 2009. Did a large impact reorient the Moon? Icarus200, 358–366.

Williams, J.G., Boggs, D.H., 2008. Lunar core and mantle. What does LLR see? In:Proceedings of the 16th International Workshop on Laser Ranging.

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Zuber, M.T., Smith, D.E., Lemoine, F.G., Neumann, G.A., 1994. The shape and internalstructure of the Moon from the Clementine mission. Science 266, 1839–1843.