Icarus 275 (2016) 132–142

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Icarus

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

Tidal dissipation in the lunar magma ocean and its effect on the early

evolution of the Earth–Moon system

Erinna M.A. Chen a , Francis Nimmo b , ∗

a Department of Physics, University of Colorado Boulder, Boulder, CO, USA b Department of Earth and Planetary Sciences, University of California, Santa Cruz, CA, USA

a r t i c l e i n f o

Article history:

Received 21 May 2015

Revised 8 April 2016

Accepted 11 April 2016

Available online 21 April 2016

Keywords:

Earth

Moon

Satellites

dynamics

Thermal histories

Tides

solid body

a b s t r a c t

The present-day inclination of the Moon reflects the entire history of its thermal and orbital evolution.

The Moon likely possessed a global magma ocean following the Moon-forming impact. In this work, we

develop a coupled thermal-orbital evolution model that takes into account obliquity tidal heating in the

lunar magma ocean. Dissipation in the magma ocean is so effective that it results in rapid inclination

damping at semi-major axes beyond about 20 Earth radii ( R E ), because of the increase in lunar obliq-

uity as the so-called Cassini state transition at ≈30 R E is approached. There is thus a “speed limit” on how fast the Moon can evolve outwards while maintaining its inclination: if it reaches 20 R E before the

magma ocean solidifies, any early lunar inclination cannot be maintained. We find that for magma ocean

lifetimes of 10 Myr or more, the Earth’s tidal quality factor Q must have been > 300 to maintain primor-

dial inclination, implying an early Earth 1–2 orders of magnitude less dissipative than at present. On the

other hand, if tidal dissipation on the early Earth was stronger, our model implies rapid damping of the

lunar inclination and requires subsequent late excitation of the lunar orbit after the crystallization of the

lunar magma ocean.

© 2016 Elsevier Inc. All rights reserved.

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1. Introduction

The Earth–Moon system has had a complex history (e.g.

Goldreich, 1966 ). The evolution of the lunar orbit involves the

interplay of tides arising on both the Earth and the Moon and

the subsequent transfer of energy from rotation and the lunar

orbit into heat via viscous dissipation. This dissipation process is

frequency- and time-dependent and can vary in magnitude for

many different reasons, such as the internal thermal structure,

the presence of a liquid magma ocean and for Earth specifically,

the properties of the liquid water oceans (if present) and the

atmosphere. Current observations of the present-day orbital and

rotational state of both the Earth and the Moon are plentiful, and

there are some geological constraints on lunar orbital evolution

over the past 0.6 Gyr or possibly up to 2.5 Ga ( Williams, 20 0 0 ). In

contrast the early evolution of the Earth–Moon system, in terms

of the orbital parameters, the relevant excitation and damping

mechanisms, and the dissipative properties of the two bodies, has

few constraints.

∗ Corresponding author. Tel.: +1 831 459 1783. E-mail address: fnimmo@es.ucsc.edu (F. Nimmo).

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http://dx.doi.org/10.1016/j.icarus.2016.04.012

0019-1035/© 2016 Elsevier Inc. All rights reserved.

Following the Moon-forming impact, the early Moon was

ikely covered by a global magma ocean ( Canup, 2004; Canup

nd Asphaug, 2001; Smith et al., 1970; Wood et al., 1970 ). Based

n geochronological techniques, the final stages of magma ocean

rystallization may have taken up to 200 Myr ( Borg et al., 1999;

arlson and Lugmair, 1988; Gaffney and Borg, 2014; Nemchin

t al., 2006; Nyquist et al., 2010 ). This extended cooling timescale

s difficult to reconcile with geophysical models of solidification

Elkins-Tanton et al., 2011; Solomon and Longhi, 1977 ). One pos-

ibility – which we will return to below – is that tidal dissipation

rimarily in the crust may have prolonged the lifetime of the

nderlying magma ocean ( Meyer et al., 2010 ).

Tidal dissipation due to viscoelastic solid-body tides has been

nvoked as a heat source for many bodies in the Solar System

Meyer and Wisdom, 2007; Peale et al., 1979; Squyres et al., 1983;

obie et al., 2005 ), the Moon included ( Meyer et al., 2010; Peale

nd Cassen, 1978 ). For a global satellite ocean, Tyler (2008) sug-

ested that a resonance between the obliquity tide and a Rossby–

aurwitz wave could generate significant ocean flow and that tur-

ulent dissipation of the energy contained in this flow could also

epresent a different and, perhaps significant, tidal heat source. In

ertain instances, this ocean-related obliquity tidal dissipation can

e larger than the corresponding solid-body tidal dissipation ( Chen

t al., 2014 ). The obliquity of the Moon is variable as the orbit

http://dx.doi.org/10.1016/j.icarus.2016.04.012http://www.ScienceDirect.comhttp://www.elsevier.com/locate/icarushttp://crossmark.crossref.org/dialog/?doi=10.1016/j.icarus.2016.04.012&domain=pdfmailto:fnimmo@es.ucsc.eduhttp://dx.doi.org/10.1016/j.icarus.2016.04.012

E.M.A. Chen, F. Nimmo / Icarus 275 (2016) 132–142 133

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xpands. At a semi-major axis of about 30 Earth radii ( R E ), there

s a large obliquity excursion due to a transition between Cassini

tates 1 and 2 ( Siegler et al., 2011; Ward, 1975 ). Peale and Cassen

1978) specifically considered dissipation during large lunar obliq-

ity excursions, but did not investigate the role of any magma

cean. Since it depends strongly on obliquity, tidal dissipation in

he lunar magma ocean may have been a significant and hitherto

nrecognized heat source when the Moon was still molten.

The goal of this paper is to present a model for the coupled

hermal-orbital evolution of the early Moon, using an approach

imilar to Meyer et al. (2010) but including the effects of tidal

issipation in the lunar magma ocean. We find that if the lunar

agma ocean is dissipative as a result of obliquity tides, then sig-

ificant inclination damping will occur. The extent of this damp-

ng is related to the magnitude of the obliquity. Thus, to allow

he Moon to maintain a primordial inclination, the magma ocean

ust have solidified prior to the large obliquity excursion asso-

iated with the Cassini state transition at around 30 R E . Since the

iming of magma ocean solidification is approximately known from

eochronological constraints (see summary in Elkins-Tanton et al.

2011) ), this argument provides a “speed limit” on the rate at

hich the early Moon can evolve outwards while maintaining a

rimordial inclination. Dissipation in the Earth controls the rate of

utward migration of the Moon. During the period when the lunar

agma ocean exists, maintaining a primordial inclination requires

hat the tidal time lag of the Earth is less than ∼ 10 s. This implies much less dissipative state than that associated with the current

alue of ∼600 s ( Munk and MacDonald, 1960 ). On the other hand, if the early Earth is characterized by

tronger tidal dissipation, the Moon evolves outwards faster, violat-

ng the “speed limit”, and experiencing rapid inclination damping.

n such a case, lunar magma ocean crystallization must be followed

y a late excitation mechanism to reproduce the observed state of

he Earth–Moon system ( Pahlevan and Morbidelli, 2015 ). Hence, in

rinciple determining the dissipative properties of the early Earth

ould permit discrimination between early and late mechanisms of

xcitation for the lunar inclination.

. Methods

The evolution of the Earth–Moon system is a coupled thermal-

rbital problem; the orbital parameters change due to tidal dissipa-

ion in the Earth and the Moon and the amount of dissipation de-

ends on their internal thermal structures, which in turn is altered

y the energy dissipated as heat. While many papers have looked

t the evolution of the Earth–Moon system ( Burns, 1986; Goldre-

ch, 1966; Meyer et al., 2010; Mignard, 1979; 1980; 1981; Peale and

assen, 1978; Touma and Wisdom, 1998; Ward and Canup, 20 0 0 ),

one of the preceding work has explicitly included tidal dissipa-

ion in a lunar magma ocean. In the case of a global ocean, it has

een recently shown that the tide due to the obliquity can force

resonant response resulting in dissipation in the ocean that may

e greater than that in the viscoelastic crust ( Chen et al., 2014;

yler, 2008 ). The Moon likely went through a period of very high

bliquities ( Ward, 1975 ). The tidal dissipation associated with this

bliquity would have had significant, previously unconsidered, ef-

ects on the orbital evolution of the Earth–Moon system.

.1. Coupled thermal-orbital evolution

Mignard (1979; 1980; 1981) derived equations describing the

ime-evolution of the lunar orbit. These equations are the aver-

ged form of Lagrange’s planetary equations assuming that the dis-

urbing function involves a constant time lag �t ( Darwin, 1908;

ignard, 1979; 1980; 1981; Touma and Wisdom, 1994 ) and in-

ludes dissipation in both the Earth and the Moon. The assump-

ion of a constant time lag is convenient, but can result in dif-

erent behavior from a constant phase lag assumption ( Efroimsky

nd Makarov, 2013; Efroimsky and Williams, 2009; Goldreich,

966; Touma and Wisdom, 1994 ). We discuss this issue briefly in

ection 4.3 . We ignore evolution of the orbital eccentricity in this

nitial analysis, because tidal dissipation in the lunar magma ocean

s likely to be dominated by obliquity tides ( Chen et al., 2014 ).

owever, the eccentricity has likely played an important role in

any facets of the Moon’s evolution, including pumping of the

rbital inclination during passage through the eviction resonance

Touma and Wisdom, 1998 ), tidal heating of the plagioclase flota-

ion crust ( Meyer et al., 2010 ), and perhaps the determination of

he Moon’s current shape ( ́Cuk, 2011; Garrick-Bethell et al., 2014;

006; Keane and Matsuyama, 2014 ).

Ignoring orbital eccentricity, the Mignard evolutionary equa-

ions are ( Meyer et al., 2010; Mignard, 1981 ):

dX

dt = C X

X 7

[−(1 + A X ) +

(UX 3 / 2 cos i + A X cos θ0

)](1)

di

dt = −C i

X 13 / 2

(U + A i

X 3 / 2

)sin i (2)

dU

dt = −C U, 0

(U − n �

n G

)+ C U, 1

X 15 / 2

(1 − UX 3 / 2 cos i

)cos i

− C U, 2 X 6

U sin 2

i (3)

d (

ω n G

)2 dt

= −C ω, 0 X 6

((ω

n G

)2 (2 + ( sin i ) 2 2

)+ U 2

(3( cos i ) 2 − 1

2

)

+ 2 U X 3 / 2

cos i

)+ C ω, 1

(2

n �

n G U −

(ω

n G

)2 − U 2

)(4)

here X = a/R E , the ratio of the semi-major axis a to the radius ofhe Earth R E , i is the orbital inclination relative to the ecliptic, θ0 s the Moon’s obliquity, ω is the rotational frequency of the Earth, G =

√ GM/R 3

E is the grazing mean motion, n � is the mean motion

f the Earth around the Sun, and U = ω/n G cos (I) with the obliq-ity of the Earth I . The constants are defined as:

X = 6 γ m M

m

μ(5)

i = 3

2 γ

m

M

(m

μ

)1 / 2 (6)

U, 0 = 3 γα

(M �

M

)2 ( R E a �

)6 (7)

U, 1 = 3 γα

(m

M

)2 (m μ

)1 / 2 (8)

U, 2 = 3 2

γ

α

(m

M

)2 (9)

ω, 0 = 3 γα

(m

M

)2 (m μ

)(10)

ω, 1 = 3 2 γ(

M �

M

)2 (m μ

)(R E a �

)6 (11)

ith

≡ k 2 E n 2 G �t E (12)

134 E.M.A. Chen, F. Nimmo / Icarus 275 (2016) 132–142

Table 1

Physical parameters utilized in the model for Earth–Moon system evolution.

Symbol Parameter Value

G Gravitational constant 6.674 × 10 −11 m 3 kg −1 s −2 M Mass of the Earth 5.97 × 10 24 kg R E Radius of the Earth 6370 km

m Mass of the Moon 7.25 × 10 22 kg R M Radius of the Moon 1737 km

M � Mass of the Sun 2.0 × 10 30 kg a � Earth–Sun distance 149.6 × 10 6 km k 2 E Love number of the Earth 0.97

α Normalized moment of inertia of the Earth 0.33

g Surface gravity of the Moon 1.62 m 2 s −1

ρ Magma ocean density 30 0 0 kg m 3

c D Bottom drag coefficient 0.002

c Normalized moment of inertia of the Moon 0.393

L Latent heat of fusion 5.0 × 10 5 J kg −1 T m Temperature of magma ocean 1200 K

T 0 Temperature of lunar surface 280 K

c p Specific heat capacity 1256 J kg −1 K −1

w

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where m is the mass of the Moon, M is the mass of the Earth, M �is the mass of the Sun, μ = (1 /m + 1 /M) −1 is the reduced mass,α is the dimensionless moment of inertia of the Earth, a � is theEarth–Sun distance, k 2 E is the tidal Love number of the Earth, and

�t E is the tidal time lag of the Earth. Note that we have corrected

typographical errors for γ , C i and C ω, 1 in Meyer et al. (2010) . Thevalues of the physical parameters are presented in Table 1 . Our

nominal model is initialized at a semi-major axis of a = 6 . 5 R E ,inclination of i = 12 ◦, and a rotation period of the Earth of 6 h.These are estimates of the conditions after passage through the

eviction resonance ( Touma and Wisdom, 1998 ) and/or disk reso-

nances ( Ward and Canup, 20 0 0 ). We explore the effect of different

initial conditions in Section 4 .

In Mignard (1981) and Meyer et al. (2010) , the Mignard tidal

parameter A is defined as

A ≡(

k 2 M �t M k 2 E �t E

)(M

m

)2 (R M R E

)5 (13)

where k 2 M is the tidal Love number of the Moon, and �t M is the

tidal time lag of the Moon and R M is the radius of the Moon.

This tidal parameter relates the amount of energy dissipation in

the Moon to that in the Earth assuming a constant time lag, vis-

coelastic dissipation model in both bodies. For viscoelastic solid-

body tides, the Mignard parameter is a constant and in (1) and

(2) above,

A X = A i = A. Tides in the lunar magma ocean may also remove energy from

and alter the orbit, but in a different manner: the tides force ocean

flow and the kinetic energy of this flow is subsequently dissipated

through turbulence ( Tyler, 2008 ). We derive the effects of magma

ocean tides on the orbit from considerations of conservation of en-

ergy and angular momentum. In order to preserve similarity to the

Mignard equations, we define two Mignard-like parameters A X and

A i for the effects of magma ocean tidal heating on the evolution of

the semi-major axis and orbital inclination respectively. Unlike in

the case of viscoelastic tides, these parameters are time-dependent.

In a Keplerian orbit, the orbital energy is given by

E = −GMm 2 a

, (14)

and tidal dissipation causes changes in the semi-major axis of the

orbit through the relation:

˙ E = GMm 2 a 2

˙ a (15)

here the dot designates a time derivative. For an inclined orbit

ith no eccentricity, the vertical component of the orbital angular

omentum is ( Chyba et al., 1989 ),

= m (GM) 1 / 2 a 1 / 2 cos i. (16)In the absence of tides raised on the Earth, this quantity is con-

erved and we obtain the following relations after taking the time

erivative of (16) and including the relation (15)

˙ a

a = 2 tan i di

dt = 2 a ̇

E

GMm . (17)

These relations must hold true independent of the energy dissi-

ation process. This approach assumes that dissipation in the pri-

ary is small, which in the case of the early Earth may be justified

posteriori (see Section 4.1 ).

In the Mignard equations, the Mignard tidal parameter conceals

he relations (17) . Ignoring dissipation in the Earth, the evolution

f the semi-major axis from (1) and (17) is

˙ = C X

(R E a

)8 aA X ( cos θ0 − 1 ) = 2 a

2 ˙ E M GMm

(18)

here ˙ E M represents dissipation in the Moon. Thus, the tidal pa-

ameter A X is related to ˙ E M by

X = 1 3 γ

(M

m

)(a

R E

)8 a ̇ E M GMm

1

( cos θ0 − 1 ) . (19)

By substituting Eq. (19) into Eq. (13) it can be shown that the

issipation rate is given by

˙ M = −3

2 k 2 M �t M

G 2 M 3 R 5 M a 9

sin 2

i (20)

hich is the standard result for dissipation due to obliquity tides

nder the assumptions that i ≈ θ0 and i is small ( Peale and Cassen,978; Wisdom, 2008 ).

Similarly, the evolution of the inclination from Eqs. (2) and

17) can be expressed as

di

dt = −C i

(R p

a

)8 A i sin i =

a ̇ E M cos i

GMm sin i (21)

nd thus,

i = −2 3 γ

(M

m

)(a

R p

)8 a ̇ E M cos i

GMm ( sin i ) 2 , (22)

hich simplifies to (13) for the dissipation given by (20) . Eqs.

19) and (22) thus allow the orbital evolution to be calculated via

qs. (1) –(4) once the magma ocean dissipation rate ˙ E M has been

pecified. We assume the only source of tidal heating ˙ E M is from

he magma ocean.

E.M.A. Chen, F. Nimmo / Icarus 275 (2016) 132–142 135

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.2. Tidal dissipation in the lunar magma ocean

Tyler (20 08; 20 09) suggested that a global liquid ocean within

satellite, specifically Europa and Enceladus, would respond reso-

antly to obliquity-related tidal forcing and that frictional dissipa-

ion of this ocean flow could provide a significant heat source. For

ost of the outer Solar System satellites, the dominant heat source

s unlikely to be ocean dissipation (see Chen et al. (2014) ); this re-

ult is primarily due to the fact that these satellites formed locally

n a disk ( Canup and Ward, 2002 ), and thus their inclinations and

he resultant obliquities are quite small. However, the Moon’s in-

lination of approximately 5 ° is currently much higher than manyf the outer Solar System satellites (except for Iapetus and Triton),

here regular satellites’ inclinations are less than 1 ° ( Yoder, 1995 ),nd the Moon likely underwent a Cassini state transition early in

ts history, where its obliquity could have been much larger than

he orbital inclination ( Ward, 1975 ). As a result, dissipation in the

agma ocean may have contributed a significant amount of heat

o the evolution of the Moon. Here we summarize a method of cal-

ulating tidal dissipation in this early magma ocean, and point the

eader to Chen et al. (2014) for further details of the method.

The behavior of a global magma ocean can be described

hrough the linear shallow-water equations on a sphere:

∂ � u

∂t + 2 cos θ ˆ r × � u = − g � ∇ η − � ∇ U + ν∇ 2 � u (23)

∂(ξη)

∂t + h � ∇ · � u = 0 (24)

here � u is the radially-averaged, horizontal velocity vector, is

he constant rotation rate, which is the same as the orbital mean

otion for a synchronous satellite, ˆ r is the unit vector in the radial

irection, g is the surface gravity, η is the vertical displacementf the surface, h is the constant ocean depth and ν represents anffective horizontal viscous diffusivity.

To approximate the effect of a rigid lid on the ocean tidal re-

ponse ( Kamata et al., 2015 , cf.), we define a damping factor ξ ,here

= k 2 − h 2 (25)

nd ξ ∈ [0, 1]. If ξ = 1 , this describes the response in the limit of weak or absent lid, while ξ ≈ 0 in the case of a very rigid lid.he models we present below are for ξ = 1 . This damping factor

s expected to be in the range 0.3–1 for likely magma ocean pa-

ameters. Including its effect results in small modifications to the

nal results (shorter ocean lifetimes, higher final inclinations) but

ot enough to change any of our conclusions.

U represents the radially-averaged forcing potential due to tides,

ere for the obliquity tide:

= −3 2 2 R 2 M θ

2 0 sin θ cos θ ( cos (φ − t) + cos (φ + t)) . (26)

Here θ is colatitude and φ is longitude. For synchronously ro-ating satellites, the forcing potential due to obliquity is resonant

ith a Rossby–Haurwitz wave ( Tyler, 2008 ), and thus flow in a

lobal ocean can be significant.

The energy dissipation associated with the obliquity tide can be

ritten as:

˙ obl =

108 π

25

ρhν8 R 10 M θ2 0

( 96 R 8

M

25 + 144 ν2 g 2 h 2

ξ 2 )

(27)

here ρ is the magma ocean density and ν is a turbulent vis-ous diffusivity given by the numerically-derived scaling law ( Chen

t al., 2014 )

= ξ

⎛ ⎝ −9 25 6 R 8 M +

√ (9

25 6 R 8

M

)2 + 4(144) (0 . 40 c D g4 R 7 M θ0 )2 2(144) g 2 h 2

⎞ ⎠

1 /

(28)

ith an associated bottom drag c D . We adopt a bottom drag formu-

ation rather than assuming a dissipation factor ( Q ) for the ocean

ecause for the latter it is unclear a priori what value should be

dopted. In contrast, for the terrestrial oceans c D ≈ 0.002 ( Egbertnd Ray, 2001; Jayne and St. Laurent, 2001; Sagan and Dermott,

982 ) and the dependence on Reynolds number is weak ( Sohl

t al., 1995 ). Ultramafic lavas have viscosities comparable to wa-

er ( ∼10 −3 Pa s; ( Lumb and Aldridge, 1991; Rubie et al., 2003 ))nd the high-Ti basalts associated with late-stage crystallization of

he magma ocean also have low ( < 100 Pa s) calculated viscosi-

ies ( Williams et al., 20 0 0 ). We initially adopt an Earth-like value

f c D and discuss this issue further in Section 4.3 . Also note that

his mechanism for dissipation only applies to synchronous bod-

es; dissipation in a deep terrestrial magma ocean is expected to

e small.

The energy dissipation in the ocean evolves in time as the

oon’s rotation period , obliquity θ0 and ocean depth h evolve.ssuming the Moon has remained synchronous, the rotational fre-

uency is equal to the orbital mean motion, which changes with

he orbital semi-major axis (cf. Eq. (1) ).

.3. Cassini state obliquity

While the current lunar obliquity can be measured, the obliq-

ity of the Moon in the past is uncertain. For this work, we assume

hat throughout its evolution, the Moon has occupied a Cassini

tate, where its spin axis, the orbit normal and an invariable pole

emain coplanar as they precess ( Colombo, 1966; Peale, 1969 ). The

bliquity is given by the angle between the orbit normal and the

oon’s spin axis.

For coplanar precession to occur, the obliquity angle θ0 mustatisfy the relation ( Bills and Nimmo, 2008; Peale, 1969 )

3

2 [ (J 2 + C 22 ) cos θ0 + C 22 ] p sin θ0 = c sin (i − θ0 ) (29)here J 2 and C 22 are the degree-2 gravity coefficients and c is the

ormalized polar moment of inertia of the Moon. The ratio p of

he orbital motion to the orbit plane precession is

p = (d orb /d t)

(30)

here orb is the longitude of the ascending node.

It is well-established that the observed gravity coefficients of

he Moon are inconsistent with a presently hydrostatic, tidally-

istorted body ( Garrick-Bethell et al., 2006; Jeffreys, 1915; Konopliv

t al., 2001; 1998 ). Because we are concerned with the Moon dur-

ng a period where it was warm and likely to have behaved like

fluid, we make the hydrostatic assumption; modifying this as-

umption does not greatly change the position at which the Cassini

tate transition occurs ( Siegler et al., 2011 ). For hydrostatic bod-

es, the normalized polar moment of inertia c can be related to

he displacement Love number h 2 via the Radau–Darwin relation

Hubbard and Anderson, 1978 )

= 2 3

(1 − 2

5

(5

h 2 − 1

)1 / 2 )(31)

For a synchronous, slowly-rotating, hydrostatic satellite, the

egree-2 gravity coefficients can be related to h 2 by

2 = 5 6

(R 3 M

2

Gm

)(h 2 − 1) (32)

136 E.M.A. Chen, F. Nimmo / Icarus 275 (2016) 132–142

C

T

t

e

p

s

a

w

T

o

t

2

t

o

p

t

o

t

b

i

m

3

6

h

t

a

n

o

w

fl

e

w

m

t

r

d

c

o

t

t

p

b

t

m

s

T

t

i

f

t

E

m

t

e

i

d

t

d

t

and

22 = 3 10

J 2 . (33)

The precession of the orbit arises due to torques from the rota-

tional flattening of the Earth and solar tides. The total precessional

frequency is the sum of the two

dorb dt

= −3 2

nJ 2 E

(R E a

)2 − 3

4

(M �

M + m )

n

(a

a �

)3 (34)

with orbital mean motion n , which for synchronous satellites is

equal to the rotation rate , and degree-2 gravity coefficient of the

Earth J 2 E . The effect of the Earth’s flattening is important when the

Moon is close to the Earth, i.e. a � 17 R E ( Goldreich, 1966 ). For ahydrostatic planet, J 2 E can be expressed as

J 2 E = 1 3

(R 3 E ω

2

GM

)k 2 E (35)

where ω is the Earth’s rotation rate and k 2 E is the secular Lovenumber which we here assume is equal to the tidal Love number,

owing to the fluid state of the early Earth. The physical constants

used for these calculations are tabulated in Table 1 .

2.4. Solidification of the lunar magma ocean

The solidification of the lunar magma ocean is a complex prob-

lem, involving fractional crystallization from a chemically- and

thermally-evolving reservoir. Our main interest with respect to the

solidification process is to determine the ocean depth h as a func-

tion of time because the tidal dissipation is dependent on this

quantity. Unlike other models, we do not seek to explain the geo-

chemistry and geochronology of the Moon’s surface ( Borg et al.,

1999; Carlson and Lugmair, 1988; Gaffney and Borg, 2014; Nem-

chin et al., 2006; Nyquist et al., 2010 ) and therefore, we choose to

model the solidification of the magma ocean as a Stefan problem

( Turcotte and Schubert, 2002 ) with bottom heating to account for

the effect of tidal dissipation. The evolution of the radial position

of the solid-liquid interface r i , in this case between the solid crust

and the liquid magma, is then

ρL d(R M − r i )

dt = −k

(∂T

∂r

)r= r i

− H (36)

where L is the latent heat of fusion, k is the thermal conductivity

of the crust, and H is the heat flux into the base of the crust. This

heat flux is assumed to be solely due to tidal dissipation in the

magma ocean such that

H ≈˙ E M

4 πR 2 M

. (37)

Assuming a conductive thermal profile in the solid crust, the

temperature gradient at the interface is given by ( Turcotte and

Schubert, 2002 ) (∂T

∂r

)r= r i

= −(T m − T 0 ) (R M − r i )

2 √ π

e −λ2 1

λ1 erf (λ1 )

(38)

where T m is the magma ocean temperature, T 0 is the surface tem-

perature, c p is the specific heat capacity of the crust, and λ1 is de-termined by

e −λ2 1

λ1 erf (λ1 ) = L

√ π

c p (T m − T 0 ) . (39)

The approach adopted here is very simple, in that it neglects

effects such as secular cooling, heat production (either tidal or ra-

diogenic) in the crust, advection of heat via volcanism, and so on.

o take into account these factors, and to also reflect the uncer-

ainties in the actual duration of the magma ocean ( Elkins-Tanton

t al., 2011; Solomon and Longhi, 1977 ), we adopt the simple ex-

edient of allowing the thermal conductivity k to vary. By doing

o, we can generate magma ocean solidification timescales in the

bsence of tidal heating ranging from 4 to 110 Myr, in agreement

ith the more sophisticated models of magma ocean solidification.

he general conclusions drawn from our orbital model only depend

n the timing of magma ocean solidification and not the details of

he process.

.5. Summary and numerical details

Our model may be summarized as follows. The orbital evolu-

ion of the Moon is updated based on Eqs. (1) –(4) , while the lunar

bliquity is calculated based on the inclination assuming it occu-

ies a Cassini state ( Section 2.3 ). Given the obliquity, the dissipa-

ion rate in the magma ocean is then calculated using the approach

f Section 2.2 , and the magma ocean thickness updated according

o Section 2.4 . Finally, the dissipation rate is fed back into the or-

ital evolution equations for the next time step. We numerically

ntegrate equations (1) –(4) using a second-order Adams–Bashforth

ethod with a maximum time step of 30 0 0 years.

. Results

Our nominal model is initialized at a semi-major axis of a = . 5 R E , inclination of i = 12 ◦, and a rotation period of the Earth of 6ours. These are estimates of the conditions after passage through

he eviction resonance ( Touma and Wisdom, 1994; 1998 ) but are

lso roughly consistent with the results of interactions with a rem-

ant proto-lunar disk ( Ward and Canup, 20 0 0 ). The initial magma

cean thickness is taken to be 100 km, since we are concerned

ith the final, slow stages of solidification after the plagioclase

otation crust was established ( Elkins-Tanton, 2012; Elkins-Tanton

t al., 2011; Shearer et al., 2006 ). Fig. 1 shows results for a model

ith tidal dissipation only in the Earth as well as fully coupled

odels incorporating lunar tidal heating.

In the absence of lunar dissipation (black curves), the inclina-

ion damps slowly as the semi-major axis increases. The obliquity

ises to a high value as the Cassini state transition is approached;

uring this period, magma ocean solidification is completed. Be-

ause dissipation in the Earth is modest, the outwards evolution

f the Moon is quite slow; at the time of magma ocean solidifica-

ion (65 Myr), a ≈ 24 R E , that is, prior to the Cassini state transi-ion at ≈30 R E . The lunar inclination remains comfortably above theresent-day value of 5.16 °. Since increase of the inclination via or-ital resonances is not thought to have occurred beyond the evic-

ion resonance ( Touma and Wisdom, 1998 ), any early excitation

odel must meet this minimum inclination criterion.

The most obvious effect of including lunar magma ocean dis-

ipation is the rapid damping of the orbital inclination ( Fig. 1 a).

he extent to which inclination is damped depends primarily on

he total amount of energy dissipated in the ocean before it solid-

fies, and thus on the obliquity evolution and ocean lifetime. Dif-

erent lines represent different dissipation rates in the Earth, and

hus different rates of outwards evolution. Low dissipation in the

arth (blue curves) results in slow outwards motion and gives the

agma ocean time to solidify prior to the onset of the Cassini state

ransition. As a result, the amount of inclination damping is mod-

st. In contrast, higher dissipation in the Earth (red curve) results

n rapid outwards motion and a magma ocean that is still present

uring the approach to the Cassini state transition. Even at rela-

ively small values for obliquity ( 1 − 2 ◦) the effect of inclinationamping is significant. In short, models that delay the evolution

hrough the Cassini state transition until after the magma ocean

E.M.A. Chen, F. Nimmo / Icarus 275 (2016) 132–142 137

Fig. 1. Results from coupled thermal-orbital models for the evolution of the (a) or-

bital inclination, which may be compared with the current lunar value of 5.16 °, indicated by the black dotted line, (b) Lunar obliquity, (c) Magma ocean depth,

(d) Magma ocean tidal dissipation, (e) Time as a function of the semi-major axis.

The models begin with the same initial conditions. The only variations are the tidal

time lag of the Earth and whether the model accounts for ocean tidal heating; the

plots are colored as follows: (black) �t E = 10 s without magma ocean tidal heating, (black-dashed) �t E = 10 0 0 s without magma ocean tidal heating, (red) �t E = 10 s with magma ocean tidal heating, (blue) �t E = 1 s with magma ocean tidal heat- ing, (blue-dashed) �t E = 0 . 1 s with magma ocean tidal heating, and (blue-dash dot) �t E = 0 . 01 s with magma ocean tidal heating. The thermal diffusivity for these models is 1 . 0 × 10 −5 m 2 s −1 which translates to a magma ocean solidification time of 11 Myr without tidal heating. These models are circled in the parameter space

in Fig. 2 . In (a), the current orbital inclination is indicated by the black-dashed line.

The evolution of the inclination with respect to semi-major axis for models with-

out magma ocean heating are the same and are indicated by the black curve. This

evolution differs in time (see (e)). In (e), the evolution of the semi-major axis with

respect to time is primarily controlled by dissipation in the Earth. The black stars

plots the evolution of the model without tidal heating with �t E = 10 s; this curve collapses on top of the evolution curve for that with tidal heating. (For interpre-

tation of the references to color in this figure legend, the reader is referred to the

web version of this article).

h

c

t

o

s

(

(

t

M

l

t

w

o

Fig. 2. Parameter space variations for Earth’s tidal time lag and the magma ocean

solidification time based on the fully-coupled model. Models vary the thermal con-

ductivity such that the thermal diffusivity varies from 1 . 0 × 10 −6 to 3 . 0 × 10 −5 m 2 s −1 , resulting in solidification times between 4 and 110 Myr without accounting for orbital and tidal dissipation effects. Green diamonds indicate parameters that result

in an inclination over 5 ° when the magma ocean solidifies, red X’s indicate models that have inclinations below. The thermal-orbital evolution for the circled models

are presented in Fig. 1 . The nonlinear evolution results in the variations in magma

ocean solidification time. (For interpretation of the references to color in this figure

legend, the reader is referred to the web version of this article).

i

w

t

b

m

t

a

E

i

E

c

l

(

4

o

b

t

t

t

i

t

t

h

T

r

as solidified can maintain an orbital inclination greater than that

urrently observed. This effect can be achieved if the Earth’s tidal

ime lag is small ( Fig. 1 ).

Dissipation in the magma ocean can prolong the magma

cean’s life (cf. Eq. (36) ). Although the semi-major axis at which

olidification is complete becomes smaller as �t E decreases

Fig. 1 c), the time to complete solidification is nonetheless longer

Figs. 1 e and 2 ). The slower outwards evolution actually increases

he amount of heat production in the ocean. The average Earth–

oon distance is smaller for smaller values of �t E , resulting in

arger tidal responses and therefore more energy dissipation.

Inclination damping is also affected by the amount of time

he magma ocean is present. Fig. 2 shows the parameter space in

hich models have an inclination exceeding 5 ° after solidificationf the magma ocean is complete. Here the magma ocean freez-

ng timescale has been varied by varying k ( Section 2.4 ). Models

ith short-lived magma oceans can preserve the orbital inclina-

ion, because the total energy dissipated in the ocean is limited

y its duration. Models with large Earth time lags or long-lived

agma oceans ( > 100 Myr) cannot preserve the inclination because

he magma ocean is still present as the Cassini state transition is

pproached, resulting in high dissipation. We note especially that

arth time lags > 10 s result in significant inclination damping even

f the magma ocean lifetime is as short as 10 My. For the smallest

arth time lags, inclination damping becomes significant again be-

ause the slow outwards evolution means that the Moon spends a

ot of time at small semi-major axes, where dissipation is strong

cf. Eq. (20) ).

. Discussion

As shown in Fig. 1 , a major constraint for our coupled thermal-

rbital evolution model is the current (relatively high) lunar or-

ital inclination. The damping of the inclination due to dissipa-

ion solely from the Earth is weak. The damping of inclination due

o satellite dissipation operates on a different, and much shorter,

imescale, especially if a magma ocean is present and the obliquity

s relatively high. In fact, this effect is so significant that we argue

hat it allows only two possible scenarios for the early evolution of

he Earth–Moon system:

• The lunar magma ocean must have solidified prior to passage

through the Cassini state transition (at around 30 R E ). In this

case, the early Earth’s effective tidal time lag was likely less

than ∼10 s, a value that is orders of magnitude less than thepresent value of ∼10 minutes ( Munk and MacDonald, 1960 ), i.e.the Earth was significantly less dissipative in its early history

when compared to its present state. • The primordial inclination of the Moon was damped via obliq-

uity tides, but the lunar inclination was subsequently re-excited

following magma ocean crystallization, for example, via colli-

sionless encounters with remnant planetesimals ( Pahlevan and

Morbidelli, 2015 ).

It has been long recognized that the Moon has anomalously

igh inclination, the “mutual inclination problem” ( Burns, 1986;

ouma and Wisdom, 1998 ), and has likely undergone a pe-

iod where the obliquity was extremely high, the Cassini state

138 E.M.A. Chen, F. Nimmo / Icarus 275 (2016) 132–142

Fig. 3. Semi-major axis evolution for an analytic orbital evolution model assuming

dissipation in the Earth alone with a constant tidal quality factor Q . The grey shaded

region is the timescale for magma ocean solidification. The red dotted line indi-

cates the semi-major axis where the obliquity of the Moon is greater than ∼2 °; if the Moon is still molten at this point, dissipation in the ocean is significant enough

to damp the inclination below the present value. Thus, Q of the early Earth is un-

likely to be as low as it is at present ( Q ∼ 12). The equivalent estimates for �t E were obtained by running our coupled model to obtain a semi-major axis of 20 R E occurring at the same time as with the analytical (constant- Q ) approach. (For inter-

pretation of the references to color in this figure legend, the reader is referred to

the web version of this article).

o

i

o

o

m

p

t

F

i

o

l

v

m

f

i

s

w

t

o

2

s

m

o

d

e

e

Q

q

(

t

q

4

M

t

transition ( Ward, 1975 ). Using a solid-body dissipation model,

Peale and Cassen (1978) looked at the Moon and concluded that

a high obliquity excursion did not significantly affect the Moon’s

long-term thermal evolution. We do not debate this conclusion;

however, it was only recently recognized that resonant flow be-

havior in the ocean due to the obliquity tide ( Tyler, 2008 ) can

result in tidal dissipation that is orders of magnitude larger than

that predicted for solid bodies even at small obliquities (less than

1 °) ( Chen et al., 2014 ).

4.1. Inclination damping

The major result of our coupled model is that dissipation in the

lunar magma ocean causes significant damping in the orbital incli-

nation ( Fig. 1 ). Inclination damping due to satellite tidal dissipation

is generally not emphasized because inclinations for regular satel-

lites of the Solar System are small and inclination tidal dissipation

is significantly less important in thermal evolution than eccentric-

ity tidal dissipation.

The ratio of the inclination damping rate due to dissipation in

the Moon to that in the Earth can be extracted from Eqs. (2) and

(17) : (di dt

)Moon (

di dt

)Earth

= XR E ˙ E M

GMm tan i

C i UX −13

2 sin i ≈ X

15 2 R E ˙ E M n g

C i ωGMmi 2 . (40)

It is not obvious from this ratio that the inclination damping

should be dominated by dissipation in the Moon. Substituting val-

ues for the constants and appropriate values from the evolution

models found in Fig. 1 , the ratio is (di dt

)Moon (

di dt

)Earth

= 1 . 75 (

˙ E M 5 TW

)(X

10

) 15 2 (

10 ◦

i

)2 (0 . 1 s �t E

)(P E

6 hr

)(41)

While a factor of 1.75 may seem relatively insignificant, this

ratio clearly changes with time. Ocean tidal dissipation decreases

with time; however, the ratio actually increases because of expan-

sion of the orbit and decrease in inclination. Later in the evolu-

tion, the value of this ratio trends between 3 and 5 depending

on model parameters. This increase in damping rate is illustrated

in the steep decrease in inclination between our orbital evolution

models with (blue and red curves) and without (black curves) lu-

nar magma ocean dissipation in Fig. 1 a.

4.2. Dissipation in the early Earth

The amount of tidal dissipation in the early Earth is unknown.

The current tidal quality factor of the Earth is small, Q E ∼ 12, duemainly to dissipation in the oceans ( Egbert and Ray, 20 0 0; Munk,

1997; Munk and MacDonald, 1960 ), which is equivalent to a tidal

time lag of �t E ≈ 1/(2 Q E ) ≈ 10 min ( Goldreich and Soter, 1966;Munk and MacDonald, 1960 ). Integrating back in time with this

present value of Q E results in an Earth–Moon system age of ∼ 1.6Gyr ( Bills and Ray, 1999; Murray and Dermott, 1999 ); therefore, it

is clear that the Q E of the Earth has evolved over time, and must

have had a period where it was significantly higher (less dissipa-

tive). Dissipation in the Earth is geologically unconstrained prior to

2.5 Ga ( Williams, 20 0 0 ).

From our model, we find that the tidal time lag of the early

Earth, specifically when the magma ocean was still present in

the Moon, must have been less than ∼10 s to preserve a pri-mordial, early-acquired lunar inclination. The time lag controls the

rate of outwards semi-major axis evolution of the Moon. As the

Moon moves outwards from the location of the eviction resonance

( 6 − 7 R E ), its obliquity grows. The magma ocean dissipation scaleswith the square of the obliquity (cf. Eq. (27)) and thus dissipa-

tion also increases as the Moon moves outwards. As dissipation

ccurs, energy is drawn from the orbit and the orbital inclination

s damped. Therefore, to retain a primordial inclination, the high

bliquity excursion related to the Cassini state transition must not

ccur while a magma ocean is present. Since the duration of the

agma ocean is (approximately) known, we therefore have an up-

er bound or “speed limit” on the rate of outwards motion, and

hus an upper bound on �t E to retain a primordial inclination.

ig. 1 shows that the obliquity of the Moon need not be at its max-

mum for the inclination damping to occur; as an estimate, if the

bliquity is larger than ∼1 − 2 ◦ then magma ocean dissipation isikely to damp the inclination to a smaller than present value. This

alue for obliquity occurs at a ≈ 20 R E in our nominal models. As discussed below, these conclusions are likely to be robust to

ost uncertainties. Indeed, the main conclusions can be derived

rom a simpler analytical model, which avoids having to spec-

fy many imperfectly-known parameter values. Fig. 3 presents the

emi-major axis evolution as a function of time for a model in

hich the constant Q of the Earth is specified and dissipation in

he Moon is neglected ( Murray and Dermott, 1999 ). For a magma

cean that remains liquid for 10–100 Myr ( Elkins-Tanton et al.,

011; Solomon and Longhi, 1977 ), a reasonable estimate for Q E uch that the semi-major axis does not exceed 20 R E is approxi-

ately � 300. This value is equivalent to an upper bound on �t E f about 10 s, compatible with the more sophisticated calculations

escribed above. It is also consistent with the results of Zahnle

t al. (2007) ; 2015 ), who suggested that an atmosphere on the

arly Earth could reduce the tidal dissipation of the Earth such that

E 300. However, the lifetime of this atmosphere was probablyuite short, likely to be around 2 Myr, but no more than 10 Myr

Abe, 1997; Abe and Matsui, 1988; Zahnle et al., 2007; 1988 ), and

idal dissipation in the solid Earth may have reduced Q E subse-

uently to lower values.

.3. Caveats

Through our coupled thermal-orbital model of the early Earth–

oon system, we have suggested that the current orbital inclina-

ion of the system provides a constraint on the dissipative behavior

E.M.A. Chen, F. Nimmo / Icarus 275 (2016) 132–142 139

o

r

o

b

v

c

n

a

m

i

l

v

o

s

t

(

t

e

s

m

b

o

n

o

o

c

a

m

2

v

R

m

(

o

a

a

i

c

u

t

l

o

a

l

i

t

i

t

o

(

s

t

n

i

b

t

w

C

t

o

t

a

Fig. 4. Similar to Fig. 2 , here with variation of the drag coefficient c D . Green dia-

monds indicate parameters that result in an inclination over 5 ° when the magma ocean solidifies, red X’s indicate models that have inclinations below. (For interpre-

tation of the references to color in this figure legend, the reader is referred to the

web version of this article).

c

f

T

w

t

a

p

o

L

6

l

o

T

G

w

c

p

i

v

e

a

i

n

t

M

o

1

f the early Earth and the lifetime of the lunar magma ocean. This

esult is related to the fact that the dissipation in a global liquid

cean subject to obliquity tides is significantly larger than solid-

ody tidal dissipation and therefore, ocean tidal dissipation can be

ery effective in damping the inclination (cf. Eq. (41) ). This result

an be interpreted in two ways. In the absence of late lunar incli-

ation excitation, it constrains the tidal time lag of the early Earth

nd the threshold value of the semi-major axis where the lunar

agma ocean is likely to have been solidified. Alternatively, if later

nclination excitation occurred, this must have happened after the

unar magma ocean crystallized.

Our conclusions are based on the assumption of a liquid, low-

iscosity lunar magma ocean. If future work suggests that the Q

f the early Earth was in fact low, one possibility is that our as-

umed lunar magma ocean rheology was incorrect. It could be that

he magma ocean formed a mushy layer, rather than a true liquid

Elkins-Tanton, 2012; Solomatov, 20 0 0 ). Because of the low viscosi-

ies of high-Ti basalts ( Williams et al., 20 0 0 ) and the monomin-

ralic nature of the anorthosite highlands, we think it likely that

eparation of solid from liquid was efficient, and that a significant

ushy layer did not develop. Nonetheless, this possibility should

e borne in mind. Accepting the assumption that a lunar magma

cean behaved as a low-viscosity fluid for some period of time, we

ow discuss the likeliest sources of uncertainty in our models.

One large uncertainty is in the calculation of ˙ E of the magma

cean. The total amount of energy dissipation scales with the value

f the effective diffusivity ν , which is poorly constrained (see dis-ussion in Chen et al. (2014) ). For the magma ocean, we have

dopted a drag coefficient value of c D = 0 . 002 , commonly used inodels for bottom friction in terrestrial oceans ( Egbert and Ray,

001; Jayne and St. Laurent, 2001 ), and mapped this value to a

iscous diffusivity through Eq. (28) . Because of its dependence on

eynolds number ( Sohl et al., 1995 ), this value of c D also implies

olecular viscosities similar to those of water and ultramafic lavas

Lumb and Aldridge, 1991; Rubie et al., 2003 ). The lunar magma

cean viscosity likely evolves in time as composition and temper-

tures change. Our numerically-derived scalings are valid for rel-

tively low effective diffusivities (cf. Eq. 29 in Chen et al., 2014 ),

.e. less than 4.5 × 10 6 m 2 s −1 at a semi-major axis of 6.5 R E . Thisonverts to a maximum model c D of 0.01 and a maximum molec-

lar viscosity of 800 Pa s ( Sohl et al., 1995 ). This limit is larger

han values for the viscosity of high-Ti magmas estimated from

ava channels on the Moon ( Williams et al., 20 0 0 ), suggesting that

ur scaling is valid. In the case where the actual magma viscosities

re higher than our modeled viscosities (but less than the scaling

imit), our results would be considered conservative estimates.

On the other hand, if the drag coefficient of the magma ocean

s in reality smaller than the value used here, the tidal time lag in

he Earth can be larger. Figs. 4 and 5 illustrate the effects of reduc-

ng the drag coefficient. The drag coefficient controls the amount of

idal dissipation. With reduced dissipation, the solidification times

f the magma ocean are closer to a case without tidal heating

compare Fig. 4 a–c). Reducing c D slightly expands the parameter

pace of successful models. Nonetheless, even for c D = 10 −5 , Earthime lags > 30 s are not consistent with maintaining an early incli-

ation for magma oceans that survive 10 My or longer. The reason

s that as the Cassini state transition is approached, the obliquity

ecomes so high that dissipation damps the inclination. As illus-

rated in Fig 5 , a reduced drag coefficient allows for further out-

ards migration, but in no case do successful models include the

assini state transition itself, because of the resulting high dissipa-

ion. Passage through the Cassini state transition while a magma

cean is present would require subsequent re-excitation of inclina-

ion.

Another uncertainty is the initial inclination of the orbit. Ward

nd Canup (20 0 0) suggest that through resonances the initial in-

lination may have exceeded 16 °. Figs. 6 and 7 show the success-ul parameter space for different values of the initial inclination.

he models that begin with inclinations of 18 ° are similar to thoseith initial inclinations of 12 °; in particular, solutions with Earth

ime lags > 30 s result in inclination damping. Minor differences

rise because the additional energy in the inclined orbit tends to

rolong the lifetime of the magma ocean, despite the higher rate

f energy dissipation associated with the higher inclination values.

astly, it is worth noting that all models with initial inclination of

° are inconsistent with the current orbital configuration unlessater inclination excitation occurs. This is consistent with models

f inclination damping due to Earth tides alone ( Goldreich, 1966;

ouma and Wisdom, 1998 ).

The evolution of the Moon’s eccentricity is still debated (e.g.

arrick-Bethell et al., 2006; Meyer et al., 2010; Ćuk, 2011 ), but here

e have neglected it entirely. The main reason for doing so is to

larify the role of the magma ocean, which is the most novel as-

ect of our work. Eccentricity tidal heating in a deep magma ocean

s minimal, while obliquity tidal heating in the same ocean can

astly exceed eccentricity tidal heating in the solid Moon ( Chen

t al., 2014 ). We have therefore chosen here to focus on obliquity

nd inclination, rather than eccentricity, but it would clearly be of

nterest to include the latter effect in future work.

A final major assumption in our models is that the Moon did

ot suffer significant inclination excitation during or following

he period under consideration ( a � 30 R E ). Simulations of theoon-forming impact suggest that the initial inclination of the

rbit was likely < 1 ° directly following lunar accretion ( Ida et al.,997 ). Proposed mechanisms to excite the lunar inclination (e.g.

140 E.M.A. Chen, F. Nimmo / Icarus 275 (2016) 132–142

Fig. 5. Similar to Fig. 4 with variation of the drag coefficient c D in the fully coupled

model, plotted here with respect to the semi-major axis at time of magma ocean

solidification. Green diamonds indicate parameters that result in an inclination over

5 ° when the magma ocean solidifies, red X’s indicate models that have inclinations below. Successful cases are not present beyond the Cassini state transition (a ≈30 R E ). (For interpretation of the references to color in this figure legend, the reader

is referred to the web version of this article).

Fig. 6. Similar to Fig. 4 with variation of the initial inclination in the fully coupled

model. Green diamonds indicate parameters that result in an inclination over 5 °when the magma ocean solidifies, red X’s indicate models that have inclinations

below. (For interpretation of the references to color in this figure legend, the reader

is referred to the web version of this article).

m

t

q

t

l

a

f

l

c

n

m

a

n

w

p

o

d

s

m

p

e

E

e

t

(

w

c

Touma and Wisdom, 1998; Ward and Canup, 20 0 0; Ćuk and Stew-

art, 2012 ) occur very early in lunar history with the inclination

subsequently decreasing monotonically over time due to tides

( Peale and Cassen, 1978 ). Later resonances may have occurred (e.g.

Ćuk, 2007 ) but do not appear likely to have excited the primordial

lunar inclination. The most likely source of late inclination exci-

tation is close encounters with planetesimals not yet incorporated

into the terrestrial planets ( Pahlevan and Morbidelli, 2015 ). Such

encounters occur with decreasing probability as time proceeds,

but may have been common during the epoch of interest. Because

of its potential importance, we discuss the potential consequences

of late inclination excitation further below.

5. Implications and conclusions

In our model for the coupled thermal-orbital evolution of the

Moon that includes dissipation produced by the lunar magma

ocean, we find that dissipation in the ocean has a strong tendency

to damp the orbital inclination. The dissipation is related to the

Moon’s obliquity. Because of the occurrence of the Cassini state

transition at ∼30 R E , the obliquity increases rapidly near a semi-major axis of ∼20 R E and it is difficult to maintain a high orbitalinclination in the presence of enhanced dissipation due to the in-

creasing obliquity. Therefore, one solution is that the magma ocean

solidified prior to ∼20 R E (with the exact distance being sensitiveto the initial conditions and details of the thermal-orbital model),

in order for the inclination to not be damped to a value lower than

that which is currently observed, i = 5 . 16 ◦.

The evolution of the semi-major axis of the Moon’s orbit is

ainly controlled in our model by dissipation in the Earth. For

his solution, we find that the time lag of the early Earth is re-

uired to be quite small (less than ∼10 s) for a lunar magma oceanhat solidifies prior to a semi-major axis of ∼20 R E . This small timeag loosely translates to a constant tidal Q for the early Earth of

pproximately 300, roughly comparable to the present-day value

or the solid Earth ( Ray et al., 2001 ). This conclusion is robust to

ikely uncertainties in both initial inclination and the drag coeffi-

ient assumed. Our result can be compared to models without lu-

ar magma ocean dissipation, where the time lag can be larger by

any orders of magnitude, even up to ∼10 0 0 s, while still yielding higher inclination than present.

It may appear paradoxical to argue that dissipation in the lu-

ar magma ocean is an important source of inclination damping,

hile at the same time requiring dissipation in the Earth – which

resumably also possessed a magma ocean – to be small. The res-

lution of this paradox is simple: the fluid dynamical resonance

riving dissipation on the Moon is only applicable to bodies in

ynchronous rotation, and is thus not relevant to the Earth. Deep

agma oceans, in the absence of the kind of the resonance pro-

osed for the Moon, are not expected to be highly dissipative ( Sohl

t al., 1995 ).

A larger challenge may be to obtain high values for Q of the

arth after crystallization of the terrestrial magma ocean. Zahnle

t al. (2007) ; 2015 ) suggested that as the Earth cooled between

he liquidus and solidus, it became significantly more dissipative

Q E < 10) due to viscous dissipation in the mantle. The cooling rate

ould be limited by the rate at which an early thick atmosphere

ould transmit heat; however, this atmosphere probably had a

E.M.A. Chen, F. Nimmo / Icarus 275 (2016) 132–142 141

Fig. 7. Similar to Fig. 6 with variation of the initial inclination in the fully coupled

model. Successful cases are plotted against the orbital semi-major axis. Once again,

successful cases are not present beyond the Cassini state transition (a ≈ 30 R E ). The initial inclination must be at least 6 ° to be consistent with the current orbital con- figuration.

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ifetime of less than 10 Myr ( Abe, 1997; Abe and Matsui, 1988;

ahnle et al., 2007; 1988 ). Additionally, surficial liquid water was

robably present during the Hadean ( Mojzsis et al., 2001; Wilde

t al., 2001 ). To avoid high dissipation in the early Earth, either

his water was not global in extent or early oceans could not have

ontributed significantly to tidal dissipation as they do at present.

An alternative solution to requiring low dissipation in the early

arth is a scenario in which the lunar inclination was excited via

ollisionless encounters with remnant bodies at a later stage, as

as been recently proposed ( Pahlevan and Morbidelli, 2015 ). In this

ase, strong damping of the primordial orbital inclination could

ave occurred, but lunar magma ocean crystallization must still

ave been complete prior to the subsequent excitation. In contrast

ith the first solution, this second solution is actually more effec-

ive with strong tides on the early Earth and rapid outwards lunar

otion. Hence, with our model, determining the dissipative prop-

rties of the early Earth could in principle distinguish between the

arious solutions to the longstanding lunar inclination problem.

cknowledgments

We thank two anonymous reviewers for their insightful com-

ents and helpful suggestions.

eferences

be, Y. , 1997. Thermal and chemical evolution of the terrestrial magma ocean. Phys.

Earth Planet. In. 100, 27–39 . be, Y. , Matsui, T. , 1988. Evolution of an impact-generated H2OCO2 atmosphere and

formation of a Hot Proto-Ocean on Earth. J. Atmos. Sci. 45, 3081–3101 .

ills, B.G. , Nimmo, F. , 2008. Forced obliquity and moments of inertia of Titan. Icarus196, 293–297 .

ills, B.G. , Ray, R.D. , 1999. Lunar orbital evolution: A synthesis of recent results. Geo-phys. Res. Lett. 26, 3045–3048 .

org, L.E. , Norman, M. , Nyquist, L.E. , et al. , 1999. Isotopic studies of ferroananorthosite 62236: A young lunar crustal rock from a light rare-earth-elemen-

t-depleted source. Geochem. Cosmochim. Acta 63, 2679–2691 . urns, J.A. , 1986. The evolution of satellite orbits. In: Burns, J.A., Matthews, M.S.

(Eds.), Satellites. The University of Arizona Press, pp. 117–158 .

anup, R.M. , 2004. Dynamics of Lunar Formation. Annu. Rev. Astron. Astrophys. 30,237–257 .

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

anup, R.M. , Ward, W.R. , 2002. Formation of the galilean satellites: Conditions ofaccretion. Astron. J. 124, 3404–3423 .

arlson, R. , Lugmair, G.W. , 1988. The age of ferroan anorthosite 60025: oldest crust

on a young Moon? Earth Planet. Sci. Lett. 90, 119–130 . hen, E.M.A. , Nimmo, F. , Glatzmaier, G.A. , 2014. Tidal heating in icy satellite oceans.

Icarus 229, 11–30 . hyba, C.F. , Jankowski, D.G. , Nicholson, P.D. , 1989. Tidal evolution in the Nep-

tune–Triton system. Astron. Astrophys. 219, L23–L26 . olombo, G. , 1966. Cassini’s second and third laws. Astron. J. 71, 891–900 .

´ uk, M. , 2007. Excitation of Lunar Eccentricity by planetary resonances. Science 318,

244 . ´ uk, M. , 2011. Lunar shape does not record a past eccentric orbit. Icarus 211, 97–100 .´ uk, M. , Stewart, S.T. , 2012. Making the Moon from a fast-spinning Earth: A giant

impact followed by resonant despinning. Science 338, 1047–1052 .

arwin, G.H. , 1908. Scientific Papers, Volume II: Tidal Friction and Cosmogony. Cam-bridge University Press .

froimsky, M. , Makarov, V.V. , 2013. Tidal friction and tidal lagging. applicability lim-

itations of a popular formula for the tidal torque. Astrophys. J. 764:26 . froimsky, M. , Williams, J.G. , 2009. Tidal torques: A critical review of some tech-

niques. Celestial Mech. Dyn. Astron. 104, 257–289 . gbert, G.D. , Ray, R.D. , 20 0 0. Significant dissipation of tidal energy in the deep

ocean inferred from satellite altimeter data. Nature 405, 775–778 . gbert, G.D. , Ray, R.D. , 2001. Estimates of M 2 tidal energy dissipation from

TOPEX/Poseidon altimeter data. J. Geophys. Res. 106, 22475–22502 .

lkins-Tanton, L.T. , 2012. Magma oceans in the inner Solar System. Annu. Rev. EarthPlanet. Sci. 40, 113–139 .

lkins-Tanton, L.T. , Burgess, S. , Yin, Q.-Z. , 2011. The lunar magma ocean: Reconcilingthe solidification process with lunar petrology and geochronology. Earth Planet

Sci. Lett. 304, 326–336 . affney, A.M. , Borg, L.E. , 2014. A young solidification age for the lunar magma

ocean. Geochem. Cosmochim. Acta 227–240 .

arrick-Bethell, I. , Perera, V. , Nimmo, F. , et al. , 2014. The tidal-rotational shape ofthe Moon and evidence for polar wander. Nature 512, 181–184 .

arrick-Bethell, I. , Wisdom, J. , Zuber, M. , 2006. Evidence for a past high-eccentricitylunar orbit. Science 313, 652–655 .

oldreich, P. , 1966. History of the lunar orbit. Rev. Geophys. 4, 411–439 . oldreich, P. , Soter, S. , 1966. Q in the Solar System. Icarus 5, 375–389 .

ubbard, W.B. , Anderson, J.D. , 1978. Possible flyby measurements of galilean satel-lite interior structure. Icarus 33, 336–341 .

da, S. , Canup, R.M. , Stewart, G.R. , 1997. Lunar formation from an impact-generated

disk. Nature 389, 353–357 . ayne, S.R. , St. Laurent, L.C. , 2001. Parameterizing tidal dissipation over rough topog-

raphy. Geophys. Res. Lett. 28, 811–814 . effreys, H. , 1915. Certain hypotheses as to the internal structure of the Earth and

Moon. Mem. R. Astr. Soc. 60, 187–217 . amata, S., Matsuyama, I., Nimmo, F., 2015. Tidal resonance in icy satellites with

subsurface oceans. J. Geophys. Res. 120. doi: 10.1002/2015JE004821 .

eane, J., Matsuyama, I., 2014. Evidence for lunar true polar wander and a pastlow-eccentricity, synchronous lunar orbit. Geophys. Res. Lett. 41. doi: 10.1002/

2014GL061195 . onopliv, A.S. , Asmar, S.W. , Carranza, E. , et al. , 2001. Recent gravity models as a

result of the lunar prospector mission. Icarus 150, 1–18 . onopliv, A.S. , Binder, A.B. , Hood, L.L. , et al. , 1998. Improved gravity field of the

Moon from lunar prospector. Science 281, 1476–1480 .

umb, L.I. , Aldridge, K.D. , 1991. On viscosity estimates for the Earth’s fluid outer coreand core-mantle coupling. J. Geomag. Geoelectr. 43, 93–110 .

eyer, J. , Elkins-Tanton, L. , Wisdom, J. , 2010. Coupled thermal-orbital evolution ofthe early Moon. Icarus 208, 1–10 .

eyer, J. , Wisdom, J. , 2007. Tidal heating in Enceladus. Icarus 188, 535–539 . ignard, F. , 1979. The evolution of the lunar orbit revisted, I.. Moon Planets 20,

301–315 .

ignard, F. , 1980. The evolution of the lunar orbit revisted, II.. Moon Planets 23,185–201 .

ignard, F. , 1981. The evolution of the lunar orbit revisted, III.. Moon Planets 24,189–207 .

ojzsis, S.J. , Harrison, T.M. , Pidgeon, R.T. , 2001. Oxygen-isotope evidence from an-cient zircons for liquid water at the Earth’s surface 4,300 Myr ago. Nature 409,

178–181 .

unk, W. , 1997. Once again: once again – tidal friction. Prog. Oceanog. 40, 7–35 . unk, W.H. , MacDonald, G.J.F. , 1960. The Rotation of the Earth: A Geophysical Dis-

cussion. Cambridge University Press, New York . urray, C.D. , Dermott, S.F. , 1999. Solar System Dynamics. Cambridge University

Press, New York .

http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0001http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0001http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0002http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0002http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0002http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0003http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0003http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0003http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0004http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0004http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0004http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0005http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0005http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0005http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0005http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0005http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0006http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0006http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0007http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0007http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0008http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0008http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0008http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0009http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0009http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0009http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0010http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0010http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0010http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0011http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0011http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0011http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0011http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0012http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0012http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0012http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0012http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0013http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0013http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0014http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0014http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0015http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0015http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0016http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0016http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0016http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0017http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0017http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0018http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0018http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0018http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0019http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0019http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0019http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0020http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0020http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0020http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0021http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0021http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0021http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0022http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0022http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0023http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0023http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0023http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0023http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0024http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0024http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0024http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0025http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0025http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0025http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0025http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0025http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0026http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0026http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0026http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0026http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0027http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0027http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0028http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0028http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0028http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0029http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0029http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0029http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0030http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0030http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0030http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0030http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0031http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0031http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0031http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0032http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0032http://dx.doi.org/10.1002/2015JE004821http://dx.doi.org/10.1002/2014GL061195http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0035http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0035http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0035http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0035http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0035http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0036http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0036http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0036http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0036http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0036http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0037http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0037http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0037http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0038http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0038http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0038http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0038http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0039http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0039http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0039http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0040http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0040http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0041http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0041http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0042http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0042http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0043http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0043http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0043http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0043http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0044http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0044http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0045http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0045http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0045http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0046http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0046http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0046

142 E.M.A. Chen, F. Nimmo / Icarus 275 (2016) 132–142

T

T

T

T

T

W

W

W

W

Y

Z

Z

Nemchin, A. , Whitehouse, M.J. , Pidgeon, R. , et al. , 2006. Oxygen isotopic signatureof 4.4–3.9 Ga zircons as a monitor of differentiation processes on the Moon.

Geochem. Cosmochim. Acta 70, 1864–1872 . Nyquist, L.E. , Shih, C.Y. , Reese, Y.D. , et al. , 2010. Lunar crustal history recorded in

lunar anorthosites. In: Proceedings of 41st Lunar and Planetary Science Confer-ence . Abstract 1383.

Pahlevan, K. , Morbidelli, A. , 2015. Collisionless encounters and the origin of the lu-nar inclination. Nature 527, 4 92–4 94 .

Peale, S.J. , 1969. Generalized Cassini’s laws. Astron. J. 74, 4 83–4 89 .

Peale, S.J. , Cassen, P. , 1978. Contribution of tidal dissipation to lunar thermal history.Icarus 36, 245–269 .

Peale, S.J. , Cassen, P. , Reynolds, R.T. , 1979. Melting of Io by tidal dissipation. Science203, 892–894 .

Ray, R.D. , Eanes, R.J. , Lemoine, F.G. , 2001. Constraints on energy dissipation in theEarth’s body tide from satellite tracking and altimetry. Geophys. J. Int. 144,

471–480 .

Rubie, D. , Melosh, H. , Reid, J. , et al. , 2003. Mechanisms of metalsilicate equilibrationin the terrestrial magma ocean. Earth Planet Sci. Lett. 205, 239–255 .

Sagan, C. , Dermott, S.F. , 1982. The tide in the seas of Titan. Nature 300, 731–733 . Shearer, C.K. , Hess, P.C. , Wieczorek, M.A. , et al. , 2006. Thermal and magmatic evolu-

tion of the Moon. Rev. Mineral. Geochem. 60, 365–518 . Siegler, M.A., Bills, B.T., Paige, D.A., 2011. Effects of orbital evolution on lunar ice

stability. J. Geophys. Res.: Planets 116. doi: 10.1029/2010JE003652 .

Smith, J.V. , Anderson, A.T. , Newton, R.C. , et al. , 1970. Petrologic history of the mooninferred from petrography, mineralogy and petrogenesis of Apollo 11 rocks. In:

Proceedings Apollo 11 Lunar Science Conference, pp. 965–988 . Sohl, F. , Sears, W.D. , Lorenz, R.D. , 1995. Tidal Dissipation on Titan. Icarus 115,

278–294 . Solomatov, V.S. , 20 0 0. Fluid dynamics of a terrestrial magma ocean. In: Canup, R.M.,

Righter, K. (Eds.), Origin of the Earth and Moon. The University of Arizona Press,

pp. 323–338 . Solomon, S.C. , Longhi, J. , 1977. Magma oceanography: 1. Thermal evolution. Proc.

Lunar Sci. Conf. 8, 583–599 . Squyres, S.W. , Reynolds, R.T. , Cassen, P.M. , et al. , 1983. The Evolution of Enceladus.

Icarus 53, 319–331 . Tobie, G. , Mocquet, A. , Sotin, C. , 2005. Tidal dissipation within large icy satellites:

Applications to Europa and Titan. Icarus 177, 534–549 .

ouma, J. , Wisdom, J. , 1994. Evolution of the Earth-Moon system. Astron. J. 108,1943–1961 .

ouma, J. , Wisdom, J. , 1998. Resonances in the early evolution of the Earth-Moonsystem. Astron. J. 115, 1653–1663 .

urcotte, D.L. , Schubert, G. , 2002. Geodynamics, 2nd ed. Cambridge University Press,New York .

yler, R.H. , 2008. Strong ocean tidal flow and heating on moons of the outer planets.Nature 456, 770–772 .

yler, R.H., 2009. Ocean tides heat Enceladus. Geophys. Res. Lett. 36 . L15205, doi:

10.1029/20 09GL03830 0 . ard, W. , Canup, R. , 20 0 0. Origin of the Moon’s orbital inclination from resonant

disk interactions. Nature 403, 741–743 . ard, W.R. , 1975. Past orientation of the lunar spin axis. Science 189, 377–379 .

Wilde, S.A. , Valley, J.W. , Peck, W.H. , et al. , 2001. Evidence from detrital zircons forthe existence of continental crust and oceans on the Earth 4.4 Gyr ago. Nature

409, 175–178 .

Williams, D.A ., Fagents, S.A ., Greeley, R., 20 0 0. A reassessment of the emplacementand erosional potential of turbulent, low-viscosity lavas on the moon. J. Geo-

phys. Res. 105 (E8), 20189–20205. doi: 10.1029/1999JE001220 . illiams, G.E. , 20 0 0. Geological constraints on the Precambrian history of Earth’s

rotation and the Moon’s orbit. Rev. Geophys. 38, 37–60 . Wisdom, J. , 2008. Tidal dissipation at arbitrary eccentricity and obliquity. Icarus 193,

637–640 .

ood, J.A. , Dickey, J.S.J. , Marvin, U.B. , et al. , 1970. Lunar anorthosites and a geo-physical model of the Moon. In: Proceedings of the Apollo 11 Lunar Science

Conference, pp. 965–988 . oder, C.F. , 1995. Astrometric and geodetic properties of earth and the Solar System.

In: Ahrens, T.J. (Ed.), Global Earth Physics: A Handbook of Physical Constants.American Geophysical Union, pp. 1–31 .

ahnle, K. , Arndt, N. , Cockell, C. , et al. , 2007. Emergence of a habitable planet. Space

Sci. Rev. 129, 35–78 . ahnle, K.J. , Kasting, J.F. , Pollack, J.B. , 1988. Evolution of a steam atmosphere during

Earth’s accretion. Icarus 74, 62–97 . Zahnle, K.J. , Lupu, R. , Dobrovolskis, A. , et al. , 2015. The tethered Moon. Earth Planet

Sci. Lett. 427, 74–82 .

http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0047http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0047http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0047http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0047http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0047http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0048http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0048http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0048http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0048http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0048http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0049http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0049http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0049http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0050http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0050http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0051http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0051http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0051http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0052http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0052http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0052http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0052http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0053http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0053http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0053http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0053http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0054http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0054http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0054http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0054http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0054http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0055http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0055http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0055http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0056http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0056http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0056http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0056http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0056http://dx.doi.org/10.1029/2010JE003652http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0058http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0058http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0058http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0058http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0058http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0059http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0059http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0059http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0059http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0060http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0060http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0061http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0061http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0061http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0062http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0062http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0062http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0062http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0062http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0063http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0063http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0063http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0063http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0064http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0064http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0064http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0065http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0065http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0065http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0066http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0066http://refhub.elsevier.com/S0019-1035(16)30058-6/sbref0066http://refhub.elsevier.com/S0019-1035(16)30058-6/