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Contraction or expansion of
the Moon’s crust during
magma ocean freezing?
Linda T. Elkins-Tanton1,† and David Bercovici2
rsta.royalsocietypublishing.org
Research
Cite this article: Elkins-Tanton LT, Bercovici D.
2014 Contraction or expansion of the Moon’s
crust during magma ocean freezing? Phil.
Trans. R. Soc. A 372: 20130240.
http://dx.doi.org/10.1098/rsta.2013.0240
One contribution of 19 to a Discussion Meeting
Issue ‘Origin of the Moon’.
Subject Areas:
solar system
Keywords:
lunar crust, magma ocean,
lithospheric deformation
Author for correspondence:
Linda T. Elkins-Tanton
e-mail: [email protected]
†
Present address: School of Earth and Space
Exploration, Arizona State University, Tempe
AZ, USA.
1 Department of Terrestrial Magnetism, Carnegie Institution
for Science, Washington DC, USA
2 Department of Geology and Geophysics, Yale University,
New Haven, CT, USA
ET, 0000-0003-4008-1098
The lack of contraction features on the Moon has
been used to argue that the Moon underwent limited
secular cooling, and thus had a relatively cool initial
state. A cool early state in turn limits the depth
of the lunar magma ocean. Recent GRAIL gravity
measurements, however, suggest that dikes were
emplaced in the lower crust, requiring global lunar
expansion. Starting from the magma ocean state, we
show that solidification of the lunar magma ocean
would most likely result in expansion of the young
lunar crust, and that viscous relaxation of the crust
would prevent early tectonic features of contraction
or expansion from being recorded permanently.
The most likely process for creating the expansion
recorded by the dikes is melting during cumulate
overturn of the newly solidified lunar mantle.
1. Introduction
Since the first Apollo samples were returned in 1969,
the Moon has been posited to have begun in at least
a partially molten state [1,2]. The Moon’s surface,
however, does not show large-scale global features
created by contraction [3,4]. Earlier geodynamic models
of lunar contraction suggest that the lunar magma ocean
was constrained in depth, because freezing of a deep
magma ocean would accumulate more crustal strain
than is observed on the Moon [5,6]. More recent work
by Zhang et al. [7] indicated that magma ocean depth
2014 The Authors. Published by the Royal Society under the terms of the
Creative Commons Attribution License http://creativecommons.org/licenses/
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source are credited.
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ρs − ρl
Vl ,
ρl
(1.1)
where Vl is the initial melt volume. The associated surface strain would then simply be
R (ρs − ρl )Vl
=
.
R0
4π R30 ρl
(1.2)
The solidifying Moon is not, however, a simple bulk system that transfers its entire volumetric
change directly to surface strain. The critical difference is that while crystallization of mafic
phases (whose solids are usually denser than their coexisting liquid) results in contraction,
crystallization of plagioclase (which is less dense than the coexisting liquid) is associated with
expansion. Thus, freezing of an assemblage with sufficient plagioclase will result in expansion
during solidification (see also Kirk & Stevenson [9], who invoke expansion during a late gabbroic
melting episode). Thus, two significant modifications have to be made to the simple assessments
of lunar contraction or expansion during freezing:
1. The relative densities of solid and liquid are sensitive to liquid composition, particularly
TiO2 , FeO and SiO2 contents. Therefore, tracking the composition of the evolving magma
ocean liquid and the corresponding equilibrium composition of the solidifying minerals
over a number of solidification steps is critical to calculate contraction or expansion of the
body.
2. The plagioclase flotation crust, in which the strains might be recorded, is not formed until
after about 80% of the magma ocean is solidified. Before that point, the solids are overlain
by a liquid magma layer and will not record strain.
Here, we present compositional models of the freezing magma ocean that indicate either
contraction or expansion is possible in the flotation crust. Further, we derive wavelengths and
timescales for the folding that would predate faulting in a contracting lunar flotation crust, and
we show that relaxation would occur so rapidly that any sign of contraction would likely be lost.
2. Models of solidification
(a) Density evolution during magma ocean freezing
We consider three lunar magma ocean scenarios. In two cases (labelled LMO1 and LMO3), the
magma ocean has the same bulk composition as was used in Elkins-Tanton et al. [10], but with
slightly different and equally plausible mineral assemblages during solidification. In the third
case, LMO2, the lunar primitive upper mantle composition from Longhi [11] is used.
The solidifying mineral assemblage and the point at which the flotation crust forms are critical
to analysing the possible crustal strain that resulted from solidification, and they depend upon
both the bulk composition of the lunar magma ocean and whether the solidification process is
fractional, bulk or a mixture. Here, we assume pure fractional solidification produced by ongoing
.........................................................
V =
2
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 372: 20130240
was not a constraint on contraction later in lunar history during thermal evolution following
magma ocean solidification.
Gravity data from the GRAIL mission have recently been interpreted as showing dike
intrusions into the lunar lower crust, which require 0.6–4.9 km expansion of the Moon during
crustal formation [8]. Here, we test whether the Moon would be expected to expand or contract
during its earliest history, and whether these dikes may have been created during that time.
The lack of contraction features on the Moon has been used to argue that the Moon underwent
limited secular cooling, and thus had a relatively cool initial state, indicating in turn that the lunar
magma ocean had limited depth. Extensive contraction during cooling from a hot initial state had
been expected to result in thrust faults and scarps, as on Mercury.
The net change in planet volume owing to freezing the magma ocean is
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Table 1. Lunar magma ocean bulk compositions, renormalized.
3
46.2
44.7
43.4
48.5
43.8
44.3
46.3
44.6
48.5
TiO2
0.40
0.17
0.30
0.39
0.40
0.30
0.00
0.30
0.17
0.40
Al2 O3 4.0
3.9
3.7
7.6
5.0
6.0
3.8
7.0
3.9
5.0
FeO
12.0
7.6
13.9
13.0
12.9
13.1
12.7
12.5
12.4
12.0
MgO 33.1
38.3
33.5
29.2
29.0
32.2
35.6
27.8
35.1
30.0
3.2
3.4
6.1
3.8
4.5
3.2
5.5
3.3
3.8
0.52
0.40
0.30
0.30
0.00
0.37
0.50
0.47
0.30
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
CaO
3.0
..........................................................................................................................................................................................................
Cr2 O3 0.30
..........................................................................................................................................................................................................
Mg# 83.1
90
81.1
80.0
80.0
81.4
83.3
79.9
83.5
81.6
..........................................................................................................................................................................................................
Table 2. Mineral assemblages used to model a lunar magma ocean solidifying from the bottom upwards.
composition
LMO1 (%)
Elkins-Tanton
et al. [10]
LMO2 (%)
Longhi
[11]
LMO3 (%)
Elkins-Tanton
et al. [10]
from 3.8 (1000 km depth)
to 3.0 GPa (700 km depth)
..........................................................................................................................................................................................................
olivine
100
100
100
..........................................................................................................................................................................................................
from 3.0 to 0.6 GPa (110 km)
..........................................................................................................................................................................................................
orthopyroxene
90
80
100
olivine
10
20
0
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
from 0.6 to 0.5 GPa (90 km)
..........................................................................................................................................................................................................
olivine
20
20
20
clinopyroxene
20
20
20
plagioclase
50
45
50
orthopyroxene
10
15
10
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
from 0.5 GPa to the surface
..........................................................................................................................................................................................................
plagioclase
40
45
40
clinopyroxene
20
20
20
orthopyroxene
30
30
30
oxides
10
5
10
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
separation owing to settling or ascent of crystals. Unfortunately, bulk magma ocean compositions
are unknown for any planet, and a fairly wide range of compositions have been proposed for the
Moon (tables 1 and 2).
The pressure boundaries and mineral assemblages predicted for a fractionally solidifying
lunar magma ocean are broadly consistent among researchers. There are currently no complete
experimental fractional crystallization studies of relevant lunar compositions on which to base
these mineral assemblages. Previous studies, including those on which we base our analysis, used
theoretical calculations of phase stability [12,13,15–18,20–22].
.........................................................
SiO2 47.1
..........................................................................................................................................................................................................
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 372: 20130240
Elkins-Tanton Longhi Ring-wood & Morgan Buck &
Taylor Wänke &
Warren O’Neill Snyder
et al. [10]
[11]
Kesson [12] et al. [13] Toksöz [14] [15] Dreibus [16] [17]
[18]
et al. [19]
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Previous workers [10] investigated the constraints a given bulk composition exerts on the
solidifying phase assemblages. They found that in the absence of tightly constraining phase
equilibrium data, there is significant leeway—without violating bounds on the bulk magma
ocean composition—in both the ratios of phases within a solidifying assemblage (e.g. 40–60%
plagioclase is allowed when the magma ocean is between 0.80 and 0.92 solidified) and with the
stage at which the solidifying assemblage changes (e.g. plagioclase begins to crystallize when 70%
to more than 80% of the magma ocean has solidified).
In our model, the mineral assemblages allow solidification to proceed to 92% before
expending any of the major oxides. At high degrees of solidification, the standard mineral
assemblages shown in table 2 would no longer pertain, and the liquid would approach a highly
evolved KREEP composition enriched in incompatible elements. No satisfactory model for the
solidifying assemblage of a very late-stage liquid exists, thus the final 8% is assumed to solidify
in bulk.
In each case, the magma ocean is assumed to be 1000 km deep, and to retain 1% interstitial
liquid at each step of solidification which proceeds in equal volume increments δV; as shown
by Elkins-Tanton et al. [10], values of δV = Vl /1000 (where recall Vl is the total initial magma
ocean volume) provide sufficient model convergence. Other details of the solidification model
are described fully in Elkins-Tanton et al. [10]. At each step of solidification, the fractional
change in volume produced by solidification is calculated by the ratio of liquid density to
solid density. The liquid density is calculated first at as a function of composition, temperature
and pressure. Composition and temperature dependence are from Lange & Carmichael [23].
Pressure dependence, which includes only the first derivative of V with respect to temperature
T, is from [24]. Further updates are from Liu & Lange [25] and from Ochs & Lange [26,27].
These calculations are expected to be accurate only to about 2 GPa, so here we use them for
1 atm calculations only, and continue the density to depth using a third-order Birch–Murnaghan
equation of state, after the technique of Delano [28].
To calculate the density of solids, first their composition is calculated in equilibrium with the
coexisting magma ocean composition using experimental distribution coefficients [29,30], and
then densities are calculated based on their temperature, pressure and composition (databases
are given in [30]). The molar volume of the mineral phase is at one atmosphere, and the model
temperature is assumed to be the solidus temperature at the depth of solidification. The pressure
effect on density is then calculated separately for each phase using appropriate moduli and a
third-order Burch–Murnaghan equation of state, which is solved iteratively for the molar volume
at a given pressure and temperature by adjusting the pressure P from the Burch–Murnaghan
equation until it matches the model pressure. For details of the calculation, see Elkins-Tanton
et al. [29] and for updated thermophysical parameters, see the electronic supplementary material
for Elkins-Tanton [30].
In each of the models, the evolving liquid becomes richer in iron and titanium as solidification
proceeds, and the densities of the resulting mineral assemblages also increase (figure 1). The slight
differences in the solidifying mineral assemblages used for models LMO1 and LMO3 cause the
solid densities to diverge towards the end of solidification, even though the models have the same
bulk composition. This divergence shows the sensitivity of the system to parameters whose true
values we do not know.
The final liquid densities (figure 1) are not simply a product of iron enrichment (table 2). The
final liquid for LMO3 is far denser than the other model liquids, in part because its high iron is
not balanced with the high quantities of low-density alumina in the LMO1 liquid, nor with the
higher MgO in LMO2 liquid. Note also that the smaller fraction of oxides being removed from
LMO2 has produced high chromium in that liquid, also affecting its density. Thus, the density
(volume) of the fractionating solids is not always constant with respect to the density (volume)
of the liquids, because their relationship depends upon the partitioning of each element between
the liquid and solid. Thus, a simple Fe–Mg or Fe–Si model will not make an acceptable prediction
for the volume change of a solidifying Moon.
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1700
LMO1
LMO3
LMO2
LMO3
LMO2
1500
radius (km)
1400
1300
1200
1100
1000
900
800
3000
3200
3400 3600 3800
density of
evolving liquid (kg m–3)
3000
3200 3400 3600 3800
density of fractionating
solid assemblage (kg m–3)
4000
Figure 1. Density of evolving magma ocean liquids and solids given at the depth of solidification, assuming solidification
proceeds from the bottom of a 1000 km deep magma ocean to surface. Because density is given at the depth and pressure
of crystallization, the floating lid exists above the top of these data. (Online version in colour.)
Table 3. Magma ocean liquids after 92% solidification, in wt%. Mg# is defined as molar (Mg/[Mg+Fe]).
model
LMO1
SiO2
5.34
Al2 O3
31.05
FeO
30.51
MgO
3.50
CaO
26.08
Sm
0.0008
Nd
0.025
Lu
0.0001
Hf
0.0006
TiO2
2.52
Cr2 O3
0.98
Mg#
36
LMO2
3.00
28.21
25.75
12.66
25.67
0.0007
0.022
0.0001
0.0005
0.55
4.15
70
LMO3
3.00
26.47
37.04
7.78
22.64
0.0007
0.022
0.0001
0.0005
2.22
0.82
48
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
The late-stage liquids shown in table 3 are also geochemically pertinent. These liquids
represent the candidate predecessors to KREEP, the last, incompatible-element-enriched melts
from the magma ocean that appear in trace amounts in some lunar samples [31]. The liquids
in table 3 are the final 8% of the magma ocean, and KREEP itself is likely to be a later-stage
fractionate, but its original composition can only be modelled from samples. The liquids shown
here, however, are sufficiently different from each other that constraints from natural samples
may be applied to models using these liquids.
Figure 2 shows the cumulative radius change as the body solidifies. The models show
planetary contraction of about 13 km during the olivine-only lowest layer of solidification. During
the second layer of solidification, consisting of olivine and orthopyroxene, models LMO1 and
LMO2 indicate continued contraction of an additional approximately 7 km, whereas model LMO3
shows expansion of about a kilometre. None of these radius changes would be recorded in the
lunar crust, however, because no solid lid has formed on top of the magma ocean, so the liquid
uppermost layer simply conforms to whatever contraction or expansion is happening beneath it.
A solid lid prior to plagioclase flotation is unlikely on the Moon. Even a thin nebular
atmosphere may hold the surface above its solidus [32] and prevent formation of a quench crust
(thus, no quench crust would be expected on larger planets such as the Moon or the Earth).
Should surface temperatures exist that allow a quench crust, this crust would be denser than
.........................................................
LMO1
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1600
5
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1.0
LMO1
LMO2
0.9
6
LMO3
plagioclase + oxides + opx + cpx
fraction solidified
0.6
0.5
0.4
0.3
0.2
olivine only
0.1
0
–25
–20
–15
–10
total radius change (km)
–5
0
Figure 2. Cumulative radius change with solidification. (Online version in colour.)
its underlying magma and would rapidly founder [33,34]. A possible quench crust on the Moon
is critically different from a lava lake on the Earth: the lava crust clings to the solid sides of the
lake, but on the Moon, its density would cause it to sink. Thus, on the Moon, the first solid lid
would be the plagioclase flotation crust.
All these models assume that plagioclase begins to form at 80% solidification. We further
assume that 5 km of plagioclase flotation is required to make a competent lid (as was assumed
in [10]), and this requires about 2% additional solidification beyond the 80% point (this depends,
of course, upon the fraction of plagioclase in the solidifying assemblage, and assumes perfect and
instantaneous flotation; the choice of 5 km as a competent lid is arbitrary). This is the critical
stage, after the plagioclase lid forms and can record strain. Models LMO2 and LMO3 show
approximately 4 km expansion over the remaining solidification, whereas LMO1 shows just a
hundred metres of further contraction. Models LMO1 and LMO3 produce approximately 46 km of
flotation crust, and LMO2 produces approximately 48 km. These estimates are somewhat higher
than the current best average crustal thickness estimates, between 34 and 43 km [35].
The differences in volume change over the last 20% of solidification among the models are
produced in part by slight differences in mineralogy, but in larger part by the different relative
densities of the late-stage liquids. In some cases, even pyroxenes become less dense than the
coexisting liquid, which is of possible significance, because there is an enigmatic pyroxene fraction
in much of the anorthosite flotation lid on the Moon [36]. The lunar magma ocean models shown
here indicate that pyroxene may in some cases naturally float into the plagioclase lid as it forms,
and not require any more complex explanation.
We conclude that expansion or contraction of the solidifying Moon was sensitive to both
bulk composition and solidifying assemblages, to a degree that we cannot address because these
compositions are not sufficiently constrained for any planet. Further, the expansion or contraction
that happens after a solid crust is formed is the only expansion or contraction that will be
recorded, and in the case of magma ocean processes, this encompasses only the last 20–30% of
solidification, and may be too small to be recorded.
Gravity data from the GRAIL mission have recently been interpreted as showing dike
intrusions into the lunar lower crust, which would require 0.6–4.9 km expansion of the Moon
during crustal formation [8], consistent with the models presented here. The amount of
.........................................................
olivine + orthopyroxene
0.7
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plagioclase + olivine + opx + cpx
0.8
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(a) Folding during contraction
As a magma ocean freezes, its floating crust could conceivably shorten and undergo folding
during planetary contraction, and these folding instabilities would lead to features such as thrusts
and scarps. Because the crust is still hot and accumulating, we treat it as a viscous layer, bounded
above and below by relatively inviscid medium (vacuum or tenuous atmosphere above, magma
below). The crustal layer is of thickness h, and we allow the origin plane z = 0 to be at the middepth, so the top and bottom are at z = ±h/2. If the layer is deflected vertically during folding at
some horizontal position x by a vertical displacement w, then the force balance on a thin vertical
segment of the layer implies (removing the hydrostatic balance)
0=
∂
∂x
w+h/2
w−h/2
σxz dz − (ρm − ρa )gw,
(3.1)
where σxz is the shear-stress on the vertical sides of the segment, ρm is the magma-ocean density,
ρa is the ‘atmospheric’ density. We have assumed there is no load to the crustal layer other than
its own deflection (i.e. no volcanic masses). The net torque on this segment is
0=
where
w+h/2
w−h/2
σxz dz −
∂
∂x
w+h/2
w−h/2
σxx z dz,
∂ 3w
h
− z + (ρc − ρa )gw + τ0 − 2μz 2 ,
σxx = − Pa + ρc g
2
∂x ∂t
(3.2)
(3.3)
where the term in [..] is the hydrostatic pressure at some depth z inside the crust, Pa is the
atmospheric pressure at an undeflected surface z = h/2, ρc is crustal density, τ0 is a uniform
imposed normal stress (in the x-direction owing to contraction) and the last term on the righthand side of (3.3) is the non-uniform deviatoric stress, owing to curvature of the layer [37], in
which μ is crustal viscosity. After carrying through the integration in (3.2) on σxx as given by (3.3),
the torque equation becomes (to first order in w)
0=
w+h/2
w−h/2
σxz dz + Π h
∂w μh3 ∂ 4 w
+
,
∂x
6 ∂x3 ∂t
(3.4)
where
Π = Pa + ρc gh/2 − τ0 .
(3.5)
Substituting (3.4) into (3.1), we arrive at the governing equation for the deflection w
0 = ρgw + Π h
where ρ = ρm − ρa .
∂ 2 w μh3 ∂ 5 w
,
+
6 ∂x4 ∂t
∂x2
(3.6)
.........................................................
3. Crustal folding and relaxation
7
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contraction or expansion is not a constraint on the original depth of the magma ocean; as
demonstrated here, a wide range of expansion or contraction values can be obtained for a single
magma ocean depth.
These models demonstrate that either contraction or expansion is possible within the range of
suggested bulk compositions and solidifying assemblages. If, in the limiting case that the Moon
experienced contraction, would thrust faults be expected? In §3, we derive the growth rate and
wavelength of the folding that would result from contraction and precede thrust faulting, and
discuss whether any vestige of this process would be preserved for detection in the present day.
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(i) Non-dimensional equations and folding growth rate
8
Non-dimensionalizing space and time according to
Πh x
ρg
and t =
μρgh t
6Π 2
(3.7)
(3.6) becomes (after dropping the primes)
0=w+
∂ 5w
∂ 2w
.
+
∂x2
∂x4 ∂t
(3.8)
Using normal mode analysis, we assume w ∼ eikx+st in that case (3.8) yields a simple relation for
the growth rate s
s=
k2 − 1
.
k4
(3.9)
The growth rate is√of course negative for k < 1 and goes to 0 for k → ∞. But s is maximum at
wavenumber km = 2 at which the growth rate is sm = 14 .
Thus, the dimensional wavelength for the fastest growing folding mode is
λm = π
2Π h
,
ρg
(3.10)
and the minimum growth time for folding (i.e. inverse of the growth rate for the fastest growing
folding mode) is
tm =
2μρgh
.
3Π 2
(3.11)
(ii) Contraction stresses
The wavelength and time-scale for folding instability (equations (3.10) and (3.11)) depend on
the values of Π and hence the choice for τ0 . There are two end-member values for τ0 , which
would be (i) the compressive stresses are due to the magma ocean contracting faster than the
crust can shorten and thicken, and (ii) the stresses occur as the shortening/thickening keeps pace
with magma ocean contraction. In the first case, we can estimate the lateral compressive stress
by considering a cylindrical axisymmetric geometry and how the hoop stress relates to radial or
vertical load; in this case, the momentum equation in the radial direction is
−
1
∂P ∂τrr
+
+ (τrr − τθθ ) − ρc g = 0,
∂r
∂r
r
(3.12)
where r is radial distance from the centre of the planet, and we have ignored shear stresses,
because motion is only radial for net contraction. (Because the folding theory is in two
dimensions, this analysis is best compared with an axisymmetric geometry, but the result would
be the same if we used a spherically axisymmetric geometry.) Because the crustal displacement w
is independent of z and hence r, then τrr = 0, and hence the lateral contraction stress is
∂P
r.
τ0 = τθθ = − ρc g +
∂r
If the system is hydrostatic then τ0 = 0.
(3.13)
.........................................................
x=
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λm = π
ρc rh
ρ
and tm =
2μρh
3ρc2 gr2
,
(3.14)
respectively. Using ρc ≈ 2700 kg/m3 , ρ/ρc ≈ 1, g = 1.6 m/s2 , r = 1740 km, h = 20 − 100 km, then
λm ≈ 800 − 2000 km and tm < 0.2 years.
Condition 2 is if contraction of the crust keeps pace with contraction of the magma ocean.
In this case, τ0 = τθθ = 2μṙ/r, where ṙ is given by the contraction rate. The contraction rate
is determined by the fact that the net planetary mass M = 43 π r3 ρ̄ is conserved, where ρ̄ is
the planet’s mean density, which changes during freezing and contraction (see §2a); we thus
˙ ρ̄. In this case, Π = ρc gh/2 + 2 μρ̄/
˙ ρ̄. Using ρc =
˙ ρ̄, and therefore τ0 = − 2 μρ̄/
obtain ṙ/r = − 13 ρ̄/
3
3
−3
−2
18
2700 kg m , g = 1.6 m s and h = 100 km, μ ≈ 10 Pa s, then τ0 is only comparable to ρc gh/2
˙ ρ̄ > 10−2 yr−1 , i.e. if the contraction timescale is shorter than about 100 years, which is
if ρ̄/
˙ ρ̄ and thus, Π ≈ ρc gh/2. In this case, the wavelength
unlikely; thus, we assume ρc gh/2 23 μρ̄/
and timescale for the least-stable folding mode are
ρc
λm = π h
ρ
and tm =
8μρ
3ρc2 gh
,
(3.15)
respectively. Using the same parameters as above then λm ≈ 60 − 300 km, and tm ≈ 200 −
1000 years.
Although the folding times for either case are short, it is important to compare them with the
relaxation time over which the warped crust flattens viscously.
(iii) Crustal viscous relaxation
The viscous folding theory assumes that the crustal layer is folded without changing its net
thickness h. However, gravitational sliding would cause this thickness to adjust spatially in order
to erase the deflection w. A complete folding theory would allow for variations in h, however, that
is beyond the scope of this simple analysis. Nevertheless, we can estimate the relaxation time for
w given pressure gradients and gravitational sliding owing to the folding instability.
The balance of forces parallel to the crustal surface on a segment of crust that is dx wide and h
thick is
∂ w+h/2
(3.16)
n̂ · σ · n̂ dz − ρc gh dxẑ · n̂,
0 = dx
∂x w−h/2
where σ is the stress tensor (and assuming stress-free top and bottom), and n̂ is the unit vector
∂w
parallel to the surface given by z = w(x), i.e. n̂ = x̂ + ẑ ∂w
∂x (assuming ∂x 1). Using σ = −PI + 2μė
then to first order in w, we arrive at
0=−
∂
∂x
w+h/2
w−h/2
P dz + 2hμ
∂ 2 vx
∂w
,
− ρc gh
∂x
∂x2
(3.17)
where vx is the horizontal velocity of a segment of crust (which we assume is independent of z
given the free-slip top and bottom). The hydrostatic pressure at position (x, z) is approximately
ρw
gz,
P(x, z) = P0 − ρc −
h
(3.18)
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Condition 1 is if there is no support from hydrostatic pressure from below because contraction
is too fast, and then we obtain τ0 = −ρc gr, which is the largest possible contraction stress. In
this case, Π = Pa + ρc g(h/2 + r) ≈ ρc gr, because Pa ≈ 0 given the Moon’s lack of atmosphere, and
h r. Then, the wavelength and timescale for the least stable folding mode are
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∂w
∂ 2 vx
+ 2μ 2 ,
∂x
∂x
(3.19)
where ρ̄ = 12 (ρm + ρa ). Integrating in x implies
∂vx (ρc − ρ̄)gw + τ0
=
,
∂x
2μ
(3.20)
where τ0 is the background in-plane stress. As the crustal deflection relaxes viscously,
conservation of mass requires that
∂hvx
(ρc − ρ̄)gh
hτ0
∂w
=−
=−
w−
,
∂t
∂x
2μ
2μ
(3.21)
where the first term on the right-hand side represents the decay of the deflection, whereas the
second term represents the uniform thickening owing to contraction stresses τ0 < 0, as discussed
above. The relaxation timescale is thus
tr =
2μ
.
(ρc − ρ̄)gh
(3.22)
Given the above parameter values this timescale is about 300–1500 years, which is only a bit
larger than the longest growth time for the folding instability, and is also very short. This implies
the folding instability is fairly rapid and is established quickly, but then relaxes away almost
as quickly. Thus, given the short timescales for folding and relaxation, it is very unlikely that
compressional features would persist in a floating crust on a contracting lunar magma ocean.
4. Discussion and conclusions
Based on both extensive petrologic evidence and modelling, the Moon accreted with a magma
ocean of some depth, and the magma solidified via fractional crystallization, differentiating
its silicates into a heterogeneous cumulate mantle, late-stage incompatible element-enriched
materials, and an anorthositic flotation crust [1,2,38–43].
No crust exists to record strain owing to volumetric change within the Moon until the
anorthositic crust is developed to some thickness, which is not likely to be obtained until after
at least 80% solidification by volume. In some cases, during the last stages of magma ocean
solidification, both pyroxenes and plagioclase become less dense than the coexisting liquid. The
simultaneous flotation of both phases may explain the enigmatic pyroxene fraction in much of
the anorthosite flotation lid on the Moon [36].
Expansion or contraction during solidification is controlled sensitively by the thermochemical
parameters of the solidifying phases compared with the liquid; the underlying solid mantle will
remain at its solidus temperature and not thermally contract until solidification is complete and
the conductive cooling front reaches the mantle. Unfortunately, bulk silicate composition and
solidifying assemblages are not sufficiently well known on any planet to make accurate forward
predictions of expansion or contraction. The analysis of magma ocean solidification shown here,
however, favours expansion over contraction. If the lunar crust did contract, then the folding
timescale and relaxation timescale are both very short, on the order of hundreds to thousands of
years, and thus any folds caused by contraction would relax away and not appear in the present
day as either faults or folds.
Andrews-Hanna et al. [8] suggest that intrusions in the lunar lower crust imaged with GRAIL
gravity data require 0.6–4.9 km expansion of the Moon during crustal formation, consistent
with the models presented here. The amount of contraction or expansion is not a constraint
.........................................................
0 = −(ρc − ρ̄)g
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where P0 is determined by the boundary condition that P(x, h/2 + w) = Pa − ρa gw, where Pa is
the atmospheric pressure at z = h/2. (Equation (3.18) comes about by saying that at the surface
P(x, h/2 + w) = Pa − ρa gw, and at the base of the crust P(x, −h/2 + w) = Pa + ρc gh − ρm gw, which
assumes that the depth z = −h/2 is a constant pressure level in that the magma ocean always
equilibrates pressure below this depth). In this case, (3.17) becomes
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