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Earth and Planetary Science Letters 271 (2008) 181–191

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Earth and Planetary Science Letters

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Linked magma ocean solidification and atmospheric growth for Earth and Mars

L.T. Elkins-Tanton ⁎Massachusetts Institute of Technology, Department of Earth, Atmospheric, and Planetary Sciences, Building 54-824, 77 Massachusetts Avenue, Cambridge MA 02139, United States

⁎ Tel.: +1 617 253 1902.E-mail address: [email protected].

0012-821X/$ – see front matter © 2008 Elsevier B.V. Aldoi:10.1016/j.epsl.2008.03.062

A B S T R A C T

Article history:

Editor: R.W. Carlson

Keywords:magma oceanMarsEarthatmosphereclement conditions

olution energetic impact, radiodecay, and core formation may have created oneor more whole or partial silicate mantle magma oceans. The time to mantle solidification and then to clementsurface conditions allowing liquid water is highly dependent upon heat flux from the planetary surfacethrough a growing primitive atmosphere. Here we model the time to clement conditions for whole andpartial magma oceans on the Earth and Mars, and the resulting silicate mantle volatile compositions.Included in our calculations are partitioning of water and carbon dioxide between solidifying mantlecumulate mineral assemblages, evolving liquid compositions, and a growing atmosphere. We find that smallinitial volatile contents (0.05 wt.% H2O, 0.01 wt.% CO2) can produce atmospheres in excess of 100 bars, andthat mantle solidification is 98% complete in less than 5 Myr for all magma oceans investigated on both Earthand Mars, and less than 100,000 yr for low-volatile magma oceans. Subsequent cooling to clement surfaceconditions occurs in five to tens of Ma, underscoring the likelihood of serial magma oceans and punctuatedclement conditions in the early planets. Cumulate mantles are volatile-bearing and stably stratified followingsolidification, inhibiting the onset of thermal convection but allowing for further water and carbon emissionsfrom volcanoes even in the absence of plate tectonics. Models thus produce a new hypothetical starting pointfor mantle evolution in the terrestrial planets.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Early in terrestrial planet formation the heat of accretion, radio-decay, and core formationmay have melted the silicate portions of theplanets either wholly or partially (Wood et al., 1970; Solomon, 1979;Wetherill, 1990). Though whole-mantle magma oceans remainpossible but unproven, there is a great likelihood that giant impactorslate in accretion created hemispheric or shallow planetary magmaoceans (Tonks and Melosh, 1993; Reese and Solomatov, 2006).

For calculating the cooling rate and time to solidification of apartially or wholly molten terrestrial planet, the most important gasesentering the early atmosphere are water and carbon dioxide. Thesetwo components, both relatively common in the potential planetarydisk building blocks of the terrestrial planets, are greenhouse gasesand will significantly slow planetary cooling and retain a hightemperature at the surface of the magma ocean. The importance ofan early water atmosphere has been recognized and studied by Abe(1993, 1997), Abe and Matsui (1988), Matsui and Abe (1986), andZahnle et al. (1988).

Here we add carbon dioxide and a further treatment of internalplanetary physics and chemistry to the calculations of magma oceansolidification, atmospheric growth, and subsequent mantle overturnto gravitational stability for the Earth and Mars. Carbon dioxide adds

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significantly to planetary surface heat retention during solidificationand may play a critical role in the oxygen fugacity of the planetaryinterior. The calculations on mantle overturn to stability producenovel predictions concerning likely initial mantle density andcompositional profiles in both planets. The interior locations ofradiogenic heat sources and reservoirs of water and carbon exertcontrols over the onset of mantle convection, melting source regioncompositions, and early degassing from the solid planet.

Results presented here model 500, 1000, and 2000-km deepmagma oceans on Earth, and 500 and 2000-km deep magma oceanson Mars. On Mars, a 2000-km magma ocean is likely a whole-mantlemagma ocean. We show theoretical results for three initial volatilecontents: 0.5 wt.% H2O and 0.1 wt.% CO2; 0.05 wt.% H2O and 0.01 wt.%CO2; and, to model a case in which all water has reacted with metalliciron, 0.0 wt.% H2O and 0.6 wt.% CO2 (Table 1).

2. Models

2.1. Volatile delivery to the growing planet

The terrestrial planets are likely to have been accreted fromchondritic material and planetesimals built from chondrites. Wood(2005) reports up to 20 wt.% of water in chondrites, and Jarosewich(1990) reports ~3 wt.% water in achondrites, though most are drier.Enstatite chondrites match the oxygen isotope composition of theEarth, but smaller fractions of the wide compositional range of othermeteorite compositions (see Drake and Righter, 2002, and references

Table 1Physical parameters used in models

Parameter and reference Symbol Value, Mars Value, Earth Units

Initial water contentof magma ocean

H2Omagma 0.5, 0.05, 0 0.5, 0.05, 0 wt%

Initial carbon dioxidecontent of magma ocean

CO2magma 0.1, 0.01, 0.6 0.1, 0.01, 0.6 wt%

Planetary radius R 3,396 6,378 kmHeat of fusion of silicates H 418,700 418,700 J/kgHeat capacity of silicates Cp 1,256.1 1,256.1 J/kgKStephan–Boltzmann constant σ 5.67×10−8 5.67×10−8 J/

m2K4secGravitational acceleration G 3.7 9.7 m/s2

Absorption coefficient of waterat pressure p0 (Yamamoto, 1952)

k0,H2O 0.01 0.01 m2/kg

Absorption coefficient of carbondioxide at pressure p0(Pujol and North, 2003; Howardand Kasting, pers. comm.)

k0,CO2 0.05 or0.0001

0.05 or0.0001

m2/kg

Reference pressure p0 101,325 101,325 PaThermal expansivity α 3×10−5 3×10−5 K−1

Slope of the adiabat 1.3×10−4 3.3×10−4 K/kmThermal diffusivity κ 1×10−6 1×10−6 m2/sInitial depth of magma ocean RM 2,000, 500 2,000, 1000,

500km

able 2olid mineral assemblages for cumulate mantle

inerals EARTH MO depth MO depth MARS MO depth MO depth

P (GPa) b500 km(b~19 GPa)

N501 kmand≤40 GPa

P (GPa) b500 km(b6 GPa)

N501 kmand≤24 GPa)

livine 0–1 0.40 0.50 0–1 0.50 0.50linopyroxene 0.25 0.25 0.25 0.25rthopyroxene 0.20 0.20 0.20 0.20lagioclase 0.15 0.05 0.05 0.05

livine 1–2.5 0.40 0.50 1–2 0.50 0.50linopyroxene 0.25 0.25 0.20 0.25rthopyroxene 0.25 0.20 0.20 0.20pinel 0.10 0.05 0.10 0.05

livine 2.5–15 0.40 0.50 2–14 0.60 0.50linopyroxene 0.25 0.20 0.15 0.20rthopyroxene 0.25 0.20 0.10 0.20arnet 0.10 0.10 0.15 0.10

adsleyite-olivine)

15–18 0.35 0.40 – 0.40

linopyroxene 0.30 0.25 – 0.25ajorite 0.35 0.35 – 0.35

ingwoodite-olivine)

18–22 – 0.45 14–24 – 0.45

ajorite – 0.55 – 0.55

erovskite N22 – 0.95 N24 – –

agnesiowustite – 0.05 – –

182 L.T. Elkins-Tanton / Earth and Planetary Science Letters 271 (2008) 181–191

therein) or volatile-rich material from greater radii in the planetarydisk may have been added later in planetary formation (e.g., Raymondet al., 2006; O'Brien et al., 2006).

The likelihood of volatile additions to a magma ocean from giantimpacts may be examined in two ways. First, volatiles bound in thesolid state may be delivered to the solid or melted portion of theplanet during impact. Impact models show that when planetesimalscollide, melting occurs internally preferentially to externally, and thetwo bodies mix intimately (e.g., Canup, 2004; Davison et al., 2008).This scenario makes partial volatile retention likely, and the volatileestimates used here are reasonable.

Second, the impactor may be devolatilized upon impact (O'Keefeand Ahrens, 1977; Canup, 2004; Ni and Ahrens, 2005; Pahlevan andStevenson, 2007). In this second case, retention of the atmospherebecomes the pertinent question. Both Vickery and Melosh (1990) andNi and Ahrens (2005) find that a Mars-sized impactor will eject aportion of the atmosphere above the plane tangent to the Earth at thepoint of impact (the “hemisphere blowoff” model). Genda and Abe(2003) studied a planet without oceans, and found that a Mars-sizedimpactor retained 70%of its atmosphere, and the Earth 90%. This planettherefore has a more massive atmosphere following the impact than ithad before. This atmospheremay reequilibratewith themagma ocean,dissolving volatiles up to the saturation limits of the magma.

2.2. Reactions between water and iron in the magma ocean

Early accretion probably occurred in a highly reducing environ-ment, as evidenced by the large metallic cores of the terrestrialplanets, in which case later accretion had to deliver the volatiles seenin the planets today (Ringwood, 1979; Wänke, 1981; Hunten et al.,1987; Dreibus et al., 1997). Wänke and Dreibus (1994) suggested thatwater would react with metallic iron in accreting material to form FeOor Fe2O3, allowing hydrogen alone to escape into the atmosphere. Inthis case, only a small fraction of the initial water would remain in ahypothetical magma ocean.

Examining higher-pressure processes, Rubie et al. (2004) demon-strate that at high temperatures and pressures in terrestrial magmaoceans, iron moves into a metallic phase preferentially to the oxidizedphase, eliminating the possibility of significant oxidation. In deepmagma oceans, therefore, metallic iron may remain stable, leavingwater in the magma ocean (see also Ohtani et al., 2005). Righter andDrake (1999) further argue thatwater in amagmaocean is necessary to

explain trace element characteristics of the Earth, and they argue foreven higher water contents than the conservative numbers used here.

The final (oligarchic) accretion of the terrestrial planets probablyproceeded from planetesimals or embryos that were themselvesalready differentiated. During planetesimal differentiation core for-mation may depend upon a low oxygen fugacity, or it may dependsimply on the ability of iron to sink quickly through any potentiallyoxidizingmagma slurry and aggregate into a core before oxidation canoccur. Using a Stokes law for sinking iron and a simple diffusion law foroxygen, it can be shown that even on small planetesimals iron sphereslarger than a centimeter will form a core without reacting with waterin themagma they sink through. In theseways, terrestrial planetsmayaccrete from differentiated planetesimals that contain water andcarbon dioxide. This scenario may also be required to explain the non-zero water content of some achondrites (Jarosewich, 1990).

2.3. Processes of magma ocean solidification

Magma ocean fluid dynamics differ significantly from solid mantledynamics, and are closely modeled by the dynamics of atmospheres(Priestley, 1957, 1959; Solomatov, 2000). Heat is transported from theinterior to the surface by cold thermal plumes descending from theupper boundary, accompanied by more diffuse return flow from theinterior. Heat radiation from the planetary surface to space is inhibitedby the atmosphere's emissivity (Abe and Matsui, 1988). Processes ofmagma ocean solidification are discussed in Abe (1993, 1997),Solomatov (2000), and Elkins-Tanton et al. (2003, 2005a,b).

In these models the magma ocean is expected to solidify from thebottom upward, because the slope of the adiabat is steeper than theslope of the solidus and thus they first intersect at depth. Water andcarbon dioxidewill enter solidifyingminerals in small quantities, will beenriched in solution in magma ocean liquids as solidification proceeds,and will degas into the growing atmosphere. At pressures andtemperatures of magma ocean crystallization no hydrous or carbonateminerals will crystallize (Wyllie and Ryabchikov, 2000; Ohtani et al.,2004), although reducing conditions may stabilize graphite (Hirsch-mann andWithers, 2008), not included in thesemodels (Table 2, Fig. 1).Bulk silicate mantle compositions for Mars and the Earth are from

TS

M

OCOP

OCOS

OCOG

W(βCM

R(γM

PM

Fig.1.Mineral assemblages assumed to solidify from a 2000-km deep terrestrial magmaocean. For other models, see Table 2.

183L.T. Elkins-Tanton / Earth and Planetary Science Letters 271 (2008) 181–191

Bertka and Fei (1997) and Hart and Zindler (1986), and averagechondritic trace elements from Anders and Grevesse (1989).

Mineral compositions are calculated in equilibrium with themagma ocean composition using experimental distribution coeffi-cients (Elkins-Tanton et al., 2003), andmineral densities are calculatedbased on their temperature, pressure, and composition (updateddatabases are given in the Supplementary data in the Appendix). Themodel tracks the evolving magma ocean liquid and atmospheric massand composition. One percent interstitial liquid is retained throughoutsolidification in each model. All solids are assumed to retain theirsolidus temperatures during the entirety of magma ocean solidifica-tion because the speed of solidification is fast compared to thermaldiffusion in the solid.

Nominally anhydrous minerals can contain a dynamically andpetrologically significant amount of OH, as much as 1000–1500 ppmfor olivine and orthopyroxene (e.g., Bolfan-Casanova and Keppler,2000; Forneris and Holloway, 2003; Koga et al., 2003; Bell et al., 2004;Aubaud et al., 2004; Hauri et al., 2006; and see Supplementary data inthe Appendix). Carbon dioxide, in contrast, cannot partition intomantle minerals in as significant quantities (Keppler et al., 2003). Asmagma ocean liquids evolve they become more enriched in bothwater and carbon dioxide, but with the nominal initial volatile loadsused in these models none of the minerals in the magma oceancumulates approach saturation. Under the assumption in thesemodels, to reach saturation in an Earth-sized planet, the magmaocean must begin with at least 10 wt.% of water (Elkins-Tanton and

Seager, 2008). Later cumulates, however, will form under higheroxygen fugacities than did early cumulates in a low-water magmaocean. Partition coefficients and saturation limits for hydroxyl andcarbon are given in Tables 1S and 2S in the Appendix.

The solidus used for Earth is fitted to experimental data oncontinental peridotite composition KLB-1 from (Takahashi, 1986;Herzberg and Zhang, 1996; Tronnes and Frost, 2002) and is given as:

TEarthsol ¼ �1:160� 10�7r3 þ 0:0014r2 � 6:382r þ 1:444� 104 ð1Þ

where r is radius in the terrestrial mantle in km. The analogous solidusfor Mars is fitted to data from Longhi et al. (1992) and given as:

TMarssol ¼ �1:963� 10�10r4 þ 1:694� 10�6r3 � 0:00533r2

þ 6:844r � 830� 6 0:2Xliq þ 0:025� ��1

h ið2Þ

where Xliq is the remaining liquid volume fraction of the originalmagma ocean. The final termmoves the solidus to lower temperaturesas the magma ocean liquids evolve; this term is fitted to solidustemperatures from pMELTS (Ghiorso et al., 2002) based on evolvedmagma compositions from ourmodels (Eq. (1) inherently includes thesolidus movement to lower temperatures, ending at ~600 °C).

Surface temperature is calculated by following the adiabat upwardfrom the top of the fully-solidified cumulate layer. The adiabat ismoved to lower temperatures to account for latent heat of crystal-lization when lying in regions between the solidus and liquidus; weassume that latent heat is distributed linearly between solidus andliquidus. The insulating atmosphere produces a surface temperaturethat remains above the solidus of the magma ocean liquids throughN95% of the solidification process, maintaining a liquid surface to theplanet, as similarly calculated by Abe (1997) and Kasting (1988).

2.4. Volatile movement during magma ocean solidification

The model degasses volatiles in excess of saturation at each timestep, maintaining a homogeneous magma ocean saturation at thesurface. At pressure both water and carbon dioxide remain in solutionin silicate melts at quantities greater than 10 wt.% (e.g., Papale, 1997),though at a given pressure water will remain in silicate liquids ingreater concentration than will carbon dioxide. Carbon dioxide andwater saturation data are taken from Papale (1997) and yield thefollowing equations for the atmospheric partial pressures of waterpH2O and carbon dioxide pCO2 [Pa] as a function of the magma oceandissolved water H2Omagma and carbon dioxide CO2magma [wt.%]contents (Fig. 2):

pH2O ¼ H2Omagma � 0:30

2:08� 10�4

� � 10:52

Pa½ �; and ð3Þ

pCO2 ¼CO2magma � 0:05

2:08� 10�4

� � 10:45

Pa½ �: ð4Þ

Volatiles are enriched in evolving magma ocean liquids as solidsform and liquid volume decreases.

Efficient degassing requires rapid convective velocities to carrysupersaturated liquid to pressures low enough that volatiles exsolveinto bubbles. Bubbles must then rise to the surface quickly enoughthat they burst into the atmosphere before being carried back downby convecting fluids. This is a relevant expression for convectivevelocity (Priestley, 1957, 1959; Solomatov, 2000):

v ¼ 0:6agLFqCP

� �13

ð5Þ

where α is thermal expansivity, g is gravity, L is the convective lengthscale, F is heat flux from the planetary surface, ρ is fluid density, andCp is heat capacity. This expression indicates that magma ocean

Fig. 2.Water and carbon dioxide saturation curves assumed for magma ocean liquids, asa function of atmospheric partial pressure. Curves correspond to Eqs. (3) and (4).

184 L.T. Elkins-Tanton / Earth and Planetary Science Letters 271 (2008) 181–191

convective velocities are expected to be on the order of 0.5m/s. For themagma ocean depths modeled here, complete circulation of magmaocean liquid past the surface will be completed in about 1 to 3 weeks.This timescale is less than the length of the time steps in the models,and so all magma ocean liquid will move close enough to the surfaceto degas at every time step of the model.

Using an equation for crystal settling from Solomatov et al. (1993)to instead calculate which bubbles will remain entrained in flow andwhich will rise out of the flow to the surface, we find that at expectedmagma ocean viscosities onMars and the Earth of 0.1 to 1 Pa s (Liebskeet al., 2005), only bubbles smaller than a millimeter will remainentrained in flow. All larger bubbles will rise and escape. Viscositymust reach 104 Pa s before centimetric bubbles will remain entrained.

To first order, then, it is reasonable to remove volatiles in excess ofsaturation from the liquid uniformly at each step (this has theadditional consequence of creating undersaturated liquids at depth).This initial assumption breaks down when a conductive lid is presenton the planet, which is only likely to occur when buoyant mineralsfloat (as in the anorthosite flotation crust on the Moon), or at the veryend of magma ocean solidification when a high crystal fraction willsupport a quench crust, as discussed below. Neither quench norflotation crusts are expected in these models.

2.5. Dynamics of solidification: Conductive lids and layered convection

During solidification the formation of a stagnant conductive lid onthemagma ocean of a planet-sized body is unlikely. First, even amodestinsulating atmospherewillmaintain themagma ocean surface above itsliquidus initially, and above its solidus for much of solidification (Abe,1997; and results presented here). Degassing from the magma oceanthus maintains the magma ocean's liquid surface.

In the absence of such insulation, the surface of the magma oceanwill quench to solid against the cold of near-space or a thinatmosphere. This solid quench material will be denser than themagma ocean liquids beneath. The solid material will therefore beunstable and prone to foundering (Walker et al., 1980; Walker andKiefer, 1985; Elkins-Tanton et al., 2005b). On terrestrial volcanicmagma pools solid crusts may persist through support at the edges ofthe pool, but on a planetary magma ocean gravitational forces it

dominate and any solid lid should founder. This lid is vulnerable toconvective stresses on its bottomboundary, and disruption by impacts.Such a lid may only persist is if it is relatively whole over the entireplanet and thick enough to resist breaking up from convective stressesand impacts. A lid tens of kilometers thick is required for this degree ofstrength, but evolving fromno lid to a complete, thick lidwould appearto be highly unlikely in the active environment of a magma ocean.

The only likely way to form a conductive lid on a magma ocean isby flotation of buoyant phases. This is likely to occur only on small, dryplanets like the Moon; on larger planets plagioclase does not becomestable until the magma ocean has solidified to a high degree, close tothe planetary surface, where high crystal fractions will preventflotation. Additionally, water suppresses the crystallization of plagi-oclase, further delaying its appearance in the magma ocean (e.g.,Goldsmith, 1982).

Within the magma ocean, layering is also possible. A number ofcommon mantle minerals reach neutral buoyancy with respect totheir coexisting mafic liquids at specific pressures. The prospect ofneutrally-buoyant minerals has lead a number of investigators tosuggest that a magma ocean would convect in layers divided bymineral septa (Franck, 1992; Morse, 1993).

Olivine and pyroxene sink with respect to their coexisting maficsilicate liquids at shallow depths, but at pressures greater than about 7.5and 10 GPa both minerals become positively buoyant and thus wouldfloat at this pointof neutral buoyancy (Stolper et al.,1981). In drymagmaocean in the models presented here, olivine and pyroxene are neutrallybuoyant at onMars at ~7.5 GPa, and at ~9.5 GPa on the Earth. In magmaoceans with water, the neutral buoyancy points are driven to higherpressures by the lower densities of the water-bearing liquids. At stillhigher pressures majorite may have a density crossover with coexistingliquids, but silicate liquid compressibilities at high pressure (particularlythosewith addedwater) are not well known.With the parameters usedhere, majorite remains denser than its coexisting liquid.

A primary consideration for septa formation is whether mineralgrains will remain entrained in flow, or whether they will sink or riseaccording to their density. At expected magma ocean viscosities of 0.1to 1 Pa s (Liebske et al., 2005), the settling vs. entrainment expressionof Solomatov et al. (1993) predicts that grains must be less than amillimeter in diameter to remain entrained in flow; those larger willsettle or rise. Those of about a millimeter will have a persistent risingor sinking velocity on the order of 10−4 m/s. With convective velocitiesat least three orders of magnitude greater than the Stokes velocity ofthe grains, the grains will likely not reach the septa location beforesolidification of that region is complete.

A secondary consideration, if indeed a septum forms, is whetherconvectivemotions will allow it to persist. Solidification of themagmaocean is expected to take place primarily in focused downwellingplumes from the cooling surface. Crystal fractions will increase as thecool plume sinks and thusmoves closer to its solidus. These plumes areexpected to sink to the bottom boundary layer of the magma ocean,where solid minerals will be deposited. Convective return flowwill bein the form of diffuse upwellings, since the core–mantle boundary willbe at thermal equilibrium and the driving force for convection the coldsurface boundary.

The depth of neutral buoyancy for both olivine and pyroxene is~250 km in these models for the Earth. Pyroxene, the first buoyantphase in the progression of magma ocean cumulates, begins crystal-lizing at a depth of ~470 km. Even more extreme on Mars, pyroxenebegins to solidify at ~1500-km depth, with its neutral buoyancy pointat ~600-km depth. Thus pyroxene and olivine would be rising to apoint of neutral buoyance that lies in the center of the radius range ofthe remaining convecting magma ocean, where vertical convectivevelocities will be the highest and therefore most likely to penetratethe septum and mix magma ocean liquids. Rapidly moving, colddownwelling plumes are therefore likely to disrupt any mineral septa,mixing liquids back to homogeneity.

Table 3Atmospheric degassing, final atmospheric pressure, and solidification times

Initial volatile contents ofmagma ocean: H2O, CO2

Earth Mars

500 km 1000 km 2000 km 500 km 2000 km

Fraction of initial volatile content degassed into initial atmosphere: H2O, CO2

0.05, 0.01 0.79, 0.84 0.75, 0.81 0.70, 0.78 0.84, 0.87 0.80, 0.840.5, 0.1 0.93, 0.96 0.92, 0.95 0.91, 0.95 0.95, 0.97 0.93, 0.960, 0.6 0, 0.98 0, 0.98 0, 0.97 0, 0.99 0, 0.98

Final atmospheric pressure (sum of CO2 and H2O partial pressures)(bar)0.05, 0.01 90 150 240 30 700.5, 0.1 1030 1890 3150 330 8000, 0.6 1090 2000 3350 350 850

Time to reach 98% solidification, for k0,H2O=0.01 and k0,CO2=0.05 or 0.001 (divided byslashes) (Myr)0.05, 0.01 0.01/0.01 0.03/0.03 0.07/0.06 0.004/0.003 0.02/0.010.5, 0.1 0.6/0.5 1.4/1.2 2.8/2.4 0.2/0.1 0.7/0.50, 0.6 1.2/0.2 2.7/0.4 5.3/0.8 0.6/0.03 2.8/0.1

Volatile content of liquids remaining at 98% solidification: H2O, CO2 (mass %)0.05, 0.01 1.1, 0.2 1.3, 0.2 1.5, 0.2 0.7, 0.1 1.0, 0.20.5, 0.1 3.1, 0.4 4.1, 0.6 5.3, 0.7 1.9, 0.2 2.7, 0.40, 0.6 0, 0.9 0, 1.2 0, 1.5 0, 0.6 0, 0.8

185L.T. Elkins-Tanton / Earth and Planetary Science Letters 271 (2008) 181–191

Layered convection delimited by mineral septa is thereforeunlikely in a magma ocean on a body the size of a terrestrial planet.A likely outcome of these calculations is that solidification beneath thepressure of neutral buoyancy is likely to be a combination of fractionaland batch solidification. Note in addition that in the range of pressureover which olivine and pyroxene are positively buoyant, garnet is co-crystallizing and is denser than the coexisting liquid. Thus garnet maysettle while olivine and pyroxene remain entrained, creating addi-tional compositional differentiation (Elkins-Tanton et al., 2003).

2.6. Heat flux and time scales

A solidification time scale can be calculated using heat flux F fromthe planet expressed as the following balance between total planetaryheat flux on the left and the sum of latent heat of crystallization of anincrement of the magma ocean and secular cooling of the planet:

4kR2F ¼ drdt

qH4kr2 þ 43kqCp

ddt

T R3 � r3� �� � ð6Þ

expressed in terms of planetary radius R, velocity of solidificationfront V, solid density ρ, heat of fusion H, radius of solidification r, heatcapacity Cp, and change in solid temperature dT (dependent uponsolidus slope). Total heat flux F can be calculated as Stefan–Boltzmannradiation from the atmosphere

F ¼ er T4 � T4l

� � ð7Þ

where T is the surface temperature [K], T∞ is the blackbodyequilibrium temperature of the planet heated only by the Sun, σ isthe Stephan–Boltzmann constant, and emissivity ε is a function ofoptical depth τ⁎ (Hodges, 2002; Pujol and North, 2003):

e ¼ 2s4þ 2

ð8Þ

where s4 ¼ Ps4i . The atmosphere is treated as a grey radiative

emitter with no convective heat transport. Emissivity is calculatedseparately for each atmospheric gas as a function of its atmosphericpartial pressure, as described in Abe and Matsui (1985). For each ithcomponent

s4 ¼ 3Matm

8kR2

� k0g3p0

� 12

ð9Þ

whereMatm is themass of the specific atmospheric gas, R is the planetaryradius, and k0 is the atmospheric absorption coefficient of a specific gas atpressure p0. [The mass of a gas in the atmosphere can be calculated fromEqs. (1) and (2) using the relation Matm=(4πpxR2/g), where px is thepartial atmospheric pressure of a given gas.] By inserting Eqs. (7) and (8)into Eq. (6), the rate of solidification for each radial increment can becalculated, and a total time from liquid to solid mantle summed.

Though Pujol and North (2003) give k0 (the atmospheric absorptioncoefficient) for carbon dioxide as 0.05 m2/kg, indicating that carbondioxide has five times the absorption of water, Howard and Kasting(unpublished) find that in the 8–13 μm region of these models carbondioxide absorption maybe less than 1% that of water. Consequently weinvestigate the sensitivity of these models to changes in k0,CO2 (Table 1).

Cooling subsequent to mantle solidification is calculated solvingthe transient heat conduction equation in a spherical geometry for thesolid planet, using the temperature distribution that results from solidcumulate gravitational overturn. The solid planet radiates heat fromits surface through the atmospheric mass produced during degassingand solidification, as described in Eqs. (6), (7), and (8).

If the atmosphere were modeled as convective, which it perhapswould be given its densities and temperatures, cooling would be fasterand the effective radiative transfer level moved well above the planet'ssurface (see Results and discussion). No speciation in the atmosphere is

calculated. No atmospheric stripping or hydrodynamic escape occurs inthese models, and no gravitationally bound atmosphere is assumed toexist before the magma ocean stage. A further simplification is theneglect of the increasing transparency of the atmosphere with low heatfluxes (below ~250 W/m2 a water atmosphere is largely transparent toradiation due to condensation and collapse [Abe et al., 2000]). Becausethe atmosphere is thin during the majority of planetary solidification,these times calculated for solidification are reasonable. During latercooling to clement conditions, when the atmosphere is thick and likelyto be convective, the models provide maximum times.

3. Results and discussion

Magma ocean solidification and subsequent planetary evolutionproceeds through three major phases. First, the magma oceansolidifies, partitioning volatiles between solid cumulates, evolvingliquids, and a growing primordial atmosphere. This step producescumulates with density that increases with radius, and are thereforegravitationally unstable to overturn (Elkins-Tanton et al., 2003,2005b). In step two, the unstable solidifiedmantle cumulates overturnto a stable configuration. The overturn process creates a mantle that isgravitationally stable and therefore resistant to the onset of thermalconvection. Hot cumulates that formed deep in the magma ocean riseto shallower depths during overturn and may melt adiabatically,producing the earliest basaltic crust. The surface of the planet istherefore heated. In step three, the planet conducts heat through itssolidified mantle and radiates it to space through the primordialatmosphere formed in step one.

3.1. Solidification and production of a primordial atmosphere

The degassing of small initial volatile contents produces sig-nificant atmospheres. For Earth, a 2000-km deep magma oceanbeginning with 0.05 wt.% of water and 0.01 wt.% of carbon dioxideproduces an atmosphere of ~240 bars (Table 3). The highestatmospheric pressure produced in these models is ~3400 bars,from a 2000-km deep magma ocean on the Earth that began with0.6 wt.% CO2. Liu (2004) estimated that if all the surface and upper-mantle water and CO2 on Earth were converted into a primitiveatmosphere, it would produce 560 bars of H2O and 100 bars of CO2,intermediate to our models.

186 L.T. Elkins-Tanton / Earth and Planetary Science Letters 271 (2008) 181–191

Solidification rates limited by radiative heat loss through theatmosphere may be very fast, with 98% by volume of a 500-km deepterrestrial magma ocean solidifying in as little as 10,000 yr (Fig. 3,Table 3). The models with the most massive primordial atmospheresrequire 3 to 5 million years to solidify to 98%. The solidification ratespredicted in these models allow almost complete solidification beforesolid-state gravitationally-driven cumulate overturn (Elkins-Tantonet al., 2005a,b).

The times to solidification predicted in these models are largely inagreement with results from Abe (1997), Zahnle et al. (1988), and Abeet al. (2000), though thosemodels did not use complete compositionalinformation for the silicate mantle, nor model any mineralogy orinternal dynamics of the solidified planet. Zahnle et al. (1988) presentsa complex model including a treatment of planetary growth bygradual accretion of volatile-rich bodies over ~50 Ma, and heat lossfrom the planet through a convecting atmosphere. Their most

Fig. 3. Solidification of a magma ocean vs time. Terrestrial and martian magma oceansare plotted using the same time scale. Though the 0.5 wt.% H2O, 0.1 wt.% CO2 and the0.6 wt.% CO2 models have the same bulk wt.% of volatiles, the CO2-rich compositiondegasses faster and more completely and the high greenhouse value of CO2 slowssolidification considerably. Note that initial volatile content is far more important thandepth of magma ocean. The differences in the shapes of the curves between Earth andMars reflect different mineral assemblages fractionating from the evolving magmaocean.

Fig. 4. Sensitivity analysis of time to magma ocean solidification to carbon dioxideatmospheric absorption coefficient k0,CO2, for three atmospheric compositions.Calculations for a 2000-km deep terrestrial magma ocean.

comparable planetary solidification time to those from our modelsmay be the time taken to cool the planetary surface from 1225 to600 °C. This Zahnle et al. (1988) cooling time appears to be about 1Ma.Our maximum atmospheric pressures and time to a solid planet arecomparable, though we make different assumptions about processesof planetary accretion and atmospheric dynamics.

In these models carbon dioxide can have the largest effect oncooling time, since it degasses more completely and faster thanwater.The relative strengths of water and carbon dioxide as greenhousegases, however, may be argued over the parameter ranges of thesemodels (Pujol and North, 2003; Howard and Kasting, unpublished).Consequently in Fig. 4 and Table 3 we show the sensitivity of thesemodels to changes in k0,CO2 for a 2000-km deep terrestrial magmaocean. In the mixed water and carbon dioxide atmospheres, changingk0,CO2 by a factor of 50 changes magma ocean solidification times byonly a factor of two or less. For the atmospheres lacking water,however, changing k0,CO2 from 0.05 to 0.001 (a factor of 50) shortenssolidification by a factor of seven, and changing from 0.05 to 0.0001 (afactor of 500) shortens solidification by a factor of twenty.

These results further indicate that a significantly thicker atmo-sphere than exists today on Mars or even the Earth can be createdthrough solidification of a magma ocean with low initial volatilecontent. Though Mars is relatively dry today, as much as 99% of Mars'original volatiles are thought to have been lost by 3.8 GPa throughescape through impacts, sputtering by solar wind, and hydrodynamicescape (Catling, 2004). The high initial atmospheric masses predictedhere are consistentwith these rates of escape. For the Earth, a 0.05wt.%initial water content, 500-km deep magma ocean degasses3.8×1020 kg of water, or about a third of an ocean, into the atmosphere(79% of the initial volatile content is degassed). The 0.5 wt.% initialwater content, 2000-km deep magma ocean, degasses 1.4×1022 kg ofwater, or about 11 oceans, into the atmosphere (91% of the initialvolatile content is degassed). These masses are comparable to themasses predicted in Genda and Abe (2003), indicating that anatmosphere can be created through one magma ocean solidification

Fig. 5. Water content of silicate cumulate mantles resulting from equilibriumpartitioning between evolving magma ocean liquids and fractionating solids. Fourmodels are shown in each figure: Two initial volatile contents each for Earth and Mars.During cumulate overturn to gravitational stability, some radial shells of the mantleobtain lateral heterogeneity as upwelling and downwelling material with the samedensity stalls at neutral buoyancy. These regions are indicated with the shaded boxesthat span the two compositions. The inset in part A shows the partitioning of waterbetween the nominally anhydrous minerals and the interstitial liquids, each of whichcontributes about half of the total. This inset model is for the lower volatile content,2000-km depth Earth magma ocean cumulates. Also in part A are two boxes spanning600–2000-km depth, the approximate perovskite stability region for Earth (the Marsmodels do not extend to perovskite stability). These boxes denote the saturation limitsof the perovskite assembly, with the ~10 ppm level used in these models, and a possible100 ppm saturation limit (saturation limits for all other mineral assemblages extendpast the scale limits of the figure). Note that parts of the overturned cumulates fromboth Earth models exceed the saturation limits for perovskite, since they originallyformed at shallower depths with larger water contents. Thus during overturn thesesinking, water-rich cumulates may dewater and trigger melting at depth.

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event of equivalent mass to that created by degassing duringaccretionary events.

The high surface water predicted from small initial magma oceanwater contents suggests that initial magma ocean volatile contentshigher than those considered here are unreasonable because theyimply initial atmospheric pressures in the thousands of bars and timesrequired for cooling to clement conditions that approach hundreds ofmillions of years. These very long cooling times and very high surfacepressures would be in conflict with the evidence for liquid surfacewater at 4.4 Ga found in zircons (Valley et al., 2002) and with rates ofatmospheric loss to attain today's atmospheric mass, indicating thatthe lower volatile contents used in these models may be morerealistic. These models therefore support the idea of serial hemi-spheric or shallow magma oceans punctuated by periods of clementsurface conditions in which initial crust building may occur.

3.2. Gravitational overturn of solidified cumulates

The second major and probably unavoidable phase of planetaryevolution from a magma ocean begins near or at the end ofsolidification when the solidified cumulates overturn gravitationallyto a stable density profile. Cumulates fractionally solidified from amagma ocean are gravitationally unstable primarily because ofmagnesium and iron exchange within the evolving magma oceanliquids. Later-crystallizing cumulates, nearer the planetary surface, areenriched in iron in comparison to earlier, deeper cumulates. Agravitationally unstable stratigraphy will overturn via Rayleigh–Taylorinstabilities to a stable profile with intrinsically densest materials atthe bottom, on timescales that depend on the rate of thermally-activated creep (Solomatov, 2000; Elkins-Tanton et al., 2003). The timeof overturn depends upon a competition between the rate ofsolidification and the time of onset of gravitational overturn. Overturntime is dependent upon the reciprocal of layer thickness squared(Hess and Parmentier, 1995; Elkins-Tanton et al., 2003), which doesnot approach the time to solidification until the mantle is almostcompletely solidified. Numerical models indicate that the major phaseof overturn is complete in 2 to 4 million years, far longer thansolidification but still geologically rapid (Elkins-Tanton et al., 2005a).

During overturn hot cumulates rise from depth and meltadiabatically to produce the earliest crust (Elkins-Tanton et al.,2005b). Degassing and solidification of the lava releases a pulse ofheat and a small addition to the atmosphere. During cumulateoverturn to gravitational stability, some radial shells of the mantleobtain lateral heterogeneity as upwelling and downwelling materialwith the same density stalls at neutral buoyancy (Fig. 5). Thisheterogeneity contradicts the common assumption of radial symme-try in mantle composition; some radial shells of a cumulate mantleproduced through magma ocean solidification will have radially-symmetric composition, but other radial shells will have compositionsthat vary in latitude or longitude at a given radius. These laterally-heterogeneous regions create variable source regions for latermagmatism.

The significant atmospheric pressure also allows erupting magmasto retain some volatile content as they solidify. At 100 bars pressurelava can retain 1 wt.% water. These models predict that the earliestbasaltic crusts will have variable water content (depending upon thecomposition of the source and the extent of melting) up to andpossibly beyond 1 wt.% since some of these models predict atmo-spheric pressure in excess of 100 bars.

3.3. Compositions and densities in the resulting cumulate mantle

Though the great majority of volatiles are degassed into theatmosphere, a geodynamically significant quantity is sequestered inthe solid cumulates, as much as 750 ppm by weight OH in modelsbeginningwith 0.5 wt.%water, and aminimumof 10 ppm byweight in

the driest cumulates of models beginning with just 0.05 wt.% water(Fig. 5). Even these small water contents significantly lower themelting temperature of mantle materials, facilitating later volcanism,as discussed in Abe et al. (2000).

Wänke and Dreibus (1994) estimate that 36 ppm by weight ofwater is left in the Martian mantle today, based on 100% degassingduring accretion but eventual retention in a 130-m thick surface layerof oxidizedmaterial added at the very end of accretion. Using a similarmodel for the Earth Dreibus et al. (1997) estimate a bulk Earth contentof 840 ppm by weight water, of which 420 ppm by weight representcrustal content. These values are similar to the values obtained in thispaper for cumulates resulting from magma ocean solidification (20 to800 ppm for Earth, and 20 to 400 ppm for Mars, Fig. 5). In the absenceof plate tectonics, therefore, mantle volatile content should becontrolled by magma ocean processes.

Fig. 6. Cumulate mantle density and trace element compositions following overturn of a 2000-km deep terrestrial magma ocean. Note the extreme density stability of the lowestlayer; this will likely sink through any underlying cumulates and thus may represent the D″ layer in the present mantle, as it is likely stable over the age of the Earth.

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The idea thatmost of the Earth's volatiles were delivered in a seriesof events after core formation, or a “late veneer” (e.g., Wänke, 1981)may be consistent with these calculations. Delivery of additionalmaterial implies one or more impacts of some size, which likelyformed magma ponds or shallow magma oceans, and solidificationand atmospheric partitioning would proceed as described here.

Overturn creates a stably-stratified, compositionally differentiatedsilicate mantle. On Mars the model results are consistent with thewide range of inferred source compositions for the Martian meteor-ites, and an incompletely mixed mantle. On Earth, midocean ridgebasalts indicate awell-mixedmantle, at least at shallow depths, but aninteresting effect of cumulate overturn concerns the Earth's internaltrace element and water budgets. During overturn, shallow, densecumulates, rich in water and incompatible elements, will sink to thebottom of the mantle (Fig. 6). Their extreme density contrast withimmediately overlying cumulates, 500 kg/m3, may prevent their beingreentrained in thermal convection. To overcome this density contrast,a temperature change of thousands of degrees is necessary, whichwould melt this thin, dense layer, and thus further stabilize it at depthbecause this melt is likely negatively buoyant with respect to theremaining cumulates (Stolper et al., 1981; Ohtani and Maeda, 2001;Bercovici and Karato, 2003; Agee, 2008). Thus these densestcumulates may hold a significant trace element budget at the core–mantle boundary, consistent with a D″ prime layer and the Nd deficitdescribed by Boyet and Carlson (2005) and Bennett et al. (2007). The2000-km deep model predicts a 50- to 80-km thick trace elementenriched layer dense enough to potentially last the age of the Earth.

Additionally, or alternatively, as dense cumulate from near thesurface sink to gravitational stability, they sink through a depth atwhich they will transform to perovskite. Perovskite appears to have asignificantly smaller water saturation level than do upper-mantleminerals. During overturn, therefore, water will be liberated fromsinking cumulates as they reach the stability regions of higher-pressure phases (Fig. 5). This water may trigger melting, and the meltis likely to be negatively buoyant and sink, carrying incompatibleelements and volatiles to depth in the lowermantle and again forminga trace element rich boundary layer at the core–mantle boundary.

This melting event would also act to homogenize the mantle andameliorate the density stratification above the deepest layer. In thisway mantle overturn on planets large enough to stabilize perovskite

may immediately eliminate some of the resistance to thermalconvection that density stratification causes, and thus speed up theonset of convection before a very thick conductive lid forms. Earlyconvection may be one important aspect of the development of platetectonics, which requires that no thick conductive lid be present. Thusthe internal pressure range of a planet may be an important control onwhether a stable density stratification remains after a magma oceanphase, or whether convection will begin quickly.

3.4. Conductive cooling of the cumulate mantle and radiative coolingfrom the planetary surface

The final stage in these models is cooling of the solidified, stableplanet. The compositionally stable cumulate mantle resists the onsetof thermal convection long past the endpoints of these models(Zaranek and Parmentier, 2004). Planetary cooling proceeds byconduction through the mantle to the planetary surface and, inthese models, radiation through the atmosphere.

Taking the final temperature profile of the overturned mantle andallowing conduction to proceed and heat flux to continue through theatmosphere, cooling of the planetary surfaces follows paths that passclose to the water critical point after 10 to 50 Ma (Fig. 7), as predictedby Liu (2004). Because no atmospheric loss and no convection aremodeled, these times to clement surface conditions are likely to bemaxima, and it is here that the largest difference results betweenthese gray emitter atmospheres and convective atmospheres such asthat in Zahnle et al. (1988). Zahnle et al. (1988) shows in their Fig. 13the cooling times for their models and those of Abe andMatsui (1988),who also used a grey radiative atmosphere. The final cooling of thesolidified planet from Abe and Matsui (1988), and in these models, isfar more sluggish than cooling in the convective atmosphere of Zahnleet al. (1988), by an order of magnitude. The final cooling from thesolidified state to ~300 °C is estimated as ~5Ma in Zahnle et al. (1988),while here it is in excess of 50 Ma. The discrepancy is highest in thisfinal cooling period because the atmospheres are at their thickest, andconvection is most likely.

In all models here the partial or wholemagma oceans are solidifiedwithin ~5 Myr (Fig. 3). The atmospheric pressures and temperaturesin these models indicate that after each whole or partial magma oceanevent terrestrial planets were close to conditions for either water

Fig. 7. Atmospheric water partial pressure vs. planetary surface temperature for Earth andMars. Solid lines at left: Phase diagram for water; Dashed lines: Carbon dioxide. Each figureshows model results for two initial volatile contents and two initial magma ocean depths, as for Fig. 3. Magma ocean solidification begins at t0 and proceeds to 98% solidification at t1(step 1 as described in this paper). After t1 the gravitationally unstable solid mantle cumulates overturn to stability, bringing hot, buoyant material to the planetary surface andproducing a first basaltic crust through decompression melting (step 2). Cooling the solid planet thus begins at t2, reflecting the newly hot surface after overturn, and proceeds to t3through conductive cooling to the planet's surface and radiation through the atmosphere, which is assumed to retain its mass and avoid convection (step 3). The solid square on the t2to t3 cooling pathmarks the temperature reached in 10Ma, and t3 is reached in 50Ma (see discussion on Zahnle et al. (1988), wherewith a convective atmosphere clement conditionsare reached in about 5 Ma. The possibility of critical fluids on the planetary surface is demonstrated, as well as the relatively rapid approach to clement conditions and surface water.Current surface conditions for Venus (V), Earth (E), and Mars (M), are shown.

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oceans or for thick water atmospheres within about 50million years atthe longest. Below a planetary surface heat flux of ~250W/m2 a wateratmosphere is largely transparent to radiation due to condensationand collapse (Abe et al., 2000), and Kasting (1991) argues similarlythat CO2 cannot maintain a greenhouse on Mars; these lines ofreasoning further indicate that surface cooling will escalate and bringplanetary surfaces to clement conditions within ten or a few tens ofmillions of years.

In all the models presented for Earth and Mars the final atmo-spheric pressure from degassing a magma ocean is close to the criticalpoint of water, indicating the likelihood of supercritical fluids on thesurface of planets as they cool, and raising the question of their effecton the composition and weathering of the earliest basaltic crusts.

4. Conclusion

Magma ocean solidification and subsequent planetary evolutionproceeds through three major phases. First, the magma ocean solid-ifies, partitioning volatiles between solid cumulates, evolving liquids,and a growing primordial atmosphere. This step produces cumulates

with density that increases with radius, and are therefore gravita-tionally unstable to overturn, as discussed in Elkins-Tanton et al.(2003, 2005b). In step two, the unstable solidified silicate mantlecumulates overturn to a stable configuration. The overturn processcreates amantle that is gravitationally stable and therefore resistant tothe onset of thermal convection. Hot cumulates that formed deep inthe magma ocean rise to shallower depths during overturn and maymelt adiabatically, producing the earliest basaltic crust. The surface ofthe planet is therefore reheated. In step three, the planet conductsheat through its solidified mantle and radiates it to space through theprimordial atmosphere formed in step one.

The significant conclusions from this study include:

1. Layered convection, delineated bymineral septa, is unlikely to formin a planetary-scale magma ocean.

2. The only conductive lid likely to exist on a magma ocean is formedby flotation of buoyant phases, and is unlikely to form on planets aslarge as or larger than Mars.

3. These models indicate that with very small initial volatile contentsof less than a weight percent, the solidification of one partial-

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mantlemagma ocean can produce significant atmospheres, into thehundreds or even thousands of bars of water plus carbon dioxide.

4. Degassing produces a water-rich initial atmosphere that passesnear the critical point of water while cooling, raising the possibilityof supercritical fluid atmospheres during planetary formation.Crusts that experienced supercritical fluids may still exist on theancient surfaces of Mars and Mercury.

5. The entire process of solidification, overturn, and cooling toclement conditions may occur within 5 to 10 Ma for Martian orterrestrial magma ocean. This short cooling interval indicates thatserial magma oceans of varying depths were probably unavoidableon the early terrestrial planets, can be responsible for a finite butgeodynamically critical water content in the silicate mantles of allterrestrial planets, and were likely punctuated by periods ofclement surface conditions in which initial crust building mayhave occurred.

6. The resulting cumulate mantles are stably stratified with respect todensity, and therefore resistant to thermal convection; this stable,initially non-convecting mantle is in strong contrast to thetraditional image of a vigorously convecting early mantle.

7. The compositional heterogeneity of differentiated mantle cumu-lates are consistent with the inferred source regions for the SNCmeteorites on Mars

8. In terrestrial models, a small volume of late-solidifying, densecumulates sink to the core–mantle boundary, carrying a high waterand trace element inventory. The resulting layer, ~50-km thick, maybe stable over the age of the Earth and may form the D″ layer andexplain observed trace element anomalies (Boyet and Carlson, 2005).

Acknowledgements

The author gratefully acknowledges the discussions with andsuggestions of E. Marc Parmentier, and the support of the MarsFundamental Research Program. James Kasting, Marc Hirschmann andan anonymous reviewer gave constructive reviews that substantiallyimproved the paper. James Kasting kindly provided unpublished dataand discussion on carbon dioxide absorption coefficients.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.epsl.2008.03.062.

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