Terrestrial Planet Evolution in the Stagnant-Lid Regime:Size Effects and the Formation of Self-Destabilizing Crust

Joseph G. O’Rourke1 , Jun Korenaga

Department of Geology and Geophysics, Yale University, New Haven, CT 06520, USA

Abstract

The ongoing discovery of terrestrial exoplanets accentuates the importance of studying planetary evolution fora wide range of initial conditions. We perform thermal evolution simulations for generic terrestrial planets withmasses ranging from that of Mars to 10M⊕ in the stagnant-lid regime, the most natural mode of convection withstrongly temperature-dependent viscosity. Given considerable uncertainty surrounding the dependency of mantlerheology on pressure, we choose to focus on the end-member case of pressure-independent potential viscosity, whereviscosity does not change with depth along an adiabatic temperature gradient. We employ principal componentanalysis and linear regression to capture the first-order systematics of possible evolutionary scenarios from a largenumber of simulation runs. With increased planetary mass, crustal thickness and the degree of mantle processingare both predicted to decrease, and such size effects can also be derived with simple scaling analyses. The likelihoodof plate tectonics is quantified using a mantle rheology that takes into account both ductile and brittle deformationmechanisms. Confirming earlier scaling analyses, the effects of lithosphere hydration dominate the effects of planetarymass. The possibility of basalt-eclogite phase transition in the planetary crust is found to increase with planetarymass, and we suggest that massive terrestrial planets may escape the stagnant-lid regime through the formation ofa self-destabilizing dense eclogite layer.

Keywords: Terrestrial planets, Interiors, Extra-solar planets

1. Introduction

Plate tectonics is only observed on Earth and is likelyimportant to Earth’s uniquely clement surface condi-tions (e.g., Kasting and Catling, 2003). Other terres-trial planets in the Solar System (i.e., Mercury, Mars,and Venus) are generally considered to feature a rigidspherical shell encompassing the entire planet, withhot mantle convecting beneath the shell (e.g., Schubertet al., 2001). This mode of mantle convection is knownas stagnant-lid convection. In fact, stagnant-lid con-vection may be most natural for planetary mantles be-cause the viscosity of constituent materials is stronglytemperature-dependent (Solomatov, 1995). The discov-ery of many extrasolar terrestrial planets with mass 1 to10M⊕ (e.g., Rivera et al., 2005; Udry et al., 2007; Quelozet al., 2009; Mayor et al., 2009; Leger et al., 2009; Char-bonneau et al., 2009; Borucki et al., 2011) makes under-standing planetary evolution in the stagnant-lid regimeespecially critical.

Parametrized models of stagnant-lid convection havelong been applied to planets in our Solar System inan effort to infer likely planetary evolution scenariosfrom limited observational constraints (e.g., Stevensonet al., 1983; Spohn, 1991; Hauck and Phillips, 2002;Fraeman and Korenaga, 2010). Previous studies ofmassive terrestrial planets are more theoretical in na-ture, focusing on two broad questions. First, the effects

of planetary mass on the likelihood of plate tectonicshave been studied through scaling analyses and simpleparametrized convection models (Valencia et al., 2007;O’Neill and Lenardic, 2007; Korenaga, 2010a; van Heckand Tackley, 2011). Second, the evolution of planetsin the stagnant-lid regime has been contrasted withevolution with plate tectonics in the hope of identi-fying atmospheric signatures that would indicate theregime of mantle convection for a distant planet (e.g.,Kite et al., 2009). Mantle dynamics in the stagnant-lidregime, however, can be more complex than previouslythought owing to the effects of mantle processing andcrustal formation, and the scaling law of stagnant-lidconvection that takes such complications into accounthas been developed only recently (Korenaga, 2009). Itis thus warranted to take a fresh look at the fate of mas-sive terrestrial planets in the stagnant-lid regime and toexplore the general effects of initial conditions includingplanetary mass.

This study extends a parametrized model of stagnant-lid convection recently applied to Mars (Fraeman andKorenaga, 2010) to terrestrial planets of various masses,including massive planets that evolve in the stagnant-lid regime that are termed “super-Venus” planets. Thismodel incorporates the effects of compositional buoy-ancy and dehydration stiffening on mantle dynamics(Korenaga, 2009), which are rarely accounted for ex-

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cept in simulations of plate tectonics. Unlike in previousstudies, sensitivity analyses are extensively performedto quantify the relationship between initial conditionsand modeling results. Principal component analysis isused to simplify the interpretation of a large number ofsimulation results. Simple scaling analyses are also con-ducted to derive a theoretical basis for major modelingresults. Moreover, the likelihood of plate tectonics isquantified by tracking the viscosity contrast across thelithosphere during each simulation.

The purpose of this study is to investigate paths alongwhich generic terrestrial planets may evolve and to esti-mate whether massive terrestrial planets are relativelymore or less likely to escape the stagnant-lid regime.Throughout this paper, “Mars” and “Venus” shouldbe considered shorthand for generalized 0.107M⊕ and0.815M⊕ terrestrial planets, respectively. The evolu-tion of a particular planet is likely to diverge from thepredictions of these simple parametrized models. Fewconstraints are available beyond planetary mass and ra-dius for extrasolar terrestrial planets. But for terres-trial planets in our Solar System, more data are avail-able from decades of observations and spacecraft visits.Here, we explore hypothetical planetary evolution withthe simplest assumptions on mantle dynamics, therebyserving as a reference model on which additional com-plications may be considered if necessary.

2. Theoretical Formulation

Parametrized convection models are used to simulatethe evolution of Mars, Venus, and putative super-Venusplanets for a wide range of initial conditions. Equa-tions used to track the thermal and chemical evolu-tion of terrestrial planets are taken from Fraeman andKorenaga (2010) with some modifications. Earth-like,peridotite mantle compositions are used to parametrizemelting behavior. Although continuous evolution in thestagnant-lid regime is assumed, a simple model of litho-spheric weakening is also considered to evaluate the like-lihood of plate tectonics occurring at some point duringplanetary evolution.

2.1. Governing Equations

Mars, Venus, and super-Venus planets are assumedto begin as differentiated bodies with a mantle andcore. Energy conservation yields two governing equa-tions. First, the energy balance for the core is (Steven-son et al., 1983)

[4πR2i ρc(Lc+Eg)

dRidTcm

− 4π

3R3cρcCcηc]

dTcmdt

= 4πR2cFc,

(1)where Rc and Ri are the radii of the core and inner core,respectively; ρc is the density of the core; Lc is the latentheat of solidification associated with the inner core; Egis the gravitational energy liberated per unit mass of theinner core; ηc is the ratio of Tcm, the temperature at thecore side of the core/mantle boundary, to the average

core temperature; Cc is the specific heat of the core;and Fc is the heat flux out of the core. The formulationof core cooling is identical to that of Stevenson et al.(1983).

Second, the energy balance for the mantle is (Hauckand Phillips, 2002)

4π

3(R3

m −R3c)

(Qm − ρmCmηm

dTudt

)− ρmfmLm

= 4π(R2mFm −R2

cFc), (2)

where Rm is the radius of the mantle; Qm is the vol-umetric heat production of the mantle; ρm is the den-sity of the mantle; Cm is the specific heat of the man-tle; ηm is the ratio of the average temperature of themantle to Tu, the potential temperature of the man-tle (a hypothetical temperature of the mantle adiabati-cally brought up to the surface without melting); fm isvolumetric melt production with associated latent heatrelease, Lm; and Fm is the heat flux across the man-tle/crust boundary.

Some of the above parameters are universal con-stants, but most are planet-specific. Many importantparameters are also time-varying. In particular, mantlemelt is extracted to form crust, causing Rm to decreasewith time. Likewise, Qm decreases with time because ofradioactive decay with some approximated average de-cay constant, λ (Stevenson et al., 1983), and extractionthrough mantle processing.

Figure 1 illustrates the assumed thermal and chem-ical structure in our model. Over time, melting pro-cesses an upper region of the original primitive man-tle (PM) to form the crust and the depleted mantlelithosphere (DML). In parallel, the mantle lithosphere(ML), which is always thicker than the DML, developsas a conductive thermal boundary layer underlying thecrust. As part of the DML can potentially delaminateand be mixed with the convecting mantle, the composi-tion of the convecting mantle can be more depleted thanthat of the PM. The mantle below the DML is thus re-ferred to as the source mantle (SM), the composition ofwhich is initially identical to the composition of the PMbut can deviate with time. The history of these layersstrongly depends on convective vigor, effects of mantlemelting, and initial conditions.

2.2. Stagnant-Lid Convection with Mantle Melting

Standard parameterizations are used for mantle rhe-ology and the vigor of convection. Mantle viscosity isa function of mantle potential temperature and the de-gree of hydration as (Fraeman and Korenaga, 2010)

η(Tu, CWSM ) = A exp

[E

RTu+ (1− CWSM )log∆ηw

],

(3)where A is a constant factor calculated using a referenceviscosity η0 at a reference temperature T ∗u = 1573 K;E is the activation energy; R is the universal gas con-stant; and ∆ηw is the viscosity contrast between wet

2

T

z

sublithospheric

mantle

hcrust

Fs

Fm

Fc

hm

hDML hML

Ts

TcrustT'u

Tu

Tcm

crust

(thermal)

lithosphere

mantle

core

Figure 1: Cartoons showing the assumed thermal and chemical structure of terrestrial planets taken from Fraeman and Korenaga (2010).In general, terrestrial planets are divided into a crust, mantle, and core, as shown in the left panel. Mantle that has been processedby melting and stays in the thermal boundary layer is depleted mantle lithosphere (DML). The thickness of the DML must alwaysbe equal to or less than the thickness of the mantle lithosphere (ML). The section of the mantle below the thermal boundary layer isthe sublithospheric mantle. The right panel shows the horizontally-averaged temperature distribution. Key model parameters are alsoindicated.

and dry mantle. We use an activation energy of E =300 kJ mol−1, which is appropriate for diffusion creepand dislocation creep within a Newtonian approxima-tion (Christensen, 1984; Karato and Wu, 1993; Kore-naga, 2006). The normalized water concentration inthe source mantle, CWSM , has an initial value of oneand decreases toward zero as mantle melting causesdehydration. To write Eq. 3, we make the major as-sumption that mantle viscosity is not strongly pressure-dependent, which is consistent with early studies of theevolution of large rocky planets (e.g., Valencia et al.,2006) and some theoretical predictions (Karato, 2011)but in contrast to recent work (e.g., Papuc and Davies,2008; Stamenkovic et al., 2011, 2012). Our choice isthus further explained in the discussion section.

Two non-dimensional parameters characterize ther-mal convection with the above viscosity formulation(Solomatov, 1995). First, the internal Rayleigh numberserves to quantify potential convective vigor (Fraemanand Korenaga, 2010)

Rai =αρmg(T ′u − Tc)h3mκη(Tu, CWSM )

, (4)

where α is the coefficient of thermal expansion; κ is thethermal diffusivity; Tc and T ′u are, respectively, the tem-perature at the bottom of the crust (called “Moho tem-

perature”) and the mantle potential temperature de-fined at the top of the mantle; and hm is the thickness ofthe mantle. Second, the Frank-Kamenetskii parameteris defined as (Solomatov, 1995; Fraeman and Korenaga,2010)

θ =E(T ′u − Tc)

RT 2u

. (5)

With these two parameters, the average convectivevelocity beneath the stagnant-lid may be calculated as(Solomatov and Moresi, 2000)

u = 0.38κ

hm

(Ra

θ

)1/2

. (6)

To include the effects of compositional buoyancy anddehydration stiffening, the Nusselt number, which is anon-dimensional measure of convective heat flux, mustbe calculated with a local stability analysis at each timestep (Korenaga, 2009). The symbolic functionality maybe expressed as

Nu = f(Ra,E, Tu, Tc, hl, hm,∆ρ,∆ηm), (7)

where ∆ρ and ∆ηm are the density and viscosity con-trasts between the source mantle and depleted mantle,respectively, and hl is the thickness of the depleted man-

3

tle lithosphere. The thickness of a thermal boundarylayer in the mantle is then easily calculated using

hML =hmNu

. (8)

The chemical evolution of the mantle strongly affectsterrestrial planet evolution. To first order, partial melt-ing of the mantle can be considered to begin at a depthwhere the temperature exceeds the solidus of dry peri-dotite, as long as the mantle is not significantly wet(Hirth and Kohlstedt, 1996). The initial pressure ofmelting is (Korenaga et al., 2002)

Pi =Tu − 1423

1.20× 10−7 − (dT/dP )S, (9)

where (dT/dP )S is the adiabatic mantle gradient, whichis roughly constant for the pressure range relevant tomantle melting. Therefore, Pi should be approximatelyconstant for any terrestrial planet with Earth-like man-tle composition. Melting stops when the convective up-welling reaches the base of the mantle lithosphere. Thatis, the final pressure of melting is given by

Pf = ρLg(hc + hML), (10)

where hc is the thicknesses of the crust; g is gravita-tional acceleration; and ρL is the density of the litho-sphere. For convenience, we use the Martian ρm as ρL,noting that ρL should remain roughly constant whereasρm, an averaged mantle parameter, increases with plan-etary mass because of pressure effects. If Pf < Pi, thenmelting occurs in the melting zone between Pi and Pf ,with thickness dm and average melt fraction equal to

φ =Pi − Pf

2

(dφ

dP

)S

, (11)

where (dφ/dP )S is the melt productivity by adiabaticdecompression. Volumetric melt productivity is finallyparametrized as

fm =2χdmuφ

hm4πR2

m, (12)

where χ ∼ 1 if the upwelling mantle is cylindrical andall downwelling occurs at the cylinder’s edge (Soloma-tov and Moresi, 2000; Fraeman and Korenaga, 2010).The crustal temperature profile is calculated as in Frae-man and Korenaga (2010), with the modification thatcrustal material with a temperature above Tcrit = 1273K is considered to be buoyant melt that migrates withinone time step immediately below the planet’s surface,producing a relatively cooler crust and a larger man-tle heat flux. This modification is not important forMartian cases, because the Moho temperature does notreach the threshold except for some extreme cases, butbecomes essential to achieve a realistic crustal thermalprofile for larger planets.

2.3. Likelihood of Plate Tectonics

Thermal evolution models featuring stagnant-lid con-vection are not applicable to planets on which platetectonics occurs. If a suitable weakening mechanismexists, the lithosphere may be broken into plates andrecycled into the mantle. Many aspects of plate tec-tonics on Earth, however, are not captured in currentmathematical models (Bercovici, 2003). Quantifyingthe conditions under which plate tectonics is favoredover stagnant-lid convection is likewise difficult, and theeffect of planetary mass on the likelihood of plate tec-tonics has been controversial (Valencia et al., 2007; Ko-renaga, 2010a; van Heck and Tackley, 2011). Recentstudies suggest, however, that the effects of planetarymass on yield and convective stresses may be dominatedby uncertainties in other important planetary parame-ters, such as internal heating and lithosphere hydration(Korenaga, 2010a; van Heck and Tackley, 2011).

This study uses a simple scaling that is consistentwith current understanding of rock mechanics (Kore-naga, 2010a), though the possibility of different litho-sphere weakening mechanisms (e.g., Landuyt et al.,2008) cannot be excluded. We assume that plate tec-tonics can occur if convective stress exceeds the brittlestrength of lithosphere given by

τy = c0 + µρgz, (13)

where c0 is the cohesive strength, µ is the effective fric-tion coefficient, and z is depth (Moresi and Solomatov,1998). Experimental data indicate that the cohesivestrength is negligible under lithospheric conditions, i.e.,c0/(µρz) � 1 (Byerlee, 1978). We use another non-dimensional parameter (Korenaga, 2010a):

γ =µ

α(Tu − Ts), (14)

where the relevant temperature difference is the differ-ence between the mantle potential and surface temper-atures. In the parameterized convection model formu-lated in the previous section, we separately consider thecrust and the mantle, but when discussing the likeli-hood of plate tectonics using the scaling of Korenaga(2010b), it is more convenient to treat the crust andmantle together, assuming that crustal rheology is sim-ilar to mantle rheology.

Detailed scaling analyses (Korenaga, 2010a,b) showthat the effective viscosity contrast across the litho-sphere can be parameterized as

∆ηL = exp(0.327γ0.647θtot), (15)

where θtot is the Frank-Kamenetskii parameter definedusing the total temperature difference explained above,i.e.,

θtot =E(Tu − Ts)

RT 2u

. (16)

A transition from plate-tectonic to stagnant-lid convec-tion can take place if the above viscosity contrast ex-

4

ceeds a critical value

∆ηL,crit = 0.25Ra1/2i,tot, (17)

where Rai,tot is defined to incorporate surface temper-ature as

Rai,tot =αρmg(Tu − Ts)(hc + hm)3

κη(Tu, CWSM ). (18)

For each simulation, if ∆ηL/∆ηL,crit ≤ 1 at any time,then plate tectonics may have been favored at somepoint during the evolution of a given planet. The sat-isfaction of this criterion may strongly depend on thevalue of µ, so a wide range of values should be tested.For silicate rocks, plausible values of µ range from 0.6to 0.7, according to both laboratory studies (Byerlee,1978) and measurements of crustal strength on Earth(e.g., Brudy et al., 1997). Surface water, however, maylower these values substantially via thermal crackingand mantle hydration (Korenaga, 2007).

3. Numerical Models

The parametrized model described above was usedto calculate thermal histories of Mars- and Venus-likeplanets and 1 to 10M⊕ super-Venus planets, where the⊕ subscript denotes parameters for Earth, for a du-ration of 4.5 Gyr using numerical integration with atime step of 1 Myr. A wide parameter space was ex-plored by varying the initial mantle potential tempera-ture, Tu(0); the initial core/mantle boundary tempera-ture, Tcm(0); the initial volumetric heat production, Q0;the reference mantle viscosity, η0; and the viscosity con-trast between dry and wet mantle, ∆ηw. Previous workfor Mars demonstrated that simulation results were notvery sensitive to the degree of compositional buoyancyand other parameters (Fraeman and Korenaga, 2010).Table 1 lists model constants common to all simulations,and Table 2 lists planet-specific ones.

3.1. Application to Mars- and Venus-like Planets

For Venus, the following sets of initial conditions wereused: Initial mantle potential temperature, Tu(0) =1400, 1550, 1700, 1850, and 2000 K; initial core/mantleboundary temperature, Tcm(0) = 3500, 4000, and4500 K; reference viscosity, η0 = 1018, 1019, and 1020 Pas; and dehydration stiffening, ∆ηw = 1, 10, and 100.For Mars, initial core/mantle boundary temperatureswere 2250, 2500, and 3000 K. In addition, five differentvalues were tested for the amount of internal heatingQ0. Compositional buoyancy was set as (dρ/dφ) = 120kg m−3 for all simulations. The fraction of light el-ements in the core was fixed at 0.2 for all simulationruns to avoid inner core solidification (Schubert et al.,1992; Fraeman and Korenaga, 2010). Simulations wereperformed for all permutations of the above initial con-ditions, although unrealistic simulation results were dis-carded in the following way: For both Venus and Mars,

Constant Value Units Ref.λ 1.38× 10−17 s−1 [1]k 4.0 W m−1 K−1 [1]α 2× 10−6 K−1 [1]κ 10−6 m2 s−1 [1]ρL 3527 kg m−3 [1]Lm 6.0× 105 J kg−1 [2]

Lc + Eg 1.0× 10−6 J kg−1 [2]Cm 1000a J kg−1 K−1 [3]Cc 850a J kg−1 K−1 [3]ηm 1.3a N/A [1]ηc 1.2a N/A [1]

(dT/dP )S 1.54× 10−8 K Pa−1 [4](dφ/dP )S 1.20× 10−8 Pa−1 [4]

Table 1: Summary of universal constants used in all simulations.References: 1. Stevenson et al. (1983), 2. Fraeman and Korenaga(2010), 3. Noack et al. (2011), 4. Korenaga et al. (2002). aMarshas Cm = 1149; Cc = 571; ηm = 1.0; and ηc = 1.1 (Fraeman andKorenaga, 2010).

inner core growth was disallowed and total surface heatflux at the present was required to be positive. Further-more, the condition hc(tp) < 500 km was imposed todisregard results with unrealistic crustal growth. Noneof the 675 simulations for Venus failed these criteria,but 30 of the 675 simulations for Mars were discarded.

The appropriate magnitude of radiogenic heating ispoorly constrained in general, especially for terrestrialexoplanets. Even for Earth, the abundance of radio-genic heating is controversial. Geochemical constraintssupport a low Urey ratio, the ratio of internal heatproduction to surface heat flux, but this is known toconflict with the cooling history of Earth unless a non-classical heat-flow scaling for mantle convection is as-sumed (Christensen, 1985; Korenaga, 2008). A Ureyratio close to one has thus long been preferred from ageophysical perspective (Davies, 1980; Schubert et al.,1980, 2001) and can be used to provide an upper boundfor the initial concentration of radioactive elements inEarth’s chemically undifferentiated mantle. Assuminga present-day surface heat flux of 46 TW (Jaupart et al.,2007), an extreme upper bound for Earth is Q0 ≈3.5×10−7 W m−3.

A recent petrological estimate on the thermal historyof Earth is actually shown to favor a low Urey ratio(∼0.3) with a non-classical heat-flow scaling (Herzberget al., 2010), indicating that geochemical constraints onthe heat budget may be robust. In the thermal evo-lution models of Kite et al. (2009), for example, con-centrations of 40K, 232Th, 235U, and 238U taken fromRingwood (1991) and Turcotte and Schubert (2002)were considered, corresponding to values forQ0 between1.2 × 10−7 and 8.7 × 10−8 W m−3 for Venus. We thuschose to use the following values for initial volumetricradiogenic heating: Q0 = 0.5, 0.75, 1.0, 1.25, and 1.75×10−7 W m−3 (in the case of Venus). The default in-termediate value is 1.0 ×10−7 W m−3. For other plan-

5

Constant Mars Venus 1M⊕ 2M⊕ 4M⊕ 5M⊕ 6M⊕ 8M⊕ 10M⊕ Unitsg 3.70 8.87 10.0 13.6 18.6 20.7 22.6 26.0 29.1 m s−2

Ts 220 730 300 300 300 300 300 300 300 KRp 3390 6050 6307 7669 9262 9821 10295 11072 11696 kmRc 1550 3110 3295 3964 4723 4986 5206 5564 5848 kmρm 3527 3551 4476 4951 5589 5845 6078 6497 6873 kg m−3

ρc 7200 12500 12961 14882 17594 18698 19708 21530 23174 kg m−3

Pcm 19 130 151 284 556 697 842 1144 1463 GPaPc 40 290 428 821 1639 2067 2508 3431 4406 GPa

Table 2: Summary of planet-specific constants for Mars, Venus, and seven super-Venus planets. Martian values were taken from Fraemanand Korenaga (2010) and references therein. Venusian values can be found in Spohn (1991) and Noack et al. (2011). Super-Venus valueswere calculated in this study from simple interior models following Seager et al. (2007).

ets, Q0 was multiplied by ρm/ρm,♀, where ♀ indicatesthe Venusian value, to maintain constant element abun-dances in more or less compressed mantles.

3.2. Application to Super-Venus Planets

One-dimensional profiles of massive terrestrial exo-planets were generated to calculate planet-specific con-stants used in the above stagnant-lid convection model.Many interior structure models exist for massive solidexoplanets, ranging from simple to very complex (Va-lencia et al., 2006; Seager et al., 2007; Sotin et al., 2007;Wagner et al., 2011). To study the first-order effects ofplanetary mass on stagnant-lid convection, a relativelysimple structure with an Fe(ε) core and a MgSiO3 man-tle is assumed, as in Seager et al. (2007) and Kite et al.(2009). The resulting interior density and pressure dis-tributions neglect several obvious factors such as tem-perature effects, but yield results remarkably similar tothose from more complex models.

Three equations are solved to calculate m(r), themass contained within radius r; P (r), the pressure dis-tribution; and ρ(r), the density distribution. A self-consistent internal structure must satisfy the materialspecific equations of state

P (r) = fEOS(ρ(r), T (r)), (19)

the equation of hydrostatic equilibrium

dP (r)

dr=−Gm(r)ρ(r)

r2, (20)

and the conservation of mass equation for a sphericalmass distribution

dm(r)

dr= 4πr2ρ(r), (21)

where G is the gravitational constant; T (r) is the radialtemperature profile; and fEOS represents a material-specific equation of state (Seager et al., 2007).

Equations of state are numerically calculated usingconstants from Table 3 to sufficient resolution so thatρ(r) can be determined to within ±1 kg m−3. For adesired MP , equations (20) and (21) are numericallyintegrated from the center of a planet with the inner

0 0.5 1 1.5 20

5

10

15

20

25

30

Radius [R

]

Density [

1000 k

g/m

3]

10M

8M

6M

5M

4M

2M

1M

PREM

Figure 2: Interior density distributions for super-Venus planetswith MP = 1, 2, 4, 5, 6, 8, and 10M⊕. The material equations ofstate and the equations of conservation of mass and hydrostaticequilibrium were numerically integrated to build simple planets,and the interior boundary condition was adjusted until the result-ing planet had the desired mass. These simple models are used tocalculate averaged values for mantle density, ρm; core density, ρc;surface gravity, g; central pressure, Pc; and core/mantle bound-ary pressure, Pcm. For comparison, the density distribution forEarth from the Preliminary Reference Earth Model (PREM) ofDziewonski and Anderson (1981) is also plotted.

boundary conditions M(0) = 0 and P (0) = Pc, wherePc is a guessed central pressure. The outer boundarycondition is simply P (RP ) = 0. Errors associated withignoring temperature effects are limited to a few per-cent (Seager et al., 2007). With this method, the choiceof Pc determines the RP at which the outer boundarycondition is satisfied. These calculations are iteratedwith the bisection method until Pc is found such thatm(RP ) = MP to within 0.1%. The equation of statefor Fe(ε) is used until m(r) = 0.325MP , mandating a32.5% core mass fraction. The MgSiO3 perovskite to en-statite phase transition is assumed to occur at 23 GPa,although the transition pressure increases to ∼25 GPaat ∼800 K (e.g., Ita and Stixrude, 1992). Neglectingthis phase transition would produce unrealistically highnear-surface densities.

Pressure, mass, and density distributions were cal-

6

Material K0 (GPa) K′

0 K′′

0 (GPa-1) ρ0 (kg m-3) EOSFe(ε) 156.2 6.08 N/A 8300 VMgSiO3(pv) 247 3.97 -0.016 4100 BME4MgSiO3(en) 125 5 N/A 3220 BME3

Table 3: Material constants used to generate interior structure models, taken from Seager et al. (2007). Using three different equationsof state, P (ρ) is calculated to high resolution for each material. The Vinet and 3rd and 4th order Birch-Murnaghan equations of stateare abbreviated V, BME3, and BME4, respectively.

culated for planets with MP = 1, 2, 4, 5, 6, 8, and10M⊕. From these models, averaged densities for thecore and mantle were calculated, along with Pc and Pcm.Surface gravitational accelerations were calculated us-ing g = GMP /R

2P for each planet. These constants

are reported in Table 2. The density distributions forthese planets are shown in Fig. 2, along with the densityprofile for Earth from the Preliminary Reference EarthModel (PREM) of Dziewonski and Anderson (1981).Compared to PREM, the scheme for calculating internalstructures used in this study overestimates the densityof the core and underestimates the radius of the coreof an Earth-mass planet, but returns RP (M⊕) ≈ R⊕despite ignoring details of mineral composition, phasetransitions, and temperature effects. In more massiveplanets, the enstatite to perovskite phase transition oc-curs at much shallower depths because a higher surfacegravity causes a greater increase in pressure with depth.Furthermore, Pc increases much more rapidly than Pcmwith increasing planetary mass.

Most of the Martian and Venusian initial conditionscan be used for super-Venus thermal evolution mod-els, but some must be modified appropriately. For ex-ample, the core/mantle temperature for super-Venusplanets should increase along the mantle adiabatic tem-perature gradient: for 5 and 10M⊕ super-Venus plan-ets, initial core/mantle boundary temperatures are in-creased by roughly 350 and 900 K, respectively, from theinitial conditions for Venus. These temperatures stillcorrespond to a so-called “hot start,” which is likelyfor terrestrial planets because of the large magnitudeof gravitational potential energy released during accre-tion (Stevenson et al., 1983). Only three simulationsfor the 5M⊕ planet, and no simulations for the 10M⊕planet, failed the requirements on crustal thickness, sur-face heat flux, and inner core growth.

4. Results

Thermal evolution simulations were performed forMars- and Venus-like planets and super-Venus planets.The following sections summarize the results, beginningwith a few representative examples for Mars and Venus.Principal component analysis, as described in the ap-pendix, was applied using all simulation results to iden-tify major model behaviors. We also tried to quantifyrelations between input and output parameters and, de-spite the complexity of our model formulation, a linear

function of initial conditions is found to reasonably ap-proximate many output parameters of interest.

4.1. Sample Thermal Histories for Mars- and Venus-like Planets

Sample thermal histories for Mars and Venus areshown in Fig. 3. These models span the entire rangeof initial radiogenic heating values with all other ini-tial conditions set to intermediate values. In particular,Tu(0) = 1700 K, µ0 = 1019 Pa s, and ∆µw = 100. ForVenus and Mars, respectively, Tcm(0) = 4000 and 2500K and Ts = 220 and 730 K. Initially very hot coresare assumed here because core segregation is expectedto release a large amount of gravitational potential en-ergy. This excess heat is released into the mantle dur-ing the first hundred million years of planetary evolu-tion. Thereafter, mantle dynamics controls core cool-ing. Whereas internal heating has a great effect on sur-face heat flux, mantle temperatures only differ to within±200 K for the sampled range of Q0. Mars evolves witha consistently lower potential temperature than Venus.Because Mars also has a relatively shallow mantle, theMartian core is cooled down more efficiently.

Crustal thickness is an important, potentially observ-able constraint for planetary evolution models. Marsand Venus, with different magnitudes of radiogenicheating, have very different crustal formation histories.Both planets start with no initial crust, but quickly pro-duce some through mantle melting. For Venus, Mohotemperatures quickly reach the melting point of basaltfor all initial internal heating choices. Crustal pro-duction occurs for the first ∼1 Gyr of evolution, withthicker crust for higher internal heating. For Mars,crustal production is gradual and crustal temperaturesare much lower, with increased internal heating causingan longer period of crustal formation and increased to-tal crustal production. Both Mars and Venus undergosubstantial mantle processing, indicating that the deepinterior serves as a significant source of endogenous wa-ter, especially during the first ∼1.5 Gyr of their evolu-tion.

4.2. Sensitivity Analyses for the Evolution of Mars- andVenus-like Planets

Figures 4 and 5 summarize the results of 1320 simula-tions for Venus and Mars, respectively. Present-day val-ues for selected output parameters are plotted againstcrustal thickness for both planets. Several correlations

7

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Figure 3: Sample histories for Venus (top) and Mars (bottom). From left to right, red curves signify Moho temperature, surface heat flow,crustal thickness, and normalized mantle water content. Likewise, blue curves represent mantle potential temperature, mantle heat flux,depleted mantle lithosphere thickness, and fraction of processed source mantle; green curves show core/mantle boundary temperature,core heat flux, and lithosphere thickness. Solid, dashed, and dotted lines indicate Venusian Q0 = 0.5, 1.0, and 1.75 ×10−7 W m−3,respectively. Default initial conditions are Tu(0) = 1700 K, η0 = 1019 Pa · s, ∆ηw = 100, and (dρ/dφ) = 120 kg m−3. Venus and Marshave Tcm(0) = 4000 and 2500 K, respectively. Because crustal melting causes highly discontinuous surface heat flux, a moving averagewith a 75 Myr span was used for plotting purposes.

are readily apparent. For Venus, thicker crust is as-sociated with higher Moho temperature, more mantleprocessing, higher mantle heat flux, and quicker crustalformation. More specifically, Moho temperature in-creases with crustal thickness in a linear fashion untilhc ≈ 75 km, after which Moho temperatures remainnear the critical value for basalt melting. Simulationswith crustal melting have highly discontinuous surfaceand mantle heat fluxes, but such discontinuous nature ismerely an artifact owing to our particular numerical im-plementation, so average values of Fs and Fm over thefinal 100 Myr of planetary evolution are used for all sub-sequent analyses. In contrast, Moho temperatures forMars only approach the critical value for basalt meltingin simulations with the thickest crust. Unlike for Venus,a decrease in present-day mantle heat flux accompaniesan increase in crustal thickness for Mars.

Principal component analysis facilitates the interpre-tation of the correlations between output parameters.For Venus, two principal components account for most(>65%) of the variance of the planetary parameters af-ter 4.5 Gyr of thermal evolution. Calculations of theprinciple components returns coefficients with valuesbetween -1 and 1 that are associated with each modeloutput parameter. Appendix A contains a table of prin-cipal component basis vectors for Venus. Comparingnumerical values of select coefficients may reveal corre-lations with physical explanations. Arrows representingthe eigenvectors associated with these principal compo-nents are plotted in Figs. 4 and 5. These arrows indicatethe axes along which the majority of the variance inthe model output primarily lies. No preferred polarityexists for the principal component eigenvectors; plot-ting these arrows with a 180◦ rotation would be equallyvalid.

The first principal component represents the most

dominant correlations among present-day planetary pa-rameters, which are characterized mainly by the thick-nesses of the crust and mantle lithosphere layers, asthey are associated with large coefficients: hc (0.34), hl(-0.36), and hML (-0.39). Because the sign of the coef-ficient for hc is opposite to the sign of the other two co-efficients, the thicknesses of the crust and mantle litho-sphere are anti-correlated. In other words, thick crust isassociated with thin depleted mantle lithosphere and athin thermal boundary layer and vice versa, since prin-cipal components have no preferred polarity. An ini-tially hotter mantle produces thicker crust and thickerdepleted lithosphere. Because mantle viscosity is lowerfor hotter mantle, however, the depleted lithosphereis more likely to be destabilized, resulting in a thin-ner lithosphere (and thus thermal boundary layer) forthicker crust. Other coefficients in the first principalcomponent indicate the effects of crustal thickness onother model parameters, including the first-order cor-relations observed during inspection of Fig. 4. For in-stance, thick crust is associated with high Moho tem-perature and high surface and mantle heat fluxes. Thickcrust also indicates a high degree of mantle processingand a corresponding low present-day mantle water con-tent. Finally, the large negative coefficients for bothtc,10% and log10(∆tc,tot) indicate that thick crust tendsto form early and quickly.

The second principal component elucidates the ef-fects of planet temperatures on other model parameters,since large coefficients are associated with Tu (0.45) andTcm (0.44). Unsurprisingly, high mantle potential andcore/mantle boundary temperatures are associated withhigh Moho temperature, since Tc has a coefficient of0.25. Moreover, high interior temperatures correspondto thick crust and a high degree of mantle processing,which would cause the present-day mantle water con-

8

0 50 100 150 200 250600

800

1000

1200

1400

hc(t

p) [km]

Tc(t

p)

[K]

(a)

0 50 100 150 200 2501400

1600

1800

2000

2200

hc(t

p) [km]

Tu(t

p)

[K]

(b)

0 50 100 150 200 25020

40

60

80

100

hc(t

p) [km]

Fs(t

p)

[mW

/m2]

(c)

0 50 100 150 200 25020

30

40

50

60

hc(t

p) [km]

Fm

(tp)

[mW

/m2]

(d)

0 50 100 150 200 2500

0.5

1

1.5

2

hc(t

p) [km]

Vpro

c(t

p)/

Vsm

(e)

0 50 100 150 200 2506

7

8

9

10

hc(t

p) [km]

log

10(t

c,t

ot)

[Gyr]

(f)

Figure 4: Summary of parameter values at the present for 675 simulations of the thermal evolution of Venus. Arrows are projectionsof the principal component basis vectors that emanate from a point representing the averaged simulation results, indicating axes thataccount for the vast majority of the data set’s variance. The red arrow represents a larger percentage of cumulative variance (41%)than the green arrow (27%). Panels show (a) Moho temperature, (b) mantle potential temperature, (c) surface heat flux, (d) mantleheat flux, (e) fraction of mantle processed by melting, and (f) total time for crust to grow from 10% to 95% of its present thickness asfunctions of crustal thickness. Because crustal melting causes highly discontinuous surface and mantle heat fluxes, the model outputsare the averaged values for the final 100 Myr of planetary evolution.

centration to be very low. Note, however, that high(present-day) interior temperatures do not correspondto thick crust in case of the first principal component(Fig. 4b). In this space of crustal thickness and uppermantle temperature, the first and second principal com-ponents are nearly orthogonal, thus explaining the over-all spread of simulation results. With principal compo-nent analysis, we can visualize how the most dominanttrend (represented by the first principal component) isaffected by secondary factors and how these secondaryfactors manifest in different parameter spaces. An im-portant point is that the overall variability of plane-tary evolution can be compactly represented by a smallnumber of principal components; that is, the effectivedimension of the model space is actually small.

The principal components for Mars are very similarto those for Venus, with some notable exceptions. Thefirst principal component again represents the effectsof strongly correlated Moho temperature and crustalthickness, and thus explains the largest portion of thevariance in the model output. As for Venus, a thin,cold crust is associated with thick depleted mantle litho-sphere, a thick thermal boundary layer, a low surfaceheat flux, and a low degree of mantle processing. Un-like Venus, however, the surface and mantle heat fluxesin the first principal component are anti-correlated (seealso Fig. 5).

Despite the complexity of our thermal evolutionmodel, some present-day parameters are found to be

predicted with reasonable accuracy for Venus and Marsusing a linear function. A general formula for this func-tion is

Bi = Ai,0 +Ai,1Tu,n(0) +Ai,2Tcm,n(0) +

Ai,3(log10(η0))n +Ai,4(log10(∆ηw))n +Ai,5Q0,n, (22)

where Bi is the value of the desired output parame-ter after 4.5 Gyr, constants Ai,0 through Ai,5 are esti-mated using the least-squares method for each Bi, andthe subscript n indicates that the input parameters arenormalized and mean subtracted.

Table 4 lists constants for Venusian Bi that have rel-atively high correlation coefficients between predictedand actual simulation results. Figure 6 shows contourplots with predicted values of mantle potential temper-ature, crustal thickness, and duration of crustal forma-tion for given initial internal heating and mantle poten-tial temperature. While present-day mantle potentialtemperature and crustal thickness depend strongly onboth initial mantle potential temperature and the mag-nitude of internal heating, the total duration of crustalformation is primarily a function of initial mantle poten-tial temperature (see Table 4 for more complete infor-mation on parameter sensitivity). Figure 6 also demon-strates a reasonable correspondence between the pre-dicted and actual values of these model outputs for allof the simulations. This way of summarizing simula-tion results allows us not only to see the sensitivity ofmodel outputs to initial parameters but also to quickly

9

0 100 200 3000

500

1000

1500

hc(t

p) [km]

Tc(t

p)

[K]

(a)

0 100 200 3001200

1400

1600

1800

2000

hc(t

p) [km]

Tu(t

p)

[K]

(b)

0 100 200 30010

20

30

40

hc(t

p) [km]

Fs(t

p)

[mW

/m2]

(c)

0 100 200 3005

10

15

20

25

hc(t

p) [km]

Fm

(tp)

[mW

/m2]

(d)

0 100 200 3000

1

2

3

hc(t

p) [km]

Vpro

c(t

p)/

Vsm

(e)

0 100 200 3007

7.5

8

8.5

9

9.5

hc(t

p) [km]

log

10(t

c,t

ot)

[Gyr]

(f)

Figure 5: Summary of parameter values at the present for 645 simulations of the thermal evolution of Mars. Arrows are projectionsof the principal component basis vectors that emanate from a point representing the averaged simulation results, indicating axes thataccount for the vast majority of the data set’s variance. The red arrow represents a larger percentage of cumulative variance (42%)than the green arrow (23%). Panels show (a) Moho temperature, (b) mantle potential temperature, (c) surface heat flux, (d) mantleheat flux, (e) fraction of mantle processed by melting, and (f) total time for crust to grow from 10% to 95% of its present thickness asfunctions of crustal thickness. Because crustal melting causes highly discontinuous surface and mantle heat fluxes, the model outputsare the averaged values for the final 100 Myr of planetary evolution.

reproduce major modeling results without redoing sim-ulation.

4.3. Evolution of Super-Venus Planets

We investigate the evolution of super-Venus planetsto explore the effects of planetary mass on stagnant-lidconvection. For simplicity, surface temperatures for allsuper-Venus planets are assumed to be 300 K, thoughin reality this temperature may vary with time and ishighly dependent on atmospheric composition and onthe distance to and the luminosity of the central star.

4.3.1. Sample Thermal Histories

Super-Venus planets with MP = 1, 5, and 10M⊕ wereevolved to study the effects of increasing planetary masson a variety of parameters, particularly crustal produc-tion. For all three planets, Q0 was scaled to the Venu-sian value of 1.0×10−7 W m−3, Tu(0) = 1700 K, andTcm(0) = 4000, 4350, and 4900 K, respectively. De-hydration stiffening and compositional buoyancy wereboth incorporated as usual. Figure 7 shows the resultsof these simulations. As with Venus and Mars, the tran-sient “hot start” in the core is lost in the first ∼100 Myr.After this initial cooling, mantle dynamics controls corecooling. Because the mantle heats up for the first ∼1Gyr and then cools only very slowly, core cooling isprecluded for the first ∼2 Gyr. As suggested by simplescaling laws (Stevenson, 2003), mantle cooling paths formassive super-Venus planets are roughly parallel.

Figure 7 also shows how the thicknesses of the crust,mantle lithosphere, and depleted mantle lithospherevary with time. With increasing planetary mass, crustalthickness decreases. The simple scaling analyses belowindicate that more massive planets have greater meltproduction. The observed increase in mantle potentialtemperature with planetary mass only accentuates thiseffect. The increased melt volume, however, is not suf-ficient to create a thicker crust on a larger planet. The1M⊕ planet in the stagnant-lid regime ceases crustalproduction soon after 1 Gyr as mantle potential tem-perature drops below a critical value. The increasedinterior temperatures for the more massive planets al-low longer durations of crustal production. For the first∼2 Gyr of thermal evolution, the thickness of the de-pleted mantle lithosphere is close to that of the mantlethermal lithosphere, reflecting the continuous delamina-tion of excess depleted mantle lithosphere. Decreasedcrustal production with increasing planetary mass cor-responds to a smaller degree of mantle processing anda higher content of residual mantle water.

4.3.2. Sensitivity Analyses

The output of 1347 simulations for 5 and 10M⊕super-Venus planets are shown in Figs. 8 and 9. Threesimulations for the 5M⊕ super-Venus planet were ex-cluded because they did not meet the requirements thathc < 500 km and that inner core growth did not oc-cur. As for Mars and Venus, present-day parameters of

10

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Figure 6: Top panels show predicted values for Venus after 4.5 Gyr given initial volumetric radiogenic heating and initial mantle potentialtemperature, the initial conditions to which the output is most sensitive. Bottom panels show the correspondence between predicted andactual simulation results. The dashed line represents perfect predictive power. Default initial conditions are Tcm(0) = 4000 K, η0 = 1019

Pa · s, ∆ηw = 100, and (dρ/dφ) = 120 kg m−3. Panels show predicted values of (a) mantle potential temperature, (b) crustal thickness,and (c) total time for crust to grow from 10% to 95% of its present thickness.

interest are plotted against present-day crustal thick-ness. The principal component eigenvectors, explainedbelow, are projected onto each plot, emanating from theaverage simulation output. The table in the appendixcontains the principal component basis vectors for the10M⊕ planet.

These scatter plots reveal similarities between theevolution of both massive planets. For instance, Mohotemperature increases with crustal thickness in a linearfashion before reaching the critical value for basalt melt-ing. With increasing planetary mass, the critical crustalthickness at which this transition occurs decreases. Forrelatively thick crust, Moho temperatures remain nearthe critical value for basalt melting. For both super-Venus planets, an increase in crustal thickness is asso-ciated with an increase in present-day mantle potentialtemperature, mantle heat flux, and degree of mantleprocessing. The total duration of crustal formation de-creases with increasing present-day crustal thickness.Again, correlations between model parameters may bestudied in more detail with principal component analy-sis.

For both planets, as for Venus and Mars, the firstprincipal component is characterized by a strong cor-relation between Moho temperature and crustal thick-ness, explaining the general trends observed in Figs. 8and 9. A decrease in both quantities is associated withan increase in the thicknesses of the depleted mantlelithosphere and the thermal boundary layer, a decreasein surface and mantle heat fluxes, an increase in the

duration of crustal formation, and a decrease in thedegree of mantle processing. The second principal com-ponent illuminates the effect of correlated interior tem-peratures. As expected, increasing mantle potential andcore/mantle boundary temperatures causes an increasesin crustal thickness, the total duration of crustal forma-tion, and the degree of mantle processing.

Many present-day model parameters of interest canbe represented as a linear function of initial conditions(Table 5). Compared to the case of Venus, a greaternumber of parameters are found to be approximatedreasonably well by this approach. The effects of melt-ing at the base of the crust undoubtedly remain a largesource of nonlinearity in the model output for all terres-trial planets more massive than Mars. A more elabo-rate numerical implementation to deal with exceedinglyhigh crustal temperatures may reduce such nonlinearity,though we did not explore this possibility.

4.4. Scaling of Crustal Thickness and Mantle Process-ing

We conduct simple scaling analyses to better under-stand the cause of decreasing crustal thickness and adecreasing degree of mantle processing with increasingplanetary mass.

4.4.1. Crustal Thickness

A number of parameters govern the scaling of crustalthickness with planetary mass. Increased melt produc-tion, for instance, is the first requirement for thicker

11

Bi Ai,0 Ai,1 Ai,2 Ai,3 Ai,4 Ai,5 Units Corr.Tu 1772 43.6 11.3 89.9 69.4 30.4 K 0.95Tcm 3128 88.5 32.5 137.0 104.8 48.8 K 0.94hc 115 26.4 22.2 -2.98 -7.84 18.0 km 0.92hl 53.6 -2.17 -5.51 17.5 17.8 -6.07 km 0.91hML 84.1 -4.82 -4.87 19.2 14.9 -6.87 km 0.94Fs 50.7 3.96 2.38 -1.87 -0.76 13.5 mW m−3 0.98Fm 29.6 3.40 1.51 -1.91 -1.55 2.92 mW m−3 0.81

log10(u) 0.89 0.11 0.01 -0.15 -0.07 0.04 - 0.82Vproc/Vsm 1.04 0.00 0.06 -0.01 0.07 0.16 - 0.74

log10(∆tc,tot) 8.42 -0.15 -0.16 0.22 0.20 0.01 - 0.87

Table 4: Coefficients for the best-fit linear function (Eq. 22) relating parameter values after 4.5 Gyr for parameters with correlationcoefficients > 0.70 to a given set of initial conditions for Venus. Correlation coefficients quantifying the correspondence between theactual and predicted output parameters were calculated using normalized and mean subtracted input and output parameters. Theaverage values of the input parameters are Tu(0) = 1700 K, Tcm(0) = 4000 K, log10(η0) = 19, log10(∆ηw) = 1, and Q0 = 1.05 × 10−7

W m−3. For the best-fit function, the input parameters are mean subtracted and normalized by 212 K, 408 K, 0.82, 0.82, and 4.30 ×10−8 W m−3, respectively.

Bi Ai,0 Ai,1 Ai,2 Ai,3 Ai,4 Ai,5 Units Corr.Tu 1882 42.1 13.6 93.9 57.7 46.6 K 0.92Tcm 3331 94.1 41.7 132.2 79.0 74.7 K 0.92hc 71.1 12.4 8.31 1.47 -0.64 14.6 km 0.89hl 17.5 -1.31 -1.71 5.35 4.87 -3.97 km 0.74hML 32.0 -2.89 -2.42 7.57 5.00 -5.08 km 0.84Fs 131.1 12.9 7.56 -8.81 -4.12 32.1 mW m−3 0.95Fm 105.6 11.8 6.57 -9.08 -6.87 17.7 mW m−3 0.89

log10(u) 1.65 0.10 0.03 -0.18 -0.07 0.07 - 0.84Cwsm 0.57 0.00 -0.01 -0.01 -0.06 -0.07 - 0.78

Vproc/Vsm 0.61 0.00 0.02 0.02 0.11 0.13 - 0.83tc,10% 0.20 -0.15 -0.10 0.10 0.08 -0.03 Gyr 0.76

log10(∆tc,tot) 8.57 -0.09 -0.12 0.17 0.19 0.03 - 0.79

Table 5: Coefficients for the best-fit linear function (Eq. 22) relating parameter values after 4.5 Gyr for parameters with correlationcoefficients > 0.70 to a given set of initial conditions for a 5M⊕ super-Venus. Correlation coefficients quantifying the correspondencebetween the actual and predicted output parameters were calculated using normalized and mean subtracted input and output parameters.The average values of the input parameters are Tu(0) = 1701 K, Tcm(0) = 4351 K, log10(η0) = 19, log10(∆ηw) = 1, and Q0 = 1.73 ×10−7 W m−3. For the best-fit function, the input parameters are mean subtracted and normalized by 212 K, 408 K, 0.82, 0.82, and 7.07× 10−8 W m−3, respectively.

crust. From Eq. 12, volumetric melt production for aplanet may scale as

fmfm,⊕

=

(dmdm,⊕

)(u

u⊕

)(φ

φ⊕

)(hm,⊕hm

)(AmAm,⊕

)≈(M

M⊕

)δ,

(23)where the subscript ⊕ denotes values for an Earth-massplanet and Am stands for the mantle surface area.

We can approximate δ using the representative in-terior models of Valencia et al. (2006), for which R ∝M0.262, ρm ∝ M0.196, and g ∝ M0.503. First, considerthe thickness of a melting region, dm = zi - zf . Since zf= Pf/(ρLg) is approximately constant for any planet,

dmdm,⊕

≈ g⊕g

=

(M

M⊕

)−0.503, (24)

where a roughly constant mantle to core thickness ratiois assumed, although planetary mantles grow slightly

more than cores with increasing planetary mass. Next,

u

u⊕=hm,⊕hm

(Ra

Ra⊕

) 12

. (25)

The Rayleigh number for a massive planet scales as

Ra

Ra⊕=

(∆Tu

∆Tu,⊕

)(η(Tu)

η(Tu,⊕)

)(g

g⊕

)(ρ

ρ⊕

)(hmhm,⊕

)3

.

(26)Assuming that the first and second terms on the righthand side are roughly equal to unity, we have

Ra

Ra⊕≈(M

M⊕

)1.485

(27)

and thusu

u⊕≈(M

M⊕

)0.481

. (28)

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Figure 7: Sample histories for 1M⊕ (solid lines), 5M⊕ (dashed lines), and 10M⊕ (dotted lines) super-Venus planets. Red, blue, andblack curves, respectively, signify (a) crust, mantle potential, and core/mantle boundary temperatures, (b) surface, mantle, and core heatflows, (c) crust, depleted mantle lithosphere, and mantle lithosphere thicknesses, and (d) normalized mantle water content and fractionof processed source mantle. Default initial conditions are Q0 = 1.0 ×10−7 W m−3 (scaled with ρm), Tu(0) = 1700 K, η0 = 1019 Pa · s,∆ηw = 100, and (dρ/dφ) = 120 kg m−3. The 1, 5, and 10M⊕ planets have Tcm(0) = 4000, 4350, and 4900 K, respectively. Becausecrustal melting causes highly discontinuous surface and mantle heat fluxes, a moving average with a 75 Myr span was used for plottingpurposes.

Because hc is usually much smaller than RP ,

AmAm,⊕

=

(RP − hc

RP,⊕ − hc,⊕

)2

≈(RPRP,⊕

)2

=

(M

M⊕

)0.524

.

(29)Finally, the rest of the scaling relations may simply beassumed as

φ

φ⊕≈ 1 (30)

andhm,⊕hm

≈(M

M⊕

)−0.262. (31)

Hence, δ ≈ 0.240 and (fm/fm,⊕) ≈ (M/M⊕)0.240.Because hc ≈ fm × ∆t/(4πR2

P ), where ∆t is the du-ration of crust growth, an increase in melt produc-tivity with mass does not guarantee an increase incrustal thickness with mass. As planetary mass, andthus radius, increases, a larger volumetric melt produc-tion is required to produce a certain crustal thickness.Specifically, crustal thickness would only increase withmass for δ > 0.524, assuming that ∆t is roughly con-

stant. Therefore, although melt productivity increaseswith planetary mass, this simple scaling analysis indi-cates that crustal thickness should decrease with scaling(hc/hc,⊕) ≈ (M/M⊕)(0.240−0.524) = (M/M⊕)−0.284 .

Panel (a) of Fig. 10 is a plot of model output present-day crustal thickness as a function of planetary massfor simulations of Mars, Venus, and seven super-Venusplanets. While initial conditions strongly affect simu-lation results, the model outputs generally follow thissimple scaling. Smaller planets can have thicker crustthough they tend to be characterized by lower mantletemperatures.

4.4.2. Mantle Processing

The scaling of mantle processing with planetary massfollows easily from the above analysis. A simplifiedequation for the volume of processed mantle is

Vproc ≈fmφ

∆t, (32)

where ∆t is a duration for crustal growth.Thus, the amount of processed mantle scales with

13

0 50 100 1500

500

1000

1500

2000

hc(t

p) [km]

Tc(t

p)

[K]

(a)

0 50 100 1501400

1600

1800

2000

2200

2400

hc(t

p) [km]

Tu(t

p)

[K]

(b)

0 50 100 15050

100

150

200

250

300

hc(t

p) [km]

Fs(t

p)

[mW

/m2]

(c)

0 50 100 15050

100

150

200

250

hc(t

p) [km]

Fm

(tp)

[mW

/m2]

(d)

0 50 100 1500

0.5

1

1.5

hc(t

p) [km]

Vpro

c(t

p)/

Vsm

(e)

0 50 100 1507.5

8

8.5

9

9.5

10

hc(t

p) [km]

log

10(t

c,t

ot)

[Gyr]

(f)

Figure 8: Summary of parameter values at the present for 672 simulations of the thermal evolution of a 5M⊕ super-Venus. Arrows areprojections of the principal component basis vectors that emanate from a point representing the averaged simulation results, indicatingaxes that account for the vast majority of the data set’s variance. The red arrow represents a larger percentage of cumulative variance(41%) than the green arrow (28%). Panels show (a) Moho temperature, (b) mantle potential temperature, (c) surface heat flux, (d)mantle heat flux, (e) fraction of mantle processed by melting, and (f) total time for crust to grow from 10% to 95% of its present thicknessas functions of crustal thickness. Because crustal melting causes highly discontinuous surface and mantle heat fluxes, the model outputis the averaged values for the final 100 Myr of planetary evolution.

planetary mass as

VprocVproc,⊕

≈(fmfm,⊕

)(φ⊕φ

)≈(M

M⊕

)ξ, (33)

so ξ ≈ δ ≈ 0.240.The volume of a super-Venus planet scales as

V

V⊕=

(R

R⊕

)3

≈(M

M⊕

)ζ, (34)

so ζ = 0.786. Therefore, (Vproc/V ) ∝ (M/M⊕)−0.546.Although the amount of processed mantle material in-creases with planetary mass, the fraction of processedmantle decreases with increasing planetary mass be-cause the mantle volume increases more rapidly thanthe amount of processed material. Panel (b) in Fig. 10confirms that the fraction of processed mantle does in-deed decrease with increasing planetary mass accordingto this scaling law, although initial conditions stronglyaffect the simulation results.

4.5. Viscosity Contrasts During Stagnant-Lid Convec-tion

The viscosity contrast across the lithosphere istracked during each thermal evolution simulation, alongwith the critical viscosity contrast above which a planetis locked in the stagnant-lid regime. Figure 11 showsthe output of 595 simulations for Mars, Venus, and twosuper-Venus planets for which Q0, Tu(0), and µ were

varied over a wide range. In particular, all permuta-tions of Q0 = 0.5, 1.0, and 1.75 ×10−7 W m−3 (scaledas usual with ρm) and Tu(0) = 1400, 1700, and 2000 Kwere considered for a range of µ between 0.0 and 0.9.

From this plot, several conclusions may be drawn.First, for values of the frictional coefficient associatedwith dry silicate rocks, µ ∼ 0.7 to 0.8, plate tectonics isnever favored. Second, increasing planetary mass doesnot substantially affect the likelihood of plate tectonics.Third, the effects of choosing different initial conditionsare amplified for greater planetary mass. Finally, al-though choosing extreme initial conditions can changethe viscosity contrast by orders of magnitude, the effectof the friction coefficient is far more important.

4.6. Formation of an Eclogite Layer

At depth, crustal rock may undergo a phase transi-tion to eclogite. To extend the simple analysis fromearlier, we write the thickness of the crust in the eclog-ite stability field as he = hc - de, where de is the depthof the phase boundary. Likewise, we consider δe and δa,the fractions of the crust in and above, respectively, theeclogite stability field. Because δe+δa = δe,⊕+δa,⊕ = 1,we may write

δe = δe,⊕ + δa,⊕

(1− δa

δa,⊕

). (35)

14

0 50 100 1500

500

1000

1500

2000

hc(t

p) [km]

Tc(t

p)

[K]

(a)

0 50 100 1501600

1800

2000

2200

2400

hc(t

p) [km]

Tu(t

p)

[K]

(b)

0 50 100 1500

200

400

600

hc(t

p) [km]

Fs(t

p)

[mW

/m2]

(c)

0 50 100 1500

100

200

300

400

500

hc(t

p) [km]

Fm

(tp)

[mW

/m2]

(d)

0 50 100 1500

0.2

0.4

0.6

0.8

1

hc(t

p) [km]

Vpro

c(t

p)/

Vsm

(e)

0 50 100 1507.5

8

8.5

9

9.5

10

hc(t

p) [km]

log

10(t

c,t

ot)

[Gyr]

(f)

Figure 9: Summary of parameter values at the present for 675 simulations of the thermal evolution of a 10M⊕ super-Venus. Arrows areprojections of the principal component basis vectors that emanate from a point representing the averaged simulation results, indicatingaxes that account for the vast majority of the data set’s variance. The red arrow represents a larger percentage of cumulative variance(42%) than the green arrow (30%). Panels show (a) Moho temperature, (b) mantle potential temperature, (c) surface heat flux, (d)mantle heat flux, (e) fraction of mantle processed by melting, and (f) total time for crust to grow from 10% to 95% of its present thicknessas functions of crustal thickness. Because crustal melting causes highly discontinuous surface and mantle heat fluxes, the model outputis the averaged values for the final 100 Myr of planetary evolution.

The fraction of the crust above the eclogite stabilityfield may scale as

δaδa,⊕

=de/hc

de,⊕/hc,⊕=

(hc,⊕hc

)(dede,⊕

)≈(M

M⊕

)ε.

(36)If we assume that pressure increases hydrostaticallywith depth and that the critical pressure below whichthe phase transition occurs is a constant, then de ∼ 1/g.So, ε = 0.284 - 0.503 = -0.219. Therefore, the fraction ofcrust in the eclogite stability field should increase withplanetary mass as

δe = δe,⊕ + δa,⊕

[1−

(M

M⊕

)−0.219]. (37)

In thermal evolution models, the heat conductionequation is numerically solved to calculate crustal tem-peratures. An approximate temperature profile canalso be calculated using a steady-state approximationas (Turcotte and Schubert, 2002)

T (z) = Ts +Fskz − Qc(tp)

2kz2, (38)

where Qc, the volumetric crustal heat production, iscalculated as

Qc(tp) = Q0e−λtp

(Vproc(tp)

Vc(tp)

), (39)

where tp is 4.5 Gyr and Vc is the volume of the crust.The boundary condition T (hc) = Tc is used to calculatesurface heat flux for specified Moho and surface temper-atures and magnitude of internal heat production. Fi-nally, Eq. 38 is used to calculate the temperature profilethroughout the entire thickness of the crust. Represen-tative temperature profiles for a planet can be used toapproximate the fraction of crust that lies within theeclogite stability field.

Figure 12 shows representative temperature profilesfor Venus and 5 and 10M⊕ super-Venus planets, calcu-lated using representative crustal thicknesses, degreesof mantle processing, and Moho temperatures from theprevious sensitivity analyses. A range of internal radio-genic heating was also considered. The stability field ofeclogite is taken from Philpotts and Ague (2009) andis drawn assuming a hydrostatic pressure increase withdepth. Panel (d) in Fig. 12 summarizes the effects ofinitial conditions on the fraction of crust in the eclog-ite stability region after 4.5 Gyr and shows the scalingfrom Eq. 37. For Mars, Venus, and seven super-Venusplanets, 54 thermal evolution simulations were run tostudy all permutations of the initial conditions Tu(0) =1700 and 2000 K and Venus-equivalent Q0 = 0.5, 1.0,and 1.75 ×10−7 W m−3. As predicted, the fraction ofeclogite crust increases with planetary mass.

15

(a) "Venus"

Temperature [K]

Depth

[km

]

200 400 600 800 1000 1200

0

50

100

150

(b) 5M

Temperature [K]

Depth

[km

]

200 400 600 800 1000 1200

0

20

40

60

80

100

(c) 10M

Temperature [K]

Depth

[km

]

200 400 600 800 1000 1200

0

20

40

60

0 2 4 6 8 100

0.2

0.4

0.6

0.8

(d)

Planet Mass [M/M

]

Fra

ction o

f E

clo

gite C

rust

Figure 12: Crustal temperature profiles for (a) Venus and (b) 5M⊕ and (c) 10M⊕ super-Venus planets, calculated assuming representativecrustal thicknesses, degrees of mantle processing, and Moho temperatures. The green shaded area is the approximate stability field foreclogite, drawn using the phase diagram from Philpotts and Ague (2009) and assuming a hydrostatic pressure increase with depth. Blacksolid and red dashed lines represent Venus-equivalent Q0 = 1.0 × 10−7 and 1.75 × 10−7 W m−3, respectively. Panel (d) shows thefraction of crust in the eclogite phase for Mars, Venus, and seven super-Venus planets. Circles and triangles represent Tu(0) = 1700 and2000 K, respectively. Blue, black, and red symbols represent Venus-equivalent Q0 = 5.0 × 10−8, 1.0 × 10−7, and 1.75 × 10−7 W m−3,respectively.

5. Discussion

5.1. Pressure Effects on Mantle Rheology

The rheological behavior of the mantles of largerocky planets is difficult to predict. While thecore/mantle boundary pressure for Earth is ∼135 GPa(e.g., Dziewonski and Anderson, 1981), pressures withinthe silicate mantles of large rocky planets likely exceed1 TPa (e.g., Valencia et al., 2006). Above the tran-sition to post-perovskite at ∼120 GPa, Earth’s man-tle is primarily made of MgSiO3-perovskite and MgO(e.g., Murakami et al., 2004). In contrast, much ofthe mantles of super-Venus planets will be dominatedby post-perovskite and perhaps, above 1 TPa, a mix-ture of MgO and SiO2 (Umemoto et al., 2006). Un-fortunately, we lack experimental measurements of theproperties of planetary materials under these extremeconditions. Until such data are available, conjecturesabout the rheology of the silicate mantles of large rockyplanets will remain controversial. Thermal evolution

simulations are very sensitive to assumed rheologicalbehaviors, so investigating the implications of variouspossible assumptions is essential.

The viscosities of most planetary materials increasewith pressure when examined at relatively low pressures(e.g., Karato, 2008). Simple extrapolation of this trendpredicts extreme increases in viscosity within massiveterrestrial planets. Extensions of known perovskite rhe-ology, for instance, imply an increase of >15 orders ofmagnitude as pressure increases to 1 TPa in an adia-batic mantle (Stamenkovic et al., 2011). Specifically,a viscosity profile may be calculated as (Stamenkovicet al., 2012)

η(P, T ) = η0 exp

[E

R

(1

T− 1

T ∗

)+

1

R

(PV ∗

T

)], (40)

where η0 is a reference viscosity at the reference tem-perature T ∗ = 1600 K and V ∗ is an activation volume.Figure 13 shows calculated viscosity profiles for η0 =

16

0 2 4 6 8 100

50

100

150

200

250(a)

Planet Mass [M/M

]

hc(t

p)

[km

]

0 2 4 6 8 100

0.5

1

1.5

2

2.5

Planet Mass [M/M

]

Vpro

c(t

p)/

Vsm

(b)

Figure 10: Summary of 54 simulations of the evolution of Mars,Venus, and seven super-Venus planets, showing the correspon-dence between simulation results and simple scaling laws for theeffects of planetary mass on (a) crustal thickness and (b) mantleprocessing. Circles and triangles represent Tu(0) = 1700 and 2000K, respectively. Blue, black, and red symbols represent Venus-equivalent Q0 = 5.0 × 10−8, 1.0 × 10−7, and 1.75 × 10−7 Wm−3, respectively. Dashed black lines show the scaling relations(a) hc ∝ (M/M⊕)−0.284 and (b) (Vproc/V ) ∝ (M/M⊕)−0.546,with each curve fixed to intersect the average output from thesimulations for the 2M⊕ super-Venus planet.

1021 Pa s and V ∗ = 2.5, 1.7, and 0.0 cm3 mol−1 forthe convecting, adiabatic mantle within a 10M⊕ super-Venus planet, following Stamenkovic et al. (2012). Ourparameterized formulation for stagnant-lid convectionis based on numerical modeling with the incompress-ible fluid approximation using temperature-dependentbut pressure-independent viscosity (e.g., Solomatov andMoresi, 2000; Korenaga, 2009). Therefore, we are as-suming pressure-independent constant potential viscos-ity, where the effect of temperature increase along anadiabatic gradient exactly balances the effect of pres-sure on viscosity, which requires a non-zero, positiveactivation volume. With the linear temperature gra-dient in Fig. 13, the assumption of constant potentialviscosity corresponds to V ∗ = 0.22 cm3 mol−1.

Strongly pressure-dependent viscosity can cause dra-matically different behavior to emerge from parameter-ized convection models, including sluggish lower mantleconvection and even the formation of a conductive lidabove the core/mantle boundary (Stamenkovic et al.,2012). If convection were effectively suppressed in the

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Figure 11: Summary of 595 simulations of the evolution of Mars,Venus, and two super-Venus planets, showing the minimum ra-tio of actual viscosity contrast to critical viscosity contrast andthus the likelihood of plate tectonics being favored at some pointduring 4.5 Gyr of planetary evolution. Each planet was evolvedfrom six different sets of initial conditions (three values for bothradiogenic heating and mantle potential temperature) for manydifferent values of µ, the effective friction coefficient. Points plot-ted above the indicated line represent simulations for which theactual viscosity contrast never dipped below the critical value fora transition to plate tectonics. Below the indicated line, whichoccurs only for µ < 0.3, plate tectonics may have been favored atsome point. For dry silicate rocks, µ ∼ 0.7 to 0.8.

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Figure 13: Internal temperature and viscosity profiles for a 10M⊕super-Venus planet, calculated as in Stamenkovic et al. (2012).The black curve shows internal temperature as a function of depthin the convecting mantle. Green, red, and blue dashed lines rep-resent viscosity profiles calculated using Eq. 40 for V ∗ = 0, 0.22,1.7, and 2.5 cm3 mol−1, respectively.

lower mantle of large rocky planets, melt productionwould be significantly decreased and the likelihood ofplate tectonics might decrease along with convectivevigor as planetary mass increased. The thermal conduc-tivity and expansivity of the mantle are also predictedto increase and decrease, respectively, with depth be-cause of increasing pressure, but the effects of thesechanges on mantle dynamics are dwarfed by the pu-tative increase in mantle viscosity (Stamenkovic et al.,

17

2011, 2012).Although an increase in viscosity under greater pres-

sures seems intuitive, the straightforward application oflimited, low-pressure experimental data may not accu-rately describe the rheology of massive terrestrial plan-ets. In fact, four mechanisms, including a transitionfrom vacancy to interstitial diffusion mechanisms, maycause a viscosity decrease with depth above a pres-sure of ∼0.1 TPa (Karato, 2011), as post-perovskiteand additional high-pressure mineral phases dominatemantle rheology. According to this study, viscositiesin the deep interiors of super-Venus planets may beless than the viscosity of Earth’s lower mantle, poten-tially by as much as 2-3 orders of magnitude. More-over, the depth-dependence of viscosity within Earth’sperovskite-dominated mantle is still debated becauseEarth’s viscosity profile has not been well-constrained.

Despite basic consensus that Earth’s lower mantle isprobably more viscous than the upper mantle, the mag-nitude of the viscosity contrast remains controversial.Early studies of Earth’s topography and geoid suggesteda ∼300-fold increase in viscosity between Earth’s up-per and lower mantle (Hager and Richards, 1989). Onthe other hand, analyses of post-glacial rebound predictan order of magnitude less of viscosity increase (e.g.,Kaufmann and Lambeck, 2002) and gravity data areconsistent, albeit loosely, with uniform or only slightlydepth-dependent mantle viscosity (Soldati et al., 2009).A joint inversion of these data sets predicts that Earth’sinternal viscosity increases by ∼2 orders of magnitudethroughout the mantle (Mitrovica and Forte, 2004), butit is noted that such inversions are known to sufferfrom severe nonuniqueness (e.g., King, 1995; Kido andCadek, 1997). In any case, considering the tremendousuncertainty surrounding viscosity profiles within Earthand putative super-Venus planets, it remains legitimateto investigate how large rocky planets might evolve inthe limiting case of pressure-independent potential vis-cosity.

5.2. Escaping the Stagnant-Lid Regime

Terrestrial planet evolution strongly depends on theregime of mantle convection. Assuming that brittle fail-ure limits the strength of the lithosphere, our simula-tions indicate that the effects of lithosphere hydrationdominate the effects of planetary mass on yield andconvective stresses. That is, the increase in convec-tive vigor with planetary mass only makes plate tecton-ics marginally more likely. Modeling results for super-Venus planets, however, suggest two additional mecha-nisms for escaping the stagnant-lid regime. First, mas-sive terrestrial planets in the stagnant-lid regime featurecrustal temperature profiles that enter the stability fieldof eclogite after crust grows beyond a critical thickness.If a sufficiently large fraction of the total crustal thick-ness is composed of eclogite, the entire crust could begravitationally unstable and susceptible to founderingbecause eclogite is intrinsically denser than mantle peri-dotite.

On Earth, the phase transition from (metamor-phosed) basalt to eclogite primarily occurs in sub-duction zones. Hydration may thus be importantto allowing this phase transition to occur relativelyrapidly (e.g., Ahrens and Schubert, 1975), although thistype of kinetic calculation strongly depends on diffu-sion data that are not well-constrained (e.g., Namikiand Solomon, 1993). Eclogite is also formed duringcontinent-continent collisions such as the Eurasian andIndian plate collisions (Bucher and Frey, 2002). Fur-thermore, the high density of eclogite is theorized tohave caused delamitation, foundering, and recyclingof relatively thick oceanic lithosphere on Earth dur-ing the Archaean (Vlaar et al., 1994). Finally, eclog-ite may be produced in large mountain ranges throughmagmatic differentiation (Ducea, 2002) and pressure-induced phase transition (Sobolev and Babeyko, 2005)in thick continental crust. Evidence for the strong influ-ence of recent eclogite production and foundering on thetopography of the central Andes Mountains has beengathered through geodynamics, petrology, and seismol-ogy (e.g., Kay and Abbruzzi, 1996; Beck and Zandt,2002; Sobolev and Babeyko, 2005; Schurr et al., 2006),as synthesized in a numerical study (Pelletier et al.,2010). Because massive terrestrial planets have rela-tively high surface gravity, the phase transition to eclog-ite will occur at a comparatively shallow depth, makingeclogite the stable mineral phase for a large fractionof the crust. The formation of a thick eclogite layerthen could cause lithosphere foundering or intermittentplate tectonics, as has been proposed in episodic sub-duction mechanisms for Venus (Turcotte, 1993; Fowlerand O’Brien, 1996).

Representative temperature profiles for massive ter-restrial planets pass through the eclogite stability fieldfor plausible initial conditions. In fact, radiogenic heat-ing and thus crustal temperatures should be greaterthan calculated with Eq. 38, which would increasethe speed of the phase transition to eclogite, becausecrustal construction mostly occurs early in planetaryhistory. For super-Venus planets with masses greaterthan ∼4M⊕, eclogite may be the stable phase for themajority of the crust unless the initial mantle potentialtemperature or magnitude of internal heating is verylow. So, crust material may undergo a phase tran-sition to eclogite at relatively shallow depths as thecrust grows during thermal evolution in the stagnant-lidregime, forming a thick eclogite layer that could subse-quently founder. The buoyant stress from the presenceof eclogite scales as ∆ρghe, where ∆ρ ∼ 100 kg m−3 isthe difference between the densities of eclogite and themantle. On the other hand, the lithospheric strengthscales as µρLgde, which can only be overcome when thedepth scale of the eclogitic layer becomes large enough(at least locally, for example, by foundering). As longas crustal production continues on large rocky planets,eclogite formation and foundering could occur period-ically, possibly yielding a regime of mantle convectionresembling intermittent plate tectonics. Although we

18

suggest that this process is plausible, pursuing its dy-namics in detail is left for future studies.

High surface and crustal temperatures may also causeperiodic transitions from the stagnant-lid regime to aform of mobile-lid convection. In this work, massive ter-restrial planets in the stagnant-lid regime with surfacetemperatures held constant at 300 K tend to have veryhot crusts. If high surface temperatures exist alongsidehigh crustal temperatures, a transition from stagnant-lid convection to a mobile-lid regime can occur (Reeseet al., 1999). Feedback between a changing mantle con-vection regime and a periodic atmospheric greenhouseeffect driven by varying amounts of volcanism, for in-stance, may be very important to the evolution of Venus(Noack et al., 2011). As surface temperature dependson atmospheric mass and the composition and lumi-nosity of the central star, however, this possibility ofescaping the stagnant-lid regime may not be as robustas the first mechanism based on the formation of self-destabilizing crust

5.3. Limitations of Parameterized Models

Any parameterized model suffers shortcomings.Steady-state evolution is assumed, for instance, whichpoorly captures transient events that occur early inplanetary evolution such as large impacts (Agnor et al.,1999) and the crystallization of a magma ocean (e.g.,Solomatov and Stevenson, 1993). Fundamental as-sumptions such an adiabatic temperature gradient inthe mantle and pressure-independent potential viscos-ity are controversial, and different approaches such asmixing length theory (Wagner et al., 2011) may be nec-essary to calculate the thermal structure of planetaryinteriors if they are not valid. However, our simpli-fied simulations only aim to illuminate the first-order,relative effects of planetary mass on terrestrial planetevolution. Recreating the thermal history of a particu-lar planet would require the introduction of many addi-tional complications. One-dimensional models only re-turn globally averaged values for important quantities,for instance, but mantle plumes, which may upwell fromthe core/mantle boundary when the core heat flux ispositive, are likely important to local magmatism andsurface features on terrestrial planets like Mars (e.g.,Weizman et al., 2001) and Venus (e.g., Smrekar andSotin, 2012). Furthermore, applying a parameterizedapproach to Venus, where the precise quantity of mag-matism is an key output, requires more computationallyintensive simulations to benchmark the relevant scalinglaws.

6. Conclusions

Terrestrial planet evolution is complicated. Althoughplate tectonics is observed on Earth, the stagnant-lidregime of mantle convection may be most natural forterrestrial planets; at least, it is most common in ourSolar System. Thermal evolution models in this study

yield first-order, relative conclusions about the evolu-tion of generic terrestrial planets in the stagnant-lidregime. Principal component analysis of simulation re-sults conducted with a wide range of initial conditionscaptures the relationships between the large number ofparameters that describe the interior of a planet. De-pending on initial conditions, these planets may haveevolved along a variety of paths, featuring differentcrustal thicknesses and temperatures, interior tempera-tures, and degrees of mantle processing. To producespecific histories consistent with spacecraft data ob-tained from Mars and Venus, complications must beadded to these simple models.

Properties of massive terrestrial exoplanets are poorlyconstrained, so questions about the effects of planetarymass on the likelihood of plate tectonics and other im-portant planetary parameters await definitive answers.In this study, we explored what might happen if in-ternal viscosity is not strongly-pressure dependent, thealternative to which has been explored previously us-ing parameterized models. Although convective vigorincreases with planetary mass, the likelihood of platetectonics is only marginally improved. Simple scalinganalyses indicate that mantle melt productivity shouldincrease with planetary mass. Because the increase inmantle processing is slow, however, crustal thicknessand the relative fraction of processed mantle actuallydecrease with increasing planetary mass, as thermalevolution simulations confirm. Surface gravity increaseswith planetary mass, so pressure in the crust of mas-sive terrestrial planets increases relatively rapidly withdepth. Plausible temperature profiles favor a phasetransition to gravitationally unstable eclogite duringnormal crustal formation, whereas the basalt to eclog-ite transformation rarely occurs aside from subductionon Earth. Therefore, thick eclogite layers, along withmobile, hot crustal material, may be important to theevolution of massive terrestrial planets.

7. Acknowledgments

CT Space Grant and the George J. Schultz Fellow-ship from Yale University’s Silliman College supportedJ. O’Rourke. Constructive comments from two anony-mous reviewers considerably improved the content andclarity of this manuscript.

8. References Cited

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Beck, S.L., Zandt, G., 2002. The nature of orogeniccrust in the central Andes. J. Geophys. Res. 107, 2230,doi:10.1029/2000JB000124.

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Parameter V1 V2 Av. SD S1 S2 Av. SDTc [K] 0.29 0.25 1237 106 0.36 0.03 1309 112Tu [K] -0.14 0.45 1772 132 0.03 0.45 1948 138Tcm [K] -0.10 0.44 3128 215 0.06 0.43 3475 225hc [km] 0.34 0.23 115 43.1 0.34 0.22 63.0 16.6hl [km] -0.36 0.20 53.6 29.1 -0.34 0.17 11.1 5.83hML [km] -0.39 0.20 84.1 27.7 -0.37 0.21 20.7 6.89

Fs [mW m−2] 0.28 0.24 50.7 14.7 0.37 0.14 204 61.6Fm [mW m−2] 0.29 0.01 29.6 6.58 0.37 0.07 173 46.7

log10(u) 0.28 -0.17 0.89 0.26 0.30 -0.14 2.00 0.30Cwsm -0.19 -0.36 0.37 0.13 -0.12 -0.36 0.62 0.10

Vproc/Vsm 0.17 0.39 1.04 0.25 0.12 0.38 0.50 0.16tc,10% [Gyr] -0.26 -0.04 0.12 0.23 -0.27 0.19 0.13 0.21

log10(∆tc,tot) -0.34 0.18 8.42 0.42 -0.18 0.36 8.54 0.38λi/Σλi 0.41 0.27 - - 0.42 0.30 - -

Table A.1: Principal component basis matrix for Venus (V) and a 10M⊕ super-Venus planet (S) for the model output after 4.5 Gyr ofplanetary evolution. Two eigenvectors account for over 65% of the variance in the normalized and mean subtracted simulation results.The fractions of the cumulative variances for which each principal component accounts, calculated by dividing the principal componenteigenvalue by the sum of the eigenvalues for all principal components, are in the bottom row. Output parameters were mean subtractedand normalized using the listed average and standard deviation values.

Appendix A. Statistical Analysis of SimulationResults

Parameterized evolution models involve quite a fewmodel parameters. It is important to understand howsimulation results depend on a particular choice ofmodel parameters by testing a variety of situations, butat the same time, it becomes difficult to grasp the in-flated amount of numerical data. Principal componentanalysis (PCA) can be used to assess the effective di-mensionality of a given data space. Our intention hereis to use PCA to extract major features and trends froma large number of simulation results. Each sensitivityanalysis consists of n simulations with m output pa-rameters, comprising a data set Dm

n . Some parameters,such as ∆ηw and u, exhibit orders of magnitudes of vari-ation, and we consider their logarithms because PCA isdesigned for linear data sets. We normalize the data setas

Pmn =Dmn − µm

σm, (A.1)

where µm is the average value of the m-th output pa-rameter,

µm =1

n

n∑i=1

Dmi , (A.2)

and σm is the standard deviation of the m-th outputparameter,

σm =

[1

n

n∑i=1

(Dmi − µm)2

]1/2. (A.3)

Because the normalized data have zero mean, thecovariance matrix CP = PTP can be decomposed asCP = AT ·diag[λ1 . . . λm] ·A, where λi, the eigenvalues,are ordered so that λ1 ≥ λ2 ≥ . . . ≥ λm. The cor-responding eigenvectors are the principal components,

which account for a progressively decreasing percentageof data variance. Principal components accounting forat least 65% (an arbitrary threshold) of the total vari-ance are selected for examination to reveal importantaspects of simulation results.

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