arX

iv:1

010.

5317

v1 [

astr

o-ph

.EP]

26

Oct

201

0

Thermal evolution and lifetime of intrinsic magnetic fields of

Super Earths in habitable zones

C. Tachinami1

Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Meguro, Tokyo 1528551(Japan)

ctchnm@geo.titech.ac.jp

H. Senshu2

Planetary Exploration Research Center, Chiba Institute of Technology, 2-17-1 Tsudanuma, Chiba 2750016(Japan)

and

S. Ida1

Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Meguro, Tokyo 1528551(Japan)

ABSTRACT

We have numerically studied the thermal evolution of various-mass terrestrial planets in hab-itable zones, focusing on duration of dynamo activity to generate their intrinsic magnetic fields,which may be one of key factors in habitability on the planets. In particular, we are concernedwith super-Earths, observations of which are rapidly developing. We calculated evolution oftemperature distributions in planetary interior, using Vinet equations of state, Arrhenius-typeformula for mantle viscosity, and the astrophysical mixing length theory for convective heattransfer modified for mantle convection. After calibrating the model with terrestrial planets inthe Solar system, we apply it for 0.1–10M⊕ rocky planets with surface temperature of 300 K(in habitable zones) and the Earth-like compositions. With the criterion for heat flux at theCMB (core-mantle boundary), the lifetime of the magnetic fields is evaluated from the calculatedthermal evolution. We found that the lifetime slowly increases with the planetary mass (Mp)independent of initial temperature gap at the core-mantle boundary (∆TCMB) but beyond a crit-ical value Mc,p (∼ O(1)M⊕) it abruptly declines by the mantle viscosity enhancement due to thepressure effect. We derived Mc,p as a function of ∆TCMB and a rheological parameter (activa-tion volume, V ∗). Thus, the magnetic field lifetime of super-Earths with Mp > Mp,c sensitivelydepends on ∆TCMB, which reflects planetary accretion, and V ∗, which has uncertainty at veryhigh pressure. More advanced high-pressure experiments and first-principle simulation as well asplanetary accretion simulation are needed to discuss habitability of super-Earths.

Subject headings: TERRESTRIAL PLANETS, THERMAL EVOLUTION, MAGNETIC FIELD,

1. Introduction

Many of exoplanets so far detected may be gasgiants with masses >

∼ 100M⊕, because massiveplanets are more easily to be detected. However,

recently several super-Earths with masses of a fewto ten M⊕ have been discovered by improved ra-dial velocity measurements (e.g., Udry et al. 2007)or microlensing observations (e.g., Beaulieu et al.

1

2006). On-going radial velocity (Mayor et al.2009) and microlensing (Gould et al. 2010) sur-veys and theoretical studies (e.g., Ida & Lin 2004,2008, 2010) strongly suggest ubiquity of super-Earths in extra solar planetary systems.

Space transit surveys such as CoRoT andKepler will also detect many super-Earths. Infact, CoRoT have detected the minimum masstransiting planet (CoRoT-7b) (Leger et al. 2009;Queloz et al. 2009). With an assumed compo-sition, a mass-radius relationship of the super-Earths gives planetary masses from transit obser-vational data. On the other hand, if the planetarymasses are obtained by follow-up radial velocityobservations, the mass-radius relationship can beused to estimate the planetary composition, al-though there is some ambiguity depending on theamount of H2O (Sotin et al. 2007). Valencia et al.(2006) used Birch-Murnaghan equation of state(EOS) for rocks and metals to obtain a mass-radius relationship of various-mass terrestrialplanets under some conditions (core ratio, sur-face temperature, and etc.). Sotin et al. (2007)also considered ocean planets which contain 50%of H2O. They used the EOS including thermalpressure to describe P − V − T relations for icesunder extremely high pressure and obtained amass-radius relationship for both terrestrial andocean planets.

Because super-Earths should exist also in hab-itable zones, the aspects related to the habitabil-ity of super-Earths are being discussed. Plane-tary habitability is often discussed in terms of thestability of liquid water on the planetary surface(Kasting et al. 1993). Assuming planets that aremassive enough to maintain dense atmosphere, arange of orbital radius in which liquid water is sta-ble is called a ”habitable zone.”

In addition to the existence of liquid water, evo-lution of amount and composition of planetaryatmosphere may also be an important factor forhabitability. It is believed that most fraction ofthe present atmosphere of the Earth was formedby impact degassing (e.g., Abe & Matsui 1985)and it consisted of CO2 and H2O with more than100 bars. The plate tectonics on the Earth hasremoved huge amount of CO2 from the Earth’satmosphere on Gyr timescales (Tajika & Matsui1992).

Valencia et al. (2007) and O’Neill & Lenardic

(2007) investigated the possibility of plate tec-tonics on the surface of super-Earths. The platetectonics would significantly affect amount andcomposition of planetary atmosphere throughcarbonate-silicate cycle with degassing and weath-ering. It also has stabilizing effect of planetary sur-face temperature, since temperature dependenceof weathering rate of carbonate works as a nega-tive feedback mechanism for surface temperaturechange (Tajika & Matsui 1992). Valencia et al.(2007) argued that super-Earths could invokeplate tectonics, because terrestrial planets largerthan Earth may have a thinned surface thermalboundary layer and increased yield stress. Onthe other hand, O’Neill & Lenardic (2007) showedthat super-Earths would have stagnant-rid stylemantle convection without plate tectonics. It isnoted that these studies a priori assumed ther-mal structure of super-Earths, as the studies on amass-radius relation did.

Although the thermal effect on the mass-radiusrelation may be negligible, plate tectonics shoulddepend on planetary thermal evolution history.Dynamo activity to generate a magnetic fieldalso depends on the thermal evolution. Plane-tary magnetic field prevents stellar winds fromsplitting the planetary atmospheres. The mag-netic field also prevents cosmic rays from pene-trating to the planetary surface. Thus, it maybe one of the most important factors for land-based life to be maintained. It is widely acceptedthat the Earth’s magnetic field is attributed to adynamo effect (e.g., Glatzmaier & Roberts 1995;Kuang & Bloxham 1997; Kageyama & Sato 1997)in the metallic core. The intrinsic magnetic fieldmay be sustained if convective fluid motions inthe core is vigorous enough, in other words, theheat flux through the core surface is large enough.Thus, detailed study on thermal evolution of thecore is needed to evaluate generation of the mag-netic field.

Using the box model (see below), Papuc & Davies(2008) modeled the thermal evolution of thevarious-mass terrestrial planets on geologicaltimescales to discuss the evolution of planetarysurface activity, i.e., the plate velocity and the de-gassing rate. Their results showed that the super-Earths may have dense atmosphere in their earlyhistory, since larger planets have higher degassingrates. Since they were concerned with planetary

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surface activity, they focused on treatment of heatgeneration of radio activity in the mantle and sur-face heat flow, leaving treatment of cores simple.We will show that careful treatment of the coresuch as the effects of inner core nucleation andincrease in heat capacity due to high compressionand gravitational energy stored in the core, whichPapuc & Davies (2008) neglected, are importantfor the study of dynamo activity (Note that theevaluation of the surface activity is hardly affectedby the careful treatment of the core).

In a series of papers, Schubert, Stevenson andtheir colleagues developed a ”box” model for ther-mal evolution of the terrestrial planets in our Solarsystem (e.g., Schubert et al. 1979; Stevenson et al.1983). In their model, the thermal structure is de-scribed by two boxes that correspond to the man-tle and the core. The temperature variation ineach box is neglected and the temperature distri-butions in the Earth are represented by three dis-tinct temperatures of the core, the mantle, andthe planetary surface. The heat flow is evalu-ated by thermal conduction through the thermalboundary layers at CMB and the planetary surfacewith the thermal Boundary Layer Theory (BLT)(Stevenson et al. 1983). In the BLT, the heatflux is determined by the thickness of the thermalboundary layer, which is given by local Rayleighnumber. Because this model is easily treated, itprovides a powerful tool to explore general trendsof thermal evolution of terrestrial planets.

For Earth’s thermal evolution, we can use ob-servational constraints such as surface heat flowand inner core size. With the calibrated model,unknown parameters such as initial temperaturedistribution of Earth’s interior (Yukutake 2000,also see section 2.7) or potassium abundance inthe core (Nimmo et al. 2004) can be constrained.Rheological parameters and impurity abundancein the core are also estimated (see section 3.3).The existence of the magnetic field for Mercuryand early decay of the magnetic field for Venus andMars are consistent with calculations with rea-sonable choice of initial temperature or impurityabundance in the cores by BLT (Stevenson et al.1983) and MLT (Appendix B).

Gaidos et al. (2010) simulated thermal evolu-tion of various sized super-Earths by the BLT toevaluate the magnetic activity of super-Earths.They concluded that massive rocky planets (>

2.5M⊕) can not sustain magnetic field, becausethey found an inversion of gradients of the melt-ing and adiabatic curves in the core under suchhigh pressure. Since the outer core is solid in theinverse state, the cooling of the inner liquid core,in which dynamo operates, is inhibited. Althoughthe possibility of the inversion raised by the paperis a very important, the conclusion depends onhigh pressure material properties such as metingand adiabatic curves that need to be confirmed. Inthe present paper, we point out another importantfactor to inhibit dynamo activity in super-Earths,drastic increase in the mantle viscosity due to pres-sure effect. Even if the inversion in the core doesnot occur, the enhanced mantle viscosity quicklyterminates dynamo activity in the core.

In the present paper, we are concerned withlifetime of magnetic fields of super-Earths. Thereis no observational constraint for super-Earthsto calibrate parameters for a model and ini-tial/boundary conditions. Here, clarifying keyquantities for generation of the magnetic field,we derive planetary mass dependence of lifetimeof magnetic activity and how it depends on themodel parameters and initial/boundary condi-tions. To reduce unknown parameters, we con-sider super-Earths in nearly circular orbits inhabitable zones. The analysis on the key rheo-logical parameters will provide new motivationsfor high pressure experiments and first principlesimulations, since state-of-arts high pressure ex-periments have already reached the pressure atthe bottom of Earth’s mantle. We will point outthat initial temperature distribution sensitivelyaffects the lifetime, which gives new motivationsfor theories of planet accretion from planetesimalsand core formation.

Here, we develop a thermal evolution model todiscuss the existence of the intrinsic magnetic fieldin terrestrial planets with various masses. Sincewe are concerned with heat flux across the CMB,we calculate detailed radial temperature distribu-tion in both core and mantle. We use the MixingLength Theory (MLT) to calculate the heat flow.The MLT is commonly used to study stellar inte-rior. We use the modified version for low Reynoldsnumber flow in solid planets that have radial dis-continuities in their interior (Sasaki & Nakazawa1986; Abe 1995; Senshu et al. 2002; Kimura et al.2009, and references therein). The modified

3

MLT is useful for calculations of super-Earths thatmay have additional higher-pressure phase tran-sitions such as a post-post perovskite transition(although we do not consider it in the presentpaper) and early-stage planets that may haveconvection-barriers at upper/lower mantle bound-ary (e.g., Honda et al. 1993) or density crossoverat melt/solid boundary Labrosse et al. (2007). Wewill also show the results for the two-layer con-vection case in which convective flow does notpenetrate the spinel-perovskite transition at up-per/lower mantle boundary, while most of calcu-lations are in the cases of one-layer convection. Wecompare the MLT with the conventional BLT indetails and show that they are in good agreementwith each other in the case of one-layer convection(Appendix A).

We will show that the lifetime is rather shorterfor super-Earths than for Earth-mass planets fornominal parameters of solid state. The mecha-nism to suppress dynamo activity in super-Earthsfound in this paper is independent of that inGaidos et al. (2010), so that it may be likely thatsuper-Earths are magnetically inactive. We alsopoint out that the choice of initial conditions andrheological parameters highly affect the thermalevolution of the planets. In section 2, we explainour numerical model. The numerical results willbe shown in section 3. Finally we discuss the hab-itability for terrestrial planets in view of the in-trinsic magnetic field.

2. Numerical Model

We follow thermal evolution of a planet thatis a cooling process from a hot early state dueto accretion from planetesimals and core-mantledifferentiation, for various-mass terrestrial planetsby using a one-dimensional spherically symmetricmodel.

As described below, there are unknown param-eters for rheological properties and initial condi-tions of planets. Furthermore, even in our So-lar system, the terrestrial planets have differentcompositions (a water-rock-iron ratio) and surfacetemperature that is regulated by orbital radius.In extrasolar planetary systems, more variationsin the water-rock-iron ratio and surface temper-ature should exist due to different metal abun-dance of host molecular clouds as well as differ-

ent pressure/temperature state of disks and plan-etary atmosphere and planetary formation pro-cesses. These unknown variations could make thestudy on the mass dependence meaningless.

In order to reduce the uncertainty, in a “nom-inal” case (see section 2.8), we consider planetswith surface temperature (Tsurf) of 300K and themantle/core mass ratio (ζm/c) of 7 : 3, which is thesame as that for the Earth. The planets may cor-respond to extrasolar terrestrial planets in nearlycircular orbits in habitable zones. In the nomi-nal case, the other unknown parameters in mantlerheology, core impurity and initial temperature arecalibrated with data of the Earth.

The restriction to the nominal case enables usto derive clear planetary mass dependences. Wealso discuss how these dependences change by dif-ferent choice of the parameters. In Appendix B,we also carry out the runs with different Tsurf andζm/c to calculate thermal evolution of Mercury,Venus and Mars. The results show that our modelcan be applied for these planets too, if we use rea-sonable non-nominal parameters. Systematic sur-vey for thermal evolution of super-Earths in thenon-nominal cases is left for future works.

The thermal evolution is calculated by the fol-lowing methods:

1. The radial density distribution is calculatedby using VINET equation of state takinginto account pressure dependence (section2.1). Since this distribution is almost in-dependent of evolution of temperature dis-tribution and the inner core growth as ex-plained below, we use the distribution calcu-lated in the initial state (t = 0) throughoutthe entire thermal evolution.

2. Given a temperature distribution in the in-terior at time t, the radius of the inner solidcore is calculated by the melting tempera-ture with pressure and composition depen-dences (section 2.2). Sulfur is consideredas the impurity in the core and its con-centration in the outer liquid core is self-consistently calculated with the condensa-tion of the inner core (section 2.2).

3. The heat transfer throughout the mantleis calculated by the astrophysical mixinglength theory modified for solid planets (we

4

discuss its validity and usefulness in section2.3). The mantle viscosity in the heat trans-fer equation is estimated by using Arrheniustype formulation (section 2.5). By subtract-ing the energy loss during the timestep (∆t),we obtain a new temperature distribution att+∆t and go back to step 2. Time evolutionis calculated by iterations of step 2 and 3.

In the following subsections, we will explain eachstep in more detail.

2.1. Density profile

The hydrostatic stratification is calculated by

dP

dr= ρ(r)g(r);

dr

dm=

1

4πr2ρ(r), (1)

where P, r, ρ, g, and m are pressure, radius, den-sity, gravitational acceleration, and mass insidethe radius r, respectively. Vinet EOS(Vinet et al.1987) is given by

P = 3K01− x

x2exp(φ(1 − x)), (2)

where

φ =3

2(K ′

0 − 1), (3)

x = ρ0/ρ, and ρ0,K0 and K ′0 are density, bulk

modulus, and its pressure differentiation at zeropressure, respectively.

Birch-Murnaghan EOS is ”finite strain” EOS,in which pressure is expressed by Taylor series ex-pansions of finite strain, and it is often used for thecalculation of interior of solid planets. However,the finite strain EOS do not accurately representsthe volume variation under very high compres-sion (if pressure exceeds the bulk modulus of zeropressure, the expansion never converges). VinetEOS is derived from a general inter-atomic po-tential energy function. For simple solids, VinetEOS provides more accurate representations of thevolume variations with pressure, under very highpressure. Since we are concerned with interior ofsuper-Earths under very high pressure, we adoptVinet EOS rather than Birch-Murnaghan EOS,following (Valencia et al. 2007).

Since thermal contraction is small enough (itis less than 1% in physical size for 100K changein the average temperature of Earth’s interior),

we neglect the temperature dependence in VinetEOS. The density profile in the planet is calculatedby numerically solving equations (1) and (2).

The compositions are assumed as olivine andγ-spinel for upper mantle, perovskite and post-perovskite for lower mantle, Fe and FeS for outercore, and Fe for inner core with the propertiesgiven in Table 1. As temperature decreases, theinner solid core grows and sulfur moves from theinner core to the outer core (see section 2.2). Ingeneral, the inner core growth changes the vol-ume of the whole core, because the parameters,ρ0,K0 and K ′

0, are different between Fe and FeS.However, since we numerically found that the vol-ume change of the whole core is very small, weneglect it. Figure 1 shows the result of numeri-cal calculation of the density profile for 0.1 to 10Earth-mass planets. This result is different in theradii of the core and the mantle from the results byValencia et al. (2006) and Sotin et al. (2007) by afew %, which may be due to different choice ofparameter values in the EOS. But, this differencedoes not affect the thermal evolution.

2.2. Thermal evolution of the core

The temperature distribution of the core is de-termined as follows:

1. The inner solid core: We assume that eachpart of the inner core memorizes the temper-ature at which it solidified, because of ineffi-cient heat transfer due to conduction in thesolid core.

2. The outer liquid core: We assume that theliquid core has an adiabatic temperature dis-tribution by vigorous convection.

3. Time evolution: The radius of the inner coreand the temperature at the CMB is deter-mined by total energy of the inner and outercores as described below, and the total en-ergy is given as a function of time with inte-grating heat flux at the CMB.

The adiabatic temperature gradient in theouter core is given by (Sohl & Spohn 1997;Yukutake 2000; Valencia et al. 2006)

∂T

∂r=

ρgγGKs

T, (4)

5

where γG and Ks are Gruneisen parameter andbulk modulus of the liquid core. Depth varia-tion of γG is calculated as γG = γG0(ρ0/ρ)

q (theparameter values used are summarized in Table1). The density at 0 pressure ρ0OC, bulk modu-lus K0OC, and its pressure derivation K ′

0OC of theouter core are given by impurity concentration xS

as:

xFeS = xSZFe + ZS

ZS(5)

ρ0OC =

(

1− xFeS

ρFe+

xFeS

ρFeS

)−1

(6)

K0OC =1

ρ0OC

11−xFeS

ρFe

1KFe

+ xFeS

ρFeS

1KFeS

(7)

K ′

0OC = −1+ρ0OCK0OC

(

1− xFeS

ρFe

1 +K ′Fe

K2Fe

+xFeS

ρFeS

1 +K ′FeS

K2FeS

)

,

(8)where xFe, xFeS, ZFe, and ZS are mass fraction ofFe and FeS, molar weights of Fe and S, respec-tively.

The inner core nucleation decelerates coolingof the core by release of gravitational energydue to the change in the density distributionand by release of latent heat (Stevenson et al.1983; Gubbins et al. 2004). The light elementsare kicked into the outer core, resulting in de-pression of a melting point of the outer core(Stevenson et al. 1983; Yukutake 2000). Theboundary between inner and outer cores is lo-cated at the intersection between adiabatic andmelting curves in the core. We use a Lindeman’sequation for the melting curve of pure iron,

Γ(ρ) = Γ0

(

ρ0ρ

)2/3

exp

{

2γ0q

[

1−

(

ρ0ρ

)q]}

.

(9)We also consider the depression of a melting pointby concentration of light elements. We define themelting point of Fe-FeS alloy as

Tmelt = (1 − 2xS)Γ(ρ), (10)

and the factor (1 − 2xS) expresses the depressionof the melting point due to dissolution of lightelements (Usselman 1975; Stevenson et al. 1983).Assuming that the outer core is well mixed by con-vection,

xS = x0SMc

Mc −Mic, (11)

where Mic and Mc are the inner core mass andtotal mass of the inner and outer cores and x0S isthe initial impurity concentration. In the nominalcase, we adopt x0S = 0.1.

Given the inner core radius, we can calculatethe total energy of the core (Ecore) which is sumof the gravitational energy (Eg), latent heat (El),and thermal energy (Eth). As described above, thetemperature at the CMB is given as a function ofthe radius of the inner core. As a result, we canobtain Ecore as a function of the temperature atthe CMB. Conversely, the radius of the inner coreand the temperature at the CMB are given as afunction of Ecore.

The energies are given by

Eg = −

∫ ric

0

4πr3ρic(r)gic(r)dr −

∫ rc

ric

4πr3ρoc(r)goc(r)dr,

El = LMic,

Eth =

∫ rc

0

4πr2ρ(r)Cp(r)T (r)dr,

(12)where L is the latent heat released by solidifica-tion of unit mass of iron, which is assumed tobe constant of 1.2× 106 J/kg (Anderson & Duba1997), and not to depend on the impurity concen-tration in the outer core, and Cp is specific heatwith constant pressure. Both the gravitational en-ergy and the latent heat are released after the in-ner core starts to solidify. Gravitational energyis also released by thermal contraction, which willbe discussed in section 2.6. The total energy Ecore

decreases with the rate that is equal to the heatflux at the bottom of the mantle (see section 2.3).Detailed calculations of the energies are given inAppendix C.

2.3. Heat transfer throughout mantle

The mantle is cooled by irradiation from theplanetary surface and heated by heat flow from thecore and internal radioactivity (see below). Theheat transfer equation is:

ρCp∂T

∂t=

1

r2∂

∂r

{

r2kc

(

∂T

∂r

)

+ r2kv

[(

∂T

∂r

)

−

(

∂T

∂r

)

s

]}

+ρQ,

(13)where kc is thermal diffusion coefficient, Q is ra-dioactive heat production rate, (∂T/∂r)s is theadiabatic temperature gradient, and the first andsecond terms in the right hand side represent con-ductive and convective fluxes.

6

To evaluate the convective flux in the man-tle, we use the astrophysical mixing length theory(MLT) modified for solid planets (Sasaki & Nakazawa1986; Abe 1995; Senshu et al. 2002; Kimura et al.2009, and references therein), rather than the con-ventional parameterized convection model (PCM;e.g., Sharpe & Peltier 1979) or the commonly usedboundary layer theory (BLT; e.g., Stevenson et al.1983).

The PCM is very simple (Appendix A). How-ever, it uses the values of kc and Rayleigh num-ber Ra that represents the whole mantle, whichare difficult to evaluate for real mantle because ofhuge spatial variation of the mantle viscosity. As aresult, although the PCM can be applied to studyoverall trend of thermal evolution, it may not beaccurate enough for evaluation of heat flux acrossthe CMB (FCMB), which we are concerned within the present paper. In the BLT, since the heatflux is expressed by quantities only in the thermalboundary layer (Appendix A), the BLT has bet-ter resolution for evaluation of FCMB. Since themodified MLT also uses local values of physicalquantities, it quantitatively agrees with the BLTfor wide range of parameters, while the PCM doesnot agree with the BLT and the MLT for the casesin which viscosity variation is large in the man-tle, as shown in Appendix A. As explained below,since the MLT is more easily to be applied forsuper-Earths, we use the MLT.

In the early Earth, the upper/lower mantleboundary could have worked as a barrier for con-vection (Honda et al. 1993). Tentative stagnancyat the upper/lower mantle boundary is also sug-gested for some subduction slabs in the presentEarth (Wortel & Spakman 2000). The densityoverturn at melt/solid boundary in deep magmaocean in the early Earth may have also workedas the barrier (Labrosse et al. 2007). In super-Earths, post-post perovskite transition in deepmantle at high pressure could also work as a bar-rier (Umemoto et al. 2006).

As explained below, the modified MLT is eas-ily applied for mantle convection with barriers,without tuning of parameters for each barrier. Al-though most of our calculations in the present pa-per only consider the surface boundary and CMB(in some runs we consider the upper/lower mantleboundary at spinel-perovskite transition as well),we use the modified MLT for future extensions of

calculations with various convection barriers. Inthe following papers, we will consider the effectsof other barriers.

In the MLT, the coefficient for convective heattransfer is given by

kv =

0 for

(

∂T

∂r

)

>

(

∂T

∂r

)

s

ρ2Cpαgℓ4

η

[(

∂T

∂r

)

−

(

∂T

∂r

)

s

]

for

(

∂T

∂r

)

<

(

∂T

∂r

)

s

(14)

where η and ℓ are viscosity and the mixing length,respectively. Here, the velocity of fluid blobs isevaluated by Stokes velocity rather than free fallvelocity in the original MLT, in order to applythe model to low Reynolds number flow in themantle. In the astrophysical context such as stel-lar interior, the density scale height is usuallyadopted as ℓ. For calculation of thermal evo-lution of the Earth, it is proposed that a dis-tance (D) from the closest barrier such as theCMB or the top of the mantle layer is appro-priate for ℓ (Sasaki & Nakazawa 1986; Abe 1995;Senshu et al. 2002; Kimura et al. 2009, and ref-erences therein). The detailed comparison withthe PCM and BLT in Appendix A shows thatℓ = 0.82D is the best choice. We adopt ℓ = 0.82Dfor all the runs in the present paper.

With this choice, as approaching a barrier, kvrapidly decreases in proportion to ℓ4 and the con-ductive term dominates in Eq. (13). As a result,thermal boundary layers, in which the conductiveheat transfer dominates, are automatically repre-sented. Thereby, the modified MLT is easily ap-plied for calculation for thermal evolution of theproto-Earth or super-Earths.

The mantle viscosity is discussed in section 2.5.The heat flux and temperature at the CMB de-termined from the calculation of the mantle heattransfer is used as a boundary condition for ther-mal evolution of the mantle. The total energy inthe core is interpolated from the result of section2.2. It decreases with time, according to the cal-culated heat flux at the CMB.

2.4. Internal heat source

For thermal evolution of terrestrial planets ongeological timescales, long-lived radiogenic ele-ments (40K, 232Th, 235U, and 238U) are impor-

7

tant heat sources in the mantle. The estimatedamounts of these elements in the Earth are com-piled in table 2 (van Schmus 1995). Here we as-sume the same abundances of the radiogenic ele-ments in the mantle in super-Earths as those inthe Earth and that the elements are distributeduniformly throughout the mantle. The heat pro-duction rate at time t, Q(t), is given by Q(t) =HU exp(−λ(t − t⊕)) where U , H , t⊕ and λ arethe abundance, heat production rate, and decayconstant of the element, respectively.

2.5. Temperature and pressure depen-

dency of mantle viscosity

The mantle viscosity is one of the most im-portant physical parameters to simulate thermalevolution, since it determines the heat trans-fer efficiency in the mantle (see Eqs. [13] and[14]). The viscosity sensitively depends on tem-perature and pressure, and both of temperatureand pressure widely vary throughout the man-tle. Here, we adopt Arrhenius type formulationfor temperature- and pressure-dependent viscositymodel (Ranalli 2001):

η(T, P ) =1

2

[

1

B1/nexp

(

E∗ + PV ∗

nRT

)]

ǫ(1−n)/n,

(15)where R, ǫ, n, B, E∗ and V ∗ are universal gas con-stant, strain rate, creep index, Barger coefficient,activation energy, and activation volume of man-tle, respectively. We use different values of theseparameters for upper and lower mantles. The min-eral properties we use are listed in Table 3. Notethat the prescription for the mantle viscosity mayinclude uncertainty. The formula is based on thetheoretical rate equation for creep law of rocks.In this formula, the most important parameter tostudy thermal evolution of super-Earths is the ac-tivation volume (V ∗), since V ∗ determines the de-pendence of the viscosity on pressure and the pres-sure in the mantle can be increased by orders ofmagnitude as the planetary mass increases. Theactivation volume is related to atomic volume, butthe exact values under extremely high pressure isnot well determined. In the nominal case, we useV ∗ = 10×10−6m3mol−1, but we also test a smallervalue of V ∗ = 3 × 10−6m3mol−1. In section 3.4,we will discuss how the conclusion in the presentpaper depends on a formula for the viscosity.

2.6. Release of gravitational energy by

thermal contraction

Although the thermal contraction is negligiblefor physical radius, the gravitational energy re-leased by the thermal contraction cannot be ne-glected (it is about 50kJ/kg for 100K change ofthe core in an Earth-mass planet). In our model,the released energy is regarded as increase in thespecific heat of the core (Yukutake 2000):

∆Cp =αP

ρ. (16)

The gravitational energy released by thermal con-traction is more effective in deeper regions. For anEarth-mass planet, ∆Cp/Cp is as large as 50% atthe CMB in our calculation.

2.7. Initial conditions

Initial temperature distribution in the mantle isdetermined by the procedure following Yukutake(2000), as illustrated in Fig. 2: (1) An adiabatis drawn from the bottom of the surface bound-ary layer with 1500 K down to the top of theboundary layer at the CMB (the obtained ten-tative temperature is denoted by T2), assumingefficient thermal convection, (2) the initial tem-perature at the bottom of the CMB is assumed tobe TCMB = T2 +∆TCMB, where ∆TCMB is deter-mined by step 6, (3) the adiabat is drawn from theCMB with temperature TCMB to the surface, (4)the initial temperature distribution in the man-tle is given by the average of the two adiabatsobtained by steps 1 and 3, (5) core temperatureis determined by the procedure given in section2.2 with TCMB, and (6) the amount of ∆TCMB

(∼ 1000K) is determined by the requirement thatthe predicted surface heat flux and inner core ra-dius for an Earth-mass planet are comparable tothe observed values for the present Earth. We usethis value for all the cases with various planetarymasses. Note that the temperature distribution isquickly relaxed to an equilibrium distribution, aslong as we use the initial conditions created by theabove procedures.

2.8. Simulation parameters

We summarize parameters for the “nominal”case:

8

• Boundary conditions

– surface temperature: Tsurf = 300K

– a mass ratio between mantle and core:ζm/c = 7 : 3

– the CMB is a barrier for convection,while convection penetrates the up-per/lower mantle boundary

• Initial conditions

– impurity fraction: x0S = 0.1

• Rheological conditions

– activation volume: V ∗ = 10×10−6m3mol−1

We first investigate planetary mass depen-dence for planets with the above nominal pa-rameters. For Earth-mass planets, we adopt∆TCMB = 1000K in most runs, because thenominal case with ∆TCMB = 1000K reproducesthe present Earth. We also systematically studydependences of the results on ∆TCMB, because∆TCMB is not well determined. In some runs, theupper/lower mantle boundary is treated as a bar-rier for convection. We also carry out calculationswith different values of Tsurf , ζm/c, and x0S to re-produce the results that are consistent with thecurrent magnetic fields of Mercury, Venus, andMars in Appendix B (we do not systematicallysurvey the dependences on these parameters).

2.9. Definition of lifetime of planetary in-

trinsic magnetic field

To drive dynamo action, liquid metallic coremust be in active convection state. FollowingStevenson et al. (1983), we adopt the thresholdheat flux in the core for generation of dynamo ac-tion (conducted heat flux along the core adiabaticthermal structure) as,

Fcrit = kc

(

∂TCMB

∂r

)

S

= kcρgγGKs

TCMB. (17)

We define the lifetime of magnetic field as a pe-riod during which the core heat flux exceeds thethreshold value.

3. Numerical Results

3.1. Thermal evolution of the Earth

We now show the evolution of temperature dis-tribution calculated by the procedures in section 2.Figure 3a shows the evolution of thermal structureof an Earth-mass (M = 1M⊕) planet for the nom-inal case. Cooling of the mantle slows down withtime, since the decrease in temperature enhancesthe mantle viscosity (eq. [15]) and hence depressesthe efficiency of heat transfer in the mantle. Thisimplies that the initial mantle temperature dis-tribution hardly affects the thermal evolution ontimescales longer than Gyr, as long as the ini-tial temperature is high enough (Stevenson et al.1983). However, since the core works as a heatbath for the mantle, the initial TCMB would affectthe thermal evolution of the mantle.

Figures 3b, c and d show the time evolutionof the surface heat flux (Fsurf), the heat fluxacross the CMB (FCMB) and the inner core ra-dius (Ric), respectively, in the nominal case. Weadopt ∆TCMB = 1000K. The inner core emergesat 2.2 Gyr. Its radius reaches 1200km at 4.5Gyrthat agrees with the observed value of the presentEarth, as expected. With higher/lower valuesof TCMB, the inner core radius at 4.5Gyr issmaller/larger. The growth rate decreases withtime, because not only geometric effect but alsothe increase in impurity concentration in the outercore depresses the melting point of the outer core.After the emergence of the inner core, the heat fluxreduction becomes more slowly because the innercore growth releases gravitational energy and la-tent heat that work as internal heat sources. Theheat flux remains larger than the critical valuegiven by Eq. (17), which is expressed by the dot-dashed line in Fig. 3d, during first 12 Gyr and thelifetime of dynamo activity is expected to be 12Gyr for this nominal case.

Some paleomagnetism data suggests that themagnetic field of the Earth is enhanced to be thepresent level at ∼ 2 Gyr (e.g., Hale 1987). It mightbe due to formation of the inner core because nu-cleation of the inner core provides additional heatsource. Paleomagnetism data may include a largeuncertainty. If more detailed data is provided, itwill constrain the condition of generation of intrin-sic magnetic field.

9

Figures 4 and 5 show the results of the two-layer convection. In this calculation, we set theupper/lower mantle boundary as a barrier for con-vection. The other boundary conditions and themodel parameters are the same as those in thenominal case. The mixing length is shorter inthe entire regions of mantle and cooling is slowerthan in the one-layer case. As is shown in Fig-ures 4, if we adopt initial ∆TCMB = 1000K asin the case of one-layer convection, the inner corecan not grow to 1200km because of the low heattransfer efficiency of the layered convection andcore temperature is somewhat higher than thatobtained by the one-layer convection. We also car-ried out a calculation with initial ∆TCMB = 800K.Figure 5a shows the evolution of thermal profile.A thermal boundary layer at upper/lower man-tle boundary is clearly established. Because thelower initial ∆TCMB is compensated with the inef-ficient two-layer convection, evolution of heat fluxthrough CMB (FCMB) and lifetime of magneticfield (11Gyr in this case) are similar to those inthe one-layer convection case.

If the two-layer convection is assumed only inthe Archean and Hadean (t < 2Gyr), with thesame initial ∆TCMB (1000K) the surface heatflux at 4.5Gyr is Fsurf ∼ 0.12Wm−2, whichis somewhat higher than the observed value(∼ 0.09Wm−2), since the thermal energy beneathupper/lower mantle boundary have been storeduntil 2Gyr and then supplied to upper mantle af-ter 2Gyr. However, the evolution of FCMB is notso different between one- and two-layer convec-tion. The lifetime of magnetic field is about 12Gyrs.

Thus, if we tune the initial ∆TCMB with thepresent observed values of Fsurf and Ric, the ex-pected lifetime of magnetic field is not affected bythe mode (one-layer or two-layer) of mantle con-vection.

3.2. Thermal evolution of Mercury, Venus,

and Mars

The existence of the magnetic field for Mercuryand early decay of the magnetic field for Venus andMars were addressed by Stevenson et al. (1983),using the box model with different parameter val-ues such as surface temperature and a mantle-coremass ratio from those in the nominal case. Thevalidity of these parameter values is discussed in

Appendix B. Adopting the same parameter val-ues as Stevenson et al. (1983), we have performedsimulations for Mercury, Venus, and Mars withour model. As discussed in Appendix B, ourmodel produces the results that are not inconsis-tent with the magnetic activity of Mercury, Venus,and Mars.

3.3. Thermal evolution of super-Earths

For super-Earths, we use the nominal param-eters (the surface temperature is 300K and themantle/core mass ratio is 7:3), assuming that theirorbits are nearly circular and in habitable zones.We also assume one-layer convection throughoutthe mantle. Detailed study on the effects of phasetransitions is left to future works. Figure 6a showsevolution of temperature distribution for a planetwith mass Mp = 5M⊕. Compared with the caseof Mp = 1M⊕ in Figure 3a, a thicker thermalboundary layer is established on the CMB withinfirst few Gyrs, since the viscosity of the bottom ofthe mantle is higher. The increase in the viscos-ity due to higher pressure dominates the decreasedue to higher temperature (Eq. [15]). Thus, FCMB

is lower than the case of Mp = 1M⊕ (Figs. 6band 3b). On the other hand, the effective heatcapacity rapidly increases with Mp. Figures 15in Appendix C show that for fixed TCMB, ther-mal energy Eth ∝ M2

p , while the core surface areaScore increases with Mp only weakly (Score ∝ Mp).Therefore, the core for higherMp cools much moreslowly. It is shown that Ric does not grow at all for10Gyr. On the other hand, Fsurf is not so differentfrom that in the case ofMp = 1M⊕. The heat bathof surface heat flow is radiogenic elements in themantle and that is proportional toMp. To balanceheat generation and cooling, Fsurf should be pro-

portional to M1/3p , provided Rp ∝ M

1/3p . Thereby

Fsurf changes by a factor of only 51/3 ∼ 1.7.

As discussed in the above, thermal evolutionof super-Earths differs from that of Earth-massplanets in many aspects. Here, we focus on evo-lution of heat flux through CMB, FCMB, and in-ner core radius, Ric, in order to study magneticactivity of super-Earths. Figures 7 show the evo-lution of FCMB (left column) and Ric (right col-umn) with various initial ∆TCMB for the case of(a)Mp = 1M⊕, (b) 2M⊕, (c) 5M⊕, and (d) 10M⊕.Solid, dot, dashed, and long-dashed lines repre-sent the results with initial ∆TCMB = 1000K,

10

2000K, 5000K and 10000K, respectively. In allcases, V ∗ = 10× 10−6m3/mol.

The results in the left column show that FCMB

is generally higher for higher initial ∆TCMB. Thedependence is more pronounced for relatively largeMp cases. For Mp = 1 M⊕, the dependenceis very weak. We found the dependence is alsovery weak for Mp < 1 M⊕. For Mp

<∼ 1 M⊕,

the temperature dependence of the mantle vis-cosity (Eq. [15]) dominates over the pressure de-pendence. Then, FCMB is high when the coretemperature is high, and FCMB declines as thecore cools. Thus, the heat flux is self-regulatedto be quickly relaxed independent of the initialvalues. On the other hand, as will be shownlater, when ∆TCMB

<∼ 1000(Mp/M⊕), the pres-

sure dependence is more effective. Then, the self-regulation does not work and the dependence ofFCMB on initial ∆TCMB is retained for more than20 Gyrs. The threshold flux for driving dynamoaction is marked by an black lines in each case.The decline of the threshold value is due to de-crease of core surface temperature (Eq. [17]). Theduration for FCMB > Fcrit determines lifetime ofmagnetic field generation.

Papuc & Davies (2008) obtained FCMB ∝

M2/3p , whereas our results shows FCMB ∝ Mp

provided that ∆TCMB is sufficiently high. Thedifference may come from the assignment of spe-cific heat of the core. Papuc & Davies (2008)assumed constant specific heat of the core, Cp =1000JkgK−1 for all sized planets. As we discussedin section 2.6, however, thermal contraction re-sults in increase in the effective Cp and the effectis more pronounced for larger Mp. In our calcula-tions that include this effect, the core tends to coolless efficiently and the dependency of FCMB on Mp

is stronger than that obtained by Papuc & Davies(2008).

The right column shows the growth of innersolid cores for ∆TCMB = 1000, 2000, 5000 and10000K. In the case of Mp = 1 M⊕, an innercore is nucleated at 2-3 Gyrs, almost indepen-dent of initial ∆TCMB, since the core cooling isself-regulated. For Mp = 2M⊕, the inner coregrowth depends on ∆TCMB for ∆TCMB

>∼ 2000K.

For such high ∆TCMB, since the core has largerthermal energy initially and the heat flux is notself-regulated, it takes more time for the core tem-perature to become below the nucleation temper-

ature. For Mp>∼ 5M⊕, core hardly cools on 20

Gyrs, the inner core does not grow from the ini-tial state. In these cases the inner core size isdetermined by a relationship between adiabaticcurve and melting curve of iron. The inner coreof super-Earths (Mp > M⊕) have never nucleatedfor ∆TCMB = 10000K. For massive planets, theincrease in the viscosity due to higher pressure isovercome only by very high initial temperature.The high ∆TCMB also delays nucleation of the in-ner solid core. As a result, there is trade off be-tween heat flux and inner core growth through therelation between melting point of iron core andtemperature- and pressure-dependency of mantleviscosity.

Figures 8 show dependence of evolution ofFCMB on Mp for fixed values of ∆TCMB. For∆TCMB = 1000K, we have already mentionedthat FCMB is rather lower for Mp = 5M⊕ thanfor Mp = M⊕ (Figs. 6b and 3b), because theincrease in the viscosity due to higher pressuredominates the decrease due to higher temperaturefor Mp = 5M⊕. This trend is clearly shown inFig. 8a.

However, this is not always the case. If thecore temperature is high enough (in other words,∆TCMB is high enough), or if pressure is lowenough (Mp is small enough), the viscosity shoulddecrease with increase in Mp due to the tempera-ture effect. For ∆TCMB = 10000K (Fig. 8d), FCMB

is approximately proportional to Mp. Even for∆TCMB = 1000 K, FCMB increases with Mp forlow mass regime (Mp < 1M⊕). Thus, FCMB hasa peak at some value of Mp for a given value of∆TCMB. Figures 8b and c show that the criticalplanet mass (Mp,c) at which FCMB takes the max-imum value is 2M⊕ for ∆TCMB = 2000K and 5M⊕

for 5000K. We empirically found that

Mp,c ≃∆TCMB

1000KM⊕. (18)

Since the mantle viscosity depends on the acti-vation volume, V ∗ (Eq. [15]), and the values ofV ∗ may have uncertainty at high pressure, wealso performed calculations with a smaller valueof V ∗. Figures 9 show the results with V ∗ =3 × 10−6m3mol−1. Due to the weakened pres-sure effect, Mp,c is increased by a factor of a few.In Figures 9, the viscosity is artificially increased(Eq. [15] by a factor of 6000) in order to compen-

11

sate the smaller value of V ∗ and reproduce Earth’sobserved values. Note that the artificial increasedoes not affect Mp,c.

The critical planetary mass Mp,c is approxi-mately derived by the Mp-dependence of the man-tle viscosity at CMB. Since we empirically foundthat PCMB ∼ (Mp/M⊕)P⊕CMB and TCMB ∼

5∆TCMB, Eq. (15) is reduced to

ηCMB(Mp, TCMB) ∝ exp

E∗ +(

Mp

M⊕

)

P⊕V∗

nRTCMB

.

(19)When the argument of exponential is larger thanunity, the viscosity rapidly increases with Mp todepress FCMB, because FCMB ∝ η−1/3. For Mp <Mp,c, we found that FCMB ∝ Mp. When the ar-gument exceeds some critical value (C > 1), theviscosity enhancement eventually overwhelms thefactors for the positive Mp-dependence of FCMB.Thus, Mp,c is given by the value of Mp with whichthe argument of exponential is ≃ C,

Mp,c ≃5nCR∆TCMB − E∗

V ∗

M⊕

P⊕CMB

≃

5nCR∆TCMB

V ∗

M⊕

P⊕CMB

∼

C∆TCMB

10000K

(

V ∗

10× 10−6m3mol−1

)−1

M⊕, (20)

which explains the dependences on ∆TCMB and V ∗

that we found numerically. (If we adopt C ∼ 10,the numerical factor is also explained.)

3.4. Lifetime of intrinsic magnetic fields

The lifetime of the intrinsic magnetic fieldsis calculated for ∆TCMB = 1000, 2000, 5000 and10000 K with a fixed value of V ∗ = 10 ×

10−6m3mol−1. The results are summarized inFig. 10. It is clearly shown that the lifetime de-clines forMp

>∼ Mp,c by the increase in the mantle

viscosity due to the pressure effect that we dis-cussed in details in the previous subsection. Theresults for V ∗ = 3× 10−6m3mol−1 show a similarproperty.

In Fig. 10, the lifetime weakly increases withMp for Mp

<∼ Mp,c. The dependence is explained

as follows. The lifetime is approximately given byτlife ∼ Eth/(SCMBFCMB), where Eth is thermalenergy of the core and SCMB is surface area of

the core. According to our calculation, SCMB ∝

M1/2p rather than M

2/3p due to self-compression.

Figures 7 and 13 show that FCMB ∝ Mp and Eth ∝

M2p for a fixed TCMB. As we mentioned in section

3.3, TCMB ∼ 5∆TCMB. Thus, for a fixed ∆TCMB,it is predicted that τlife ∼ Eth/(SCMBFCMB) ∝

M2−1/2−1p = M

1/2p , which is consistent with the

numerical results in Fig. 10.

When Mp > Mp,c, the higher mantle viscositydue to the effect of higher pressure significantly de-presses heat transfer at the bottom of the mantle.The suppressed heat flux cannot maintain the vig-orous core convection. As a result, the magneticfield lifetime is rather shorter for Mp > Mp,c.

Figure 10 also shows that the lifetime forMp < Mp,c does not depend on ∆TCMB at all.The initial temperature is high enough to over-come the pressure-dependency even for the caseof ∆TCMB = 1000 K, resulting in the effectiveself-regulation of FCMB. As a result, the lifetimedoes not depend on the initial value of ∆TCMB.

3.5. Strength of magnetic fields

The strength of magnetic fields is as impor-tant as their lifetime to discuss habitability of theplanets. Here we evaluate the strength of themagnetic fields, using the scaling law derived byChristensen et al. (2009), For planets with suffi-ciently rapid spins, Christensen et al. (2009) de-rived the magnetic strength at the core surface as

, Bc ∼ 0.5µ1/20 ρ1/6c F 1/3

conv, (21)

where ρc is average density of the core andµ0 is permeability. If the magnetic momentis dipole-dominant and the dipole moment is∝ 1/r3 (Gaidos et al. 2010), the strength of mag-netic dipole at the planetary surface is Bsurf =Bc(rc/rp)

3.

With Fconv = FCMB−Fcond, we calculated Bsurf

from our simulation results. Figure 11 shows thecalculated Bsurf at t = 5Gyr for various Mp and∆TCMB. The strength monotonically increases ifthe initial ∆TCMB is sufficiently high. The rela-

tively weak dependence (B ∝ M1/3p ) comes from

the adopted the scaling law, B ∝ F1/3conv, and the

numerically obtained relation, FCMB ∝ Mp. If∆TCMB is not high enough, the pressure effectis dominant and the strength is significantly sup-pressed for Mp

>∼ Mp,c.

12

4. Conclusion and Discussion

We have developed a numerical model to sim-ulate thermal evolution of various-mass terrestrialplanets in habitable zones. The density distri-bution of the planetary interior is calculated byVinet EOS taking into account pressure depen-dence. Using the interior structure model, we cal-culate heat transfer through mantle, using the as-trophysical mixing length theory modified to man-tle convection. The modified mixing length theoryis easily applied to multi-layer convection that maybe dominated convection mode in super-Earths.We have calibrated the modified mixing lengththeory with the conventional parametrized con-vection model and the boundary layer theory, insimple one-layer convection cases.

With nominal parameters of surface temper-ature Tsurf = 300K, a mantle-core mass rationζm/c = 7 : 3, initial core impurity x0S of 10 wt%,and initial temperature gap at CMB ∆TCMB =1000K, our model for M = 1M⊕ reproduces sur-face heat flow and inner core radius of the presentEarth. With different parameter values suitablefor Mercury, Venus, and Mars, our model also re-produces the results that are not inconsistent withpresent magnetic activity of these planets.

With this model, we calculated thermal evolu-tion of terrestrial planets with mass Mp = 0.1–10M⊕ in habitable zones, using the nominal pa-rameters, to study lifetime of intrinsic magneticfield that is one of the important factors for theplanets to be habitable. We found from the nu-merical calculations that the lifetime is maximizedat

Mp,c ∼∆TCMB

1000K

(

V ∗

10× 10−6m3mol−1

)−1

M⊕,

(22)where V ∗ is activation volume of mantle material.Planets with smaller masses cool more rapidly, sothat they cannot maintain core heat flux to gen-erate dynamo long enough. For Mp > Mp,c, therapid increase in the mantle viscosity caused byhigh pressure significantly depresses heat trans-fer throughout the mantle and hence that in thecore. As a result, dynamo cannot last long. Al-though the temperature effect tends to decreasethe mantle viscosity as planetary mass becomeslarge, the pressure effect to increase the viscosityoverwhelms the temperature effect for Mp > Mp,c.

With the numerically obtained empirical relation,TCMB ∼ 5∆TCMB, we can analytically deriveEq. (22) from the Arrhenius-type formula for themantle viscosity that we adopt (Eq. [15]).

We found that while the lifetime of magneticfields does not depend on ∆TCMB for Mp < Mp,c,it sensitively depends on ∆TCMB for Mp > Mp,c

because Mp,c ∝ ∆TCMB (Eq. [22]). The initial∆TCMB, that is, the initial temperature profileof planetary interior, is one of the most uncer-tain parameters, because it highly depends onthe processes of planetary formation and differ-entiation of the planetary interior. As is shownby SPH simulations, if a planet undergoes gi-ant impacts, its metallic core is heated as highas several tens thousands K for Mp ∼ 1 M⊕

(Canup 2004). On the other hand, if a planetaccreted from small planetesimals without giantimpacts, the initial temperature profile is deter-mined by the balance between gravitational en-ergy buried by planetesimals and thermal trans-fer efficiency through rocky mantle. The pro-cess includes crystallization of magma ocean anddepends on the mechanical property of moltenmantle (Abe & Matsui 1986; Zahnle et al. 1988;Senshu et al. 2002). Thus, to evaluate the lifetimeof magnetic fields, in particular for super-Earthsthat are likely satisfy Mp > Mp,c, detailed anal-ysis for accretion and early thermal evolution ofterrestrial planets are needed.

It is also found that higher initial tempera-ture profile delays the inner core nucleation. Forsuper-Earths, in order to maintain magnetic fieldmore than 10 Gyr, the initial temperature hasto be high enough to overwhelm the pressure-dependence. However, in that case, the temper-ature of the core center never reaches its con-densation temperature and the inner core cannotgrow. Some geo-dynamo simulations suggest thatthe presence of the inner core stabilizes the dipolemoment of geomagnetic field (Sakuraba & Kono1999). It is also suggested that because thermallydriven convection is not sufficient to drive dynamoaction against the ohmic dissipation within thecore of Earth (Gubbins et al. 2003), the compo-sitional convection induced by light elements re-leased to the outer core by solidification of the in-ner core plays an essential role in dynamo genera-tion (Stevenson et al. 1983; Gubbins et al. 2004).Since our results (Figures 6) show that inner

13

core is not nucleated and compositional convec-tion does not occur for Mp

>∼ 5M⊕, dipole mag-

netic fields of super-Earths might not be stable.

The existence of magnetic field of extrasolarplanets could be directly detected by the polar-ization observation of the photon from transitingplanets or detection of H+

3 trapped by the mag-netic fields in the future. Another possibility ofthe detection of planetary magnetic field is, al-though it is indirect, observation of compositionof planetary atmosphere or atmospheric tail. Ifthe planet has intrinsic magnetic field, its atmo-sphere could keep H2O molecules for long period.Venus may have lost H2O molecules on a shorttime scale (Bullock & Grinspoon 2001). Thus ifwater series molecules, such as H2O, H3O, andHO, were detected in the planetary atmosphere,it would indicate the existence of intrinsic mag-netic field, although super-Earths might be ableto sustain the HXO molecules in the atmosphereby their high gravity even without the protectionby magnetic fields.

We need to elaborate our thermal evolutionmodel, by considering details of mantle convectionmode that is affected by phase transition betweenγ-spinel to perovskite (Christensen & Yuen 1985)at upper/lower mantle. We also should take intoaccount further mineral transitions suggested byab initio calculations (Umemoto et al. 2006) thatmay appear in super-Earths, because they mayaffect internal density structure and the mantleconvection mode.

The abrupt enhancement in the mantle vis-cosity due to the pressure effect relies on theArrhenius-type formula for the mantle viscosity weadopt here. The critical mass beyond which thepressure effect dominates is inversely proportionalto activation volume (Eq. [22]). Thus, detailedrheological properties affect habitability of super-Earths. The values of the activation volume arenot clear at such high pressure as in deep mantle insuper-Earths. The mechanism to inhibit dynamoactivity in super-Earths proposed by Gaidos et al.(2010) also depends on high pressure materialproperties (melting and adiabatic curves), whichalso need to be confirmed. These provide newmotivations to high pressure experiments and firstprinciple simulations. Super-Earths provide goodlinks between astronomy and high-pressure mate-rial science.

Acknowledgment

The authors thank to useful discussions withDiana Valencia, Masahiro Ikoma, and HidenoriGenda. This work is partly supported by GlobalCOE program ”From the Earth to Earths”.

Appendix A. Comparison among Nu-Ra re-

lationship model, thermal boundary layer

model and mixing length theory model.

In evaluation of thermal transfer of mantle con-vection, we compare the modified mixing lengththeory (Sasaki & Nakazawa 1986; Abe 1995) withthe conventional parameterized convection model(PCM; e.g., Sharpe & Peltier 1979) and com-monly used thermal boundary layer model (BLT;e.g., Stevenson et al. 1983).

The original mixing length theory (MLT; e.g.,Vitense 1963; Spiegel 1963) is often used in thethermal transfer within the stellar interior to sim-ulate the stellar evolution. Sasaki & Nakazawa(1986) modified the mixing length theory for verylow Reynolds number convection in which the ver-tical flow is characterized by the Stokes velocitydetermined by a balance between buoyant forceand resident force of viscosity rather than by freefall velocity. In the modified version, a distance(D) from the closest barrier such as the CMB orthe top of the mantle layer is adopted for the mix-ing length ℓ, while in the original theory, the den-sity scale height is usually adopted for ℓ.

The PCM uses the empirical Nu-Ra relation-ship,

Nu = ζ

(

Ra

Rac

)1/3

, (23)

where Nusselt number represents the ratio be-tween total heat flux and heat flux only due toconduction without convection,

Nu =Ftotal

Fcond, (24)

Fcond = k∆T

d, (25)

and Rayleigh number is a dimensionless numberrepresenting the strength of convection,

Ra =ρgα∆Td3

κη, (26)

14

where g, α,∆T and d are gravitational accelera-tion, thermal expansion, temperature differencebetween top and bottom and thickness of con-vective region, respectively, and Rac is criticalRayleigh number (∼ 650) for thermal convection.Because when Ra ∼ Rac, Nu must be ∼ 1, ζis O(1). Sotin et al. (1999) derived ζ ∼ 1.5-2.0through 3D fluid dynamical simulation althoughthe value of ζ is somewhat lower in high Ra re-gion. We here adopt ζ = 1.7.

From eqs. (23) to (25), total heat flux througha fluid layer is represented by Rayleigh number as

Ftotal = ζ

(

Ra

Rac

)1/3

Fcond. (27)

This model is very simple, but Ra is “mean” valueof the whole mantle that is difficult to evaluate forreal mantle in which viscosity changes by orderof magnitude throughout the mantle. In particu-lar, it may not have enough resolution to evaluateFCMB that we are concerned with in the presentpaper.

In the BLT, heat flux is evaluated in theboundary layer. The thickness of boundary layeris estimated by an assumption that the layeris marginally stable against thermal instability.Then, the local Rayleigh number of the thermalboundary layer (Ral) is nearly equal to the criticalRayleigh number for thermal instability, that is,

Ral =

(

ρgα

κη

)

l

δ3∆TTB ∼ Rac, (28)

where δ is thickness of thermal boundary layer and∆TTB is temperature difference between the bot-tom and the top of the boundary layer, and sub-script “l” denotes the values in the thermal bound-ary layer. Thus, the heat flux through the layer iscalculated as

Ftotal = k∆TTB

δ, (29)

where δ is calculated as

δ = ζ′[(

κη

ρgα

)

l

Rac∆TTB

]1/3

, (30)

and the factor ζ′ ∼ O(1) is determined as follows.If κ, η, ρ, g, and α are constant, ∆TTB ∼ ∆T/2, sothat

Ftotal =1

24/3ζ′

(

RalRac

)1/3

Fcond. (31)

To be consistent with 3D fluid dynamical simula-tion by Sotin et al. (1999), we set 24/3ζ′ = 1/1.7,that is, ζ′ = 0.23.

Since the heat flux is expressed by quantitiesonly in the thermal boundary layer (eqs. [27] and[28]), which is localized in the mantle, the BLThas better resolution than the PCM, in particular,for evaluation of FCMB. However, since the valuesof viscosity change by order of magnitude even inthe thin thermal boundary layer, it is not clearwhich value has to be chosen as a representativevalue of the viscosity in Eq. [30]. For the terrestrialplanets in our Solar system, observational data canbe used to constrain the uncertainty.

Since the modified MLT uses local values ofphysical quantities (Eq. [13]), it quantitativelyagrees with the BLT for wide range of parametersas shown below. There is no uncertainty for choiceof a representative value of viscosity in the MLT,while choice of the mixing length has uncertainty.For calculation of thermal evolution of the Earth,it is proposed that a distance (D) from the closestbarrier such as the CMB or the top of the man-tle layer is appropriate for ℓ (Sasaki & Nakazawa1986; Abe 1995; Senshu et al. 2002; Kimura et al.2009, and references therein). Through compari-son with the calibrated PCM and BLT, we adoptℓ = 0.82D as shown below.

To compare these models, we calculate the heatflux in the case of radially constant η with the indi-vidual calibrated models. Internal heat generationdue to radioactive elements is neglected. Figure 10shows the heat flux at the base of the mantle asa function of Ra, obtained by each model. Thevalues are normalized by Fcond, that is, Nusseltnumber. Although the MLT does not assume therelation of Nu ∝ Ra1/3, it produces the relation.To match the absolute values, we set ℓ = 0.82D.The maximum value of ℓ is proportional to d andheat flux is proportional to ℓ4 (Eq. [14]) The sen-sitive dependence on d is canceled out to result inthe rather weak dependence, Nu ∝ Ra1/3 ∝ d,because we found that [(∂T/∂r) − (∂T/∂r)s] de-creases with increase in ℓ (Eq. [14]). Analyticalargument for it is found in Abe (1995).

We also examined a case in which the viscosityis strongly temperature-dependent,

η(T ) = η0 exp[log(η1/η0)(1 − T )], (32)

where η0 and η1 are viscosity at the top (T = 0)

15

and the bottom (T = 1) of convective region. Fig-ure 13 shows Nusselt number obtained by PCM,BLT and MLT as a function of Ra. In the PCM,Ra is a mean value for a whole mantle. The repre-sentative viscosity is evaluated using average tem-perature of mantle, that is, T = 0.5 if mantle isthermally equilibrated because the PCM assumeconstant heat flux throughout mantle. In the BLTand MLT, the heat flux is evaluated by local quan-tities. The BLT and MLT produce the same heatflux within 1% in all cases, while the results by thePCM deviate from those by the BLT and the MLTfor high Ra or high η0/η1. These results showthat MLT is as good as BLT to calculate ther-mal evolution of terrestrial planets. Since MLTis more easily to be applied for super-Earths thatmay have barriers for convection in their mantle(section 2.3), we adopt MLT.

Appendix B. On the magnetism of planets

in Solar system

In order to confirm the validity of our model,we show that our model produces thermal evo-lution for individual terrestrial planets in theSolar system that is not inconsistent with theircurrent magnetic activity, with appropriate non-nominal parameter values, in a similar way toStevenson et al. (1983). Currently, Earth andMercury have self-generating magnetic fields in-duced by dynamo action, while Venus and Marsdo not (although some parts of the Martiancrust have remnant magnetic field in the past(Acuna et al. 1999)).

To apply our model to Mercury, Venus andMars, we need to use non-nominal parameter val-ues:

• Mercury: ζm/c = 3 : 7 (a significantlylarge metallic core) and Tsurf = 440K.These are observed values. We also testedsmaller values of x0S = 0.01, 0.05 accordingto Stevenson et al. (1983). We also testedhigher mantle viscosity than Eq. (15) bymultiplying viscosity increase factor ∆η =100

• Venus: Tsurf = 737K, while the nominal val-ues are used for ζm/c and x0S. We also testedhigher mantle viscosity as well as in the caseof Mercury. Note that two-layer convection

model is used for Venus, because the spinel-perovskite transition also could work as abarrier for Venusian mantle.

• Mars: Tsurf = 210K. ζm/c is nominal valueand x0S = 0.1, 0.15, 0.2. The standard for-mula, Eq. (15), is used for mantle viscosity.

∆η is viscosity increase factor due to lack ofwater in the case of Mercury and Venus. It issuggested by experiment that dry rock has factorof 100 higher viscosity than that of hydrated rocks.Thereby, we multiply ∆η = 100 in the case ofVenus and Mercury.

The lifetime of magnetic fields calculated by ourmodel is shown in Fig. 14. In order to be consis-tent with current Mercury, Venus and Mars, thelifetime must be longer than 4.5Ga for Mercuryand shorter than 4.5Ga for Venus and Mars. Be-cause Martian crust of age ∼ 4Gyr retains pale-omagnetic field, the lifetime of Martian magneticfield may be longer than 0.5 Gyr.

Figures 14 show that for Mercury, the lifetimeis longer than 4.5Ga for relatively small values ofx0S (∼ 0.01 − 0.05) except for extremely small∆TCMB (< 200 − 300K). The relatively long life-time is resulted by nucleation of inner core due tolower solidification temperature corresponding tosmall values of x0S. If the nominal value of x0S

is used, the lifetime is short. The small value ofx0S for Mercury was discussed by Stevenson et al.(1983).

The predicted lifetime of magnetic field forVenus is quite short for relatively high mantle vis-cosity (∆η > 100). Observation suggests thatVenus is lack of H2O. That may be due to run-away greenhouse effect of H2O itself and conse-quent dissipation by UV dissociation and heatingof the molecules. Because melting temperature ofthe mantle viscosity is lowered by H2O, relativelyhigh mantle viscosity is more likely, although wedo not know exact values of Venus’ mantle viscos-ity.

The predicted lifetime of magnetic field for forMars is longer than 1 Gyr but shorter than 4.5Gyr,if initial ∆TCMB is ∼ 10 − 500K. If Mars hasnever undergone giant impacts that cause signif-icant heating of metallic core, such low initial∆TCMB is likely.

Thus, with non-nominal parameters that reflect

16

distance from the Sun and accretion history of in-dividual planets, our model can produce the re-sults that are not inconsistent with the currentterrestrial planets in the Solar system. However,in order to clarify intrinsic physics in generationof magnetic field of extra solar terrestrial planets,we focus on the results with the nominal parame-ters (Tsurf = 300K, ζm/c = 7 : 3, and x0S = 0.1),which correspond to the parameters of terrestrialplanets with the same compositions as the Earthin habitable zones.

Appendix C. Energy in the core

Figure 15 shows the thermal and gravitationalenergy, released latent heat, and their sum as afunction of TCMB for the nominal cases with plan-etary mass M = 1, 2, 5 and 10M⊕, which are cal-culated by the procedures in section 2.2. We setthat each value is zero at the temperature at theinitiation of inner core growth. As is shown in thisfigure, the loss of thermal energy occupies aboutone third of the total energy loss of the core forthe case of M = 1M⊕. Released latent heat corre-sponds to about one fifth of the total energy loss,which depends on TCMB because of the nonlineardensity-dependency of the melting temperature ofmetal.

The gradient of the total energy in Fig. 15 cor-responds to an effective specific heat of the core.The total heat capacity is twice larger than thespecific heat of thermal energy solely just after theinner core initiation (TCMB ≃ 4100K), while theirvalues converge as temperature decreases. This isbecause impurity concentration increases with thetemperature decrease in the outer core. Inner coregrowth is moderated by the depression of meltingtemperature of outer core due to the concentra-tion of impurities into outer core. Released grav-itational energy and latent heat become smallerthan thermal energy as the temperature decreases.Note that the gravitational energy released by thethermal contraction of the core also works as re-sistance to cooling of the core (see section 2.2).

The ratio of gravitational energy, latent heatand thermal energy is varied with planetary mass.Thermal energy is more dominant than other en-ergies for more massive planet. It means that thegravitational energy and latent heat are not mainenergy source to drive dynamo action within cores

of massive super-Earths. This is mainly because ofthe change in slope of adiabatic curve within core.The higher gravity causes steeper adiabatic ther-mal structure, and then core posses large amountof thermal energy inside it for the case of massiveplanets. This is also the reason why the effectivespecific heat of core is increased as planetary massincreases.

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This 2-column preprint was prepared with the AAS LATEX

macros v5.2.

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Table 1: Physical properties of mantle and core components we adopted(Valencia et al. 2007)

material ρ0 K0 K ′0 γ0 q θ0 Refs.

(kgm−3) (GPa)ol 3347 126.8 4.274 0.99 2.1 809 a

wd+rw 3644 174.5 4.274 1.20 2.0 908 apv+fmw 4152 223.6 4.274 1.48 1.4 1070 appv+fmw 4270 233.6 4.524 1.68 2.2 1100 b

Fe 8300 164.8 5.33 1.36 0.91 998 c,dFeS 5330 126 4.8 1.36 0.91 998 c,d

a (Stixrude & Lithgow-Bertelloni 2005), b (Tsuchiya et al. 2004), c (Williams & Knittle 1997), d(Uchida et al. 2001)

Table 2: Parameters of radiogenic elements we adopted (van Schmus 1995)

element U(ppb) H(µWkg−1) λ(yr−1)K40 28.0 29.17 5.54×10−10

Th232 76.4 26.38 4.95×10−11

U235 0.14 568.7 9.85×10−10

U238 20.1 94.65 1.551×10−10

Table 3: Parameter of the viscosity in upper mantle and lower mantle (Ranalli 2001)

B(Pa−ns−1) n E∗(103Jmol−1) V∗(10−6m2mol−1) ǫ(s−1)upper mantle 3.5×10−15 3.0 430 10 10−15

lower mantle 7.4×10−17 3.5 500 10 10−15

Table 4: Physical property of upper mantle, lower mantle, and core (Yukutake 2000)

kc(W mK−1) Cp(J kg−1K−1) α(K−1)upper mantle 5 1250 3.6×10−5

lower mantle 10 1260 2.4×10−5

outer core 40 840 1.4×10−5

20

Fig. 1.— Radial density profiles for 0.1, 0.2, 0.5, 1, 2, 5, 10 M⊕ planets (with inner core of 6 wt% of eachcore) obtained by our model in the nominal case.

21

Fig. 2.— The schematic diagram of the procedure to obtain initial temperature distributions. For moredetail, see text.

22

(a) (b)

(c) (d)

Fig. 3.— Time evolution of (a) the temperature profile, (b) the surface heat flux, (c) the inner core radiusand (d) the heat flux through CMB for an Earth-mass planet with ∆TCMB = 1000K. (a)Time evolution ofinternal temperature distribution in the case of ∆TCMB = 1000K with 1 M⊕. The planet radius of 6385kmand core radius of 3375km. The surface heat flux declines to ∼ 0.08 Wm−2 and the inner core grows up to1200 km at 4.5 Gyr after, which is nearly equal to the present observed value of the inner core radius of theEarth. The core heat flux monotonically decreases with time but its time derivative discontinuously changesat the initiation of the inner core at around 2.2 Gyr. The solid curve represents the threshold flux (Fcond)to maintain dynamo activity in the outer core (Eq. [17]).

23

(a) (b)

(c) (d)

Fig. 4.— Same figures as Figs. 3 except for two-layered mantle convection model with ∆TCMB =1000K.

24

(a) (b)

(c) (d)

Fig. 5.— Same figures as Figs. 3 except that two-layered mantle convection model with ∆TCMB = 800K isconsidered instead of one-layered model with ∆TCMB =1000K.

25

(a) (b)

(c) (d)

Fig. 6.— Same figures as Figs. 3 except for Mp = 5M⊕.

26

(a)

(b)

(c)

(d)

Fig. 7.— Evolution of the core heat flux (left column) and inner core radius (right column) for M = (a)1, (b) 2, (c) 5, (d) and 10M⊕ with various initial ∆TCMB. In the left column, Fcrit for each ∆TCMB is expressed by thinner line with the same type. Some lineswith different initial ∆TCMB are overlapped by each other. For these parameters, the initial conditions do not affect the evolution sinceself-regulation of mantle heat transfer works due to temperature dependence of the mantle viscosity. Inner cores never nucleate in thecases of ∆TCMB = 2000, 5000, and 10000K for Mp = 5M⊕ and ∆TCMB = 5000 and 10000K for Mp = 10M⊕.

27

(a) (b)

(c) (d)

Fig. 8.— Evolution of the core heat flux for ∆TCMB = (a)1000K, (b) 2000K, (c) 5000K, (d) 10000K withMp = 1, 2, 5, 10M⊕, respectively.

28

(a)

(b)

(c)

(d)

Fig. 9.— Same as Fig. 7except V ∗ = 3× 10−6m3mol−1

29

(a) (b)

Fig. 10.— Lifetime of magnetic fields as a function of planetary mass (Mp) with various ∆TCMB for (a)V ∗ = 10× 10−6m3mol−1 and (b) V ∗ = 3× 10−6m3mol−1.

(a) (b)

Fig. 11.— Strength of magnetic fields after 5 Gyr as a function of planetary mass (Mp) with various ∆TCMB

for (a) V ∗ = 10× 10−6m3mol−1 and (b) V ∗ = 3× 10−6m3mol−1.

Fig. 12.— Temporal averaged Nusselt numbers obtained by PCM, BLT and MLT models. Open triangles,circles and squares represent the results of PCM, BLT and MLT models, respectively.

30

(a) (b)

(c) (d)

Fig. 13.— Temporal averaged Nusselt numbers obtained by PCM, BLT and MLT models. The viscosity ischanged from the bottom to the top with the ranges of (a) ∆η= 10, (b) 102, (c) 103 and (d) 104, respectively.

31

(a)

(b)

(c)

Fig. 14.— Lifetime of magnetic fields of (a) Mercury, (b) Venus and (c) Mars obtained through our simula-tions. The free parameters are initial temperature gap at CMB ∆TCMB and initial impurity concentrationin core x0, viscosity increase factor ∆η, respectively. (a)ζm/c = 3 : 7, Tsurf = 440K, and squares, circles, andtriangles are corresponding to models with x0s = 0.01, 0.05,and 0.1, respectively. The filled and open symbolsrepresent models with the standard viscosity and a higher viscosity multiplied by ∆η = 100. (b)ζm/c = 7 : 3,Tsurf = 737K and x0 = 0.1. The filled and open symbols represent the same meaning as is in the case ofMercury. The square and triangles are models in which one- and two-layered mantle convection are assumed.(c)ζm/c = 7 : 3, Tsurf = 210K. squares, triangles and circles are corresponding to models with x0 = 0.1, 0.15and 0.2.

32

1M⊕ 2M⊕

5M⊕ 10M⊕

Fig. 15.— Individual core energies as a function of TCMB for the nominal case with M = 1, 2, 5, 10M⊕ andx0S = 0.1. Dashed, dotted, and dot-dashed curves represent gravitational energy, latent heat, and thermalenergy, respectively. Solid curve shows the total energy. The gradient of total energy corresponds to thespecific heat of the core.

33