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Terrestrial Interiors

C. Sotin Propulsion l-aboratory, Caffirnia Institute of Techno,loSy

J. M. Jackson Caffirnia Institute of Technology

S. Seager Massachus etts Institute of Tbchnology

The discovery and study of has always motivated the question of the existbnce and of terrestrial exoplanets, especially habitable . mass and radius measurements (yielding average density) are possible for a growing number of exoplanets, including terrestrial planets. The mass and radius provide a constraiiit for interior models and, via models, enable interpretation of a planet's bulk composition. This chapter describes the fundamental equations for calculating the interior structure of terres- trial planets (including -rich, -rich, and -rich planets). A detailed description of interior structure models is given, with emphasis on the equation of state. High- experimental measurements are needed to understand the internal structure and of massive terrestrial exoplanets. Interpretation of low-mass planet interiors are fundamentally limited by degeneracies, because there a1e only two data points per planet: mass and radius. For example, large terrestrial planets having a massive primordial atrnosphere of H, and IIe cannot be distinguished from planets having dominant internal H2O layers based on the mass and radius alone. The occurrence of on exoplanets more massive than is a controversial question, and, although unanswered, this chapter addresses the relationship between plate tectonics and thermal . A,s more and more low-mass exoplanets are being discovered, mass vs. radius statistics will build up. The hope for terrestrial exoplanet mass and radius measurements is that unique populations will emerge from statistics, helping us to understand planet formation and evolution.

1.. INTRODUCTION J) is related to the planet's ability to retain a large primordial of H and He. Whether or not a planet retains The search for terrestrial exoplanets is one of the most substantial amounts of H and He depends primarily on a exciting challenges of the twenty-first century. For the first planet's mass. Wuchterl et al. (2000), for example, propose time have the chance to uncovet alarge sample that a planet less than 15 times the mass of Earth (Md will of planets that are predominantly rocky or icy and not the be terrestrial in nature, i.e., without significant H and He. easier-to-detect giant planets, which are composed mostly Beyond the terrestrial and groupitrgs, each of H and He. Terrestrial planets are of major scientific sig- category can actually be subdivided into two subsets. The nificance because they are the planets suitable for life as we terrestrial planets include both silicate-dominated planets like know it and amenable for future observational searches for Earth and , as well as -like planets, which are atmospheric gases. In the conventional sense enriched in iron (i.e., depleted in ). The Mercury-type a habitable planet is one with some surface liquid water, exoplanet is of interest because finding many close to the because all life on Earth requfues liquid water. In contrast host will help further understand Mercury's formation. to terrestrial planets, gtant and -sized planets en- The giant planet category includes not only the FI/He- shrouded by gas envelopes have no solid or liquid surfaces to dominated and but also icy planets and support life as we know it, and their temperatures just below Neptune. The ice giants have a much smaller H-He envelope the deep atmosphere rapidly become too hot for life to exist. than Jupiter and Saturn (10-1 5Vo by mass). Note that how The planets are conveniently divided into water-rich Uranus and Neptune are is not accurately known. two categories: the terrestrial planets (Mercury, Venus, Earth, Although their are -rich, there is a ) located in the inner solar system, and the giant planets tradeoff for their interior bulk compositions between ice + (Jupiter, Saturn, Uranus, Neptune) located in the outer solar an H/He envelope and a combination of /iron, much less system. The difference between teffestrial. and giant planets ice, and a more massive FI/FIe envelope. Extrapolating to icy

375 316 Exoplanets

planets with insignificant H-He envelopes compared to Ura- total mass of the planet: O, Fe, Mg, and Si explain 9570 nus and Neptune are exoplanets of signiflcant astrobiological of the mass of Earth. If Ni, S, Al, and Ca are added, then interest: planets or water worlds, which could also be 99.9Vo of the mass is taken into account (Thble 1).The latter called super Ganymedes or super . Under the right at- three minor elements add a lot of complexities in the system. mospheric mass and interior temperature conditions, some of For example, experimental studies suggest that the oxidation these planets are likely to harbor internal liquid water . state of iron and exsolution of metallic iron particles from Icy exoplanets with thin atmospheres can be considered as iron-bearing silicates is strongly affected by the presence of lnge analogs of the large icy of Jupiter (, Al (Frost et al., 2004) and extreme and tempera- , ) and Saturn (). tures (Jackson et al., 2009).If S is added to Fe, the melting The bulk interior composition of an exoplanet can be con- temperature decreases with increasing pressure and the phase strained with planet mass and radius measurements together relations become very complex (Chen et al., 2008). Although with planet interior equations. The planet mass is measured there is considerable debate sunounding the identity of the via measurements (see chapter by Lovis and light element and crystal structure of the Fe-domin ant al- Fischer) and the planet radius via measurements (see loy in Earth's core, the effect on total planet mass is small. chapter by Winn). Some exoplanets are discovered with the Based on the available data, adding Ni, S, Al, and Ca to their radial velocity method and arc Iater found to transit, while closest major element results in a less than l%o error in the others are discovered by the transit technique and followed model-calculated total mass. Therefore, the composition of up for mass measurements via radial velocity. each layer can be described with four elements: O, Fe, Mg, Conceptually the equations that describe a terrestrial and Si (see Sotin et aI., 2001).In the , Al is equally planet interior ate the conservation of mass, hydrostatic divided between Mg and Si for charge conservation and Ca equilibrium, energy transport, and the equation of state. The is added to Mg. In the core, Ni and S are added to Fe. most basic assumptions common to most models are that the Due to Earth's seismically active interior, the average terrestrial planet interior is formed from three basic materials structure of Earth (Fig. 1) is well known from global Earth (iron, silicate, and water) with some mixture and of various models (Dziewonski and Anderson, 1981). Earth is mainly phases, and that the planets have differentiated interiors. The composed of two layers: the iron-rich core (one-third of the new territory for terrestrial exoplanet interior models lies in mass) and the silicate mantle (two-thirds of the mass). These the equation of state, because terrestrial exoplanets more two layers differentiated very early in Earth's evolutionary massive than Earth (super ) can have interior pressures because the melting temperature of iron alloys is much higher. For example, a planet ten times more massive lower than that of silicates and their density is higher. It is than Earth can have internal pressures three times higher envisaged that this differentiation processes occuffed into (e.9., Sotin et a1.,2007), which implies that new high-pressure the by segregation and into by mineral phases may exist. Super Earths are loosely defined Rayleigh-Thylor instabilities (Chambers, 2005). When the to be planets between 1 and 10 Mo that are predominantly protoplanets collided to form the terrestrial planets, their iron rocky or icy (see the Appendix of the chapter by Seager and cores would have merged. The change in gravitational energy Lissauer for exoplanet definitions). during the processes controls the temperature differ- This chapter describes the concepts and equations needed ence between the core and the mantle (e.g. , Solomon, 1919). to understand and model terrestrial planet interiors. Applica- tion to the mass-radius relationship for a variety of terrestrial planet types and a detailed presentation of the equation of state different given. for materials is This chapter also fo- TABLE 1. Mass fraction of the major elements contained in cuses on terrestrial planet internal dynamics by providing EH enstatite (values from Javoy, 1995). the equations describing subsolidus convection in plan etary mantles and investigating the controversial question of plate Element Enstatite Model tectonics. We conclude with an outlook for the future of Mass Fraction charactenzing terrestrial exoplanet interiors based on the o 30.28 limited observational data (two data points per planet) and Fe 33.39 anticipated planet harvests. Si 19.23 Mg 12.2r 2. INTERIOR STRUCTURE Total 95.17

2.1. Bulk Composition and Structure Ni 2.02 Ca 1.01 AI 0.93 The minerals that compose the different layers of a planet S 0.85 are determined according to the elements that are present, the Total 99.92 pressure, and the temperature. Previous studies (e.g. , Sotin et al., 2007, and references therein) have shown that only This example shows that 8 elements account for more than 99.970 a limited number of elements are necessary to describe the of the total mass.

tI Sotin et al.: Terrestrial Planet Interiors 377

3. The upper mantle. This layer is mostly composed of olivine, ortho- and clino- pyroxenes, and garnet (e.g., Vacher et al., 1998). By neglecting Ca and Al, the upper mantle is assumed to be simply made of olivine ([Mg,Fe]rSiO+) a14 orthopyroxene enstatite ( [Mg,Fe] 2Si2Ou). This simplifi cation might imply an overestimate of the planet radii, because (1) garnet is,denser than the other silicates, and (2) we ne- glect the transition zone as a distinct layer containing high- pressure olivine and garnet phases. This neglect introduces only -0.2Vo effor in the model-calculated total planet mass (see Sotin et al., 2001). 4. The . With a laterally varying thickness from ,|d" oceanic (-6 km) to continental (-20 to 90 k-), the crust represents about 0.47o of Earth's mass. We therefore neglect the crust when calculating the mass-radius relationship. In addition, two other planet layers might be considered but are omitted here because they also do not contribute significantly to the total planet mass and radius for Earth considered as an exoplanet: (1) The (oceans), which represents only 2.5 1f of Earth's mass. The oceans may contain more than 507o of the total HrO. (2) The at- mosphere, which is a very thin layer representing only 1G6 Fig. L. Interior structure of Earth. Seismic data,laboratory experi- of Earth's mass. ments, and numerical simulations are used in order to simulate The thickness of each layer depends on its elementary the interior structure and dynamics of Earth. Pressure and depth composition. The input parameters are the total mass of the of major interfaces are indicated on the left. The description of planet, the composition of the star (Fe/Si and Mg/Si), and each layer is'described in the text. the Mg content of the mantle (Mg# - Mg/[Mg + Fe]). To- ) gether with the Fe/Si ratio, the Mg content and total planet mass determines the size of the core. Another approach (e.g., Valencia et aJ., 2006) is to fix the size of the core. Both ap- The differentiation processes control the amount of iron that proaches give similar results. The Mg# is largely unknown is left in the silicate mantle. This explains why the martian except for Earth (-0.9) and Mars (-0.7). Its value depends mantle is enriched in iron compared to that of Earth. Earth's on the degree of differentiation of a planet. The more dif- mantle is divided into an upper mantle and a lower mantle. ferentiated the planet is, the larger the value of Mg#, because Partial melting in the mantle migrates to the surface to form more Fe settles into the planettr! core. the oceanic crust. Similar differentiation processes produce The abundances of the rock-forming elements Si and Mg, the continental crust. In the simulations for Earth described as well as Fe, can be measured in stellar atmospheres [see below, the layers have the following characteristics: Fig. 2 and, e.g., Huang et al. (2005)1. The relevant issue 1. The core. The solid inner core (radius of 1220 km) for terrestrial planet models is whether or not the range of is nestled inside a liquid outer core,(radius out to 3480 km). Si, Mg, and Fe found in needs to be considered. It is These two layers are merged into one single iron-rich core, interesting to find that the variation of Mg, Si, and Fe in because the chemical differences between the two layers solar-type stars is not large enough to cause significant dif- lead to minimal changes in total planet mass as calculated ferences in model-calculated planet mass. Assuming a solar from the model. or stellar composition for the bulk composition of a planet 2. The lower mantle. Bound by the major seismic dis- is still debated (McDonough and , 1995). Although CI continuities below (P - 135 GPa, radius = 3480 km) and chondrites have a solar composition, it has been suggested above (P - 23 GPa, radius = 5701 km), the lower mantle is that the composition of Earth may be similar to that of EH composed primarily of -707o (Mg,Fe)SiO3 perovskite ("pv"), enstatite chondrites (Javoy, 1995; Mattern et al., 2005). In -20Vo (Mg,Fe)O periclase, and -lUVo CaSiO, pe?ovskite. this case, rocks are more Si-rich and the ratio Mg/Si and Fel We merge CaSiOr-pv into (Mg,Fe)SiO3-pv. We note the Si are lowered to 0.734 and 0.878, respectively (Table 2). likely existence of postperovskite near Earth's core-mantle- A precise description of the effect of varying Mg, Fe, and boundary (Murakami et al., 2004), and the occurrence of Si ratios will be given in section 3.1 but it can be already a high-spin to low-spin crossover in (Mg,Fe)O under deep noted that varying molar ratios Fe/Si and Mg/Si in the range mantle conditions (Badro et a1.,2003; Sturhahn et a1.,2005). discussed above causes only small differences on computed Due to very limited equation of state dataunder these condi- planetary radii (0.3Vo on average). tions for low-spin (Mg,Fe)O and postperovskite, we exclude The next step in modeling a terrestrial planet interior is to t these phases in our model. transform elementary composition into mineralogical com- 318 Exoplanets

TABLE 2. Comparison of the solar (e.g., Cox et al., 1999) and EH enstatite chondritd composition (values are 1,8 based on Earth values). 1.6

1.4 Solar EH Enstatite Chondrites

1.2 (Fe/Si) 0.977 0.986 0.878 0.909 (Me/Si) cr) r.072 1.131 0.734 0.803 (D 1 LL 08 Two models are endmember models described in the text. For each composition, the first column indicates the solar ratio and 06 the second the corrected ratio when the minor elements (Ca, Al, 04 Ni) are replaced by the major elements as described in the text.

08 1 12 1.4 18 2 o o .= planetary differentiation suggest that the formation of the o ot- +, core of a planet during its accretion is the consequence of cU) Mg/Si t-a tU o TL parttal melting caused by the transfer of kinetic energy into 1 heat when the planet becomes Iarge enough. The melting than that silicates Fig. 2. The composition of the stars with orbiting planets com- temperature of iron alloys being lower of pared to the solar composition (square) and the enstatite endmem- and their density largeg Rayleigh-Thylor instabilities of the ber composition (filled square). The circle is the average value of dense iron-rich liquids lead to the formation of the iron cores. the relevant abundance ratios of all the stars in the study and the The Iarger the planet, the more iron can be segregated into cross shows the uncertainty. Domains for different values of the the core since the amount of gravitational energy available is core mass fraction are delimited by plain lines (I7o), dotted lines proportional to the mass of the planet. Although the informa- (20Vo), and dashed lines (407o). tion from the solar system gives only two points (Mars and Earth), the trend is at least observed with smaller Mg# for Mars than for Earth. The upper bound for the metallic core can be calculated if all the iron segregated out of the silicates position for the mantle. If ylnn and yuna are the iron content and migrated to form the core (Fig. 2). of each phase in the lower and upper mantle, respectively Returning to the composition of stellar atmospheres, (Thble 1), then measured abundances define a domain bounded by an Fe/Si ratio and Mg/Si ratio between 0.6 and I.7, and 0.8 and 2, respectively. Some stars a^re magnesium rich and are close to Mg #1'nffi"l -l-yrvr -l-yuu (1) the forsterite limit while other stars are close to the enstatite \^'^D / silicates limit (Grasset et al., 2009). The average stellar abundance ratios ate (Fe/Si = 1.1; Mg/Si = 1.3). By comparison, the We assume magnesium and are only found in the Sun is enriched in Si compared to most of the stars. How- upper and lower mantle so that the ratio Mg/Si for each ever, the uncertainties on stellar compositions are large. The mantle must equal the input ratio. If x* and Xuvr are the abundance in a given element X is usually the comparison of proportion of perovskite and olivine in the lower and upper the atomic ratio of the amount of this element to the amount mantle (Table 1), respectively, then of hydrogen (H) abundance with the solar ratio

€ Xru - 1- xvwl2 (2) rd-,osGt);,osGo)" (3)

For a given value of xrr*, the amount of iron is fixed in the The atomic ratios Fe/Si and Mg/Si can therefore be calculated mantle with equations (1) and (2).Iron being present in the using the following e ations silicate mantles and in the core, the ratio Fe/Si links the mass (and the size) of the metallic core (Mr) with XLvr. But there is only one set (xpy , Mt) that provides the correct mass fg =f9,otr.l-tsil andfYg^| =fYg^|10[Ms]-[s,] (4) for the whole planet. This is illustrated in Fig. 2, where the [si/* [si/" \si/* \si/o different lines illustrate different compositions depending on the value of both the Mg# and the relative amount of The composition of stars hosting exoplanets (Beirao et olivine. For example, if there is no core, the ratio Fe/Si of a al., 20(5; Gilli et al., 2006) is an example that provides the pure enstatite mantle is equal to (l-Mg#) whereas it is equal values of the (Fe/Si) and (Mg/Si) ratios plotted in Fig. 2. to (2-2xMg#) for a pure olivine (dunite) mantle. Models of These alues are compatible with planet core mass frac- Sotin et al.: Terrestrial Planet Interiors 379 tion between 207o and 40Vo and Mg# between 0,6 and 0.9 to plate tectonics. The relationship between convection and (Fig. 2). The average of the stellar values is (Fe/Si; Mgl plate tectonics is an active domain in Earth and is Si) = (1.0; 1.3).The solar Fe/Si ratio is similar whereas the discussed in section 4. For the purpose of modeling the solar Mg/Si is smaller (1.1). If an enstatite composition is planetary radius and the interior structure, it suffices to model assumed for the protoplanets that accreted to form Earth the temperature profile by a large temperature difference at (Javoy, 1995,; Mattern et a1., 2005), then the Mg/Si ratio is each layer interface and adiabatic temperature gradients in the even smaller and the mantle composition would be very close convective mantle and in the convective core. As discussed to the pyroxene end member (no olivine) with a 307o core in previous papers (Valencia et a1.,2006; Sotin et al., 2007), mass fraction. The exact composition of terrestrial planets the effect of temperature on the determination of the radius compared to that of iron- and rock-forming elements in their is small. On the other'hand, the temperature profile controls host star is debated. The uncertainties on the stellar values the amount of partial melt in the mantle, which in turn af- of [Fe], [Mg], and tsil are high, in the range 0.03 to 0.1, fects degasstng (Elkins-Tanton and Seage4 2008), viscosity meaning that the abundance ratios also span a relatively and convection (see section 5), and hence planet evolution. wide range. The elemental abundance uncertainties can be The adiabatic temperature profile is given by converted into uncertainties of 0.3 to 0.5 for the Fe/Si and Mg/Si ratios (Fig. 2). These values are on the order of the dT uT yt difference between solar composition and EH composition (6) pO (Thble 2), at least for the Mg/Si ratio. Uncertainty about the dP pCp iron content is the most critical for determining the mass- radius relationship because the molar mass of iron is much with y and @ the Grtineisen and the seismic parameters, larger that that of Mg and Si. respectively

2.2. Equations (7)

ltrS relation between the mass of a planet (M) and its radius (R) is derived from four equations: ,mass. conserva- K, tion, energy transport (generally described by an adiabat, o - -gl- (8) together with temperature variations across boundary lay- \PdP ers), , and the equation of state (the In equations (6)-(8) T, P, p, o(, Cn, and Ks are temperature relationship between density, pressure, and temperature for (K), pressure (Pa), density (kg/m3), thermal expansion co- a given material in thermodynamic equilibrium). efficient (K-t;, heat capacrty (JlkglK), and adiabatic bulk The expression for the total mass and densrty distribution is modulus (GPa), respectively. The temperature within each layer can be computed using equations (6)-(8). Within the necessary acsuracy of predicting compositions of exoplanets, M - +nJ*r2p(r)dr (s) the parameters are relatively well constrained for the candi- date phases (Thble 3) at pressure values relevant to Earth's case, with some exceptions. By applying equations (6)-(8) to where the density (p) depends on composition (section 2.I), a much larger pressure domain, expect important devia- temperature, and pressure. we tions of the computed thermal profiles compared to the real The temperature profile is determined by the equations profiles. But at the present time, it is not possible to provide describing heat transfer. Earth's mantle behaves like a highly an accurate model for planets much more massive Earth, viscous fluid over geologic timescales and heat transfer is than dominated by subsolidus convection. The temperature of because very few experimental and theoretical constraints exist for silicates under these conditions (Mosenfelder et aI., Earth's mantle is at most a few hundreds of degrees lower 2009; Umemoto et al., 2006). than melting temperature (Mosenfelder et a1.,2009).In more The pressure (P) is computed using the hydrostatic equi- massive planets, the temperature in the silicate mantle is librium equation controlled by the competition between the heating due to the decay of long-lived radiogenic elements and the cooling by subsolidus convection that is more and more efficient as dP(r) = -p(r)g(r) (9) temperature increases--and viscosity decreases (Tozer, I97 2). dr Although there is an extensive literature on convection processes, three-dimensional spherical models of thermal The remaining equation is the equation of state (EOS), convection in a volumetrically heated fluid with isothermal which links density to temperature and pressure. Two dif- boundaries are still limited by computer performance (see ferent approaches are commonly used in Eanh for section 4). Farth's internal dynamics are chalactenzed by describing the pressure and temperature dependences of hot plumes forming at the core-mantle boundary and cold materials (e,9., Jackson, 1998). One method introduces the sheets of oceanic lithospfnere recycled into the mantle due effect of temperature in the parameters that describe the min- 3 80 Exoplanets eral's isothermal EOS and is achieved using the third-order The third-order Birch-Murnaghan (BM) EOS is usually Birch-Murnaghan EOS with the thermal effect incorporated chosen for the upper mantle where the pressure range is lim- - using the mineral's thermal expansion coefficient ited to less than 25 GPa. Other EOS can be used, such as the Vinet EOS, which provides the same result as BM and MGD at low pressure, because the parameters entering into these equations are well-constrained from laboratory experiments. The Mie-Grtineisen-Debye (MGD) formulation is prefened for the lower mantle and core, as it permits a self-consistent approximation on the vibrational properties of the phases in the absence of limited experimental data. As noted rn Sotin et al. (2007), the electronic pressure term should become important at higher pressures, requiritrg a precise description. K?,0 = Ko * ep (r - rr) Within the temperature range of Earth's lower mantle, the thermal pressure term in the Mie-Griineisen-Debye formula- Ki,o - K6 tion provides estimates close to EOS derived from select ab initio calculations and shock experiments (Thompson, 1990). Pt,o In order to solve for pressure, temperature, and density in the interior model, an iterative process is implemented until 0T,o convergence is attained [see detailed descriptions rnValencia et aI. (2006) and Sotin et aI. (2007)1. As noted before, the The relation between pressure, temperature, and density is input parameters are mass, composition (Mg/Si and Fe/Si), then described using the eight pa^rameters known at ambient and proportion of iron in the mantle (Mg#). The model starts pressure, Ts, po, Ko, Ki,o, ap, ar, br, and co the reference with estimated values for the core. At the end of the iteration temperature, density, bulk modulus, pressure, and temperafure process, the Fe/Si ratio is calculated and compared to the derivatives of bulk modulus, and thermal expansion coef- stellar value. A new core mass fraction is estimated using flcients respectively (Thble 3). the bisection method. The code, as described rn Sotin et al. The second approach dissociates static pressure and ther- (2007), iterates the computation of temperature, density, and mal pressure by implementing the Mie-Griineisen-Debye pressure profiles until the difference between the computed formulation Fe/Si and the stellar Fe/Si is less than 0.5Vo. Super Earths are more massive than Eafth and the valid- ity of the EOS at much higher pressures and temperatures is P(p,r) - P'h,ro)l+ A questionable (Grasset et al., 2009; Valencia et al., 2009). The Vinet and MGD formulation appear to be valid up to 200 GPa p) (e.g., Seager et a1.,2007).Above -200 GPa, elecfronic pressure P (p, ro) - becomes an important component thatcannot be neglected. At ;",IftPo0), very high pressure (P > 10 TPa), first-principles EOS such as the Thomas-Fermi-Dirac (TFD) formulation can be used (e.9., Fortney et al., 20071' Seager et al., 2007). The pressure at the core-mantle boundary (CMB) of terrestrial planets 5 and 10 times more massive than Earth is equal to 500 GPa and 1 TPa, respectively (Sotin et aI., 2007).In this interme- diate pressure range, one possibility is to use the AMOS t't'dtl(.,_t code, which was developed to process shock experiments f" (Thompson et al., 1990). The study by Grasset et al. (2009) compares the density-pressure curves of iron and forsterite using the MGD, TFD, and ANEOS formulations (Fig. 3). The TFD formulation, as expected, predicts values of densi- ties much too small at low pressure. We emphasize that the TFD theory is for a pressure-ionrzed gas of noninteracting electrons; it cannot describe chemical bonds. In contrast, the ANEOS seems to fit the MGD at low pressure and the TFD at very high pressure. Therefore, the ANEOS appears to be where Ts, po, Ko, K6, 0oo, fl, y0, and q are the reference a good choice in the intermediate pressure range from 0.2 temperature, density, isothermal bulk modulus, pressure to 10 TPa. However, laboratory experiments approaching derivative of bulk modulus, reference Debye temperature, this very high pressure range are required to confirm which number of atoms per chemical formula, and scaling expo- phases are present at these high pressures and to determine nents, respectively (Table 3). the density of material. Sotin et al.: Terrestrial Planet Interiors 381

TABLE 3. Parameters describing the equation of state, bulk composition of the layers, mineralogical transformations, and thermal profiles in the planets (values based on Earth values).

Variable Layer I Layer 2 Layer 3

Equation of state EOS2 EOS2 EOSl Phases Iron-rich phase Layer composition (Vo) 100 Components Fe FeS Phase composition (Vo) 80 20 tr Density af ambient 8340 4900 conditiott Po :.3a ;^') o 4.,. Reference temperature (K) To 300 300 300 300 300 oti Reference bulk modulus rq 135 254.7 r57 128 105.8 U (GPa) +)C) Pressure derivative of the .r7 , d r\o 6,0 4.3 4.0 4.3 8.5 a:) bulk modulus +.ro Debye temp. (K) tr o( parameter (10-s K-1) 0g/a, 474/5.5 )zetz.z 936/3.8 7 57 /3.0 7 1013.2 o 5)d Gruneisen parameterl 5 d o parameter (10-s K-2) yoftr 1.364{A 2.23/NA 1.45A[A r.1J/0.74 1.0094[A E] Power exponent/ cx parameter (K) qlcr O.91ATA 1.83/I'.IA 3.0/l{A 0.54l-0.5 1.Oa{A Temperature derivative of bulk modulus -0.045 -0.02 -0.019 -0.016 -0.026 (GPa/K) References lrl,[2] [3], [7] l4l,ul tsl t6l Es Upper temp. drop (K) AT 800 300 t200 FE References t7l t7l UI f,il

References: lll Uchida et al. (2001); l2l Kavner et al. (2001); [3] Mosenfelder et al. (2O09) (BM3S model); l4l Hemley et al. (1992), Sinogeikin and Bass (2000), Sinogeikin et aI. (2000); l5l Duffy et al.'(1995), Bouhifd et al. U996); [6] Vacher et a/. (1998), and lackson (2002), Jackson et al. (2007); Ul Poirier (2000); l8l Irifune (1987). NA: Not applicable.

2.3. Earth and Solar System Terrestrial Planets tions (Fig. 2) compared to the difference between the solar composition and the terrestrial composition, the stellar compo- The planets Mars, Venus, and Earth are obviously better sition can be assumed as a good approximation. However, one known than any terrestrial-like exoplanet. In order to validate must note that this does not work for both Mercury and the the model approach described above, previous studies (e.g., , which are enriched and depleted in iron, respectively. Valencia et aI., 2006; Sotin et al., 2007) have compared the For Earth, the locations of the interfaces between layers are observed radius to the computed values obtained with the well constrained but the composition of the deep layers relies numerous approximations. For the three solar system plan- on the assumption that the bulk composition is chondritic. ets, the reference elementtry composition (Mg/Si and Fe/Si) ' In the results described in Fig. 4, the Fe/Si, Mg/Si, and Mg# is the solar composition (Table 2). The difference between are those of Model 1 given in Thble 2. The model fit is quite these extreme models is surprisingly less than IVo tn radius good for Earth's pressure and density, two profiles that are (Table 3), which is quite small. In order to investigate the provided by seismic inversion (Dziewonski and Anderson, influence of chemical composition, values of the radius have 1981). The value of the model radius is 43 km (0.77o) larger been computed using a nonsolar composition coffesponding than the value of 6371 km. Considering the assumptions made to tho composition of enstatite chondrites (Javoy, 1995, 1999), for this simulation, and in particular the fact that only four which represents another end member. In this case the amount elements are taken into account (Si, Mg, Fe, O), the model of Mg is much smaller, leading to a smaller value of the Mg/ flt is considered to be quite good. The temperature profile Si ratio (Fig. 2).The resulting difference is that planet radius computed for Earth is, as expected, close to the real one since is much less than l%o (Sotin et al., 2007). Similarly, if the information from Earth was used to calculate the temperature value if the Mg# is changed from 0.9 to 0.7, the change in proflle. It is interesting to note that the value of Fe/Si that computed planet radius varies by less than l7o (Sotin et al., would provide an exact flt to the radius is 1.10 (Thble 4), a 2007). Considering the uncertainties in the stellar composi- value that is I27o larger than the solar value. 382 Exoplanets

1000 3000 4000 5000 6000 7000 ) 2000 'iri!:l """"" "i" il t"-*'- TJ _j--i_ .:..i...... t... j-*-.'.' 1 it i It 4000 . .,-_i.

I i 4"tr ....Yt 12000 (f) cf) t !i I It * tl E E ,/ il 1 0000 o) tj I I' o, lz iil '''I l jz ; il i,i it ;:i ii 8000 ': i: ffi it r.l // tll .= .= .;--t.-i i"i" ''''I"'ltl T v) U) ) c 6000 c ii I r,t, tlil o it o .'l I \/ il o t:; , ti o I t il 4000 ! ii I il c iti : ti o i:i I L_-, L..,* :.:::'!:ri t tl *-*J- il I 2000 ir I it

It!i il r1 0 !t (r ri ) 400 350 (o 300 (L (, 250 o 200 f cf) 150 E a@ o) o 100 lz (L 50 .= c@ 0 oo 1000 2000 3000 4000 o 1 0000 o a o Fig. 4. Comparison between predictions and PREM for Earth tL with Fe/Si -0.987, Mg/Si - 1.136, and Mg# = 0.9. The value of the radius is 6414 km, which rs 0.67o larger than Earth's mean radius. The computed and measured profiles of both density and pressure cannot.be distinguished except in the solid inner core 100 1000 10000 for the density profile. The numbers I,2, and 3 refer to the core, Pressure (GPa) lower mantle, and upper mantle, respectively.

Fig. 3. Comparison between the different equations of state (EOS) described in the text for a pressure range up to 100 TPa. After Grasset et al. (2009). than Earth, leading to different internal composition.Apply- ing the model developed for Earth, therefore, does not yield accurate radii, giving a radius I2Vo too large for Mercury and 87o too small for the Moon. There are three scenarios that have The same model has been applied to Mars and Venus. For been proposed to explain Mercury's high density (Cameron et Venus, the Earth-like model (Mg# = 0.9) gives a radius only al., 1988): the equilibrium condensation scenario, planetary 5 km (0.1 Vo) lnger than the observed value. It suggests thet evaporation of the crust and part of the mantle, and a major the Earth-like model applies very well to Venus, an observation planetary collision. The vaportzation scenario (Fegley and that was already made by Anderson (1980). For Mars, taking Cameron, 1987) predicts elementary composition of the sur- a terrestrial value for the Mg# gives a radius that is 40 km face that will be tested by two forthcoming missions: (I.57o) smaller than the measured value (Thble 4). However, MESSENGER mission and ESA s Bepi Colombo mission. It the measurements made on SNC suggest that the is interesting to note that if Mercury had the same Fe/Si ratio Mg# for Mars is smaller than Earth's value (Dreibus and as the other terrestrial planets, its size would be 640 km less Wc)nke, 1985). If one takes Mg# - 0.7 then the radius is only than Mars. This suggests that even without vaporization, the 23 km (0.77o) smaller (Table 4).A smaller value of the Mg# planet closest to the Sun would have been quite small. For the means that less iron differentiated into an iron core, which is Moon, the situation is also different because the Moon formed in agreement with estimates of the core radius (Yoder et al., as Earth had already differentiated into an iron-rich core and a zD3).Values of Mg# on the order of 0.75 have been calculated silicate mantle. Therefore the Fe/Si ratio for the Moon is much in order to explain the chemical composition of the basaltic lower than the solar value. A value of 0.22 for the Fe/Si ratio meteorites supposed to have come from Mars (Dreibus and is found in order for the model to match the exact value of the Wcinke, 1985). Moon's radius (Thble 3). In this case, the Moon would have Mercury and the Moon, two objects much smaller than an iron core 405 km in radius. Such a value is in agreement Earth, are bodies that had very different accretion with values proposed by Kusl

TABLE 4. Comparison of the radius observed for the solar system bodies and the predictions given by the models.

Planetary Radius Best Fit

Name MassAvI6 Measured Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4

Mercury 0.055 2437 2705 2723 2706 27 t5 8 8 7.5 7.5 Mars 0.107 3389 3349 3366 3342 3357 0.78 0.84 0.7 r 0.79 Venus 0.81 605 1 6056 607 r 6008 6032 0.96 1.03 0.80 0.85 Earth 1 637 r 6414 6447 6379 6405 1.10 T.T9 0.92 0.99 Moon 0.0123 1738 1600 r642 r59r r6:2r 0.22 0.48 0.30 0.30

The different models coffespond to solar composition (models t and 2) and EH composition (models 3 and 4). Models 1 and 3 have a value of the Mg# equal to 0.9, whereas models 2 and 4 have a values of the Mg# equal to 0.7. The right part of the table indicates the required value of Fe/Si (Earth-like planet) in order to get the value of the measured radius for each body. Results from Sotin et al. (2007).

2.4. Exoplanets: Super Earths, Super Mercuries, (N4/N4CI)0.178 if one takes b as being equal to 0.274. A 10-M@ ) and Icy Planets planet would have a density 507o larg.or (Thble 6). The surface acceleration (not taking into account the centrifugal We now turrl to a review of exoplanet interior composition acceleration) increases as (M/Mo)l-2*b, which is equal to calculations. Although the topic is not as mature as interior (N4/-Mo)0.4s2. Therefore the gravity acceleration of a 10-Mo structure modeling for solar system terrestrial planets, a num- planet is about three times larger. It means that the pressure ber of fundamental issues can still be described. gradient in such planets is three times larger than on Earth, One goal in exoplanet interior modeling is to provide re- which has implications for the depth of phase transitions lationships between mass and radius. The studies by Valenciu and mineralogical transformations, which are pressure- et al. (2006) and Sotin et al. (2007) do this for planets up to dependent. Both the core and the mantle thickness increase 10 Mo. The mass-radius relationship can be written with increasing mass. On the other hand, the thickness of the upper mantle decreases because of the larger pressure gradient described above (Fig. 6). R/R*- u (rur/vr*)o (t2) The radius of the core depends on the EOS that one takes for iron and its state depends on the iron phase diagram at where (a,b) is equal to (1 ,0.274) for planets less than 10 M@. pressures relevant to the center of exoplanets. There is a Values of different studies are very close to each other (Ta- lack of experimental data to constrain the EOS of iron and ble 5), One can note that the more massive the planet, the silicates at pressures above 500 GPa (e.g. , Poirier, 2000; smaller the value of parametet "b" because the compression Valencia et al., 2009). Such pressures exist within planets due to the pressure effect becomes more and more important. that are only a few Earth masses. For example, the central Also, Grasset et aI. (2009) propose a more complex equation pressure of a 5-Mo planet is 1700 GPa (Sotin et al., 2007). ((b)) where the parameter depends on the mass of the planet. Therefore more reliable results have to wait for laboratory The values that they give are very close to those proposed by experiments that will confirm ab initio calculations (Volcadlo Valencia et al. (2006) although the EOS and the parameters et al., 1997; AW, 2009). Several studies have investigated the in those EOS were different. The mass-radius relationship for melting temperature of iron alloys in order to assess the state terrestrial planets can be plotted (Fig. 5). A 10-Mo planet is of Earth's iron core (e.g., Williams et al., 1987; Anderson and about twice as large and a 100-Ms planet is only 3.2 times Ahrens, 1994, 1996; Boehler 1993; AW,2009). The melting as large (Fig. 5). For an example of a study that considers temperature of Fe-FeS alloys is smaller than that of silicates mass-radius relationships for planet masses >10 Mo, see at the same pressure. It suggests that cores would be liquid Seager et aI. (2007). For a description of why mass-radius (Fig. 7). However, the large uncertainties of the composi- relationships for exoplanets of different internal composition tion of metallic cores (e.g., Poirier 1994) and the lack of follow a similar functional fonn, see the detailed discussion experimental databeyond 300 GPa make any conclusion very rn Seager et al. (2001) tentative. The state of the iron core is a critical parameter in Another goal of exoplanet interior modeling (using the order to assess the existence of an internal that mo$eling approach described in section 2) is to investigate would protect the atmosphere from bombardment of energetic the mass dependence of several planetary characteristics such stellar ions that would efficiently erode it. as mean density, radius of the core, depth of the upper-lower Another parameter that is important to consider when mantle interface, equilibrium heat flux, and other properties. simulating the thermal evolution of terrestrial planets is the The mean density being the mass divided by the volume, radioactive heat flux. On Earth, about 507o of the heat flux its dependence varies as (MAvIe)1-3xb, which is equal to comes from the decay of the long-lived radioactive elements 384 Exoplanets

TABLE 5. Coefficients a and b of the law (R/Rs) = a(M/I{e)b for the different cases explained in the text.

MAvIe Terrestrial Planets Ocean Planets (50 wt%o) Super Mercury Range of mass ab ab a 0.01-1* 1.00 0.306 r.258 0.302 1-10* 1.00 0.27 4 t.262 0.27 5 1-101 1.00 0.267-0.272 -0.30 1-100+ 1.01 0.252-0.285 r.253 0.239-0.212

*Sotin et al. (2007). fVakncia et al. (2006). {Grasset et al. (2009).

40K 235U, 238U, 232Th. such as , and The other 507o is the Calculations for a planet that contains 50Vo HrO by mass cooling of the planet, which depends on the efficiency of (Fig. 5) suggest that the radius would be more than 25Vo mantle convection (see section 3). Assuming that the amount larger compared to a planet with an Earth-like interior com- of radioactive elements is proportional to the mass, then position (Sotin et al., 2007). Grasset et al. (2009) expanded the radiogenic heat flux varies as (MAvle)j-4sz (Fig. 8 and the calculations and provided relationships for any amount Table 6). The radiogenic heat flux is equal to the radiogenic of HrO. With anHr[He envelope of about I}Vo by mass, and heating rate divided by the surface area. The heat flux scaling assuming that Uranus and Neptune are otherwise composed results from the radiogenic heating rate being proportional to of water, the Uranus and Neptune HrO mass fraction would planet mass and the planet mass being proportional 16 ft0.274. need to be as large as 7 5Vo in order to flt the mass and radius. The existence of icy planets such as more massive versions The choice of EOS (equation ( 1 1)) for the high-pressure of Ganymede is very likely. It has been proposed (Kuchner phase of ice is discussed in Sotin et al. (2007). It must be 2003; Ldger et al., 2004) that some of these planets could turn noted that they use a value of q - 1 instead of the surpris- into water planets. These planets would have formed in the ing negative value published by Fei et al. (1993) in order to outer stellar system at a distance where HrO ice can condense compute adiab attc profiles at pressures larger than 60 GPa. and would have then migrated by a type 2 mtgration toward Another implication of the presence of the thick H2O layer their star. The recent discovery of OGLE-2005-BLG-390Lb is that the pressure at the interface with the silicates is larger (Beaulieu et al., 2006) is a possible example. than the pressure at the upper-lower mantle interface. There- fore the upper mantle layer would not exist in such planets. If the surface temperature becomes larger than 300 K, ice melts and an ocean forms. Because the melting temperature of the low-pressure phase of ice (ice I) decreases with increas- ing pressure, the depth of the ocean is limited by the melting temperature of the high-pressure phase of ice. Sotin et al. (2007) have investigated the thickness of an ocean layer as a function of the mass of the planet. For aHrO mass fraction of 50%o, the thickness varies from 47 5 km to 50 km for planets 1 and 10 times Earth's mass, respectively. The smaller thick- ness of the ocean with increasing mass is due to the pressure gradient effect described earlier. Other parameters such as heat flow, surface temperature, solutal convection, latitudinal circulation, superrotation, and others may affect the thickness of the ocean. As noted in Ldger et al. (2004), for such large amounts of HrO the liquid ocean is not in contact with the silicate shell. It means that if volcanism exists in the silicate 100 shell, this volcanism will be in contact with high-pressure ice and not with liquid, which gives conditions quite different from those existing on Earth's seafloor. Fig. 5. Mass-radius curves for planets with Earth-like composi- Application of this model to the large icy of Jupiter tion and for water planets. The amount of HrO varies from }Vo (almost Earth-like) to l00%o. The large icy satellites of Jupiter and and Saturn gives very good results, although these moons Saturn have about 50Vo water by mass. The two circles coffespond have masses much lower than Earth's mass. The icy moons to CoRoT-7b and GJ 12I4b, the smallest planets for which both Ganymede, Callisto, and Titan have an HrO mass fraction mass and radius have been determined; the size of the circle ap- close to 507o (Sotin and Tbbie, 2004).If a three-layer struc- proximately represents the uncertainty in interior composition. ture is assumed (silicate mantle and alayer with the density Sotin et al.: Terrestrial Planet Interiors 385

TABLE 6. Values of the core radius, upper mantle thickness, nantle thickness, and radioactive flux for planets with different masses.

Mass Relative Density Radius Core Radius Upper Mantle Mantle Radioactive Heat to Earth (kg/m3) (km) (km) Thickness (km) Thickness Flux (mWm2)

1 55 15 6422 3074 658 3348 47 2 6239 7866 3698 494 4168 64 3 6706 8838 4t04 416 4734 77 6 7 587 10730 4870 310 5860 106 10 8309 r2308 5486 248 6822 r33

of ice VI covered by a layer with the density of ice I), the Here we review the major concepts and definitions rele- calculated radii of the large icy moons match the actual radii vant for plate tectonics. The different assumptions and pa- by better than IVo. Sotin et al. (2007) have also applied this rameters used in each study are first described. The definition model to Uranus and Neptune. These outer planets have of,then1al boundary layer and lithosphere is clarifled. It is a thick hydrogen-rich outer layer that is taken care of by confirmed that scaling laws predict that the more massive reducing both the radius and the mass by 1 R* and 2 W, the planet, the more likely the occuffence of plate tectonics. respectively- (Hubbard et aI., 1995; Podolak et a1.,2000). The No scaling laws exactly describe the numerical calculations curves on Fig. 5 show that if composed predominantly of H, reported in the study by O'Neill and Lenardic (2007), where He, and II'O, these planets must contain more than 50Vo mass the vigor of convection of a fluid heated frorn both within fraction of H2O. Not shown is that Uranus and Neptune may and below is examined. The lack of information on scaling alternatively have interiors dominated by rock, with much laws describing velocity and the ratio of driving to less ice and a more massive FIAIe envelope. resistance is emphasized. Finally, simulations of the thermal So far iron-rich, silicate-rich, and icy planets have been oVolution of terrestrial planets (e.g ., Papuc and Davies,2003) described in detail, without regard to how to identify them. are described. In reality, deducing exoplanet interior composition is very difficult. Only two datapoints are avatlable per planet: mass 3.L. Mantle Convection Processes in and radius. Even with perfect measurements there is simply Terrestrial Exoplanets not enough information:to uniquely identiff a planetis interior composition. Two exceptions are at the density extremes; gi- A planet's internal dynamics are driven by solid-state ant planets of low density must be composed almost entirely convective heat transfer in the mantle. Heat sources include of H and He, and any planet of extremely high density must radiogenic due to the decay of the long-lived be iron-dominated because iron is the densest cosmically radioactive blements 40K, 2321J,235U, and 242Th, and the initial abundant substance. See section 5 for further discussion of heat stored in the planet during accretion and differentiation. interior composition degeneracies. The ,convective processes in the mantle control the thermal evolution of the planet (e.g. , Schubert et al., 2001), In a fluid 3. DYNAMICS that is heated from within, cooled from the top (cold surface temperature), and heated from below (hot corq), cold plumes The likelihood of plate tectonics on exoplanets larger than form at the upper cold thermal boundary layer and hot plumes Earth can be assessed using either scaling laws or numerical form at the hot thermal boundary layer, which coffesponds to models describing mantle thermal convection. TWo papers, the core-mantle boundary (e.g., Sun et al., 2007). The efficien- Valencia et al. (2007) and O'Neill and Lenardic (2007), came cy of heat fransfer is mainly confrolled by the mantle viscosity, to opposite conclusions based on scaling laws and numerical which depends on a number of parameters i4cluding, but not calculations, respectively. Valencia et AL (2007) conclude limited to, mineral composition of the mantle, temperature, that as planetaw mass increases, the shear stress available pressure, and grain size. Laboratory experiments (Davaille to overcome resistance to plate increases while the and Jaupart, L993) and numerical studies (Solomatov and plate thickness decreases, thereby enhancing plate weakness. Moresi,2000; Grasset and Parmentier, 1998) have shown the These effects contribute favorably to the subduction of the major role of temperature-dependent viscosity. Scaling laws lithosp$ere, an essential component of plate tectonics. On the have been derived and then employed to predict the thermal other hand, the numerical simulations described by O'Neill evolution of exoplanets (e.g ., Valencia et al., 2007; Papuc and and Lenardic (2007) suggest that increasing planetary radius Davies,2008). Scaling laws can also predict the velocity and acts to decrease the ratio of driving to resisting stresses. hence the shear sfress acting below the lithosphere. If the shear They conclude that super-stze:d Earths are likely to be in an stress is large enough, it may overcome the resistance of the episodic or stagnant lid regime. lithosphere and produce faulting. This would trigger the plate 386 Exoplanets tectonics regime as described in both studies by O'NeilI and ( 13) Lenardic (2007) and Valencia et al. (2007). dl-n('q)laT Scaling laws based on the stability of the thermal boundary layers are used to describe the heat that can be removed by convective processes. Laboratory experiments (e.g., Davaille where the viscosity n strongly depends on temperature. The and Jaupart, 1993) and numerical simulations (Grasset and viscosity of Earth's mantle can be determined by studying the Parmentier 1998) have shown that the temperature differ- postglacial rebound and the temperature of the mantle (T,o) ence across the upper thermal boundary layer (ATrsL) is is estimated around 1350"C at 80 km depth from studies of proportional to a viscous temperature scale (ATn) defined by mantle rocks sampled by kimberlites (e.g., Bertrand et al.,

Terrestrial Planets Ocean Planets 4567 567 2500

o o 2000 (L (,IL 2500 (, 1 500 2000 t-o o f, a 1 500 a 1 000 @ Cn o 1 000 o IL (L 500 s00

0 0 25000 25000

20000 cf) 20000 cf) E E o) o) 1 5000 IZ 1 5000 .\z .= .= 1 0000 @ 1 0000 a c c o o 5000 o 5000 o 0 0 1 6000 1 4000 1 4000 1 2000 12000

1 0000 E E 1 0000 Y .:< 8000 8000 a U) J o 6000 E 6000 $ o 4000 t 4000 t 2000 2000 0 0 8000 8000 7000 6000 \< 6000 Y o o t- 5000 4000 o 4000 G ot- o o- 3000 o- E 2000 E 2000 F_o g 1 000 0 0

Mass/M6 Mass/Ms

Fig. 6. Pressure, density, radius, and temperature of planets with Earth-like composition and water planets (with 50Vo HrO by mass). The mass of exoplanets varies from I to l0 Mo. From Sotin et al. (2007). Sotin et al.: Tbrrestrial Planet Interiors 387

1986). It seems relevant to use q (1350"C) = I02r Pa s-1 as an anchor point for the viscosity law. Laboratory experiments ATn = (1s) suggest that the deformation rate of a solid is a thermal$ acti- vated process that canbe described by an Arrhenius-type law Different viscous laws are used to describe how viscosity ql.r-r] depends on temperature. Valencia et al. (2006) use the ex- a ro) pression Tl = lo(T/Ts)-3o, which gives a viscous temperature n - Aeff or n - IoeR[r ;4) scale equal to T-/30. The two viscous temperature scales are equivalent at low temperature but differ significantly atlarge where Q is the activation energy (Q = E + P AV), R the gas temperatures with the Arrhenius-type law, giving a viscous constant, and (no,To) - (IQzt Pa s-1; 1350'C). The viscous temperature scale l57o and 807o larger at a temperafure equal temperature scale' is therefore to 1350'C and 2250"C, respectively. As illustrated in Fig. 9, the thermal boundary layer is under the conductive lid. Its thickness 6 is controlled by the value of the thermal boundary layer Rayleigh number (Rarsl) deflned as

Rur"r=# (16) 6O Go rL= where cr is the coefficient of thermal expansion, p is the den- _9F sity, ATot = 295 ATn - 2.25 RT,?/Q, and r is the thermal io--(I) diffusivity. The value of the thermal boundary layer Rayleigh Etr. number ,fi9 depends on the boundary conditions but numerical simulations and laboratory experiments suggest that a value of 20 is adequate. This value must not be confused with the value of the critical Rayleigh number for convection to occur, which is around 1000. For a given value of the mantle tem- perature T*, one can calculate the viscosity (equation (14)), the thickness of the thermal boundary la5rer (equation (4)), and the temperature across the thermal boundary layer. This provides the heat flux (q) that is transferred by convection 65 OUUU Eg lI +, (17) ooc E 4000 ocoo- obF

10 100

@ Pressure (GPa) EX -Y LL .9 +' -cOt t-o I Fig. 7. Comparison between temperature profiles and melting curves of H2O and iron alloys. Melting domains of iron (Wil- 4a liams et al., 1987; Anderson and Ahrens, 1994, 1996; Boehler tE6 1993; AW et aI., 2002) and ice VII (Mishima and Endo, 1978; Fei et al., 1993; Frank et aI., 2004) are plotted as a function of pressure. Above these domains, the component is always liquid. 1 (a) Eath-like planets: Thermal profiles for 1,5, and 10 M6 are Mass above the melting curve of the FeS component in the core (P,T) domain. (b) Ocean-planets: The transition from liquid to ice VII occurs at low pressure (between 1 and 2 GPa depending on the Fig. 8. Equilibrium heat-flux, radius, and thickness of the upper surface temperature). The iron core is probably liquid, but what- mantle thickness for different values of planet mass. The equilib- ever the , it is colder and at lower pressure than rium heat flux is the internal radiogenic heating rate divided by the for the Earth-like case. surface area. Scaling relationships are nofinahzedto Earth values. 3 8 8 Exoplanets

where k is the thermal conductivity. In order to compare the The set of equations (13)-(18) can be used in order to vigor of convection for different planet sizes, it is assumed predict the thermal evolution of an exoplanet (Papuc and - that there is equilibrium between internal heating (radiogenic Davies, 2008). The determination of the Rayleigh number heating) and heat flux. As described above, the surface heat (equation (18)) is used to estimate velocities and stresses flux varies almost as the square root of mass (Mo.+sz;. One below the lithosphere and to assess the likelihood of plate must note that in this surface heat flux expression, the secular tectonics. cooling is omitted. Taking an Earth-like planet without plate tectonics and 3.2. Scaling Laws: Relations Between the Yigor of the concomitant efficient heat removal as a reference, the Convection and Plate Tectonics equilibrium heat flux would be 47 mWm2, which is the heat removed by convection if the viscosity is equal to 1 .7 x The lithosphere breaks when the stresses induced by 101e Pa s-1. Such a low viscosity corresponds to a mantle convection become larger than the yield stres s (Moresi and temperature of 1925 K (1650'C). As discussed in Papuc and Solomatov, 1998). This approach was used by O'Neill and Davies (2008), such a high value of the temperature produces Lenardic (2001).In order to compare with the study of Va- a lot of partial melt in the mantle. This temperature may lencia et al. (2007), scaling laws for stresses are now derived. reflect the present time state of Venus as previous numeri- The velocity (u) of the plumes and the horizontal velocity cal studies suggested (e.g., Arkani-Hamed, 1994; Sotin and within the boundary layers depend on the vigor of convection Labrosse, 1999; Moresi and Solomatov, 1998). (Schubert et al., 2001). Numerical simulations suggest that Then the thickness of the lithosphere (D) can be calcu- the velocity scales to the Rayleigh number (equation (18)) as lated, assuming continuity of temperature and heat flux at the conductive lid/thermal boundary layer interface (Fig. 9). tc u - o.r2 Razt3 ( 1e) It is interesting to note that the mass dependence is the same (b-D) for the thickness of both the thermal boundary layer and the lithosphere thickness (M-0.3e). The lithosphere thickness Applied to an Earth-like planet without plate tectonics, decreases with increasing mass because the heat flux (equa- equation (19) gives a value on the order of 1 cm/yr, which tion (I7)) is larger for planets of increasing mass (Fig. 10). is compatible with values of plate velocities on Earth. The Rayleigh number can be calculated using (b-D) as the By equilibrating stresses on the lithosphere, the normal thickness of the convective layer where b is the thickness stress (o) applied on the lithosphere by the convection pro- of the mantle cess can be written

crpslTto, (b - D)' L Ra- ( 18) o-x- (20) KNG;) D

where t is the convective shear stress affecting the base of the lithosphere on a distance L, which is the width of the convective cell. This length (L) is also the distance between Temperature hot plume and cold plume, or half the distance between 0.25 0.50 T 0.75 0

01

02 Driving/Resistance (V) t' 03 = 1M/M6,)o

-c 04 o- E05 Driving/Resistance = (M/Me)o'u 06 p = (M/Me)o*o 0.7 ;- 2 i- ; -; ; 8 ;-;; 0.8 '

0.9 Fig. L0. Ratio of driving forces to resistance (yield strength) for two plate tectonic studies (O'Neill and Lenardic, 2007; Valencia et al., 2007). The ratio is much higher in Valencia et al.'s study due to alarger shear stress and alarger distance between plumes. Fig. 9. Schematic view of the temperature profile in a fluid heated Note that the trend for the lithosphere thickness is similar for from within and from below. both studies.

!- Sotin et aI.: Tetestrial Planet Inteiors 389 the cold plumes if no hot phimes are present in the case of lid state. When the convection pattem turns from active lid convection processes in a volumetrically heated fluid (e.g., regime into the conductive lid regime, the amount of heat Parmentier et al., 1994). The shear stress can be written that can be removed is much less. Little is said tn O'Neill (Schubert et al., 20Ol) and Innar ic (2007) about the change in heat flux. However, their Fig. 1 shows that the conductive lid is thick and that (2r) the heat flux must quite small. In the case of mixed heating, one can use the isoviscous scaling laws proposed by Sotin and l-abrosse (1999) Followrng Valencia et al. (2006), the length of the plate (L) can be defined as the time it takes for the thermal boundary (2s) layer to reach its thickness 6 and it is found to increase with mcreasmg mass

T T-T | =,u62 = 140'32 (22) _ -T (26) K q o#_ o. 3446k#(e)*,, (*u)',,

Therefore, one can determine the value of the normal stress, where which balances the shear stress, and it is found that it in- crps(Tl T. D)' creases with increasing mass Ra- - )(b - KN(L)

g=,t =1408o (23) 6 and rr _ H(u-D)' ll(r r - The convective stress (equation (23)) has to be compared k(Tl -T.) with the yield stress, which increases with increasing mass because of the pressure gradient effect mentioned above. The with H the volumetric heating rate in Wm3. In order to solve ratio of these two stresses increases with increasing mass for T., Trr,, D and 9,4 fourth equation is needed, which is the (Fig. 10) as Mo'83. temperature difference across the boundary layer given by The next part of the present study is to scale the numeri- equation (13). When these equations are used, the heat flux cal experiments of O'Neill and'Lenardic (2007) (in order to is a factor of 4lower for the conductive lid regime than that compare with Valencia et al., 2007). First, one must list the for the active tid regime and the temperature of the convective hypotheses of their simulations, which include the following: mantle increases to reach a value larger than 1500 K. The heat 1. Although the viscosity law is not given rn O'Neill and flux is smaller than the heat'flux necessary to transfer just Lenardic (2007), they use the model described in the study the radiogenic heating. It results in an increase of the mantle by Moresi and Solomatov (1998), in which the viscous law temperature until the equilibrium is reached. The heat flux is the Frank-Kamenetskii approximation described by predicted by this scaling is almost identical to the one pre- dicted by scaling laws for fluids with temperature-dependent viscosity because the viscosity variations in the convective (24) domain are small. When this scaling is applied, the ratio of driving to resistance still increases with increasing where tlo is the viscosity at the surface temperature (To = mass (-Mo.z6) but with a much smaller coefficient (Fig. 10). 300 K) and y is a constant set to 14 (Moresi and Solomatov, The fact that the two-dimensional numerical experiments 1998). Therefore, the viscous temperature scale is equal to of O'NeiIl and Lenardic (2007) predict the transition from (Tr-To)/y. Since the basal temperature seems to be the same the conductive lid regime into the active lid regime (plate whatever the scaling factor, the viscous temperature scale tectonics) at different mass than the predictions of scaling and therefore the temperature difference in the cold thermal laws is not surprising. First, scaling laws are one-dimensional, boundary layer is the same and equal to 235 K. whereas the brittle failure of the lithosphere may be easier at 2. The aspect ratio has been set up to 4. In their Fig. I, either upwelling plumes or downwelling plumes that can only the conductive lid regime leads to a much higher internal be realistically accounted for by three-dimensional models. temperature than that in the plate tectonics regime. The Second, the scaling laws describing the horizontal velocity number of cold plumes is equal to 7 and no hot plume can in the thermal boundary layer (equation (19)) are based on be seen. This is in agreement with the work by Parmentier simulations for isoviscous fluids. It is therefore important and Sotin (2000) for a volumetrically heated fluid and with to determine the scaling laws for fluids having complex the predictions of the scaling laws. velocities. Third, the length of the plate plays a major role 3. The heat flux is set up such that the ratio of internal in the scaling but it is poorly constrained. For example, the heating to basal heating ts 64-36Vo for the reference active- terrestrial plates are longer than the proposed scaling (equa- 390 Exoplanets

tion (22)) and small-scale convection has been proposed as TABLE 7. Low-mass exoplanets with known radius. the oceanic plate thickens (e.9., Davaille and Jaupart, 1994). The conclusion about plate tectonics is that the scaling Mass (Me) Radius (R@) Distance to the Star (AU) of the driving to resistance forces is not yet avatlable to de- CoRoT-7b 4.8 (<11) r.72 0.017 termine whether planets more massive than Earth are likely GJ 436b 22.8 4.92 0.0287 to have plate tectonics. Further simulations are required and GJ 1214b 6.55 2.68 0.0144 the effects of different parameters including the , the thickness of the upper mantle, and the presence of wa- ter, the viscosity, and composition of the mantle need to be 4.1. Exoplanet Internal Structure: The Case investigated. of CoRoT-7b

4. RECENT HIGHLIGHTS Searching for transiting planets, the Convection and Transit (CoRoT) (Rouan et al., 1999; Borde Out of the hundreds of known exoplanets, only a few have et a1.,2003) reported a planet with a radius equal to 1.7 Ro a mass less than 10 Mo Gig. 11). The HARPS instrument (Ldger et al., 2009) and a mass lower than 11 Mo. This dis- found a planet with a of 1.9 Mo (Mayor et covery was followed by HARPS observations to reflne the al., 2009).In this GJ 581 system, three other planets were value of the planet mass to around 4.8(*0.8) Mo (Queloz et already known to around this M dwarf, including two al., 2009). This planet, known as CoRoT-7b, has a period with a minimum mass less than 10 Mo. Another important of 0.854 days around a G9V star. Such a planet is on the discovery is the detection of OGLE-2005-390-Lb by gravi- Earth-like planet mass-radius curve (Fig. 5). If the radius is tational lensing around an M star (Beaulieu et al., 2006). I.72 R@,the mass should be 7.24 M@ according to the model This planet would have a mass equal to 5 .5!Z.l times Earth's for Earth-like planet composition described in section 2. mass. It its star at about 5 AU, a distance where the Considering an uncertainty of 0.13 on the radius, the mass blackbody temperature is about 50 K. While measurements uncertainty would be around 0.45 Mo (Grasset et a1.,2009). of both mass and radius is available for almost 100 planets CoRoT-7b has a lower mass than needed for an Earth-like (Fig. I2), only three of these planets have masses much composition, which suggests the presence of low-density lower than 100 Mo (Figs. 12 and 13): GJ 436b, GJ I2I4b, material. Due to expected for a planet and CoRoT-7b (Table 7). CoRoT:-7b lies on the Earth-like in an 0.8-d orbit about a Sun-like star, it is hard to imagine composition curve, whereas GJ 436b and GJ I2I4b have how CoRoT:7b could have retained a thick atmospheric layer much lower densities, which make them consistent with ice of H, and He because the planet orbits so close to its host K giants. However, these two planets could instead have FV star: 0.0112 AU (see also the discussion on degeneracies in He envelopes surrounding a predominantly rocky interior. section 4.2).If there is little to no atmosphere, it would imply the presence of HrO at depth. To further expand on CoRoT: 7b's density, aplanet with (mass; radius) = (4.8; I.72)shas a density of 5.2I. However, an Earth-like composition planet that size would have a density of L85 due to the effect of compression from pressure. With these numbers, the surface 1 0000 gravity is equal to 16 mls2. -c f. G 1 LU 000 € + o $++ o

(u 100 ?' x. t8 xxffx -c o y.Y, l- t G a 16 LIJ a .t\ X 15 (U 10 xx o XX o tE 10 1 o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t @ f Distance to the Star (AU) 1) \'h to

10 100 1000 10000 Fig. 11. Mass vs. semimajor axis for all planets observed by transits (filled diamonds) and for the smallest planets observed by Mass Relative to Earth any detection method (stars). Only three planets less than 100 Mo have been observed by transit and radial velocity (CoRoT-7b, GJ 1214b, and GJ 436b). Fig. 12. Mass vs. radius for planets observed by transit.

i Irl t Sotin et aI.: Terrestria! Planet Interiors 39I

CoRoT:7b is so close to its star that it must be locked in at t = 4.5 G.y. and 510 TW at t = 0. The equivalent heat flux synchronous rotation. The dayside could have temperatures is equal to 115 and 345 mWm2, respectively. If convection as high as 2000 K whereas the nightside could be very cold processes can remove more heat than'produced, then the depending on how the atmosphere carries the heat all around planet can cool down. Otherwise it would keep heating up the planet. Such a high temperature is quite incompatible with until convection processes are vigorous enough to remove the presence of H2Oa Also, the escape tate of gaseous species more heat than radiogenic heating (or other internal heat must,be quite high. Building an interior model for CoRoT-7b sources). With a surface temperature of 2000 K, it is likely is a difficult task if one does not know the atmospheric com- that a ocean would be,present on the dayside. Also, position. For illustration, an-Earth-like composition planet the tidal dissipation may provide an additional heating source. size of CoRoTTb would have an iion core about 5100 km in Plate tectonics is difficult to imagine in such a context, radius and a silicate mantle about 6000 km thick with a thin because the sfrong heating implies that the ptrates may not upper mantle of only 300 km, which is more than two times exist or would be too ductile. CoRoT:1b may be volcanically thinner than Earth's upper mantle. Such a planet would have active due to tidal stresses and may resemble,Io, Jupiter's a radiogenic heating rute equal to I70 TW (24TW for Earth) rnost volcanically active moon.

4.2. Degeneracies: Example of GJ I2l4'

Tn-e above discussion on CoRoT-7b illusffates the issue of degeneracy since it is not possible to determine which light component is present to explain CoRoT-7b's density. The interior composition degeneracy can also be illusfrated with GJ l2l4b, discovered by Charbonneau et aI. (2009). The planetary mass and radius are in agreement with a com- position of primarily water enshrouded by a FI/IIe envelope. GJ I2I4b has a low enough density (p = 1870 + 400 kg m-r; that it cannot be composed of rocky and iron material alone (Fig. 13). In fact, the planet cannot be composed of pure water ice (Fig. 5). The planet almost certainly contains a gas component. Rogers and Seager (2010) have investigated three dif- ferent possibilities for the interior composition that match the GJ L2l4b rneasured mass and radius. One possibility is a planet with an iron core; silicate mantlo, and water outer layer; topped by a thick FI/IIe envelope up to 6.8Vo of the planet mass. A second possibility is a planet with no inte- rior water, but with an iron core, silicate mantle, ffid more massive FI/IIe envelope. A thfud possibility is a water planet that requires at least 47Vo water by mass and has a massive steam atmosphere. The identification of the gas component would provide important constraint. Specifically, a measured scale height from ffansmission spectra (see chapters by Winn, Burrows and Orton, and Meadows and Seager) can in principle distin- guish between an envelope or atmosphere dominated by light elements such as H and He and an,atmosphere composed of water vapor. The details of the interior composition, horvever, will likely remain unknown. For a funher description of interior degeneracies and how to quantify them, see, e.8., Valencia et al. (2007), Adams et al. (2008), Zeng and Seager (2008), and Rogers and Seager (2010).

4,3. Laboratory Measurements

Although our knowledge of the phase behavior and the Fig. L3. Mass-radius curves for different types of planets (from coffesponding EOS at very high pressures and temperatures Grasset et al., 2009). is still limited, there have been some recent breakthroughs 392 Exoplanets

that allow us to better chancteraze the interior structure of with a thick FI/FIe atmosphere or envelope based on a mass terrestrial planets more massive than Earth. Recent progresses and radius measurement alone. An atmosphere measurement in high-pressure experiments, especially at advanced radiation could help discriminate between the two via the scale height: sources, have produced more accurate EOS for candidate A steam atmosphere would have a much smaller scale height materials within such planets (Angel et al., 2009; Jackson, than aHlHe atmosphere. A major cautionary note is that the 2010). For example, the electronic spin crossover occurring atmospheres of solar system planets do not really correlate in the Fe2* component of (Mg,Fe)O has been shown to be with interior composition. Careful work must be done to associated with a density increase of -37o (Badro et al., understand which types of planet interiors can be constrained 2003; Lin et al., 2005), occurring over a wide temperature further with which kind of atmosphere measurements. interval (Sturhahn et al., 2005). Such an effect has been A second path forward from the interior composition arnbi- shown to have dynamical implications (Bower et aI., 2009). guity is statistics. With enough planets with a measured mass The volume-compression of (Mg,Fe)O with various iron and radius (perhaps dozens or more), the hope is that specific concentrations have been measured with high precision up planet populations in the mass-radius diagram (Fig. 13) will to pressures of -130 GPa, showing complex EOS behavior emerge. With distinct planet populations, characteristics of ter- near the spin transition that cannot be explained by Birch- restrial planets in general can be identified, even if the actual Murnaghan or Vinet formalisms (Lin et al., 2005; Fei et aI., composition of individual planets cannot. With even more 2007; Zhuravlev et al., 2009). At the core-mantle boundary optimism we hope to see distinct exoplanet populations in the region in Earth (P - 115-135 GPa and T - 33004300 K), mass-radius-period parameter to be able to form general silicate perovskite is likely to transform into postperovskite statistical statements about planet formation and migration. with a -l.S%o density increase (llfiurakami et al., 2004). A1- NASA's Kepler has presented early results though this region occupies only -37o of the total volume of (Borucki et al., 2010) for 300 planet candidates orbiting faint Earth, similar regions inside larger terrestrial planets occupy (V = I2-I4) host stars. A major challenge for Kepler is more than 507o of the total volume (Fig. 6). Experiments follow-up radial velocity for mass measurements and planet approaching such conditions have also shown that there may verification; many Kepler planet candidate host stars may exist significant postperovskite-like structural variations and simply be too faint. Borucki et al. state that half of the 2010 chemically heterogeneous phase assemblages (Murakami et announced planet candidates might be false positives. There al., 2004; Mao et al., 2004; Oganov et al., 2005; Tschauner is, however, still value in planet radius as a function of period. et aI., 2008; Catalli et al., 2009; Wicks et al., 2010). At Planets over a certain size cannot be predominantty rocky, extremely large compressions, first-principles calculations regardless of their mass. A relevant early Kepler result that show that postperovskite disassociates into MgO and SiO2 should stand regardless of the false positive rate is that for at -10.5 Mbar (1.05 TPa) at 5000 K (Umemoto et al., 2006). periods shorter than 30 days, the majority of exoplanets are This pressure is easily achieved in the mantle of terrestrial found to be Neptune-sized and smaller, and not Jupiter-sized. planets or icy planets more than 8 Mo. In the area of terres- The link between thermal convection and plate tecton- trial-like cores, the isothermal EOS of hcp-structured iron has ics is still an actwe topic in . When did plate been measured with similar precision to I97 GPa at 298 K, tectonics start? Has it been present since the end of accre- yielding &r = 163.4 + 7.9 GPa, and dMP - 5.38 + 0.16 tion or did it happen much later in the ? (Dewaele et al., 2006). Recent progress tn ab initio calcula- Numerical simulations combined with better knowledge of tions using quantum Monte Carlo techniques suggests that the physical properties of mantle rocks might help answer hcp-structured iron has a relatively high melting temperature these questions. The continuously improvittg computing at Eath's inner-core outer-core boundary (P = 330 GPa) of power makes possible simulations with smaller and smaller 6900 + 400 K (Sola and AW, 2009). grid sizes, which become appropriate for describing the thermal boundary layers of Earth-like planets. The strongly 5. FUTURE PROSPECTS temperature- dependent characteri stic s of mantle minerals and rocks, such as viscosity, are also better handled with smaller A major limitation for terrestrial planet interior compo- grid size and therefore a larger number of grid points. Ai- sitional chanctenzation is the fact that only two data points curate knowledge of the appropriate phase assemblages and are available per planet: mass and radius. Even with perfect their EOS at very high pressures and temperatures is still measurements there is simply not enough information to limited; however, rapid progress is occurring (section 4.3). uniquely identify the planet's interior composition. (One As described in section2, the postperovskite phase is present exception is iron-dominated planets, because a very dense at conditions near Earth's core-mantle boundary. On larger planet has no other alternative.) terrestrial planets, this phase or its disassociated phase may There are two possible paths forward. One path involves be the main solid phase(s) in the sili cate mantle. observations and interpretation of an exoplanet atmosphere There are many space telescope concepts to build on the to help break the interior composition degeneracy for a exquisite Kepler . These concepts focus on tran- specific exoplanet. For example, an apparent massive water siting exoplanet surveys of stars that are much brighter than planet, with an interior dominated by water and a thick steam Kepler's and that are therefore amenable to radial velocity atmosphere, cannot be distinguished from a silicate planet follow-up observations for planet confirmation and mass Sotin et al.: Terrestrial Planet Interiors 393 measurements. The Transiting Exoplanet Survey Beaulieu J.-P., Bennett D. P., Fouquet P., Williams A., Dominik (TESS) study, led by P.I. .George Ricker, aims to observe M., et al. (2006) Discovery of a cool planet of 5.5 Earth masses over 2 mlllion stars brighter than about I = 13 for transit- throngh gravitational microlensing. Nature, 439, 437440. ing planets with periods less than 2 months. Some planets Beirao P, Santos N. C., Israelian G., and Mayor M. (2005) Abun- dances of Na, Mg and A1 for stars exoplanets. orbiting M stars will be down to Earth size and orbiting in with Asfron. Astrophys., 4 3 8, 25I-256. the habitable zone of the host stars. PlAnetary Tiansits and Bertrand P, Sotin C., Mercier J-C., and Takatrashi E. (1986) From Oscillations of stars (PL{TO) is under study by ESA. The the simplest chemical system to nafural one: Garnet peridotite goal is to monitor 100,000 main-sequence stars brighter baromefty; Contrib. Mineral. Petrol., 93, 168-,178. than aboutV - 11, using an ensemble of 28 identical small, Boehler R, (1993) Temperatures in the Earth's core from melting- very-wide-field telescq)es, assqmbled on a single platform point measurements of iron'at high static pressure,p. Nature, and all looking at the same 25" diameter fleld, Even more 363,534-536. exoplanet trilhsit mission concepts are under study, leading Borde P., Rouan D., and L6ger A. (2003) Exoplanet detection us to believe that alarge pool of transiting terrestrial planets capability of the CoRoT space rnission. Astron. Astrophys., with measured mass and radius is a very likely possibility 405, II37-II44. in the future. Borucki W. J,, Koch D., ,Basri G., Batalha N., Brown T., et al. (2010) Kepler planet-detection mission: The search for terrestrial exoplanets is just starting. The Introduction and first results. Science, 327, 977-980. ,: last 20 have shown that planetary systems are very Bouhifd M. A., Andrault D., Fiquet G., and Richet P. (1996) Ther- common. The next 20 :s may shed light on the unique- mal expansion of forsterite up to the melting point. Geophys. ness of Earth-like planets. Res. Lett., 23, LI43-II46, Bower D. J., Gurnis M., Jackson J, M., and Sturhahn W; (2009) Acknowledgments. This work has been ca:ried out at the Jet Enhanced convection and fast plutnes in the lower mantle l,aboratory-California Propulsion Institute of Technology, under induced by the spin transition in ferropericlase. Geophys. Res. contract with the National Aeronautics and Space Administration. Lett., 36, LI,0306, DOI: 10.102912009GL0377 06. C.S. acknowledges JPt R&TD support. J.M.J. acknowledges the Cameron A. G-. W., Fegley B., BenzW., and Slattery W. L. (1988) National Science Foundation (0711542 and CSEDI 0855315) and The strange density of Mercury: Theoretical considerations. Caltech for partial support of this work. In Mercury (F.Vilas et al., eds,), pp. 692-708. Univ. of Ari- zona, Tucson REFERENCES Catalli K., Shim S.-H., and Prakapenka V. (2009) Thickness and Clapeyron slope of the post-perovskite boundary. Nature, 462, Adams E. R., Seager S., and Elkins-Tanton L. (2008) Osean 782-'795. planet or thick atmosphere: On the mass-radius relationship Chambers J. E. (2005) Planet formation. ln Treatise on Geochem- for solid exoplanets with massive atmospheres. Astrophys. J., istry, Vol. I: Meteorites, , and Planets (A. M. Davis 673, 1160-1 164. et p1., eds.), pp. 461475. Elsevier, Oxford. Alfe D. (2009) Temperature of the inner-core boundary of the Charbonneau D. and 18 colleagues (2009) A super-Earth transiting Earth: Melting of iron at high pressure frorn first-principles a nearby low-mass star. Nature, 462, 891-894. coexistence simulations. Phys. Rev. B, 79, 060101(R). Chen B., Li J., and Hauch S. A. (2008) Non-ideal liquidus curve Alfe D., Price G. D., and Gillan M. J. (2002) Iron under Earth's . in the Fe-S system and Mercury's snowing core. Geophys. Res. core conditions: Liquid-state thermodynamics and high- Lett., 3 5, L0720I, DOI: L0.I02912008GL0333 1 1. pressure melting curve from ab'initio calculations. Phys. Rev. Cox A. N.,,ed. (1999) Allen's Astrophysical Quantities, 4th edition. 8., 65, 1651 18, DOI: 10.1 103/PhysRevB.65.1651 18. Springer-Verlag, New York. 714 pp. Anderson D. L. (1980) Tectonics and composition of Venus. Davaille A. and Jaupart C. (1993) Transient high Rayleigh number Geophys. Res. Lett., 107-'i,02. thermal convection large 7, with viscosity variations. "I. Fluid Andersoq W. W. and Ahrens T. J. (L994) An equation of state for Mech., 253, I4L-I66. : liquid iron and implications for the Earth's core. uI. Geophys. Davaille A. and Jaupart C. (1994) Onser of thermal convection Res., 99, 42734284. in fluids with temperature-dependent viscosity Application Anderson W. W. and Ahrens T. J. (1996),Shock temperature and of the oceanic mantle. ,I. Geophys. Res., 99, 19853-19866.- melting in iron sutrfides core pressures. Res., Dewaele A., Loubeyre P, Occelli F., Mezoua^r Dorogokupets at "I. Geophys. M., 101(83), 5627-5642. P. I., and Torrent M. (2006) Quasihydrostatic equation of state Angel R. J. and Jackson J. M. (2002) Elasticity and equation of of iron above 2 Mbar. Phys. Reu. Lett., 97, 215504. state of orthoenstatite, MgSiOz. A*. Mineral., 87,558-561. Drejbus G. and W[nke H. (1985) Mars: A volatile rich planet. Angel R. J., Jackson J. M., Speziale S,, and Reichmann H. J. (2009) Meteoritics, 20, 367 -382. Elasticity measurements on minerals: A review. Eun J. Mineral., Duffy T. S .,Zha C. S., Downs R. T., Mao H. K., and Hemley R. J. 2 1(3 ), 525 554 DOI: 10. 1 127 10935-12211200910021,-1925. (1995) Elasticity of forsterite to 16 GPa and the composition Arkani-Hamed J. (1994) Effects of the core cooling on the inter- of the upper mantle. Nature, 378, l7Vl73.

nal dynamics and thermal evolution of terrestrial planets. .,l'. Dziewonski A. M. and Anderson D. L. (1981) Preliminary refer- Geophys. Res., 99, I2I09-I2II9. ence Earth model. Phys. Earth Planet. Inten, 25, 297-356. Badro J., Fiquet G., Guyot F., Rueff J.-P., Struzhkin V. V., Vanko Elkins-Tanton L. T. and Seager S. (2008) Ranges of atmospheric G., and Monaco G. (2003) Iron partitioning in Earth's lower mass and composition of Super=Earth exoplanets. Asfrophys.

, mantle: Toward a deep lower mantle discontinuity. Science, J., 685, 1237-1246. 300,789-7,9I. Fegley B. and Cameron A. G. W. (1987) A vaponzation model 394 Exoplanets

for iron/silicate fractionation in the Mercury . Earth DOI: 1 0. 1 029 /2009GL0378 I 5. Planet. Sci. Lett., 82, 207-222. Javoy M. (1995) The integral enstatite model of the FeiY., Mao H., and Hemley R. (1993) Thermal expansivity, bulk Earth. Geophys. Res. Lett., 22, 2219-2222. modulus, and melting curve of H2o ice vII to 20 GPa. J. Chem. Javoy M. (1999) Chemical Earth models. C. R. Acad. Sci. Paris, Phys., 99, 5369-5313. 329,537-555. Fei Y., Zhang L., Corgne A., Watson H. C., Ricolleau A., Meng Y., Kavner A., Duffy T. S., and Shen G. (2001) Phase srability and and Prakapenka V. (2007) Spin transition and equations of state density of FeS at high pressures and temperatures: Implica- (Mg, of Fe)O solid solutions. Geophyt. Res. Lett., 34, Ll730l . tions for the interior structure of Mars . Earth Planet. ,Sci. Fortney J. J., Marley M. S., and Barnes J. w. (2007) Planetary radii Lett., 185,25-33. across five orders of in mass and stellar insolation: Kuchner M. J. (2003) Volatile-rich earth-mass planets in the habit- Application to transrts. Astrophyt. J., 659, l66I-16i2. able zone. Astrophys. Lett., 596(1), "I. L105-L108. Frank M. R., Fei Y., and Hu J. (2004) Constraining the equation Kuskov O. L. and Kronrod V. A. (200I) Core sizes and internal of state of fluid HzO to 80 GPa using the melting curve, bulk structure of Earth's and Jupiter's satellites. Icarus, I5l, 20+227 , modulus, and thermal expansivity of ice VII. Geochim. Cos- L6ger A., Selsis F., Sotin C., Guillot T., Despois D., Mawet D., mochim. Acta, 68, 278I-2790. Ollivier M., Labbque A., Calette C., Brachet F., Chazelas B., Frost D. J., Liebske C., Langenhorst F., McCammon C. A., and Lammer H. (2004) A new family of planets? "ocean- Tr@nnes R. G., and Rubie D. C. (2004) Experimental evidence planets." Icerlts, 169, 499-504. for the existence of iron-rich in the Earth's lower mantle. L6ger A., Rouan D., Schneider J., Alonso R., Samuel B., et al. Nature, 428, 409412. (2009) Transiting exoplanets from the CoRoT space mission: Gilli G., Israelian G., Ecuvillon A., Santos N. C., and Mayor vm. CoRoT-Exo-7b: the first Super-Earth with measured M. (2006) Abundances of refractory elements in the atmos- radius. Astron. Astrophys., 506, 287-302. pheres of stars with extrasolar planets. Asfron. Astrophys., Lin J.-F., Struzhkin V. V., Jacobsen S. D., Hu M. Y., Chow P., et 449, 723-136. al. (2005) Spin transition of iron in magnesiowtistite in the Grasset O. and Parmentier E. M. (1998) Thermal convection in Earth's lower mantle. Nature, 436, 377480, DOI: 10.1038/ a volumetrically heated, infinite Prandtl number fluid with nature03825. strongly temperature-dependent viscosity: Implications for Lunine J. I., Macintosh B., and Peale S. (2009) The detection and planetary evolution. J. Geophys. Res., 103, 18171-18181. characterizatron of exoplanets. Phys. Tbday, May 2009, 46-5r. Grasset o., Schneider, J., and Sotin C. (2009) A study of the ac- Mao W. L., Shen G., Prakapenka V. B., Meng Y., Campbell A. J., curacy of mass-radius relationships for silicate-rich and ice-rich et al. (2004) Ferromagnesian postperovskite silicates in the D" planets up to 100 Earth. Astrophyt. J., 693, 722-733. layer of the Earth. Proc. Natl. Acad. Sci., 101(45), I586i-15869,

Hemley R. J., Stixrude L., Fei Y., and Mao H. K. (1992) Con- DOI: 10.107 3/pnas.0407 135101 . straints on lower mantle composition from P-V-T measurements Mattern E., Matas J., Ricard Y., and Bass J. (2005) Lower mantle of (Fe,Mg)Sio3-perovskite and (Fe,Mg)o. In High Pressure composition and temperature from mineral and ther- Research: Applications to Earth and Planetary Sciences (Y. modynamic modelling. Geophys. J. Intl., 160(l), 9i3-990, Syono and M. H. Manghnani, eds.), pp. 183-189. Geophysi- DOI: 10. 1 IIIlj.t365-1246X. cal Monograph Series, vol. 6J, American Geophysical Union, Mayor M. and 12 colleagues (2009) The HARPS search for Washington, DC. southern extra-solar planets? XVm. An Earth-mass planet in Hubbard w. B., Podolak M., and stevenson D. J. (1995) The the GJ 581 . Astron. Astrophys., 507,487494. interior of Neptune. In Neptune and (D. P. Cruikshank, McDonough w. F. and Sun S.-S. (1995) The composition of the ed.), pp. 109-138. Univ. of Arizona, Tucson. Earth. Chem. Geol., 120, 223-253. Huang C., Zhao G., Zhang H. W., and Chen Y. Q. (2005) Chemi- Mishima o. and Endo S. (1978) Melting curve of ice VII. J. Chem. cal abundances of 22 extrasolar planet host stars. Mon. Nofl R. Phys., 68, 44174418.

Astron. Soc., 363, 7l-78. Moresi L. and Solomatov V. (1998) Mantle convection with a Irifune T. (1987) An experimental investigation of the pyroxene- brittle lithosphere: Thoughts on the global tectonic styles of garnet transformation in a pyrolite composition and its bearing the Earth and Venus. Geophys. "I. Intl., 133, 669-682. on the constitution of the mantle. Phys. Earth Planet. Inten, Mosenfelder J. L., Asimow P. D., Frost D. J., Rubie D. C., and 45,32+336. Ahrens T. J. (2009) The MgSiO3 system at high pressure: Ther- Jackson I. (1998) Elasticity, composition and temperature of the modynamic properties of perovskite, postperovskite, and melt

Earth's lower mantle: A reappraisal. Geophys. J. Intl., 134, from global inversion of shock and static compression data. J. 29r-3rt. Geophys. Res., I I 4( B 1), DOI: 10. 1 02912008JB005900. Jackson J. M. (2010) Synchrotron-based spectroscopic techniques: Murakami M., Hirose K., Kawamura N., Sat& N., and onishi Y. Mossbauer and high-resolution inelastic scattering. In High- (2004) Post-perovskite phase transition in MgSio r. Science, Pressure Crystallography: From Fundamental Phenomena to 304, 955-959. Technological Appli,cations (E. Boldyreva and P. Dera, eds.), oganov A. R., Martofidk R. , Laio A., Raiteri P., Parrinello M. in press. Springer Science, DoI: 10. 1007 /978-90-48 r-9258_5. (2005) Anisotropy of Earth's D" layer and stacking faults in Jackson J. M., Sinogeikin s. v., and Bass J. D. (2007) Sound the MgSiO3 post-perovskite phase. Nature, 438, II42-II44. velocities and single-crystal elasticity of orthoenstatite to O'Neill C. O. and Lenardic A. (2007) Geological consequences of 1073 K at ambient pressurc. Phys. Earth Planet. Inten, 16l, super-sized Earths. Geophys. Res. Lett., 34, L19204. l-12, DOI: 10.1 016/j.pepi.(2006) 11.002. Papuc A. M. and Davies G. F. (2008) The internal activity and Jackson J. M., Sturhahn w., Tschauner o., Lerche M., and Fei y. thermal evolution of Earth-like planets. Icarus, 195, 44i45& (2009) Behavior of iron in (Mg,Fe)Sio, post-perovskite as- Parmentier E. M. and Sotin C. (2000) 3D numerical experiments semblages at Mbar pressures. Geophyt. Res. Lett., 36, LI0301, on thermal convection in a very viscous fluid: Implications for

,l. Sotin et al.: Tbrrestrial Planet Interiors 395

the dynamics of a thermaf boundary layer at high Rayleigh sun D., Tan E., Helmberger D., and Gurnis M. (2007) Seismo- number. Phys. Fluids, 12, 609-617. logical support for the metastable superplume model, sharp Parmentier E. M., Sotin C., and Travis B. J. (1994) Turbulent 3D features, and phase changes with the lower mantle. Proc. Natt. thermal convection in an infinite Prandtl number, volumetri- Acad. Sci., 104(22), 9151-9155. cally heated fluid: Implications for mantle dynamics. Geophyt. Thompson S. L. (1990) ANE)S - Analytic Equations of State for J. Intl., I16, 24I-25I. Shock Physics codes. Sandia Natl. Lab. Doc. SAND89-295L, Podolak M., Podolak J. I., and Marley M. S. (2000) Further in- Albuquerque, New Mexico. vestigations of random models of Uranus and Neptun e. Planet. Tozer D. C. (1972) The present thermal state of the terrestrial Space Sci., 48, 143-I5I. planets. Phys. Earth Planet. Inten, 6, I82-I97. Poirier J.-P. (1994) Physical properties of the Earth's core. C. R. Tschauner O., Kiefer B., Liu H., Sinogeikin S., Somayazulu M., Acad. Sci., 318, 34I-350. and Luo S.-N. et al. (2008) Possible structural polymorphism Poirier J.-P. (2000) Introduction to the Physics of the Earth in Al-beating magnesiumsilicate post-perovskite. Am. Mineral., Interior Znd edition. Cambridge Univ., Cambridge. 264 pp. 93, 533-539. Queloz D. and 39 colleagues (2009) The CoRoT:7 planetary system: Uchida T., Wang Y., Rivers M. L., and Sutton S. R. (2001) Sta- Two orbiting super-Earths. Astron. Astrophys., 506, 303-3 19. bility field and thermal equation of state of e-iron determined Rogers L. A. and Seager S. (2010) Three possible origins for the by synchrotron X-ray diffraction in a multianvil apparatus. ,I. gas layer on GJ 12I4b. Astrophys. J., 716, 1208-1216. Geophys. Res., 106, 21799-21810. Rouan D., Baglin A., Barge P., Copet E., Deleuil M., L6ger A., Umemoto K., wentzcovitch R. M., and Allen P. B. (2006) Dis- Schneider J., Toublanc D., and Vuillemin P. (1999) Searching sociation of MgSiO: in the cores of gas giants and terristrial for exosolar planets with the CoRoT space mission. Phys. exoplanets. Sci ence, 3 I l, 983-986. Chem. Earth, 24, 567-57I. Vacher P., Mocquet A., and Sotin C. (1998) Computation of Schubert G., Turcotte D. L., and Olson P. (2001) Mantle Convec- seismic profiles from mineral physics: The importance of tion in the Earth and Planets. Cambridge Univ., Cambridge. the non-olivine components for explaining the 660 km depth Seager S., Kuchner M., Hier-Majumder C., and Milrtzer B. (2007) discontinuity. Phys. Earth Planet. Inten, 106,275-298. Mass-radius relationships for solid exoplanets. Astrophys. J., valencia D., o'Connel R. J., and and Sasselov D. D. (2006) 669, 1279-1297. Internal structure of massive terrestrial planets. Icarus, I8l, Sinogeikin S. V. and Bass J. D. (2000) Single-crystal elasticity 545-554. of pyrope and MgO to 20 GPa by Brillouin scattering in the Valencio D., o'Connel R. J., and Sasselov D. D. (2007) Inevita- diamond cell. Phys. Earth Planet. Inter., 120, 43-62. bility of plate tectonics on Super-Earths. Astrophys. .,I. Lett., Sinogeikin S. V., Jackson J. M., O'Neill B., Palko J. W., and Bass 670, L45-L49. J. D. (2000) Compact high-temperature cell for Brillouin scat- Valencio D., o'Connel R. J., and Sasselov D. D. (2009) The role of tering measurements. Rev. Sci. Instruments, 71(I), 20I-206. high-pressure experiments on determining super-Eafth proper- Sola E. and Alfe D. (2009) Melting of iron under Earth's core ties. Astrophys Space Sci., 322, 135-139. conditions from diffusion Monte Carlo free energy calcu- Volcadlo L., de Wijs G. A., Kresse G., Gillan M., and Price G. D. lations. Phys. Rev. Lett., 103(7), DOI: 10.11O3/PhysRev- (1997) First principles calculations on crystalline and liquid iron Lett.103.078501. at Earth's core conditions. Faraday Disct,tss., 106, 205-217. Solomatov V. S. and Moresi L. N. (2000) Scaling of time- Wicks J. K., Jackson J. M., and Sturhahn W. (2010) Very low dependent stagnant lid convection: Application to small-scale sound velocities in iron-rich (Mg,Fe)O: Implications for the convection on Earth and other terrestrial planets. .,I. Geophys. core-mantle boundary region. Geophys. Res. Lett., 37, LI5304, Res., 105, 2I795-2I8I7 . DOI: 10.1029 /2010cL043689. (1979) Solomon S. C. Formation, history and energetics of cores in Williams Q., Jeanloz R., Bass J., Svendsen B., and Ahrens T. J. the terrestrial planets. Phys. Earth Planet. Inter., 19, 168-182. (1987) The melting curve of iron to 250 gigapascals A Sotin C. and Labrosse S. (1999) Three-dimensional thermal con- constraint on the temperature at Earth's centet Science, -236, vection in an iso-viscous, infinite Prandtl number fluid heated 181-184. from within and from below: applications to the transfer of Wuchterl G., Guillot T., and Lissauer J. J. (2000) Giant planet heat through planetary mantles. Phys. Earth Planet. Inten, formation. In and Planets (V. Mannings et al., eds.), I 12, I7I-190. p. 1081. Univ. of Arizona, Tucson. Sotin C. and Tobie G. (2004) Internal structure and dynamics of Yoder C. F., Konopliv A. S., Yuan D. N., Standish E. M., and the large icy satellites. C. R. Acad. Sci. Phys., 5, 769-780. Folkner W. M. (2003) Fluid core size of Mars from detection Sotin C., Grasset o., and Mocquet A. (2007) Mass-radius curve of the solar tide. Science, 300, 299-303. for extrasolar Earth-like planets and ocean planets . Icarus, Zeng L. and Seager S. (2008) A computational tool to interpret tgt,337-35r. the bulk composition of solid exoplanets based on mass and Stevenson D. J. (1999) Life-sustaining planets in interstellar space? radius measurements. Publ. Astron. Soc. Pac., 120, 983-99I. Nature, 400, 32. Zhuravlev K. K., Jackson J. M., Wolf A. S., Wicks J. K., Yan Sturhahn W., Jackson J. M., and Lin J.-F. (2005) The spin state of J., and Clark S. M. (2009) Isothermal compression behavior iron in rninerals of Earth's lower mantle. Geophys. Res. Lett., of (Mg,Fe)o using as a pressure medium. Phys. chem. 32, L12307 , DOI: 10.1029/2005GL022802. Minerals, 37, 465474, DOI: 10. 1007/s00269-009-0347 -6.