Structure and evolution of super-Earth to super-Jupiter exoplanets: I. heavy element enrichment in the interior
aa r X i v : . [ a s t r o - ph ] F e b Astronomy & Astrophysics manuscript no. planet11 c (cid:13)
ESO 2018November 2, 2018
Structure and evolution of super-Earth to super-Jupiterexoplanets: I. heavy element enrichment in the interior
I. Baraffe , G. Chabrier , T. Barman ´Ecole normale sup´erieure de Lyon, CRAL (CNRS), 46 all´ee d’Italie, 69007 Lyon,Universit´e de Lyon, France (ibaraffe, [email protected]) Lowell observatory, 1400 West Mars Hill Road, Flagstaff, AZ 86001, USA ([email protected])Received /Accepted
ABSTRACT
Aims.
We examine the uncertainties in current planetary models and we quantify their impact on the planet coolinghistories and mass-radius relationships. These uncertainties include (i) the differences between the various equationsof state used to characterize the heavy material thermodynamical properties, (ii) the distribution of heavy elementswithin planetary interiors, (iii) their chemical composition and (iv) their thermal contribution to the planet evolution.Our models, which include a gaseous H/He envelope, are compared with models of solid, gasless Earth-like planets inorder to examine the impact of a gaseous envelope on the cooling and the resulting radius.
Methods.
We find that for a fraction of heavy material larger than 20% of the planet mass, the distribution of theheavy elements in the planet’s interior affects substantially the evolution and thus the radius at a given age. For planetswith large core mass fractions ( > ∼ ∼
10% effecton the radius after 1 Gyr.
Results.
We show that the present mass and radius determinations of the massive planet Hat-P-2b require at least 200M ⊕ of heavy material in the interior, at the edge of what is currently predicted by the core-accretion model for planetformation. As an alternative avenue for massive planet formation, we suggest that this planet, and similarly HD 17156b,may have formed from collisions between one or several other massive planets. This would explain these planet unusualhigh density and high eccentricity. We show that if planets as massive as ∼ M J can form, as predicted by improvedcore-accretion models, deuterium is able to burn in the H/He layers above the core, even for core masses as large as ∼
100 M ⊕ . Such a result highlights the confusion provided by a definition of a planet based on the deuterium-burninglimit. We provide extensive grids of planetary evolution models from 10 M ⊕ to 10 M Jup , with various fractions of heavyelements. These models provide a reference to analyse the transit discoveries expected from the CoRoT and Keplermissions and to infer the internal composition of these objects.
Conclusions.
Key words.
Planetary systems stars: individual: GJ436, Hat-P-2, HD149026
1. Introduction
The number of newly discovered exoplanets transiting theirparent star keeps increasing continuously, revealing a re-markable diversity in mean densities for planet massesranging from Neptune masses to several Jupiter masses.A large fraction of these transits exhibit a mean densitysignificantly larger than that of an object essentially com-posed of gaseous H/He, like brown dwarfs or stars, indi-cating a composition substantially enriched in heavy ele-ments. The first compelling evidence for such a significantenrichment was provided by the discovery of a Saturn massplanet, HD149026b, with such a small radius that 2/3 ofthe planet’s mass must be composed of elements heavierthan He (Sato et al. 2005). Another remarkable discoveryis the case of GJ 436b, a ∼
22 M ⊕ Neptune-like planetwith a radius comparable to that of Uranus or Neptune(Gillon et al. 2007a). Such a radius implies an inner struc-ture composed by more than 90% of heavy elements. Thatexoplanets can be substantially enriched in heavy mate-
Send offprint requests to : I. Baraffe rial such as rock or ice is not a surprise, since this isa well known property of our own Solar System planets.Moreover, the presence of an icy/rocky core and of overso-lar metallicity in the envelope is an expected consequenceof the most widely accepted planet formation scenario, theso-called core-accretion model (Alibert et al. 2005a, andreferences therein).Given this expectation, it becomes mandatory to takeinto account heavy element enrichment in planetary modelsdevoted to the analysis and the identification of current andforthcoming observations of extra-solar planets. Many ef-forts are now devoted to the modeling of massive terrestrialplanets, essentially composed of solid material (Valencia etal. 2006; Sotin et al. 2007; Seager et al. 2007) and jovianplanets with H/He envelope and a substantial metal enrich- Under usual planet formation conditions, the word ”rocks”refers primarily to silicates (Mg-, Si- and O-rich compounds)whereas the term ”ice” includes collectively H O, CH and NH ,water being the most important of these three components. Aswill be discussed in §
3, the term ”ice” may be inappropriate insome cases, as water could be under a liquid or gaseous form. Baraffe, Chabrier & Barman: From super Earth to super Jupiter exoplanets. ment (Baraffe et al. 2006; Guillot et al. 2006; Burrows etal. 2007; Fortney et al. 2007). Because of remaining uncer-tainties in the input physics describing the planetary struc-tures, and of many unknown quantities such as the totalamount of heavy elements, their chemical composition andtheir distribution within the planet’s interior, these mod-els are based on a number of assumptions and thus retainsome, so far unquantified, uncertainties. The main goal ofthe present paper is to analyse and quantify these uncer-tainties and to explore as precisely as possible the impact ofthe heavy material contribution on the planet structure andevolution. We will focus on planets with mass larger than10 M ⊕ , the expected limit for the gravitational capture ofa gaseous H/He envelope and atmosphere (Mizuno 1980;Stevenson 1982; Rafikov 2006; Alibert et al. 2006), whichhas a significant impact on the evolution. Below this limitmass, the objects are essentially solid bodies with no or ateneous gaseous envelope and their mass-radius relation-ship has been studied recently by several authors (Valenciaet al. 2006, 2007; Sotin et al. 2007; Seager et al. 2007). Thisstudy is motivated by the level of accuracy on planetarymass and radius measurements which is now reached withground-based (HARPS, VLT) and space-based instruments(HST, SPITZER). Observations are expected to reach anunprecedented level of precision in the near-future withCOROT, KEPLER, and on a longer term with GAIA. Thelatter project will measure distances and thus stellar radiiwith high precision, removing one of the main sources ofuncertainty in planetary radius measurements. This racefor precision is motivated by the possibility to infer withthe best possible accuracy the inner composition of an exo-planet, with the aim to better understand planet formationand to identify the presence of astrobiologically importantmaterial such as liquid water. In this context, it is crucialto quantify the uncertainties in the structure and evolu-tion planetary models used to analyse these observations.In section § §
3, we examine the main input physicsand assumptions used in structure and evolutionary modelsavailable in the literature. In section §
4, we analyse quan-titatively the impact of these assumptions. We focus onspecific cases such as HD149026b and GJ436b in § §
6. Our various planetary models,covering a wide mass range and including different levels ofheavy element enrichments, are presented in §
7. Discussionand perspectives follow in §
2. Uncertainties and assumptions in the modellingof extra-solar planets
According to a recent study devoted to the structure of ourgiant planets (Saumon & Guillot 2004), Jupiter should havea total amount of heavy elements ranging from 8 to 39 M ⊕ , i.e a metal mass fraction Z ∼ ⊕ and a maximum envelope metalmass fraction Z env = 12%. For Saturn, the same study sug-gests a total mass of heavy elements ranging from 13 to 28M ⊕ , i.e Z ∼
13% to 29%, with a maximum core mass of22 M ⊕ and a maximum Z env ∼ ∼ ⊕ above which gas accretion begins (Mizuno 1980; Stevenson 1982; Rafikov 2006). During this gas accretionphase, planetesimals are still accreted and are either de-stroyed in the gas envelope or are falling onto the core,leading to further increase of the core mass. The fate ofthese accreted planetesimals, disrupted in the envelope oraccreted onto the dense core, depends on their size, an un-known parameter in current planet formation models, andon the envelope mass. This general picture thus predictsthat planets should have a dense core of heavy material(rock, water/ice) of, at least, a few Earth masses and canshow different levels of heavy element enrichment in theirenvelope, depending on the accretion history (amount ofgas accreted, size of planetesimals, etc...). Current mod-els based on the core-accretion scenario are able to matchthe core mass and the envelope metal enrichment derivedfor Jupiter and Saturn (Alibert et al. 2005b), but predictin some cases much larger heavy element enrichments inthe envelope, depending on the planet’s mass, with valuesas large as Z env > ∼
50% for Neptune-size planets ( ∼ ⊕ ) (Baraffe et al. 2006).Despite this widely accepted picture, current extra-solarplanet models often simply assume that all heavy elementsare located in the central core and that the envelope iseither metal-free, Z env = 0, (Burrows et al. 2007; Fortneyet al. 2007), or has a solar metallicity , Z env = Z ⊙ (Guillotet al. 2006). This simplification is based on the assumptionthat whether the heavy elements are located in the core orin the envelope should not affect the planet’s evolution. Thevalidity of such an assumption, however, has never beenexamined, as will be done in the present paper. An other simplification found in planet modelling is the useof temperature-independent EOS, assuming that the heavyelement material is at zero-temperature or at a uniform,low temperature (Seager et al. 2007; Fortney et al. 2007).This is certainly a good assumption when examining thestructure of terrestrial-like planets, composed essentially ofsolid (rocky/icy) material (Valencia et al. 2006; Sotin et al.2007). This assumption, however, is not necessarily valid forthe evolution of more massive planets. For these objects, thethermal and gravitational energy contributions of the coreto the planet’s cooling history are usually ignored and it isimportant to examine the impact of such a simplification.
Finally, a conventional assumption in planet modelling isto assume that the interiors (at least the gaseous envelope)of giant planets are homogeneously mixed, due to the dom-inant and supposedly efficient transport mechanism pro-vided by large-scale convection. Accordingly, the internaltemperature gradient is given by the adiabatic gradient,since superadiabaticity is negligible. The validity of this as-sumption, however, has been debated for the interior of ourown jovian planets for already some time (Stevenson 1985)and has been questioned again more recently in the contextof transiting extra-solar planets (Chabrier & Baraffe 2007).It should be kept in mind, however, that, even though the Note also that different envelope helium mass fractions havebeen assumed, with Y = 0 .
25 (Burrows et al. 2007) or Y =0.28(Fortney et al. 2007).araffe, Chabrier & Barman: From super Earth to super Jupiter exoplanets. 3 planetary internal heat flux can only be carried out by con-vection (Hubbard 1968), the assumption of large-scale, fullyadiabatic convection in planetary interiors has never beenproven to be correct and even slightly inefficient convec-tion can have a major impact on the planet’s structure andevolution (Chabrier & Baraffe 2007).
3. Equations of state for heavy elements
As quickly mentioned in the introduction, the generic term”ice” may often be inappropriate to describe the thermo-dynamic state of water under planetary conditions. Indeed,depending on the temperatures prevailing in the parent pro-toplanetary nebula at the initial location of the planet em-bryo, water could possibly be initially under the form ofsolid ice, but it could also be under the form of liquid orvapor. If initially solid, it may also melt or vaporize underthe conditions prevailing in the planet’s interior, or mayalso dissociate under the form of ionic melts at high pres-sures and temperatures (Schwegler et al. 2001). In fact,the phase diagram of heavy elements under the pressureand temperature conditions characteristic of the consid-ered planet interiors is largely unknown, as only part ofthis diagram is presently accessible to high-pressure exper-iments or computer numerical simulations. Water, for in-stance, the dominant component after H and He, is knownto exhibit a complex phase diagram with many stable ormetastable (amorphous) phases and several triple and crit-ical points, because of the high flexibility of the hydrogenbonding. The melting line of water at high pressure andhigh temperature has been probed experimentally up to P = 3 . × dyn cm − (0.35 Mbar) and T = 1040 K.H O has been found to remain solid at larger pressures andtemperatures, suggesting that the melting line increases (in T ( P )) at higher temperatures and extends up to at least T > ∼ O might be present at some depth inthe interiors of these objects, as predicted for ocean-planets(Selsis et al. 2007), while supercritical H O is more likelyto be liquid or gaseous in the hotter interiors of Saturn-like or larger planets. As mentioned above, water is alsofound from shock-wave experiments and first-principles cal-culations to dissociate into H O + + OH − ion pairs above ∼ For Earth-like planets, models can test a variety of com-plex heavy element compositions inspired by the knowl-edge of the structure of our own Earth (Valencia et al.2006, 2007; Sotin et al. 2007). Moreover, the EOS of ma- terials which may be found in planetary interiors (water,iron, dunite or olivine, etc...) are reasonably well deter-mined at zero-temperature or at 300 K (see Seager et al.2007). Unfortunately, for larger planets, the exploration ofinternal compositions is restricted to a few materials forwhich EOS are available and cover a large enough rangeof pressures and temperatures. The two most widely usedEOS in this context (Saumon & Guillot 2004; Baraffe etal. 2006; Fortney et al. 2007; Burrows et al. 2007) areANEOS (Thompson & Lauson 1972) and SESAME (Lyon& Johnson 1992), which describe the thermodynamicalproperties of water , rocks (olivine or dunite, i.e Mg SiO ,in ANEOS; a mixture of silicates and other heavy elementscalled ”drysand” in SESAME) and iron . Figure 1 shows acomparison between these two EOS for the three aforemen-tioned materials for two temperatures, namely T = 300 Kand T = 6000 K. Comparison is also shown with the zero-temperature EOS presented in Seager et al. (2007) for wa-ter, perovskite (MgSiO ) and iron. These authors use fitsto experimental data at low pressure ( P < ∼ P > ∼
100 Mbar), where the contribution of the degenerateelectron fluid becomes dominant. As stressed by these au-thors, the main difficulty is to bridge the gap in the pressureregime 2 Mbar < ∼ P < ∼
100 Mbar (2 × - 10 dyne cm − ),which is the most relevant for planetary interiors. Theirmodel EOS represents some improvement upon ANEOSand SESAME, in particular for water, but is valid only atzero-temperature.The presently used EOS models (ANEOS andSESAME), and thus the inferred planet internal structures,thus retain a significant degree of uncertainty in the ex-perimentally unexplored high-P and high-T domains. Inthese regimes, both EOS models, as well as the one used bySeager et al. (2007), are based on interpolations betweenexperimental data at low or moderate density/temperatureand well-known asymptotic limits, in general the Thomas-Fermi or more accurate density-functional type models, inthe very high density, fully ionized limit. One can only hopethat these interpolations do not depart too much from re-ality, as might be the case, for instance, if first-order phasetransitions, which imply density and entropy discontinu-ities, occur in the regions of interest. For the static prop-erties, this assumption is probably reasonable at high tem-peratures, but might be more questionable near meltinglines. When addressing the transport properties, like e.g.the thermal diffusivity or kinematic viscosity, the results inthe interpolated regime are definitely more doubtful.As shown in Figure 1, at low (room) temperature, thevarious EOS agree reasonably well. For water, the agree-ment lies within less than 5% at P =0.1 Mbar and withinless than 16% at P = 100 Mbar. For ”rocks”, keeping inmind that this term refers to different compositions in thevarious EOS, the agreement is comparable, although thedifference can reach 27% at P=0.1 Mbar. For iron, the threeEOS agree well at low pressure but differences as large as ∼
20% can occur at P=100 Mbar between SESAME andSeager et al. (2007). Such cool temperature conditions, how-ever, are more relevant to Earth-like planets than to theones explored in the present study.Figure 1 also shows, for ANEOS and SESAME, the vari-ation of P ( ρ ) with temperature, for conditions more suit-able to our planetary interior conditions. The thermal con- Baraffe, Chabrier & Barman: From super Earth to super Jupiter exoplanets.
Fig. 1.
EOS for water, rock and iron for two temperatures: T=300K for ANEOS (red solid line) and SESAME (bluedash-dotted curve); T=6000K for ANEOS (red short-dashed curve) and SESAME (blue dotted curve). The long-dashed(magenta) curves correspond to the zero-temperature EOS of Seager et al. (2007; their Table 3). The inset in the lowerpanel (water) shows a zoom of the P ( ρ ) relation for log P < . P . ∼ P ( ρ ) between the T = 300 K and T = 6000 K isotherms, for iron and water,respectively. The differences keep increasing significantlyfor T > P ( ρ ) is the relevant quantity for the structure ,the relevant one for the evolution is the entropy. Figure2 portrays the P- and T-dependence of the entropy, forthree isotherms and isobars, for water, for the ANEOS andSESAME EOS, under conditions relevant to the planetsof present interest. The values for the H/He fluid (EOSof Saumon et al. 1995; hereafter SCVH EOS) are also dis-played for comparison. We see that, whereas the EOS agreereasonably well under Jovian-planet conditions, as expectedas they reach the asymptotic, high-P, high-T regime accu-rately described by Thomas-Fermi-Dirac or more accuratedensity-functional theories, the difference can be substan-tial for conditions characteristic of the Neptune-mass do-main. Indeed, for this latter case, most of the interior liesin the interpolated regime where guidance from either ex-periments or numerical simulations is lacking. These dif-ferences between the EOS, of course, are amplified for the quantities involving the derivatives of the entropy, such asthe adiabatic gradient or the specific heat. Our goal is to examine the impact on the planetary modelsand the inferred mass-radius relationships due to uncer-tainties in (i) the distribution of heavy elements within theplanet interiors, (ii) their dominant chemical composition,(iii) the different EOS describing their thermodynamicalproperties and (iv) their thermal contribution. To achievethis goal, we have implemented the ANEOS and SESAMEEOS for water, rock and iron (or what is so-denominated)in our planetary evolution code. Both EOS provide all ther-modynamic quantities relevant to the evolution of planets,including internal energy, entropy (in ANEOS) or free en-ergy (in SESAME), and all relevant derivatives. ANEOSalso provides Rosseland and conductive opacities. Whenthe SESAME EOS is used, conductive opacities are calcu-lated according to Potekhin (1999). The evolutionary cal-culations, including the presence of a dense core, proceed asdescribed in Baraffe et al. (2006): the structure equationsare integrated from the center to the surface; at the coreboundary, the EOS is switched to the one characteristic ofthe gaseous envelope, and the change in chemical composi- araffe, Chabrier & Barman: From super Earth to super Jupiter exoplanets. 5
Fig. 2.
Pressure (in dyne cm − ) and temperature (in K)dependence of the entropy (in erg/g/s), as obtained for wa-ter for the ANEOS EOS (solid line) and the SESAME EOS(dash-line). The entropy for a H/He fluid, with Y = 0 . T and P domainscharacteristic of Neptune-like and Jupiter-like planet inte-riors are indicated.tion yields a density jump, but continuity in pressure andtemperature is enforced.To account for the thermodynamic contributions ofheavy elements in the H/He envelope, we have tested twodifferent procedures.(1) Following the method described inChabrier et al. (1992) and used in Baraffe et al. (2006),we first mimic the presence of metals with mass fraction Z by an equivalent helium fraction Y equiv = Y + Z in theH/He SCVH EOS, with Y the real helium mass fraction inthe envelope. As mentioned in Chabrier et al. (1992), thisapproximation is reasonable as long as the mass fractions Y and Y equiv are small compared with unity and both ρ Z and ρ He are large compared with ρ H , where ρ i denotes themass density of the i-component at pressure P . (2) The sec-ond, more general approach to describe the EOS of a mix-ture of various species in the absence of a reliable theory isto apply exactly the additive volume law (hereafter AVL),which is exact in the ideal gas limit, without restriction onthe species mass fractions and densities (see Fontaine et al.1977 and Saumon et al. 1995 for extensive discussions ofthe validity of the AVL). In this approach, the interactionsbetween the three different fluids, namely hydrogen, heliumand the heavy element component, are not taken into ac-count (but interactions between particles in each of these fluids are treated properly) and the EOS of the mixtureis simply the mass-weighted interpolation of each speciescontribution at constant intensive variables, P and T , plusthe ideal entropy of mixing for the entropy term. Withinthe ideal volume law, the mass density of the mixture of aH/He fluid with a helium mass fraction Y plus some heavyelement material with mass fraction Z at pressure P andtemperature T thus reads:1 ρ ( P, T ) = (1 − Z ) ρ H/He ( P, T ) + Zρ Z ( P, T ) (1)The extensive variables, e.g. the internal energy and specificentropy, thus read: U ( P, T ) = (1 − Z ) U H/He ( P, T ) +
Z U Z ( P, T ) (2) S ( P, T ) = (1 − Z ) S H/He ( P, T ) +
Z S Z ( P, T ) + S mix ( P, T )(3)where the EOS of the
H/He component is given by theSCVH EOS while the one of the Z -component is describedby either the ANEOS or the SESAME EOS. All along thepresent calculations, we have taken the cosmic helium frac-tion Y =0.275. The details of the calculation of the con-tribution due to the ideal entropy of mixing are given inAppendix A. This term is found to contribute non negligi-bly to the total entropy S . Depending on the mixture andthe P - T range, it can amount to 10%-20% of S . This isconsequential when calculating the internal structure of aplanet, whose interior is essentially isentropic, at a givenage.For the evolution, the relevant quantities are the vari-ation of the entropy with time and its derivatives w.r.t P and T , which give the adiabatic gradient: ∇ ad = ( ∂ log S∂ log P ) T / ( ∂ log S∂ log T ) P (4)Given the fact that the internal composition of theplanet ( N H , N He , N Z , where N i denotes the number of par-ticles of species i ), remains constant along the evolution,the only P - and T -dependence of the mixing entropy termarises from the variation of the degree of ionization andthus of the abundance of free electrons. The degree of ion-isation of the heavy element component, however, is un-known, and its variation with temperature has been ignoredin the present calculations. Therefore, only the variationwith temperature of the number of electrons provided byH and He contribute to the variation with time of the mix-ing entropy. Within this limitation, the variation of S mix with P and T is found to be negligible for planets in themass range of interest, for all levels of metal enrichment inthe envelope, so that the entropy of mixing term does notcontribute significantly to the evolution .
4. Effect of the different treatments anddistributions of heavy elements on the planetcooling history
In this section, we analyse the impact of the localisationof the heavy elements within the planet on its structureand evolution. The heavy elements are distributed eitherin the core or in the gaseous envelope, and we considerin the present section mass fractions with Z ≤ Baraffe, Chabrier & Barman: From super Earth to super Jupiter exoplanets. encompasses the level of enrichment of our giant planetsJupiter and Saturn and of previous theoretical studies de-voted to the analysis of transit planets (Guillot et al. 2006;Burrows et al. 2007). The effect of larger fractions of heavyelements (
Z > ⊕ ) and a Jovianmass planet (1 M J = 318 M ⊕ ). As a test case, we restrictthe analysis in this section to planets irradiated by a Sun at0.045 AU, since the observational determination of the ra-dius is presently accessible only to transiting, short-periodplanets. We use outer boundary conditions derived from agrid of irradiated atmosphere models with solar metallic-ity (Barman et al. 2001). These models take into accountthe incident stellar flux, defined by F inc = f ( R ⋆ a ) F ⋆ , with R ⋆ and F ⋆ the radius and the flux of the parent star, re-spectively, and f the redistribution factor. The Barmanet al. (2001) models use f =2, corresponding to the inci-dent flux being redistributed only over the dayside of theplanet. Recent observations of the day-night contrast of ex-oplanets with Spitzer (see e.g. Knutson et al. 2007) and at-mopsheric circulation models (see the discussion in Marleyet al. 2007), however, seem to favor a redistribution overthe entire planet’s surface, i.e f =1. For the specific caseof HD209458b, Baraffe et al. (2003) found out that such avariation of F inc by a factor 2 has a significant effect on theouter atmospheric profile, but a small effect on the planetevolution. In a forthcoming paper, we will explore in moredetails the effects of the redistribution factor f , of differentlevels of irradiation and of heavy element enrichment in theatmosphere. Figure 3 shows the evolution of the radius with time fora 20 M ⊕ planet with Z =50%, for different heavy elementmaterials and for different distributions of these latter inthe planet interior. We first consider models with a 10 M ⊕ core and Z env =0, i.e. all heavy elements are located in thecentral core of the planet. The results for pure water (solidline), rock (dashed line) and iron (dash-dotted line) coresare displayed in Fig. 3a, based on the ANEOS EOS forthese materials. A comparison of these results with thoseobtained with the SESAME EOS for the same material(water and drysand) shows less than a 1% difference onR at a given time. For iron, the SESAME EOS does notprovide the free energy and thus the entropy, the adiabaticgradient and other quantities required for the evolution.However, according to the comparison between the ANEOSand SESAME EOS portrayed in Fig. 1, we do not expectsignificant differences on the evolution between the differentiron EOS.For the aforementioned global heavy element enrich-ment, Z = 50%, the nature of the core material affectsthe cooling, and thus the radius evolution at the ∼ §
5, however, that for larger heavy el-ement mass fractions, the impact of the core compositioncan be more severe. At 1 Gyr, the radius of the planet witha pure rocky (iron) core is smaller by 6% (11%) comparedto the pure water core case (see Table 1) . For water andiron, the ANEOS EOS indicates whether the material is ina solid, liquid or melt phase. For dunite, this information is not provided. In all the models, water is always found tobe in a liquid state. However, as mentioned in § ∼ T ∼ . × K and ρ ∼
30 g.cm − .Figure 3b illustrates the effect of the localisation of theheavy elements within the planet’s interior, with two lim-iting assumptions: (A) all heavy elements are in the core( Z env =0) and (B) they are distributed all over the planet’sinterior ( M core =0). In the latter case, we compare the twoprocedures mentioned in § – Under assumption (B), models based on a Y equiv , forsuch a high metal fraction, Z = 50%, lead to coolingsequences that differ drastically from the ones based onthe AVL, with the SESAME EOS. This shows that theY equiv simplification can not be used for such values of Z , as anticipated from the limitations of this assumption(see § – Case (A) and case (B), with the AVL and the SESAMEEOS, yield similar cooling sequences, with ∼
4% differ-ence at t=1 Gyr (7% at 5 Gyr). – In case B, we find a significant difference when usingthe AVL with ANEOS compared with all other cases ( ∼
30% difference in R for t ≥ § P, T ) predicted by the different EOS in the relevantpressure regime indicated in the middle panel of Fig. 4.Although all the evolution sequences are forced to startfrom the same initial entropy state, the sequence basedon the ANEOS EOS shows a much stronger variationof the entropy with time. This stems from the strongervariation of S with P and T predicted by this EOS, forthe mass range characteristic of Neptune-like planets(see Fig. 2). This is highlighted in the upper panel of Fig.4: the sequence based on the AVL with ANEOS shows ∼
10 times larger local gravothermal energy, − T dS/dt ,at the beginning of the evolution. Consequently, this se-quence contracts and cools much faster than the otherones, loosing its internal entropy at a faster rate (lowerpanel of Fig. 4) and reaching a significantly smaller ra-dius after 1 Gyr (see Fig. 3b).The reason why the difference between the entropy, S ( P, T ), obtained with the ANEOS and SESAME EOS(Fig. 2 and Fig. 3) is inconsequential when all theZ-material is located in the core, whereas it yieldssignificantly different cooling sequences when the Z-component is mixed with the H/He envelope stems from araffe, Chabrier & Barman: From super Earth to super Jupiter exoplanets. 7
Table 1.
Radius and central thermodynamic properties of a planet of mass M P = 20 M ⊕ at 1 Gyr with a totalheavy element mass fraction Z = ( M Z /M P )=50%. Results for different heavy element compositions (water=W, rock=R,iron=I) and EOS (aneos=a, sesame=s, SCVH=SC) and different heavy element distributions are given. M core EOS (core) Z env
EOS(env) R p T c ρ c ( M ⊕ ) env. ( R J ) (K) (g cm − )10 W-a 0 SC 0.890 1.1 10 a ) 0.953 1.2 10 a ) 0.876 1.2 10 a Value of Y equiv . the fact that the evolution of a planet with a core is dom-inated by the entropy variation of the H/He envelope, | dS/dt | HHe ≫ | dS/dt | Z , a consequence of the smallercompressibility, [ ρ ( dP/dρ )] − , and smaller specific heatof the Z-material compared with the H/He one becauseof its much larger mean molecular weight (see § ⊕ core surrounded by a gaseous en-velope with a Z env =16.6% heavy element mass fraction,which corresponds to a total heavy element enrichment of50%. First of all, we note that, for this value of Z , the cool-ing sequence base on the Y eff formalism (dash-dot line) isvery similar (within < ∼ Y eff approximation for the treatment of a multispecies H/He/ZEOS can be used relatively safely up to Z ≈ M core = 8 M ⊕ , Z env =16.6% (withthe AVL and the SESAME EOS or with a Y equiv ) and theones with M core = 0, Z env =50% (solid line) differ by lessthan 4%. Therefore, for such levels of Z env , the models areless sensitive to the treatment of the Z-element in the enve-lope. As mentioned previously, the sequence based on theANEOS EOS (short-dash line) yields a significantly fastercooling sequence, with a ∼ −
10% smaller radius at 1Gyr, compared with all other sequences. This large uncer-tainty on the EOS will be the major culprit for preventingaccurate determinations of the exoplanet internal composi-tion from their observed radius.We have also explored the effects of the heavy elementdistribution for smaller total enrichments,
Z < <
2% global effect on the radius at a givenage, depending whether the heavy elements are all locatedin the core or are distributed throughout the entire planet,using either an effective He abundance or the AVL withthe SESAME EOS for the thermodynamics of the heavycomponent. Models based on the ANEOS EOS (with heavyelements distributed over the entire planet) still predict thesmallest radii at a given time, with a maximum ∼ T ( P ) and ρ ( P ) profilesfor the various models of our 20 M ⊕ planet at an age of 4Gyr. The first conclusions to be drawn from these tests for aNeptune-mass planet are the following:(i) for a core mass less than 50% of the planet’s mass, avariation of the core composition from pure water to pureiron, the maximum expected difference in mean densities,yields a difference on the radius of less than ∼
10% after 1Gyr. Yet, such a difference is accessible to the observationaldetermination of some transit planet radii, if (when) othersources of uncertainties, in particular on the heavy materialEOS, were (will be) under control.(ii) for a metal-fraction in the envelope Z env < ∼ equiv effective helium fraction in the SCVHEOS. This approximation become more dubious, and evenwrong, above this limit.(iii) if Z < ∼ < ∼ except when using the ANEOSEOS, which yields a ∼ difference . Within this limitfor Z, and given all the other uncertainties in planet mod-elling, the impact of heavy elements on the evolution ofNeptune-mass planets can be mimicked reasonably well byconsidering that the heavy elements are all located in thecore.(iv) For larger heavy element enrichments ( Z > ∼ equiv or AVL) can significantly affectthe cooling and thus radius determination for a given age(more than 10% difference after 1 Gyr). These uncertain-ties, unfortunately, hamper an accurate determination ofthe detailed composition of the heavy element material inthe planet’s interior. In this section, we extend the analysis done in the previ-ous section to a template 1 M J planet. We first analysethe effect of core composition for a core mass of 159 M ⊕ ,corresponding to a total heavy element enrichment Z =50%(see Fig. 6a). We find slightly larger effects than for theNeptune-mass case, with a 7% (15%) difference in radiusbetween the pure water and the pure rock (iron) core cases, Baraffe, Chabrier & Barman: From super Earth to super Jupiter exoplanets.
Fig. 3.
Effect of the composition and internal distributionof heavy elements on the radius evolution for a planet ofmass M P = 20 M ⊕ with a total heavy element mass fraction Z = ( M Z /M P )=50%. (a): Models with M core = 10 M ⊕ of water (solid line), rock (dashed-line) and iron (dottedline), and Z env =0. (b): Models with no core and heavy el-ements distributed over the entire planet. Dash-dotted line:mixture of H/He + Z described by the SCVH EOS withY equiv =0.275 + 0.5=0.775. Long-dashed line: mixture ofH/He + water (SESAME EOS) using the additive volumelaw. Short-dashed line: mixture of H/He + water (ANEOSEOS) using the additive volume law. For comparison, themodel with M core = 10 M ⊕ of water and Z env =0 is shownby the solid line. (c): More realistic models with a 8 M ⊕ core of water (ANEOS EOS) and a Z env = 16.6 % heavy ele-ment enrichment in the envelope. Dash-dotted line: mixtureof H/He + Z env described by the SCVH EOS with Y equiv =0.275 + 0.166 = 0.441. Long-dashed line: mixture of H/He+ water (SESAME EOS) using the additive volume law.Short-dashed line: mixture of H/He + water (ANEOS EOS)using the additive volume law. Also displayed is the modelwith M core = 10 M ⊕ of water and Z env =0 (solid line).respectively. Whether such massive cores can indeed formfor Jovian-mass planets will be considered in § < ∼
3% differences on R for water and < ∼
2% for rock. For these planets, the iron core does not un-dergo a phase transition and iron always remains liquid, ac-cording to the ANEOS EOS. Indeed, central temperaturesand densities are significantly larger than for Neptune-massplanets, with T c ∼ K and ρ c ∼
157 g.cm − at 1 Gyr fora 1 M J planet with an iron core (see Table 2). Fig. 4.
Effect of the EOS on the inner profile as a func-tion of fractional mass for a planet of 20 M ⊕ with amass fraction Z =50% of heavy elements distributed overthe entire planet. In all panels, the curves correspond to:a mixture of H/He + Z described by: the SCVH EOSwith Y equiv =0.275 + 0.5=0.775 (solid line); a mixture ofH/He + water (SESAME EOS) using the additive vol-ume law (dash-dotted line); a mixture of H/He + water(ANEOS EOS) using the additive volume law (dashed line). Upper panel:
Inner profile of the local gravothermal en-ergy, − T dS/dt (in erg/g/s), at the beginning of the evo-lution.
Middle panel : Pressure profile (in dyne/cm ) at 1Gyr. Lower panel : Specific entropy profile (in erg/g/K) at1 Gyr.We explore the effects of the heavy element distribu-tion with Z = 50% (Fig. 6b) and Z = 20% (Fig. 6c). Thelatter case is comparable to the expected level of enrich-ment in Jupiter and Saturn (Saumon & Guillot 2004). Inthe Z = 50% case, both the various thermodynamic treat-ments and localizations of the heavy elements yield signif-icantly different cooling behaviors (see Fig. 6b). The se-quence based on the Y equiv approximation, notably, differsfrom the other ones, as expected from our previous studyfor Neptune-mass planets. At 1 Gyr, the radii obtained withmodels based on the AVL with ANEOS and SESAME, re-spectively, differ by ∼ R between the case withM core =159 M ⊕ , Z env =0 and the case with Z distributedthroughout the entire planet with ANEOS. Interestinglyenough, for the present Jovian conditions, models basedon the ANEOS or SESAME EOS, when metals are mixedthroughout the entire planet, yield smaller differences than araffe, Chabrier & Barman: From super Earth to super Jupiter exoplanets. 9 Table 2.
Radius and central thermodynamic properties of a planet of mass M P = 1 M J at 1 Gyr for two heavy elementmass fractions Z = ( M Z /M P )=50% and Z = 20%. The labels are the same as in Table 1. As mentioned at the end of § Z =50%, the solution with M core =0 and Z = Z env is similar to a solution with M core ∼
10 M ⊕ and the rest ofthe heavy material distributed in the envelope. Z M core EOS (core) Z env
EOS(env) R p T c ρ c ( M ⊕ ) env. ( R J ) (K) (g cm − )0.5 159 W-a 0 SC 0.861 5.3 10 a ) 0.782 7.7 10 a ) 0.994 3.9 10 a Value of Y equiv . Fig. 5.
Internal temperature (K) and density (g cm − ) pro-files for a 20 M ⊕ planet, irradiated by a Sun at 0.045 AU,at an age of 4 Gyr. Solid line: M core = 0 . M p , Z env =0;long-dash line: M core = 0, Z = 0 .
5, AVL SESAME EOS;short-dash line: M core = 0, Z = 0 .
5, AVL ANEOS EOS.for the Neptune-mass planet case, after about 1 Gyr, i.e.at high pressure and moderately high temperature. Indeed,the inner pressure and temperature conditions are very dif-ferent between Neptune-mass and Jupiter-mass planets (seeFigures 5 and 7). For the former ones, P ranges between 10 and 10 dyne.cm − over more than 99% of the totalmass (see Fig. 4) while T ranges from ∼ K. Forthe latter ones, the typical domains are P = 10 − dyne.cm − and T = 10 − K. As mentioned in § P and T dependence of the entropy betweenthe different EOS are more pronounced under the interiorconditions of Neptune-mass planets than for Jupiter-massplanets (see Fig. 2).If, instead of the extreme M core = 0 or Z env = 0 cases,we take a more realistic model with a 10 M ⊕ core, com-parable to what is expected in Jupiter or Saturn, and wedistribute the rest of heavy elements homogeneously in theH/He rich envelope, we find essentially the same evolutionas when the heavy elements are distributed throughout thewhole planet, with no core. This suggests that for mas-sive, metal-rich planets , the evolution is better describedby models which assume that all metals are distributedover the entire planet , since this yields results similar tothe ones obtained with a more realistic distribution, thanby models which assume that all heavy elements are in thecore, with a metal-free, Z = 0 envelope.For a more moderate heavy element enrichment, Z =20% (see Fig. 6c), the treatment of heavy elements in theentire planet, based on an Y equiv or on the AVL withANEOS or SESAME, is found to be less consequential, forthe present Jupiter-mass planet case. The different meth-ods to describe the EOS yield less than 2% differences on R at a given age. The effect of the distribution of heavyelements (all in the core versus all distributed over the en-tire planet) is slightly more consequential, with up to 4%difference on R .Figure 7 portrays the internal T ( P ) and ρ ( P ) profilesfor the various models of our 1 M J planet at an age of 4Gyr.The conclusions derived from these tests for a Jupiter-mass planet can be summarized as follows:(i) for a core mass less than 50% of the planet’s mass, avariation of the core composition between pure water andpure rock (iron) yields an effect on the radius of < ∼ Fig. 6.
Effect of the composition and the distribution ofheavy elements on the radius evolution of a planet of 1 M J (318 M ⊕ ). (a): Total heavy element enrichment Z = 50%.Models with M core = 159 M ⊕ of water (solid line), rock(dashed-line) and iron (dotted line), and Z env =0. (b): Total heavy element enrichment Z = 50%. The solid linecorresponds to a model with M core = 159 M ⊕ of waterand Z env =0. The other curves correspond to models withno core and heavy elements distributed over the entireplanet: mixture of H/He + Z described by the SCVH EOSwith Y equiv =0.275 + 0.5=0.775 (dash-dotted line); mix-ture of H/He + water (SESAME EOS) using the additivevolume law (long-dashed line); mixture of H/He + water(ANEOS EOS) using the additive volume law (short-dashedline). (c): Total heavy element enrichment Z = 20%. Solidline: model with M core = 63 . ⊕ of water and Z env =0;dash-dotted-line: no core and mixture of H/He + Z de-scribed by the SCVH EOS with Y equiv =0.275 + 0.2 =0.475; long-dashed line: no core and mixture of H/He +water (SESAME EOS) with the AVL and Z env =0.20; short-dashed line: no core and mixture of H/He + water (ANEOSEOS) using the AVL and Z env =0.20.(ii) for a global metal-enrichment Z < ∼ R at agiven age).(iii) For significant heavy element enrichment ( Z ∼ Fig. 7.
Internal temperature and density profiles for a 1 M J planet at an age of 4 Gyr. Solid line: M core = 0 . M p ;long-dash line: M core = 0, Z = 0 .
5, AVL SESAME EOS;short-dash line: M core = 0, Z = 0 .
5, AVL ANEOS EOS.with about a 10 M ⊕ core and the rest of the material in theenvelope, while a model with a metal-free envelope and allheavy elements in the core differs substantially from thesesequences. As mentioned in § − T dSdt ) core to the cooling of the planet. Asan other extreme, Burrows et al. (2007) arbitrarily assumethat the specific entropy of the core material is the sameas that of the H/He envelope, therefore overestimating theheat release of the dense core.According to the ANEOS or SESAME EOS, the typicalheat capacity of water in the cores of the presently studiedNeptune-mass and Jovian-mass planets is C v ∼ × -5 × erg g − K − , for typical temperatures T ∼ × K and pressures P ∼ − dyne cm − (seeFigs. 5 and 7). This is about 1/3 the specific heat ofH/He in the envelope. This corresponds to variations ofthe specific heat from the high-temperature nearly ideal gaslimit, ∼ R /A (where A is the species mean atomic weightand R is the perfect gas constant), to regimes which in-clude potential-energy contributions associated with trans- araffe, Chabrier & Barman: From super Earth to super Jupiter exoplanets. 11 lational and librational modes (3 R /A maximum for threetranslational and three librational modes per molecule). Ifthe species enters the solid phase, the specific heat decreasesrapidly (Debye regime) and eventually vanishes. The spe-cific heat of rock or iron is even smaller (larger atomic mass)and the contribution of cores made of such materials to thecooling history of planets is negligible. For planets in themass range 20 M ⊕ - 1 M J , as explored in the previous sec-tion, with core masses less than 50% of the planet’s mass,and with cores made up of water, the release of gravitationalenergy ( P dVdt ) and of internal energy ( dUdt ) never exceeds 40% of the total released energy, T ( dSdt ). For Neptune-massplanets, the dominant contribution from the core is due toits contraction, i.e the P dVdt work, during the first 0.5 - 1Gyr of evolution, and by the release of its internal energy atolder ages. For jovian-mass planets, the contraction of thecore provides the dominant contribution to its gravother-mal energy during the entire planet evolution.The assumption of zero-T or uniform low temperatureaffects the structure of the cores, because of the temper-ature dependence of the P ( ρ ) relations predicted by cur-rent EOS (see § dUdt ) core , is forced to be zero, andthe small compressibility of water at low T drastically un-derestimates the true contraction of the core during theplanet evolution, and thus the release of gravitational en-ergy. We have analysed this assumption by imposing a con-stant and uniform temperature of 300 K in the water coreof the planet models analysed in the previous section, us-ing ANEOS and SESAME. For the jovian-mass planet (1 M J ) with M core =159 M ⊕ , the effect on the radius is neg-ligible (less than 2%) after 1 Gyr. On the Neptune-massplanet (20 M ⊕ ), the effect is larger, the largest effect beingfound with the SESAME EOS for the model with a 10 M ⊕ core, that yields a 6% effect on R after 1 Gyr. The largesttemperature variations of the EOS are indeed found at lowpressure ( P < K c ∝ ρ / ,and the conductive flux to dominate the convective one, F cond = − K c ∇ T > F conv . It starts to dominate earlierfor rock and iron cores, compared to water cases. Theseresults, however, are hampered by the uncertainties in theconductive opacities calculated with the ANEOS EOS orwith Potekhin (1999) for such materials. We have checkedthe effect of such an uncertainty on the cooling history inthe case of the 20 M ⊕ planet with M core =10 M ⊕ of waterwith the SESAME EOS, as this sequence provides a casewith the largest energetic contribution from the core, for en-richments Z ≤
50 %. Over the entire evolution, heat is pre-dicted to be transported by convection within the core andthe temperature gradient is given by the adiabatic gradient.We have arbitrarily decreased the conductive opacities sothat conduction now dominates over convection. The corethus becomes isothermal. The effect remains small on the radius evolution (maximum 3% on R at a given age com-pared to the convective case). Therefore, the uncertaintyin the heat transport efficiency of the core has a smallerimpact than neglecting the temperature dependence of thecore material and its global energetic contribution.To conclude this section, the present calculations showthat neglecting the thermal and gravitational contributionsfrom the core, by assuming zero-T or low uniform T, in cur-rent planet modelling, leads to a maximum effect of ∼
5. Extreme metal-enrichment: the two specificcases of HD149026b and GJ436b
In this section, we analyse extreme cases of heavy elementenrichments (
Z >
The discovery of the Saturn-mass planet HD149026b (Satoet al. 2005) revealed an unexpectedly dense planet, witha mass M P = 0 . ± . M J and a radius R P = 0 . ± . R J , i.e. a mean density ¯ ρ = 1 . ± .
35 g.cm − . Forcomparison, Saturn has a mass of 0.3 M J but a radius of ∼ R J , i.e. ¯ ρ = 0 .
66 g.cm − . The transit planet is orbitinga G-type star at an orbital distance a = 0 .
042 AU, about230 times closer to its Sun than Saturn. The age of thesystem, ∼ (2 ± .
8) Gyr, is also younger than our SolarSystem. It is important to understand the nature and theorigin of this puzzling planet, and in particular to knowwhether current planet formation scenarios, in particularthe core accretion model, can explain it. This requires aknowledge of its structure and composition, which can onlybe inferred from theoretical models. Several authors havetried to infer the inner structure of this planet under theusual assumptions described in §
2. The models assume thatalmost all heavy elements are in the core, and the H/Heenvelope is either free of metals (Burrows et al. 2007), or ismoderately enriched, with Z = Z ⊙ (Ikoma et al. 2006) or Z = 0 .
045 (Fortney et al. 2007). Ikoma et al. (2006) havealso investigated a case with Z env = 0 .
37. Fortney et al.(2007) use a zero-temperature EOS for the core and thusignore its heat content contribution. They find negligibleeffect when using a nonzero temperature EOS for the icemixture given by Hubbard & Marley (1989). This latterEOS provides pressure-density relations appropriate for thedescription of warm adiabatic mixtures ( T ∼ ) K, butdo not explicitly account for the temperature dependence.Ikoma et al. (2006) also use this EOS and assume a uniform Fig. 8.
Evolution of a planet with the characteristics ofHD149206b ( m P =116 M ⊕ ∼ . M J ) with different heavyelement distributions. Solid line: model with M core =80 M ⊕ of water (ANEOS EOS) and Z env =0. The othercurves correspond to models with heavy elements dis-tributed over the entire planet. Dash-dot line: mixture ofH/He + Z described by the SCVH EOS with an effectivehelium abundance, Y equiv =0.275+0.69=0.965, which cor-responds to Z = 0 . i.e
80 M ⊕ of heavy elements. Long-dash line: mixture of H/He + water (SESAME EOS) usingthe AVL with Z =69%. Short-dash line: mixture of H/He +water (ANEOS EOS) using the AVL with Z =69%. Dottedline: Mixture of H/He + water (ANEOS EOS) using theAVL with Z =51%, i.e
60 M ⊕ of heavy elements.temperature in the core and a specific heat C v ∼ ergg − K − to account for the core heat release. According toall these models, the total mass of heavy elements in theplanet lies in the range ∼ ⊕ , i.e Z ∼
35% - 80%.We will test the impact of these main assumptions donein current structure models of HD149026b, namely the dis-tribution of all heavy elements in a core and the use ofzero-T EOS. Since the present study focusses on the sen-sitivity of the structure and evolution for a given setup ofatmosphere models, we use, as outer boundary condition tothe inner structure, atmosphere models with solar compo-sition, Z ⊙ , irradiated by a G-star at 0.04 AU, even thoughthe results are expected to change to some extent with thecomposition of the atmosphere (see e.g Burrows et al. 2007).This issue will be explored in a forthcoming paper. We ob-tain a good match of the planet’s radius with a model witha water core of 80 M ⊕ and a metal-free envelope, Z env = 0.Figure 8 displays this model (solid line), calculated with theANEOS EOS in the core. The use of the SESAME EOS forwater in the core yields a 3% smaller radius, still providinga good match to the observed value. Figure 8 also showsthe impact of the heavy element distribution, with modelswhere the 80 M ⊕ of heavy elements are distributed over theentire planet. As anticipated from the studies conducted in § equiv in the SCVH EOS to handle theheavy element contribution yields, for such a high metalfraction, an evolution that differs drastically from the other ones, even though, coincidentally, it gives a good solution atthe age of the system . As expected from the tests performedin §
4, models with 80 M ⊕ of heavy elements mixed withH/He throughout the entire planet, using the AVL withthe SESAME (long-dash line) or the ANEOS (short-dashline) EOS for water, yield denser structures. The latter oneyields a radius ∼
25% smaller than the observed value, aneffect larger than changing the core composition from waterto rock ( ∼
10% effect on R ), for the same core mass of 80M ⊕ , and similar to the one obtained when changing from apure water to a pure iron core, a rather unlikely solution.Instead of the two aforedescribed extreme heavy ele-ment distributions, we have also examined a more realisticmodel with a 20 M ⊕ core, similar to what is predicted forSaturn, and the remaining 60 M ⊕ mixed within the H/Heenvelope, using the AVL both with SESAME and withANEOS. As anticipated from the tests performed in § ⊕ of heavyelements, instead of 80, mixed with H/He throughout theentire planet, described with the ANEOS EOS for waterand the AVL formalism (dotted line in Fig. 8), whereaswhen using the SESAME EOS for water, 70 M ⊕ are re-quired. As mentioned above and as shown in our previ-ous studies, such models are equivalent to models with asmall core mass and a significantly enriched envelope. Thisshows that for a given heavy element material, water in thepresent case, the effect of modifying its internal distribu-tion (everything in the core or a fraction redistributed inthe H/He envelope) has by itself a large impact, yieldingan uncertainty on the amount of heavy material required toreproduce the observed radius of 80-60=20 M ⊕ . As shownin the previous sections, assuming all heavy elements to bein the core yields structures less dense than when these el-ements are mixed throughout the envelope, so that thesemodels require a larger amount of heavy material to matcha given radius.Finally, as done in § ⊕ water core, using ANEOS and SESAME EOS. Thelargest effect is found with this latter EOS, the model witha uniform low temperature yielding a radius ∼ The first Neptune-mass transiting planet has been discov-ered recently (Gillon et al. 2007a), with a mass M P =22 . ± . ⊕ and a radius R P = 25200 ± ± . R J . The planet is orbiting an M-type star of ∼ M ⊙ at an orbital distance a ∼ .
028 AU. According to themodels of Fortney et al. (2007), Gillon et al. (2007a) sug-gest that the planet is composed predominantly of ice witha thin H/He envelope of less than 10% in mass. Based onSpitzer observations, Gillon et al. (2007b) and Deming etal. (2007) determine a slightly larger radius, with R P =0.386 ± . R J . We adopt this value in the following and we as-sume an age for the system of 1-5 Gyr, as it is essentiallyunconstrained by the observations. Models available in theliterature to determine the inner structure of this planet araffe, Chabrier & Barman: From super Earth to super Jupiter exoplanets. 13 use temperature-independent EOS to describe the core andthus ignore its thermal contribution (Fortney et al. 2007;Seager et al. 2007; Adams et al. 2007). We will test thisassumption.We have calculated models characteristic of GJ436b.We use solar metallicity atmosphere models and, given thelow luminosity of the parent star, we neglect presently theirradiation effects. Recent determinations of GJ436b irra-diation induced temperature (i.e. brightness temperature)indeed suggest that the evolution is not likely to be signifi-cantly altered by irradiation (Demory et al. 2007). A modelwith a core of 21 M ⊕ made of water and a metal-free en-velope, Z env =0, provides a good match to the observedradius, as illustrated in Fig. 9 (solid and dashed lines). TheANEOS EOS for water (solid line) yields a slightly larger( ∼ ⊕ . Given the high mass fraction of heavy ma-terial of this planet, more than 80%, the freedom to varyits distribution is limited. Assuming a core of 21 M ⊕ withZ env =0 or a slightly smaller core with some heavy elementenrichment in the envelope, a more realistic solution, e.g.a model with M core = 20 M ⊕ and Z env =0.38, yield lessthan 8% variations on the radius, still within the observederror bars. According to the tests performed in § ⊕ planet, larger values of Z env ( Z env > ∼ § ∼
6% effect with SESAME. In thelatter case, however, the model lies outside the observa-tional error bars (see dotted line in Fig. 9). As seen in Fig.10, for planets with such a large fraction of heavy mate-rial, and conversely with such a modest gaseous H/He frac-tion ( < ∼ T dS/dt , remains small, aproper calculation should take into consideration the corecontribution. Note also that even a modest H/He fractionaffects the radius determination, as illustrated in Table 3by comparing the present results with the value of the ra-dius corresponding to a pure 22.6 M ⊕ icy planet with nogas envelope, as derived from the fitting formulae of Seageret al. (2007).Importantly enough, taking into account the thermaland gravitational energetic contributions from the core be-comes even more crucial if the irradiation effects fromthe parent star are important. Fig. 9 shows the evolu-tion of the same planet model, with M core = 21 M ⊕ ofwater (SESAME EOS), but with an incident stellar flux F inc = ( R ⋆ a ) F ⋆ , where F ⋆ is the flux from the parentstar, six times larger than for GJ436a. This corresponds to Fig. 9.
Evolution of a planet characteristic of GJ436b( M P =22.6 M ⊕ ,) with different heavy element distributions.Models with no irradiation: Solid line: model with M core =21 M ⊕ of water (ANEOS EOS) and Z env =0; short-dashline: model with M core = 21 M ⊕ of water (SESAME EOS)and Z env =0; dotted line: test model with M core = 21 M ⊕ of water (SESAME EOS) and Z env =0, assuming an uni-form temperature of 300 K in the core. Effect of irradia-tion, with F inc = 6 × F inc for GJ436b (see text): long-dashline: same as short-dash line with irradiated atmospheremodels; dash-dot line: same as dotted line with irradiatedatmosphere models Fig. 10.
Various contributions to the gravothermal energy, E g = − T dS/dt , normalized to the total value, for a planetcharacteristic of GJ436b ( M P =22.6 M ⊕ ), with a 21 M ⊕ core of water with the SESAME EOS. Solid line: globalcontribution from the core; dash-dot line: contraction workcontribution from the core; long-dash line: thermal contri-bution from the core; short-dash line: global contributionfrom the H/He 1.6 M ⊕ envelope. Table 3.
Radius of a 22.6 M ⊕ planet at 2 Gyr with dif-ferent water core masses, EOS and levels of irradiation,characterised by a stellar incident flux, F inc . Two cases areconsidered: no irradiation (F inc =0) and F inc =6 x F inc ofGJ436b. Comparison is done with the pure water (ice) caseof Seager et al. (2007). Results based on the assumption ofa uniform core temperature of T=300K are also given. M core F inc EOS (core) R p ( M ⊕ ) ( R J )21 0 aneos 0.39621 0 aneos T=300K 0.37921 0 sesame 0.38121 0 sesame T=300K 0.35822.6 0 Seager 0.28521 6 x F GJ436inc aneos 0.45221 6 x F
GJ436inc aneos T=300K 0.42121 6 x F
GJ436inc sesame 0.44621 6 x F
GJ436inc sesame T=300K 0.397 a parent star about 50% hotter. In that case, neglectingthe temperature dependence of the EOS in the core and itscontribution to the planet’s cooling yields a ∼
11% (7%)smaller radius with SESAME (ANEOS) EOS (dash-dot vslong-dash curves). This stems from the larger planet in-terior temperature and entropy in the irradiated sequencecompared with the non-irradiated one. In the irradiatedcase, the core temperature ranges from 5000 to ∼ × K(from the bottom of the H/He envelope to the center), to becompared with 3000 to ∼ K in the non-irradiated case,characteristic of the temperatures expected in the rocky/icypart of Neptune and Uranus (Guillot 2005).
6. Evolution of super Jupiter planets: Hat-P-2band deuterium burning planets
The final part of our study is devoted to ”super-Jupiter”extra-solar planets, with masses M P ≫ M J . It is mo-tivated by the growing number of discoveries of massiveextra-solar planets, in the mass regime overlapping the oneof low-mass brown dwarfs, issued from a different forma-tion mechanism. These discoveries feed the heated debateconcerning the definition of a planet and the possibilityto distinguish planets from brown dwarfs of similar mass.In this context, one of the most remarkable discoveries isthe transiting super-Jupiter object HAT-P-2b (also namedHD147506b), with a mass M p = 9.04 M J and a radius R p = 0.982 R J (Bakos et al. 2007). Loeillet et al. (2007) reanal-ysed the orbital parameters of the system and find similarvalues, M p = 8.64 M J , R p = 0.952 R J . In that case, themass-radius relationship offers a unique information to inferthe gross composition of the object and to determine its realnature, low-mass gaseous brown dwarf or very metal-richmassive planet. Bakos et al. (2007) suggest that the meandensity of this planet is only marginally consistent withmodel predictions for an object composed predominantlyof H and He, and requires the presence of a large core, withM core > ∼
100 M ⊕ . Here, we calculate more thoroughly the in-ternal structures consistent with the radius determination,along the lines described in the previous sections. We findthat a total amount of ∼ ⊕ of a water(rock) component is required to explain the radius at thepresent age, as illustrated in Fig. 11. The rather large pre- Fig. 11.
Evolution of a super-Jupiter planet with the char-acteristics of HAT-P-2b ( M p = 9 M J ) for different distribu-tions of heavy elements. Irradiation effects from the par-ent star are taken into account, adopting a time-averageorbital distance a = 0 .
077 AU, for the proper eccentricity e =0.52 and semimajor axis of the relative orbit a rel =0.0677AU. Solid line: M core = 600 M ⊕ of water (ANEOS EOS)and Z env =0. Long-dashed line: M core = 350 M ⊕ of wa-ter (ANEOS EOS) and Z env =0. Dash-dotted line: 600 M ⊕ of heavy elements with 100 M ⊕ in the core (water, withANEOS EOS) and 500 M ⊕ in the envelope, i.e. Z env =0 .
18, mimicked by an Y equiv =0.275 + 0.18. Short-dashedline: H/He brown dwarf with Z = Z ⊙ .dicted range of heavy material enrichment stems from thelarge observational error bars. As seen in the figure, a browndwarf (Baraffe et al. 2003), i.e. a gaseous H/He object withsolar metallicity, is predicted to have at this age a radiusmarginally consistent at the 2 σ limit with the observations(short-dashed line) and thus can be excluded at the > ∼ ∼ M J , well above the opacity-limit for frag-mentation, m ∼ M J (Whitworth & Stamatellos 2006),the expected minimum mass for brown dwarf formation.These two distinct astrophysical populations should thenoverlap over a substantial mass domain.In this context, it is interesting to explore the fate of planets massive enough to ignite deuterium-fusion in theircentral parts, i.e. with M p > ∼ M J (Saumon et al. 1996,Chabrier et al. 2000). Indeed, recent calculations of plane-tary population synthesis based on improved core-accretionmodels of planet formation (Alibert et al. 2005a) and a widevariety of initial conditions, predict the formation of super-massive planets, up to ∼ M J , with rocky/icy coremasses up to several 100 M ⊕ (Ida & Lin 2004; Mordasiniet al. 2007, 2008). If planets with a massive core canform above the aforementioned deuterium burning mini-mum mass, a key question is to determine whether or notthe presence of the core can prevent deuterium burning tooccur in the deepest layers of the H/He envelope. Guidedby the results of Mordasini et al. (2008), we have consid- araffe, Chabrier & Barman: From super Earth to super Jupiter exoplanets. 15 ered a 25 M J planet with a 100 M ⊕ core. Independentlyof the composition of the core material (water or rock),deuterium-fusion ignition does occur in the layers above thecore and deuterium is completely depleted in the convec-tive H/He envelope after ∼
10 Myr. The same conclusionholds for a core mass of several 100 M ⊕ . These results high-light the utter confusion provided by a definition of a planetbased on the deuterium-burning limit.
7. Planet evolutionary models and mass-radiusrelationships
Some of the internal structures determined in the previoussections in order to match the observed radii and inferredmean densities of transit planets are rather unusual, andthe possibility to form such compositions must be exam-ined in the context of our current understanding of planetformation. According to current models of planet forma-tion, which include migration (Alibert 2005a, 2006;), up to ∼
30% of the heavy material contained in the protoplane-tary disk can be incorporated into forming giant planets(Mordasini et al. 2007). The maximum mass for a sta-ble protoplanetary disk is M D < ∼ . M ⋆ , so that, for a 1 M ⊙ parent star of solar composition, Z ≃ M Z ≈ . × . × . × (3 . × ) ≈
200 M ⊕ of heavymaterial can be accreted onto the planet. Present planetformation models (Mordasini et al. 2007) reach about thislimit for very massive planets, for a 1 M ⊙ parent star.Therefore, in principle , according to these calculations, aheavy element mass fraction >
50% can not be excluded,even for Jovian-type planets. This requires, however, accre-tion rates from the planet’s feeding zone significantly largerthan the values typical of the early ”runaway growth” ac-cretion phase, ∼ − M ⊕ yr − (Ward 1996). It also im-plies migration of the planet’s embryo, or some substantialorbit eccentricity. Indeed, in the absence of migration oreccentricity, tidal interactions between the planet and thedisk are supposed to lead to the opening of a gap oncethe planet has reached about a Saturn total (gas+solid)mass, ∼
100 M ⊕ , for the minimum mass solar nebulae con-ditions, after which planetesimal accretion decreases dra-matically (Lin & Papaloizou 1986). Note that, if the plan-ets formed originally at large orbital distances and migratedinwards, they are expected to have a significant content ofheavy material, given the larger available mass reservoir.Furthermore, short-period planets are expected to have alarger fraction of heavy material than planets located fur-ther away for two reasons. First of all, more impacts fromlow eccentricity orbit planetesimals are expected. Secondof all, the closer the orbit of the planet, the larger its or-bital speed, ( GM ⋆ /a ) / , compared to its escape velocity,(2 GM ⋆ /R ⋆ ) / , making planetesimal ejection less efficient(Guillot 2005).A second concern is the possibility to have the type ofheavy element distributions examined in the present paper.As mentioned earlier, the fact that gas accretion is triggeredonce the core mass has reached about 6 to 10 M ⊕ seemsto be a rather robust result (Mizuno 1980, Stevenson 1982,Pollack et al. 1996, Alibert et al. 2005a, Rafikov 2006). Forlarger enrichments, the rest of heavy elements should thusbe mixed with the H/He envelope. As shown by Stevenson(1982), the maximum amount of heavy material (compared with the H/He medium) which can be redistributed uni-formly by convection throughout the planet from an ini-tially stably stratified configuration is of the order of theplanet’s mass. So in principle, a planet with no or small corebut all or most of the heavy material being redistributedthroughout the gaseous envelope is possible. As shown inour study, for Jovian type planets (see § Z = 50%, if we assume a core of 10 M ⊕ ,comparable to what is expected in Jupiter or Saturn, andwe distribute the rest in the envelope, we find essentiallythe same evolution as when the heavy elements are dis-tributed throughout the whole planet, with no core. Thissupposes, of course, that large-scale convection remains effi-cient enough to redistribute homogeneously the heavy ma-terial in the H/He envelope (Chabrier & Baraffe 2007). Notethat this result also holds for lower mass planets with simi-lar enrichment, Z = 50%, if the core mass remains < ∼ M p .Therefore, the enrichment in heavy material and theinternal compositions explored in the present calculationshave at least some reasonable theoretical foundation andcan not be excluded a priori. These arguments can be ex-amined for the case of Hat-P-2b, for which the observedmass-radius relation requires a total mass of heavy ele-ments of at least 200 to 300 M ⊕ , i.e. a mean mass-fraction Z > ∼ F e/H ] = 0 .
1) star, with M ⋆ = 1 . M ⊙ . The maximumtotal amount of heavy element material available in theparent disk was thus about 900 M ⊕ . So the required con-tent of heavy elements would be close to the aforementioned ∼
30% upper limit of accretion efficiency. If both the massand the density of Hat-P-2b are confirmed, this object thuslies at the edge of what is predicted to be possible withinthe current standard core-accretion scenario. We suggest,however, an alternative formation scenario, namely thatthe formation of Hat-P-2b involves collision(s) with one orseveral other massive planets. Besides forming big cores,collisions will lead to a substantial loss of the gaseous en-velope, thus to a larger relative fraction of heavy elements.Furthermore, gravitational scattering among planets gen-erally results in a tight orbit with a large eccentricity forone of the planets, which could explain Hat-P-2b’s largeeccentricity, and to the ejection of the other planet(s) ordebris to interstellar medium. Such scattering processes be-tween planets seem to provide a viable and possibly dom-inant scenario to explain the observed eccentricity distri-bution of exoplanets (Chatterjee, Ford & Rasio 2007). Wethus speculate that Hat-P-2b was formed from such colli-sion processes. Note that a scenario based on giant impactshas also been suggested to explain the large heavy elementcontent of HD149026b (Ikoma et al. 2006).Finally, in the present calculations, the heavy material issupposed to be composed entirely of one single component,water, rock (silicates) or iron. This is of course a simplifyingassumption, as the inner composition of the planets is ex-pected to possess various fractions of each of these compo-nents. The water to silicate fraction, in particular, will varydepending whether the object has formed inside or beyondthe ice line. Migration, however, will affect this fraction,as the migration process yields a larger collision rate of theplanet embryo with rocky planetesimals and thus a decreas-ing abundance of volatiles. All ratios M ice /M rock from 0 to1 are thus probably possible. In any event, although the ex- act composition of the heavy material component may havesome implication on the mass-radius relationship for Earth-like planets (Valencia et al. 2006; Sotin et al. 2007; Seageret al. 2007), the present study shows that, for planets with agaseous H/He envelope of mass fraction M HHe > ∼ M p ,current uncertainties on the EOS and the heavy elementdistribution lead to larger uncertainties on the planet’s ra-dius determination than the effect due to different internalcompositions (see Tables 1-2 and § In tables 4-5 , we present a subset of our grid of planetarymodels from 20 M ⊕ to 1 M J with different levels of heavyelement enrichment, Z = Z ⊙ , 10%, 50 % and 90%. For thepurpose of the present paper, this grid is restricted for themoment to solar metallicity atmosphere models with twoexternal atmospheric conditions, namely: (i) no irradiation(non-irradiated planets) and (ii) irradiation effects from aSun at 0.045 AU, which is a typical incident irradiation formost of the transit planets discovered up to now. The effectof different levels of irradiation and different atmosphericcompositions will be explored in a forthcoming paper. Wehave compared our models in this mass range with the mod-els of Fortney et al. (2007). We find an excellent agreementfor jovian-mass planets using the same assumptions, i.e same core masses and level of irradiation. Small differencesoccur for Neptune-mass planets (Fortney’s 17 M ⊕ model)for a large core mass fraction ( M core = 10 M ⊕ ). This isdue most likely to the zero-temperature EOS assumptionadopted in Fortney et al. (2007) for the core EOS, as shownearlier in this paper (see § water using the SESAME EOS. The departuresfrom this case due to different heavy material compositions(rock, iron), distributions within the planet’s interior andEOS have been quantified in the previous sections of thispaper (see Tables 1-2 for an illustration and § § Z = 0 .
02, have the same composition as thatof a brown dwarf, but such objects with masses as low as afew Neptune-masses are of course not realistic. These mod-els are an extension to planetary masses of the models ofBaraffe et al. (2003). Note that the 20 M ⊕ planet modelwith an enrichment as low a Z = 0 .
02 and with, conse- The complete grid of models, from 10 M ⊕ to 10 M J , is avail-able on http://perso.ens-lyon.fr/isabelle.baraffe/PLANET08 quently, a low mean density, expands under the effect ofirradiation. This model is not included in Tab. 5, beingmeaningless. – For all masses and Z = 10%, all heavy elements arelocated in the core, since for such low Z , we have shownthat their distribution has only a modest effect on theradius. – For larger enrichments ( Z = 50%-90%), we make a dis-tinction between planets below M p < ∼
20 M ⊕ , hereafterdenominated as ”light planets”, and more massive plan-ets. Because all planets are expected to have a core ofabout ∼
10 M ⊕ , the following distributions seem to bethe most realistic: – For light planets, all heavy material is located in thecore. – For more massive planets, we have shown that dis-tributing the heavy elements over the entire planetis similar to distributing them partly in a core of atmost ∼
10% of the mass of the planet and partlyin the envelope. We thus adopt such a distributionfor these objects: the heavy material is distributedover the entire planet, using the AVL with SESAMEEOS. The uncertainties due to the EOS (ANEOSversus SESAME) for this type of distribution havebeen quantified in § § – For planets with masses > M J , we only provide modelswith Z =10%, since enrichments as large as 50% or 90%correspond to an amount of heavy material greater thanthe quantity available in protoplanetary disks aroundsolar type stars.Figure 12 shows the mass-radius relationships for plan-ets in the mass range 20 M ⊕ - 1 M J , for different levelsof irradiation and heavy element enrichments. The modelsare compared to the observed mass-radius data of transitingplanets. Note that in the case of irradiation (right panel),the theoretical planetary radius is not corrected from theeffect of the atmospheric extension. Such a correction adds ∼
4% to the calculated radius for this level of irradiation(Baraffe et al. 2003).
8. Discussion and perspectives
In this paper, we have explored the uncertainties in currentplanetary structure and evolutionary models arising fromthe treatment of the heavy material component in the in-terior of these planets. The study covers a mass range from10 M ⊕ to 1 M J . Our main results can be summarized asfollows: – The ideal mixing entropy contribution arising from theheavy Z-material is found to be inconsequential on theplanet’s evolution , within the limit that the variation ofthe degree of ionization of these elements along the evo-lution is presently ignored. This mixing entropy, how-ever, represents about 10 to 20% of the total, H/He + Zelements, entropy, and thus modifies the internal isen-trope. A proper calculation of the planet’s structure fora given entropy should thus include this contribution. – For a metal mass fraction in the envelope Z env < ∼ equiv effective helium mass fraction inthe SCVH EOS. Above this limit, this approximation araffe, Chabrier & Barman: From super Earth to super Jupiter exoplanets. 17 Fig. 12.
Mass-radius relationships for planets in themass range 10 M ⊕ - 1 M J at different ages, as indi-cated in the panels, and different levels of heavy ele-ment enrichment: Z = Z ⊙ (solid lines); Z =10% (short-dash lines); Z =50% (dash-dot lines); Z ∼ pont/TRANSITS.htm).becomes more and more incorrect and yields erroneouscooling sequences. – For core mass fractions less than 50% of the planet’smass, a variation of the core composition between purewater and pure rock (iron) yields a difference on theradius of less than 7% (15%) after 1 Gyr, for all theplanet masses of interest. – For a total mass fraction of heavy elements
Z < ∼ – For heavy material enrichments
Z > – For metal-rich (
Z > ∼ light planets ( < ∼
20 M ⊕ ), sincethe planets are expected to have a massive ∼
10 M ⊕ core, it seems realistic to put all the heavy material inthe core. – For massive metal-rich planets ( M p > ∼
50 M ⊕ and Z >
Table 4.
Radii of planets (in R J ) in the mass range 20 M ⊕ -1 M J for different levels of heavy element enrichment Z (seetext, § Z M p / M ⊕ R . R R Table 5.
Same as table 4 for irradiated models by a Sunat 0.045 AU.
Z M p / M ⊕ R . R R models which assume that the heavy elements are dis-tributed throughout the entire planet than by modelswith all heavy elements in the core and none in thegaseous envelope. The former models yield results sim-ilar to the ones obtained with a more realistic distri-bution, namely a ∼
10 M ⊕ core and the rest of heavymaterial distributed in the envelope. – The temperature dependence of the heavy material EOSand the release of gravitational and thermal energy ofthe core have negligible effects on the cooling historyof massive planets (saturnian and jovian masses), inde-pendently of the core mass. For Neptune mass planets,these effects are significant ( ∼
10% difference on the ra-dius after 1 Gyr) for extreme heavy element enrichments(
Z > P ≫ T ≫ ∼ −
10 M ⊕ , these heavy material EOSrelated uncertainties are the major culprit for preventingaccurate determinations of the exoplanet internal composi-tion from the observed mass and radius. ”Accurate” meansin this context at a level better than yielding < ∼
10% varia-tions on the radius. For Earth or super-Earth planets withno gaseous envelope, a radius measurement accuracy bet-ter than 5% is expected to allow to distinguish icy fromrocky internal compositions (Valencia et al. 2007). However,as shown in § <
30% by mass)H/He contribution, varying the distribution of the heavymaterial within the planet has by itself a larger impact,not mentioning the one due to the uncertainty in the EOS.Therefore, for planets above ∼
10 M ⊕ , massive enough toaccrete an H/He envelope, it seems difficult to determine precisely the internal composition with current structuremodels, as the effect of the heavy material composition onthe radius is blurred by the presence of a gaseous enve-lope. Even a modest fraction ( ∼ ⊕ mass planet retaining a 10% H/He envelope is ∼ e = R eq / ( R eq − R pol ),where R eq and R pol denote the equatorial and polar ra-dius, respectively, with future transit observations (Seager& Hui 2002; Barnes & Fortney 2003). This in turn leads tothe determination of the rotation rate and thus of the cen-trifugal potential, providing a more stringent constraint onthe internal structure of the planet. (v) Last but not least,the perspective, on the observational front, of direct plan-etary atmosphere observations (LYOT project, GEMINI,ELT, DARWIN/TPF or their precursors) and transit detec-tions (CoRoT, Kepler) will improve our knowledge of theirsurface composition and radius measurements and provideimportant constraints on the planet’s content in heavy ma-terial.Finally, we have shown in this paper that massive ( > ∼ several Jupiter mass) planets may form from two differ-ent avenues, namely the standard core accretion scenario(Mordasini et al 2008), and giant impacts between massiveplanets or planet embryos. This latter process is likely toyield very metal-enriched and thus very dense massive plan-ets, with a finite eccentricity, as a result of planet scatter-ing. Objects like HD 149026b, Hat-P-2b or the very recentlydiscovered HD 17156b (Gillon et al. 2007c) could be the il-lustrations of this latter planet formation mechanism. Theobservation of Hat-P-2b, together with the numerous obser-vations of free floating brown dwarfs of a few Jupiter-masses araffe, Chabrier & Barman: From super Earth to super Jupiter exoplanets. 19 (Caballero et al. 2007) shows that planets and brown dwarfshave a substantial (about one order of magnitude in mass)mass domain overlap. As we have shown in this paper,planets massive enough to exceed the deuterium-burningmass limit will indeed ignite this reaction at the bottomof their H/He rich envelope, at the top of the core. Thisis one more evidence, if it were still necessary, that usingthe deuterium-burning limit as a criterion to distinguishplanets from brown dwarfs has no valid foundation.The complete grid of models, from10 M ⊕ to 10 M J , is available onhttp://perso.ens-lyon.fr/isabelle.baraffe/PLANET08 Acknowledgements.
The authors thank D. Saumon for useful discus-sions during the elaboration of this work and our referee, J. Fortney,for his valuable comments. Part of this work was done as I.B and G.Cwere visiting the University of Toronto and the Max-Planck Institutfor Astrophysics in Garching; these authors thank these departmentsfor their hospitality. The financial support of Programme National dePhysique Stellaire (PNPS) and Programme National de Plan´etologieof CNRS/INSU (France) is aknowledged.
References
Adams, E.R., Seager, S., Elkins-Tanton, L. 2007, ApJ, in press (astro-ph/0710.4941)Alibert, Y., Mordasini, C., Benz, W., Winisdoerffer, C. 2005a, A&A,434, 343Alibert, Y., Mousis, O., Mordasini, C., Benz, W. 2005b, ApJ, 626,L57Alibert, Y., et al., 2006, A&A, 455, L25Bakos, G.A., Kovacs, G., Torres, G., et al. 2007, ApJ, submitted,astro-ph/7050126Baraffe, I., Alibert, Y., Chabrier, G., Benz, W. 2006, A&A, 450, 1221Baraffe, I., Chabrier, G., Barman, T., Allard, F. & Hauschildt, P.,2003, A&A, 402, 701Barman, T., Hauschildt, P., Allard, F. 2001, ApJ, 556, 885Barnes, J.W., Fortney, J. 2003, ApJ, 588, 545Bodenheimer, P., Laughlin, G., Lin, D. N. C. 2003, ApJ, 592, 555Burrows, A., Hubeny, I., Budaj, J., Hubbard, W.B. 2007, ApJ, 661,502Caballero, J. et al., 2007, A&A, 470, 903Chabrier, G., Saumon, D., Hubbard, W.B. & Lunine, J., 1992, ApJ,391, 817Chabrier, G., Baraffe, I., Allard, F. & Hauschildt, P. 2000, ApJ, 542,L119Chabrier, G., Baraffe, I. 2007, ApJ, 661, L81Chatterjee, S., Ford, E. & Rasio, F., 2007, arXiv:astro-ph/0703166Deming, D., Harrington, J., Laughlin, G. et al. 2007, ApJ, 667, L199Demory et al. 2007, A&A, 475, 1125Fontaine, G., Graboske, H.C. Jr., Van Horn, H.M 1977, ApJS, 35, 293Fortney, J., Saumon, D., Marley, M., Lodders, K., Freedman, R.S.2006, ApJ, 642, 495Fortney, J., Marley, M. S., Barnes, J.W. 2007a, ApJ, 659, 1661Fortney, J., Marley, M.S. 2007, ApJ, 666, L45Gillon, M., Pont, F., Demory, B.-O. et al. 2007a, A&A, 472, L13Gillon, M., Demory, B.-O.; Barman, T. et al. 2007b, A&A, 472, L51Gillon, M., et al. 2007c, arXiv:0712.2073Guillot, T., 2005, Ann. Rev. of Earth and Planetary Sciences, 33, 493Guillot, T., Santos, N. C., Pont, F., Iro, N., Melo, C., Ribas, I. 2006,A&A, 453, L21Guillot, T. 2005, Ann. Rev. of Earth and Planetary Sciences, 33, 493Hubbard, W.B., MacFarlane, J.J. 1980, JGR, 85, 225Hubbard, W.B., Marley, M. 1989, Icarus, 78, 102Hubbard, W. B., Podolak, M., Stevenson, D. J. 1995, “Neptune andTriton” book, p. 109Ida, S., Lin, D. N. C. 2004, ApJ, 604, 388Ikoma, , M., Guillot, T., Genda, H., Tanigawa, T., Ida, S. 2006, ApJ,650, 1150Knutson, H.A., Charbonneau, D., Allen, L., Fortney, J., et al. 2007,Nature, 447, 183Lin, D. & Papaloizou, J., 1986, ApJ, 309, 846Lin, J-F., Gregoryanz, E., Struzhkin, V., Somayazulu, M., Mao, H. &Hemley, R., 2005, Geoph. Res. Lett., 32, 11306Linden, P.F. & Shirtclife, T.G., 1978, J. Fluid Mech., 87, 417 1978 Loeillet, B., Shporer, A., Bouchy, F., et al. 2007, A&A Letter, sub-mitted, (astro-ph/0707.0679)Lyon, S.P., Johnson, J.D. 1992, LANL Rep. LA-UR-92-3407 (LosAlamos; LANL)Marley, M., Fortney, J., Seager, S., Barman, T. 2007, in “Protostarsand Planets V”, eds. B. Reipurth, D. Jewitt, and K. Keil,University of Arizona Press, p.733Mizuno, Z. 1980, Prog. Th. Phys., 64, 544Mordasini, C., Alibert, Y., Benz, W. 2007, in ”Extreme SolarSystems”, ASP Conf. Series, eds. D. Fischer, F. Rasio, S. Thorsettand A. Wolszczan, Santorini, June 2007Mordasini, C., Alibert, Y., Benz, W., Naef, D. 2008, A&A, in prepa-rationPodolak,M., Hubbard, W.B., Stevenson, D.J. 1991, “Uranus”, Tucson,AZ, University of Arizona Press, 1991, p. 29Pollack, J.B., et al. 1996, Icarus, 124, 62Potekhin, A. 1999, A&A, 351, 787Rafikov, R., 2006, ApJ, 648, 666Sato, B., Fischer, D. A., Henry, G.W. et al. 2005, ApJ, 633, 465Saumon, D., Hubbard, W. B., Burrows, A., Guillot, T., Lunine, J. I.,Chabrier, G. 1996, ApJ, 460, 993Saumon, D., Chabrier, G., VanHorn, H.M. 1995, ApJS, 99, 713Saumon, D., Guillot, T. 2004, ApJ, 609, 1170Seager, S., Kuchner, M., Hier-Majumder, C.A., Militzer, B. 2007, ApJ,669, 1279Seager, S.& Hui, L. 2002, ApJ,, 574, 1004Selsis, F., et al., 2007, Icarus, 191, 453Schwegler E., Galli, G., Gygi, F., Hood, R. 2001, Ph. Rev. Letter, 87,265501Stevenson, D.J. 1982, Ann. Rev. of earth and planetary sc., 10, 257Stevenson, D.J. 1985, Icarus, 62, 4Sotin, C., Grasset, O., Mocquet, A. 2007, Icarus, 191, 337Thompson, S.L., Lauson, H.S. 1972,
Improvements in the chartD Radiation-hydrodynamics CODE III: revised analytic equa-tion of state , Technical Report SC-RR-61 0714, Sandia NationalLaboratoriesValencia, D., O’Connell, R.J., Sasselov, D. 2006, Icarus, 181, 545Valencia, D., Sasselov, D. D., O’Connell, R.J. 2007, ApJ, 665, 1413Ward, W., 1996, in ”Completing the Inventory of the Solar System”,ASPC Proceedings, volume 107, T.W. Rettig and J.M. Hahn,Eds., 337Whitworth, A. & Stamatellos, D., 2006, A&A, 458, 817
Appendix A: Calculation of the mixing entropy
In this appendix, we calculate the expression for the idealmixing entropy of a two-component system composed of aH/He mixture, identified as one component, on one side andof a heavy material component of mass fraction Z = M Z /M on the other side. In the following, the subscript ”1” denotesthe H/He component while the subscript ”2” refers to theZ-component. By definition, the ideal entropy of mixingreads: S mix k B = N ln N − N ln N − N ln N − N e ln N e , (A.1)where N = N + N denotes the total number of parti-cles, including free electrons, in the fluid, N i = N i + N ie denotes the total number of particles of component i , with N i the number of nuclei of component i and N ie the numberof electrons provided by the component i . Developing eq.(A.1) yields: S mix k B = N ln(1 + N N ) + N ln(1 + N N )+ N ln N + N ln N − N ln N − N ln N − N e ln N e = N ln(1 + N N ) + N ln(1 + N N ) − N e ln N e + N e ln N e + N e ln N e + S (1) mix k B + S (2) mix k B where S ( i ) mix k B = N i ln N i − N i ln N i − N ie ln N ie (A.2)denotes the ideal mixing entropy of the component i , in-cluding various ionic, atomic or molecular species as wellas electron contributions (see SCVH), and N e = N e + N e is the total number of electrons provided by the H/He and Zcomponents. These ideal mixing entropy contributions arealready included in the SCVH EOS for the H/He compo-nent (their eq.(53)) and in the appropriate EOS for theZ-component. Removing these two contributions, we ob-tain the ideal entropy of mixing which arises only from themixture of the H/He and Z components: S mix k B = N ln(1 + N N ) + N ln(1 + N N ) − N e ln N e + N e ln N e + N e ln N e (A.3)The specific entropy, i.e. the entropy per unit mass isgiven by ˜ S mix k B = S mix M k B = N M S mix N k B (A.4)where M = M + M is the total mass.We define a mean atomic mass ¯ m i = M i /N i and meancharge 1 + ¯ Z i = N i /N i for each component, where we haveused the electroneutrality condition N ie = N i ¯ Z i , so that M i N i = ¯ m i / (1 + ¯ Z i ). For the H/He component, the quanti-ties ¯ m and ¯ Z are given in terms of the relative numberfractions x of H , H, He, He + , He ++ and e − by the SCVHEOS (their eqs.(33)-(35)), with: M N = (2 x H + x H + x H + ) m H + ( x He + x He + + x He ) m He (A.5)and ¯ Z = x (1) e − x (1) e (A.6)where x (1) e = N e N = 11 + βγ x H + + βγ βγ ( x He + +2 x He ) (A.7)denotes the number-concentration of free electrons in theHHe mixture as defined in SCVH, m H and m He denote theatomic mass of hydrogen and helium, respectively, and thegrec symbols α, β, γ are defined by eqs.(54)-(56) of SCVH.For the Z -component, the mean mass corresponds to theatomic mass of the compound under consideration. Themean charge ¯ Z is unknown. We have carried out calcula-tions for the two limiting cases of fully neutral ( ¯ Z = 0)and fully ionised ( ¯ Z = Z nuc ) heavy material, where Z nuc denotes the nuclear charge of the compound. Note the following typos in the SCVH paper. In eqs. (45)-(46), the entropy ratios on the rhs of the equations should be S H S and S He S and not SS H and SS He . In eq.(56) for the parameter δ , the fraction in front of the bracketed terms on the rhs of theequation should be and not . After some algebra, the ideal specific entropy of mixingof the (HHe)/Z mixture can be written: ˜ S mix k B = S mix Mk B = X ¯ m (1 + ¯ Z ) × ( ln(1 + β ′ γ ′ ) + β ′ γ ′ ln(1 + β ′ γ ′ ) − x (1) e ln(1 + δ ′ ) − β ′ γ ′ x (2) e ln(1 + δ ′ ) ) where X = M /M ≡ M HHe /M and β ′ = ¯ m ¯ m Z − Z , γ ′ = 1 + ¯ Z Z , δ ′ = N e N e = β ′ ¯ Z ¯ Z = x (2) e x (1) e β ′ γ ′ (A.8)with Z = M Z /M and x ( i ) e = ¯ Z i / (1 + ¯ Z ii