THERMAL EVOLUTION AND STRUCTURE MODELS OF THE TRANSITING SUPER-EARTH GJ 1214b

Under the assumption the deep envelope layers convect efficiently, it is the radiative atmosphere atop the convective region that serves as the bottleneck for interior cooling, just as in Neptune, Uranus, and giant planets generally (e.g., Hubbard 1977). Observations of GJ1214b's atmosphere are consistent with a water-dominated composition as well as with an H/He atmosphere with clouds or hazes (Bean et al. 2010). Given our current ignorance of the composition of the atmosphere, we consider the two likely end-member cases, either an H/He-dominated atmosphere or a pure steam atmosphere, and find that the opacity is dominated by water vapor in either case (Miller-Ricci & Fortney 2010). For planetary structure, we assume chemical equilibrium in the atmosphere, thereby ignoring possible alterations of the T–P profile through photoionization.

For planetary evolution, a grid of model atmospheres is generally used as the upper boundary condition, see, for instance, Fortney et al. (2007). These grids relate the specific entropy (s) of the convective interior, surface gravity (g) of the planet, and the intrinsic effective temperature (T_int) from the interior. We have computed such a grid from T_int = 175 K down to 30 K, with the correct limiting behavior down to 0 K (an exhausted interior) across surface gravities from 100 to 1500 cm s⁻². The grid is computed at 50× solar metallicity, under the assumption of complete redistribution of absorbed stellar flux (meaning f = 1/4; see Fortney et al. 2007; Miller-Ricci & Fortney 2010), and no clouds. Similar models, which describe the technique in more detail, are found in Fortney et al. (2007). The opacity database is described in Freedman et al. (2008). We note that this very high metallicity is realistic given that Neptune and Uranus are 30–60 times solar in carbon (see Guillot & Gautier 2009, for a review).

We use this grid for all evolution calculations of our models, whether they posses thin H/He atmospheres or pure steam atmospheres. This is certainly a broad brush treatment for a wide range of possible atmospheres, but given our current ignorance regarding the planet's atmosphere, we feel our treatment is justified. The importance and utility of the coupled model atmosphere/interior cooling calculation is that it allows us to estimate T_int and P_ad as a function of time. For instance, in the recent work of Rogers & Seager (2010), the value of T_int was not calculated, but was extrapolated from evolutionary models of Baraffe et al. (2008), for higher mass objects. Generally, we find a 15 K lower T_int, meaning a colder interior, than Rogers & Seager (2010) used. While T_int may change from 175 down to 30 K during evolution, we find that the effective temperature T_eff remains nearly constant within 562–557 K, which is close to the zero-albedo, planet-average equilibrium temperature of 555 K.

Metal-rich⁴ planets such as super-Earths are generally suspected to harbor a variety of materials. We aim to represent this variety in a simplified manner by confining silicates and iron into a "rocky" core, and H, He, and water to an envelope. For core material, we use the P–ρ relation for rocks by Hubbard & Marley (1989) which describes an adiabatic mixture around 10⁴ K of 38% SiO₂, 25% MgO, 25% FeS, and 12% FeO. Such kind of rocks' mass fraction of Si, Mg, and Fe is, respectively, about 0.5, 0.62, and 1.05 times that of the bulk Earth (McDonough & Sun 1995). For H/He envelopes, we use the interpolated hydrogen and helium EOS developed by Saumon et al. (1995). For water we use H₂O-REOS, which was applied to Jupiter (Nettelmann et al. 2008), Uranus and Neptune (Fortney et al. 2010), and in a slightly modified version to CoRoT-7b (Valencia et al. 2010). This water EOS comprises various water EOS appropriate for different pressure–temperature regimes. It includes the melting curve and phase Ice I (Feistel & Wager 2006), the saturation curve and liquid water (Wagner & Pruß 2002), vapor and supercritical molecular water (SESAME 7150; Lyon & Johnson 1992), and for T ⩾ 1000 K and ρ ⩾ 2 g cm⁻³ supercritical molecular water, ionic water, superionic water, plasma, ice XII, and ice X based on FT-DFT-MD simulations (French et al. 2009). At pressures below 0.1 GPa, and/or temperatures below 1000 K, H₂O-REOS relies on Sesame EOS 7150 (Lyon & Johnson 1992). For mixtures of hydrogen, helium, and water, we use H-REOS for hydrogen (Nettelmann et al. 2008), H₂O-REOS for water, and an improved version (Kerley 2004) of the helium Sesame EOS 5761 (Lyon & Johnson 1992). Other materials are not considered here.⁵

Mass escape caused by stellar energy input is known to occur from the highly irradiated atmosphere of the hot Jupiter HD 209458b (Vidal-Madjar et al. 2003; Yelle 2004; Erkaev et al. 2007; Murray-Clay et al. 2009), and shown to have a significant impact on the current composition of super-Earth CoRoT-7b (Valencia et al. 2010; Jackson et al. 2010). While the sun-like star CoRoT-7 irradiates the planet with a present X-ray and ultra-violet energy flux (XUV) $F_{\rm XUV}=5\times 10^5\:\rm erg\,cm^{-2}\,s^{-1}$ (Valencia et al. 2010), GJ 1214 is supposed to be an inactive M star (Charbonneau et al. 2009). Assuming it obeys the empirical relation for the surface energy flux of M stars, $F_{\star,\rm XUV}/F_{\star,\rm bol}=(1\hbox{--}100)\times 10^{-5}$ (Scalo et al. 2007), then with L_⋆ = 0.0033 L_☉, R_⋆ = 0.211 R_☉, the energy flux $F_{\rm XUV}=F_{\star,\rm XUV}(R_p^2/4a_p^2)$ received by GJ 1214b is only (0.8–80) erg cm⁻² s⁻¹, i.e., (0.16–16) × 10⁻⁴ that of CoRoT-7b, at the current time. We then expect XUV irradiation to have a comparatively lesser, but still important, influence on the atmospheric mass through time.

We use the energy limited escape model of Erkaev et al. (2007) to investigate the mass-loss history of the planet. With a heat absorption efficiency ε = 0.4, and a correction factor K_tide = 0.95 accounting for the height decrease of the Roche-lobe boundary through tidal effects (Erkaev et al. 2007), the energy-limited mass escape rate $\dot{M}_{\rm esc} = (\varepsilon \, F_{\rm XUV} R_p^3)\,(G\,M_p\,K_{\rm tide})^{-1}$ (following Valencia et al. 2010) is (0.012–2.8) × 10⁸ g s⁻¹. This is ≈(0.02–6) × 10⁻³ the mass loss of present (rocky) CoRoT-7b, with a value of (1–100) × 1.84 × 10⁶ g s⁻¹, for the fiducial GJ 1214b values M_p = 6.55 M_⊕ and R_p = 2.678 R_⊕. Correcting M_esc ∼ r²₁R_p for the altitude r₁ > R_p where the XUV flux is absorbed (Lammer et al. 2003), the actual value can further rise by a factor of 10. Within 1 Gyr, this mass loss accumulates to (1–100) × 10⁻⁴ M_⊕. For a low-mass atmosphere of only about 1% of GJ 1214b's total mass (which is quite possible for a small mean molecular weight atmosphere: see Section  3.1), the fraction of the atmosphere lost during the 3–10 Gyr lifetime of this planet can be large enough to influence its cooling behavior. In particular, according to the estimates above and assuming constant mass loss over time, the initial atmosphere can have been larger by (1–100) × (0.5–1.5)%. We therefore must include mass loss in our evolution calculations of H/He envelope models, as described below.

Three classes. We consider three classes of hydrostatic two-layer interior models of present GJ 1214b assuming a homogeneous envelope above a rocky core. The three classes differ in the materials constituting the envelope. Class I models have an H/He envelope with solar He mass fraction Y = M_He/(M_H + M_He) = 0.27. Class II models have a pure water envelope ("water worlds"), and class III models an H/He/H₂O envelope with variable water mass fraction Z₁, and also Y = 0.27. The rock core mass of present GJ 1214b models is found by the condition to match the radius R_p = 2.678 R_⊕ (recently confirmed through both optical and infrared photometry; Sada et al. 2010) for a given planet mass M_p = 6.55 M_⊕ and surface thermal boundary condition. We do not consider different possible R_p–M_p pairs within the observational error bars $\sigma _{M_p}=\pm 1.0\,M_{\oplus }$ and $\sigma _{R_p}=\pm 0.13\,R_{\oplus }$ as such work has already been presented by Rogers & Seager (2010). For the thermal structure of the atmosphere, we apply the solar composition model atmosphere between 20 mbar and 10 bar to model classes I and III, and to model class II the water model atmosphere between 20 mbar and 1 bar from Miller-Ricci & Fortney (2010). At pressures higher than respectively 10 bar (classes I and III) or 1 bar (class II), we assume an isothermal temperature—a reasonable assumption (Miller-Ricci & Fortney 2010) given our general understanding of highly irradiated atmospheres.

Calculating the structure. We choose 20 mbar as the low-pressure boundary of our models. This is a choice of convenience, since our water EOS ends at this pressure, but it is also realistic. The wide-band optical transit radius for the planet is at ∼10 mbar (E. Miller-Ricci 2010, private communication). This is consistent with the cloud-free atmosphere calculations of Fortney et al. (2003) for HD 209458b, as well.

The high-pressure boundaries of the model atmospheres of, respectively, 1 and 10 bar are chosen within the isothermal part of the atmosphere, before it transitions to the adiabatic interior at some pressure P_ad. For present time (t₀) structure models, we consider the transition pressure P_ad(t₀) a variable parameter and investigate the response of the core mass and the cooling time on the choice of P_ad(t₀).

Since the model atmospheres predict almost constant equilibrium abundances of H, He, and H₂O, we derive the mass density ρ(P, T(P)) in the atmosphere T(P) from an EOS table for constant composition. Given M_p, R_p, and the P–ρ relations according to the EOS ρ(P, T(P)) in the atmosphere, and constant s(P, T, ρ(P, T)) in the adiabatic interior, we obtain internal profiles m(r), T(r), and P(r) by integrating the equation of hydrostatic equilibrium dP/dr = −Gmρ/r² from the surface toward the center. Mass conservation $M_p=\int _0^{R_p} dr\: 4\pi r^2\:\rho (r)$ is ensured by the proper choice of the rock core mass M_core.

The rock core assumption. We assume the rocky core to be appropriately described by an EOS of homogeneous, adiabatic "rocks" at all times (compare Rogers & Seager 2010: differentiation into an iron core and a silicate layer of $\rm Mg_{0,1}Fe_{0,1}SiO_3$ at uniform temperature). Differentiation will have occurred when temperatures in the primitive rock core rose above the melting temperature of iron during formation, and will have affected the thermal evolution. In the Earth, solidification of the inner iron core still causes a buoyancy of light elements driving convection of the outer iron core, and supports—together with gravitational energy release from core shrinking—subsolidus convection of the silicate mantle. If the melting line of iron rises steeply with temperature as indicated by ab initio data (see Valencia et al. 2010, for an overview), also the central part of a several Earth mass core of GJ 1214b might transition from liquid to solid iron due to high pressure up to ∼15 Mbar, supporting the assumption of an adiabatic interior. Due to the poorly constrained iron mass fraction of GJ 1214b and uncertainties in the iron melting line, the deep interior could potentially be fully liquid or solid and isothermal. Since the EOSs of rocky materials at high pressure above few Mbar are not well known and the effect of temperature on the P–ρ relation is negligible (Seager et al. 2007), we believe the P–ρ relation of the rock EOS used is appropriate for our purpose of determining the core mass and its contribution to the cooling time. We denote by T_core the temperature at the core–mantle boundary and assume its time derivative T_core/dt to be representative for the whole core.

Calculating the evolution. Choosing P_ad(t₀) for class I and II models, and P_ad(t₀) and Z₁ for class III models uniquely defines the core mass of resulting interior models. A selection of six such present time interior models is shown in Table 1. For class I and II models, we calculate the cooling curve by first generating ∼50 profiles with decreasing transition pressures P_ad(t₀) ⩾ P_ad > 20 mbar values, thereby increasingly warmer interiors. For each of these intermediate profiles, core mass and composition are conserved. As the interior becomes warmer with decreasing P_ad, the planet radius R_p rises. In order to obtain the cooling curve R_p(t), we integrate the energy balance equation

$$\begin{equation} L_{\rm eff}-L_{\rm eq}= -\frac{dE_{\rm int}}{dt} \end{equation} \tag{ 1 }$$

backward in time, starting with the present time structure models. In Equation (1), L_eff = 4πR²_pσT⁴_eff is the net luminosity the planet radiates into space, and L_eq = 4πR²_pσT⁴_eq is the stellar energy absorbed. The difference L_eff − L_eq ∼ T⁴_int is provided by our atmosphere grid T_int(g, s), see Section  2.1, and sets the intrinsic luminosity the planet can radiate away from the interior through its atmosphere at a given gravity and internal entropy. Given T_int, we can then derive the time interval dt it needs to lose the intrinsic energy dE_int,

$$\begin{equation} \frac{dE_{\rm int}}{dt} = \int _{M_{{\rm core}}}^{M_{p}}\hspace{-8.53581pt} dm\: \frac{T\,ds}{dt} + c_v\,M_{{\rm core}}\frac{dT_{{\rm core}}}{dt} - L_{{\rm radio}}. \end{equation} \tag{ 2 }$$

Expression (2) accounts for the heat loss δq = T(m)ds of each envelope mass shell dm, the heat loss of the core due to cooling, and the energy gain L_radio of rocky core material due to decay of radioactive elements (see below).

Table 1. GJ 1214b Structure Model Input and Output Data

Label	Mcore	Pad(t0)	Z1	Envelope Material	$\dot{M}$	k2	Tint	Age
	(M⊕)	(bar)			(107 g s−1)		(K)	(Gyr)
Ia	6.464	300	0	H/He	1.84	0.0183	42.7	3.1
Ib	6.434	800	0	H/He	1.84	0.0170	31.8	7.2
IIa	0.873	80	1	Water	...	0.5769	51.9	3.05–3.16
IIb	0.203	300	1	Water	...	0.7374	37.6	9.23–9.26
IIIa	4.432	120	0.85	H/He/water	...	0.1434	48.4	2.92–3.22
IIIb	4.016	400	0.85	H/He/water	...	0.1857	35.6	9.21–10.10

Note. These structure models have R_p = 2.678 R_⊕ and M_p = 6.55 M_⊕.

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Experimental data for the specific heat c_v of warm, compressed rocks at P > 2 Mbar are not available. Ab initio calculations for iron at a few Mbar and several thousand K (Earth's core conditions) predict $c_v\approx 0.5\:\rm J\,K^{-1}\,g^{-1}$ (Alfè et al. 2001), while $c_v=1.0\:\rm J\,K^{-1}\,g^{-1}$ was formerly applied by Guillot et al. (1995) to the core of Jupiter and by Valencia et al. (2010) to the silicate-iron interior of CoRoT-7b. We aim to bracket the uncertainty in c_v by choosing $c_v=0.5\hbox{--}1.0\:\rm J\,K^{-1}\,g^{-1}$. For given T_int, the time interval required to lose E_int rises with c_v.

Radiogenic heat. As we will see, our class I and III planet models have large rocky cores. Modeling the radiogenic heat from the rocky portion of the interior is important to accurately calculate the thermal evolution. For L_radio we consider the isotopes ²³⁸U, ²³⁵U, ²³²Th, and ⁴⁰K. We adopt isotopic abundances and element abundances of meteorites for the elements U, Th, K, and Si⁶ as given in Anders & Grevesse (1989). With respective half-life times in Gyr of 4.468, 0.704, 14.05, and 1.27, and respective decay energies in MeV of 4.27, 4.679, 4.083, and 1.33,⁷ we find a radioactive energy release for 1 M_⊕ of meteoric material on Earth today of 2.3 × 10¹³ J s^-1/1 M_⊕ (=: L_radio,meteor(t₀)) and 4.56 Gyr ago of 2.7 × 10¹⁴ J s^-1/1 M_⊕ (=: L_radio,meteor(0)). The dominant contribution during this time interval is mostly due to β⁻ decay of ⁴⁰K into ⁴⁰Ca. Extension of the radioactive decay law to 10 Gyr ago would increase L_radio to 20 × L_radio,meteor(0), an unrealistically high value for rock material in the young universe. Of course, for small core masses (0.1M_p), cooling of the core does not significantly affect the model's cooling time, but for large core mass models it does, and in particular the choice of the initial value L_radio(0) matters. Therefore, we define a cosmological luminosity L_cosmo := L_radio,meteor(0) that we require each cooling model of GJ 1214b to start with, within 10%. Models with long cooling times (10 Gyr) will then have L_radio(t₀) ≈ 0.1 × L_radio,meteor(t₀), and models with short cooling times (3 Gyr) will have L_radio(t₀) ≈ 2 × L_radio,meteor(t₀). By this choice of L_cosmo we avoid extremely high or low initial values.

Evolution with mass loss. For a given planet radius, low mean molecular weight atmospheres or envelopes are also of low mass, and hence such envelopes may lose a larger relative mass fraction than large mean molecular weight atmospheres. We take into account mass loss only for the H/He envelope models, e.g., models Ia,b in Table 1. For each neighbored pair of interior profiles as given by P_ad, we calculate the mass dM lost during a time interval dt self-consistently using a Newton–Raphson scheme to find the root of the function $f(dt(dM))=dM-\dot{M}_{{\rm esc}}dt(dM)=0$.

Other contributions. We did not include the effect of tidal heating due to tides raised on the planet by the star. This additional amount of energy would tend to delay the cooling time, in particular in the past when the eccentricity e, and hence the tidal heating dE/dt ∼ (e²/a⁶_p) (to second order in e, e.g., Batygin et al. 2009), was larger than today. On the other hand, inclusion of tidal migration $\dot{a}_p$ of the planet would have reduced the amount of stellar irradiation received at early times (Miller et al. 2009). A proper treatment of both effects is beyond the scope of this paper since the orbital eccentricity is not well constrained (Charbonneau et al. 2009). Instead, we make the standard assumption that the formation of GJ 1214b yielded an initial heat of accretion that was not released before the planet arrived at its current location.

The cooling time of our thermal evolution models is the time between the present state and the time when the derivative dT_eff/dt approaches −∞, which corresponds to a hot start that is insensitive to the initial conditions. While such a treatment of young planets is common (e.g., Baraffe et al. 2003), it actually ignores the process of planet formation which has been shown to alter the luminosity during the first tens of millions of years (Fortney et al. 2005). This is important for mass determinations from cooling tracks, whereas in our case it just might induce an error of ∼10 Myr to the calculated cooling time.

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Love number k₂. We follow the call by Ragozzine & Wolf (2009) to tabulate the tidal Love number k₂ values of representative models. This quantity is a planetary property which solely depends on the internal density distribution. If known, k₂ imposes an additional constraint on interior structure models. Physically, k₂ quantifies the quadrupolic gravity field deformation at the surface in response to an external perturbing body of mass M, which can be the parent star, another planet, or a satellite. M causes a tide-raising potential W(r) = ∑_n=2W_n = (GM/a)∑_n=2(r/a)ⁿP_n(cos ψ), where a is the distance between the centers of mass (planet and body), r is the radial coordinate of the point under consideration inside the planet, ψ is the angle between the planetary mass element at r and the center of mass of M at a, and P_n are Legendre polynomials. Each external potential's pole moment W_n(r) induces a change V_n(r) in the corresponding degree of the planet's potential, where V_n(R_p) = k_n W_n(R_p) defines the Love numbers. For the calculation of k₂ we follow the approach described in Zharkov & Trubitsyn (1978).

To first order in the expansion of the planet's potential, k₂ is proportional to the gravitational moment J₂ (see, e.g., Hubbard 1984, Section 4). That is why measuring k₂ will provide a constraint for extrasolar planets that is equivalent to J₂ for the solar system planets. In particular, k₂ is known to be a measure of the central condensation of an object, which can be parameterized by the core mass within a two-layer model approach. However, as for J₂, the inverse problem—the deduction of the internal density distribution of the planet from k₂—is non-unique.

We stress that the Love number k₂ is a potentially observable parameter if the planet's eccentricity is non-zero. It can be obtained with the help of transit light curves that contain information about the tidally induced apsidal precession of close-in planets (Ragozzine & Wolf 2009), or for specific two-planet systems can also be derived from measuring the orbital parameters (Batygin et al. 2009), given that the planets are in apsidal alignment and on coplanar orbits (Mardling 2010).

For our H/He envelope+rock core models, we find a narrow core mass fraction range of 0.975–0.995 for a wide pressure range 50 bar ⩽ P_ad(t₀) ⩽ 4000 bar (Figure 1(a)). Despite its low mass fraction, the H/He envelope extends over 0.37 R_p (Figure 1(b)) independent of the envelope mass. This is because of its high temperature, which increases from 1030 K at P_ad(t₀) to 2400–4200 K at ∼35–90 kbar at the envelope–core boundary. At these conditions, hydrogen is molecular throughout the envelope according to the SCvH-i EOS.

Not all of the models shown in Figures 1(a) and (b) are consistent with an age τ_⋆ = 3 to 10 Gyr of the star GJ 1214. If P_ad(t₀) decreases (increases), internal entropy rises (falls) and the planet will need less (more) time to cool down to this state. This behavior is illustrated by the cooling curves in Figure 1(c), according to which an age of 3 Gyr requires P_ad(t₀) ⩾ 300 bar; and a much longer cooling time of 7.2 Gyr is obtained for P_ad(t₀) = 800 bar, whereas P_ad(t₀) = 100 bar would give a cooling time below 1 Gyr. With a mass loss rate according to F_XUV/F_bol = 10⁻⁵ as assumed for the cooling curves in Figure 1(c), a cooling time of 10 Gyr cannot be obtained through a further increase of P_ad beyond 800 bar if L_radio(0) is not to drop below 0.9 × L_cosmo. The colder the interior, the lower T_int as predicted by our model atmosphere grid, and hence the intrinsic energy then can be transported through the radiative atmosphere. Enhancing P_ad(t₀) from 300 to 800 bar lowers T_int from 42.7 to 31.8 K (Table 1). For even colder interiors, the atmosphere is no longer capable of radiating away the heat generated by radioactive decay, which would contradict the assumption of such a cold interior.

The cooling time increases with the mass loss rate (Figure 1(d)). For stellar XUV radiation F_XUV/F_bol = 10⁻⁵to10⁻⁴ as typical for quiet M stars (Scalo et al. 2007), this enhancement is small (see Figure 1(d)) and we obtain essentially the same range of P_ad(t₀) and hence structure models. Of GJ 1214b's initial total mass (initial H/He envelope mass), only 0.0054%–0.013% (0.41%–0.74%) is lost if F_XUV/F_bol = 10⁻⁵, and about 0.1% (6%) if F_XUV/F_bol = 10⁻⁴. On the other hand, for a permanent, strong irradiation F_XUV/F_bol = 10⁻³ as observed for young, active M Stars listed in the ROSAT catalog, GJ  1214b would have lost 1.8% (53%). These numbers are in agreement with the rough estimates in Section  2.3. In the last case, existence of a thin H/He atmosphere to date becomes less likely since it begins to require fine-tuning of the initially accreted H/He envelope mass. We have found cooling tracks with an age of 10 Gyr or more only if P ⩾ 800 bar and the mass loss rate is high, or L_radio(t) ≪ L_cosmo, which we do not favor.

Models with 300 bar ⩽ P_ad ⩽ 800 bar as constrained by our evolution calculations have M_core = 6.434–6.464 M_⊕ and k₂ = 0.0170–0.0183 implying a high degree of central condensation. The presence of an iron core could even enhance the central condensation. Therefore, although the H/He layer is low in mass, it significantly strengthens the property of central condensation compared to a closer to zero-mass atmosphere planetary object such as the Earth, the theoretical k₂ value of which is ∼0.3 (Zhang 1991).

Class I models are closest to giant planets that formed within the snowline of the disk and did not have enough time and/or material in their surrounding to accrete a massive H/He envelope. Class I structure models can best be compared to "case I" models by Rogers & Seager (2010), where the difference in composition assumptions can be reduced to the core (undifferentiated rock core versus differentiated iron–silicate–water–ice core in their models), and a slightly different envelope He abundance of, respectively, 0.27 and 0.28.

Our obtained H/He mass fraction range, 1.3%–1.8%, is due to the uncertainty in T_int. This range is much smaller than theirs (9 × 10⁻³%–6.8%), which includes uncertainties from the 1σ errors of M_p and R_p contributing an uncertainty of 0.4%–1.6%, from their T_int values used contributing up to 1.5%, and from the uncertainty in core composition (pure iron, iron–silicates–water, or pure water). Since iron is included in our rock EOS, we consider (1.8%+1.6%=) 3.4% a reliable upper limit of the H/He mass fraction if observations reveal a low-mean molecular weight atmosphere. Recent observations of GJ 1214b's atmosphere in the 0.78–1 μm wavelengths range with the Very Large Telescope facility's UT1 telescope suggest a mean molecular weight of 5 g mol⁻¹ or more and thus disfavor an H/He-dominated atmosphere (Bean et al. 2010). On the other hand, the presence of clouds or hazes in such an atmosphere could mimic a short scale height, hence high mean molecular weight, and cannot be excluded by current observations and model atmospheres.

The derived core mass of water envelope+rock core models responds much more strongly to a change of the onset of the adiabatic part of the envelope than models with H/He envelope. At P_ad(t₀) ≈ 400 bar, the core mass reaches zero: a pure water planet (Figure 2(a)). Deeper isothermal regions would only be possible if some amount of the pure water planet would be replaced by lighter elements such as methane or ammonia. Warming up the deep envelope by an outward shifting of the onset of the adiabatic region below 50 bar is accompanied by a strong rise in core mass. A solar water:rock ratio of 2.5 occurs for P_ad = 5 bar—when the deep interior is extremely hot. Smaller ratios would be possible only if the isothermal region is allowed to disappear. We next consider whether such models are consistent with the cooling time.

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Because of the relatively small core mass fraction of class II models, varying the specific heat of the core has a negligible effect on the cooling time. For 80 bar ⩽ P_ad(t₀) ⩽ 300 bar the resulting cooling times are consistent with τ_⋆, see Table 1 and Figure 2(c). Those models have M_core = 0.20–0.87 M_⊕, and a water:rock ratio of 6.5–31. This is much higher than the solar ice to rock ratio (I:R)_☉ ∼ 2.5, which would give a cooling time shorter than 3 Gyr.

In contrast to models Ia,b, models IIa,b are weakly centrally condensed as parameterized by their high k₂ values of, respectively, 0.57 and 0.74.

Class II models are different from those by Fu et al. (2010) who consider cold water planets with ice or liquid ocean layers above silicate/iron cores. GJ 1214b is not that cold. Our models also differ from the water steam atmosphere models of CoRoT-7b by Valencia et al. (2010) who find water envelope + silicate/iron core models where water in present CoRoT-7b contributes at most 10% to the total mass and is in the vapor phase or supercritical molecular phase. They also differ from the water steam atmosphere models by Rogers & Seager (2010) who describe the deep interior by a water–ice EOS, whereas according to the phase diagram (Figure 3; updated from French et al. 2009; Valencia et al. 2010), water would be in the plasma phase in GJ 1214b.

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In our class II models, water transitions from the vapor phase to supercritical molecular water still in the atmosphere, becomes dissociated into an ionic fluid at about 4000 K and 0.2 Mbar, and finally fully dissociated and ionized for T ⩾ 5100 K and P ⩾ 1 Mbar (Figure 2(b)) forming a plasma with electronic conductivity ${\ge} 100\:\rm \:\Omega\, cm^{-2}$ (Redmer et al. 2011).

Under these circumstances, a planetary interior can be able to maintain a dynamo generating a dipolar magnetic field. In Uranus and Neptune, the magnetic field may be generated in a thin shell (Stanley & Bloxham 2006) of possibly ionic water (Nellis et al. 1988). This view of the cold (T_eff ≈ 59 K) outer planets Uranus and Neptune is supported by the Neptune adiabat in Figure 3, while in GJ 1214b, since it is warmer, the fluid conductive envelope would extend down to the small core, somewhat akin to Jupiter, and therefore preferably lead to a dipolar field, according to the field geometry considerations by Stanley & Bloxham (2006).

This structure type resembles the outer envelope of Uranus and Neptune if Z₁ < 0.4, or their inner envelope if Z₁ > 0.7 according to three-layer Uranus and Neptune models of Fortney & Nettelmann (2010). A strong enrichment in metals of the outer H/He layer in Uranus and Neptune is necessary to match the gravity field data, and also some admixture of light elements in the deep interior.

In Section  3.2, we have seen that water envelope+rock core models of GJ 1214b yield supersolar I:R ratios of more than 2.6×solar. Lowering this ratio can be achieved by replacing water with hydrogen and helium. The two limiting cases of this implementation are a structure where an H/He layer is on top of a water layer, or a homogeneous mixture of H/He and H₂O. We find that in the first case, such differentiated three-layer models (H/He, water, rock) cannot have 1×(I:R)_☉ and be in agreement with τ_⋆, the reason of which is the following. Class I and II models require a radiative atmosphere down to 80 bar or more at present in order to meet τ_⋆. If composed solely of H/He, the atmosphere extends over about 0.4 R_⊕. In order to match a remaining radius r(M_p) ≈ 2.3 R_⊕, the core mass fraction of the remaining water+core body is of the order of 20%–50% (see Valencia et al. 2010, Figure 8). We find I:R<0.9 and M_core = 3.5–4.2 M_⊕ for this case of differentiated models. Increasing P_ad increases the depth of the thin H/He atmosphere, thereby lowering the I:R ratio even more. Increasing the planet's mean density within the 1σ error bars of M_p and R_p allows for I:R up to at most 0.56×(I:R)_☉.

Consequently, the only way to obtain a solar I:R ratio is to limit the radius of the H/He atmosphere by enhancing its mean molecular weight (Miller-Ricci et al. 2009) through admixture of water. Here we consider the case of equal metallicity in the radiative atmosphere and in the adiabatic envelope (our class III models) as parameterized by the water mass fraction Z₁. Figure 4(a) shows the change of the I:R ratio of single models. The water to core mass ratio rises moderately up to Z₁ = 0.9, passes 1×(I:R)_☉ at Z₁ = 0.95, and then rises rapidly up to the values found for class II models. This behavior depends very weakly on the choice of P_ad(t₀).

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For class III models, the resulting H/He mass fraction of the planet (see Figure 4(a)) is about 2–3 times larger than in case of an H/He layer on top of a water layer. It reaches the maximum for Z₁ ≈ 0.80 (the core mass must not be too large, requiring a high metallicity, and also Z₁ not too close to 1). We find a planetary H/He mass fraction ⩽5.9% if P_ad(t₀) = 400  bar as in Figures 4(a) and (b), and slowly rising with P_ad(t₀) up to 7% (colder envelopes reduce the core mass). However, a colder present time interior would take longer than 10 Gyr to cool. For the cooling curve calculations we choose a metallicity Z₁ = 0.85, which gives an H/He mass fraction close to the maximum value but also a core mass below 2/3 M_p, hence a real alternative to classes I and II. Figure 4(c) shows that the isothermal region of present GJ 1214b must end between 120 and 400 bar to give consistency with a cooling time of 3–10 Gyr.

With Z₁ = 0.85, T_core = 5730 K, and P_core = 1.4 Mbar (Figure 4(b)), model IIIb resembles the interior of Uranus and Neptune in composition and temperature (Fortney & Nettelmann 2010). Lower in total mass, the pressure does not rise up to 5–7 Mbar as in the outer solar system giant planets, so that water will not adopt the superionic phase according to the phase diagram of water, but remain in a fluid state in GJ 1214b (Figure 3). This property bolsters our assumption of a homogeneous mixture of water with hydrogen and helium.

Our GJ 1214b interior models rely on a separation of the interior into a rock core and one homogeneous envelope of the same composition as in the visible atmosphere. In contrast, giant and terrestrial planets in the solar system are not successfully described by such a two-layer structure but require the assumption of various internal layer boundaries to be consistent with the atmospheric He abundance and the gravity field data (giant planets: see Gudkova & Zharkov 1999; Saumon & Guillot 2004; Nettelmann et al. 2008; Fortney & Nettelmann 2010), long-term spacecraft tracking data (Mars: see Konopliv et al. 2006), and with seismic data (Earth).

4.1.1. Class III: H/He Phase Separation?

4.1.2. Class II: Incomplete Differentiation?

4.1.3. Class I: Choked off Giant Planet Formation?

From interior and atmosphere models that are consistent with the observationally derived parameters T_eq, M_p, and R_p after 3–10 Gyr of cooling, the most certain conclusion we can draw about the composition of GJ 1214b is a metallicity of 94%–100%. The lower limit can further shrink somewhat if the planet in reality is dry, and the mass fraction of water in our class III models then resembles a mixture of H/He and rocks. In contrast, the mass fractions of water, used as a proxy for the ices H₂O, CH₄, NH₃, and H₂S, is essentially unconstrained (0%–97%) as is the mass fraction of rocks (3%–99%).

Two further observables we can hope to attain in the near future are the Love number k₂ from transit timing variations or from the shape of the transit light curves (Ragozzine & Wolf 2009), and second the mean molecular weight in the atmosphere from transmission spectroscopy, in particular from the wavelength dependence of star light absorption in the planetary atmosphere during transit (Miller-Ricci & Fortney 2010).

4.2.1. Love Number k₂

The k₂ values of models from our classes I–III differ greatly from each other and thus we consider k₂ a useful quantity to discriminate between atmospheres of different mean molecular weight. The trend of decreasing k₂ value with increasing core mass that is known for two-layer models of Jupiter-mass giant planets with a core and one envelope (Batygin et al. 2009), or for n = 1 polytropic planets in general (Kramm et al. 2011), is confirmed by our two-layer models of GJ 1214b. An illustration of the k₂(M_core) behavior is shown in Figure 5, which contains the same data as Table 1 and also some intermediate points as well as two solutions of the H/He envelope models with, respectively, P_ad(t₀) = 1 and 2 kbar, which are too cold as explained in Section  3.1.

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However, when the H/He atmosphere becomes thin and its mass low enough (<2%), this planet begins to more closely resemble a relatively homogeneous rock body than a core+envelope planet, and k₂ rises again. This degeneracy of k₂ that is otherwise well known for multi-layer models is—at first glance—a surprising finding in two-layer super-Earth models. As Figure 5 suggests, pure H atmosphere models of GJ 1214b (Rogers & Seager 2010) would approach the k₂ values of solar system bodies such as the Moon, i.e., higher than that of H/He atmospheres.

Moon, Mars, and Earth can be considered as remnant protoplanetary cores (M_core = M_p) that did not accumulate sufficient mass to accrete a significant envelope. On the other hand, because measured k₂ values of Earth (see Ray et al. 2001), Mars (Yoder et al. 2003; Konopliv et al. 2006), and Moon (Zhang 1992) cannot be explained by a homogeneous rock interior but require the assumption of a dense, iron-rich core (e.g., Zharkov et al. 2009), they can also be considered as objects with small (iron) core and large (silicate) mantle (M_core ≪ M_p). This places Mars and Earth close to our GJ 1214b models with H/He/Z envelope in the k₂–M_core diagram, implying that k₂ is a degenerate quantity with respect to composition, too. Separation into layers of different composition owing to phase differentiation and phase transitions in rocks (Valencia et al. 2007) has likely also occurred in GJ 1214b. Such an advanced treatment of a core would enhance the level of central condensation and result in even lower minimum k₂ values below 0.017. The same trend is expected for inclusion of solid-body effects, which will be important for colder super-Earth planets and those with less extended, less massive atmosphere. Theoretical k₂ values for the terrestrial objects are in good agreement with the observations when making the simplifying assumption of an elastic interior (Zhang 1992; Yoder et al. 2003; in which case the shear modulus becomes frequency-independent, i.e., k₂ a static Love number), while their observed moments of inertia are close to 0.4 indicating a nearly homogeneous interior, the theoretical Love number k₂ of which would approach 1.5 if the body were fluid and compressible.

4.2.2. Mean Molecular Weight

Class III models with 1×solar I:R ratio are not excluded by our thermal evolution calculations. Those models require Z₁ = 0.95, corresponding to a mean molecular weight $\mu =13.4\rm \,g\,mol^{-1}$ and 440×solar O:H particle number ratio (assuming a solar O:H of 0.85 × 10⁻³ according to Anders & Grevesse 1989). However, we do not consider such models a realistic description of GJ 1214b. For rapid translations between metallicity and atmospheric μ values from transmission spectroscopy observations, we present the simple relations between μ, O:H, and Z₁ in Figure 6.

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In particular, our class III models with Z₁ = 0.85 have $\mu =8.9\rm \,g\,mol^{-1}$, a 270× solar atmospheric O:H ratio, 0.2 times solar bulk I:R ratio, and ≈5.8% of M_p is H/He. For Z₁ < 0.55, our class III models become severely rock-dominated with M_core > 0.9 M_p, $\mu <4.4\rm \,g\,mol^{-1}$ and O:H <92×solar. A 50×solar O:H ratio as of our model atmosphere grid applied to the thermal evolution calculations (see Section  2.1) implies Z₁ = 0.38. For $\mu =5\rm \,g\,mol^{-1}$ as suggested from the first transmission spectrum observations by Bean et al. (2010), Figure 6 gives Z₁ = 0.7 and according to Figure 4, GJ 1214b could be mostly rocky with an I:R ratio of 0.15, where the H/He/H₂O atmosphere contributes ∼18% to the planet's mass and 40% to its radius.