IN SITU ACCRETION OF HYDROGEN-RICH ATMOSPHERES ON SHORT-PERIOD SUPER-EARTHS: IMPLICATIONS FOR THE KEPLER-11 PLANETS

We simulate the radially one-dimensional structure and quasi-static evolution of the atmosphere of a protoplanet embedded in a protoplanetary disk. The atmosphere's contraction (or expansion) results in its mass gain (or loss). The detail of the model is described in Ikoma & Genda (2006). The numerical integration is done with the code that we developed and used in Hori & Ikoma (2010, 2011). Below is a brief summary of the model.

The planet consists of a compressive atmosphere with solar elementary abundance on top of an incompressive "rocky" body with density of 3.9 g cm⁻³. Choice of value of the rocky density and incorporation of rocky small compressibility have tiny impacts on the structure and mass of the atmosphere. We integrate a usual set of four equations describing the spherically symmetric, quasi-hydrostatic structure of a self-gravitating atmosphere, which includes the equations of hydrostatic equilibrium, mass conservation, radiative/convective energy transfer, and energy conservation (i.e., time change in entropy). The equation of state for the atmospheric gas that we use in this study is from Saumon et al. (1995).

The atmosphere is assumed to be equilibrated with the disk at the smaller of the Bondi and Hill spheres (see Ikoma & Genda 2006); namely, the atmospheric density and temperature are equal to those of the ambient disk gas there. On the other hand, the bottom of the atmosphere corresponds to the interface between the atmosphere and the rocky body. The atmosphere is heated by the underlying rocky body at the bottom. In this study, the energy flux from the rocky body is given at the atmospheric bottom as an inner boundary condition, as described below.

In this paper, we consider grain-free atmospheres, unless otherwise noted, to investigate upper limits to the masses of the H–He atmospheres that the SEs gain via the in situ accretion. Thus, we assume that the atmospheric opacity includes only the gas opacity, which is taken from Freedman et al. (2008) in this study. We suppose that the protoplanet formed via giant impacts in a dissipating disk. Each simulation of the atmospheric growth starts with an arbitrarily hot state (i.e., a given high luminosity). The initial choices of the luminosity do not affect our results because the initial cooling of the atmosphere occurs fast relative to the disk dissipation. No continuous energy supply by planetesimal bombardment is assumed. This assumption also leads to finding upper limits of the atmospheric mass.

In this study, we include two effects newly: we assume that disk dissipation occurs concurrently with the atmospheric growth. Furthermore, we incorporate the effect of the heat capacity of the rocky body. While both effects were not included in previous studies, they have crucial impacts on atmospheric growth in the situations considered in this study, as shown below.

The disk-gas density, ρ_d(a, t), is assumed to decrease in an exponential fashion, namely,

$$\begin{equation} \rho _d (a, t) = \rho _d^0 (a) \exp {(-t/\tau _d)}, \end{equation} \tag{ 1 }$$

where a is the semimajor axis, t is time, and τ_d is the dissipation time that is regarded as a parameter; the initial disk density, ρ⁰_d(a), is taken from Hayashi (1981) in this study. According to modern theories of disk evolution (see Calvet et al. 2000 for a review), the overall dissipation first occurs via viscous diffusion in ∼10⁷ yr. Once the gas density becomes so low in the inner disk that the stellar UV penetrates through the disk, photoevaporation occurs around the gravitational radius of several AU. From that time on, the inner disk is separated from the outer disk and evolves via viscous diffusion separately. Since this is equivalent to the fact that the disk size decreases by a factor of 10, the diffusion timescale (being proportional to the square of disk size) for the inner disk decreases by a factor of 100 to become ∼10⁵ yr. Based on a simple analytical argument, namely, solving the one-dimensional viscous-diffusion equation for surface gas density, Σ_d,

$$\begin{equation} \frac{\partial \Sigma _d}{\partial t} = \frac{1}{a} \frac{\partial }{\partial a} \left[3 a^{1/2} \frac{\partial }{\partial a} \left(\Sigma _d \nu a^{1/2} \right) \right] \end{equation} \tag{ 2 }$$

with Σ_d = 0 at the inner and outer edges, one finds that the inner disk dissipates in such a way that ρ_d∝exp (− π²t/10⁵ yr), which corresponds to τ_d ∼ 10⁴ yr. This is consistent with results from recent numerical simulations of disk evolution (e.g., Gorti et al. 2009).

Decrease in disk density results in cooling of the atmosphere (Ikoma & Genda 2006). The rocky body below the atmosphere also cools down. Detailed treatment of the rocky body's thermal evolution is beyond the scope of this paper. Instead, we evaluate its contribution by inputting the luminosity from the rocky body in the form of L_rock = −C_rockM_rockdT_b/dt + L_radio as an inner boundary condition for the atmospheric structure, where C_rock is the specific heat of silicate (=1.2 × 10⁷ erg K⁻¹ g⁻¹), M_rock is the mass of the rocky body, T_b is the temperature at the atmospheric bottom, and L_radio is the luminosity due to the radioactive decay of chondrites, 2 × 10²⁰(M_rock/M_⊕) erg s⁻¹ (see Guillot et al. 1995). In the present case, since the decline of T_b is caused by that of ρ_d,

$$\begin{equation} \frac{dT_b}{dt} = \frac{d \ln \rho _d}{dt} \frac{dT_b}{d \ln \rho _d} = - \frac{1}{\tau _d} \frac{dT_b}{d \ln \rho _d}. \end{equation} \tag{ 3 }$$

Furthermore, from the analytical solution of the radiative atmosphere (Ikoma & Genda 2006), T_b is related to the disk temperature, T_d, as dT_b/dln ρ_d = T_d/4. Hence, we calculate L_rock as

$$\begin{equation} L_{\rm rock} = \frac{M_{\rm rock} C_{\rm rock} T_d}{4 \tau _d} + L_\mathrm{radio}. \end{equation} \tag{ 4 }$$

Examples of atmospheric growth are shown in Figure 1. In the simulations, M_rock is assumed to be 4 M_⊕. The dotted (blue) line shows the atmospheric growth when the disk gas density is constant through the simulation. In this case, after gradual growth, runaway gas accretion starts at t ∼ 3 Myr, resulting in forming an atmosphere that is much more massive than the atmospheres of interest in this study.

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The dashed (green) and solid (red) lines show the atmospheric mass evolution with concurrent disk dissipation with τ_d = 1 × 10⁵ yr; the former and latter represent the cases with C_rock = 0 (i.e., no contribution of the rocky body's cooling to heating the atmosphere) and C_rock = 1.2 × 10⁷ erg K⁻¹ g⁻¹, respectively. As shown by the two lines, the atmospheric growth levels off at some point in these cases because the disk is assumed to disperse before the runaway gas accretion would happen if it were not for disk dispersal. After the leveling-off, the atmosphere is found to be eroded. The simulations were stopped when the disk-gas density became low enough that the mean free path of the ambient gas molecules was longer than the planetary radius;

$$\begin{eqnarray} \rho _d &=& \frac{k T_d}{G M_\mathrm{rock} \sigma } = 1.4 \times 10^{-19} \left(\frac{M_\mathrm{rock}}{1\,M_\oplus }\right)^{-1} \hspace{64pt}\nonumber\\ && \times \left(\frac{T_d}{100 \,\mathrm{K}} \right) {\mathrm{g\ cm}^{-3}} \hspace{64pt} \end{eqnarray} \tag{ 5 }$$

$$\begin{equation} = 10^{-10} \rho _d^0 (a) \left(\frac{M_\mathrm{rock}}{1\,M_\oplus }\right)^{-1} \left(\frac{T_\mathrm{d}}{100\,\mathrm{K}} \right) \left(\frac{a}{1\,\mathrm{AU}}\right)^{11/4}, \end{equation} \tag{ 6 }$$

where k is the Boltzmann constant, G is the gravitational constant, and σ is the collisional cross-section of the molecule (=2.5 × 10⁻¹⁶ cm²).

The atmospheric erosion during disk dissipation has been newly found in this study. The erosion is due to atmospheric expansion. The expansion occurs because disk depressurization enhances the outward pressure gradient near the outer boundary, which pushes the atmosphere outward. Then, the atmospheric gas seeps out from the protoplanet's gravitational sphere, which continues until the atmosphere is equilibrated with the ambient, depressurized disk. The decrease in atmospheric mass due to this effect (the dashed, green line) is, however, small, relative to the decrease due to heatup by the cooling rocky body (the solid, red line). As described in Section 2.1, the disk dissipation cools the surface of the rocky body, resulting in heat supply to the atmosphere. This lifts up the atmosphere, which results in further atmospheric erosion. This is why the atmospheric erosion is more significant when the rocky heat capacity is incorporated.

In Figure 2, the final mass of the H–He atmosphere, M_{H + He}, which is divided by the rocky body's mass, M_rock, is shown as a function of M_rock for different values of the disk's temperature, T_d, and dissipation time, τ_d. While those two parameters are focused on in this study because upper limits of M_{H + He} are of special interest, the sensitivity of M_{H + He} to the initial disk density, ρ⁰_d, and the atmospheric opacity is briefly seen below.

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First, for a given set of the parameters, M_{H + He} increases with M_rock in the small-M_rock regime. Also, it is found that there is a critical value of the rocky body's mass, M*_rock, beyond which the atmosphere becomes quite massive eventually. This is because the atmospheric accretion enters the runaway state before the disk is depleted significantly. Thus, the atmospheric growth has two distinctly different outcomes, depending on M_rock. The atmospheric growth beyond the critical point is discussed in Section 3.

The three red lines of different types show the sensitivity of M_{H + He} to τ_d. For a given M_rock, larger τ_d results in larger M_{H + He}. The sensitivity is not large; M_{H + He} increases by a factor of less than 10 for two-order-of-magnitude increase in τ_d. Also, M*_rock increases, as τ_d decreases. This is because a short lifetime of the disk (i.e., small τ_d) allows only massive protoplanets to start runaway accretion. Difference in τ_d also makes only a small change in M*_rock and also M_{H + He} at the critical point (<0.1M_rock). It is noted that while physically small, the sensitivities of M_{H + He} may suffice to validate observationally inferred masses of the atmospheres, as discussed in Section 3.

The results for different three disk temperatures are shown by the solid green (T_d = 940 K), red (T_d = 550 K), and blue (T_d = 200 K) with τ_d = 100 kyr in Figure 2. It is found that T_d has a significant impact on M_{H + He} and M*_rock. Qualitatively, high T_d results in small M_{H + He} and, therefore, large M*_rock. This is basically because atmospheres made from warmer gas (more exactly, higher-entropy gas) are gravitationally less bound. However, the impact of the outer boundary conditions on the atmospheric mass is known to be negligibly small in the case of massive atmospheres embedded in cool disks like proto-envelopes of gas giants. This is because most of the atmospheric mass is concentrated in the deep region of the atmosphere whose structure is insensitive to the outer boundary conditions (Mizuno 1980; Stevenson 1982; Wuchterl 1993; Ikoma et al. 2001). In contrast, in the case of low-mass atmospheres embedded in warm disks which are appropriate to SEs with short orbital periods, our results in Figure 2 reveal that the impact is significantly large.

The initial disk density, ρ⁰_d, has only a small impact on M_{H + He} and M*_rock. For example, the yellow line shows the result for the case of ρ⁰_d being 10 times larger than that used for the red solid line. Since high density corresponds to low entropy, the atmosphere embedded in denser disks tends to be more massive for the same reason described above. However, it is realized that the atmosphere before being eroded is massive enough that the structure in its deep part is insensitive to disk properties.

Finally, the values of M_{H + He} and M*_rock that we have derived are upper and lower limits, respectively, for a given set of T_d and τ_d in the sense that no grain opacity and no additional energy input are assumed. In the case represented by the purple line, the effect of grain opacity is evaluated. To do so, we have adopted Semenov et al.'s (2003) protoplanetary-disk grain-opacity model. This would be an extreme case of large opacity, namely, a lower limit to M_{H + He} because large opacity slows down contraction of the atmosphere (Ikoma et al. 2000). Thus, the actual solutions are in between the lines of grain-free (red solid line) and grain-rich (purple line) atmospheres.

The SEs orbiting Kepler-11 are thought to have relatively thick H–He atmospheres, as described in the Introduction. In this section, we apply our theory to the planets and, thereby, get some implications for their composition and origin. Specifically, we compare the atmospheric masses derived from modeling with those which we have calculated in this study.

The comparison is made in Figure 3: the x-axis is the planetary total mass, M_p (i.e., M_rock + M_{H + He}); the y-axis is the percentage of M_{H + He} relative to M_p. The black crosses represent M_{H + He} derived by the structure modeling of the planets, which account for the observed masses and radii. The data were kindly provided by Lopez et al. 2012 who calculated them using a modified version of the modeling method from Lissauer et al. (2011); the effect of the rocky heat capacity having been incorporated in the thermal evolution, which yielded a slight decrease in M_{H + He} (Lopez et al. 2012).

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The curves show the results of our calculations in this study for four different cases. The temperatures 940 K and 550 K correspond to those at Kepler-11b's and 11f's semimajor axes (a = 0.088 and 0.25 AU), respectively, in an optically thin disk (Hayashi 1981). Points indicated by filled squares are the critical points (see Section 2.3). Beyond the critical points, we have also drawn extrapolated lines (hereafter called super-critical lines) by assuming that H–He atmospheres accrete on the non-growing rocky bodies with the critical masses; mathematically, the super-critical lines are expressed by y = (1 − M*_rock/x) × 100%, where x = M_rock + M_{H + He} and y = M_{H + He}/(M_rock + M_{H + He}). To check the validity of the super-critical lines, we have simulated the super-critical (i.e., runaway) accretion whose rate is not higher than supply limit of the disk gas due to viscous diffusion, 3πΣ_dν_d, where Σ_d and ν_d are the surface density and viscosity of the disk gas, respectively. We have adopted the α-prescription for the disk viscosity; τ_d = 10 kyr and 1 Myr correspond to α ≃ 10⁻² and 10⁻⁴, respectively. It is shown that the numerical solutions are certainly on the super-critical lines. In the case of T_d = 550 K and τ_d = 10 kyr, for example, M_rock = 13.0 M_⊕ at the point on the super-critical line, which is close to M*_rock = 12.9 M_⊕; nevertheless, M_{H + He}/M_p is larger by a factor of ∼5. This demonstrates that a slight difference in M_rock around M*_rock results in a large difference in M_{H + He}.

Recent N-body simulations of the formation of hot SEs (Terquem & Papaloizou 2007; Ogihara & Ida 2009) suggest that disk dissipation often causes collisions between embryos of SEs. The major accretion of atmospheres occurs after the collisions. In the late stages of disk evolution, photoevaporation separates the inner disk from the outer disk; then, the inner disk dissipates on a timescale of as short as 10⁴ yr (Gorti et al. 2009). The dashed lines in Figure 3 correspond to the cases of τ_d = 1 × 10⁴ yr. Comparing the lines with the crosses, one finds that the values of M_{H + He} inferred from modeling are much larger than those derived in this study, except for Kepler-11b and 11c. Furthermore, because the atmospheres are assumed to contain no grains, the derived atmospheric masses are maximal. Thus, the predicted atmospheres of the Kepler-11 planets are not accounted for by the in situ accretion in such a quickly dissipating disk.

Slower disk dissipation may be preferable in this respect. The green solid line shows the case of τ_d = 1 Myr and T_d = 550 K. While the corresponding line for T_d = 940 K is not drawn, the difference between the solid and dashed lines is similar to that for T_d = 550 K. In this case, as far as Kepler-11b to 11e are concerned, the values of M_{H + He} inferred from modeling seem to be consistent with those derived in this study. Kepler-11f lies above the line; only the high-mass end is close to the line. Furthermore, a cooler disk would be compatible with the M_{H + He} inferred for all the planets. As a reference, we show the result for T_d = 200 K by a blue solid line. However, it would be worth noting that Kepler-11d and 11e are near the critical points in the super-critical regime. As seen above, such cases are rare because slightly large cores result in much more massive M_{H + He}.

Before concluding this study, we discuss the loss of the atmospheres. The Kepler-11 planets are orbiting relatively close to their host star which is a G dwarf aged 8 ± 2 Gyr. Since the star is old, the current irradiation level of X-ray and EUV (XUV) is low. In the past, however, the planets should have been exposed to intense stellar XUV. Stellar-XUV irradiation results in loss of planetary atmospheres.

Here we estimate the mass of the H–He atmosphere that a planet with a given density loses via the energy-limited hydrodynamic escape (e.g., Watson et al. 1981). We integrate the equation

$$\begin{equation} \dot{M}_{\rm loss} = \frac{3 \epsilon F_{\rm XUV}}{4 G \bar{\rho } K_{\rm tide}}, \end{equation} \tag{ 7 }$$

where is the heating efficiency, F_XUV is the incident flux of stellar XUV, $\bar{\rho }$ is the planetary bulk density, and K_tide is a correction factor for the Roche lobe effect (Erkaev et al. 2007; Lecavelier des Etangs et al. 2004). We adopt = 0.4 (see Valencia et al. 2010), the fitting formula of F_XUV from Ribas et al. (2005), and K_tide = 0.92 appropriate for the Kepler-11 system.

Figure 4 shows the mass that the planet loses for 10 Gyr as a function of semimajor axis and planetary bulk density, together with the measured values of a and $\bar{\rho }$ for Kepler-11b to 11f. It is revealed that despite different distances to the host star, all the planets have lost similar amounts of atmospheric gas, which are of the order of 0.1 M_⊕. Even for Kepler-11b ($M_p = 4.3 {+2.2 \atop -2.0}\, M_\oplus$) and 11f (M_p = 2.3 ± 1.2 M_⊕) which are the two lightest planets among them, the amounts of mass loss correspond to ∼1%–10% of the planetary masses. (Although their early-stage inflated structure being neglected in this estimate, the results are similar to those from more detailed simulations by Lopez et al. 2012). Thus, the consideration of mass loss does not affect the implications obtained above for Kepler-11c to 11f. As for Kepler-11b, the condition becomes more severe.

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3.3.1. Degassing

3.3.2. Water-dominated Planets