ATMOSPHERE-INTERIOR EXCHANGE ON HOT, ROCKY EXOPLANETS

Our focus is the chemical evolution of the surface. To obtain physical insights into chemical evolution, we use a simple model and make idealizations. We use simple models of the winds and currents. We do not resolve the details of atmosphere-ocean coupling. Instead, we emphasize how the winds and currents interact with the chemical evolution of the surface.

Following Léger et al. (2011), we model a thin, hemispheric melt pool as the conduit between the solid interior and the atmosphere (Figure 1). We go beyond Léger et al. (2011) by considering horizontal convection in the magma pool, by calculating the winds driven by vapor-pressure gradients, and—most importantly—by considering the effects of fractional evaporation on residual-melt density. Because of these differences, we find that dynamic, compositionally primitive surfaces are likely. This is in contrast to Léger et al. (2011), who conclude that melt pools should have compositionally evolved CaO–Al₂O₃ surfaces. Our approach to winds is anticipated by Castan & Menou (2011), who consider winds in a pure-Na atmosphere on a hot rocky exoplanet and show that winds have little effect on surface temperature. We approximate the solid silicate interior of hot super-Earths as isentropic, consistent with mantle convection models (e.g., van Summeren et al. 2011) which show that solid-state convection can subdue large inter-hemispheric contrasts in interior temperature. To predict silicate-atmosphere compositions, we use the MAGMA code (Fegley & Cameron 1987; Schaefer & Fegley 2004, 2009, 2010; Schaefer et al. 2012; Figure 2). We assume negligible H₂O in the silicates.

Because magma planets are at close orbital distance, they are—and will remain—intrinsically easier to detect and characterize than true Earth analogs. Already, phase curves, albedo constraints, and time variability have been reported (Rouan et al. 2011; Rappaport et al. 2012, 2014; Demory 2014; Dragomir et al. 2014; Sheets & Deming 2014; Sanchis-Ojeda et al. 2015; Vanderburg et al. 2015; Demory et al. 2016a, 2016b). Therefore, there is now a pressing need for self-consistent theory relating magma-planet geophysics to data.

We assume that the melt pool's extent is set by a transition between fluid-like and solid-like behavior at a critical crystal fraction. Crystallizing magma acquires strength when increasing crystal fraction allows force to be transmitted through continuous crystal chains—"lock-up." Lock-up occurs at a melt fraction of ∼40% (Solomatov 2015). 40% melt fractions are reached at a lock-up temperature T_lo ≈ 1673 K (Katz et al. 2003; for a peridotitic composition; Appendix A). At T_lo, viscosity increases >10¹⁰-fold. Because of this large viscosity contrast, we treat material inside the pool as liquid, and refer to material colder than T_lo as "solid."

The pool angular radius, θ_p, is (in radiative equilibrium)

$$\begin{eqnarray}&&{\theta }_{p}\approx {\cos }^{-1}{\left(\displaystyle \frac{{T}_{{lo}}}{{T}_{{ss}}}\right)}^{4},\end{eqnarray} \tag{ 1 }$$

where T_ss = ${T}_{* }{(1-\alpha )}^{1/4}\sqrt{{r}_{\ast }/2a}({4}^{1/4})$ is the substellar temperature and the pool is centered on the substellar point.⁶ Here, a is the semimajor axis, T_* is the star temperature, r_* is the star radius, and we assume basalt-like albedo α = 0.1. (The optical albedo of liquid basalt is unknown, but we assume it to be similar to the albedo of solid basalt). Equation (1) applies if flow in the pool is sluggish, the atmosphere is optically thin, and atmospheric heat transport is negligible (Castan & Menou 2011; Léger et al. 2011)—all good assumptions (Sections 2.2, 2.3). Most planets with magma pools have θ_p > 60°.

Crystal fraction increases with depth. This is partly because pressure favors crystallization (Sleep 2007). Also, temperatures deep in the planet's solid mantle are smoothed-out by convective heat transport to the nightside, so that the mantle below the pool has lower entropy than the pool (van Summeren et al. 2011). These effects give a pool-depth estimate d_p ∼ O(10) km (Appendix B). Pool depth is further reduced by within-pool circulation (Section 2.2). Typically the melt pool is a shallow, hemispheric, magma ocean.

Magma flow is driven by gradients in T, X, or crystal fraction (Figure 3). An estimate of magma speed (v_p) in the near-surface of the pool can be obtained by neglecting viscosity and inertia (this will be justified a posteriori). Then, balancing the pressure-gradient and rotational forces (geostrophic balance) yields

$$\begin{eqnarray}&&{v}_{p}\sim \displaystyle \frac{{\rm{\nabla }}P}{f{\rho }_{l}},\end{eqnarray} \tag{ 2 }$$

where ∇P = ${g}^{\prime }{\rho }_{l}{\delta }_{T}/L$ = ${g}_{p}{\rm{\Delta }}{\rho }_{l}{\delta }_{T}/L$, where g' = ${g}_{{pl}}{\rm{\Delta }}{\rho }_{l}/\rho$ being the reduced gravity, g_pl is the planet surface gravity, Δρ_l is the density contrast across the fluid interface at the bottom of the density boundary layer, f ≈ 2Ω sin(θ_p/2) is the Coriolis parameter evaluated at θ_p/2 (halfway between the substellar point and the latitudinal limits of the pool), here, Ω = 2π/p, with p the planetary period, δ_T is the thickness of the density boundary layer (where the density contrast can be due to T, X, or crystal fraction), and $L={\theta }_{p}R$ is the pool center-to-edge distance measured along the planet's surface (planet radius is R). This model ignores currents at the equator, which are not directly affected by Coriolis deflection.

Consider flow driven by ∇T. Substellar radiative equilibrium temperatures can exceed T_lo by >10³ K. Magma expansivity is ∼10⁻⁴ K⁻¹ (Ghiorso & Kress 2004), and so Δρ_l/ρ_l ∼ 10%. The pool thermal boundary layer grows diffusively for as long as magma takes to move from the substellar point to the pool edge. Therefore ${\delta }_{T}\approx \sqrt{{\kappa }_{T}L/{\rm{\Xi }}{v}_{p}}$, where κ_T is the diffusivity of heat (which we take to be the molecular thermal diffusivity, 5 × 10⁻⁷ m² s⁻¹; Ni et al. 2015), and Ξ ("ageostrophic flow fraction") is the scalar product of the magma-velocity unit vector and a unit vector that is directed from the center to edge of the pool. Now we can replace ∇P with ${g}^{\prime }{\rho }_{l}{L}^{-1}\sqrt{\kappa L/{\rm{\Xi }}{v}_{p}}$ and solve for the thermal-flow timescale τ_T:

$$\begin{eqnarray}&&{\tau }_{T}\approx \displaystyle \frac{L}{{\rm{\Xi }}{v}_{p}}\approx {\kappa }^{-1/3}{\left(\displaystyle \frac{{L}^{2}f}{{\rm{\Xi }}{g}^{\prime }}\right)}^{2/3},\end{eqnarray} \tag{ 3 }$$

not counting any wind-driven circulation. A more sophisticated scaling (Vallis 2006)—including the variation of f with θ, which is appropriate for meridional flow—yields

$$\begin{eqnarray}&&{\tau }_{T}\approx {\kappa }^{-1/3}{\left(\displaystyle \frac{{{Lf}}^{2}}{\beta {g}^{\prime }}\right)}^{2/3}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\tau }_{T}\approx 15\,\mathrm{yr}{\left(\displaystyle \frac{\kappa }{{10}^{-6}{{\rm{m}}}^{2}{{\rm{s}}}^{-1}}\right)}^{-1/3}{\left(\displaystyle \frac{r}{{r}_{\oplus }}\right)}^{1/4}{\left(\displaystyle \frac{{\theta }_{p}}{\pi /2}\right)}^{1.3}{\left(\displaystyle \frac{1\mathrm{day}}{p}\right)}^{2/3},\end{eqnarray} \tag{ 5 }$$

where $\beta =(2\pi /{pr})\cos ({\theta }_{p}/2)$, i.e., evaluated at $\theta =\tfrac{1}{2}{\theta }_{p}$. κ = 10⁻⁴ m² s⁻¹ is possible with magma-wave breaking, but waves are likely small (Appendix C). Equation (5) is equal to Equation (3) with ${\rm{\Xi }}=L\beta /f$. Equations (3) and (5) yield the same results within 21% for Kepler's hot rocky exoplanets and Ξ ∼ 1; we use the results of Equation (5).

Magma current speed in the surface branch of the overturning circulation is v_p ∼ L/τ_T ∼ 0.02 m s⁻¹. The ratio of inertial to rotational forces (Rossby number Ro ≡ v_p/Lf ≈ 10⁻³) and the ratio of viscous to rotational forces (Ekman number ${Ek}\equiv \mu /{\rm{\Omega }}{d}_{p}^{2}\ll 1$) both turn out to be small, and so our neglect of inertia and viscosity in Equation (2) is justified a posteriori.

Magma-pool heat transport F_o (column W m⁻²) is small. F_o is given by the product of v_p and the θ-gradient in boundary-layer column thermal energy,

$$\begin{eqnarray}&&{F}_{o}\approx \displaystyle \frac{{\delta }_{T}}{{\tau }_{T}}\,{c}_{p,l}{\rho }_{m}({T}_{{ss}}-{T}_{{lo}}),\end{eqnarray} \tag{ 6 }$$

where ${c}_{p,l}$ is melt heat capacity (10³ J kg⁻¹ K⁻¹) and ρ_l is melt density (2500 kg m⁻³), giving

$$\begin{eqnarray}&&{F}_{o}\approx 30\,{\rm{W}}\ {{\rm{m}}}^{-2}\left(\displaystyle \frac{{T}_{{ss}}-2000\,{\rm{K}}}{{T}_{{ss}}-{T}_{{lo}}}\right),\end{eqnarray} \tag{ 7 }$$

10⁴× less than insolation (Figure 9). F_o is also small for pools with gradients in crystal fraction and in X. This is because the Δρ for freezing, for compositional differences, and for temperature gradients are all O(10%), and combining all three effects (∼tripling g') only decreases the overturn time by a factor of two (Equation (5)). Including the enthalpy of crystallization in Equation (6) would not alter this conclusion. Magma oceans are much less efficient at evening-out temperatures than are liquid-water oceans. This is because, for hot planets, radiative re-equilibration (scaling as T⁴) defeats advection (which scales as T¹; Showman et al. 2010). Because F_o is small, heat transport by melt-pool meridional overturning circulation cannot affect orbital phase curves (Esteves et al. 2015; Hu et al. 2015). Furthermore, an initially global surface magma pool cannot be sustained by melt-pool overturning circulation: instead, global surface magma pools beneath thin atmospheres will rapidly shrink to regional pools. Small F₀ implies that ocean surface elevation smoothly tapers to near-zero at θ_p—so that levees of frozen, overspilt magma (Gelman et al. 2011) are not tall.

The thermal structure set up by horizontal convection (Figure 3; Rossby 1965; Hughes & Griffiths 2008) consists of a cool, deep layer at nearly constant potential temperature, fed by narrow downwellings near the pool edge, and topped by a thin thermal boundary layer of thickness δ_T:

$$\begin{eqnarray}&&{\delta }_{T}\approx {\kappa }^{1/3}{\left(\displaystyle \frac{{f}^{2}L}{\beta {g}^{\prime }}\right)}^{1/3}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{\delta }_{T}\approx 15\,{\rm{m}}{\left(\displaystyle \frac{\kappa }{{10}^{-6}}\right)}^{1/3}{\left(\displaystyle \frac{r}{{r}_{\oplus }}\right)}^{\sim 1/8}{\left(\displaystyle \frac{{\theta }_{p}}{\pi /2}\right)}^{2/3}{\left(\displaystyle \frac{1\mathrm{day}}{p}\right)}^{1/3}.\end{eqnarray} \tag{ 9 }$$

Downwellings ventilate the subsurface interior of the pool with material that has been chilled near the pool edge to ≈T_lo. Because the material just below δ_T has T ≈ T_lo, d_p will not be much greater than δ_T. We assume d_p = 10 δ_T; d_p = 3 δ_T is possible. Either option gives a pool depth that is >100× shallower than in the absence of overturning circulation (Equation (20); Léger et al. 2011).

In this subsection, we show the following:

• 1.  
  atmospheric pressure P adjusts to the pressure in equilibrium with local surface temperature,
• 2.  
  atmospheric transport is $\propto \partial P/\partial \theta$;
• 3.  
  atmospheric energy transport and atmospheric mass transport both fall rapidly near the pool edge.

Busy readers may skip the remainder of this subsection.

Silicate atmospheres for T_ss < 3000 K (corresponding to all Kepler's planets) are thin enough to be in vapor-equilibrium (Heng & Kopparla 2012; Wordsworth 2015). Tenuous vapor-equilibrium atmospheres consisting of a single component equilibrate with local surface temperature on horizontal distances of ∼10 atmospheric scale heights H = RT/μg ≈ (8.3 J mol⁻¹ K⁻¹ × 2400 K)/(∼0.03 kg mol⁻¹ × 15 m s⁻²) ≈ 50 km for hot rocky exoplanet silicate atmospheres, where R is the gas constant, μ is molar mass, and g is planet surface gravity (Ingersoll 1989; Castan & Menou 2011). The appropriateness of the single-component assumption is discussed in Section 4.1. μ is set to the mean of μ_SiO and μ_Mg; SiO and Mg dominate at intermediate stages of fractionation (Schaefer & Fegley 2009). Because H ≪ planet radius (r), the approximation that pressure P adjusts to surface temperature T_s locally is well-justified. With local adjustment, pressure gradients are everywhere directed away from the substellar point. Because T_s is low on the nightside, nightside pressures are close to zero. Because the dayside atmosphere expands into near-vacuum on the nightside, wind speeds are v ∼ $\sqrt{{RT}/\mu }$ ∼ 1 km s⁻¹ (sound speed), where T ($\sim \overline{T}$) is the hot-zone atmospheric temperature. At least for the analogous case of Io, surface friction is ineffective in braking the flow (Ingersoll et al. 1985). Inertial forces modestly exceed rotational forces (v/Lf ≈ 4 for p = 3 days), so that winds are also directed away from the substellar point. Winds transport mass from an evaporation region near the substellar point to cold traps. Because cold traps are outside the pool, the loss of mass from the pool requires atmospheric flow past θ_p. Therefore, the temperature at θ_p regulates loss from the pool (given the approximation of local adjustment)—this temperature is T_lo. At T_lo, the pressure P_eq in equilibrium with silicate after 20 wt% fractional evaporation from an initial composition of Bulk-Silicate Earth is 1 × 10⁻² Pa (∼5 × 10⁻⁴ kg m⁻² for Super-Earths; Schaefer & Fegley 2009). The flux over the pool perimeter is thus m_f = vP_eq/g ∼ 0.5 kg s⁻¹ m⁻¹. For hemispheric pools, this corresponds to a column loss rate of m_f/ρ_lr ∼ 5 × 10⁻⁴ m yr⁻¹ averaged throughout the pool (ρ_l ≈ 2500 kg m⁻³, r ∼ 10⁷ m). So the time for the pool to be depleted of the constituent that is dominant in the vapor is

$$\begin{eqnarray}&&{\tau }_{d}\sim \displaystyle \frac{{d}_{r}{gr}\,{f}_{c}{\rho }_{l}}{v\,{P}_{\mathrm{eq}}({X}_{s},{T}_{{lo}})}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\approx 2\times {10}^{5}\,\mathrm{years}\\ &&\,\times \left(\displaystyle \frac{{P}_{\mathrm{eq}}({\rm{B}}.{\rm{S}}.{\rm{E}}.@20 \% ,{T}_{{lo}})}{{P}_{\mathrm{eq}}({X}_{s},T)}\right)\left(\displaystyle \frac{{d}_{r}}{100\,{\rm{m}}}\right)\left(\displaystyle \frac{{f}_{c}{f}_{g}}{1}\right),\end{eqnarray} \tag{ 11 }$$

where d_r is the effective depth of mixing (i.e., the column depth of melt that can be obtained by depressurization and melting on timescale τ_d), f_g is a geometric correction equal to 1 if the pool only samples material vertically beneath it, and f_c is the mass-fraction of the component in the magma.

τ_d falls as T rises because P_eq (and thus mass flux) increases super-exponentially with T. P_eq(T) (for an initial composition of Bulk-Silicate Earth, 1400 K < T < 2800 K), is well fit by

$$\begin{eqnarray}&&{\rm{log10}}({P}_{\mathrm{eq}})\approx {k}_{1}\exp ({k}_{2}{T}_{s}),\end{eqnarray} \tag{ 12 }$$

where k₁ = {−34.7, −60.1, −60.2} and k₂ = {−0.00112, −0.00122, −0.00101} for small (<0.01), intermediate (0.5), and large (>0.9) fractions of silicate vaporization, respectively, and P_eq is in bars (Figure 2). Pool mean surface temperatures $\overline{T}$ are typically 500 K higher than pool-edge temperature T_lo. Column-integrated atmospheric mass flux scales with the gradient in the pressure and with the surface pressure. Because the surface pressure increases exponentially with local temperature, atmospheric transport near the center of the pool is much faster than atmospheric transport over the edge of the pool. For example, for CoRoT-7b, a basic calculation shows (Figure 6) that the trans-atmospheric transport over the evaporation-condensation boundary within the pool is 10³ kg s⁻¹ (m perimeter)⁻¹, which is 5000× greater than transport over the edge of the pool. Therefore the pool can internally differentiate—develop compositional boundary layers—faster than pool material can be lost by atmospheric flow over the edge of the pool. This justifies our quasi-steady-state approximation for mass exchange within the pool.

In addition to the strong T dependence, τ_d is sensitive to X_s. As fractional evaporation progressively removes the more volatile components, P_eq falls (Figure 2).

τ_d is proportional to d_r. d_r can vary from less than pool thermocline depth δ_T (10 m) up to planet radius r (10⁷ m). Although as little as 10⁸ years are required to remove Na from an Earthlike planet (f_g = 2; d_r ≈ 10⁷ m; f_c ∼ 0.0035, Table 2), 4 × 10¹⁰ years are required for full planet evaporation (f_g = 2; d_r ≈ 10⁷ m; f_c ∼ 1).

Atmospheric heat flux (W m⁻²) is given by

$$\begin{eqnarray}&&{F}_{a}\approx {l}_{v}P\left(\displaystyle \frac{v}{{gL}}\right)\approx {l}_{v}{k}_{1}\exp ({k}_{2}{T}_{s})\left(\displaystyle \frac{v}{{gL}}\right)\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{F}_{a}\approx 60\,{\rm{W}}\ {{\rm{m}}}^{-2}\left(\displaystyle \frac{P({T}_{s})}{1\,\mathrm{Pa}}\right){\left(\displaystyle \frac{{R}_{\oplus }}{R}\right)}^{\sim 3/2},\end{eqnarray} \tag{ 14 }$$

where l_v = 6 × 10⁶ J kg⁻¹ is magma's vaporization enthalpy (neglecting sensible heat transport and wind kinetic energy, which are both <10% of l_v). F_a equals absorbed insolation at 3500 K. For T ≪ 3500 K, F_a is small compared to insolation (Castan & Menou 2011).

Loss of mass to cold traps is supplemented by escape to space. Atmospheric escape is 10⁸ kg s⁻¹ for a planet around a 1 Gyr old Sun-like star—if escape is EUV-flux-limited and 100% efficient (Valencia et al. 2010). Efficiencies for rocky planets are poorly constrained (Murray-Clay et al. 2009; Ehrenreich et al. 2015; Owen & Alvarez 2015; Tian 2015). Even at 100% efficiency, escape to space corresponds to a hemisphere-averaged vertical breeze of 10 m s⁻¹ (4 × 10⁻³ m yr⁻¹ of melt-column ablation); this does not greatly alter the conclusion that P ≈ P_eq, and so liquid remains stable (Ozawa & Nagahara 1997). Hydrostatic Roche-lobe overflow is minor for Super-Earths (Rappaport et al. 2013). Summing escape to-space and loss to cold traps, Earth-sized planets cannot be wholly processed in the age of the universe, but smaller planets can disintegrate (Section 4.4).

Surface composition for melt pools is set by competition between the pool's overturning circulation (which refreshes the surface on timescale τ_T, Section 2.2), and flow in the atmosphere, which produces a chemically differentiated boundary layer on a timescale τ_X. Consider a pool with two chemical components, volatile A and refractory Z. "A" evaporates from the surface preferentially near the substellar point, and condenses on the surface beyond the evaporation-condensation boundary—the angular distance (θ₀) where the sign of net evaporation changes. If ρ_A > ρ_Z , then the residual melt in the evaporation zone becomes buoyant, and the residual melt in the condensation zone will sink into the pool interior. If ρ_A < ρ_Z, then the residual melt in the condensation zone becomes buoyant, and the residual melt in the evaporation zone will be unstable to sinking. Therefore, regardless of whether A or Z is denser, the magma pool will (given time) acquire a variegated surface, with parts of the surface having evolved X_s and parts of the surface being close to the initial composition. (Here we assume that there is a large density difference, and that θ_p > θ₀.) Diffusive exchange with the deeper layers of the pool will initially resupply/remove constituents faster than fractional evaporation E, and so the time to form a chemically variegated surface is set by the diffusion-ablation balance,

$$\begin{eqnarray}&&{\tau }_{X}\approx \left(\displaystyle \frac{{\kappa }_{X}}{\overline{{E}_{e}}/{\rho }_{l}}\right)\left(\displaystyle \frac{{\rho }_{l}}{\overline{{E}_{e}}}\right)={\kappa }_{X}\left(\displaystyle \frac{{\rho }_{l}^{2}}{{\overline{{E}_{e}}}^{2}}\right),\end{eqnarray} \tag{ 15 }$$

where κ_X is molecular diffusivity and $\overline{{E}_{e}}$ is the mass-loss rate due to fractional evaporation. The first term in brackets in Equation (15) is the compositional boundary-layer thickness δ_X = ${\kappa }_{X}{\rho }_{l}/\overline{E}$, and the second term in brackets is the compositional boundary-layer processing speed. κ_X is 10⁻⁹–10⁻¹⁰ m² s⁻¹ for liquid silicates at hot magma-pool temperatures (Karki et al. 2010; de Koker & Stixrude 2011). The Lewis number (${\kappa }_{X}/{\kappa }_{T}\,$) is  <0.01. Increasing κ_X slows down chemical boundary-layer development—because chemical diffusion refreshes the surface with fresh material. By contrast, increasing κ_T speeds up horizontal convection—because upwelling via thermal diffusion is needed to close the circulation (Equation (5)).

When τ_X/τ_T ≪ 1, chemical differentiation between the surface boundary layer and the interior of the magma pond may occur (Figure 3(c)). If chemical fractionation occurs, then the atmosphere effectively samples the chemically fractionated skin layer δ_X. By contrast, when τ_X/τ_T ≫ 1, the atmosphere effectively samples a well-mixed pool (Figure 3(b)).

${\tau }_{X}/{\tau }_{T}\ll 1$ is neither necessary nor sufficient for a buoyant lag to form. If the chemical boundary-layer density ρ_δ exceeds the initial density ρ₀, then the incompletely differentiated skin layer can founder and be replaced at the surface by material of the starting composition. Alternatively, if the chemical boundary-layer density ρ_δ is less than the initial density ρ₀, the entire pool might evolve into a (well-mixed) lag that is compositionally buoyant with respect to the mantle, even if τ_X/τ_T ≫ 1.

Pool depth stays steady during evaporation. This is because melt-back of the stratified solid mantle at the base of the pool keeps pace with evaporative ablation of the pool surface.⁷

$$\begin{eqnarray}&&{\tau }_{\kappa }\approx 2.32\sqrt{{\kappa }_{T}{\tau }_{d}}\ll \displaystyle \frac{E}{{\rho }_{m}}{\tau }_{d}.\end{eqnarray} \tag{ 16 }$$

Melt-back dilutes the fractionating pool with unfractionated material, slowing compositional evolution. (We assume that any solid phases that form at the base of the pool by reactions between the liquid and the ascending solid rock, such as spinels, are swept up into the pool and remelt).

Using the atmospheric model (Appendix D) to compute E, we find that molecular diffusion within the solid mantle is too slow to delay whole-pool fractional evaporation:

$$\begin{eqnarray}&&\displaystyle \frac{{\rho }_{m}\,{f}_{c}{\delta }_{p}}{E}\ll 2.32\sqrt{{\kappa }_{S}{\tau }_{d}},\end{eqnarray} \tag{ 17 }$$

where κ_S (≲10⁻¹⁴ m² s⁻¹) is a molecular diffusivity in crystalline silicates (Brady & Cherniak 2010).⁸

Suppose that the pool's composition is uniform and in steady state, and the pool is much less massive than the time-integrated mass lost to trans-atmospheric transport. Then, mass balance requires that the material lost from the pool must have the same composition as the melt-back input to the pool—i.e., bulk-silicate composition (Wang et al. 1994, p. 348; Ozawa & Nagahara 2001; Richter 2004). The pool adjusts to a composition X_b (buffer) that satisfies this condition. Because of the wide range of volatilities for the component oxides (Figure 2), X_b is CaO/Al₂O₃-dominated, with tiny proportions of Na₂O and K₂O, and a low total atmospheric pressure (steady, uniform, evolved surface composition; Figure 2).

If $\rho ({X}_{b})\lt \rho ({X}_{0})$, then X_p = X_b is a steady state. However, if $\rho ({X}_{b})\gt \rho ({X}_{0})$, then the intrinsically dense material in the pool may be unstable to finite-amplitude perturbations in the pool-base. X_p can then be intermittently reset to X₀ by pool drainage into the mantle.

The pool can drain by infiltrating, by diking, or by forming one or more approximately spherical blobs (diapirs). Diapirism is rate-limited by the time needed to concentrate a hemispheric shell into a spherical diapir (Honda et al. 1993; Reese & Solomatov 2006). In numerical simulations (Honda et al. 1993), diapirs form on a timescale

$$\begin{eqnarray}&&{\tau }_{f}\approx \displaystyle \frac{27{\eta }_{m}}{8\pi {\omega }^{2/3}G{\rho }_{0}^{2}{r}^{2}}{\left(\displaystyle \frac{\rho ({X}_{b})}{{\rho }_{0}}-1\right)}^{-1}{\left(\displaystyle \frac{{r}^{3}}{{(r-{d}_{p})}^{3}}-1\right)}^{-2/3}\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{\tau }_{f}\approx 50\,\mathrm{Kyr}\,\left(\displaystyle \frac{{\eta }_{m}}{{10}^{18}\,\mathrm{Pa}\,{\rm{s}}}\right),\end{eqnarray} \tag{ 19 }$$

where η_m is the mantle viscosity (∼10¹⁸ Pa s; Zahnle et al. 2015), G is the gravitational constant, r is the planet radius, and ω ≈ 0.3 is the fraction of the hemispheric shell's volume that contributes to the diapir. To obtain Equation (19), we assume ρ(X_b)/ρ₀ ≈ 1.1. In order to drain, the diapir density must exceed the solid density, which might occur through compositional evolution, or through partial freezing by ventilation by cool currents at the base of the pool. τ_f ≈ τ_κ (Equation (16)). Diapir volume will be ${V}_{d}=2\pi {\delta }_{p}\omega {r}^{2}$ giving diapir radius O(10²) km. As the diapir sinks, a maximum of ${V}_{d}{\rho }_{m}\,{\rm{\Delta }}\rho \,G\,{M}_{{pl}}/r\sim {10}^{26}$ J of gravitational potential energy is converted to heat, enough to warm the planet's interior by 0.01 K per event—delayed differentiation. (Alternatively, melting of the adjacent mantle from viscous dissipation as the diapir sinks may entrain lighter fluid and halt the descent of the diapir). Although a single diapir is the most linearly-unstable mode (Ida et al. 1987, 1989), smaller diapirs (or dikes, or magma solitons) may be more realistic, reducing ω. As $\omega \to 0$, composition becomes steady. For $\omega \ne 0$, composition is unsteady.

Rocky-planet destruction by evaporation has five steps: (1) loss of atmophiles (H, C, N, S, P, halogens, noble gases), oceans, and continents; (2) loss of Na+K, (3) loss of Fe+Mg+Si, (4) loss of the residual Al+Ca(+Ti); (5) loss of the metal core. We consider steps 2, 3, and 4, for a range of bulk-silicate (mantle + crust) compositions (Table 2). We do not consider carbide-rich planets (e.g., Wilson & Militzer 2014). The main density-determining components are SiO₂, CaO, Al₂O₃, MgO, FeO(T), Na₂O, and K₂O (Figure 2). (FeO(T) can stand for FeO, Fe₂O₃, Fe₃O₄, or a mixture, depending on oxidation state). Cations are lost from the melt in the order Na > K > Fe > (Si, Mg) > Ca > Al.

Output from the MAGMA code for fractional evaporation of melt at 2000 K (Bulk-Silicate Earth composition) is shown in Figure 4. Initially the magma outgasses a Na/K rich mix which (upon condensation) has a low density. The residue density does not change much at this stage because Na₂O and K₂O are minor components of the melt. Next to outgas is an Fe-rich mix. Upon condensation, this material is denser than the magma underlying it; the Fe-rich material will sink to the base of the pool. Continuing beyond ∼22 wt% fractional evaporation, the density of the residual magma increases. This will lead to small-scale compositional convection within the evaporation-zone compositional boundary layer, but the boundary layer as a whole still has lower density than the initial density (ρ₀). The still-buoyant boundary layer continues to evolve to a more-fractionated composition (the gas is now dominated by Mg, Si, and O). Beyond 81 wt% fractional vaporization (point A in Figure 4), the density of the now-well-mixed boundary layer exceeds ρ₀. That is because the boundary layer is now CaO-rich (∼20% molar) and CaO is dense (Appendix A). We define ρ_h as the maximum density at >70 wt% fractionation. Beyond A, the boundary layer sinks into the underlying magma. Fractional evaporation beyond A is not possible.⁹ The surface reaches a dynamic steady state, with parts of the surface covered by fresh material, and parts of the surface covered by evolved material that is nearly ready to sink. Atmospheric pressure will not be much less than the pressure above an unfractionated melt. The relevant reservoir for long-term net pool compositional evolution is the entire silicate mantle (assuming dense material drains efficiently into the solid mantle).

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Fractional vaporization of an FeO-rich exoplanet (Elkins-Tanton & Seager 2008a) proceeds differently (Figure 5). At 0 wt% fractionation, the melt is ∼10% denser than the unfractionated melt of Bulk-Silicate Earth. The FeO-rich pool will initially develop a stably stratified layer due to loss of Fe. At 60 wt% fractionation, density starts to rise and small-scale convection develops within the boundary layer (Figure 5). At 95 wt% fractionation, the extent of convection reaches its maximum—but convection is still confined within the boundary layer. The boundary layer stays buoyant; re-equilibration with the deep interior is inhibited. The relevant reservoir depth for fractionation is δ_X or δ_T for τ_T ≫ τ_X, or the entire pool for τ_T ≪ τ_X. Because τ_d (δ_X), τ_d(δ_T) and τ_d(δ_p) are all ≪1 Gyr (Equation (11)), FeO-rich planets are vulnerable to surface-composition evolution. This leads to an extremely low-pressure atmosphere (Figure 2). Results for additional compositions, and additional details, are shown in Figures 14–17.

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Vulnerability to stratification is proportional to (ρ₀–ρ_h). (ρ₀–ρ_h) is shown in Figure 7 for a range of pool $\overline{T}$ and mantle compositions (Table 2). High [FeO] makes stratification more likely because FeO is both dense and volatile. When [FeO] is high, ρ₀ is high. For a high-[FeO] world, the near-complete loss of FeO before 70 wt% fractionation greatly decreases residual-magma density relative to ρ₀. Therefore the Ca-driven "uptick" in density at high fractionation does not reach ρ₀ (Figure 5). Therefore, (ρ₀–ρ_h) is high (Figure 7). If [FeO] is low, then stratification is impossible. If [FeO] is high, then stratification is possible.

To calculate the time τ_X required for compositional evolution (Figure 8), we need to know the mean evaporation rate ($\overline{E}$) within the pool. To find $\overline{E}$, we use a 1D atmospheric model (Appendix D), which largely follows Castan & Menou (2011) and Ingersoll (1989). Typical 1D model output is shown in Figure 6.

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By combining Equations (5)–(11) and (23)–(26), we can predict the surface composition and atmospheric pressure for each magma planet. Predictions for specific magma planets are shown in Table 3. Figures 7–9 show the expected steady-state outcomes.

• 1.  
  If hot rocky exoplanets are FeO-rich, then pool compositions will be steady, uniform, and dominated by CaO and Al₂O₃. We refer to this state as a "buoyant shroud." Atmospheres will be ≲O(1) Pa (Figure 2). Venting of Na and K from the pool will be minor, and escape to space may be reduced.
• 2.  
  If hot rocky exoplanets are FeO-poor, then pool composition will be time-variable, and probably variegated. Averaged over timescales >τ_f, atmospheres will be thick. Escape to space will not be limited by the supply of atmosphere (Figure 2).
• 3.  
  τ_T increases as $\overline{T}$ increases because hotter pools are wider—with increased L(θ_p) and f(θ_p).

Table 3.  Predictions of Atmospheric Pressure and Surface Composition for Selected Magma Planets

Planet	p (day)	Tss (K)	Melt-pool Radius (°)	Mean Pool Temperature $\overline{T}$ (K)	Pool Depth (δp × 10) (m)	(θp > θ0)?	Pool-overturn Timescale (s)	Atmosphere Timescale (s)	Pss at 95 wt% Fractionation (Pa)	Pss at 20 wt% Fractionation (Pa)
Not known to be disintegrating:
CoRoT-7b	0.85	2425	77°	2146	100	Y	108	107	100	102
55 Cnc e	0.74	2671	88°	2263	200	Y	109	106	101	103
Kepler-36b	13.8	1476	0°	n.a.	n.a.	n.a.	n.a.	n.a.	<10−8	10−5
Kepler-93b	4.73	1569	0°	n.a.	n.a.	n.a.	n.a.	n.a	<10−7	10−4
Kepler-78ba	0.36	3056	102°	2423	200	Y	109	102	102	104
Kepler-10b	0.84	3007	96°	2464	200	Y	109	103	102	104
Disintegrating:
KIC 12557548b	0.65	2096	66°	1919	100	Y	108	107	10−2	100
K2-22b	0.62	2135	68°	1945	100	Y	109	106	10−2	100
KOI-2700b	0.91	1780	39°	1730	60	Y	108	1011	10−5	10−2

Notes. Dominant timescales are underlined. Timescales calculated for 20 wt% fractionation. An initial composition corresponding to that of Bulk-Silicate Earth is assumed. Pressures <O (1) Pa are too low for disintegration (Perez-Becker & Chiang 2013).

^aThe assumptions made in obtaining the atmosphere and ocean timescales are only marginally satisfactory for planets as hot as Kepler-78; see Section 3.3.

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[FeO(T)] measures the extent to which a planet's building-blocks have been oxidized. (For this paper, [FeO(T)] = [FeO] for the Bulk-Silicate Earth composition, the Bulk-Silicate Mercury composition, and the Bulk-Silicate Mars composition, and [FeO(T)] = [Fe₂O₃] for the Coreless Exoplanet composition.) The most likely oxidant is water.

FeO(T) is not expected from condensation-from-vapor of an solar-composition protoplanetary disk: fast condensation of solar-composition gas at the snowline (145–170 K) forms large quantities of iron (Fe + FeS), Mg-silicates, and H₂O, but FeO(T) in silicates is not expected because of kinetic effects (Krot et al. 2000; Lewis 2004; Podolak & Zucker 2004; Moynier & Fegley 2015). However, FeO_x production occurs readily via metal-water reactions (Lange & Ahrens 1984; Dreibus & Wanke 1987; Rosenberg et al. 2001),

$$\begin{eqnarray*}&&{\mathrm{Fe}}^{0}+{{\rm{H}}}_{2}{\rm{O}}\ \to \ {\mathrm{Fe}}^{2+}{\rm{O}}+{{\rm{H}}}_{2}\,\,(\mathrm{Reaction}\,\ 1),\end{eqnarray*}$$

on planetesimals and larger bodies, as well as via water-rock reactions, such as (Früh-Green et al. 2004; McCollom & Bach 2009; Klein et al. 2013)

$$\begin{eqnarray*}6{\mathrm{Fe}}_{2}{\mathrm{SiO}}_{4}\ +\ 11{{\rm{H}}}_{2}{\rm{O}}\ & \to & 2{\mathrm{Fe}}_{3}{{\rm{O}}}_{4}\ +\ 5{{\rm{H}}}_{2}\\ & & +\ 3{\mathrm{Fe}}_{2}{\mathrm{Si}}_{2}{{\rm{O}}}_{5}{(\mathrm{OH})}_{4}\,\,(\mathrm{Reaction}\ 2),\end{eqnarray*}$$

and by condensation of silicates from a water-enriched vapor, for example, a plume produced by impact into a water-rich target planetesimal (Fedkin & Grossman 2016). FeO(T) produced by Reactions 1 and 2 remains in the silicate mantle. Because of accretion energy, growing planets get hotter. Above ∼900 K, Reaction 2 is thermodynamically unfavorable. However, iron oxidation can continue if accreted material includes both Fe-metal and H₂O (e.g., via Reaction 1; Kuwahara & Sugita 2015). [FeO(T)] production is also favored by partitioning of metallic Fe into the core and by H-escape (Wade & Wood 2005; Frost et al. 2008; Zahnle et al. 2013).

Because Reactions 1 and 2 involve water, they occur more readily where water content is enhanced: e.g., beyond the snowline or in objects that accrete material from beyond the snowline. Consistent with this, rocky objects that formed further out in the solar nebula have more mantle FeO(T) (Figure 7; Rubie et al. 2011, 2015). With 18 wt% FeO in silicates, Mars is (just) unstable to stratification under fractional vaporization (although only for relatively cool magma; Figure 7). FeO-rich silicate compositions (>25 wt%) are also obtained for the parent body of the CI-class meteorites, the parent body of the CM-class meteorites, and some angrite-class meteorites (Anders & Grevesse 1989; Lewis 2004; Keil 2012). Planets with even more FeO are theoretically reasonable, and Ceres may be a solar system example (McCord & Sotin 2005; Elkins-Tanton & Seager 2008a; Ciesla et al. 2015; Rubie et al. 2015). [FeO] can affect hot rocky-planet atmospheric thickness and time variability (Figures 2, 7). Because atmospheric thickness and magma-planet variability are potentially observable, this suggests a route to constrain hot rocky-planet composition. Such routes are valuable because mass and radius measurements only weakly constrain the composition of exoplanet silicates (Rogers & Seager 2010; Swift et al. 2012; Gong & Zhou 2012; Fogtmann-Schulz et al. 2014; Dorn et al. 2015).

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Furthermore, because water is less available inside the snowline, [FeO] probes the hot rocky planet's birth location relative to the snowline (and thus migration distance). Assuming a radial temperature gradient similar to the solar nebula, and further assuming that only a negligible fraction of Fe-silicates from the birth molecular cloud (Jones 1990; Min et al. 2007) survive to be incorporated into hot rocky exoplanets, evidence for mantle [FeO] in hot rocky exoplanets is evidence against in situ accretion (Chiang & Laughlin 2013). Evidence for mantle [FeO] in hot rocky exoplanets might be used to test inside-out planet formation (Chatterjee & Tan 2014). This is because [FeO] formation in the Solar System occurred on >km-sized objects, but in the Chatterjee & Tan (2014) model, >km-sized objects are assembled close to the star from dehydrated boulders. If hot rocky exoplanets formed through migration of objects of planetesimal size or larger from beyond the snowline (Cossou et al. 2014; Raymond et al. 2014, p. 595), then we expect high [FeO] on hot rocky exoplanets.

Our model omits or simplifies geologic processes that are not well understood even for the rocks of Earth. For example, two-phase (mush) effects, such as magma solitons, filter-pressing, and fingering instabilities (Scott & Stevenson 1984; Katz et al. 2006; Solomatov 2015), are not included in our model. Melt-residue separation at modest temperatures (low melt fractions) will yield Ca-rich, Al-rich melts. Ca/Al-rich melts have low T_lo and tend to favor overturning during fractional vaporization. This overturning-promoting effect is less strong for the high melt fractions (hot rocky exoplanets) that are considered here. However, even at high melt fraction, olivine seperation from melt might still be important, and the effect of this process on surface composition could be a target for future work (Asimow et al. 2001; Suckale et al. 2012).

Bulk-rock compositions are assumed to be similar to Solar System silicates, consistent with rocky-planet densities, stellar spectra, and white-dwarf data (Lodders et al. 2009; Jura & Young 2014; Adibeykan et al. 2015; Dressing et al. 2015; Gaidos 2015; Thiabaud et al. 2015). Some accretion simulations predict strongly varying Mg/Si (Carter-Bond et al. 2012; Carter et al. 2015). Future work might compute (ρ₀–ρ_h) for a broad range of silicate composition (Figure 7).

We assume a well-stirred mantle. Well-stirred mantles are predicted for large, hot planets by simple theories. Simple theories are undermined by ¹⁴²Nd and ¹⁸²W anomalies, which show that the Early Earth's mantle was not well-stirred (Debaille et al. 2013; Carlson et al. 2014; Rizo et al. 2016). This is not understood.

We do not discuss loss of atmophiles (i.e., primary and outgassed atmospheres), oceans and crust (Lupu et al. 2014). These steps should complete in <1 Ga even if the efficiency of EUV-driven atmospheric escape is low, although CO₂ might resist ablation through 4.3 μm-band cooling (Tian et al. 2009; Tian 2015). Thick remnant atmospheres are not ruled out by density data (e.g., Ballard et al. 2014). Figure 19 shows results of a sensitivity test for crustal evaporation.

We also do not pursue the question of what happens to the magma-pool circulation on planets where the magma-pool circulation is dominated by the mass loading of atmospheric condensates (Tokano & Lorenz 2016). Such mass loading could throw the magma-pool circulation into reverse, but only on strongly atmosphere-dominated planets where the magma-pool circulation is much less efficient at transporting heat than is the atmosphere.

The atmospheric model has several limitations. (1) We assume that the temperature of evaporating gases is equal to the substellar temperature. (2) The model assumes a single atmospheric species. Although a single species typically dominates the silicate atmosphere (Schaefer & Fegley 2009), real atmospheres have multiple species. This is not a big problem in the evaporation zone (where the total saturation vapor pressure exceeds the total pressure of the overlying atmosphere). However, in the condensation zone, less-refractory atmospheric constituents form a diffusion barrier to condensation, and so condensation-zone atmospheric pressures will be higher in reality than in our model. This could increase P at θ_p. (3) Feedbacks between X_s and P(θ) during surface compositional evolution are neglected. (4) Magnetic effects (Batygin et al. 2013; Rauscher & Menou 2013; Koskinen et al. 2014; Lopez-Morales et al. 2011) are not included. Thermal ionization reaches ∼10⁻³ (fractional) at 3100 K for Bulk-Silicate Earth. (5) We find the initial winds, not the more complex wind pattern that may subsequently develop after compositional variegation develops. (6) Our model lacks shocks (Heng 2012).

Io's atmosphere is the best solar system analog to magma-planet atmospheres. Over most of Io's dayside, the approach of Ingersoll (1989) matches Io-atmosphere data (Jessup et al. 2004; Walker et al. 2012). Walker et al. (2010) find (in a sophisticated model that includes both sublimation and volcanic loading) that Io's atmosphere's evaporation zone extends 45°–105° from the substellar point, which is more than predicted by Ingersoll (1989).

Crystallization is inevitable near the edge of a compositionally evolved pool because T_lo is only just above the CaO–Al₂O₃ solidus (Berman 1983; Mills et al. 2014). If crystals are small enough to remain entrained in the melt, then evolved fluid near the margin of the pool will increase in density and sink (increased buoyancy forcing in Equation (5)). If crystallization increases viscosity, then the drainage of dense, evolved material into the solid mantle will slow, increasing the likelihood that the surface will have time to evolve into a CaO–Al₂O₃-rich composition. Lag formation is hard to avoid if the entire melt pond freezes.

We do not consider light scattering by rock clouds (Juhász et al. 2010). Cloud grain-size depends on P (and thus T_s), and so feedback involving silicate-dust clouds may modulate magma-pool activity (Rappaport et al. 2012).

Thermo-chemical boundary layers can have a complex substructure, which we simplify. Thermal stirring will tend to mix the boundary layer in the small-scale convection zone θ > $\theta ({T}_{s}=\overline{T})$, inhibiting compositional boundary-layer development. Therefore, the development of a buoyant layer segregation is most likely if the compositional boundary layer is developed before $\theta \gt \theta ({T}_{s}=\overline{T})$. This effect reduces the critical τ_X/τ_T by a factor of ∼2 (within the gray bars in Figures 8 and 9).

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We do not consider how condensed volatiles might return to the pool by gravity-current flow of viscous condensates from the permanent nightside back into the light (Leconte et al. 2013). For example, the Na from Earth corresponds to a hemispheric sheet of thickness 80 km, thick enough to flow back onto the light-side (Leconte et al. 2013). Sublimation of this Na flow might allow an Na-dominated (and thus much thicker) atmosphere to persist even after Na has been removed from the silicate mantle. Analogous flow may occur for S, which is the dominant volatile on Io's surface and makes up 250 ppm of Earth's mantle (McDonough & Sun 1995). Escape to-space is necessary to finally remove these materials from the planet.

For T_s > 3500 K, atmospheric heat transport melts the nightside, and the magma pool covers the entire surface—a magma sea. This limit is reached for accreting planets (Lupu et al. 2014; Hamano et al. 2015; Kuwahara & Sugita 2015), planets around post-main-sequence stars, and the hard-to-observe rock surfaces of planets with thick volatile envelopes (e.g., Howe & Burrows 2015; Chen & Rogers 2016; Owen & Morton 2016).

Only a few relevant albedo measurements exist (Gryvnak & Burch 1965; Adams 1967; Nowack 2001; Petrov 2009), and so albedo-composition feedbacks are not considered in detail in this paper. High surface albedo would reduce atmospheric P and mass fluxes, and so magma currents are relatively more important on high-albedo worlds. Limited data (Zebger et al. 2005) indicate increasing UV/VIS reflectance with increasing CaO/SiO₂ ratio in the system CaO–Fe₂O₃–SiO₂. Taken at face value, these results suggest CaO-rich surfaces could explain data that have been interpreted as indicating high magma-planet albedos (e.g., Rouan et al. 2011; Demory 2014). Per our calculations, CaO-rich surfaces indicate FeO-rich initial compositions. In turn, this is a strike against in situ planet formation. This argument is necessarily speculative because of the paucity of relevant lab data. Motivated by the connection between magma-planet albedo and planet formation theory, gathering more laboratory data relevant to magma-planet albedos could make these arguments more rigorous.

On Earth, horizontal convection fills the deep sea with cool fluid (Lodders & Fegley 1998; Hughes & Griffiths 2008). Therefore, Earth's sea-floor temperature is ∼1°C, even though Earth's mean sea-surface temperature is 18°C. Similarly, magma-pool downwellings irrigate the magma-solid interface ("sea-floor") with cool magma. Cool-magma downwellings set a low and uniform top temperature boundary condition on the dayside mantle circulation. For a dayside-spanning shallow magma pool, the mean temperature at the upper boundary of a convecting-mantle simulation is the mean of the pool-edge temperature T_lo and the antistellar-hemisphere surface temperature T_AS, i.e., ∼850 K (Figure 10). This is because T_lo is the characteristic temperature for the magma-solid interface, and magma floods the dayside, while the nightside remains very cold. Because mantle convection (in the absence of tidal heating) will transport much less heat than magma currents, this is a constant-T boundary condition. These low temperatures make 1:1 synchronous rotation more likely because pseudo-synchronous rotation requires a hot (> T_lo) interior (Makarov 2015). The "tectonic refrigeration" effect is large (Figure 10). Convection models (van Summeren et al. 2011; Foley et al. 2012; Lenardic & Crowley 2012; O'Rourke & Korenaga 2012; Tackley et al. 2013; Noack et al. 2014; Miyagoshi et al. 2014) could investigate the effects of a hemispherically uniform upper temperature boundary condition on mantle convection.

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Planet density decreases with increasing planet radius in the range 1 < r_⊕ < 4 (Wu & Lithwick 2013; Weiss & Marcy 2014). This density trend can be simply explained by models in which small planets consist of varying proportions of Earth-composition cores and H₂-rich envelopes (Hadden & Lithwick 2014; Lissauer et al. 2014; Lopez & Fortney 2014; Dressing et al. 2015; Wolfgang et al. 2015). H₂-rich envelopes might form by nebular accretion (Bodenheimer & Lissauer 2014; Jin et al. 2014; Lee et al. 2014; Inamdar & Schlichting 2015; Ogihara et al. 2015), or by outgassing of H₂-rich material (Sleep et al. 2004; Elkins-Tanton & Seager 2008b; Sharp et al. 2013). The hottest rocky exoplanets either fail to develop a H₂-rich envelope, or lose H₂ in <1 Gyr (Chiang & Laughlin 2013; Chatterjee & Tan 2014; Lopez & Fortney 2014; Schlichting 2014; Ogihara et al. 2015). The upper limit on H₂ outgassing via iron oxidation, corresponding to [FeO(T)] = 48.7 wt% (Table 2), is 1.7 wt% (Rogers et al. 2011). 1.7 wt% H₂ is not very much greater than the amount required to explain the radii of low-mass Kepler planets (Lopez & Fortney 2014; Howe et al. 2014; Howe & Burrows 2015; Wolfgang & Lopez 2015). Therefore, if outgassing explains the observed radii of Kepler Super-Earths, then their rocky cores must contain abundant FeO(T). Upon H₂ removal via photo-evaporation and impact erosion (Ehrenreich & Desert 2011; Schlichting et al. 2015), the rocky cores of the most-oxidized planets may develop stratified surfaces (Figure 4), forming a CaO–Al₂O₃-rich lag. This is speculative because the percentage of oxidation needed to match Kepler data depends on the molecular weight of the outgassed atmosphere, and that percentage is poorly constrained. The lag might be detectable through its effect on albedo (e.g., Demory 2014) and atmospheric composition.

The molten surfaces of disintegrating rocky planets are the single most likely source for time-variable dust plumes orbiting KIC 12557548, KOI-2700, K2-22, and WD 1145+017 (Rappaport et al. 2012, 2014; Sanchis-Ojeda et al. 2015; Vanderburg et al. 2015). Planet disintegration involves vapor pressures ≳O(1) Pa (Perez-Becker & Chiang 2013). Because evolved-surface-composition worlds usually have lower vapor pressures (Figure 2), they usually resist disintegration. Therefore, observed disintegrating planets require active surface-interior exchange, or entrainment of solids by escaping gas.

Smaller-radius worlds (e.g., KIC 12557548b, KOI-2700b, K2-22b) are more likely (relative to other worlds with the same temperature) to be atmosphere-dominated in our model. This is because the same gradient in atmospheric pressure with angular separation from the substellar point (∂P/∂θ) corresponds (for smaller-radius worlds) to a larger mass flux $\tfrac{\partial M}{\partial x}\propto {\rm{\nabla }}(P/{rg})\propto {r}^{-3}$ (assuming r ∝ M^1/4; Seager et al. 2007). Larger atmospheric mass fluxes favor variegated, time-variable surfaces.