Exoplanet secondary atmosphere loss and revival

Significance

Earth and Venus have significant atmospheres, but Mercury does not. Thousands of exoplanets are known, but we know almost nothing about rocky exoplanet atmospheres. Many rocky exoplanets were formed by a sub-Neptune-to-super-Earth conversion process during which planets lose most of their H₂-rich (primary) atmospheres and are reduced in volume by a factor of >2. Does such a gas-rich adolescence increase or decrease the likelihood that super-Earths will subsequently exhibit a H₂-poor (secondary) atmosphere? We show that secondary atmospheres exsolved from the magma ocean are unlikely to be retained by super-Earths, but it is possible for volcanic outgassing to revive super-Earth atmospheres. For M-dwarf planetary systems, super-Earths that have atmospheres close to the star likely were formed with abundant volatiles.

Abstract

The next step on the path toward another Earth is to find atmospheres similar to those of Earth and Venus—high–molecular-weight (secondary) atmospheres—on rocky exoplanets. Many rocky exoplanets are born with thick (>10 kbar) H₂-dominated atmospheres but subsequently lose their H₂; this process has no known Solar System analog. We study the consequences of early loss of a thick H₂ atmosphere for subsequent occurrence of a high–molecular-weight atmosphere using a simple model of atmosphere evolution (including atmosphere loss to space, magma ocean crystallization, and volcanic outgassing). We also calculate atmosphere survival for rocky worlds that start with no H₂. Our results imply that most rocky exoplanets orbiting closer to their star than the habitable zone that were formed with thick H₂-dominated atmospheres lack high–molecular-weight atmospheres today. During early magma ocean crystallization, high–molecular-weight species usually do not form long-lived high–molecular-weight atmospheres; instead, they are lost to space alongside H₂. This early volatile depletion also makes it more difficult for later volcanic outgassing to revive the atmosphere. However, atmospheres should persist on worlds that start with abundant volatiles (for example, water worlds). Our results imply that in order to find high–molecular-weight atmospheres on warm exoplanets orbiting M-stars, we should target worlds that formed H₂-poor, that have anomalously large radii, or that orbit less active stars.

The Solar System has three planets—Earth, Venus, and Mars—that have atmospheres derived from H₂-free solids (secondary atmospheres), and four giant planets whose atmospheres are derived from protoplanetary nebula gas (H₂-dominated primary atmospheres) (1, 2). However, this clean separation in process and outcome is apparently unrepresentative of the known exoplanets.

The two most common types of known exoplanet, the rocky super-Earth–sized planets (planet radius R_pl < 1.6 R_⊕, where “⊕” is the Earth symbol, planet density ρ_pl > 4 g/cc; “super-Earths”) and the gas-shrouded sub-Neptunes (R_pl = 2 to 3 R_⊕), are divided by a valley in (planet radius)–(orbital period) space in which planets are less common (Fig. 1) (e.g., ref. 3). The radius valley can be understood if, and only if, a substantial fraction of planets that are born with thick (>10 kbar) H₂-dominated primary atmospheres lose those atmospheres and shrink in radius to become rocky super-Earths (4). This radius-shrinking process, which carves out the radius valley, may be the way that most super-Earths form. There is no evidence that Earth and Venus underwent this process, and primary atmospheres are not thought to have contributed to the origin of major volatile elements in Earth (5). Both Venus and Earth have secondary volatile envelopes (composed of solid-derived volatiles, including H₂O and CO₂) that are much less massive than the atmospheres of sub-Neptunes. Large surface reservoirs of H₂O, C species, and N species are essential to life on Earth and to Earth’s habitable climate (6, 7).

Fig. 1.

The exoplanet abundance histogram (gray band, for orbital periods <100 d, corrected for detection biases; from ref. 3). Two classes of small exoplanet are seen: volatile-rich sub-Neptunes and rocky super-Earth–sized exoplanets. Sub-Neptune-to-super-Earth conversion is implied by the data and may be the way that most super-Earths form.

Does forming with a thick primary atmosphere (sub-Neptune) but ending up as a rocky super-Earth favor the rocky planet ending up with a secondary atmosphere? Do primary atmospheres, in dying, shield high–molecular-weight species from loss during the early era of atmosphere-stripping impacts and intense stellar activity, allowing those constituents to later form a secondary atmosphere? Or does the light (H₂-dominated) and transient primary atmosphere drag away the higher–molecular-weight species? Getting physical insight into the transition from primary to secondary atmospheres is particularly important for rocky exoplanets that are too hot for life. Hot rocky exoplanets are the highest signal/noise rocky targets for upcoming missions such as James Webb Space Telescope (JWST) (8) and so will be the most useful for checking our understanding of this atmospheric transition process.

Secondary atmospheres are central to the exoplanet exploration strategy (9, 10). Previous work on the hypothesis that primary atmospheres played a role in forming secondary atmospheres includes that of Eucken in the 1940s (11), Urey (12), Cameron and coworkers (e.g., ref. 13), Sasaki (14), and Ozima and Zahnle (15). Secondary atmosphere formation on exoplanets in the absence of a primary atmosphere has been investigated by, e.g., Elkins-Tanton and Seager (16), Dorn et al. (17), and references therein.

Gas Survival during Planetary Volume Reduction.

During sub-Neptune-to-super-Earth conversion, we suppose the planet contains both nebular-derived H₂ and also high-μ species (derived from the planet-forming solid materials) that could form a secondary atmosphere—if retained. Retention of volatiles is controlled by atmospheric loss and atmosphere-interior exchange (Fig. 2). For both sub-Neptunes and super-Earths, silicates (both magma and solid rock) apparently make up most of the planet’s mass (e.g., refs. 18 and 19).

Fig. 2.

Processes (italics) and reservoirs (upright font) in our model. Atmosphere-interior exchange is central to the transition from primary to secondary atmospheres. Timescales are approximate.

Atmosphere loss is more difficult for high–molecular-weight volatiles than for H₂ because higher–molecular-weight volatiles such as H₂O are more easily shielded within the silicate interior and are also more resistant to escape.

This can be understood as follows. First, a basic control on atmosphere loss is the ratio, λ, of gravitational binding energy to thermal energy:

$$\begin{array}{l}\lambda =\left(\mu /k{T}_{ua}\right)\left(G{M}_{pl}/\left(R+z\right)\right)\\ \approx 2.3\mu \left({10}^{4}K/T_{ua}\right)\left(M_{pl}/6M_{\oplus }\right)/(\left(R+z\right)/2R_{\oplus }),\end{array}$$ [1]

where μ is molecular mass, k is Boltzmann’s constant, T_ua is upper-atmosphere temperature, G is the gravitational constant, M_pl is planet mass, R is the radius of the silicate planet, and z is atmosphere thickness. When λ ≲ 2, the upper atmosphere flows out to space at a rate potentially limited only by the energy available from upper-atmosphere absorption of light from the star (20). If the upper atmosphere absorbs 100 W/m² of light from the star, the upper limit on loss is ∼1,000 bars/My. This hydrodynamic outflow ejects the tenuous upper atmosphere at high speed, but the dense lower atmosphere remains close to hydrostatic equilibrium. When λ > 10, the hydrodynamic outflow shuts down. For primary atmospheres, the mean molecular weight μ_avg ∼ 2 Da (H₂), while for secondary atmospheres, μ_avg is ≳10× higher favoring secondary atmosphere retention. Moreover, many secondary-atmosphere constituents (e.g., CO₂) are much more effective coolants than H₂, so for secondary atmospheres T_ua is lower (e.g., refs. 21 and 22). Atmospheric thickness z is also smaller for high-μ_avg atmospheres due to their smaller scale height, raising λ. For all atmospheres, proximity to the star increases T_ua (lowering λ) and also increases the upper atmosphere outflow rate when the λ > 2 condition is satisfied. Moreover, closer to the star impacts occur at higher velocities that are more erosive (23). Thus, we expect massive worlds far from the star to retain atmospheres and low-mass worlds closer to the star to lose them (10). This expectation, while physically valid, offers little guidance as to whether or not super-Earths will have atmospheres. To go further, we need to consider the effect of evolving atmospheric composition on loss rate (Eq. 1) and also track the shielding of volatiles within silicates (magma oceans and solid rock) (Fig. 2).

The atmospheres of young sub-Neptunes are underlain by magma oceans (e.g., ref. 24 and SI Appendix, Fig. S1). Equilibrium partitioning of a volatile i between the atmosphere (where it is vulnerable to escape) and the magma ocean (where it is shielded from escape) is given by the following:

$$\begin{array}{l}{c}_i=e_i+p_i{s}_i{m}_{magma}+\mathrm{other}\hspace{0.17em}\text{reservoirs}\\ =e_i+\left[\left(e_i/A_{mai}\right)g_{mai}\left({\mu }_{avg}/{\mu }_i\right)\right]s_i{m}_{magma}+\mathrm{other}\hspace{0.17em}\text{reservoirs},\end{array}$$ [2]

where c_i is the total inventory of the volatile, e_i is the mass in the atmosphere, p_i is the partial pressure of i at the magma–atmosphere interface, A_mai (and g_mai) are the area of (and gravitational acceleration at) the magma–atmosphere interface, μ_avg is the mean μ of the atmosphere, s_i is the solubility coefficient (mass fraction pascal⁻¹) of the volatile in the magma, and m_magma is the mass of liquid in the magma ocean. Solubility in magma is higher for many secondary-atmosphere constituents (e.g., CH₄, H₂S, H₂O) than for H₂. When the atmosphere contains both H₂ and a more-soluble, high-μ constituent, the H₂ takes the brunt of atmosphere loss processes that might otherwise remove the high-μ species. Differential solubility also protects the more-soluble species from impact shock (25). Other reservoirs include volatiles stored in the crystal phase. This is an unimportant reservoir early on when the planet has a massive magma ocean, but becomes important later, because these volatiles can be released by volcanic degassing. We neglect volatiles that are stored in the iron core because they do not contribute to the observable atmosphere.

Atmospheric loss typically drives magma ocean crystallization (SI Appendix, Fig. S1) (except for planets that either orbit uncommonly close to their star, or undergo strong tidal heating, bare-rock planets do not have a global molten-rock layer). The greenhouse effect from the atmosphere keeps the magma ocean liquid for longer, which delays partitioning of the volatile into crystals. This partitioning is described by the following:

$${\mathit{X}}_{\mathit{xtl}}\mathrm{=}{\mathit{D}}_{\mathit{i}}{\mathit{X}}_{\mathit{i}},$$ [3]

where X_xtl is the concentration of i in the crystal phase, D_i is a solid-melt distribution coefficient/partition coefficient, and X_i is the concentration of i in the magma. Crystallization enriches the residual melt in volatiles (i.e., D_i << 1; e.g., ref. 26).

One last effect can aid retention of high–molecular-weight volatiles. If outflow to space is slow, then the high-μ species can sink back toward the planet (diffusive separation driven by buoyancy) (e.g., refs. 27 and 28 and SI Appendix, section 1g).

Eqs. 1–3 open two paths from a primary to a secondary atmosphere. A secondary atmosphere can be exsolved as the magma ocean crystallizes. Alternatively, the atmosphere can be revived by volcanic degassing long after the mantle has almost entirely solidified (Fig. 2). We explore these paths below.

Model of the Transition from Primary to Secondary Atmospheres.

We model the effect of the loss of a thick H₂-dominated atmosphere on the retention (or loss) of a hypothetical volatile, s. Species s has molecular mass 30 Da, which is within a factor of <1.5 of the molecular mass of all known major secondary-atmosphere constituents. In the uppermost atmosphere, species s is modeled to split into fragments of mass 15 Da for the purpose of calculating whether or not s is effectively entrained by escaping H. We assume that reactions between s and H₂ do not affect the mass inventory of either species; in effect, s is chemically inert. We assume that deviations from thermochemical equilibrium driven by photochemistry in the uppermost atmosphere are reset to thermochemical equilibrium at the high temperature and pressure of the magma–atmosphere interface. While idealized, the model includes many effects that have not previously been incorporated into a model of atmosphere loss, such as a realistic rock melting curve, differential solubility effects, etc. (SI Appendix).

Planet equilibrium temperature T_eq (in kelvin), for zero planet albedo, is given by the following:

$$T_{eq}=278{\left(F/F_{\oplus }\right)}^{1/4},$$ [4]

where F is insolation (in watts per square meter) and the Sun’s insolation at Earth’s orbit, F_⊕, is 1,361 W/m². Upper-atmosphere temperatures T_ua > T_eq are essential for a super-Earth atmosphere to flow out to space: This is possible because upper atmospheres efficiently absorb light at wavelength ≲100 nm, but do not readily reemit this light.

We adopt s_H2 = 2 × 10⁻¹² Pa⁻¹ (ref. 29 and SI Appendix, section 1e), which for initial atmospheric pressure P_{atm, init} = 50 kbar gives a total mass of H₂ (atmosphere plus dissolved-in-magma) of 7 × 10²³ kg (2% of planet mass). We neglect He, so our model primary atmosphere is slightly more soluble in magma and has a slightly lower molecular weight than in reality. We assume the crystal-melt partition coefficients are D_H2 = 0 (for simplicity) and D_s = 0.02 (SI Appendix, section 1f). Sub-Neptunes have a global shell of magma that freezes during conversion to a super-Earth (SI Appendix, Fig. S1). Planet thermal structure (below the photosphere, which is isothermal by assumption) is as follows (SI Appendix, Fig. S2). The temperature at the top of the magma ocean, T_mai, is given by the atmosphere adiabat:

$$\left(T_{mai}/T_{RCB}\right)={\left(P_{atm}/P_{RCB}\right)}^{\left(\gamma -1\right)/\gamma },$$ [5]

where T_RCB is the temperature at the radiative–convective boundary within the atmosphere (RCB), P_RCB is the pressure at the RCB, and γ is the adiabatic index. We assume T_RCB/T_eq = 1, that the temperature jump at the magma–atmosphere interface is small, and that the magma ocean is isentropic. (If T_RCB/T_eq = 1.5 in a 1000 K orbit, then the planet cooling timescale is only <1 ky). This basic model absorbs the planet cooling rate and the radiative opacity of the atmosphere into a single parameter, P_RCB (SI Appendix, section 1c); for more sophisticated models, see, e.g., ref. 30. As P_atm decreases, T_mai cools, and the magma crystallizes. Crystallization starts at great depth and the crystallization front sweeps slowly upward (e.g., ref. 26). Volatiles enriched near the crystallization front due to the low solubility of volatiles in crystals will be stirred by fast magma currents (speed up to 10 m s⁻¹; ref. 31) to the near surface, where they form bubbles that pop and add gas to the atmosphere. Stirring and bubbling can degas the mantle down to at least ∼100-GPa depths (e.g., refs. 31 and 32 and references therein). A smaller portion of s will go into the crystals. This portion is shielded within the rock and available for later volcanic outgassing. We do not include liquid volatiles (e.g., clouds) or fluid–fluid immiscibility. We emphasize results for T_eq < 1150 K, cold enough for silicates to condense (33). The model is intended for super-Earths too hot for life.

Volcanic outgassing after the magma ocean has completely crystallized is guided by the results of ref. 34 (SI Appendix, Fig. S3 and section 1h).

We approximate diffusive separation of H₂ and heavy gases as zero during sub-Neptune-to-super-Earth conversion. During conversion, escape proceeds too quickly for the s to settle out (28) (SI Appendix, section 1g). On worlds that are cool enough for life, diffusive separation is important (SI Appendix, section 1g). On worlds that are cool enough for life, diffusive separation can allow high-μ species to be retained in the atmosphere while H escapes, including in cases where s is a reducing species that is dissociated into easily escaping H plus a heavy atom (e.g., H₂O → 2H + O; ref. 35). The secondary atmospheres that may be exsolved as the magma ocean crystallizes form more quickly (36) than volcanically outgassed atmospheres. Volcanically outgassed atmospheres are produced on solid-state mantle homogenization timescales (τ > 1 Gy; e.g., refs. 37 and 38). Because volcanic degassing of terrestrial planets is so slow/inefficient (e.g., ref. 39), we consider magma-ocean exsolution separately from volcanic outgassing. Specifically, we set rerelease of s from the solid mantle to zero so long as the exsolved atmosphere is present. With these approximations, for a given insolation L, planet mass M_pl, initial dose of s, and P_atm at which that dose is applied, the output depends on how much atmosphere has been removed, but not on the speed (or process) of removal. In other words, the equations are time independent.

The Small Planet Evolution Sequence.

We first model atmospheric evolution during magma ocean crystallization (Figs. 3 and 4). We show results for 6 M_⊕ (∼1.6 R_⊕), corresponding to the largest (and therefore highest signal/noise) planets that commonly have densities consistent with loss of all H₂ (40). We use a total mass of high-μ volatile (M_s) = 3 × 10²¹ kg. This corresponds to the near-surface C inventory of Earth and Venus, scaled up to 6 M_⊕ (and it is 2× the mass of Earth’s ocean).

Fig. 3.

The small planet evolution sequence, for 6 M_⊕. The black triangles correspond to total initial inventory of high-μ species. (A) Solubility of high-μ species in magma (s_s) = 10⁻¹¹ Pa⁻¹, insolation normalized to Earth’s insolation (F/F_⊕) = 240 (planet equilibrium temperature 1095 K). (B) s_s = 10⁻¹¹ Pa⁻¹, F/F_⊕ = 3 (planet equilibrium temperature = 360 K). (C) s_s = 10⁻⁹ Pa⁻¹, F/F_⊕ = 240. (D) s_s = 10⁻⁹ Pa⁻¹, F/F_⊕ = 3.

Fig. 4.

(A) How the mass of volatiles shielded within the solid mantle depends on F and s_s. (B) How atmospheric mean molecular weight (after crystallization of the magma ocean is complete) depends on F and s_s.

We drive the model by decreasing P_atm. In Fig. 3, as P_H2 (blue dashed lines) falls, the magma ocean crystallizes, so that remaining volatiles go into the atmosphere (green), go into the solid mantle (maroon), or escape to space (black). We find that the main controls on the transition on exoplanets from primary to secondary atmospheres are F and the solubility (s_s) of the high-μ atmosphere constituent.

Fig. 3A shows a case with low s_s (10⁻¹¹ Pa⁻¹), for a planet close to its host star (F/F_⊕ = 240; T_eq = 1100 K; typical for Kepler super-Earths). The magma ocean stays fully liquid until the atmospheric mass is reduced by 90%. (Release of dissolved-in-magma H₂ by bubbling is a negative feedback on atmospheric loss.) Crystallization begins at P_atm = 2 kbar and completes at ∼100 bars. s is passively entrained (either as atoms or molecules) in the escaping H₂. We stop the run at P_atm = 1 bar, so we do not track the removal of the last bit of the exsolved atmosphere. In this limit of small s_s, only 6 × 10¹⁶ kg × (s_s/10⁻¹¹ Pa⁻¹) × (D_s/0.02) is shielded within the solid mantle. Most of the s is lost to space before crystallization begins. The outcome is a bare rock with a volatile-starved solid mantle, incapable of much volcanic outgassing.

Fig. 3B shows results for the same s_s as in Fig. 3A and a cooler orbit (F/F_⊕ = 3, T_eq = 370 K, intermediate between Mercury and Venus in our Solar System). In the cooler orbit, some crystals are present initially. With no fully liquid stage, crystallization can start to shield s within crystals as soon as atmospheric loss starts. The s available for later volcanic outgassing in the cooler orbit case is 50 times greater.

Raising s_s to 10⁻⁹ Pa⁻¹ (equivalent to 1 wt% solubility for 100 bars of partial pressure) favors secondary atmosphere occurrence (Fig. 3 C and D). More s is dissolved in the magma, and so more s partitions (during crystallization) into the rock, where it is shielded. Because s is much more soluble in the magma than H₂, very little s is initially in the atmosphere. Therefore, relatively little s is carried away to space during H₂ removal and so more s is available for exsolution during the final crystallization of the magma ocean. This can create a high-μ_avg atmosphere, which is easier to retain. Protection by differential solubility is enhanced by the hot orbit (L/L_⊕ = 240; Fig. 3C) as opposed to the cool orbit (L/L_⊕ = 3; Fig. 3D), because the magma ocean is long-lived for this case, and so the volatiles are safely dissolved for longer. As a result, in the hot orbit case, enough high-μ species remain at final magma ocean crystallization to create an atmosphere with high-μ_avg (the green line crosses the blue line in Fig. 3C). Such a high-μ_avg atmosphere is easier to retain. The small amount of H₂ that remains in the atmosphere at this point can be lost by diffusion-limited escape.

Together, F and s_s have strong effects on the chance of secondary atmosphere occurrence. Volatiles that are shielded within rock are available for late volcanic outgassing. For high s_s, the mass of shielded volatiles tends toward the product of the initial inventory of s and the solid–liquid partition coefficient D_s (Fig. 4A). This is because almost all of the s is in the magma until the magma ocean has almost completely crystallized. The effect of F/F_⊕ on the mass of shielded volatiles is relatively modest for s_s > 3 × 10⁻¹¹ Pa⁻¹. For F/F_⊕ ≤ 25 and low s_s < (μ_s/μ_H2)s_H2, the amount of the high–molecular-weight species that is shielded within rock is proportional to s_s. For low s_s, the mass of shielded volatiles decreases rapidly for hot orbits. In hot orbits, the H wind can carry away s for a long time before the magma ocean cools enough for solidification (and shielding).

When μ_avg is high, loss to space is less likely (Eq. 1 and surrounding discussion). Fig. 4B shows how atmospheric μ_avg (after crystallization of the magma ocean is complete) depends on F and s_s. For s_s > 10⁻¹¹ Pa⁻¹, a greater fraction of the s_s is stored in the magma than is H₂. As a result, the atmosphere becomes more s-rich as the magma ocean crystallizes. The enrichment is especially strong for hot orbits, because for hot orbits there is more s available to be exsolved: Less s has escaped to space before crystallization completes. For s_s < 10⁻¹¹ Pa⁻¹, most of the s is stored in the atmosphere, and so the atmospheric s mixing ratio is near-constant during atmosphere loss. [At F/F_⊕ < 3, diffusive separation favors high-μ_avg atmospheres (SI Appendix, section 1g).]

Cooler orbits favor volcanic outgassing but hotter orbits permit exsolved high–molecular-weight atmospheres. To determine which effect is more important for the chance of seeing a secondary atmosphere on a super-Earth, we turn to a time-dependent model.

Where Are Planets Today on the Small Planet Evolution Sequence?

We map planet evolution onto time and host-star mass (Fig. 5). The atmosphere loss that converts sub-Neptunes into super-Earths could be driven by photoevaporation, impact erosion, or accretion energy (e.g., refs. 19, 23, and 41). Here, we consider photoevaporation due to X-ray and extreme UV flux (XUV). F_XUV plateaus at ∼10⁻³ × total F for planets around young stars, switching to a power-law decay at <0.1 Gy for solar-mass stars (F_XUV = 3 × 10⁻⁶ F at Earth today) (SI Appendix, section 1a). The plateau of high F_XUV/F is longer at red dwarf (M) stars (≥0.3 Gy long for ≤0.5 M_☉). Therefore, we expect that (for a given T_eq) planets orbiting M-stars will have lost more atmosphere (e.g., ref. 42).

Fig. 5.

Time-dependent results for planets orbiting a solar-mass star. (SI Appendix, Fig. S6 shows the results for a 0.3 M_☉ star.) (A) Atmospheric pressure vs. time for s_s = 10⁻⁹ Pa⁻¹. (From Top to Bottom) F/F_⊕ = {49, 283, 347, 422, 720}, corresponding to planet equilibrium temperature (T_eq) = {735, 1140, 1200, 1275, 1440} K. (B) As A, but for s_s = 10⁻¹¹ Pa⁻¹.

F_XUV drives atmospheric loss (rate dM_atm/dt) (SI Appendix, section 1b). Nebula-composition atmospheres do not cool efficiently, leading to high T_ua that favors hydrodynamic escape (43). For nebular-composition atmospheres, we use the following:

$$d{M}_{atm}/dt\mathrm{=}\epsilon R_{pl}{\left(R\mathrm{+}z\left(t\right)\right)}^2{F}_{XUV}/\left(G\hspace{0.17em}{M}_{pl}\right),$$ [6]

where ε is an efficiency factor. For high-μ atmospheres, we adopt the loss fluxes of a CO₂ atmosphere (44) as an example of a strong coolant. The model of ref. 44 (and ref. 45) includes dissociation of CO₂ under high XUV levels, and the escaping material is atomic C and O. The model of ref. 44 predicts low ε for CO₂ atmospheres, and negligible hydrodynamic escape for F_XUV < 0.6 W⋅m⁻² (=150× the value on Earth today) (SI Appendix, Fig. S4). At intermediate compositions, we interpolate using a logistic function (SI Appendix, section 1b).

Fig. 5 shows atmosphere thickness vs. time for a solar-mass star and P_{atm, init} = 50 kbar. High-μ_avg atmospheres can only persist at a narrow range of F. [A similar pattern is seen for low-mass stars (SI Appendix, Fig. S6).] Worlds far from the star receive a low XUV flux and stay as sub-Neptunes. Worlds in hotter orbits lose their atmospheres completely. They may undergo a rapid increase in μ_avg (blue to yellow in Fig. 5A), but the resulting high-μ_avg atmosphere is almost always short-lived. Why is this? According to the pure-CO₂ model of ref. 44, dM_atm/dt for high–molecular-weight atmospheres on super-Earths has a threshold at ∼150 × F_XUV/F_XUV,⊕, below which loss is much slower. However, in most cases the primary atmosphere is lost when F_XUV is still very high, so any exsolved secondary atmosphere is swiftly lost. For low s_s, the atmosphere has a composition that is always H₂-dominated (by number), so exsolved high-μ species are lost in the H wind (Fig. 5B).

We conclude that exsolved atmospheres are rare. This conclusion is robust because we use parameters that are favorable for an exsolved atmosphere. For example, we use a high solubility-in-magma for the high-μ species, but our high-μ escape-to-space parameterization is for a species (CO₂) whose solubility-in-magma is low.

Revival of Secondary Atmospheres by Volcanic Outgassing.

Volcanic outgassing can regenerate the atmosphere of a planet. We do not know whether or not this process actually occurs on rocky exoplanets. We use a basic model to explore this process. We use a time-dependent rate-of-volcanism model for super-Earths (ref. 34 and SI Appendix, section 1h) that is tuned to the rate of CO₂ release at Earth’s midocean ridges (12 ± 2 bars/Gy; ref. 46). The rate of outgassing is adjusted downward to account for loss of volatiles during sub-Neptune-to-super-Earth conversion, assuming 50% of worlds have s_s = 10⁻⁹ Pa⁻¹, and 50% of worlds have s_s = 10⁻¹¹ Pa⁻¹. We neglect atmosphere reuptake to form minerals (e.g., ref. 47), because our focus is on worlds that are too hot for aqueous weathering. This omission is conservative relative to our conclusion that volcanically outgassed atmospheres on hot rocky exoplanets are uncommon. Fig. 6 (gray curve) outlines the region within which, in our model, volcanic outgassing will build up a secondary atmosphere. The line of revival sweeps toward the star over time, because the rate of volcanic degassing falls off more slowly with time than does the star’s XUV flux. Volcanic revival of the atmosphere is difficult for planets around M = 0.3 M_☉ stars, but easier for rocky planets around solar-mass stars. The results are sensitive to changes in s_s, and to the choice of XUV flux models (SI Appendix, Figs. S9–S11 and S17). SI Appendix, Fig. S17 also shows results for volcanism at a constant Earth-scaled outgassing rate. Fig. 6 also shows the atmosphere presence/absence line for rocky worlds that start with no H₂, and with all high–molecular-weight volatiles in the atmosphere (red curves). Such “intrinsically rocky” worlds retain residual secondary atmospheres over a wider range of conditions than do worlds that start as sub-Neptunes. The line of atmosphere loss for these residual atmospheres sweeps further away from the star with time.

Fig. 6.

Secondary atmosphere presence/absence model output for 6 M_⊕ (higher planet mass favors atmosphere retention). The red lines and gray lines show atmosphere presence/absence contours for two different scenarios. The red lines show atmosphere retention thresholds after 3.0 Gy for the case where all volatiles are in the atmosphere initially and there is no primary atmosphere; the 16th and 84th percentiles are shown, for varying XUV flux (by ±0.4 dex, 1 σ; ref. 48) relative to the baseline model following the results of refs. 49 and 50 (SI Appendix, section 1a). The red lines move away from the star over time (red arrows). The gray lines show the 16th and 84th percentiles for exhibiting an atmosphere after 3.0 Gy for the case where volcanic outgassing rebuilds the atmosphere from a bare-rock state. The solid gray lines are for stagnant-lid tectonics, and the dashed gray lines are for plate tectonics. The lines of atmospheric revival sweep toward the star over time (gray arrows) because the rate of volcanic degassing falls off more slowly with time than does the star’s XUV flux. In each case, the atmosphere/no-atmosphere threshold is 1 bar. The black symbols show known planets that may be tested for atmospheres using JWST (51). For any individual planet, star-specific XUV-flux estimates, star age, and the planet’s mass, should be combined to make a more accurate estimate than is possible using this overview diagram. SI Appendix, Figs. S8, S16, and S17 show further details.

Discussion

Our model results are sensitive to the rate of XUV-driven mass loss. The XUV flux of young solar-mass stars varies between stars of similar age by a factor of 3 to 10 (e.g., ref. 52). Stars with low XUV flux are more likely to host planets with atmospheres (SI Appendix, Fig. S11). Escape of N₂ or H₂O is likely faster than the ref. 44 loss rate estimate (which is for pure-CO₂ atmospheres) that is used in our model (based on the results of refs. 22 and 53). Improved knowledge of escape rate will require more escape rate data and XUV flux data (e.g., refs. 54 and 55). Currently, XUV flux data as a function of star mass and star age are limited, and upcoming space missions such as SPARCS will gather more data (55, 56).

Our model considers only thermal loss. However, solar wind erosion can remove atmospheres that are already thin (57). A possible example is Mars over the last ∼4 Ga. Planetary dynamos can under some circumstances suppress solar-wind erosion (e.g., ref. 58). Mars had a dynamo prior to 4 Gya, and Earth (but not Venus) has one today.

Solubility of gas in magma varies between species. Carbon-bearing volatile species are very insoluble in magma. H₂O’s solubility in basaltic magma at H₂O partial pressure 0.03 GPa is ∼2 wt%; linearizing, this gives solubility s_s = 7 × 10⁻¹⁰ Pa⁻¹ (e.g., ref. 59). HCl is even more soluble in magma than H₂O (60), so a Cl-rich initial composition would have a greater chance of forming an exsolved high-μ atmosphere, although we still predict it would be short-lived. The effects on atmosphere prevalence of 100-fold reduction in solubility are shown in SI Appendix, Figs. S9–S11. Better constraints on solubility at T > 2000 K are desirable (e.g., ref. 61).

Volcanism on super-Earths should wane over gigayears, according to models. We need more data to test these models. In models, the rate of decrease of volcanism depends on whether or not the planet has plate tectonics, on planet mass, and on mantle composition (e.g., refs. 17 and 62–64 and SI Appendix). Two effects, of uncertain relative importance, are ignored in our volcanism model. The first is a buffering effect: If the mantle is volatile-rich, then magma is produced more easily (65), but if the mantle is volatile-poor, then the melt rate is reduced (reducing the rate of volatile loss). The second is a potential fine-tuning issue: If volcanic degassing is very rapid, then volatiles will be released from the protective custody of the mantle before atmosphere loss process have lost their bite.

Overall, our choices of M_pl, D_s, and s_s tend to favor the existence of volcanically outgassed atmospheres. Even with these choices, atmospheres are usually not stable for planets at T_eq > 500 K around M-stars. So, our conclusions are broadly unfavorable for atmospheres on rocky exoplanets at T_eq > 500 K around M-stars. However, this conclusion is moderated by the possibility (discussed next) that worlds with abundant H₂O exist close to the star.

Observational Tests.

Atmospheres on rocky exoplanets can now be detected (e.g., refs. 66 and 67). Theory predicts that retaining an atmosphere should be harder on planets orbiting low-mass stars, and the present study extends that prediction to super-Earths that form as sub-Neptunes. A test of this theory would have major implications for habitable zone planets. If this prediction fails, that would suggest that M-star rocky exoplanets formed more volatile-rich than rocky exoplanets orbiting Sun-like stars (68).

It is possible that some planets form with more volatiles than can be removed by loss processes (69). Some models predict formation of hot Super-Earths with 1 to 30 wt% H₂O, either by accretion of volatile-rich objects (for example, extrasolar analogs to CI/CM chondrites), or by planet migration (e.g., refs. 69 and 70). Such volatile-rich worlds are hard to distinguish from bare-rock planets using current data. XUV-driven loss can remove at most a few wt% of an M = 6 M_⊕ planet’s mass for T_eq < 1000 K. Our model implies that a planet in the “no atmosphere” zone of Fig. 6 with a JWST detection of H₂O-dominated atmosphere is more volatile-rich than Venus and Earth. Possible volatile-rich worlds include planets that have radii ∼0.2 R_⊕ greater than expected for Earth composition (SI Appendix, Fig. S12) (71).

Fig. 6 enables the following testable predictions. Since atmospheres close to the star can only persist if the initial H₂O inventory is high, N₂/NH₃ should be diluted to very low mixing ratio for such atmospheres. If a super-Earth–sized planet has an atmosphere, then planets at greater semimajor axis in the same system should also have an atmosphere. Starting out as a sub-Neptune is unfavorable for atmosphere persistence, so systems where the planets formed intrinsically rocky should have statistically more atmospheres. Multiplanet systems enable strong tests because uncertainty in time-integrated stellar flux cancels out.

Conclusions

A large fraction of rocky exoplanets on close-in orbits (closer to their star than the habitable zone) were born with thick (>10 kbar) H₂-dominated (primary) atmospheres but have since lost their H₂. In our model, these T_eq > 400 K exoplanets almost never transition smoothly to worlds with high–molecular-weight atmospheres (Fig. 7). Instead, the high–molecular-weight species are usually carried away to space by the H wind. Volcanic outgassing is an alternative source for a high–molecular-weight atmosphere. Revival of a bare-rock planet by volcanic outgassing gets easier with time, because atmosphere loss slows down rapidly with time, but atmosphere supply by volcanism decays slowly. However, volcanic outgassing is also enfeebled by early loss of biocritical volatiles via the H wind. Overall, for a given initial dose of high–molecular-weight species, atmospheres are less likely on hot rocky exoplanets that were born with thick H₂-dominated atmospheres. Many uncertainties remain, the most important of which is the initial planet volatile content. Within our model, for planets that orbit solar-mass stars, super-Earth atmospheres are possible at insolations much higher than for planet Mercury. For planets that orbit ∼0.3 M_☉ stars, secondary atmospheres at much higher insolation than planet Mercury in our solar system are unlikely unless the planet formed H₂-poor, or includes a major (∼1 wt%) contribution of solids from beyond the water ice line (water world).

Fig. 7.

Graphical summary. The left column corresponds to atmosphere revival (gray lines in Fig. 6), and the right column corresponds to residual atmospheres (red lines in Fig. 6). For worlds in T_eq > 400 K orbits that start as sub-Neptunes, formation of a high-μ atmosphere is unlikely, unless the planet starts with abundant high-μ volatiles.

Data Availability.

All of the code for this paper, together with instructions to reproduce each of the figures and supplementary figures, can be obtained via the Open Science Framework at https://osf.io/t9h68 or by emailing the lead author.