Exoplanet secondary atmosphere loss and revivalEdwin S. Kitea,1 and Megan N. BarnettaaDepartment of the Geophysical Sciences, University of Chicago, Chicago, IL 60637

Edited by Mark Thiemens, University of California San Diego, La Jolla, CA, and approved June 19, 2020 (received for review April 3, 2020)

The next step on the path toward another Earth is to find atmo-spheres similar to those of Earth and Venus—high–molecular-weight (secondary) atmospheres—on rocky exoplanets. Many rockyexoplanets are born with thick (>10 kbar) H2-dominated atmo-spheres but subsequently lose their H2; this process has no knownSolar System analog. We study the consequences of early loss of athick 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, andvolcanic outgassing). We also calculate atmosphere survival for rockyworlds that start with no H2. Our results imply that most rocky exo-planets orbiting closer to their star than the habitable zone that wereformed with thick H2-dominated atmospheres lack high–molecular-weight atmospheres today. During early magma ocean crystalliza-tion, high–molecular-weight species usually do not form long-livedhigh–molecular-weight atmospheres; instead, they are lost to spacealongside H2. This early volatile depletion also makes it more difficultfor later volcanic outgassing to revive the atmosphere. However, at-mospheres should persist on worlds that start with abundant vola-tiles (for example, water worlds). Our results imply that in order tofind high–molecular-weight atmospheres on warm exoplanets orbit-ing M-stars, we should target worlds that formed H2-poor, that haveanomalously large radii, or that orbit less active stars.

exoplanets | atmospheric evolution | planetary science

The Solar System has three planets—Earth, Venus, andMars—that have atmospheres derived from H2-free solids

(secondary atmospheres), and four giant planets whose atmo-spheres are derived from protoplanetary nebula gas (H2-domi-nated primary atmospheres) (1, 2). However, this clean separationin process and outcome is apparently unrepresentative of theknown exoplanets.The two most common types of known exoplanet, the rocky

super-Earth–sized planets (planet radius Rpl < 1.6 R⊕, where“⊕” is the Earth symbol, planet density ρpl > 4 g/cc; “super-Earths”) and the gas-shrouded sub-Neptunes (Rpl = 2 to 3 R⊕),are divided by a valley in (planet radius)–(orbital period) spacein which planets are less common (Fig. 1) (e.g., ref. 3). The ra-dius valley can be understood if, and only if, a substantial fractionof planets that are born with thick (>10 kbar) H2-dominatedprimary atmospheres lose those atmospheres and shrink in ra-dius to become rocky super-Earths (4). This radius-shrinkingprocess, which carves out the radius valley, may be the waythat most super-Earths form. There is no evidence that Earthand Venus underwent this process, and primary atmospheres arenot thought to have contributed to the origin of major volatileelements in Earth (5). Both Venus and Earth have secondaryvolatile envelopes (composed of solid-derived volatiles, includingH2O and CO2) that are much less massive than the atmospheresof sub-Neptunes. Large surface reservoirs of H2O, C species, andN species are essential to life on Earth and to Earth’s habitableclimate (6, 7).Does forming with a thick primary atmosphere (sub-Neptune)

but ending up as a rocky super-Earth favor the rocky planetending up with a secondary atmosphere? Do primary atmo-spheres, in dying, shield high–molecular-weight species from lossduring the early era of atmosphere-stripping impacts and intensestellar activity, allowing those constituents to later form a secondary

atmosphere? Or does the light (H2-dominated) and transient pri-mary atmosphere drag away the higher–molecular-weight species?Getting physical insight into the transition from primary to sec-ondary atmospheres is particularly important for rocky exoplanetsthat are too hot for life. Hot rocky exoplanets are the highest signal/noise rocky targets for upcoming missions such as James WebbSpace Telescope (JWST) (8) and so will be the most useful forchecking our understanding of this atmospheric transition process.Secondary atmospheres are central to the exoplanet explora-

tion strategy (9, 10). Previous work on the hypothesis that pri-mary atmospheres played a role in forming secondary atmospheresincludes that of Eucken in the 1940s (11), Urey (12), Cameron andcoworkers (e.g., ref. 13), Sasaki (14), and Ozima and Zahnle (15).Secondary atmosphere formation on exoplanets in the absence of aprimary atmosphere has been investigated by, e.g., Elkins-Tantonand 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 con-tains both nebular-derived H2 and also high-μ species (derivedfrom the planet-forming solid materials) that could form a sec-ondary atmosphere—if retained. Retention of volatiles is con-trolled by atmospheric loss and atmosphere-interior exchange(Fig. 2). For both sub-Neptunes and super-Earths, silicates (bothmagma and solid rock) apparently make up most of the planet’smass (e.g., refs. 18 and 19).Atmosphere loss is more difficult for high–molecular-weight

volatiles than for H2 because higher–molecular-weight volatiles

Significance

Earth and Venus have significant atmospheres, but Mercurydoes not. Thousands of exoplanets are known, but we knowalmost nothing about rocky exoplanet atmospheres. Many rockyexoplanets were formed by a sub-Neptune-to-super-Earth con-version process during which planets lose most of their H2-rich(primary) atmospheres and are reduced in volume by a factorof >2. Does such a gas-rich adolescence increase or decrease thelikelihood that super-Earths will subsequently exhibit a H2-poor(secondary) atmosphere? We show that secondary atmospheresexsolved from the magma ocean are unlikely to be retained bysuper-Earths, but it is possible for volcanic outgassing to revivesuper-Earth atmospheres. ForM-dwarf planetary systems, super-Earths that have atmospheres close to the star likely wereformed with abundant volatiles.

Author contributions: E.S.K. designed research; E.S.K. and M.N.B. performed research; andE.S.K. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.

Data deposition: All of the code for this paper, together with instructions to reproduceeach of the figures and supplementary figures, can be obtained via the Open ScienceFramework at https://osf.io/t9h68.1To whom correspondence may be addressed. Email: kite@uchicago.edu.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2006177117/-/DCSupplemental.

First published July 21, 2020.

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such as H2O are more easily shielded within the silicate interiorand 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 tothermal energy:

λ = (μ=kTua)(GMpl/(R + z))≈ 2.3μ(104K/Tua)(Mpl/6M⊕ )/((R + z)/2R⊕ ), [1]

where μ is molecular mass, k is Boltzmann’s constant, Tua isupper-atmosphere temperature, G is the gravitational constant,Mpl is planet mass, R is the radius of the silicate planet, and z isatmosphere thickness. When λ K 2, the upper atmosphere flowsout to space at a rate potentially limited only by the energyavailable from upper-atmosphere absorption of light from thestar (20). If the upper atmosphere absorbs 100 W/m2 of lightfrom the star, the upper limit on loss is ∼1,000 bars/My. Thishydrodynamic outflow ejects the tenuous upper atmosphere athigh speed, but the dense lower atmosphere remains close tohydrostatic equilibrium. When λ > 10, the hydrodynamic outflow

shuts down. For primary atmospheres, the mean molecularweight μavg ∼ 2 Da (H2), while for secondary atmospheres, μavgis J10× higher favoring secondary atmosphere retention. More-over, many secondary-atmosphere constituents (e.g., CO2) aremuch more effective coolants than H2, so for secondary atmo-spheres Tua is lower (e.g., refs. 21 and 22). Atmospheric thicknessz is also smaller for high-μavg atmospheres due to their smaller scaleheight, raising λ. For all atmospheres, proximity to the star increasesTua (lowering λ) and also increases the upper atmosphere outflowrate when the λ > 2 condition is satisfied. Moreover, closer to thestar impacts occur at higher velocities that are more erosive (23).Thus, we expect massive worlds far from the star to retain atmo-spheres and low-mass worlds closer to the star to lose them (10).This expectation, while physically valid, offers little guidance as towhether or not super-Earths will have atmospheres. To go further,we need to consider the effect of evolving atmospheric compositionon loss rate (Eq. 1) and also track the shielding of volatiles withinsilicates (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

Fig. 1. The exoplanet abundance histogram (gray band, for orbital periods <100 d, corrected for detection biases; from ref. 3). Two classes of small exoplanetare 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 theway that most super-Earths form.

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Fig. 2. Processes (italics) and reservoirs (upright font) in our model. Atmosphere-interior exchange is central to the transition from primary to secondaryatmospheres. Timescales are approximate.

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partitioning of a volatile i between the atmosphere (where it isvulnerable to escape) and the magma ocean (where it is shieldedfrom escape) is given by the following:

ci = ei + pisimmagma + other  reservoirs= ei + [(ei=Amai)gmai(μavg/μi)]simmagma + other  reservoirs, [2]

where ci is the total inventory of the volatile, ei is the mass in theatmosphere, pi is the partial pressure of i at the magma–atmosphere interface, Amai (and gmai) are the area of (and grav-itational acceleration at) the magma–atmosphere interface, μavgis the mean μ of the atmosphere, si is the solubility coefficient(mass fraction pascal−1) of the volatile in the magma, andmmagma is the mass of liquid in the magma ocean. Solubility inmagma is higher for many secondary-atmosphere constituents(e.g., CH4, H2S, H2O) than for H2. When the atmosphere con-tains both H2 and a more-soluble, high-μ constituent, the H2takes the brunt of atmosphere loss processes that might other-wise remove the high-μ species. Differential solubility also pro-tects the more-soluble species from impact shock (25). Otherreservoirs include volatiles stored in the crystal phase. This isan unimportant reservoir early on when the planet has a massivemagma ocean, but becomes important later, because these vola-tiles can be released by volcanic degassing. We neglect volatilesthat are stored in the iron core because they do not contribute tothe observable atmosphere.Atmospheric loss typically drives magma ocean crystallization

(SI Appendix, Fig. S1) (except for planets that either orbit un-commonly close to their star, or undergo strong tidal heating,bare-rock planets do not have a global molten-rock layer). Thegreenhouse effect from the atmosphere keeps the magma oceanliquid for longer, which delays partitioning of the volatile intocrystals. This partitioning is described by the following:

Xxtl = DiXi, [3]

where Xxtl is the concentration of i in the crystal phase, Di is asolid-melt distribution coefficient/partition coefficient, and Xi isthe concentration of i in the magma. Crystallization enriches theresidual melt in volatiles (i.e., Di << 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 speciescan sink back toward the planet (diffusive separation driven bybuoyancy) (e.g., refs. 27 and 28 and SI Appendix, section 1g).Eqs. 1–3 open two paths from a primary to a secondary at-

mosphere. A secondary atmosphere can be exsolved as themagma ocean crystallizes. Alternatively, the atmosphere can berevived by volcanic degassing long after the mantle has almostentirely solidified (Fig. 2). We explore these paths below.

Model of the Transition from Primary to Secondary Atmospheres.Wemodel the effect of the loss of a thick H2-dominated atmosphereon the retention (or loss) of a hypothetical volatile, s. Species s hasmolecular mass 30 Da, which is within a factor of <1.5 of themolecular mass of all known major secondary-atmosphere con-stituents. In the uppermost atmosphere, species s is modeled tosplit into fragments of mass 15 Da for the purpose of calculatingwhether or not s is effectively entrained by escaping H. We assumethat reactions between s and H2 do not affect the mass inventoryof either species; in effect, s is chemically inert. We assume thatdeviations from thermochemical equilibrium driven by photo-chemistry in the uppermost atmosphere are reset to thermo-chemical equilibrium at the high temperature and pressure of themagma–atmosphere interface. While idealized, the model in-cludes many effects that have not previously been incorporatedinto a model of atmosphere loss, such as a realistic rock meltingcurve, differential solubility effects, etc. (SI Appendix).

Planet equilibrium temperature Teq (in kelvin), for zero planetalbedo, is given by the following:

Teq = 278(F=F⊕ )1=4, [4]

where F is insolation (in watts per square meter) and the Sun’sinsolation at Earth’s orbit, F⊕, is 1,361 W/m2. Upper-atmospheretemperatures Tua > Teq are essential for a super-Earth atmo-sphere to flow out to space: This is possible because upper at-mospheres efficiently absorb light at wavelength K100 nm, butdo not readily reemit this light.We adopt sH2 = 2 × 10−12 Pa−1 (ref. 29 and SI Appendix,

section 1e), which for initial atmospheric pressure Patm, init = 50kbar gives a total mass of H2 (atmosphere plus dissolved-in-magma) of 7 × 1023 kg (2% of planet mass). We neglectHe, so our model primary atmosphere is slightly more soluble inmagma and has a slightly lower molecular weight than in reality.We assume the crystal-melt partition coefficients are DH2 =0 (for simplicity) and Ds = 0.02 (SI Appendix, section 1f). Sub-Neptunes have a global shell of magma that freezes duringconversion to a super-Earth (SI Appendix, Fig. S1). Planetthermal structure (below the photosphere, which is isothermal byassumption) is as follows (SI Appendix, Fig. S2). The tempera-ture at the top of the magma ocean, Tmai, is given by the at-mosphere adiabat:

(Tmai=TRCB) = (Patm=PRCB)(γ−1)=γ , [5]

where TRCB is the temperature at the radiative–convectiveboundary within the atmosphere (RCB), PRCB is the pressureat the RCB, and γ is the adiabatic index. We assume TRCB/Teq = 1, that the temperature jump at the magma–atmosphereinterface is small, and that the magma ocean is isentropic. (IfTRCB/Teq = 1.5 in a 1000 K orbit, then the planet cooling time-scale is only <1 ky). This basic model absorbs the planet coolingrate and the radiative opacity of the atmosphere into a singleparameter, PRCB (SI Appendix, section 1c); for more sophisti-cated models, see, e.g., ref. 30. As Patm decreases, Tmai cools,and the magma crystallizes. Crystallization starts at great depthand the crystallization front sweeps slowly upward (e.g., ref. 26).Volatiles enriched near the crystallization front due to the lowsolubility of volatiles in crystals will be stirred by fast magmacurrents (speed up to 10 m s−1; ref. 31) to the near surface, wherethey form bubbles that pop and add gas to the atmosphere. Stir-ring and bubbling can degas the mantle down to at least ∼100-GPa depths (e.g., refs. 31 and 32 and references therein). Asmaller portion of s will go into the crystals. This portion isshielded within the rock and available for later volcanic outgas-sing. We do not include liquid volatiles (e.g., clouds) or fluid–fluid immiscibility. We emphasize results for Teq < 1150 K, coldenough for silicates to condense (33). The model is intended forsuper-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. Duringconversion, escape proceeds too quickly for the s to settle out(28) (SI Appendix, section 1g). On worlds that are cool enoughfor life, diffusive separation is important (SI Appendix, section1g). On worlds that are cool enough for life, diffusive separationcan allow high-μ species to be retained in the atmosphere whileH escapes, including in cases where s is a reducing species that isdissociated into easily escaping H plus a heavy atom (e.g., H2O→ 2H + O; ref. 35). The secondary atmospheres that may beexsolved as the magma ocean crystallizes form more quickly (36)than volcanically outgassed atmospheres. Volcanically outgassed

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atmospheres are produced on solid-state mantle homogenizationtimescales (τ > 1 Gy; e.g., refs. 37 and 38). Because volcanicdegassing of terrestrial planets is so slow/inefficient (e.g., ref. 39),we consider magma-ocean exsolution separately from volcanicoutgassing. Specifically, we set rerelease of s from the solidmantle to zero so long as the exsolved atmosphere is present.With these approximations, for a given insolation L, planet massMpl, initial dose of s, and Patm at which that dose is applied, theoutput depends on how much atmosphere has been removed, butnot on the speed (or process) of removal. In other words, theequations are time independent.

The Small Planet Evolution Sequence. We first model atmosphericevolution during magma ocean crystallization (Figs. 3 and 4). Weshow results for 6 M⊕ (∼1.6 R⊕), corresponding to the largest(and therefore highest signal/noise) planets that commonly havedensities consistent with loss of all H2 (40). We use a total massof high-μ volatile (Ms) = 3 × 1021 kg. This corresponds to thenear-surface C inventory of Earth and Venus, scaled up to 6 M⊕(and it is 2× the mass of Earth’s ocean).We drive the model by decreasing Patm. In Fig. 3, as PH2 (blue

dashed lines) falls, the magma ocean crystallizes, so thatremaining volatiles go into the atmosphere (green), go into thesolid mantle (maroon), or escape to space (black). We find thatthe main controls on the transition on exoplanets from primaryto secondary atmospheres are F and the solubility (ss) of thehigh-μ atmosphere constituent.

Fig. 3A shows a case with low ss (10−11 Pa−1), for a planet closeto its host star (F/F⊕ = 240; Teq = 1100 K; typical for Keplersuper-Earths). The magma ocean stays fully liquid until theatmospheric mass is reduced by 90%. (Release of dissolved-in-magma H2 by bubbling is a negative feedback on atmo-spheric loss.) Crystallization begins at Patm = 2 kbar and completesat ∼100 bars. s is passively entrained (either as atoms or mole-cules) in the escaping H2. We stop the run at Patm = 1 bar, so wedo not track the removal of the last bit of the exsolved atmo-sphere. In this limit of small ss, only 6 × 1016 kg × (ss/10−11 Pa−1) ×(Ds/0.02) is shielded within the solid mantle. Most of the s is lost tospace before crystallization begins. The outcome is a bare rockwith a volatile-starved solid mantle, incapable of much volcanicoutgassing.Fig. 3B shows results for the same ss as in Fig. 3A and a cooler

orbit (F/F⊕ = 3, Teq = 370 K, intermediate between Mercury andVenus in our Solar System). In the cooler orbit, some crystals arepresent initially. With no fully liquid stage, crystallization canstart to shield s within crystals as soon as atmospheric loss starts.The s available for later volcanic outgassing in the cooler orbitcase is 50 times greater.Raising ss to 10−9 Pa−1 (equivalent to 1 wt% solubility for 100

bars of partial pressure) favors secondary atmosphere occur-rence (Fig. 3 C and D). More s is dissolved in the magma, and somore s partitions (during crystallization) into the rock, where it isshielded. Because s is much more soluble in the magma than H2,very little s is initially in the atmosphere. Therefore, relatively

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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-μspecies in magma (ss) = 10−11 Pa−1, insolation normalized to Earth’s insolation (F/F⊕) = 240 (planet equilibrium temperature 1095 K). (B) ss = 10−11 Pa−1, F/F⊕ =3 (planet equilibrium temperature = 360 K). (C) ss = 10−9 Pa−1, F/F⊕ = 240. (D) ss = 10−9 Pa−1, F/F⊕ = 3.

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little s is carried away to space during H2 removal and so more sis available for exsolution during the final crystallization of themagma ocean. This can create a high-μavg atmosphere, which iseasier to retain. Protection by differential solubility is enhancedby the hot orbit (L/L⊕ = 240; Fig. 3C) as opposed to the coolorbit (L/L⊕ = 3; Fig. 3D), because the magma ocean is long-livedfor this case, and so the volatiles are safely dissolved for longer.As a result, in the hot orbit case, enough high-μ species remain atfinal magma ocean crystallization to create an atmosphere withhigh-μavg (the green line crosses the blue line in Fig. 3C). Such ahigh-μavg atmosphere is easier to retain. The small amount of H2that remains in the atmosphere at this point can be lost bydiffusion-limited escape.Together, F and ss have strong effects on the chance of sec-

ondary atmosphere occurrence. Volatiles that are shieldedwithin rock are available for late volcanic outgassing. For high ss,the mass of shielded volatiles tends toward the product of theinitial inventory of s and the solid–liquid partition coefficient Ds(Fig. 4A). This is because almost all of the s is in the magma untilthe magma ocean has almost completely crystallized. The effectof F/F⊕ on the mass of shielded volatiles is relatively modest forss > 3 × 10−11 Pa−1. For F/F⊕ ≤25 and low ss < (μs/μH2)sH2, theamount of the high–molecular-weight species that is shieldedwithin rock is proportional to ss. For low ss, the mass of shieldedvolatiles decreases rapidly for hot orbits. In hot orbits, the H

wind can carry away s for a long time before the magma oceancools enough for solidification (and shielding).When μavg is high, loss to space is less likely (Eq. 1 and sur-

rounding discussion). Fig. 4B shows how atmospheric μavg (aftercrystallization of the magma ocean is complete) depends on Fand ss. For ss > 10−11 Pa−1, a greater fraction of the ss is stored inthe magma than is H2. As a result, the atmosphere becomesmore s-rich as the magma ocean crystallizes. The enrichment isespecially strong for hot orbits, because for hot orbits there ismore s available to be exsolved: Less s has escaped to spacebefore crystallization completes. For ss < 10−11 Pa−1, most of thes is stored in the atmosphere, and so the atmospheric s mixingratio is near-constant during atmosphere loss. [At F/F⊕ < 3,diffusive separation favors high-μavg atmospheres (SI Appendix,section 1g).]Cooler orbits favor volcanic outgassing but hotter orbits per-

mit exsolved high–molecular-weight atmospheres. To determinewhich effect is more important for the chance of seeing a secondaryatmosphere 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 considerphotoevaporation due to X-ray and extreme UV flux (XUV).FXUV 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(FXUV = 3 × 10−6 F at Earth today) (SI Appendix, section 1a).The plateau of high FXUV/F is longer at red dwarf (M) stars(≥0.3 Gy long for ≤0.5 M☉). Therefore, we expect that (for agiven Teq ) planets orbiting M-stars will have lost more atmo-sphere (e.g., ref. 42).FXUV drives atmospheric loss (rate dMatm/dt) (SI Appendix,

section 1b). Nebula-composition atmospheres do not cool effi-ciently, leading to high Tua that favors hydrodynamic escape (43).For nebular-composition atmospheres, we use the following:

dMatm=dt = «Rpl(R + z(t))2FXUV/(G Mpl), [6]

where « is an efficiency factor. For high-μ atmospheres, we adoptthe loss fluxes of a CO2 atmosphere (44) as an example of astrong coolant. The model of ref. 44 (and ref. 45) includes dis-sociation of CO2 under high XUV levels, and the escaping ma-terial is atomic C and O. The model of ref. 44 predicts low « forCO2 atmospheres, and negligible hydrodynamic escape forFXUV < 0.6 W·m−2 (=150× the value on Earth today) (SI Appen-dix, Fig. S4). At intermediate compositions, we interpolate usinga logistic function (SI Appendix, section 1b).Fig. 5 shows atmosphere thickness vs. time for a solar-mass

star and Patm, init = 50 kbar. High-μavg atmospheres can onlypersist at a narrow range of F. [A similar pattern is seen for low-mass stars (SI Appendix, Fig. S6).] Worlds far from the star re-ceive a low XUV flux and stay as sub-Neptunes. Worlds in hotterorbits lose their atmospheres completely. They may undergo arapid increase in μavg (blue to yellow in Fig. 5A), but the resultinghigh-μavg atmosphere is almost always short-lived. Why is this?According to the pure-CO2 model of ref. 44, dMatm/dt forhigh–molecular-weight atmospheres on super-Earths has athreshold at ∼150 × FXUV/FXUV,⊕, below which loss is muchslower. However, in most cases the primary atmosphere is lostwhen FXUV is still very high, so any exsolved secondary atmo-sphere is swiftly lost. For low ss, the atmosphere has a compo-sition that is always H2-dominated (by number), so exsolvedhigh-μ species are lost in the H wind (Fig. 5B).We conclude that exsolved atmospheres are rare. This con-

clusion is robust because we use parameters that are favorable

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for an exsolved atmosphere. For example, we use a highsolubility-in-magma for the high-μ species, but our high-μescape-to-space parameterization is for a species (CO2) whosesolubility-in-magma is low.

Revival of Secondary Atmospheres by Volcanic Outgassing. Volcanicoutgassing can regenerate the atmosphere of a planet. We do notknow whether or not this process actually occurs on rocky exo-planets. We use a basic model to explore this process. We use atime-dependent rate-of-volcanism model for super-Earths (ref.34 and SI Appendix, section 1h) that is tuned to the rate of CO2release at Earth’s midocean ridges (12 ± 2 bars/Gy; ref. 46). Therate of outgassing is adjusted downward to account for loss ofvolatiles during sub-Neptune-to-super-Earth conversion, assum-ing 50% of worlds have ss = 10−9 Pa−1, and 50% of worlds havess = 10−11 Pa−1. We neglect atmosphere reuptake to form min-erals (e.g., ref. 47), because our focus is on worlds that are toohot for aqueous weathering. This omission is conservative rela-tive to our conclusion that volcanically outgassed atmosphereson hot rocky exoplanets are uncommon. Fig. 6 (gray curve)outlines the region within which, in our model, volcanic out-gassing will build up a secondary atmosphere. The line of revival

sweeps toward the star over time, because the rate of volcanicdegassing falls off more slowly with time than does the star’sXUV flux. Volcanic revival of the atmosphere is difficult forplanets around M = 0.3 M☉ stars, but easier for rocky planetsaround solar-mass stars. The results are sensitive to changes in ss,and to the choice of XUV flux models (SI Appendix, Figs. S9–S11and S17). SI Appendix, Fig. S17 also shows results for volcanismat a constant Earth-scaled outgassing rate. Fig. 6 also shows theatmosphere presence/absence line for rocky worlds that startwith no H2, and with all high–molecular-weight volatiles in theatmosphere (red curves). Such “intrinsically rocky” worlds retainresidual secondary atmospheres over a wider range of conditionsthan do worlds that start as sub-Neptunes. The line of atmo-sphere loss for these residual atmospheres sweeps further awayfrom the star with time.

DiscussionOur model results are sensitive to the rate of XUV-driven massloss. The XUV flux of young solar-mass stars varies betweenstars of similar age by a factor of 3 to 10 (e.g., ref. 52). Stars withlow XUV flux are more likely to host planets with atmospheres(SI Appendix, Fig. S11). Escape of N2 or H2O is likely faster thanthe ref. 44 loss rate estimate (which is for pure-CO2 atmo-spheres) that is used in our model (based on the results of refs.22 and 53). Improved knowledge of escape rate will require moreescape 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

A

B

Fig. 5. Time-dependent results for planets orbiting a solar-mass star. (SIAppendix, Fig. S6 shows the results for a 0.3 M☉ star.) (A) Atmosphericpressure vs. time for ss = 10−9 Pa−1. (From Top to Bottom) F/F⊕ = {49, 283,347, 422, 720}, corresponding to planet equilibrium temperature (Teq) ={735, 1140, 1200, 1275, 1440} K. (B) As A, but for ss = 10−11 Pa−1.

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Fig. 6. Secondary atmosphere presence/absence model output for 6 M⊕(higher planet mass favors atmosphere retention). The red lines and graylines show atmosphere presence/absence contours for two different sce-narios. The red lines show atmosphere retention thresholds after 3.0 Gy forthe case where all volatiles are in the atmosphere initially and there is noprimary atmosphere; the 16th and 84th percentiles are shown, for varyingXUV flux (by ±0.4 dex, 1 σ; ref. 48) relative to the baseline model followingthe results of refs. 49 and 50 (SI Appendix, section 1a). The red lines moveaway from the star over time (red arrows). The gray lines show the 16th and84th percentiles for exhibiting an atmosphere after 3.0 Gy for the casewhere volcanic outgassing rebuilds the atmosphere from a bare-rock state.The solid gray lines are for stagnant-lid tectonics, and the dashed gray linesare for plate tectonics. The lines of atmospheric revival sweep toward thestar over time (gray arrows) because the rate of volcanic degassing falls offmore slowly with time than does the star’s XUV flux. In each case, the at-mosphere/no-atmosphere threshold is 1 bar. The black symbols show knownplanets that may be tested for atmospheres using JWST (51). For any indi-vidual planet, star-specific XUV-flux estimates, star age, and the planet’smass, should be combined to make a more accurate estimate than is possibleusing this overview diagram. SI Appendix, Figs. S8, S16, and S17 showfurther details.

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are limited, and upcoming space missions such as SPARCS willgather more data (55, 56).Our model considers only thermal loss. However, solar wind

erosion can remove atmospheres that are already thin (57). Apossible example is Mars over the last∼4 Ga. Planetary dynamoscan under some circumstances suppress solar-wind erosion (e.g.,ref. 58). Mars had a dynamo prior to 4 Gya, and Earth (but notVenus) has one today.Solubility of gas in magma varies between species. Carbon-

bearing volatile species are very insoluble in magma. H2O’ssolubility in basaltic magma at H2O partial pressure 0.03 GPa is∼2 wt%; linearizing, this gives solubility ss = 7 × 10−10 Pa−1 (e.g.,ref. 59). HCl is even more soluble in magma than H2O (60), so aCl-rich initial composition would have a greater chance offorming an exsolved high-μ atmosphere, although we still predictit would be short-lived. The effects on atmosphere prevalence of100-fold reduction in solubility are shown in SI Appendix, Figs.S9–S11. Better constraints on solubility at T > 2000 K are de-sirable (e.g., ref. 61).Volcanism on super-Earths should wane over gigayears,

according to models. We need more data to test these models. Inmodels, the rate of decrease of volcanism depends on whether ornot the planet has plate tectonics, on planet mass, and on mantlecomposition (e.g., refs. 17 and 62–64 and SI Appendix). Twoeffects, of uncertain relative importance, are ignored in ourvolcanism model. The first is a buffering effect: If the mantle isvolatile-rich, then magma is produced more easily (65), but if themantle is volatile-poor, then the melt rate is reduced (reducingthe rate of volatile loss). The second is a potential fine-tuningissue: If volcanic degassing is very rapid, then volatiles will bereleased from the protective custody of the mantle before at-mosphere loss process have lost their bite.Overall, our choices of Mpl, Ds, and ss tend to favor the exis-

tence of volcanically outgassed atmospheres. Even with thesechoices, atmospheres are usually not stable for planets at Teq >500 K around M-stars. So, our conclusions are broadly unfa-vorable for atmospheres on rocky exoplanets at Teq > 500 Karound M-stars. However, this conclusion is moderated by thepossibility (discussed next) that worlds with abundant H2O existclose to the star.

Observational Tests.Atmospheres on rocky exoplanets can now bedetected (e.g., refs. 66 and 67). Theory predicts that retaining anatmosphere should be harder on planets orbiting low-mass stars,and the present study extends that prediction to super-Earths thatform as sub-Neptunes. A test of this theory would have majorimplications for habitable zone planets. If this prediction fails, thatwould 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 predictformation of hot Super-Earths with 1 to 30 wt% H2O, either byaccretion of volatile-rich objects (for example, extrasolar analogsto CI/CM chondrites), or by planet migration (e.g., refs. 69 and70). Such volatile-rich worlds are hard to distinguish from bare-rock planets using current data. XUV-driven loss can remove atmost a few wt% of an M = 6 M⊕ planet’s mass for Teq < 1000 K.Our model implies that a planet in the “no atmosphere” zone ofFig. 6 with a JWST detection of H2O-dominated atmosphere ismore volatile-rich than Venus and Earth. Possible volatile-richworlds include planets that have radii ∼0.2 R⊕ greater thanexpected for Earth composition (SI Appendix, Fig. S12) (71).Fig. 6 enables the following testable predictions. Since atmo-

spheres close to the star can only persist if the initial H2O in-ventory is high, N2/NH3 should be diluted to very low mixing ratiofor such atmospheres. If a super-Earth–sized planet has an at-mosphere, then planets at greater semimajor axis in the samesystem should also have an atmosphere. Starting out as a sub-

Neptune is unfavorable for atmosphere persistence, so systemswhere the planets formed intrinsically rocky should have statisti-cally more atmospheres. Multiplanet systems enable strong testsbecause uncertainty in time-integrated stellar flux cancels out.

ConclusionsA large fraction of rocky exoplanets on close-in orbits (closer totheir star than the habitable zone) were born with thick (>10kbar) H2-dominated (primary) atmospheres but have since losttheir H2. In our model, these Teq > 400 K exoplanets almostnever transition smoothly to worlds with high–molecular-weightatmospheres (Fig. 7). Instead, the high–molecular-weight speciesare usually carried away to space by the H wind. Volcanic out-gassing is an alternative source for a high–molecular-weight at-mosphere. Revival of a bare-rock planet by volcanic outgassinggets easier with time, because atmosphere loss slows down rap-idly with time, but atmosphere supply by volcanism decaysslowly. However, volcanic outgassing is also enfeebled by earlyloss of biocritical volatiles via the H wind. Overall, for a giveninitial dose of high–molecular-weight species, atmospheres areless likely on hot rocky exoplanets that were born with thick H2-dominated atmospheres. Many uncertainties remain, the mostimportant of which is the initial planet volatile content. Withinour model, for planets that orbit solar-mass stars, super-Earthatmospheres are possible at insolations much higher than forplanet Mercury. For planets that orbit ∼0.3 M☉ stars, secondaryatmospheres at much higher insolation than planet Mercury inour solar system are unlikely unless the planet formed H2-poor,or includes a major (∼1 wt%) contribution of solids from beyondthe water ice line (water world).

Data Availability. All of the code for this paper, together withinstructions to reproduce each of the figures and supplementaryfigures, can be obtained via the Open Science Framework athttps://osf.io/t9h68 or by emailing the lead author.

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ini!ally thick, H2-rich

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(common among exoplanets) (e.g. Earth, Venus)

wider range of condi"onspermits secondary atmospheredifficult transi"on from primary

to secondary atmosphere

rockbarerock

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Fig. 7. Graphical summary. The left column corresponds to atmosphererevival (gray lines in Fig. 6), and the right column corresponds to residualatmospheres (red lines in Fig. 6). For worlds in Teq> 400 K orbits that start assub-Neptunes, formation of a high-μ atmosphere is unlikely, unless theplanet starts with abundant high-μ volatiles.

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ACKNOWLEDGMENTS. We thank two reviewers for accurate and usefulreviews. We thank B. Fegley Jr., L. Schaefer, L. Rogers, E. Ford, and J. Bean

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SIAppendix.1.Modeldescription.Atmosphere evolution involvesmanyprocesses (Catling&Kasting2017).This is (toourknowledge) the first paper on the transition on exoplanets from primary to secondaryatmospheres.Thereforeourapproachistokeepthetreatmentofeachprocesssimple.Weconsiderstaragesfrom5Myr,whichisatypicaltimefornebulardiskdispersal,upto8Gyr. We consider star masses from 0.1-1M☉. We frequently state results for 0.3M☉,becausetheseareexpectedtobethemostcommonhoststarsforworldsdetectedbytheTESSmission(e.g.Sullivanetal.2015,Huangetal.2018).Weemphasizeplanetmass6M⊕(bare-rock radius ≈ 1.6 R⊕), because these are the largest (and therefore highestsignal/noise) worlds that commonly have densities consistent with loss of all H2(Rogers2015).Thesilicatemassfractionisheldfixedat2/3,withthebalanceconsistingoftheFe/Nimetalcore.ThissilicatemassfractionisbasedonSolarSystemdata.Ourresultsareonlyweakly sensitive to reasonablevariations in the silicatemass fraction.The totalmassofH2issmallcomparedtoplanetmass(Lopez&Fortney2014).Thenon-H2volatilemassfractionisalsoassumedtobesmall,consistentwithdata(e.g.vanEylenetal.2018).1a.Driversofatmosphereloss.Candidate drivers for the atmosphere loss that converts sub-Neptunes into super-Earthsare photoevaporation, impact erosion, and core luminosity (e.g. Inamdar & Schlichting2016,Zahnle&Catling2017,Burgeretal.2018,Ginzburgetal.2018,Owen&Wu2017,Biersteker & Schlichting 2019, Kegerreis et al. 2020, Denman et al. 2020).We focus onphotoevaporation in this study. Photoevaporation has been directly observed forsub-Neptuneexoplanets(Bourrieretal.2018),fitsalmostallofthedata(e.g.VanEylenetal. 2018), and is relatively well-understood (e.g. Murray-Clay et al. 2009).Othermechanisms are important at least for a small number of planetary systems(e.g.Owen&Estrada2020,Loydetal.2020).The rate of photoevaporation is set by the X-ray / Extreme Ultraviolet (XUV) flux, FXUV(e.g.Murray-Clay et al. 2009, but see also Howe et al. 2020). XUV is generated by hotregions near the star’s surface. These regions are heated by magnetic fields whoseimportancedeclinesas thestarshedsangularmomentum, throughthestellarwind,overtime. Thus the star’s XUV luminosity FXUV declines over time. Early in a star’s history,thestar’s XUV luminosity LXUV reaches a plateau value (referred to as “saturation”)ofbetween10-4×and10-3×thestar’stotal,bolometricluminosity(L).EstimatingLXUV involvescombiningdirectmeasurementsforstarsofsimilarspectraltypeand age, and interpolation over wavelength regions where star UV is absorbed by theinterstellarmedium.DataonLXUVissynthesizedbyLinsky(2019).Lopez&Rice(2018)andLuger & Barnes (2015) propose power-law decays for Sun-like stars. Tuetal.(2015)quantify the variability of X-ray emission for stars of Solar mass. Guinan et al. (2016)compileLXUVdataforlow-massstars.

www.pnas.org/cgi/doi/10.1073/pnas.2006177117

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We used two approaches to interpolating inLXUV/L as a function of starmass and time.Inoneapproach,weusedFigure5fromSelsisetal.(2007)fortheX-rayflux,obtainingtheXUVfluxfromtheX-rayfluxusingthetoprowofTable4inKingetal.(2018).Selsisetal.(2007) assume a constant LXUV/L of 10-3.2 during saturation. Selsis et al. (2007) do notattempttoquantifythevariationofX-rayfluxbetweenstarsofthesameageandthesamestellar mass. In the other approach, we used Jackson et al.’s (2012) fits for LXUV/LforM≥0.5M☉, patching toGuinanet al.’s (2016) fits forM <0.5M☉.Weassumed thatGuinan et al.’s (2016) fits applied forM= 0.35M☉ and extrapolated to other M-stars.WeusedstandardmodelsforLandstarradius(Baraffeetal.2015)(Fig.S13).TheLXUV/LoutputfromthetwoapproachesisshowninFig.S14.Thezonesofsecondaryatmosphereloss and revival from these approaches are shown in Fig. S9-S11. For both approaches,wealso calculated how results would vary between stars of different LXUV/L, assuminga1-standard-error scatter in LXUV/L of ±(0.4-0.5)dex (following Loydetal.2020) (Fig. 6,S16). We use an upper wavelength cutoff of 91.2 nm (13.6 eV), corresponding to thelowest-energy photon that can ionize H. However, longer-wavelength light can stillcontribute to atmospheric escape, especially as μavg rises. Future work using moresophisticated escape models (e.g. Wang & Dai 2018) might investigate the effectofswitchingonoroffvariousUVspectralbandsatlowerenergiesthan13.6eV.1b.Atmospherelossrates.Forpure-H2atmospheres,weset dMatm/dt=εRpl(R+z)2FXUV/(GMpl) (S1)(Eqn.6frommaintext)wheretheheatingefficiencyfactor,ε=0.15.ε=0.15isacommonchoice in atmospheric evolution models. Models indicate that ε= 0.1-0.2(e.g.Shematovichetal.2014).Astheplanet’satmospherelosesmassandcools, itshrinks(Lopez&Fortney2014).Thisactsasanegative feedbackon loss inpartbecausesmallerplanets intercept fewer XUV photons. To relate the atmospheremassMatm to the planetradius(R+z),weusedthetransit-radiustablesofLopez&Fortney(2014).Thehomopauseradiuscouldbebiggerthanthetransitradiusbytypically5-25%(Malsky&Rogers2020).Forourpure-high-molecular-massatmosphereendmemberlossrate,weusethelossratein molecules/sec/cm2 calculated for a pure-CO2 atmosphere on a super-Earth byTian(2009).Thisisanendmemberofahigh-μavgatmospherethatcanself-coolefficiently.Thislossrateismultipliedbyμavg,andbytheareaofthesurface-atmosphereinterface,togetthelossrateinkg/s.Ourintentistobracketthelikelydown-shiftinatmosphericlossastheatmosphereevolvestohigh-μavg.WeextrapolatebeyondtheXUV-flux limitsshowninFigure6ofTian(2009)linearlyinlossrateforverylargeXUVfluxes,andlinearlyinthelogoflossrateforverysmallXUVfluxes.High-μavgatmospheresofdifferentcompositionmayshed mass faster (or slower) than the pure-CO2 case (Johnstone 2020,Johnstoneetal.2018,Zahnleetal.2019).

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Foratmospheresofintermediatecomposition,weinterpolatebetweentheenergy-limitedformula (dMatm/dt� FXUV) to the exponential cutoff in FXUV for high-μavg atmospheres(dMatm/dt≈0belowa criticalFXUV) foundbyTian (2009). Constraints forhydrodynamicloss rates of atmospheres of intermediate composition are limited (Kulikov et al. 2007,Johnstone et al. 2018). In the absence of better constraints, we use a logistic curve inlog(escaperate)andlog(mixingratio):Xs,atm=(μavg–μp)/(μs–μp); (S2)Y=1/(1+exp(-k1log10Xs,atm–log10k2)) (S3)withk1=8andk2=0.2.WethinkthattheseparameterchoicesunderstatetheXCO2neededtosuppressescape(i.e.favorthesurvivalofhigh-μatmospheres),whichisconservativeintermsofourconclusionthatatmospheresurvivalisunlikely.NextwesetZ=log10(dMatm/dt)p-Ylog10(dMatm/dt)p-log10(dMatm/dt)s (S4) (dMatm/dt)mixed=10Z (S5)where (dMatm/dt)p is the energy-limited escape rate with ε = 0.15, (dMatm/dt)s is fromTian(2009),and(dMatm/dt)mixedisthelossrateforimpureatmospheres.1c.Planetthermalstructure.Planet thermalstructure inourmodel isdefinedbythe temperatureat theatmosphere’sradiative-convective boundary (TRCB) and the temperature at the magma-atmosphereinterface, Tmai. TRCB and Tmai are related by the thickness of the convecting (adiabatic)envelope, which (in pressure units) is (Patm – PRCB), and by the adiabatic index γ (heldconstant at 4/3). We do not consider changes in γ with changes in atmosphericcomposition.Planets form hot and cool over time. Cooling involves energy transport throughtheatmosphere by radiation and/or convection. If, at a given optically thick level oftheatmosphere, all cooling isaccomplishedbyradiation, then theradiative temperaturegradientis dT/dr=–(3κρ/15σT3)(Lint/4πr2) (S6)(e.g. Rogers et al. 2011), where κ is the local Rosseland-mean opacity, ρ is the localatmosphericdensity,Tislocaltemperature,σistheStefanconstant,and(Lint/4πr2)istheinternalluminositycorrespondingtothecoolingoftheplanet(whereristhedistancefromthe center of the planet). If the radiative dT/dr from Eqn. (S6) exceeds the convectiveadiabatictemperaturegradient,thenconvectiontakesoveranddT/drisreducedtoavalueslightlygreaterthantheadiabatic lapserate.Thus,astheplanetcools,andLintdecreases,the radiative zoneexpands (PRCBmoves togreaterdepths) (e.g.Bodenheimeret al.2018,

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Vazanetal.2018).WesetPRCB=10barsandneglectPRCBchangewithtime,whichmeansourmagma oceans are slightly longer-lived than they would be in reality. On the otherhand,weusethe1GyrTeqvaluethroughoutthecalculation,whenforcoolstarsTeqwillbehigherearlierintheplanet’slife.Thissimplificationtendstoshortenmodelmagmaoceanlifetimerelativetoreality.Weassumefullredistributionofenergyabsorbedonthedaysidetotheentire(4π)areaofthe planet. Our model planets are also isothermal with altitude above the radiative-convectiveboundary. Ineffect,weneglectthetemperaturedifferencebetweentheplanetphotosphere (level atwhich the optical depth is approximately unity) and the radiative-convective boundary. At the top of the atmosphere we assume zero albedo. This isreasonable for cloud-free atmospheres of planets orbiting cool stars, although for Solar-mass stars the albedo will be ~0.2, considering only CO2(g) and H2O(g) opacity(Plurieletal.2019).Our focus is interior to the habitable zone and for thick atmospheres. For such worldsatmospheric retentionwould result inmagmaoceans thatpersist forGyr (Hamanoetal.2013,Vazanetal.2018,Fig.S1).Thus,interiortothehabitablezone,atmospherelossistherate-limiter for magma ocean crystallization (Hamano et al. 2013, Hamano et al. 2015,Lebrunetal.2013).Inprincipleourmodelcouldleadtoanequilibriumsituationwhereafew-hundred-bar high-µ atmosphere sustains amagmaocean indefinitely.However suchoutcomes in our model (which assumes all volatiles are delivered early) require fine-tuning.During sub-Neptune-to-super-Earth conversion, planet cooling is driven by atmosphericremoval. As the overburden pressure is removed, the remaining atmosphere coolsadiabatically.Thiscausesatemperaturegradientbetweenthemagmaandtheatmosphere,andheatflowfromthemagmaintotheatmospheresoonwarmstheatmospherebecausetheatmosphereheat capacity is small compared to themagmaoceanheat capacity.ThisraisesTRCB, andbecauseTRCB/Teq>1cools theplanetquickly, thiscools theplanet.Allofthese processes occur continuously in our simplified model: we assume the magmatemperature tracks the loss of atmosphere. In effect we assume that convection swiftlyreduces super-adiabatic temperature gradients and that theboundary layer temperaturecontrastatthetopoftheliquidmagma(viscosity<10-1Pas)issmall.Moredetailedmodelsof sub-Neptune thermalevolution includeRogersetal. (2011),Howe&Burrows (2015),Vazanetal.(2018),andBodenheimeretal.(2018).Forcoolerorbitsinourmodel,somecrystalsarepresentinitially.Thisincomplete-magma-oceaninitialconditionisappropriateiftheplanetisassembledbyimpactsofobjectsthatare Mars-sized or smaller (e.g. planetesimals, pebbles, or planetary embryos), andtheplanetcoolsdowntotheadiabatbetweenimpacts.1d.Magmaoceancrystallization.Magma ocean crystallization shields dissolved volatiles from loss. The magma oceancrystallizes(losingmasstocrystals)astheplanetcools.Themagmaoceanmassatagiven

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Tmaidependsonconditionsdeepwithinthesilicatelayer(higherTandmuchhigherPthanatthemagma-atmosphereinterface).Thesilicatesolidus(Tvs.Pfor0%meltfraction)andliquidus(Tvs.Pfor100%meltfraction)arecurvedinT-Pspace.Asaresult, forasteadydecline inTmai, themagmamasswilldecrease firstquicklyandthenslowly.Totrackthiseffect,weusedthesolidus,theliquidus,andtheadiabatofAndraultetal.(2011)(Fig.S15).Weassumethatthemeltfractionincreaseslinearlywithincreasingtemperaturebetweenthe solidus and the liquidus, and neglect chemical fractionation during crystallization.WealsoneglectthereductioninthesolidusTduetovolatileenrichmentascrystallizationproceeds. More detailed models of magma ocean crystallization includeNikolauetal.(2019), Katyaletal. (2019), and Bower et al. (2019). Our model predictswholemantlemeltingforTmai≥3000KwithnegligiblemeltbelowTmai~1750K.Meltingcurvesdifferbetweenexperimentersanddifferdependingonassumedmantlecomposition(e.g.Andraultetal.2017,Miyazaki&Korenaga2019).Torelatepressuretodepth,weusethepressure-densitycurve fromDziewonski&Anderson(1981).Toget theradiusat themagma-atmosphere interface we assume (Rpl/R⊕) = (Mpl/M⊕)0.27 (Valencia et al. 2006).Weintegratedownward from themagma-atmosphere interface. Starting atPatm andTmai,wefollowtheadiabatuntilwereachthesolidus.Themeltmassisthemassofmelt(ifany)betweenthemagma-atmosphereinterfaceandthedepthofthemagmasolidus.Ifthismeltmass exceeds 2/3 of planet mass, the melt mass is set equal to 2/3 of planet mass.Theeffectoftheweightoftheatmosphereonthelocationofthesolidusisincludedinourcalculations,butturnsouttobeunimportant.1e.Redistributionofvolatilesbetweenatmosphere,magma,androcks.We assume full re-equilibration between magma and atmosphere as the atmosphere isremoved (efficient degassing). Degassing is efficient if the coolingmagma ocean is fullyconvective(Ikomaetal.2018),whichcanbeunderstoodasfollows.Whetherornotafullyliquid magma ocean undergoing whole-magma-ocean convection will degas as the H2envelopeisremovedandtheliquidcoolsdependson(a)thenumberoftimeseachmagmaparcelcyclesthroughtheupperthermalboundarylayerduringcooling,and(b)theratioofthedegassedboundary layer thickness to the thermalboundary layer thickness.Supposethedegassedboundary layer thickness is equal to themaximumdepthatwhichbubblescanform(thisassumesswiftbubbleascent,andthatsupersaturationisunimportant).Forsub-Neptunetosuper-Earthconversion,thisis(upto10skbar)/(ρmagmag)=upto50km.Thethermalboundarylayerthicknessforafullyliquidmagmaoceanwillbe�50km.Eachfluidparcelwillgothroughtheupperthermalboundarylayeratleastonceduringcooling.Wecalculate theconcentrationofdissolvedgas inmagmausingHenry’s law.Manygasesshow linear solubility inmagma.Othersdonot.Forexample,H2Odissolves inmagma inproportiontothesquarerootofpartialpressureatlowpH2O,becomingmorelinearathighpH2O(e.g.Stolper1982,Matsui&Abe1986).We adopt sH2 = 2 × 10-12 Pa-1. The basis for this is as follows. Hirschmann et al. (2012)report results from laboratory experiments on basaltic melts. Extrapolating to melts ofperidotiticcomposition,theystatethat“at1GPainthepresenceofpureH2,themolecular

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H2 concentration [in themelt] is 0.19wt%.” Fegley et al. (2020, their Table 5) compilesolubilitiesofvariousvolatilesinmoltensilicates.We ignore non-ideal solubility (due to non-ideal fugacity of the gas in the atmosphere),becauseitisnotveryimportantattherelativelylowPatmweconsider(Kiteetal.2019).Wealso ignore joint-solubility effects because relevant data are not available. Fewmeasurements have been made of the temperature dependence of volatile solubility inmagmaatT>2000K(e.g.Fegley&Schaefer2014,Guillot&Sator2011)andweneglectTdependenceofvolatilesolubility.Solid-melt distribution coefficients, D, for water partitioning into nominally anhydrousmantlemineralsvaryfrom10-4to10-1(TableS1inElkins-Tanton2008),butforthemostcommonmantleminerals are 0.02 or less. Here, “nominally anhydrous”meansmineralsthatdonothaveHintheirchemicalformula.WeuseD=0.02,whichisatthehighendofexperimentally observed range (corresponding to water portioning into pyroxene). Weneglect saturation limits. C distribution coefficients are much lower, from 2 × 10-4 to7×10-3accordingtoElkins-Tanton(2008)andwithlow(1-5ppmw)saturationlimits.1f.Magma-atmospherecoevolution.Keyinitialconditionsarethatthequantityofthehighmolecularweightvolatileinitiallyinthecrystalphaseiszero(sxtl=0),and es=Cs/(1+(μp /μs )(gmai/Amai)mmagmass) (S7)whichassumesthattheatmosphereisdominantlyp(i.e.,H2)atthezerothtimestep.We restrict ourselves toF/F⊕<103. ForF/F⊕>103, silicate vapor-pressure equilibriumatmospheresdevelop(e.g.Kiteetal.2016).Ourmodelloopsthroughthefollowingsteps:(Step1)Losesomeatmosphere.à(Step2)Post-escapeoutgassing.à(Step3)Crystallize(sequesteringsomevolatiles,butincreasingtheconcentrationofvolatilesinthemagma).à(Step4)Post-crystallizationoutgassingàLoopbacktoStep1.Step 1. Lose atmosphere. A down-step in atmospheric pressure is prescribed.Nofractionation is permitted between H2 and the high-μ species during this escape(seesection 1g). The H2 and the high-μ species are lost in proportion to their bulkabundanceintheatmosphere.Step 2. Post-escape outgassing. Themagma and atmosphere now re-equilibrate throughoutgassingofbothH2andsuntilthepartialpressuresofH2andofsareinequilibriumwiththeconcentrationsofH2andsdissolvedinthemagma.ThepartialpressureofH2dependsonthemassofH2intheatmosphereandonthemeanmolecularweightoftheatmosphere,whichisaffectedbytheamountofs intheatmosphere.Similarlythepartialpressureofsdepends on themass of s in the atmosphere and on themeanmolecular weight of the

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atmosphere,whichisaffectedbytheamountofpintheatmosphere.FromEqn.(2)inthemaintext,weobtainthecoupledequations Cs=es+es((es+ep)/(es/μs+ep/μp))k1/μs (S8) Cp=ep+ep((es+ep)/(es/μs+ep/μp))k2/μp (S9)whereCs (kg)isthetotalon-planetmassofs,Cp (kg)isthetotalon-planetmassofH2(p),esisthekgofsintheatmosphere,episthekgofH2intheatmosphere,thesecondtermontheright-hand-sideof(S8)isthemass(kg)ofsdissolvedinthemagma,thesecondtermonthe right-hand-side of (S9) is the mass (kg) of H2 dissolved in the magma, μs isthemolecularweightofs,μpisthemolecularweightofH2,and k1=(gmai/Amai)mmagmass (S10) k2=(gmai/Amai)mmagmasp (S11)wheregmaiandAmai are (respectively) thegravityat,andareaof, themagma-atmosphereinterfacefor(Rmai/R⊕)=(Mpl/M⊕)0.27,mmagmaiscalculatedasdescribedinsection1dabove,ss isthesolubilityofsinthemagma,andspisthesolubilityofH2inthemagma.Thesimultaneousequations(S10)-(S11)havetwosolutions,andthephysicalsolutionis

ep=(Cp k1 +Cs k2 +(k2 (Cp k1 μs -k1 (Cp 2k1 2μp 2+2Cp 2k1 μp μs +Cp 2μs 2 +2Cp Cs k1 k2 μp μs -2Cp Cs k1 μp 2 +4Cp Cs k1 μp μs +4Cp Cs k2 μp μs -2Cp Cs k2 μs 2 +2Cp Cs μp μs +Cs 2 k2 2 μs 2 +2Cs 2 k2 μp μs +Cs 2 μp 2 )0.5-Cs k1 μp+2Cs k2 μs +Cp k1 2 μp +Cs k1 k2 μs ))/(2(k1 μp -k2 μs +k1 2 μp -k1 k2 μs ))+(k1 k2 (Cp k1 μs -k1 (Cp 2 k1 2 μp 2 +2Cp 2 k1 μp μs +Cp 2 μs 2 +2Cp Cs k1 k2 μp μs -2Cp Cs k1 μp 2 +4Cp Cs k1 μp μs +4Cp Cs k2 μp μs -2Cp Cs k2 μs 2 +2Cp Cs μp μs +Cs 2 k2 2 μs 2 +2Cs 2 k2 μp μs +Cs 2 μp 2 )0.5-Cs k1 μp+2Cs k2 μs +Cp k1 2 μp +Cs k1 k2 μs ))/(2(k1 μp -k2 μs +k1 2 μp -k1 k2 μs )))/(k1 +k1 k2 )

(S12)

es=-(Cp k1 μs -k1 (Cp 2 k1 2 μp 2 +2Cp 2 k1 μp μs +Cp 2 μs 2 +2Cp Cs k1 k2 μp μp-2Cp Cs k1 μp 2 +4Cp Cs k1 μp μs +4Cp Cs k2 μp μs -2Cp Cs k2 μs 2 +2Cp Cs μp μs +Cs 2 k2 2μs 2 +2Cs 2 k2 μp μs +Cs 2 μp 2 )0.5-Cs k1 μp +2Cs k2 μs +Cp k1 2 μp +Cs k1 k2 μs )/(2(k1 μp -k2 μs +k1 2 μp -k1 k2 μs ))

(S13)WerecalculatetheothervariablesbyinsertingthenewesandepintoEqns.(S8)and(S9).The coupling of Eqn. (S8) and Eqn. (S9) viaμavg has two important effects that disfavorsecondaryatmospheresonexoplanets.Duringthelossoftheprimaryatmosphere,theμavgeffect increasestheamountof thehigh-μ constituent in theatmospherebya factorof15(=μs/μp),increasingtheamountofthehigh-μconstituentthatislosttospace.Moreover,asμavgrises from2 to30,because thepartialpressureof thehigh-μconstituentscaleswiththenumberratioofsintheatmosphere,fs=es/(ep+es)(μavg/μp),thenumberofmolesofsinthe atmosphere for a given saturation vapor pressure of s decreases by a factor of 15

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(=μs/μp).Thisnegative feedback reduces thenumberofworlds that smoothly transitionfromaprimarytoasecondaryatmosphere.Step3.Crystallize.Themagmamass is forcedtoa lowervalue,consistentwiththe lowerTmai associated with the atmospheric pressure that was reduced in Step 1. The freshly-producedcrystalsgainansconcentration(inmassfraction)ofDs×Xs,whereXs isthemassfractionofthevolatileinthemagma.Thecrystalsarenotremixed(byassumption)onthetimescaleofmagma-ocean crystallization. swithin crystals is relatively safe fromreleaseinto the unsafe surface environment during sub-Neptune-to-super-Earth conversion,becausesolid-statecreepspeedswithinthesolidmantleare10-9–10-8ms-1,versusupto101ms-1inthemagma.Asaresult,theconcentrationofthecrystalsisindependentofthepreviouspartitioningofthevolatilesintothesolidrock.PartitioningofH2intothecrystalsisnotconsidered.Step4. Post-crystallizationoutgassing.BecauseDs <1, themagmaocean is enriched in safterthecrystallizationstep.Thereforewere-equilibratethemagmaandtheatmospheretakingaccountofthenow-higherconcentrationofs(Xs)inthemagma.TheprocessesinvolvedinSteps1-4alloccurontimescalesshortcomparedtothemagmaoceanlifetime.1g.Calculatingevolutionovertime.To map the rocky planet evolution sequence onto time we combine the environmentaldriversfrom(1a), the lossratesfrom(1b),andthesmallplanetevolutionsequencefrom(1f).TRCB is fixed in the crystallization calculation,butTeq evolveswith starage.WedealwiththisbysettingTRCBthroughoutthecalculationequaltoTeqatafixedageof1Gyr.Thus,planetsorbitingcoolstarsinourmodelhaveshorter-livedmagmaoceansthaninreality.Diffusive separation of H2 from high-µ species is important at lower XUV flux(e.g.Huetal.2015, Wordsworth et al. 2018, Saito & Kuramoto 2018, Odert et al. 2018,Malsky&Rogers2020).However,diffusiveseparationiswashedoutatthehighXUVfluxesrequired if photoevaporation is to cause sub-Neptune-to-super-Earth conversion(Tian2015, Catling&Kasting2017).Todemonstrate this,we first define the (atomic)Hflux, F1, on the assumption that the overwhelmingmajority of the upper atmosphere iscomprisedofH: F1=εFXUV(R+z)/(4GMplm1) (S14)inunitsofmoleculess-1m-2.Here,ε=0.15istheefficiencyofconversionofXUVenergyintoatmospheric escape,R is solid-planet radius,z is atmosphere thickness (whichwe set to0.4R tocorrespondtoasmall sub-Neptune),G is thegravitationalconstant,Mpl isplanetmass,andm1isthemass(kg)ofthehydrogenatom.Fromthisweobtainthecrossovermassmc(Tian2015)

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mc=m1+kTF1/(bgX1) (S15)where T is the temperature of the heterosphere, and b = 4.8 × 1019T 0.75m-1 s-1 is thebindary diffusion coefficient, corresponding to neutral H and neutral O, used byTian(2015).IncludingtheeffectofionizationofHwoulddecreasetheeffectivevalueofb(Huetal.2015)andsostrengthenourconclusionthatdiffusiveseparationisnotimportantduringsub-Neptunetosuper-EarthconversionforKepler’ssuper-Earths.WesetT=104K,thesamevalueusedbyHuetal.(2015)andMalsky&Rogers(2020).Theflux,F2,ofthehigh-µspeciesiszerowhenmc<m2.Whenmc>m2thefluxF2isgivenby F2=F1(X2(mc–m2))/(X1(mc–m1)) (S16)Substituting in values for 6 M⊕ and atomic mass 15 Da we find FXUV = 0.7 W/m2 atcrossover, 1.3W/m2 for 50% reduction in escaping flux due to weight differences, and6.1W/m2forescapingfluxofhigh-molecularweightspeciesat90%ofthefluxiftherewasnofractionationatall.AllthreethresholdsareshownasdashedverticallinesonFig.S5andFig.S6.Inthecontextofsub-Neptunetosuper-Earthconversion,mostoftheatmospheremassisremovedathigherFXUV.Mostexoplanetseitherstayassub-NeptunesorlosetheiratmospheretooquicklyfortheXUVfluxtodroptolevelsthatpermit>10%differencesinlossrateduetofractionation.Thiscanbeunderstoodasfollows.Iftheupperatmosphereabsorbs 1W/m2 of light from the star, the upper limit on loss rate for a 6M⊕world is~10bars/Myr, falling to 1.5bars/Myr at ε = 0.15. Since sub-Neptune-to-super-Earthconversiontakes~300Myr,and~5×104barsofH2needtoberemoved(includingH2thatis initially dissolved in the magma), FXUV must be ~100 W/m2. Using the saturationthresholdofSelsisetal. (2007), (FXUV/F=10-3.2) thiscorresponds to160kW/m2,F/F⊕=120,fordistancefromthestar=0.1AUforaSun-likestar.ThisisthetypicaldistanceofaKeplersuper-Earthfromitshoststar(theabundanceof1.6R⊕~6M⊕super-Earthsdropsoffrapidlyatlargerseparations(Fultonetal.2017,Fulton&Petigura2018).AtsuchhighXUV fluxes, fractionation causes negligible differences in the loss rate relative to the no-fractionationcase.Therefore,ourapproximationofnofractionationduringsub-Neptunetosuper-Earthconversionisvalid.However, fractionation by diffusion can be important for the composition of late-stagevolcanically outgassed atmospheres. For example, the volcanic H2 outgassing flux iscalculatedtobelessthanthediffusively-limitedH2escaperateunlessH2staysataverylowlevel (e.g.Catlingetal.2001,Batalhaetal.2016,Zahnleetal.2019).Weexpect that thiseffectwouldmaintainvolcanicallyoutgassedatmospheresathighμavg,evenifoutgassingofH2byserpentinization(Sleepetal.2004)and/orvolcanicoutgassingislarge.Fractionation by diffusion is alsomuchmore important forHabitable Zone super-Earth-sizedplanets.ForFbol=1361W/m2(theSolarfluxat1AUtoday),evenatsaturationFXUVis0.9W/m2,whichisonlymarginallycapableof liftingatomicmass=15Daspeciesoutoftheatmosphere.This favorstheoccurrenceofsecondaryatmospheresonHabitableZoneexoplanets.

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Finally, exposure to moderate FXUV for Gyr may convert a H2-dominated sub-Neptuneatmosphere intoaHe-enrichedsub-Neptune,provided theplanetdoesnotstartwith toomuchH2(Huetal.2015,Malsky&Rogers2020).1h.Volcanicoutgassing.Noexoplanetshaveyetbeenconfirmedtobevolcanicallyactive.Weuseasimplemodelofthegeodynamicsandrateofvolcanismonsuper-Earth-sizedplanets(Kiteetal.2009).Themodel includes parameterized mantle convection, thermal evolution, and volcanism forboth plate tectonics and stagnant lid modes. The model is tuned to reproduce Earth’spresent-day rate of volcanism in plate tectonics mode. Kite et al. (2009) considerthreedifferentmeltingmodels. In thepresent study,weuse the results from thewidely-used melting model of Katz et al. (2003). Many more sophisticated models of rockyexoplanetthermalevolution,geodynamics,andvolcanicactivityexist(e.g.Dornetal.2018,andreferencestherein).The procedure for estimating the rate of volcanism starts from the resultsofKiteetal.(2009)(Fig.S3).ThisisanEarth-tunedparameterizedmantleconvectionandrateofvolcanismmodelthatpredictstherateofvolcanismversustimeforeitherstagnantlidmode and plate tectonicsmode. The rate of CO2 release at Earth’smid-ocean ridges(12±2 bars/Gyr; Tucker et al. 2018) is multiplied by the computed rate of volcanism,inunitsofEarth’spresent-dayrate,toobtainthepredictedrateofCO2releaseontherockyexoplanet. This is adjusteddownward in proportion to any early loss of high-molecular-weight volatiles to space. For example, if a model exoplanet has lost 70% of its high-molecular-weightvolatilestospace,thentherateofvolcanicoutgassingisreducedto30%of the volumetric volcanic flux predicted from the Earth-tuned model of Super-Earthvolcanism.Fig.6comparesthebroadzoneofworldswithatmosphereforplanetsthat formwithoutthick H2 atmospheres to the much more restricted zone of volcanically revivedatmospheresforworldsthatformwiththickH2atmospheres.Thiscalculationassumesthathalfofworldsformwithss=10-9Pa-1(forwhich~90%ofhigh-μvolatilesarelosttospaceduring the loss of the H2-dominated atmosphere), and half form with ss = 10-11 Pa-1.Atmosphericingassingisneglectedbecauseworldswell insidethehabitablezonearetoohotforweatheringreactionstosequesteratmophilesinthecrustalrocks.Volatilereleaseby volcanic outgassing is neglected, by construction, onworlds that start with all theirvolatiles in the atmosphere. Forming with a secondary atmosphere is advantageous forsubsequentlyexhibitingasecondaryatmosphere,relativetothetotal-loss-and-subsequent-revival process. The stagnant-lid andplate-tectonicspredictionsdiverge greatly after~4Gyr, when volcanism shuts down on the stagnant-lid model planet (Fig. S8). The plate-tectonicsandstagnantlidpredictionsotherwiseappearsimilarinthisEarth-scaledmodel(Kiteetal.2009).However, thisoverall similarityofplateandstagnant-lidpredictions isspecific to theEarth-tunedmodelofKite et al. (2009): at leastoneothermodelpredictsmuch stronger suppressionof volcanismon stagnant-lid super-Earth-sized rockyplanets(e.g.Dorn et al. 2018). Tidal locking, by itself, has little effect on volcanism(e.g.vanSummerenetal.2011). So long as we do not mechanistically understand

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volcanismversustimeforSolarSystemworlds(Byrneetal.2019),whyEarth’smantletooksolongtomix,norwhyEarth’smantleisnotyetcompletelyoutgassed,wethinkthatitisappropriate to use basic models to predict rate of volcanism versus time for rockyexoplanets. A robust constraint (for non-tidally-heated planets) is decline of mantletemperatureby50-200KperGyr,withaconcomitantdecline in thepotential formelting(Stevenson2003).We assume the rate of degassing is proportional to the rate of magmatism. This is asimplification.Evenlow-degreepartialmeltscaneffectivelyextractalmostallthevolatilesfrom the full mass of silicate mantle that sweeps through the partial melt zone. Thisisbecause volatiles partition strongly into the melt during partial melting (i.e., Di�1).Once thepartialmelt fraction ishighenough (~1%) for themelt to form interconnectedchannels, the volatile bearing melt rises from the melt production zone geologicallyinstantaneously(withnolossofvolatiles)viafilter-pressing,meltchannelformation,anddiking.Whether themelt gets close enough to the surface to formbubbles and/or eruptexplosively will depend on the stress state of the lithosphere (e.g. Solomon 1978).Explosive volcanic eruptions are aided by bubble formation, which is easier when thevolatilecontentofthemagmaishigh.Ifmeltcrystallizesatdepth(intrusion)thenvolatileswillgointohydratedmineralsorintotheglassphase,wheretheymightstillbereleasedtotheatmosphereovergeologictime.ExtrapolationofsimplemodelsofEarth’sthermalevolutionandoutgassingratebackintoEarth’s past leads to the expectation that Earth’s mantle was very rapidly stirred andquicklyoutgassedveryearlyinEarthhistory–incontradictiontoisotopicdatathatshowsluggish stirring. Similarly, our simple Earth-tuned model of thermal evolution andoutgassing rate indicates that 4-6×more volatiles outgas over the lifetime of the planetthan are present at the beginning – a mass balance violation. Because an excessofvolcanism over XUV-driven atmospheric loss leads (in our model) to build-up of adetectable atmosphere in <1 Gyr, the rate of volcanism is more important than thecumulativeamountofvolcanisminsettingatmospherepresence/absence.Thereforewedonotadjustourvolcanicfluxesdownwardtotakeaccountofthiseffect,reasoningthatrealplanetsoutgasmoreslowlythansimplemodelspredict.Alternatively,ourresultscouldbeinterpretedassayingthatsuper-Earthseruptawaytheirsolid-mantlevolatilesduringthefirstGyrofthestar’slife,whenXUVfluxesarestillhigh.Thiswouldleadtoanevenmoreunfavorable conclusion for the likelihood of volcanically outgassed atmospheres atdistancesmuchclosertothestarthanthehabitablezonethanispresentedinFig.6.Tocalculatetheatmospherepresence/absencelinesshowninFig.6andFig.S16,wevariedfollowing parameters. (1) XUV flux: multiplied by a factor drawn randomly fromalognormal distribution centered on 1 and with a standard deviation of 0.4 dex(Lyonetal.2019). (2)EarthCO2 outgassing rate: a quantity drawn randomly from anormaldistributioncenteredon0andwithastandarddeviationof2bars/Gyrisaddedtothenominalvalueof12bars/Gyr.(3)Solubilityofthehigh-molecular-weightspeciesinthemagma:alternatedbetweenvaluesof10-11Pa-1and10-9Pa-1.Weused1baratmosphericpressureasthethresholdforatmospherepresence/absence.

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ThemodelresultsaremoresensitivetochangesintheXUVfluxthantheyaretochangesintherateofvolcanicoutgassing.ThisisduetotheexponentialcutoffintheXUV-flux-drivenlossrateofCO2(Tian2009).1i.Supplyandobservability.The inventory of CO2 even on today’s Earth is uncertain. For example, the review ofLeeetal.(2019)reportsarangeofestimatesfortheCstoredinthenon-sedimentaryrocksofEarth’scontinentalcrust, from4.2×107GtonCto2.6×108GtonC.This increasesthelikelihood that some hot rocky exoplanets have C inventories large enough to retain anobservableatmosphere.Noexoplanetshaveyetbeenconfirmedtocontain>1wt%H2O.Modelingpaperstrackingtheformation,migration,and(forhabitable-zoneworlds)climateevolutionofworldswith>1wt%H2OincludeKite&Ford(2018),Bitschetal.(2019),andreferencestherein.AnotherwaytoincreasevolatilemassistoballastthehydrogenwithoxygenobtainedfromthereactionFeO(magma)+H2(g)àFe+H2O(g); i.e.,endogenicwaterwhichdoesnotrequireplanetmigration(Kiteetal.2020).SIReferences.Andrault,D.,Bolfan-Casanova,N.,Nigro,G.L., etal.2011,Solidusand liquidusprofilesofchondriticmantle:ImplicationformeltingoftheEarthacrossitshistory,EarthPlanet.Sci.Lett.,304,251.

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SIFigures.

Fig.S1.Howmagmaoceanmassincreaseswithatmosphericthickness.Outputfromatoymodel of sub-Neptune thermal structure (Kite et al. 2020). Dashed lines correspond tomagma ocean mass, labeled in Earth-masses of magma, for a volatile-free planet massof6M⊕. Colored lines correspond to temperatures at the magma-atmosphere interfaceof1500K (maroon), 2000K (red), 3000K (orange), and4000K (yellow).Magmaoceanmasses inexcessof2Earthmassesarenotplottedbecausethiscorrespondstoamagmapressurerangethatisnotwellexploredbyexperiments.

Fig.S2.Planetthermalstructure(reproducedfromKiteetal.,2020).Temperatureversuspressure plot, showing adiabats within the atmosphere and the magma (black line).Trheo(whiteline)correspondstothetemperatureoftherheologicaltransition(~40%meltfraction)forrock(dashedwhitelineat levelswherenorockispresent).RCB=radiative-convective boundary. Tmai = temperature at the magma-atmosphere interface. For realplanets the equilibrium temperature (Teq), the effective temperature (Teff), and thetemperatureat theRCB(TRCB),arerelatedbyTeq<Teff≲TRCB. In thispaperwemaketheapproximationthatthesethreetemperaturesareequal(SIAppendix,section1c).

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Fig. S3. (Reproduced fromKiteetal.2009, ref.34 in themain text,bypermissionof theAAS).RateofvolcanismperunitmassonmassiveEarth-likeplanetsundergoingstagnantlidconvection,normalizedtocalculatedrateonaplate-tectonicEarth,forKatzetal.(2003)meltingmodel.Dark gray shaded regions correspond tomantle temperatures associatedwithveryintensevolcanism,toohighforareliablecrustalthicknesscalculation.Contoursare at 0, 0.5, 1, 2, 5, and 10 times Earth’s present-day rate of volcanism (~4×10−19s−1,equivalent to 24 km3 yr−1, in plate-tectonics mode). For example, a 6M⊕planet on the“2”contourerupts6×2×24≈300km3yr−1.

Fig. S4. Atmospheric loss rate forapureCO2atmosphere, combiningresults for super-Earths(Tian2009)andforMars(Tianetal.2009,scaledto1AU).Theverticaldashedlinecorrespondsto150×Earth’spresent-dayXUVflux.

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Fig. S5. The same evolutionary tracks as in Fig. 5 (main text), shown in FXUV – μavgcoordinates.F/F⊕increasesfromrightmosttrack(F/F⊕=49,Teq=735K)toleftmosttrack(F/F⊕=720,Teq=1440K).Thedashedlineshighlight(fromrighttoleft)thethresholdofFXUVbelowwhichallofthehigh-molecular-massspeciesisretainedbytheplanet;theFXUVcorrespondingtoa50%reductionintheno-fractionationlossrateofthehigh-molecular-massspecies;andtheFXUVcorrespondingtoescapeofthehigh-molecular-massspeciesat90% of the rate at which no loss would occur (SI Appendix, section 1g). Right of therightmostdashedline,fractionationprotectstheconstituentsofthesecondaryatmosphere,andleftoftheleftmostdashedline, fractionationismuchlessimportant.CalculationsaredoneassuminganatomicwindofHentraininga1%mixingratioofatomsofmass15Da.

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(a)

(b) Fig. S6. (a) As Fig. 5a (main text), but for a planet orbiting a star of 0.3 Solar masses.Time-dependentresults.Atmosphericpressurevs.timeforplanetsfor(fromrighttoleft)F/F⊕={49,283,347,422,720},correspondingtoplanetequilibriumtemperature(Teq)={735,1140,1200,1275,1440}K.(b)AsFig. S5,but for aplanetorbitinga starof0.3 Solarmasses.Thesameevolutionary tracksas in the toppanel, shown inFXUV–μavg coordinates.F/F⊕increasesfromrightmosttrack(F/F⊕=49,Teq=735K)toleftmosttrack(F/F⊕=720,Teq=1440K).Thedashed lines highlight (from right to left) the threshold of FXUV belowwhich all of the high-molecular-massspeciesisretainedbytheplanet;theFXUVcorrespondingtoa50%reductionintheno-fractionation lossrateof thehigh-molecular-massspecies;andtheFXUVcorrespondingtoescapeofthehigh-molecular-massspeciesat90%oftherateatwhichnolosswouldoccur(SI Appendix, section 1g). Right of the rightmost dashed line, fractionation protects theconstituentsofthesecondaryatmosphere,andleftoftheleftmostdashedline,fractionationismuch less important. Fractionation calculations are done assuming an atomic wind of Hentraininga1%mixingratioofatomsofmass15Da.

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a) b)

c) d)

Fig.S7.DetailsofplanetevolutionforthetracksshowninFig.5a(maintext).Solar-massstar,6M⊕.Linethicknesscorrespondstoinsolation,withthethickestlinescorrespondingtothegreatestinsolation.ResultsareshownforF/F⊕=49(blue),F/F⊕=283(red),F/F⊕=347(yellow),F/F⊕=422(purple),andF/F⊕=720(green),correspondingtoTeq={735,1140, 1200, 1275, 1440} K, respectively. (a) Planet radius versus time. (b) Planetequilibrium temperature versus time. (c) Planetmass loss rate versus time (solid lines),andthemasslossratethattheplanetwouldhaveiftheatmospherewascomposedentirelyofCO2basedonthecalculationsofTian(2009)(dashedlines).Theplanetscorrespondingtothered,yellow,purple,andgreenlinesevolvetoahigh-μatmosphere,butonlyintheredcaseisthisatmospherelong-lived.(d)Thefractionofthehigh-μspecies(s)thatisshieldedwithin solid rock. This is zero for theworld that remains as a sub-Neptune (because nosolidrockforms),andverysimilarforthefourworldsthattransitiontosuper-Earths(fullmagmacrystallization).

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a)b)

c)d)

Fig. S8.Atmospheric thickness versus time for the Selsis et al. (2007)XUVmodel. (a-b)volcanic revival, ss = 10-9Pa-1, Kite et al. (2009) volcanismmodel. Results are shown forbothplate tectonics (plates), and for stagnant lidmode.There is aminor trend inplate-tectonics mode for planets around Solar-mass stars at Teq<500 K toward decreasingatmosphericthicknesswithdecreasingTeq.Thisisduetothesmallerinitialvolumeofthe(s-protecting) magma ocean at low Teq. (c-d)Start-with-all-volatiles-in-the-atmosphere,residualatmospherecase. Inthismodel, theplanetformswithnoH2(intrinsicallyrocky)andwith500barsofs,allintheatmosphere.

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Fig. S9. Deterministic calculation for atmosphere presence/absence after 3.0Gyr.Deterministic calculationwith a fixed XUV flux for each starmass, combining the fits ofJacksonetal.(2012)andGuinanetal.(2016).Thereddash-dotlinecorrespondstothelineof vanished atmospheres for planets that have all volatiles in the atmosphere initially.WorldsathotterTeqthanthislinecanretainanatmosphereifthereisnoH2initiallyandallhigh-molecular-weightvolatilesareintheatmosphereinitially.Thislinemovesawayfromthe starover time. The thickgray solid (plate tectonics) anddashed (stagnant lid) linescorrespondtothelineofatmosphericrevivalbyvolcanicoutgassingforplanetsthatloseallatmosphere during transition from a sub-Neptune to a super-Earth. These lines ofatmospheric revival sweep towards the star over time because the rate of volcanicdegassingfallsoffmoreslowlywithtimethandoesthestar’sXUVflux.Thischartassumesinitial volatile supply is independent of F and similar to Earth and Venus. Increasingvolatile supply will move thresholds to higher F. The thin gray lines show results for areductioninthesolubilityofthehigh-molecular-weightvolatileinthemagmafrom10-9Pa-1to10-11Pa-1.Selectedexoplanetsoverplotted.NotethatbecausemostoftheplanetsthatareshownaretoosmalltohaveEarth-likecompositionand6M⊕,thisplotisoptimisticforatmosphericsurvival(higherplanetmassfavorsatmosphereretention).Foranyindividualplanet, star-specific XUV-flux estimates, star age, and the planet’s mass, should becombinedtomakeamoreaccurateestimatethanispossibleusingthisoverviewdiagram.

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Fig. S10. Deterministic calculation for atmosphere presence/absence after 3.0Gyr.Deterministic calculationwith a fixed XUV flux for each starmass, following Selsis et al.(2007).Thereddash-dotlinecorrespondstothelineofvanishedatmospheresforplanetsthat have all volatiles in the atmosphere initially.Worlds at hotterTeq than this line canretainanatmosphereifthereisnoH2initiallyandallhigh-molecular-weightvolatilesareintheatmosphereinitially.Thislinemovesawayfromthestarovertime(redarrows). Thethickgraysolid(platetectonics)anddashed(stagnantlid) linescorrespondtothelineofatmospheric revival by volcanic outgassing for planets that lose all atmosphere duringtransitionfromasub-Neptunetoasuper-Earth.Theselinesofatmosphericrevivalsweeptowardsthestarover time (grayarrows)becausetherateofvolcanicdegassing fallsoffmore slowlywith time than does the star’s XUV flux. This chart assumes initial volatilesupplyisindependentofFandsimilartoEarthandVenus.Increasingvolatilesupplywillmove thresholds to higher F. The thin gray lines show results for a reduction in thesolubility of thehigh-molecular-weight volatile in themagma from10-9Pa-1to 10-11 Pa-1.Selectedexoplanetsoverplotted.NotethatbecausemostoftheplanetsthatareshownaretoosmalltohaveEarth-likecompositionand6M⊕,thisplotisoptimisticforatmosphericsurvival(higherplanetmassfavorsatmosphereretention).Foranyindividualplanet,star-specificXUV-fluxestimates,starage,andtheplanet’smass,shouldbecombinedtomakeamoreaccurateestimatethanispossibleusingthisoverviewdiagram.

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Fig.S11.Smallplanetatmospherepresence/absencediagramafter3.0Gyr.AsFig.S10,butwithareductioninXUVfluxbyafactorof10,or,equivalently,decreasingtheefficiencyofXUV-drivenlossbyafactorof10,relativetothebaselinebasedontheestimatesofSelsisetal. 2007. This increases the range of exoplanets for which a volcanically supportedatmosphere is possible. The red dash-dot line corresponds to the line of vanishedatmospheresforplanetsthathaveallvolatilesintheatmosphereinitially.WorldsathotterTeqthanthislinecanretainanatmosphereifthereisnoH2initiallyandallhigh-molecular-weightvolatilesare in theatmosphere initially.This linemovesaway from the starovertime.The thickgraysolid (plate tectonics)anddashed (stagnant lid) linescorrespond tothelineofatmosphericrevivalbyvolcanicoutgassingforplanetsthatloseallatmosphereduring transition from a sub-Neptune to a super-Earth. The lines of atmospheric revivalsweep towards the star over time because the rate of volcanic degassing falls off moreslowlywithtimethandoesthestar’sXUVflux.Thischartassumesinitialvolatilesupplyisindependent of F and similar to Earth and Venus. Increasing volatile supply will movethresholdstohigherF.Thethingraylinesshowresultsforareductioninthesolubilityofthe high-molecular-weight volatile in the magma from 10-9Pa-1 to 10-11 Pa-1. Note thatbecausemostof theplanets thatareshownare toosmall tohaveEarth-likecompositionand 6M⊕, this plot is optimistic for atmospheric survival (higher planet mass favorsatmosphereretention).Foranyindividualplanet,star-specificXUV-fluxestimates,starage,and the planet’s mass, should be combined to make a more accurate estimate than ispossibleusingthisoverviewdiagram.

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Fig.S12.PlanetdensitiesfromOtegietal.(2020).ThebluelinecorrespondstodensityforEarth-like composition, assuming scaling of mass with planet radius as (R/R⊕) =(M/M⊕)0.27. The red line corresponds to 80% of that density. Many super-Earth sizedplanets plot below the red line (for example HD3167 b), indicating either a volatileenvelope(atmosphere)oralternativelyaverylow(Mg+Fe)/SiratiorelativetothatoftheEarth.Super-EarthswiththesameorlowerdensityasEarth(e.g.HD3167b)areexcellentcandidatesforhavingelevated-molecular-weightsecondaryatmospheres.Planetsplottinginthelow-density,largeradiusclumpinthebottomrightaresub-Neptunes,retainingthickH2-dominatedatmospheres.

Fig.S13.Stellarluminosity,L,versustimefromthemodelofBaraffeetal.(2015)forstarmassesrangingfrom0.1–1.0M☉,andnormalizedtotheSun’sluminosityatthepresent-day (L☉).Contoursdrawnat intervalsof0.1M☉.Theblue linehighlights0.1M☉.Theredlinehighlights0.3M☉. Theblacklinehighlights1.0M☉.

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(a)

(b) Fig.S14.FXUVataninsolationcorrespondingtoacircularorbit0.1AUfromtoday’sSun,inunitsnormalizedtotheFXUVatEarth’sorbittoday.(a)forthemodelofSelsisetal.(2007),and (b)combining the fits of Jackson et al. (2012) andGuinan et al. (2016). The dashedlines highlight (from top to bottom) the threshold of FXUV below which all of the high-molecular-mass species is retained by the planet; the FXUV corresponding to a 50%reductionintheno-fractionationlossrateofthehigh-molecular-massspecies;andtheFXUVcorrespondingtoescapeofthehigh-molecular-massspeciesat90%oftherateatwhichnolosswouldoccur.Abovethetopdashedline,fractionationprotectstheconstituentsofthesecondary atmosphere, and below the lowest dashed line, fractionation is much lessimportant. Calculations are done assuming an atomicwind ofH entraining a 1%mixingratioofatomsofmass15Da,andplanetmass6M⊕.

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Fig. S15.Andraultetal.(2011)meltcurves.Coloredlinesaremagmaadiabats(notethattheseareonlyphysicallymeaningful formoltenmagma, i.e. fortemperatureshigherthanthebluelines).Bluelinesaremeltfractioncurves,wherethelowestoneisthesolidus.Wefind that the effect of atmospheric overburden pressure on the solidus temperature isrelativelysmall.XaxiscorrespondstodepthwithinahypotheticalEarth-massplanet(thecurves correspond to lines inP-T space, so theymap to smaller depthonhigher-gravityworlds).

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Fig. S16. As Fig. 6 (main text), substituting in an XUV-flux-vs.-time parameterizationfollowing Selsisetal. (2007). Secondary atmosphere presence/absencemodel output for6M⊕ (higher planetmass favors atmosphere retention). The dashed red lines show thelines of atmosphere retention after 3.0 Gyr for the case where all volatiles are in theatmosphereinitiallyandthereisnoprimaryatmosphere;the16thand84thpercentilesareshown, for varying XUV flux (by ±0.4 dex, 1σ; Loyd et al. 2020) relative to the baselinemodelfollowingtheresultsofJacksonetal.(2012)andGuinanetal.(2016)(SIAppendix,section1a).Theselinesmoveawayfromthestarovertime.Thegraylinesshowthe16thpercentile for exhibiting an atmosphere after 3.0 Gyr for the case where volcanicoutgassing rebuilds the atmosphere from a bare-rock state (the 84th percentile is atTeq<400K).Thesolidgray linesare for stagnant-lid tectonicsand thedashedgray linesareforplatetectonics.Thelinesofatmosphericrevivalsweeptowardsthestarovertimebecausetherateofvolcanicdegassingfallsoffmoreslowlywithtimethandoesthestar’sXUV flux. In each case the atmosphere/no-atmosphere threshold is 1 bar. The blacksymbolsshowknownplanets thatmaybe tested foratmospheresusing JWST(Kolletal.2019,Mansfield et al. 2019). For any individualplanet, star-specificXUV-flux estimates,starage,andtheplanet’smass,shouldbecombinedtomakeamoreaccurateestimatethanispossibleusingthisoverviewdiagram.

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(a)

(b)

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Fig. S17. (a) Secondaryatmospherepresence/absencediagrams for6M⊕(higherplanetmass favors atmosphere retention). The dash-dot lines correspond to the line ofatmosphere vanishing forplanets that have all volatiles in the atmosphere initially. Thesolidanddashedlinescorrespondtothelineofatmosphericrevivalbyvolcanicoutgassingforplanetsthatloseallatmosphereduringtransitionfromasub-Neptunetoasuper-Earth.The line of revival sweeps towards the star over time because the rate of volcanicdegassing fallsoffmoreslowlywith time thandoes thestar’sXUV flux.The locationsofthesecurveschangedependingonmodelassumptions.Increasingvolatilesupplywillmovethresholds tohigherL.Allowingvolatile supply to increasewithdecreasingTeq,which isrealistic, will steepen gradients with L. The gray line is for a constant Earth-scaledoutgassing rate (72bars/Gyr for6M⊕).(b).As (a),but showingchangesover time.Thegray lines are for 72bars/Gyr outgassing at different star masses; from top to bottom,{0.3,0.6,1.0}M☉.