1 Exoplanet secondary atmosphere loss and revival 2 3 Edwin S. Kite1 & Megan Barnett1 4 1. University of Chicago, Chicago, IL ([email protected]). 5 6 Abstract. 7 8 The next step on the path toward another Earth is to find atmospheres similar to those of 9 Earth and Venus – high-molecular-weight (secondary) atmospheres – on rocky exoplanets. 10 Many rocky exoplanets are born with thick (> 10 kbar) H2-dominated atmospheres but 11 subsequently lose their H2; this process has no known Solar System analog. We study the 12 consequences of early loss of a thick H2 atmosphere for subsequent occurrence of a high- 13 molecular-weight atmosphere using a simple model of atmosphere evolution (including 14 atmosphere loss to space, magma ocean crystallization, and volcanic outgassing). We also 15 calculate atmosphere survival for rocky worlds that start with no H2. Our results imply that 16 most rocky exoplanets interior to the Habitable Zone that were formed with thick H2- 17 dominated atmospheres lack high-molecular-weight atmospheres today. During early 18 magma ocean crystallization, high-molecular-weight species usually do not form long-lived 19 high-molecular-weight atmospheres; instead they are lost to space alongside H2. This early 20 volatile depletion makes it more difficult for later volcanic outgassing to revive the 21 atmosphere. The transition from primary to secondary atmospheres on exoplanets is 22 difficult, especially on planets orbiting M-stars. However, atmospheres should persist on 23 worlds that start with abundant volatiles (waterworlds). Our results imply that in order to 24 find high-molecular-weight atmospheres on warm exoplanets orbiting M-stars, we should 25 target worlds that formed H2-poor, that have anomalously large radii, or which orbit less 26 active stars. 27 28 Introduction. 29 30 The Solar System has three planets – Earth, Venus, and Mars – that have atmospheres 31 derived from H2-free solids (secondary atmospheres), and four giant planets whose 32 atmospheres are derived from H2-dominated protoplanetary nebula gas (primary 33 atmospheres) (Atreya et al. 1989, Marty et al. 2016). However, this clean separation in 34 process and outcome is apparently unrepresentative of the known exoplanets. 35 36 The two most common types of known exoplanet, the rocky super-Earth-sized planets 37 (planet radius Rpl < 1.6 R⊕, where “⊕” is the Earth symbol, planet density ρpl > 4 g/cc; 38 “super-Earths”) and the gas-shrouded sub-Neptunes (Rpl = 2 - 3 R⊕), are divided by a valley 39 in (planet radius)-(orbital period) space in which planets are less common (Fig. 1) 40 (Fulton et al. 2017). The radius valley can be understood if, and only if, a substantial 41 fraction of planets that are born with thick (>10 kbar) H2-dominated primary atmospheres 42 lose those atmospheres and shrink in radius to become rocky super-Earths (Van Eylen et al. 43 2018). This radius-shrinking process, which carves out the radius valley, may be the way 44 that most super-Earths form. There is no evidence that Earth and Venus underwent this 45 process, and primary atmospheres are not thought to have contributed to the origin of

1 46 major volatile elements in Earth (Dauphas & Morbidelli 2014). Both Venus and Earth have 47 secondary volatile envelopes (comprised of solid-derived volatiles, including H2O and CO2) 48 that are much less massive than the atmospheres of sub-Neptunes. Large surface reservoirs 49 of H2O, C species, and N species are essential to life on Earth and to Earth’s habitable 50 climate (Bergin et al. 2015, Catling & Kasting 2017). 51 52 It is not known whether or not being born with a thick primary atmosphere that is 53 subsequently lost – forming as a gas-shrouded sub-Neptune but ending up as a rocky 54 super-Earth – favors the planet ending up with a secondary atmosphere, like Venus and 55 Earth. Do primary atmospheres, in dying, shield high-molecular-weight species from loss 56 during the early era of atmosphere-stripping impacts and intense stellar activity, allowing 57 those constituents to later form a secondary atmosphere? Or does the light (H2-dominated) 58 and transient primary atmosphere drag away the higher-molecular weight species? Getting 59 physical insight into the transition from primary to secondary atmospheres is particularly 60 important for rocky exoplanets that are too hot for life. Hot rocky exoplanets are the 61 highest signal/noise rocky targets for upcoming missions such as James Webb Space 62 Telescope (JWST) (Kempton et al. 2018) and so will be the most useful for checking our 63 understanding of this atmospheric transition process. 64 65 Secondary atmospheres hold a central place in the exoplanet exploration strategy 66 (National Academy of Sciences 2018, Zahnle & Catling 2017). Nevertheless, to our 67 knowledge, this is the first study of the transition on exoplanets from primary to secondary 68 atmospheres (Urey 1951, Ozima & Zahnle 1993). 69 70 Gas survival during planetary volume reduction. 71 72 During the sub-Neptune to super-Earth conversion process, we suppose the planet 73 contains both nebular-derived H2 and also high-μ species (derived from the planet-forming 74 solid materials) that – if retained – could form a secondary atmosphere. Retention of 75 volatiles is controlled by atmospheric loss and atmosphere-interior exchange (Fig. 2). For 76 both sub-Neptunes and super-Earths, silicates (both magma and solid rock) apparently 77 make up most of the planet’s mass (e.g. Dai et al. 2019, Owen and Wu 2017). 78 79 Atmosphere loss is more difficult for high-molecular-weight volatiles than for H2 because 80 higher-molecular-weight volatiles such as H2O are more easily shielded within the silicate 81 interior and are also more resistant to escape. 82 83 This can be understood as follows. First, a basic control on atmosphere loss is the ratio, λ, of 84 gravitational binding energy to thermal energy: 85 86 λ = (μ / k Tua) (G Mpl / (R + z ) ) 4 87 ≈ 2.3 μ (10 K / Tua)( Mpl / 6 M⊕)/( (R + z ) / 2 R⊕) (1) 88 89 where μ is molecular mass, k is Boltzmann’s constant, Tua is upper-atmosphere 90 temperature, G is the gravitational constant, Mpl is planet mass, R is the radius of the silicate 91 planet, and z is atmosphere thickness. When λ ≲ 2, the upper atmosphere flows out to

2 92 space at a rate potentially limited only by the energy available from upper-atmosphere 93 absorption of light from the star (Watson et al. 1981). If the upper atmosphere absorbs 94 100 W/m2 of light from the star, the upper limit on loss rate is ~1000 bars/Myr. This 95 hydrodynamic outflow ejects the tenuous upper atmosphere at high speed, but the dense 96 lower part of the atmosphere remains close to hydrostatic equilibrium. When λ > 10, the 97 hydrodynamic outflow completely shuts down. For primary atmospheres, the mean 98 molecular weight μavg ≈ 2 Da (H2) while for secondary atmospheres, μavg is ≳10× higher 99 favoring secondary atmosphere retention. Moreover, for secondary atmospheres Tua is 100 lower, as many secondary-atmosphere constituents (e.g CO2) are much more effective 101 coolants than H2 (e.g. Kulikov et al. 2007, Johnstone et al. 2018). Atmospheric thickness z is 102 also smaller for high-μavg atmospheres due to their smaller scale height. For both secondary 103 atmospheres and primary atmospheres, proximity to the star drives λ to lower values by 104 increasing Tua and also increases the upper atmosphere outflow rate when the λ ≲ 2 105 condition is satisfied. Moreover, closer to the star impacts occur at higher velocities that 106 are more erosive (Kegerreis et al. 2020). Thus we expect massive worlds far from the star 107 to retain atmospheres and low-mass worlds closer to the star to lose them (Zahnle & 108 Catling 2017). This expectation, while physically valid, offers little guidance as to whether 109 or not super-Earths will have atmospheres. To go further, we need to consider the effect of 110 evolving atmospheric composition on loss rate (Eqn. 1) and also track the shielding of 111 volatiles within silicates (magma oceans and solid rock). 112 113 The atmospheres of young sub-Neptunes are directly underlain by magma oceans (e.g. 114 Vazan et al. 2018; Fig. S1). Equilibrium partitioning of a volatile i between the atmosphere 115 (where it is vulnerable to escape) and the magma ocean (where it is shielded from escape) 116 is set by 117 118 ci = ei + [ (ei / Amai) gmai (μavg/ μi ) ] si mmagma + other reservoirs (2) 119 120 where ci is the total inventory of the volatile, ei is the mass in the atmosphere, Amai (and 121 gmai) are the area of (and gravitational acceleration at) the magma-atmosphere interface, -1 122 μavg is the mean μ of the atmosphere, si is the solubility coefficient (mass fraction Pa ) of the 123 volatile in the magma, and mmagma is the mass of liquid in the magma ocean. The term in 124 square brackets in Eqn. (2) corresponds to the partial pressure of i at the magma- 125 atmosphere interface. Solubility in magma is higher for many secondary-atmosphere 126 constituents (e.g. CH4, H2S, H2O) than for H2. When the atmosphere contains both H2 and a 127 more-soluble, high-μ constituent, the H2 is less protected by shielding within the magma, so 128 the H2 takes the brunt of atmosphere loss processes that might otherwise remove the high- 129 μ species. Greater solubility in magma also protects high-μ species from impact shock 130 (Genda & Abe 2005). Other reservoirs include volatiles stored in the crystal phase. This is 131 an unimportant reservoir early on when the planet has a massive magma ocean, but 132 becomes important later (for mature rocky planets, such as the Earth, the rock is almost 133 entirely solid). We neglect volatiles that are stored in the iron core, which are permanently 134 entombed and do not contribute to the atmosphere. 135 136 Atmospheric loss typically drives magma ocean crystallization (Fig. S1) (except for planets 137 that either orbit uncommonly close to their star, or undergo strong tidal heating, bare-rock

3 138 planets do not have a global molten-rock layer). The greenhouse effect from the H2 keeps 139 the magma ocean liquid for longer, which delays partitioning of the volatile into crystals. 140 This partitioning is described by 141 142 Xxtl = Di Xi (3) 143 144 where Xxtl is the concentration of i in the crystal phase, Di is a solid-melt distribution 145 coefficient / partition coefficient, and Xi is the concentration of i in the magma. 146 Crystallization enriches the residual melt in volatiles (i.e. Di << 1; e.g. Elkins-Tanton 2008). 147 148 One last effect can aid retention of high-molecular-weight volatiles. If outflow to space is 149 slow enough for the high-μ species to sink to lower levels within the atmosphere (diffusive 150 separation driven by buoyancy) then the outflow to space does not remove the high-μ 151 species (e.g. Hunten et al. 1987, Hu et al. 2015; Supplementary Information 1g). 152 153 Eqns. (1)-(3) open two paths from a primary to a secondary atmosphere. A secondary 154 atmosphere can be exsolved as the magma ocean crystallizes. Alternatively, the 155 atmosphere can be revived by release of high-μ gases during volcanism associated with 156 fractional melting long after the mantle has almost entirely solidified (Fig. 2). We explore 157 these paths below. 158 159 Model of the transition from primary to secondary atmospheres. 160 161 We model the effect of the loss of a thick primary atmosphere composed primarily of H2 on 162 the retention (or loss) of a hypothetical species, s. Species s has molecular mass 30 Da, 163 which is within a factor of <1.5 of the molecular mass of all known major secondary- 164 atmosphere constituents. We assume that during atmosphere evolution reactions between 165 s and H2 do not affect the mass inventory of either species; in effect, s is chemically inert. 166 While idealized, the model includes many effects that (to our knowledge) have not 167 previously been incorporated into a model of atmosphere loss, such as a realistic rock 168 melting curve, differential solubility effects, e.t.c. (Supplementary Information). 169 170 Planet equilibrium temperature Teq (in K), for zero planet albedo, is given by 171 1/4 172 Teq = 278 (F/F⊕) (4) 173 2 2 174 where F is insolation (W/m ) and the Sun’s insolation at Earth’s orbit, F⊕, is 1361 W/m . 175 Upper-atmosphere temperatures Tua > Teq are essential for a Super-Earth atmosphere to 176 flow out to space: this is possible because upper atmospheres efficiently absorb light at 177 wavelength ≲ 100 nm, but do not readily re-emit this light. 178 -12 -1 179 We adopt sH2 = 2 × 10 Pa (Hirschmann et al. 2012; Supplementary Information 1e), 180 which for initial atmospheric pressure Patm,init = 50 kbar gives a total mass of H2 181 (atmosphere plus dissolved-in-magma) of 7 × 1023 kg (2% of planet mass). We assume the 182 crystal-melt partition coefficients are DH2 = 0 (for simplicity) and Ds = 0.02 (Supplementary 183 Information 1f).

4 184 185 We consider a range of F and Patm,init within which all worlds start with a global shell of 186 magma but none retain it following sub-Neptune to super-Earth conversion. Planet thermal 187 structure (below the photosphere, which is isothermal by assumption) is as follows 188 (Fig. S2). The temperature at the top of the magma ocean, Tmai, is given by the atmosphere 189 adiabat 190 γ 191 (Tmai/TRCB) = (Patm/PRCB) (5) 192 193 where TRCB is the temperature at the radiative-convective boundary within the atmosphere, 194 PRCB is the pressure level defining the radiative-convective boundary, and γ is the adiabatic 195 index. We assume TRCB/Teq = 1 and that the temperature at the top of the magma is equal to 196 the temperature at the bottom of the atmosphere. (If TRCB/Teq = 1.5 in a 1000 K orbit then 197 the planet cooling timescale is only < 1 Kyr). This model absorbs the planet cooling rate and 198 the radiative opacity of the atmosphere into a single parameter, PRCB (Supplementary 199 Information 1c); for more sophisticated models, see e.g. Marcq et al. (2017). As Patm 200 decreases, Tmai cools, and the magma ocean crystallizes. The magma ocean starts to 201 crystallize at great depth and the crystallization front sweeps upward over time (e.g. 202 Elkins-Tanton 2008). Volatiles enriched near the crystallization front due to the low 203 solubility of volatiles in crystals will be stirred by magma currents (which can be as fast as 204 10 m s-1, Solomatov 2015) up to the near-surface, where they form bubbles that pop and 205 add their gas to the atmosphere. Stirring and bubbling can efficiently degas the mantle to a 206 depth of ~100 GPa and perhaps deeper (e.g. Caracas et al. 2019; Solomotov 2015, and 207 references therein). A smaller portion of s will go into the crystals. This portion is then 208 available for later volcanism. We do not include liquid volatiles (e.g. clouds) or fluid-fluid 209 immiscibility, which makes the model most suitable for super-Earths too hot for life. 210 211 Volcanic outgassing after the magma ocean crystallizes completely is guided by the results 212 of Kite et al. (2009) (Fig. S3) (Supplementary Information 1h). 213 214 We approximate diffusive separation of H2 and heavy gases as zero during sub-Neptune to 215 super-Earth conversion, for which escape proceeds too quickly for the s to settle out (Hu et 216 al. 2015) (Supplementary Information 1g). On worlds that are cool enough for life, diffusive 217 separation is important (Supplementary Information 1g). The secondary atmospheres that 218 may be exsolved as the magma ocean crystallizes form more quickly (Elkins-Tanton 2011) 219 than volcanically-outgassed atmospheres, which are produced on solid-state mantle 220 homogenization timescales (τ > 1 Gyr; e.g., Gonnerman & Mukhopadhyay 2009, Saji et al. 221 2018). Because volcanic degassing of terrestrial planets is so slow/inefficient (e.g. Kaula 222 1999), we consider magma-ocean exsolution separately from re-release of s from the solid 223 mantle. Specifically, we set re-release of s from the solid mantle to zero so long as the 224 exsolved atmosphere is present. With these approximations, for a given insolation L, planet 225 mass Mpl, initial dose of s, and Patm at which that dose is applied, the output depends on how 226 much atmosphere has been removed, but not on the speed (or process) of removal. In other 227 words, the equations are time-independent. 228 229

5 230 The small planet evolution sequence. 231 232 We first model atmospheric evolution during magma ocean crystallization (Figs. 3-4). We 233 emphasize 6 M⊕ (≈1.6 R⊕) results, which correspond to the largest (and therefore highest 234 signal/noise) planets that commonly have densities consistent with loss of all H2. The initial 235 conditions are atmospheric pressure Patm,init = 50 kbar, and total mass of the high-μ volatile 21 236 (Ms) = 3 × 10 kg. This corresponds to the near-surface C inventory of Earth and Venus 237 (~100 bars of CO2-equivalent), scaled up to the 6 M⊕ exoplanets we model (and it is 2× the 238 mass of Earth’s ocean). 239 240 We drive the model by decreasing Patm. In Figure 3, as PH2 (blue dashed lines) falls, the 241 magma ocean crystallizes, so that remaining volatiles go into the atmosphere (green), go 242 into the solid mantle (maroon), or escape to space (black). We found that the main controls 243 on the transition on exoplanets from primary to secondary atmospheres are F and the 244 solubility (ss) of the high-μ atmosphere constituent. 245 -11 -1 246 Fig. 3a shows a case with low ss (10 Pa , sparingly soluble), for a planet close to its host 247 star (F/F⊕ = 240; Teq = 1100 K; typical for Kepler super-Earths). The magma ocean stays 248 fully liquid until the atmospheric mass is reduced by 90%. (Release of dissolved-in-magma 249 H2 by bubbling is a negative feedback on atmospheric loss.) Crystallization begins at Patm = 250 5 kbar, and completes at ~200 bars. Throughout the calculation, most of the s is in the 251 atmosphere. s is passively entrained (either as atoms or molecules) in the escaping H2. We 252 stop the run at Patm = 1 bar, so we do not track the removal of the last bit of the exsolved 16 -11 -1 253 atmosphere. In this limit of small ss, only 6 × 10 kg × (ss / 10 Pa ) × ( Ds / 0.02 ) is 254 shielded within the solid mantle, potentially available for later outgassing. Most of the s is 255 lost to space before crystallization begins. The outcome is a bare rock with a volatile- 256 starved solid mantle, incapable of much volcanic outgassing. 257 258 Fig. 3b shows results for the same ss as in Fig. 3a and a cooler orbit (F/F⊕ = 3, Teq = 370 K, 259 intermediate between Mercury and Venus in our Solar System). In the cooler orbit, some 260 crystals are present initially. With no fully-liquid stage, crystallization can start to shield s 261 within crystals as soon as atmospheric loss starts. The s available for later volcanic 262 outgassing in the cooler orbit case is 30× greater. 263 -9 -1 264 Raising ss to 10 Pa (equivalent to 1 wt% solubility for 100 bars partial pressure) favors 265 secondary atmosphere occurrence (Figs. 3c-3d). More s is dissolved in the magma, and so 266 more s partitions (during crystallization) into the protective custody of the solid mantle. 267 Because s is much more soluble in the magma than H2, very little s is initially in the 268 atmosphere. Therefore, relatively little s is carried away to space during H2 removal and so 269 more s is available for exsolution during the final crystallization of the magma ocean. For 270 the hot orbit case (Fig. 3c), the final extrusion of volatiles creates an atmosphere with high- 271 μavg (the green line crosses the blue line in Fig. 3c). Such a high-μavg atmosphere is easier to 272 retain. The small amount of H2 that remains can be lost by diffusion-limited escape. 273 274 The combined effects of F and ss on the chance of secondary atmosphere occurrence are 275 shown in Fig. 4.

6 276 277 Fig. 4a shows the amount of the high-molecular-weight species that is shielded within rock 278 and so available for late volcanic outgassing. For high ss, almost all of the s is in the magma 279 until the magma ocean has almost completely crystallized, so the mass of shielded volatiles 280 tends towards the product of the initial inventory of s and the solid-liquid partition 281 coefficient Ds. For F/F⊕ ≤ 25 and low ss < (μs/μH2)sH2, both the amount of the high- 282 molecular-weight species that is shielded within rock and the magma concentration of s are 283 proportional to ss. For high F and low ss, the amount of the high-molecular-weight species 284 that is shielded within rock decreases rapidly for hotter orbits. In hotter orbits the H wind 285 can carry away s for a long time before the magma ocean cools enough for s-protective 286 crystals to form. 287 288 Fig. 4b shows how atmospheric μavg (after crystallization of the magma ocean is complete) 289 depends on F and ss. When μavg is high, loss to space is less likely (Eqn. 1 and surrounding -11 290 discussion). For ss < 10 , most of the s is stored in the atmosphere, and so the atmospheric -11 291 s number mixing ratio is near-constant during atmosphere loss. For ss > 10 , most of the s 292 is stored in the magma, and a greater fraction of the ss is stored in the magma than is H2. As 293 a result, the atmosphere becomes more s-rich as the magma ocean crystallizes. At F/F⊕ < 3 294 (not shown), diffusive separation increasingly favors high- μavg atmospheres 295 (Supplementary Information 1 g). 296 297 Overall, cooler orbits favor volcanic outgassing but hotter orbits permit exsolved high- 298 molecular-weight atmospheres. These effects have opposing sign in terms of the chance of 299 seeing a secondary atmosphere on a super-Earth. To determine their relative importance 300 requires time-dependent analysis (discussed next). 301 302 Where are planets today on the small planet evolution sequence? 303 304 We map planet evolution onto time and host-star mass (Fig. 5). The atmosphere loss that 305 converts sub-Neptunes into super-Earths could be driven by high-energy photons from the 306 star (photoevaporation), by impact erosion, or by accretion energy (e.g. Owen & Wu 2017, 307 Gupta & Schlichting 2019, Kegerreis et al. 2020). Here we consider atmospheric 308 photoevaporation driven by wavelength ≲ 100 nm photons; X-ray and Extreme Ultraviolet -3 309 flux, or XUV (Supplementary Information 1a). FXUV plateaus at ~10 × total F for planets 310 around young stars, switching to a power-law decay at < 0.1 Gyr for Solar-mass stars (FXUV -6 311 = 3 × 10 F at Earth today). The plateau of high FXUV/F is extended for planets orbiting red

312 dwarf (M) stars (≥ 0.3 Gyr long for ≤ 0.5 M☉). Therefore, we expect planets orbiting M-stars 313 to have lost more atmosphere (e.g. Lammer et al. 2007). 314 315 We connect FXUV to the small planet evolution sequence via the atmospheric loss rate, 316 dMatm/dt (Supplementary Information 1b). Nebula-composition atmospheres do not cool 317 efficiently, leading to high Tua that favors hydrodynamic escape (Murray-Clay et al. 2009). 318 For nebular-composition atmospheres, we set dMatm/dt proportional to XUV flux: 319 2 320 dMatm/dt = ε Rpl (R+z(t) ) FXUV / (G Mpl) (5)

7 321 322 where ε is an efficiency factor. For high-μ atmospheres we adopt the loss fluxes of a CO2 323 atmosphere (Tian 2009) as an example of a strong coolant. Tian (2009) finds low ε for CO2 324 atmospheres and a shut-off in hydrodynamic escape below a critical XUV flux (for -2 325 Mpl = 6 M⊕) of FXUV < 0.6 W m (= 150× the value on Earth today) (Fig. S4). At intermediate 326 compositions we interpolate using a logistic function in log(fs) (Supplementary 327 Information 1b). 328 -9 329 Fig. 5 shows atmosphere thickness versus time for a Solar-mass star (with ss = 10 ). High- 330 μavg atmospheres can only persist at a narrow range of F (Fig. 5). (A similar pattern is seen 331 for low-mass stars; Figure S6). Worlds in cooler orbits (upper tracks) are too far from the 332 star to receive much XUV and stay as sub-Neptunes. Worlds in hotter orbits lose their 333 atmospheres completely. They may undergo a rapid increase in μavg (blue to yellow in Fig. 334 5a), but the resulting high-μavg atmosphere is almost always short-lived. Why is this? 335 According to the pure-CO2 model of Tian (2009), dMatm/dt for high molecular weight 336 atmospheres on super-Earths has a threshold at ~150× FXUV/FXUV,⊕, below which loss is 337 much slower. This defines a zone of protection for high molecular weight atmospheres 338 (gray shaded region in top right of Fig. S5). But in most cases the primary atmosphere is 339 lost early, when FXUV is still very high (Fig. S5). As a result, any exsolved secondary 340 atmosphere is swiftly lost. For low ss, the atmosphere has a composition that is always H2- 341 dominated (by number), so exsolved high-μ species are lost in the H wind (Fig. 5b). 342 343 We conclude that exsolved atmospheres are rare. This conclusion is robust because we use 344 parameters that are favorable for an exsolved atmosphere. For example, we use a high 345 solubility-in-magma for the high-μ species, but our high-μ escape-to-space 346 parameterization is for a species (CO2) whose solubility-in-magma is low. 347 348 Revival of secondary atmospheres by volcanic outgassing. 349 350 If both the initial H2 atmosphere and any transient exsolved high μavg atmosphere have 351 been lost, then atmosphere revival will require volcanic outgassing. A possible example of 352 volcanic revival of the atmosphere after the planet has turned into a bare rock is Mars 353 ~4 Ga (Tian et al. 2009). Fig. 6 shows the region within which volcanic outgassing could 354 build up a secondary atmosphere. We use a time-dependent rate-of-volcanism model for 355 super-Earths (Kite et al. 2009; Supplementary Information 1h) that is tuned to the rate of 356 CO2 release at Earth’s mid-ocean ridges (12±2 bars/Gyr; Tucker et al. 2018). The rate of 357 outgassing is adjusted downward to account for loss of volatiles during sub-Neptune-to- -9 358 super-Earth conversion at ss = 10 . We neglect atmosphere re-uptake to form minerals 359 (e.g., Sleep & Zahnle 2001), because our focus is on worlds that are too hot for aqueous 360 weathering. This omission is conservative relative to our conclusion that volcanically- 361 outgassed atmospheres on hot rocky exoplanets are uncommon. These results are sensitive 362 to further reductions in ss, and to the choice of XUV flux models (Figs. S10-S11). 363 364 Fig. 6b shows the balance between volcanic outgassing and hydrodynamic escape taking 365 into account the expected decline over time of super-Earth volcanism (Kite et al. 2009) -9 366 (Fig. S8, Supplementary Information 1h). This calculation assumes ss = 10 (for which

8 367 ~90% of high-μ volatiles are loss to space during the loss of the H2-dominated

368 atmosphere). Volcanic revival of the atmosphere is difficult for planets around M = 0.3 M☉ 369 stars, but easier for rocky planets around solar-mass stars. Fig. 6 also shows the 370 atmosphere presence/absence line for rocky worlds that start with no H2, and with all 371 high-molecular-weight volatiles in the atmosphere. Such “intrinsically rocky” worlds have 372 secondary atmospheres over a wider range of conditions than do worlds that start as sub- 373 Neptunes. 374 375 Discussion. 376 377 For a given initial volatile content, the parameter to which our model is most sensitive is 378 the rate of XUV-driven mass loss. The XUV flux of young Solar-mass stars varies between 379 stars of similar mass by a factor of 3-10 (e.g. Tu et al. 2015). Fig. S11 shows how our results 380 change if the efficiency of photoevaporation is reduced by a factor of ten: a much wider 381 range of planets can have volcanically supported atmospheres. Escape of N2 or H2O is likely 382 faster than the Tian (2009) loss rate estimate (which is for pure-CO2 atmospheres) that is 383 used in our model (e.g. Zahnle et al. 2019, Johnstone et al. 2018). Improved knowledge of 384 escape rate will require more escape rate data and XUV flux data (e.g. Bourrier et al. 2018, 385 Ardila et al. 2018). 386 387 Our model considers only thermal loss. However, solar wind erosion can remove 388 atmospheres that are already thin (Dong et al. 2018). A possible example is Mars over the 389 last ~4 Ga. Planetary dynamos can under some circumstances suppress solar-wind erosion 390 (e.g. Gunnell et al. 2018). Mars had a dynamo prior to 4 Gya, and Earth (but not Venus) has 391 one today. 392 393 Some gases dissolve sparingly in magma (like the low-ss case shown in Figs. 3ab), and other 394 gases dissolve readily in magma (like the high-ss case shown in Figs. 3cd). Carbon-bearing 395 volatile species are very insoluble in magma. H2O’s solubility in basaltic magma at H2O -10 -1 396 partial pressure 0.03 GPa is ~2 wt%; linearizing, this gives ss = 7 × 10 Pa 397 (e.g. Papale 1997). Figs. S9-S11 show how the revived-atmosphere zone shrinks when ss is 398 reduced 100-fold (which is similar to the effects of a 100-fold reduction in Ds). Better 399 constraints on solubility at T > 2000 K are desirable (e.g. Guillot & Sator 2011). 400 401 Volcanism on super-Earths is expected to cease 1-10 Gyr after planet formation. The slow- 402 down of volcanism is paced by the mode of mantle convection (stagnant lid vs. plate 403 tectonics), by planet mass, and by variations in radiogenic-element abundance and mantle 404 mineralogy, among other factors (e.g. Stevenson 2003, Unterborn et al. 2015, Dorn et al. 405 2018, Byrne 2019; Supplementary Information). If volcanic degassing is too easy, then 406 volatiles will be released from the protective custody of the mantle before atmosphere loss 407 process have lost their bite. If volcanic degassing is too difficult, the atmosphere will be too 408 thin to detect. Offsetting this potential fine-tuning issue is a buffering effect: if the mantle is 409 volatile-rich, then magma is produced more easily (Médard & Grove 2006), but if the 410 mantle is volatile-poor, then the melt rate is reduced (reducing the rate of volatile loss).

9 411 How these effects combine is not clear and both effects are ignored in our model. Io, which 412 is volcanically hyperactive and is ~4 Gyr old, is still outgassing SO2. 413 414 Overall, our choices of Mpl, Ds, and ss tend to increase the chance of volcanic revival of 415 atmospheres on rocky exoplanets that form with thick H2 envelopes. Because even with 416 these choices atmospheres are not stable for planets at Teq > 500 K around M-stars, our 417 conclusions are broadly pessimistic for atmospheres on rocky exoplanets at Teq > 500 K 418 around M-stars. However, this pessimistic conclusion is moderated by the possibility 419 (discussed next) that worlds with abundant H2O (waterworlds) exist close to the star. 420 421 Observational tests. 422 423 Atmospheres on super-Earth-sized exoplanets can now be detected (e.g. Demory et al. 424 2016, Kreidberg et al. 2019). Theory predicts that retaining an atmosphere should be 425 harder on planets orbiting low-mass stars and the present study extends that prediction to 426 super-Earths that form as sub-Neptunes. A test of this theory would have major 427 implications for habitable zone planets. If this prediction fails, then either 428 photoevaporation is not the main driver of atmospheric loss, or M-star rocky exoplanets 429 formed more volatile-rich than rocky exoplanets orbiting Solar-mass stars 430 (Tian & Ida 2015). 431 432 Worlds that start with more volatiles than can be removed by loss processes will have 433 atmospheres today. XUV-driven loss can remove at most a few wt% of an M = 6 M⊕ planet’s 434 mass for Teq < 1000 K. We do not know the initial complement of high-μ species for small 435 exoplanets, although they can be estimated (e.g. Bitsch et al. 2019). Theory predicts worlds 436 with 1 wt % - 30 wt% H2O (waterworlds), which are hard to distinguish from bare-rock 437 planets using current data. Our model implies that a planet in the “no atmosphere” zone of 438 Fig. 6 with a JWST detection of H2O-dominated atmosphere is a waterworld. Possible 439 waterworlds include planets, such as HD 3167 b, that have radii ~0.2 R⊕ greater than 440 expected for Earth-composition (Fig S12), consistent with a H2O atmosphere of 1 wt% of 441 planet mass (Turbet et al. 2019). 442 443 Fig. 6 enables the following testable predictions. Since atmospheres close to the star can 444 only persist if the initial H2O and C inventory is high, N2/NH3 should be diluted to very low 445 mixing ratio for such atmospheres. Volatile supply should increase with semimajor axis, so 446 if a super-Earth-sized planet has an atmosphere, then planets at greater semimajor axis in 447 the same system should also have an atmosphere. Starting out as a sub-Neptune is 448 unfavorable for atmosphere persistence, so systems containing only rocky planets (where 449 it is more likely that the planets formed intrinsically rocky) should have statistically more 450 atmospheres. Multi-planet systems enable strong tests because uncertainty in time- 451 integrated stellar flux cancels out. 452 453 Conclusions. 454 455 A large fraction of rocky exoplanets interior to the Habitable Zone were born with thick 456 (>10 kbar) H2-dominated (primary) atmospheres but have since lost their H2. In our model

10 457 these exoplanets almost never transition smoothly to worlds with high-molecular weight 458 atmospheres (Fig. 7). Instead, the high-molecular-weight species are usually carried away 459 to space by the H wind during magma ocean crystallization. Volcanic outgassing is an 460 alternative source for a high-molecular-weight atmosphere. Atmosphere revival gets easier 461 with time, because atmosphere loss processes slow down rapidly with time, but 462 atmosphere supply by volcanism decays slowly. But volcanic outgassing is also enfeebled 463 by early loss of biocritical volatiles via the H wind. Our model omits feedback between 464 mantle volatile content and the rate of volcanism. Overall, atmospheres are less likely on 465 rocky exoplanets that were born with thick H2-dominated atmospheres. The most 466 uncertainty for our model is the initial planet volatile content. Within our model, for 467 planets that orbit Solar-mass stars, super-Earth atmospheres are possible at insolations

468 much higher than for planet Mercury. For planets that orbit ~0.3 M☉ stars, secondary 469 atmospheres at much higher insolation than planet Mercury in our solar system are 470 unlikely unless the planet formed H2-poor, or includes a major (~1 wt%) contribution of 471 solids from beyond the water ice line (waterworld). 472 473 Significance statement. 474 475 Earth and Venus have significant atmospheres, but Mercury does not. Thousands of 476 exoplanets are known, but we know almost nothing about the atmospheres of rocky 477 exoplanets. A large fraction of the known rocky exoplanets were formed by a sub-Neptune 478 to super-Earth conversion process during which planets lose most of their H2-rich 479 (primary) atmospheres and are reduced in volume by a factor of >2. It remains unclear 480 whether such a gas-rich adolescence increases or decreases the likelihood that super- 481 Earths will subsequently exhibit a H2-poor (secondary) atmosphere. Here we show that 482 secondary atmospheres exsolved from the magma ocean are unlikely to be retained by 483 super-Earths, but it is possible for volcanic outgassing to revive super-Earth atmospheres. 484 Especially for M-dwarf planetary systems, super-Earths that have atmospheres close to the 485 star likely were formed with abundant volatiles (waterworlds). 486 487 Acknowledgements: B. Fegley, Jr., L. Schaefer, L. Rogers, E. Ford, and J. Bean.

11 488 Figures.

Figure 1. The exoplanet abundance histogram (gray line, for orbital periods < 100 days, corrected for detection biases; Fulton & Petigura 2018) defines two classes of small exoplanet, volatile-rich sub-Neptunes and rocky super-Earth-sized exoplanets. Sub-Neptune to super-Earth conversion may be the way that most super-Earths form. 489

12

Figure 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.

13 -11 -1 -9 -1 (a) ss =10 Pa , F/F⊕ = 240 (b) ss =10 Pa , F/F⊕ = 3

-9 -1 -9 -1 (c) ss =10 Pa , F/F⊕ = 240 (d) ss =10 Pa , F/F⊕ = 3

Figure 3. The small planet evolution sequence, for 6 M⊕. Black triangles correspond to total initial inventory of high-μ species. (a) Solubility of high-μ species in magma -11 -1 (ss) =10 Pa , insolation normalized to Earth’s insolation (F/F⊕) = 240 (planet -11 -1 equilibrium temperature 1095 K). (b) ss =10 Pa , F/F⊕ = 3 (planet equilibrium -9 -1 -9 -1 temperature = 360 K). (c) ss =10 Pa , F/F⊕ = 240. (d) ss =10 Pa , F/F⊕ = 3. 490

14 (a)

(b)

Fig. 4. (a) How the mass of volatiles shielded within the solid mantle depends on F and ss. (b) How atmospheric mean molecular weight (after crystallization of the magma ocean is complete) depends on F and ss. 491

15 (a)

(b)

Figure 5. Time-dependent results for planets orbiting a Solar-mass star. (Fig. S6 -9 shows the results for a 0.3 M☉ star.) (a) Atmospheric pressure vs. time for ss =10 -1 Pa , (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 -11 -1 ss =10 Pa . 492

16 493 (a)

494 (b) 495 Figure 6. Secondary atmosphere presence/absence diagrams for 6 M⊕ (higher planet mass favors 496 atmosphere retention). Fig. S8 shows further details. (a) The dash-dot lines correspond to the line 497 of atmosphere vanishing for planets that have all volatiles in the atmosphere initially. The solid and 498 dashed lines correspond to the line of atmospheric revival by volcanic outgassing for planets that 499 lose all atmosphere during transition from a sub-Neptune to a super-Earth. The line of revival 500 sweeps towards the star over time because the rate of volcanic degassing falls off more slowly with 501 time than does the star’s XUV flux. The locations of these curves change depending on model 502 assumptions (Figs. S9-S11). Increasing volatile supply will move thresholds to higher L. Allowing 503 volatile supply to increase with decreasing Teq, which is realistic, will steepen gradients with L. The 504 gray line is for a constant Earth-scaled outgassing rate (72 bars/Gyr). (b). As (a), but showing 505 changes over time. The gray lines are for 72 bars/Gyr outgassing at different star masses.

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19 617 Tu, L., et al., 2015, A&A, 577, L3 618 619 Tucker, J.M., Mukhopadhyay, S., & Gonnerman, H.M., 2018, EPSL, 496, 108-119. 620 621 Turbet, M., et al., 2019 arXiv:1911.08878 622 623 Unterborn C. T., Johnson J. A., Panero W. R., 2015, ApJ, 806, 139 624 625 Urey, H., 1951 GCA 1, 209-277 626 627 van Summeren, J., Conrad, C. P., & Gaidos, E. 2011, ApJL, 736, L15 628 629 Van Eylen, V., et al. 2018 MNRAS 479, 4786-4795 630 631 Vazan, A., et al. 2018, ApJ 869:163. 632 633 Watson, A., et al. 1981, Icarus 48, 150-166 634 635 Zahnle, K., & D. Catling 2017, ApJ 843:122 636 637 Zahnle, K., et al. 2019, Geochim. Cosmichim. Acta 244, 56-85. 638 639 Supplementary Information. 640 641 1. Model description. 642 643 Atmosphere evolution involves many processes (Catling & Kasting 2017). Because this is 644 (to our knowledge) the first paper on the transition on exoplanets from primary to 645 secondary atmospheres, our approach is to keep the treatment of each process simple. 646 647 We consider star ages from 5 Myr to 8 Gyr. 5 Myr is a typical time for nebular disk

648 dispersal. We consider star masses from 0.1-1 M☉, and planet masses of 6-10 M⊕ (bare-

649 rock radius ≈ 1.6-1.9 R⊕). We frequently state results for 0.3 M☉, because these are the 650 most common host stars for worlds detected by the TESS mission (e.g. Sullivan et al. 2015, 651 Huang et al. 2018). The silicate mass fraction is held fixed at 2/3, the other 1/3 of planet 652 mass consists of the Fe/Ni metal core. This silicate mass fraction is based on Solar System 653 data and is consistent with data from contaminated white dwarfs, moreover our results are 654 not very sensitive to reasonable variations in the silicate mass fraction. The total mass of H2 655 is small compared to planet mass (Lopez & Fortney 2014). The non-H2 volatile mass 656 fraction is also assumed to be small, consistent with data (e.g. van Eylen et al. 2018). 657 658 1a. Drivers of atmosphere loss. 659 660 Candidate drivers for the atmosphere loss that converts sub-Neptunes into super-Earths 661 are photoevaporation, impact erosion, and core luminosity (Inamdar & Schlichting 2016, 662 Zahnle & Catling & 2017, Burger et al. 2018, Ginzburg et al. 2018, Owen & Wu 2017, 663 Biersteker & Schlichting 2019). We focus on photoevaporation in this study. 664 Photoevaporation has been directly observed for sub-Neptune exoplanets (Bourrier et al. 665 2018), fits almost all of the data (e.g. Van Eylen et al. 2018), and is relatively well- 666 understood (e.g. Murray-Clay et al. 2009). Other mechanisms are important at least for a 667 small number of planetary systems (e.g. Owen & Estrada 2019, Loyd et al. 2019). 668 669 Photoevaporation is controlled by the X-ray / Extreme Ultraviolet (XUV) flux, FXUV (e.g. 670 Murray-Clay et al. 2009, but see also Howe & Meyer 2019). XUV is produced from hot 671 regions near the star’s surface. These hot regions are heated by magnetic fields whose

20 672 importance declines as the star sheds angular momentum, through the stellar wind, over 673 time. Thus the star’s XUV luminosity FXUV declines over time. Early in a star’s history, the 674 star’s XUV luminosity LXUV reaches a plateau value (referred to as “saturation”) of between 675 10-4× and 10-3× the star’s total, bolometric luminosity (L). 676 677 Data on LXUV is synthesized by Linsky (2019). Estimating LXUV involves combining direct 678 measurements with interpolation over wavelength regions where star UV is absorbed by 679 the interstellar medium. Lopez & Rice (2018) and Luger & Barnes (2015) propose power- 680 law decays for Sun-like stars. Tu et al. (2015) quantify the variability of X-ray emission for 681 stars of Solar mass. Guinan et al. (2016) compile LXUV data for low-mass stars. 682 683 We used two approaches to interpolating in LXUV/L as a function of star mass and time. In 684 the first approach, we used Figure 5 from Selsis et al. (2007) for the X-ray flux, obtaining 685 the XUV flux from the X-ray flux using the top row of Table 4 in King et al. (2018). -3.2 686 Selsis et al. (2007) assume a constant LXUV/L of 10 during saturation. Selsis et al. (2007) 687 do not attempt to quantify the variation of X-ray flux between stars of the same age and the 688 same stellar mass. The 1-standard-error scatter is 0.4-0.5 dex according to 689 Loyd et al. (2019), motivating our sensitivity test shown in Fig. S11. In the second

690 approach, we used Jackson et al.’s (2012) fits for LXUV/L for M ≥ 0.5 M☉, patching to Guinan

691 et al.’s (2016) fits for M < 0.5 M☉. We assumed that Guinan et al.’s (2016) fits applied for

692 M = 0.35 M☉ and extrapolated to other M-stars. For both approaches, we used standard 693 models for L and star radius (Baraffe et al. 2015) (Fig. S13). The LXUV/L output from the two 694 approaches is shown in Fig. S14. The zones of secondary atmosphere loss and revival from 695 the first approach are shown in Fig. S9 (and Fig. 6), and the zones obtained from the second 696 approach are shown in Fig. S10. 697 698 We use an upper photon-wavelength cutoff of 91.2 nm (13.6 eV), which is the lowest- 699 energy photon that can ionize H. However, longer-wavelength light can still contribute to 700 atmospheric escape, especially as μavg rises. Future work using more sophisticated escape 701 models (e.g. Wang & Dai 2018) might investigate the effect of switching on or off various 702 UV spectral bands at lower energies than 13.6 eV. 703 704 1b. Atmosphere loss rates. 705 706 For pure-H2 atmospheres, we set 707 2 708 dMatm/dt = ε Rpl (R+z) FXUV / (G Mpl) (S1) 709 710 (Eqn. 5 from main text) where the heating efficiency factor, ε = 0.15. Models indicate that 711 ε = 0.1-0.2 (e.g. Shematovich et al. 2014), and ε = 0.15 is a common choice in atmospheric 712 evolution models. As the planet’s atmosphere loses mass and cools, it shrinks (Lopez & 713 Fortney 2014). This acts as a negative feedback on loss in part because smaller planets 714 intercept fewer XUV photons. To relate the atmosphere mass Matm to the planet radius 715 (R+z), we interpolated in the transit-radius tables of Lopez & Fortney (2014). The

21 716 homopause radius could be bigger than the transit radius by typically 5-25% (Malsky & 717 Rogers 2020). 718 719 For our pure-high-molecular-mass atmosphere endmember loss rate, we use the loss rate 720 in molecules/sec calculated for a pure-CO2 atmosphere on a super-Earth by Tian (2009). 721 This is an endmember of a high-μavg atmosphere that can self-cool efficiently. Our intent is 722 to bracket the likely down-shift in atmospheric loss as the atmosphere evolves to high-μavg. 723 We extrapolate beyond the XUV-flux limits shown in Figure 6 of Tian (2009) linearly in loss 724 rate for very large XUV fluxes, and linearly in the log of loss rate for very small XUV fluxes. 725 High-μavg atmospheres of different composition may shed mass faster (or slower) than the 726 pure-CO2 case (Johnstone 2019, Johnstone et al. 2018, Zahnle et al. 2019). 727 728 For atmospheres of intermediate composition, we interpolate between the energy-limited 729 formula (dMatm/dt ∝ FXUV) to the exponential cutoff in FXUV for high-μavg atmospheres 730 (dMatm/dt ≈ 0 below a critical FXUV) found by Tian (2009). Constraints for hydrodynamic 731 loss rates of atmospheres of intermediate composition are extremely limited (Kulikov et al. 732 2007, Johnston et al. 2018). In the absence of better constraints, we use a logistic curve in 733 log (escape rate) and log (mixing ratio) : 734 735 Xs,atm = (μavg – μp) / (μs – μp); (S2) 736 737 Y = 1 / (1 + exp(-k1 log10 Xs,atm – log10k2 ) ) (S3) 738 739 with k1 = 8 and k2 = 0.2. We think that these parameter choices understate the XCO2 needed 740 to suppress escape (i.e. favor the survival of high-μ atmospheres), which is conservative in 741 terms of our pessimistic conclusions for atmosphere survival. Next we set 742 743 Z = log10(dMatm/dt)p - Y log10 (dMatm/dt)p - log10 (dMatm/dt)s (S4) 744 Z 745 (dMatm/dt)mixed = 10 (S5) 746 747 where (dMatm/dt)p is the energy-limited escape rate with ε = 0.15, (dMatm/dt)s is from 748 Tian (2009), and (dMatm/dt)mixed is the loss rate for impure atmospheres. 749 750 1c. Planet thermal structure. 751 752 Planet thermal structure in our model is defined by the temperature at the atmosphere’s 753 radiative-convective boundary (TRCB) and the temperature at the magma-atmosphere 754 interface, Tmai. TRCB and Tmai are related by the thickness of the convecting (adiabatic) 755 envelope, which (in pressure units) is (Patm – PRCB), and by the adiabatic index γ (held 756 constant at 4/3). We do not consider changes in γ with changes in atmospheric 757 composition. 758 759 Planets form hot and cool over time. Cooling involves energy transport through the 760 atmosphere by radiation and/or convection. If, at a given optically thick level of the

22 761 atmosphere, all cooling is accomplished by radiation, then the radiative temperature 762 gradient is 763 3 2 764 dT/dr = – (3 κ ρ / 15 σT ) (Lint /4 πr ) (S6) 765 766 (e.g. Rogers et al. 2011), where κ is the local Rosseland-mean opacity, ρ is the local 2 767 atmospheric density, T is local temperature, σ is the Stefan constant, and (Lint /4 πr ) is the 768 internal luminosity corresponding to the cooling of the planet (where r is the distance from 769 the center of the planet). If the radiative dT/dr from Eqn. (S6) exceeds the convective 770 adiabatic temperature gradient, then convection takes over and dT/dr is reduced to a value 771 slightly greater than the adiabatic lapse rate. Thus, as the planet cools, and Lint decreases, 772 the radiative zone expands (PRCB moves to greater depths) (e.g. Bodenheimer et al. 2018, 773 Vazan et al. 2018). We set PRCB = 10 bars and neglect PRCB change with time, which means 774 our magma oceans are slightly longer-lived than they would be in reality. On the other 775 hand, we use the 1 Gyr Teq value throughout the calculation, when for cool stars Teq will be 776 higher earlier in the planet’s life. This simplification tends to shorten model magma ocean 777 lifetime relative to reality. 778 779 We assume full redistribution of energy absorbed on the dayside to the entire (4π) area of 780 the planet. Our model planets are also isothermal with altitude above the radiative- 781 convective boundary. In effect, we neglect the temperature difference between the planet 782 photosphere (level at which the optical depth is approximately unity) and the radiative- 783 convective boundary. At the top of the atmosphere we assume zero albedo. This is 784 reasonable for cloud-free atmospheres of planets orbiting cool stars, although for Solar- 785 mass stars the albedo will be ~0.2, considering only CO2(g) and H2O(g) opacity (Pluriel et al. 786 2019). 787 788 Our focus is interior to the habitable zone and for thick atmospheres. For such worlds 789 atmospheric retention would result in magma oceans that last Gyr (Hamano et al. 2013, 790 Vazan et al. 2018, Fig. S1). Thus, interior to the habitable zone, atmosphere loss is the rate- 791 limiter for magma ocean crystallization (Hamano et al. 2013, Hamano et al. 2015, Lebrun et 792 al. 2013). In principle our model could lead to an equilibrium situation where a few- 793 hundred-bar high-µ atmosphere sustains a magma ocean indefinitely. However such 794 outcomes in our model require fine-tuning. 795 796 During sub-Neptune-to-super-Earth conversion, planet cooling is driven by atmospheric 797 removal. As the overburden pressure is removed, the remaining atmosphere cools 798 adiabatically. This causes a temperature gradient between the magma and the atmosphere, 799 and heat flow from the magma into the atmosphere soon warms the atmosphere because 800 the atmosphere heat capacity is small compared to the magma ocean heat capacity. This 801 raises TRCB, and because TRCB/Teq > 1 cools the planet quickly, this cools the planet. All of 802 these processes occur continuously in our simplified model: we assume the magma 803 temperature tracks the loss of atmosphere. In effect we assume that convection swiftly 804 reduces super-adiabatic temperature gradients and that the boundary layer temperature 805 contrast at the top of the liquid magma (viscosity < 10-1 Pa s) is small. More detailed models

23 806 of sub-Neptune thermal evolution include Rogers et al. (2011), Howe & Burrows (2015), 807 Vazan et al. (2018), and Bodenheimer et al. (2018). 808 809 For cooler orbits in our model, some crystals are present initially. This incomplete-magma- 810 ocean initial condition is appropriate if the planet is assembled by impacts of objects that 811 are Mars-sized or smaller (e.g. planetesimals, pebbles, or planetary embryos), and the 812 planet cools down to the adiabat between impacts. 813 814 1d. Magma ocean crystallization. 815 816 Magma ocean crystallization shields dissolved volatiles from loss. The magma ocean 817 crystallizes (losing mass to crystals) as the planet cools. The magma ocean mass at a given 818 Tmai depends on conditions deep within the silicate layer (higher T and much higher P than 819 at the magma-atmosphere interface). The silicate solidus (T vs. P for 0% melt fraction) and 820 liquidus (T vs. P for 100% melt fraction) are curved in T-P space. As a result, for a steady 821 decline in Tmai, the magma mass will decrease first quickly and then slowly. To track this 822 effect, we used the solidus, the liquidus, and the adiabat of Andrault et al. (2011) (Fig. S15). 823 We assume that the melt fraction increases linearly with increasing temperature between 824 the solidus and the liquidus, and neglect chemical fractionation during crystallization. We 825 also neglect the reduction in the solidus T due to volatile enrichment as crystallization 826 proceeds. More detailed models of magma ocean crystallization include 827 Nikolau et al. (2019) and Katyal et al. (2019). Our model predicts whole mantle melting for 828 Tmai ≥ 3000 K with negligible melt below Tmai ~ 1750 K. Melting curves differ between 829 experimenters and differ depending on assumed mantle composition (e.g. Andrault et al. 830 2017, Miyazaki & Korenaga 2019). To relate pressure to depth, we use the pressure-density 831 curve from Dziewonski & Anderson (1981). To get the radius at the magma-atmosphere 0.27 832 interface we assume (Rpl/R⊕) = (Mpl/M⊕) (Valencia et al. 2006). We integrate downward 833 from the magma-atmosphere interface. Starting at Patm and Tmai, we follow the adiabat until 834 we reach the solidus. The melt mass is the mass of melt (if any) between the magma- 835 atmosphere interface and the depth of the magma solidus. If this melt mass exceeds 2/3 of 836 planet mass, the melt mass is set equal to 2/3 of planet mass. The effect of the weight of the 837 atmosphere on the location of the solidus is included in our calculations, but turns out to be 838 unimportant. 839 840 1e. Redistribution of volatiles between atmosphere, magma, and rocks. 841 842 We assume full re-equilibration between magma and atmosphere as the atmosphere is 843 removed (efficient degassing). Whether or not a fully liquid magma ocean undergoing 844 whole-magma-ocean convection will degas as the H2 envelope is removed and the liquid 845 cools depends on (a) the number of times each magma parcel cycles through the upper 846 thermal boundary layer during cooling, and (b) the ratio of the degassed boundary layer 847 thickness to the thermal boundary layer thickness. Suppose the degassed boundary layer 848 thickness is equal to the maximum depth at which bubbles can form (this assumes swift 849 bubble ascent, and that supersaturation is unimportant). For sub-Neptune to super-Earth 850 conversion, this is (up to 10s kbar)/(ρmagma g) = up to 50 km. The thermal boundary layer

24 851 thickness for a fully liquid magma ocean will be ≪50 km. Each fluid parcel will go through 852 the upper thermal boundary layer at least once during cooling. Therefore, degassing is 853 efficient if the cooling magma ocean is fully convective (Ikoma et al. 2018). 854 855 We calculate the concentration of dissolved gas in magma using Henry’s law. Many gases 856 show linear solubility in magma. Others do not. For example, H2O dissolves in magma in 857 proportion to the square root of partial pressure at low pH2O, becoming more linear at high 858 pH2O (e.g. Stolper 1982, Matsui & Abe 1986). 859 -12 -1 860 We adopt sH2 = 2 × 10 Pa . The basis for this is as follows. Hirschmann et al. (2012) 861 report results from laboratory experiments on basaltic melts. Extrapolating to melts of 862 peridotitic composition, they state that “at 1 GPa in the presence of pure H2, the molecular 863 H2 concentration [in the melt] is 0.19 wt%.” Fegley et al. (2020, their Table 5) compile 864 solubilities of various volatiles in molten silicates. 865 866 We ignore non-ideal solubility (due to non-ideal fugacity of the gas in the atmosphere), 867 because it is not very important at the relatively low Patm we consider (Kite et al. 2019). We 868 also ignore joint-solubility effects because relevant data are not available. Few 869 measurements have been made of the temperature dependence of volatile solubility in 870 magma at T > 2000 K (e.g. Fegley & Schaefer 2014, Guillot & Sator 2011) and we neglect T 871 dependence of volatile solubility. 872 873 Solid-melt distribution coefficients, D, for water partitioning into nominally anhydrous 874 mantle minerals vary from 10-4 to 10-1 (Table S1 in Elkins-Tanton 2008), but for the most 875 common mantle minerals are 0.02 or less. Here, “nominally anhydrous” means minerals 876 that do not have H in their chemical formula. We use D = 0.02, which is at the high end of 877 experimentally observed range (corresponding to water portioning into pyroxene). We 878 neglect saturation limits. C distribution coefficients are much lower, from 2 × 10-4 to 879 7 × 10-3 according to Elkins-Tanton (2008) and with low (1-5 ppmw) saturation limits. 880 881 1f. Magma-atmosphere coevolution. 882 883 Key initial conditions are that the quantity of the high molecular weight volatile initially in 884 the crystal phase is zero (sxtl = 0), and 885 886 es = Cs/(1 + (μp /μs )(gmai/Amai) mmagma ss) (S7) 887 888 which assumes that the atmosphere is dominantly μp at the zeroth timestep. 889 3 3 890 We restrict ourselves to F/F⊕ < 10 . For F/F⊕ > 10 , silicate vapor-pressure equilibrium 891 atmospheres develop (e.g. Kite et al. 2016). 892 893 Our model loops through the following steps: (Step 1) Lose some atmosphere. à (Step 2) 894 Post-escape outgassing. à (Step 3) Crystallize (sequestering some volatiles, but increasing

25 895 the concentration of volatiles in the magma). à (Step 4) Post-crystallization outgassing à 896 Loop back to Step 1. 897 898 Step 1. Lose atmosphere. A down-step in atmospheric pressure is prescribed. 899 No fractionation is permitted between H2 and the high-μ species during this escape 900 (see section 1g). The H2 and the high-μ species are lost in proportion to their bulk 901 abundance in the atmosphere. 902 903 Step 2. Post-escape outgassing. The magma and atmosphere now re-equilibrate through 904 outgassing of both H2 and s until the partial pressures of H2 and of s are in equilibrium with 905 the concentrations of H2 and s dissolved in the magma. The partial pressure of H2 depends 906 on the mass of H2 in the atmosphere and on the mean molecular weight of the atmosphere, 907 which is affected by the amount of s in the atmosphere. Similarly the partial pressure of s 908 depends on the mass of s in the atmosphere and on the mean molecular weight of the 909 atmosphere, which is affected by the amount of p in the atmosphere. From Eqn. (2) in the 910 main text, we obtain the coupled equations 911 912 Cs = es + es((es + ep)/(es/μs +ep/μp )) k1/μs (S8) 913 Cp = ep + ep((es + ep)/(es/μs +ep/μp )) k2/μp (S9) 914 915 where Cs (kg) is the total on-planet mass of s, Cp (kg) is the total on-planet mass of H2 (p), 916 es is the kg of s in the atmosphere, ep is the kg of H2 in the atmosphere, the second term on 917 the right-hand-side of (S8) is the mass (kg) of s dissolved in the magma, the second term on 918 the right-hand-side of (S9) is the mass (kg) of H2 dissolved in the magma, μs is the 919 molecular weight of s, μp is the molecular weight of H2, and 920 921 k1 = (gmai/Amai) mmagma ss (S10) 922 923 k2 = (gmai/Amai) mmagma sp (S11) 924 925 where gmai and Amai are (respectively) the gravity at, and area of, the magma-atmosphere 0.27 926 interface for (Rmai/R⊕) = (Mpl/M⊕) , mmagma is calculated as described in section 1d above, 927 ss is the solubility of s in the magma, and sp is the solubility of H2 in the magma. 928 929 The simultaneous equations (S10)-(S11) have two solutions, and the physical solution is 930 2 2 2 2 2 2 931 ep = (Cp k1 + Cs k2 + (k2 (Cp k1 μs - k1 (Cp k1 μp + 2 Cp k1 μp μs + Cp μs + 2 Cp Cs k1 2 2 932 k2 μp μs - 2 Cp Cs k1 μp + 4 Cp Cs k1 μp μs + 4 Cp Cs k2 μp μs - 2 Cp Cs k2 μs + 2 Cp Cs μp 2 2 2 2 2 2 0.5 2 933 μs + Cs k2 μs + 2 Cs k2 μp μs + Cs μp ) - Cs k1 μp + 2 Cs k2 μs + Cp k1 μp + Cs k1 2 2 2 2 2 934 k2 μs ))/(2 (k1 μp - k2 μs + k1 μp - k1 k2 μs )) + (k1 k2 (Cp k1 μs - k1 (Cp k1 μp + 2 Cp 2 2 2 935 k1 μp μs + Cp μs + 2 Cp Cs k1 k2 μp μs - 2 Cp Cs k1 μp + 4 Cp Cs k1 μp μs + 4 Cp Cs k2 μp μs 2 2 2 2 2 2 2 0.5 936 - 2 Cp Cs k2 μs + 2 Cp Cs μp μs + Cs k2 μs + 2 Cs k2 μp μs + Cs μp ) - Cs k1 μp + 2 2 2 937 Cs k2 μs + Cp k1 μp + Cs k1 k2 μs ))/(2 (k1 μp - k2 μs + k1 μp - k1 k2 μs )))/(k1 + k1 k2 ) 938 (S12) 939

26 2 2 2 2 2 2 940 es = -(Cp k1 μs - k1 (Cp k1 μp + 2 Cp k1 μp μs + Cp μs + 2 Cp Cs k1 k2 μp μp - 2 Cp Cs 2 2 2 2 2 941 k1 μp + 4 Cp Cs k1 μp μs + 4 Cp Cs k2 μp μs - 2 Cp Cs k2 μs + 2 Cp Cs μp μs + Cs k2 μs + 2 2 2 2 0.5 2 942 Cs k2 μp μs + Cs μp ) - Cs k1 μp + 2 Cs k2 μs + Cp k1 μp + Cs k1 k2 μs )/(2 (k1 μp - k2 μs 2 943 + k1 μp - k1 k2 μs )) 944 (S13) 945

946 We recalculate the other variables by inserting the new es and ep into Eqns. (S8) and (S9). 947 948 The coupling of Eqn. (S8) and Eqn. (S9) via μavg has two important effects that disfavor 949 secondary atmospheres on exoplanets. During the loss of the primary atmosphere, the μavg 950 effect increases the amount of the high-μ constituent in the atmosphere by a factor of 15 951 (= μs/μp), increasing the amount of the high-μ constituent that is lost to space. Moreover, as 952 μavg rises from 2 to 30, because the partial pressure of the high-μ constituent scales with 953 the number ratio of s in the atmosphere, fs = es / (ep+es)(μavg/μp), the number of moles of s in 954 the atmosphere for a given saturation vapor pressure of s decreases by a factor of 15 955 ( = μs/μp). This negative feedback reduces the number of worlds that smoothly transition 956 from a primary to a secondary atmosphere. 957 958 Step 3. Crystallize. The magma mass is forced to a lower value, consistent with the lower 959 Tmai associated with the atmospheric pressure that was reduced in Step 1. The freshly- 960 produced crystals gain an s concentration (in mass fraction) of Ds × Xs, where Xs is the mass 961 fraction of the volatile in the magma. The crystals are not remixed (by assumption) on the 962 timescale of magma-ocean crystallization. s within crystals is relatively safe from release 963 into the unsafe surface environment during sub-Neptune-to-super-Earth conversion, 964 because solid-state creep speeds within the solid mantle are 10-9 – 10-8 m s-1, versus up to 965 101 m s-1 in the magma. As a result, the concentration of the crystals is independent of the 966 previous partitioning of the volatiles into the solid rock. Partitioning of H2 into the crystals 967 is not considered. 968 969 Step 4. Post-crystallization outgassing. Because Xs < 1, the magma ocean is enriched in s 970 after the crystallization step. Therefore we re-equilibrate the magma and the atmosphere 971 taking account of the now-higher concentration of s (Xs) in the magma. 972 973 The processes involved in Steps 1-4 all occur on timescales short compared to the magma 974 ocean lifetime. We set the number of atmosphere-loss steps such that mass is conserved to 975 better than 1 part in 104. 976 977 1g. Calculating evolution over time. 978 979 To map the rocky planet evolution sequence onto time we combine the environmental 980 drivers from (1a), the loss rates from (1b), and the small planet evolution sequence from 981 (1f). TRCB is fixed in the crystallization calculation, but Teq evolves with star age. We deal 982 with this by setting TRCB throughout the calculation equal to Teq at a fixed age of 1 Gyr. Thus, 983 planets orbiting cool stars in our model have shorter-lived magma oceans than in reality. 984

27 985 Diffusive separation of H2 from high-µ species is important at lower XUV flux (e.g. 986 Hu et al. 2015, Wordsworth et al. 2018, Saito & Kuramoto 2018, Odert et al. 2018, 987 Malsky & Rogers 2020). However, diffusive separation is washed out at the high XUV fluxes 988 required if photoevaporation is to cause sub-Neptune-to-super-Earth conversion. Our 989 calculations follow Tian (2015) and Catling & Kasting (2017). First, we define the (atomic) 990 H flux, F1, on the assumption that the overwhelming majority of the upper atmosphere is 991 comprised of H: 992 993 F1 = ε FXUV (R + z) / (4 G Mpl m1) (S14) 994 995 in units of molecules s-1 m-2. Here, ε = 0.15 is the efficiency of conversion of XUV energy into 996 atmospheric escape, R is solid-planet radius, z is atmosphere thickness (which we set to 997 0.4 R to correspond to a small sub-Neptune), G is the gravitational constant, Mpl is planet 998 mass, and m1 is the mass (kg) of the hydrogen atom. 999 1000 From this we obtain the crossover mass mc (Tian 2015) 1001 1002 mc = m1 + k T F1 /(b g X1) (S15) 1003 1004 where T is the temperature of the heterosphere, and b = 4.8 × 1019 T 0.75 m-1 s-1 is the 1005 bindary diffusion coefficient, corresponding to neutral H and neutral O, used by Tian 1006 (2015). Including the effect of ionization of H would decrease the effective value of b (Hu et 1007 al. 2015) and so strengthen our conclusion that diffusive separation is not important 1008 during sub-Neptune to super-Earth conversion for Kepler’s super-Earths. We set T = 104 K, 1009 the same value used by Hu et al. (2015) and Malsky & Rogers (2020). 1010 1011 The flux, F2, of the high-µ species is zero when mc < m2. When mc > m2 the flux F2 is given by 1012 1013 F2 = F1 (X2 (mc – m2)) / ( X1 (mc – m1)) (S16) 1014 2 1015 Substituting in values for 6 M⊕ and atomic mass 15 Da we find FXUV = 0.7 W/m at 1016 crossover, 1.3 W/m2 for 50% reduction in escaping flux due to weight differences, and 1017 6.1 W/m2 for escaping flux of high-molecular weight species at 90% of the flux if there was 1018 no fractionation at all. All three thresholds are shown as dashed vertical lines on Fig. S5 and 1019 Fig. S6. In the context of sub-Neptune to super-Earth conversion, most of the atmosphere 1020 mass is removed at higher FXUV. Most exoplanets either stay as sub-Neptunes or lose their 1021 atmosphere too quickly for the XUV flux to drop to levels that permit >10% differences in 1022 loss rate due to fractionation. 1023 1024 This can be understood as follows. If the upper atmosphere absorbs 1 W/m2 of light from 1025 the star, the upper limit on loss rate for a 6 M⊕ world is ~10 bars/Myr, falling to 1026 1.5 bars/Myr at ε = 0.15. Since sub-Neptune-to-super-Earth conversion takes ~300 Myr, 4 1027 and ~5 × 10 bars of H2 need to be removed (including H2 that is initially dissolved in the 2 1028 magma), FXUV must be ~100 W/m . Using the saturation threshold of Selsis et al. (2007), -3.2 2 1029 (FXUV/F = 10 ) this corresponds to 160 kW/m , F/F⊕ = 120, for distance from the star = 1030 0.1 AU for a Sun-like star. This is the typical distance of a Kepler super-Earth from its host

28 1031 star (the abundance of 1.6 R⊕ ~ 6 M⊕ super-Earths drops off rapidly at larger separations 1032 (Fulton & Petigura 2018). At such high XUV fluxes, fractionation causes negligible 1033 differences in the loss rate relative to the no-fractionation case. Therefore, our 1034 approximation of no fractionation during sub-Neptune to super-Earth conversion is valid. 1035 1036 However, fractionation by diffusion is important is important for late-stage volcanically 1037 outgassed atmospheres. For example, the volcanic H2 outgassing flux is calculated to be less 1038 than the diffusively-limited H2 escape rate unless H2 stays at a very low level (e.g. Catling et 1039 al. 2001, Batalha et al. 2016, Zahnle et al. 2019). This effect maintains volcanically 1040 outgassed atmospheres at high μav, even if outgassing of H2 by serpentinization (Sleep et al. 1041 2004) and/or volcanic outgassing is large. 1042 1043 Fractionation by diffusion is also much more important for Habitable Zone super-Earth- 2 1044 sized planets. For Fbol = 1361 W/m (the Solar flux at 1 AU today), even at saturation FXUV is 1045 0.9 W/m2 , which is only marginally capable of lifting atomic mass = 15 Da species out of 1046 the atmosphere. This favors the occurrence of secondary atmospheres on Habitable Zone 1047 exoplanets. 1048 1049 Finally, marinating at moderate FXUV for Gyr may convert a H2-dominated sub-Neptune 1050 atmosphere into a He-enriched sub-Neptune (Hu et al. 2015, Malsky & Rogers 2020). 1051 1052 1h. Volcanic outgassing. 1053 1054 The model results are more sensitive to changes in the XUV flux than they are to changes in 1055 the rate of volcanic outgassing. This is due to the exponential cutoff in the XUV-flux-driven 1056 loss rate of CO2 (Tian 2009). 1057 1058 No exoplanets have yet been confirmed to be volcanically active. Nevertheless, rocky 1059 exoplanet thermal evolution, geodynamics, and volcanic activity is discussed in numerous 1060 theoretical papers (e.g. Dorn et al. 2018, and references therein). We use a simple model of 1061 the geodynamics and rate of volcanism on super-Earth-sized planets (Kite et al. 2009). The 1062 model includes parameterized mantle convection, thermal evolution, and volcanism for 1063 both plate tectonics and stagnant lid modes. The model is tuned to reproduce Earth’s 1064 present-day rate of volcanism in planet tectonics mode. Kite et al. (2009) consider three 1065 different melting models. In the present study, we use the results from the widely-used 1066 melting model of Katz et al. (2003). 1067 1068 The procedure for estimating the rate of volcanism starts from the results of 1069 Kite et al. (2009) (Fig. S3). This is an Earth-tuned parameterized mantle convection and 1070 rate of volcanism model that predicts the rate of volcanism versus time for either stagnant 1071 lid mode and plate tectonics mode. The rate of CO2 release at Earth’s mid-ocean ridges 1072 (12±2 bars/Gyr; Tucker et al. 2018) is multiplied by the computed rate of volcanism, in 1073 units of Earth’s present-day rate, to obtain the predicted rate of CO2 release on the rocky 1074 exoplanet. This is adjusted downward in proportion to any early loss of high-molecular- 1075 weight volatiles to space. For example, if a model exoplanet has lost 70% of its high- 1076 molecular-weight volatiles to space, then the rate of volcanic outgassing is reduced to 30%

29 1077 of the rate predicted by comparing the volumetric volcanic flux from the Earth-tuned code 1078 to that of the Earth. 1079 1080 Fig. 6 compares the broad zone of worlds with atmosphere for planets that form without 1081 thick H2 atmospheres to the much more restricted zone of volcanically revived 1082 atmospheres for worlds that form with thick H2 atmospheres. This calculation assumes ss = -9 1083 10 (for which ~90% of high-μ volatiles are loss to space during the loss of the H2- 1084 dominated atmosphere). Atmospheric ingassing is neglected because worlds well inside 1085 the habitable zone are too hot for weathering reactions to sequester atmophiles in the 1086 crustal rocks. Volatile release by volcanic outgassing is neglected, by construction, on 1087 worlds that start with all their volatiles in the atmosphere. Forming with a secondary 1088 atmosphere is advantageous for subsequently exhibiting a secondary atmosphere relative 1089 to the total-loss-and-subsequent-revival process. The stagnant-lid and plate-tectonics 1090 predictions diverge greatly after ~4 Gyr, when volcanism shuts down on the stagnant-lid 1091 model planet (Fig. S8). The plate-tectonics and stagnant lid predictions otherwise appear 1092 similar in this Earth-scaled model (Kite et al. 2009). However, this overall similarity of 1093 plate and stagnant-lid predictions is specific to the Earth-tuned model of Kite et al. (2009). 1094 Other models predict much stronger suppression of volcanism on stagnant-lid planets 1095 (e.g. Dorn et al. 2018). Tidal locking has little effect on volcanism 1096 (e.g. van Summeren et al. 2011). So long as we do not mechanistically understand 1097 volcanism versus time for Solar System worlds (Byrne et al. 2019), why Earth’s mantle took 1098 so long to mix, nor why Earth’s mantle is not yet completely outgassed, any detailed 1099 predictions of the rate of volcanism versus time for rocky exoplanets are questionable. 1100 Robust constraints (for non-tidally-heated planets) are decline of mantle temperature by 1101 50-200K per Gyr, with a concomitant decline in the potential for melting (Stevenson 2003). 1102 1103 We assume the rate of degassing is proportional to the rate of magmatism. This is a 1104 simplification. Even low-degree partial melts can effectively extract almost all the volatiles 1105 from the full mass of silicate mantle that sweeps through the partial melt zone. This is 1106 because volatiles partition strongly into the melt during partial melting (i.e., Di ≪ 1). Once 1107 the partial melt fraction is high enough (~1%) for the melt to form interconnected 1108 channels, the volatile bearing melt rises from the melt production zone geologically 1109 instantaneously (with no loss of volatiles) via filter-pressing, melt channel formation, and 1110 diking. Whether the melt gets close enough to the surface to form bubbles and/or erupt 1111 explosively will depend on the stress state of the lithosphere (e.g. Solomon 1978). If melt 1112 crystallizes at depth (intrusion) then volatiles will go into hydrated minerals or into the 1113 glass phase, where they could still be released to the atmosphere over geologic time. 1114 Explosive volcanic eruptions are aided by bubble formation, which is easier when the 1115 volatile content of the magma is high. 1116 1117 Extrapolation of simple models of Earth’s thermal evolution and outgassing rate back into 1118 Earth’s past leads to the expectation that Earth’s mantle was very rapidly stirred and 1119 quickly outgassed very early in Earth history – in contradiction to isotopic data that show 1120 sluggish stirring. Similarly, our simple Earth-tuned model of thermal evolution and 1121 outgassing rate indicates that 4-6× more volatiles outgas over the lifetime of the planet

30 1122 than are present at the beginning – a mass balance violation. Because an excess of 1123 volcanism over XUV-driven atmospheric loss leads (in our model) to build-up of a 1124 detectable atmosphere in <1 Gyr, the rate of volcanism is more important than the 1125 cumulative amount of volcanism in setting atmosphere presence/absence. Therefore we do 1126 not adjust our volcanic fluxes downward to take account of this effect, reasoning that real 1127 planets outgas more slowly than simple models predict. Alternatively, our results could be 1128 interpreted as saying that super-Earths erupt away their solid-mantle volatiles during the 1129 first Gyr of the star’s life, when XUV fluxes are still high. This would lead to an even more 1130 pessimistic conclusion for the likelihood of volcanically outassed atmospheres at distances 1131 much closer to the star than the habitable zone than is presented in Fig. 6. 1132 1133 1i. Supply and observability. 1134 1135 The inventory of CO2 even on today’s Earth is uncertain. For example, the review of Lee et 1136 al. (2019) reports a range of estimates for the C stored in the non-sedimentary rocks of 1137 Earth’s continental crust, from 4.2 × 107 Gton C to 2.6 × 108 Gton C. 1138 1139 No exoplanets have yet been confirmed to be waterworlds. Nevertheless, waterworld 1140 formation, migration, and (for habitable-zone waterworlds) climate evolution is discussed 1141 in numerous theoretical papers (e.g. Kite & Ford 2018, Bitsch et al. 2019, and references 1142 therein). 1143 1144 Another way to increase volatile mass is to ballast the hydrogen with oxygen obtained from 1145 the reaction FeOmagma + H2(g) à Fe + H2O(g); i.e., endogenic water which does not require 1146 planet migration (Kite et al. 2020). 1147 1148 Supplementary References. 1149 1150 Andrault, D., Bolfan-Casanova, N., Nigro, G. L., et al. 2011, EPSL, 304, 251 1151 1152 Andrault, D., Bolfan-Casanova, N., Bouhifd, M. A., et al. 2017, Phys. Earth & Planet Interiors, 1153 265, 67 1154 1155 Baraffe et al. 2015 Astronomy & Astrophysics, Volume 577, id.A42. 1156 1157 Biersteker, J. B., & Schlichting, H. E. 2019, MNRAS, 485, 4454 1158 1159 Bodenheimer, P., Stevenson, D. J., Lissauer, J. J., & D’Angelo, G. 2018, ApJ, 868, 138 1160 1161 Catling, D.C., K. J. Zahnle, and C. P. McKay, Science, 293, 839-843, 2001. 1162 1163 Dziewonski, A. M., & Anderson, D. L. 1981, Phys. Earth Planet. Int., 25, 297 1164 1165 Fegley, B., & L. Schaefer 2014 Chemistry of the Earth's Earliest Atmosphere. Treatise on 1166 Geochemistry, 2nd edition, chapter 6.3. 1167 1168 Fegley, B., et al., 2020, Volatile element chemistry during accretion of the Earth, 1169 Geochemistry: Chemie der Erde, in press. 1170 1171 Ginzburg, S., et al., Super-Earths, in Formation, Evolution, and Dynamics of Young Solar 1172 Systems, Astrophys. & Space Sci. Library, v.445, edited by M. Pessah & O. Gressel, Springer. 1173 1174 Guinan et al. 2016, ApJ, 821, Issue 2, article id. 81. 1175 1176 Hamano, K. et al. 2015, ApJ, 806, Issue 2, article id. 216. 1177 1178 Howe, A. R., & Burrows, A. 2015, ApJ, 808, 150 1179 1180 Howe & Meyer 2019, arXiv:1912.08820 1181

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1282 1283 Fig. S1. How magma ocean mass increases with atmospheric thickness. Output from a toy 1284 model of sub-Neptune thermal structure (Kite et al. 2020). Dashed lines correspond to 1285 magma ocean mass, labeled in Earth-masses of magma, for a volatile-free planet mass 1286 of 6 M⊕. Colored lines correspond to temperatures at the magma-atmosphere interface 1287 of 1500 K (maroon), 2000 K (red), 3000 K (orange), and 4000 K (yellow). Magma ocean 1288 masses in excess of 2 Earth masses are not plotted because this corresponds to a magma 1289 pressure range that is not well explored by experiments.

33 1290 1291 Fig. S2. Planet thermal structure (reproduced from Kite et al., 2020). Temperature versus 1292 pressure plot, showing adiabats within the atmosphere and the magma (black line). Trheo 1293 (white line) corresponds to the temperature of the rheological transition (~40% melt 1294 fraction) for rock (dashed white line at levels where no rock is present). RCB = radiative- 1295 convective boundary. Tmai = temperature at the magma-atmosphere interface. For real 1296 planets the equilibrium temperature (Teq), the effective temperature (Teff), and the 1297 temperature at the RCB (TRCB), are related by Teq < Teff ≲ TRCB. In this paper we make the 1298 approximation that these three temperatures are equal (Supplementary Information 1c).

1299 1300 Fig. S3. (Reproduced from Kite et al. 2009). Rate of volcanism per unit mass on massive 1301 Earth-like planets undergoing stagnant lid convection, normalized to calculated rate on a 1302 plate-tectonic Earth, for Katz et al. (2003) melting model. Dark gray shaded regions 1303 correspond to mantle temperatures associated with very intense volcanism, too high for a 1304 reliable crustal thickness calculation. Contours are at 0, 0.5, 1, 2, 5, and 10 times Earth’s 1305 present-day rate of volcanism (~4 × 10−19 s−1, equivalent to 24 km3 yr−1, in plate-tectonics 3 −1 1306 mode). For example, a 6 M⊕ planet on the “2” contour erupts 6 × 2 × 24 ≈ 300 km yr .

34 1307 1308 Fig. S4. Atmospheric loss rate for a pure CO2 atmosphere, combining results for super- 1309 Earths (Tian 2009) and for Mars (Tian et al. 2009, scaled to 1 AU). The vertical dashed line 1310 corresponds to 150× Earth’s present-day XUV flux.

1311 1312 Fig. S5. The same evolutionary tracks as in Fig. 5, shown in FXUV – μavg coordinates. F/F⊕ 1313 increases from lowest track (F/F⊕ = 49, Teq = 735 K) to uppermost track (F/F⊕ = 720, 1314 Teq = 1440 K). The dashed lines highlight (from top to bottom) the threshold of FXUV below 1315 which all of the high-molecular-mass species is retained by the planet; the FXUV 1316 corresponding to a 50% reduction in the no-fractionation loss rate of the high-molecular- 1317 mass species; and the FXUV corresponding to escape of the high-molecular-mass species at 1318 90% of the rate at which no loss would occur (Supplementary Information 1g). Right of the 1319 rightmost dashed line, fractionation protects the constituents of the secondary atmosphere, 1320 and left of the leftmost dashed line, fractionation is much less important. Calculations are 1321 done assuming an atomic wind of H entraining a 1% mixing ratio of atoms of mass 15 Da.

35 1322

1323 Fig. S6. (Top panel) As Fig. 5 (top panel), but for a planet orbiting a star of 0.3 Solar masses. 1324 Time-dependent results. Atmospheric pressure vs. time for planets for (from top to bottom) 1325 F/F⊕ = {49, 283, 347, 422, 720}, corresponding to planet equilibrium temperature (Teq) = {735, 1326 1140, 1200, 1275, 1440} K. (Bottom panel). As Fig. S5 but for a planet orbiting a star of 0.3 1327 Solar masses. The same evolutionary tracks as in the top panel, shown in FXUV – μavg coordinates. 1328 F/F⊕ increases from lowest track (F/F⊕ = 49, Teq = 735 K) to uppermost track (F/F⊕ = 720, Teq 1329 = 1440 K). The dashed lines highlight (from top to bottom) the threshold of FXUV below which 1330 all of the high-molecular-mass species is retained by the planet; the FXUV corresponding to a 1331 50% reduction in the no-fractionation loss rate of the high-molecular-mass species; and the 1332 FXUV corresponding to escape of the high-molecular-mass species at 90% of the rate at which no 1333 loss would occur (Supplementary Information 1g). Right of the rightmost dashed line, 1334 fractionation protects the constituents of the secondary atmosphere, and left of the leftmost 1335 dashed line, fractionation is much less important. Fractionation calculations are done assuming 1336 an atomic wind of H entraining a 1% mixing ratio of atoms of mass 15 Da.

36 1337 a) b)

1338 1339 c) d)

1340 1341 Fig. S7. Details of planet evolution for the tracks shown in Fig. 4. Solar-mass star, 6 M⊕. 1342 Line thickness corresponds to insolation, with the thickest lines corresponding to the 1343 greatest insolation. Results are shown for F/F⊕ = 49 (blue), F/F⊕ = 283 (red), F/F⊕ = 347 1344 (yellow), F/F⊕ = 422 (purple), and F/F⊕ = 720 (green), corresponding to Teq = {735, 1140, 1345 1200, 1275, 1440} K, respectively. (a) Planet radius versus time. (b) Planet equilibrium 1346 temperature versus time. (c) Planet mass loss rate versus time (solid lines), and the mass 1347 loss rate that the planet would have if the atmosphere was composed entirely of CO2 based 1348 on the calculations of Tian (2009) (dashed lines). The planets corresponding to the yellow, 1349 purple, and green lines evolve to a high-μ atmosphere, but only in the yellow case is this 1350 atmosphere long-lived. (d) The fraction of high-μ shielded within solid rock. This is zero for 1351 the two worlds that remain as sub-Neptunes (because no solid rock forms), and very 1352 similar for the three worlds that transition to super-Earths (full magma crystallization).

37 1353

a) b)

c) d)

1354 -9 1355 Fig. S8. Atmospheric thickness versus time for (a-b) volcanic revival, ss = 10 and 1356 (c-d) start-with-all-volatiles-in-the-atmosphere. There is a minor trend in plate-tectonics 1357 mode for planets around Solar-mass stars at Teq < 500 K toward decreasing atmospheric 1358 thickness with decreasing Teq. This is due to the smaller initial volume of the (s-protecting) 1359 magma ocean at low Teq. Note that Dorn et al. (2018)’s model predicts no volcanism at all 1360 on stagnant-lid super-Earths. If this prediction is correct, that would further narrow the 1361 range of conditions under which volcanism can support an atmosphere.

38 1362 1363 Fig. S9. As Fig. 6a, but showing more details for atmosphere presence/absence after 1364 3.0 Gyr. The dash-dot line corresponds to the line of vanished atmospheres for planets that 1365 have all volatiles in the atmosphere initially. This line moves away from the star over time. 1366 The solid and dashed lines correspond to the line of atmospheric revival by volcanic 1367 outgassing for planets that lose all atmosphere during transition from a sub-Neptune to a 1368 super-Earth. The lines of atmospheric revival sweep towards the star over time because the 1369 rate of volcanic degassing falls off more slowly with time than does the star’s XUV flux. This 1370 chart assumes initial volatile supply is independent of F and similar to Earth and Venus. 1371 Increasing volatile supply will move thresholds to higher F. The gray lines reduce the 1372 solubility of the high-molecular-weight volatile in the magma from 10-9 to 10-11. Selected 1373 exoplanets overplotted. Note that because most planets are too small to have Earth-like 1374 composition and 6 M⊕, this plot is optimistic for atmospheric survival (higher planet mass 1375 favors atmosphere retention).

39 1376 1377 Fig. S10. Small planet atmosphere presence/absence diagram after 3.0 Gyr. Substituting in 1378 an XUV-flux-vs.-time parameterization following the results of Jackson et al. 2012 and 1379 Guinan et al. 2016. This increases the range of exoplanets for which a volcanically 1380 supported atmosphere is possible relative to the baseline based on the estimates of Selsis 1381 et al. 2007. The dash-dot line corresponds to the line of vanished atmospheres for planets 1382 that have all volatiles in the atmosphere initially. This line moves away from the star over 1383 time. The solid and dashed lines correspond to the line of atmospheric revival by volcanic 1384 outgassing for planets that lose all atmosphere during transition from a sub-Neptune to a 1385 super-Earth. The lines of atmospheric revival sweep towards the star over time because the 1386 rate of volcanic degassing falls off more slowly with time than does the star’s XUV flux. This 1387 chart assumes initial volatile supply is independent of F and similar to Earth and Venus. 1388 Increasing volatile supply will move thresholds to higher F. The gray lines reduce the 1389 solubility of the high-molecular-weight volatile in the magma from 10-9 to 10-11. Note that 1390 because most planets are too small to have Earth-like composition and 6 M⊕, this plot is 1391 optimistic for atmospheric survival (higher planet mass favors atmosphere retention).

40 1392 1393 Fig. S11. Small planet atmosphere presence/absence diagram after 3.0 Gyr. Reduction in 1394 XUV flux by a factor of 10, or, equivalently, decreasing the efficiency of XUV-driven loss by a 1395 factor of 10, relative to the baseline based on the estimates of Selsis et al. 2007. This 1396 increases the range of exoplanets for which a volcanically supported atmosphere is 1397 possible. The dash-dot line corresponds to the line of vanished atmospheres for planets 1398 that have all volatiles in the atmosphere initially. This line moves away from the star over 1399 time. The solid and dashed lines correspond to the line of atmospheric revival by volcanic 1400 outgassing for planets that lose all atmosphere during transition from a sub-Neptune to a 1401 super-Earth. The lines of atmospheric revival sweep towards the star over time because the 1402 rate of volcanic degassing falls off more slowly with time than does the star’s XUV flux. This 1403 chart assumes initial volatile supply is independent of F and similar to Earth and Venus. 1404 Increasing volatile supply will move thresholds to higher F. The gray lines reduce the 1405 solubility of the high-molecular-weight volatile in the magma from 10-9 to 10-11. Note that 1406 because most planets are too small to have Earth-like composition and 6 M⊕, this plot is 1407 optimistic for atmospheric survival (higher planet mass favors atmosphere retention).

41 1408 1409 Fig. S12. Planet densities from Otegi et al. (2020). The blue line corresponds to density for 1410 Earth-like composition, assuming scaling of mass with planet radius as (R/R⊕) = 0.27 1411 (M/M⊕) . The red line corresponds to 80% of that density. Many super-Earth sized 1412 planets plot below the red line (for example HD 3167 b), indicating either a volatile 1413 envelope (atmosphere) or alternatively a very low (Mg+Fe)/Si ratio relative to that of the 1414 Earth. Super-Earths with the same or lower density as Earth (e.g. HD 3167 b) are excellent 1415 candidates for having elevated-molecular-weight secondary atmospheres. Planets plotting 1416 in the low-density, large radius clump in the bottom right are sub-Neptunes, retaining thick 1417 H2-dominated atmospheres. 1418

1419 1420 Fig. S13. Stellar luminosity, L, versus time from the model of Baraffe et al. (2015) for star

1421 masses ranging from 0.1 – 1.0 M☉, and normalized to the Sun’s luminosity at the present-

1422 day L☉. Contours drawn at intervals of 0.1 M☉. The blue line highlights 0.1 M☉. The red line

1423 highlights 0.3 M☉. The black line highlights 1.0 M☉.

42 (a)

(b) 1424 Fig. S14. FXUV at an insolation corresponding to a circular orbit 0.1 AU from today’s Sun, 1425 relative to the FXUV at Earth’s orbit today. (a) for the model of Selsis et al. (2007), and (b) 1426 combining the fits of Jackson et al. (2012) and Guinan et al. (2016). The dashed lines 1427 highlight (from top to bottom) the threshold of FXUV below which all of the high-molecular- 1428 mass species is retained by the planet; the FXUV corresponding to a 50% reduction in the no- 1429 fractionation loss rate of the high-molecular-mass species; and the FXUV corresponding to 1430 escape of the high-molecular-mass species at 90% of the rate at which no loss would occur. 1431 Above the top dashed line, fractionation protects the constituents of the secondary 1432 atmosphere, and below the lowest dashed line, fractionation is much less important. 1433 Calculations are done assuming an atomic wind of H entraining a 1% mixing ratio of atoms 1434 of mass 15 Da, and planet mass 6 M⊕. Jumps and wiggles in the curves are artifacts of 1435 interpolation.

43 1436 1437 Fig. S15. Andrault et al. (2011) melt curves. Colored lines are magma adiabats. Blue lines 1438 are melt fraction curves, where the lowest one is the solidus. We find that the effect of 1439 atmospheric overburden pressure on the solidus temperature is relatively small. X axis 1440 corresponds to depth within a hypothetical Earth-mass planet (the curves correspond to 1441 lines in P-T space, so they map to smaller depth on higher-gravity worlds).

44