The longevity of water ice on Ganymedes and Europas around migrated giant planets Owen R. Lehmer 1, David C. Catling 1, Kevin J.

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1 The longevity of water ie on Ganymede and Europa around migrated giant planet Owen R. Lehmer 1, David C. Catling 1, Kevin J. Zahnle 1 Dept. of Earth and Spae Siene / Atrobiology Program, Univerity of Wahington, Seattle, WA NASA Ame Reearh Center, Moffett Field, CA ABSTRACT The ga giant planet in the Solar Sytem have a retinue of iy moon, and we expet giant exoplanet to have imilar atellite ytem. If a Jupiter-like planet were to migrate toward it parent tar the iy moon orbiting it would evaporate, reating atmophere and poible habitable urfae oean. Here, we examine how long the urfae ie and poible oean would lat before being hydrodynamially lot to pae. The hydrodynami lo rate from the moon i determined, in large part, by the tellar flux available for aborption, whih inreae a the giant planet and iy moon migrate loer to the tar. At ome planet-tar ditane the tellar flux inident on the iy moon beome o great that they enter a runaway greenhoue tate. Thi runaway greenhoue tate rapidly tranfer all available urfae water to the atmophere a vapor, where it i eaily lot from the mall moon. However, for iy moon of Ganymede ize around a Sun-like tar we found that urfae water (either ie or liquid) an perit indefinitely outide the runaway greenhoue orbital ditane. In ontrat, the urfae water on maller moon of Europa ize will only perit on timeale greater than 1 Gyr at ditane ranging 1.49 to 0.74 AU around a Sun-like tar for Bond albedo of 0. and 0.8, where the lower albedo beome relevant if ie melt. Conequently, mall moon an loe their iy hell, whih would reate a toru of H atom around their hot planet that might be detetable in future obervation. 1. INTRODUCTION One of the major reult of exoplanet dioverie i that giant planet migrate (Chamber 009). Thi wa firt dedued from hot Jupiter, and although thee are found around 0.5-1% of Sun-like tar 1

2 (Howard 013), hot Jupiter are not the only planet to migrate and giant planet migration i likely widepread. Indeed, uh migration probably ourred in the early olar ytem (Tigani et al. 005). All the giant planet in the olar ytem have a olletion of iy moon. We expet that imilar exomoon orbit giant exoplanet and that thee moon would likely migrate along with their hot planet. If a giant exoplanet were to migrate toward it parent tar, iy moon ould vaporize, imilar to omet approahing the Sun, and develop atmophere. If uh a giant planet and iy moon were to form in the habitable zone of a tar (or migrate there hortly after formation) the high XUV flux from the young tar would rapidly erode the atmophere of moon many time the ma of Ganymede (). In addition, they ould melt and maintain liquid urfae a they migrate inward, whih ould be potentially habitable environment. Suh a moon would have an atmophere primarily ontrolled by the vapor equilibrium et by the urfae temperature and the rate of hydrodynami eape to pae. A uh, the longevity of the water hell and atmophere will depend primarily on the ditane to the hot tar and the exomoon radiu and ma. Several uh bodie exit in the olar ytem, where the atmopheri thikne i determined by vapor equilibrium with a ondened phae, i.e. the Clauiu-Clapeyron relation for the relevant volatile. Let u all uh atmophere Clauiu-Clapeyron (C-C) atmophere. The N atmophere on both Triton and Pluto are example of C-C atmophere, where the urfae vapor preure i in equilibrium with the N urfae ie at the prevailing temperature for eah body. The preent Martian atmophere i another C-C atmophere ine the polar CO ie ap at ~148 K buffer the atmophere to ~600 Pa urfae preure (Leighton & Murray 1966) (ee (Kahn 1985) for an explanation over geologi timeale). For an iy exomoon migrating toward it parent tar, the atmopheri water vapor will be ontrolled by the availability of urfae water and temperature. Very deep ie and ie-overed oean are poible on thee moon given that water an aount for ~5-40% of the bulk ma of iy moon in the olar ytem (Shubert et al. 004). However, the mall ma of exomoon and relatively high tellar flux

3 a the exomoon migrate toward the tar make water vapor ueptible to eape. Auming exomoon are of omparable ize to the moon found in the olar ytem, thi tudy look at the end-member ae of how rapidly a pure water vapor atmophere will be lot hydrodynamially during exomoon migration. The migration of exomoon i eential if iy moon of Ganymede ize are to retain their urfae water for more than 1 Gyr in the habitable zone of a Sun-like tar. If a Ganymede-like iy moon formed in the habitable zone of a tar (or migrated there hortly after formation) the high XUV flux from the young tar ould rapidly erode it atmophere (Heller, Marleau, & Pudritz 015; Lammer et al. 014). Therefore, thi tudy look at the longevity of urfae water on iy moon that migrate toward their hot tar after thi period of intene XUV-driven hydrodynami eape. Hydrodynami eape i a form of preure-driven thermal eape where the upper level of an atmophere beome heated and expand rapidly, aelerate through the peed of ound, and eape to pae en mae (Hunten 1990). An important proe in atmopheri evolution, hydrodynami eape likely ourred during the formation of the terretrial atmophere (Kramer & Toltikhin 006; Kuramoto, Umemoto, & Ihiwatari 013; Pepin 1997; Toltikhin & O'Nion 1994). Moreover, hydrodynami eape ha been oberved on exoplanet uh a the ga giant HD 09458b, whih orbit a Sun-like tar at 0.05 AU and ha hot H atom beyond it Rohe lobe, preumably depoited there by hydrodynami eape (Linky et al. 010; Vidal-Madjar et al. 004). The loer a body i to it parent tar, the more effetive the hydrodynami eape, and the maller the body, the more eaily an atmophere i lot (e.g., Zahnle & Catling (017)). The longevity of an atmophere and iy hell will depend primarily on temperature, et in large part by the tellar flux available for aborption. A an ie overed exomoon move toward it parent tar, heating will aue more water vapor to enter the atmophere, hatening the lo rate. In addition, thi water vapor will provide a greenhoue effet, further warming the moon. At a ertain exomoon-tar ditane the water vapor atmophere will impoe a runaway greenhoue limit on the outgoing thermal infrared (IR) flux from the exomoon. If the aborbed tellar flux exeed thi limit, the exomoon urfae 3

4 will heat rapidly until all available water i in the atmophere a vapor. Thi limit repreent the ditane at whih all urfae water will be tranferred to the atmophere where it will be rapidly lot. Figure 1. Coneptual viualization of the 1D hydrodynami eape model. The inoming aborbed tellar flux, given by 1 4 (1 AF, heat the exomoon urfae, for Bond albedo A, and tellar flux. The ) exomoon will remain in thermal equilibrium by evaporating water vapor, loing ma via hydrodynami eape, and radiating in the thermal infrared. The thermal infrared radiation i given by i the Stefan-Boltzmann ontant, T i the urfae temperature) in a blakbody approximation. The outward radial flow veloity, u, inreae monotonially until it urpae the iothermal peed of ound,, at the ritial radiu,, where the ga i till olliional. Beyond the oni level, u ontinue to rie and r oon urpae the eape veloity v eape with atmopheri nearurfae denity balane. and outward radial urfae veloity. The exomoon urfae i at radiu u r T 4 F ( u. See equation () for the global energy. METHODS We onider three ae of hydrodynami eape: (.1) an iothermal atmophere where the atmopheri temperature i et by inoming tellar flux and equal to the effetive temperature; (.) a vapor aturated atmophere where the temperature and humidity profile of the entire atmophere are ditated by the C-C relation; and (.3) an iothermal atmophere imilar to (.1) but the urfae temperature, and the iothermal atmopheri temperature, are inreaed from the effetive temperature by 4

5 the total greenhoue warming of the water vapor atmophere. We hoe the iothermal and C-C ae beaue they preent upper and lower limit on the rate of hydrodynami eape, repetively, a deribed below. Figure 1 provide a oneptual piture of the model. For a pure water vapor atmophere around a Sun-like tar, an iothermal atmophere at the effetive temperature repreent the greatet poible temperature at the top of atmophere to drive hydrodynami eape. Water vapor radiate in the IR more effiiently than it aborb unlight, o the radiative-onvetive temperature for a pure water vapor atmophere will be le (Pierrehumbert 010; Robinon & Catling 01). A uh, the iothermal atmopheri approximation provide an upper bound on the atmopheri lo rate. In ontrat, the lowet poible temperature to drive hydrodynami eape i the aturated ae, where the temperature and preure at all height are et by the C-C relation, whih i defined by T P T 0 1 ln P / P0 RT0 / Lv (1) for referene temperature T 0 at referene preure P 0, where R i the univeral molar ga ontant, and L v i the latent heat of vaporization for water (e.g., Pierrehumbert (010), p.100). The urfae temperature i aumed to be in equilibrium with the inoming tellar flux, ooling aoiated with ma lo via hydrodynami eape, and latent heat of evaporation. If the temperature were to dereae with altitude fater than the C-C relationhip, the water vapor would ondene out reulting in a C-C urve that, when extrapolated, would reult in a urfae temperature no longer in equilibrium with inoming tellar flux and eape. Therefore, the hydrodynami lo rate of a pure water vapor atmophere i bounded by the iothermal and aturated ae, whih we will now onider in turn..1. Iothermal Cae 5

6 In the blakbody approximation, radiative ooling i given by 4 T at iothermal temperature T, allowing a traightforward formulation of eape veru radiative ooling. A uh, the firt order global energy balane for an iy exomoon i between inoming tellar flux veru the energy flux lot to vaporizing the water, lifting moleule out of the gravity well, and radiative ooling, i.e., 1 GM 1 A F Lv u T 4 r aborbed tellar flux ma lo flux 4 radiative ooling () Here F i the inoming olar flux available for aborption, A i the Bond albedo, i the Stefan- Boltzmann ontant, r i the urfae radiu where the atmopheri denity i, uh that u i the ma flux given an outward radial flow veloity at the urfae u, G i the gravitational ontant, and M i the ma of the exomoon. In addition to the C-C relationhip [equation (1)], three equation are needed to derive the teady tate, hydrodynami atmopheri lo in the iothermal approximation (e.g., Catling & Kating (017), Ch. 5). The firt i teady tate ma ontinuity, given by r r u 0 (3) where r i the radial ditane from the planet enter, u i the outward radial flow veloity, and i the atmopheri denity. Steady tate momentum onervation i expreed a u 1 p u g r r (4) with gravity g GM / r and preure p. Finally, the equation for energy balane i given by equation (). Combining equation (1), (), (3), and (4) an analyti expreion for the iothermal atmopheri ma lo rate in kg -1 i given by (ee Appendix A for the derivation): 6

7 1 3 G 3 G 3 M M exp M 3 u 0 u0 4 m 3 (5) where u 0 i the iothermal ound peed given by u kt m with Boltzmann ontant k 0 / and mean moleular weight m, and m i the mean denity of the exomoon (aumed g m -3 ). The atmopheri urfae denity,, i et the by C-C equation for the aturation vapor preure of water at the prevailing temperature. For time averaged ma lo rate, M, the lifetime of the exomoon urfae water i given by M M Water Water (6) For an upper limit, we aume the total ma of water preent on the exomoon urfae, M Water, i 40% of the bulk ma. However, even if 5% water were ued [the lower limit for Europa (Shubert, et al. 004)] from equation (6) we an ee that it would tranlate to a hange in Water by a fator of 8, ompared to 40% water. From equation (5) we ee that M, and hene Water, ha an exponential dependene on ma, o we would antiipate that the differene between 5% and 40% water i not the major fator determining Water, whih i borne out by our reult. In addition, if ubtantial water vapor i lot the bulk denity of the moon, m, may inreae over time. However, from equation (5) we ee that the exponential term ale like 1/3 M /3 with /3 m M largely determining the lo rate o the enitivity to m i mall. It i important to note that in equation (5) we have aumed the ma lo rate, M, i uffiiently mall that energy balane i dominated by radiative lo. Thi i indeed the ae for exomoon of interet in thi paper, where the low temperature water vapor atmophere lat for more than 1 Gyr. The urfae 7

8 preure are well below ~500 Pa until the runaway greenhoue limit i reahed. For bodie with rapid hydrodynami eape the numerial approah defined in Appendix A i appropriate... Saturated Temperature Profile Cae The aturated ae i derived from the ame equation a the iothermal ae [equation (1), (), (3), and (4)] but temperature i allowed to hange with altitude. The temperature at the ritial point at radiu r in Figure 1 (where the iothermal ound peed u 0 equal the radial eape peed) i et uh that numerially integrating equation (1), (), (3), and (4) from the ritial point to the urfae will reult in a urfae temperature equivalent to that in equilibrium with inoming olar flux taking into aount the evaporative ooling (ee Appendix A for detail). One the ritial temperature i known the radial outflow veloity i readily alulated and thu the ma lo rate..3. Iothermal Cae with Greenhoue Effet Conidered In Cae (.1) we let the iothermal atmopheri temperature be et by jut the inoming tellar flux and thu be equal to the effetive temperature. However, for a thik water vapor atmophere the urfae will be heated by the greenhoue effet of the overlying atmophere. In thi ae, we till ued an iothermal atmophere approximation but inreaed the atmopheri temperature by the total greenhoue warming of the atmophere at the urfae. The larger iothermal atmopheri temperature under thi regime will inreae the hydrodynami lo rate ompared to Cae (.1). To aount for the atmopheri greenhoue effet, we ued a gray, radiative, plane-parallel approximation where the total gray atmopheri optial depth in the thermal infrared at the urfae i given by 8

9 ref P (7) gp ref for ma aborption oeffiient ref at preure P ref and urfae preure P where preure broadening aue the P dependeny of the optial depth (Catling & Kating 017, p.381). Here we ued 0.05 m kg -1 and ref Pref 10 4 Pa (from Catling & Kating (017), Ch. 13). Having P in equation (7) i appropriate for thik atmophere, whih i the ae when the runway greenhoue limit i approahed. For thin atmophere P i appropriate (Catling & Kating 017, p.38). In thi tudy, the urfae preure are in the low-preure regime (le than ~500 Pa) until the runaway limit i reahed. However, the differene between P and P in equation (7) i mall at uh low preure where the total greenhoue warming i le than a few K until the runaway limit i reahed. Setting P for uh low-preure moon in equation (7) ha negligible impat on the alulated ma lo rate o we approximate the optial depth of all atmophere in thi tudy with P. One the total optial depth of the atmophere i known from equation (7), the firt order global energy balane i given by (ee Appendix B for derivation) 1 GM 1 1 v 4 r 4 A F L u T (8) and from equation (3) and (4) we derived an expreion for u (ee Appendix A) u r 1 GM 1 1 u0 exp r u0 r r (9) Equation (1), (7), (8), and (9) were olved imultaneouly to find T and u, with being given by the ideal ga law. The ma lo rate i then alulated by 4ur M (10) 9

10 Uing equation (10) the time averaged lo rate i alulated and urfae water lifetime i then obtained via equation (6). It i poible that no phyially meaningful olution exit to equation (1), (7), (8), and (9). When the initial urfae temperature, and therefore urfae preure, i large (above ~60 K for thi model), the optial depth given by equation (7) will be ignifiant. Thi will aue an inreae in urfae temperature further inreaing the urfae preure and thu the optial depth of the atmophere. The poitive feedbak between temperature, preure, and optial depth will aue equation (1), (7), (8), and (9) to have no valid olution if the initial urfae temperature, et by the inoming tellar flux, i large. The exomoon-tar ditane where thi poitive feedbak reult in no olution i the runaway greenhoue limit, and it i akin the runaway limit found by Ingeroll (1969). For all three model enario, we onidered a pure water vapor atmophere above a urfae water reervoir. We looked at iy exomoon with mae ranging from to 0.04 Earth mae between 0.9 and.0 AU from a Sun-like tar. Thi ma range inlude bodie lightly maller than Europa (0.008M Earth), and lightly larger than Ganymede (0.05M Earth). We et the Bond albedo to 0. for eah run. We hoe a Bond albedo of 0. for two reaon, the firt i that it approximately repreent the lower bound for iy moon Bond albedo in the olar ytem (Buratti 1991; Howett, Spener, & Pearl 010). In addition, a Bond albedo of 0. approximate the albedo of open oean with partial loud over (Goldblatt 015; Leonte et al. 013). Should an iy moon form urfae oean, the 0. Bond albedo give u the bet repreentation when alulating water longevity. 10

11 11

12 Figure. For all three plot, the red urve repreent the Runaway Flux where the iy moon will be loe enough to the tar that a runaway greenhoue our. The blue urve repreent ontour of urfae water lifetime (plotted in Gyr). The urfae temperature of the moon are hown by the olored bakground. In plot A and B the urfae temperature orrepond to the effetive temperature. In plot C the olored bakground how the urfae temperature, beyond the runaway limit ditane, aounting for the effet of the water vapor greenhoue. Comparing the urfae temperature in plot C to thoe in A and B the water vapor greenhoue i negligible exept very loe to the runaway limit ditane. The rate of hydrodynami eape depend on both the ma and radiu of the exomoon, a uh we plot eape veloity v. ditane to inorporate both parameter. Plot A how the iothermal analyti model (Setion.1) baed on equation (5). Plot B how the aturated ae where the atmophere wa aumed to follow the Clauiu-Clapeyron equation and wa aturated from the urfae to the ritial radiu for eape (Setion.). Plot C how the iothermal model with the greenhoue effet of water vapor onidered (Setion.3). The reult hown in all three figure are dependent on the hoen albedo. If the albedo were to be inreaed from the hoen value of 0., the effet would be a linear dereae in aborbed flux. Thi would hift the runaway limit and the ontour of oean lifetime loer to the hot tar. For a Ganymede ized moon with a Bond albedo of 0. (hown here), 0.4, and 0.8 the runaway limit our at 1.05, 0.91, and 0.5 AU repetively. 3. RESULTS For eah body in the range of mae and ditane onidered, we alulated the time averaged ma lo rate, M, uing a time tep of 10 4 year. With the water ontent of eah world aumed to be 40% of the bulk ma, the water lifetime, alulation are hown in Figure. Water, wa then alulated via equation (6). The reult of thee Figure how ontour of Water a a funtion of tellar ditane and eape veloity, whih i defined a v e GM r 1/ (11) The runaway greenhoue tar-exomoon ditane i hown with red ontour on eah plot in Figure. From Figure A we an ee that, in the analyti model, water on a Ganymede-like exomoon (with an eape veloity of ~.74 km -1 ) would perit indefinitely at a ditane beyond the runaway limit. However, the ie on a Europa ized moon would only urvive for timeale greater than 1 Gyr beyond 1

13 ~1.5 AU. Given that the iothermal and aturated ae repreent the upper and lower bound on eape rate, the true olution i likely omewhere between the two plot (Figure A and B). In Figure C, the impat of the water vapor greenhoue effet wa onidered. Under the radiative model, the greenhoue effet of a pure water vapor atmophere ontribute a few degree K of warming. However, if the body reeive uffiient tellar warming a runaway our. With a pure water vapor atmophere the urfae never rie above the freezing point of water without entering a runaway greenhoue. But if loud were to inreae the albedo, a world with a liquid water urfae may exit with a marginally table urfae temperature up to 75 K (Goldblatt et al. 013). However, uh a world may be tranient and eaily wing to either a nowball via the ie-albedo feedbak, or a runaway greenhoue tate (Goldblatt, et al. 013). 4. DISCUSSION The loely paked lifetime line in Figure reult from a trong dependene on eape veloity and therefore on ma. From equation (5) and (6), with all the ontant tripped away, we ee that there i an exponential relationhip between oean lifetime and ma, if the mean denity of the moon i held ontant, given by 1 exp M /3 Water M (1) v Gr r, o impliit For ontant denity,, the eape veloity from equation (11) i 8 1/ 3 in equation (1), 3 M v e 3 o Water ve expve e. Thi trong exponential dependene on v e an be een in Figure in both the iothermal and aturated ae. There i a threhold ma region, below whih urfae water i tranient, while moon with mae above thi region will lat for billion of year. Ganymede ized moon will perit indefinitely beyond the tar-exomoon ditane of the runaway limit. 13

14 If a ga giant planet poeed rapidly evaporating iy moon future obervation may be able to detet them. The eaped H from water would form a toru in the orbit of the moon that may produe detetable attering in the Lyman-α. However, for young, migrating planet thi H toru may be inditinguihable from aptured nebular H before it diipate. Thi degeneray ould be addreed by oberving aging ga giant planet that are jut entering the habitable zone a the hot tar brighten over time. A a Jupiter-like planet enter the habitable zone around an aging tar, hydrogen i unlikely to eape from the planet. Indeed, if we aume a Jupiter-like planet at 0.9 AU around a Sun-like tar ha an exobae temperature of 1500 K then, following Sánhez-Lavega (011), p. 88, the thermal lo of hydrogen via Jean eape from uh a planet will be ~10-37 kg -1. Thi i ~40 order of magnitude le than the lo rate from iy moon at the ame orbital ditane o any oberved H toru may be an indiation of evaporating moon. Iy moon around uh a planet are of partiular interet beaue they may provide habitable urfae ondition for hundred of million to billion of year, depending on the tellar type (Ramirez & Kaltenegger 016). A the hot tar brighten the mallet iy moon in the habitable zone would rapidly evaporate, produing the H toru, while more maive moon ould retain their urfae water for billion of year. A imilar toru-produing proe our for Io, where a plama toru around Jupiter ontain ulfur and oxygen lot by the moon that are trapped by Jupiter magneti field line (Yohioka et al. 011). Alo, O atom may linger around the iy exomoon, analogou to the O -rih olliional atmophere of Callito (Cunningham et al. 015) and ould poibly eape the moon to form a toru imilar to the eaped H. A eond, heavier omponent in the exomoon atmophere, uh a oxygen, would generally at to lower the rate of eape and water lo. However, a more ophitiated model than preented here i required to tudy eape from a multiomponent atmophere. 5. CONCLUSION 14

15 Planetary migration i likely a ommon phenomenon throughout planetary ytem (Tigani, et al. 005). In addition, all the large planet in the olar ytem have a retinue of iy moon and ga giant exoplanet may have imilar iy moon. Inward migration by a ga giant would ubjet it iy moon to inreaed tellar heating. Like a omet entering the inner olar ytem, the moon ould evaporate and reate atmophere. The longevity of uh an atmophere depend trongly on the ditane from the hot tar, and the ma and radiu of the exomoon. The maller the tar-exomoon ditane, the warmer the iy exomoon will beome. A an iy exomoon approahe a ditane of ~1.1 AU around a Sun-like tar it will enter a runaway greenhoue tate when the urfae melt. However, thi utoff i dependent on the albedo of the moon, whih wa et to 0. in thi paper. Inreaing the albedo will allow table urfae ondition at loer orbital ditane before the runaway tate i ahieved. The high temperature from a runaway tate will drive rapid hydrodynami eape and erode the water from the exomoon on very hort timeale. If the exomoon it beyond thi runaway limit the urfae water may perit muh longer. Beyond the tar-exomoon ditane of the runaway limit, there i an exponential relationhip between ma and water longevity. For an iy moon of Ganymede ize around a Sun-like tar, urfae water will likely perit indefinitely. Large moon of thi ize will maintain their atmophere for long period in the habitable zone and ould potentially maintain a liquid urfae for timeale greater than 1 Gyr. Thu, uh moon ould be habitable. However, an iy moon of Europa ize would evaporate rapidly at ~1.1 AU around a Sun-like tar, and only beyond ~1.5 AU would urfae water (a ie) on a Europa ized moon lat for more than 1 Gyr. ACKNOWLEDGEMENTS ORL, DCC, and KJZ were upported by NASA Planetary Atmophere grant NNX14AJ45G awarded to DCC. We would like the thank Tyler D. Robinon, for inightful omment and uggetion on thi tudy. 15

16 APPENDIX A: DERIVATION OF ISOTHERMAL AND SATURATED HYDRODYNAMIC ESCAPE MODELS A.1. Iothermal Model The three key equation for hydrodynami eape ontinuity, momentum, and energy an be written generally (e.g., multiple peie, et. (Kokinen et al. 013)) but we will ue a implified pherially ymmetri model with ontant mean moleular ma (ee Ch. 5 in (Catling & Kating 017) for a more omplete diuion of the topi). We aume the atmopheri denity and atmopheri flow veloity only hange in the radial diretion. A uh, the derivative for ma ontinuity and momentum onervation are omplete. Under thee aumption, the time-dependent and teady-tate ontinuity and ma onervation equation are a follow. Continuity i given by: where i the ma denity, r d r u r u 1 d, teady tate: 0 t r dr dr i the radial ditane from the planet enter, and u (A1) i the atmopheri flow veloity. Momentum onervation i given by: 1 u du dp du dp u g, teady tate: u g t dr dr dr dr (A) where p i preure, and g i gravity. If we aume an iothermal atmophere, we an relate preure and denity with the iothermal ound peed u 0 kt (A3) m where k i the Bolztmann ontant, T i the iothermal temperature, and m i the mean moleular ma of the atmophere. From the ideal ga law p u 0 (A4) 16

17 Integrating equation (A1) in the teady tate, we get the ma eape rate per teradian of r u, whih, when ombined with equation (A1) and (A), give an expreion for the iothermal planetary wind from a body with ma M: u u u g, or u u u0 0 0 du du GM (A5) u dr r u dr r r Equation (A5) i analogou to Parker olar wind equation. For a trongly bound atmophere at ome ritial ditane from the planet urfae, the right hand ide of equation (A5) reahe zero, indiating that either the flow reahe the peed of ound or du / dr 0. The uboni olution, du / dr 0, require a finite bakground preure that inhibit eape o we will fou on the tranoni olution where u u 0. The tranoni olution ha du / dr 0 at all time and i onitent with a trongly bound atmophere at the urfae and zero preure at infinity. The ritial ditane r our in equation (A5) when u u 0 whih give u: 0 u GM 0 r r (A6) Solving for r in equation (A6) we find r GM (A7) u 0 If we integrate equation (A5) from the urfae radiu r to r and ignore the u term near the urfae, where it i negligible for bodie of interet in thi tudy, we get the equation: u r 1 GM 1 1 u0 exp r u0 r r (A8) A the radial ditane from the moon inreae the ma flux, teady tate ontinuity given by equation (A1), when integrated give u, (in kg m - -1 ), dereae. The 4r u C where the ontant of 17

18 integration C i jut the total rate of ma lo (in kg -1 ) through a pherial urfae. A r goe to infinity u goe to 0 ine u 1/ r. Therefore, the outflowing wind loe kineti energy a r. Thu, the energy flux required to drive the eaping ma flux i given by the energy required to remove the ma flux from the gravity well of the moon, u GM / r. A firt-order global energy balane between inolation and ooling via ma lo i then given by: 1 GM 1 A F Lv u T 4 r aborbed tellar flux ma lo flux 4 radiative ooling (A9) where A i the Bond albedo, F i the inident tellar flux, and i the Stefan-Boltzmann ontant. The eape flux i given by u and i multiplied by the energy required for that flux to eape the planet. The energy inlude a gravitational potential energy term, and the latent heat of vaporization L v (for thi model 6 Lv.510 J kg -1 ). In equation (A9) we aume the atmophere i tranparent to both hortwave and infrared radiation. Equation (A8), and (A9) an be olved imultaneouly for the two unknown olved, we an alulate the total eaping ma rate by: u and T. One 4ur M (A10) with being alulated from 0 P / u with urfae preure P. We alulate urfae preure with equation (1) for a C-C atmophere given the urfae temperature of water, where referene parameter are at the triple point: P Pa, T K. For our model, we only onider water world with pure H O atmophere o etimating the urfae denity from the aturation vapor preure i valid (Adam, Seager, & Elkin-Tanton 008). We refer to thi approah, where equation (A8) and (A9) are olved numerially, a the Numerial Model. 18

19 Figure A1. Contour of urfae water lifetime omparing the analyti model given by equation (5), hown in red, and the numerial approah where T and are olved for imultaneouly, hown in dahed blue ontour. Plot A how the analyti model, whih doe not onider the greenhoue effet, plotted with the numerial model taking into aount the greenhoue effet of water vapor a derived in Appendix B. Both model produe idential reult until the runaway limit i approahed and the numerial model aymptote along the limit. Plot B how the analyti and numerial model a well; however, the greenhoue effet i negleted in the numerial model for thi plot. In thi ae, both method produe idential reult, a expeted, for lowly evaporating bodie with urfae water lating more than 1 Gyr. u For the lowly evaporating moon of interet in thi tudy, thoe with urfae water lating more than 1 Gyr, the eape i o low it doe not appreiably ool the moon. Thu an analyti model an be derived by negleting the ma lo flux ooling term in equation (A9). With the implified equation (A9) our iothermal temperature i imply alulated from inoming tellar flux. And, from our aumption that the exomoon have an average bulk denity of g m -3, we an alulate the urfae radiu r (3 M / (4 )) exomoon 1/3. Subtituting thee two equation into equation (A8) we find that 1 r 3 3 G 3 u u0 exp M r u0 4 exomoon /3 (A11) 19

20 By plugging equation (A11) into equation (A10), we get the analyti expreion for the ma lo due to hydrodynami eape given in equation (5). Calulating u in thi manner aume the temperature in our energy balane equation i a ontant and et olely by the inoming tellar flux and the emitted thermal flux from the urfae. For the low temperature bodie (< 73 K) we are intereted in for thi tudy, equation (A11) give idential reult a the previouly defined numerial model until the runaway limit i approahed. See Figure A1 for a omparion. A.. Saturated Model We alo modeled hydrodynami eape from a non-iothermal atmophere, the aturated ae. To model eape in the aturated ae we tart with equation (A1), (A), (A3), and (A4). Intead of uing equation (1) to relate temperature and preure, we will approximate the Clauiu-Clapeyron relation with an expreion imilar to Tenten formula, given by p p exp T / T (A1) w w for referene temperature T w and preure p w. A reaonable approximation for 50 T 400 K over water take Tw 500 K and 6 pw bar. A very good approximation for 150 T 73 K over ie take Tw 6140 K and 7 pw bar. From Wexler (1977), whoe expreion we ve approximated, the imple exponential fit i likely good to within a few perent for the temperature in our model. Thi implified expreion i deirable beaue we want to work with an analyti expreion for dt / dr. We an eliminate p from equation (A) uing equation (A3) and (A4), giving u 0

21 u du u0 d u0 dt GM (A13) dr dr T dr r We ue equation (A1) to expre dt / dr in term of d / dr u0 d Tw T u0 dt dr T T dr (A14) and equation (A1) eliminate d / dr in term of du / dr giving u our aturated wind equation u 0 T w du u 0 T w GM u u Tw T dr r Tw T r (A15) or equivalently a an expreion for du / dr 0 w w 0 w / w 1 du N u / r T / T T GM / r u dr D u u T T T (A16) where the numerator N( r, T ) i N u T GM r Tw T r 0 w (A17) and the denominator D( r, T, u ) i T w D u u0 Tw T (A18) Equation (A16) i the form we will ue to numerially integrate ur (). Equation (A15) an be written equivalently a 1 du D N u dr (A19) 1

22 atmophere, Reall from the iothermal ae that, for hydrodynami eape from a trongly bound du / dr 0. Near the urfae of the moon the numerator, N( r, T ), will be negative a the gravity term will dominate given that our atmophere i trongly bound. At ome ditane r the u / r 0 term will equal the fore of gravity, o N( r, T ) 0 at r. Sine N( r, T ) 0 and du / dr 0, from equation (A19), D( r, T, u) 0 at r a well. At the ritial point, N 0 provide a imple relation between T and r r GMm( Tw T) (A0) kt T w Similarly, D 0 relate u and T by u u T GM w 0 3 Tw T r (A1) The tranoni olution i obtained by numerially integrating equation (A16) from the ritial point to the urfae. The firt tep i to olve for du / dr at the ritial point. Thi i obtained from equation (A16) by uing L Hopital rule. 1 du N 0 dn / dr u dr D 0 dd / dr (A) The numerator beome 4 u0 TwT u 0 TwT Tw T Tw T dn GM du dr r r r u dr (A3) and the denominator beome

23 dd T T du u T T dr dr r w w u Tw T Tw T (A4) 1 If we let x u du dr /, implify equation (A3) to replae GM / r with equation (A1), and divide out the ommon fator u, then equation (A) an be written a the quadrati equation TwT 4 TwT 4 TwT x x 0 Tw T r Tw T r Tw T r (A5) The poitive root of thi equation orrepond to the aelerating flow at the ritial point. To find the ma flux lo, the firt tep i to gue an initial temperature at the ritial point, T. Given T, we know u, p i given from equation (A1), and i given by the ideal ga law. From equation (A0) and (A1), we get r and u repetively, whih allow u to olve equation (A5) for the ritial lope du dr /. Denity an then be found at the new point from ontinuity, ur u r. Given, we an olve for T and p from equation (A1) with the help of the ideal ga equation. Thi integration proeed to the urfae. The gue for T i adjuted numerially until the deired urfae temperature (in balane with inoming tellar flux and ma lo given by equation (A9) ) i ahieved. One the orret value are found, equation (A10) will give the ma lo rate. For both iothermal and non-iothermal model, the urfae temperature i aumed to be et by the inident olar flux averaged over time and hemiphere, whih i given by equation () for a rapidly rotating body. The iothermal ae repreent the warmet poible atmophere negleting greenhoue effet under the ae of hydrodynami eape. The non-iothermal ae repreent a minimum poible temperature for a water vapor atmophere at r ine it i aturated at all point baed on the urfae 3

24 temperature et from the olar flux. Thee two model repreent the extreme of atmopheri temperature profile for a water vapor atmophere, with the real olution likely omewhere between them. APPENDIX B: DERIVATION OF SURFACE TEMPERATURE ACCOUNTING FOR GREENHOUSE EFFECT AND HYDRODYNAMIC ESCAPE We would like to alulate the total urfae warming due to the greenhoue effet of a water vapor atmophere onidering the energy aborbed to drive atmopheri expanion and eape throughout the atmophere. We tart with the greenhoue effet of a hydrotati atmophere, then adapt the equation for a hydrodynami atmophere. We aume the atmophere i tranparent to hortwave radiation. From Catling and Kating (017), p. 55, for a moon with a gray, radiative, hydrotati atmophere the energy balane at the urfae i given by T 4 F 1 / (B1) net where i the total thermal infrared optial depth of the atmophere at the urfae, i the Stefan-Boltzmann ontant, and T i the urfae temperature. The time-averaged, hemipherially-averaged flux inident on the moon i given by F net 1 A F / 4 for Bond albedo A, and inident tellar flux F. In our model, we are onerned with moon in the hydrodynami regime where water vapor i lifted from the urfae of the moon and aelerate upward until it eape to pae. The total energy required to remove a ma flux of water vapor from the moon urfae i given by GM u (B) r 4

25 for urfae radiu r. In equation (B) M i the ma of the moon, G i the gravitational ontant, u i the radial outflow veloity of the atmophere at the urfae, and i the atmopheri denity at the urfae, uh that r u i the ma flux [kg m - -1 ]. In the hydrodynami atmophere of interet in thi tudy, the energy flux needed to remove the atmophere, given by equation (B), mut ome from the tellar radiation and the thermal IR flux. That i, it mut ome from the F 1 / A uh, the energy balane at the urfae will then be given by net energy input term in equation (B1). GM T F 1 / u (B3) 4 net r in the hydrodynami regime. We alo aount for the energy required to vaporize the water ma flux at the urfae, given by Lv u for latent heat of vaporization L v. Subtrating Lv u from the right-hand ide of equation (B3) and reorganizing the term we find the following energy balane of input and output: 1 GM 1 1 v 4 r 4 A F L u T (B4) Equation (B4) i the global energy balane at the urfae for an iy moon with the greenhoue effet onidered under the hydrodynami regime. It an be ompared with equation () in the main text where we aumed an atmophere that wa optially thin in the thermal infrared. REFERENCES Adam, E. R., Seager, S., & Elkin-Tanton, L. 008, The Atrophyial Journal, 673 Buratti, B. J. 1991, Iaru, 9, 31 Catling, D. C., & Kating, J. F. 017, Atmopheri Evolution on Inhabited and Lifele World (CUP, New York) Chamber, J. E. 009, Annual Review of Earth and Planetary Siene, 37, 31 5

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