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The longevity of water ice on Ganymedes and Europas around migrated giant planets

The gas giant planets in the Solar System have a retinue of icy moons, and we expect giant exoplanets to have similar satellite systems. If a Jupiter-like planet were to migrate toward its parent star the icy moons orbiting it would evaporate, creating atmospheres and possible habitable surface oceans. Here, we examine how long the surface ice and possible oceans would last before being hydrodynamically lost to space. The hydrodynamic loss rate from the moons is determined, in large part, by the stellar flux available for absorption, which increases as the giant planet and icy moons migrate closer to the star. At some planet-star distance the stellar flux incident on the icy moons becomes so great that they enter a runaway greenhouse state. This runaway greenhouse state rapidly transfers all available surface water to the atmosphere as vapor, where it is easily lost from the small moons. However, for icy moons of Ganymede's size around a Sun-like star we found that surface water (either ice or liquid) can persist indefinitely outside the runaway greenhouse orbital distance. In contrast, the surface water on smaller moons of Europa's size will only persist on timescales greater than 1 Gyr at distances ranging 1.49 to 0.74 AU around a Sun-like star for Bond albedos of 0.2 and 0.8, where the lower albedo becomes relevant if ice melts. Consequently, small moons can lose their icy shells, which would create a torus of H atoms around their host planet that might be detectable in future observations.


INTRODUCTION
One of the major results of exoplanet discoveries is that giant planets migrate (Chambers 2009).
This was first deduced from hot Jupiters, and although these are found around 0.5-1% of Sun-like stars (Howard 2013), hot Jupiters are not the only planets to migrate and giant planet migration is likely widespread. Indeed, such migration probably occurred in the early solar system (Tsiganis et al. 2005).
All the giant planets in the solar system have a collection of icy moons. We expect that similar exomoons orbit giant exoplanets and that these moons would likely migrate along with their host planet.
If a giant exoplanet were to migrate toward its parent star, icy moons could vaporize, similar to comets approaching the Sun, and develop atmospheres. If such a giant planet and icy moons were to form in the habitable zone of a star (or migrate there shortly after formation) the high XUV flux from the young star would rapidly erode the atmospheres of moons many times the mass of Ganymede (). In addition, they could melt and maintain liquid surfaces as they migrate inwards, which could be potentially habitable environments. Such a moon would have an atmosphere primarily controlled by the vapor equilibrium set by the surface temperature and the rate of hydrodynamic escape to space. As such, the longevity of the water shell and atmosphere will depend primarily on the distance to the host star and the exomoon radius and mass.
Several such bodies exist in the solar system, where the atmospheric thickness is determined by vapor equilibrium with a condensed phase, i.e. the Clausius-Clapeyron relation for the relevant volatile. Let us call such atmospheres Clausius-Clapeyron (C-C) atmospheres. The N2 atmospheres on both Triton and Pluto are examples of C-C atmospheres, where the surface vapor pressure is in equilibrium with the N2 surface ice at the prevailing temperature for each body. The present Martian atmosphere is another C-C atmosphere since the polar CO2 ice caps at ~148 K buffer the atmosphere to ~600 Pa surface pressure (Leighton & Murray 1966) (see (Kahn 1985) for an explanation over geologic timescales).
For an icy exomoon migrating toward its parent star, the atmospheric water vapor will be controlled by the availability of surface water and temperature. Very deep ice and ice-covered oceans are possible on these moons given that water can account for ~5-40% of the bulk mass of icy moons in the solar system (Schubert et al. 2004). However, the small mass of exomoons and relatively high stellar flux as the exomoon migrates toward the star makes water vapor susceptible to escape. Assuming exomoons are of comparable size to the moons found in the solar system, this study looks at the end-member case of how rapidly a pure water vapor atmosphere will be lost hydrodynamically during exomoon migration.
The migration of exomoons is essential if icy moons of Ganymede's size are to retain their surface water for more than 1 Gyr in the habitable zone of a Sun-like star. If a Ganymede-like icy moon formed in the habitable zone of a star (or migrated there shortly after formation) the high XUV flux from the young star could rapidly erode its atmosphere (Heller, Marleau, & Pudritz 2015;Lammer et al. 2014). Therefore, this study looks at the longevity of surface water on icy moons that migrate toward their host star after this period of intense XUV-driven hydrodynamic escape.
Hydrodynamic escape is a form of pressure-driven thermal escape where the upper levels of an atmosphere become heated and expand rapidly, accelerate through the speed of sound, and escape to space en masse (Hunten 1990). An important process in atmospheric evolution, hydrodynamic escape likely occurred during the formation of the terrestrial atmospheres (Kramers & Tolstikhin 2006;Kuramoto, Umemoto, & Ishiwatari 2013;Pepin 1997;Tolstikhin & O'Nions 1994). Moreover, hydrodynamic escape has been observed on exoplanets such as the gas giant HD 209458b, which orbits a Sun-like star at 0.05 AU and has hot H atoms beyond its Roche lobe, presumably deposited there by hydrodynamic escape (Linsky et al. 2010;Vidal-Madjar et al. 2004). The closer a body is to its parent star, the more effective the hydrodynamic escape, and the smaller the body, the more easily an atmosphere is lost (e.g., Zahnle & Catling (2017)).
The longevity of an atmosphere and icy shell will depend primarily on temperature, set in large part by the stellar flux available for absorption. As an ice covered exomoon moves towards its parent star, heating will cause more water vapor to enter the atmosphere, hastening the loss rate. In addition, this water vapor will provide a greenhouse effect, further warming the moon. At a certain exomoon-star distance the water vapor atmosphere will impose a runaway greenhouse limit on the outgoing thermal infrared (IR) flux from the exomoon. If the absorbed stellar flux exceeds this limit, the exomoon surface 4 will heat rapidly until all available water is in the atmosphere as vapor. This limit represents the distance at which all surface water will be transferred to the atmosphere where it will be rapidly lost.

METHODS
We consider three cases of hydrodynamic escape: (2.1) an isothermal atmosphere where the atmospheric temperature is set by incoming stellar flux and equal to the effective temperature; (2.2) a vapor saturated atmosphere where the temperature and humidity profiles of the entire atmosphere are dictated by the C-C relation; and (2.3) an isothermal atmosphere similar to (2.1) but the surface temperature, and the isothermal atmospheric temperature, are increased from the effective temperature by the total greenhouse warming of the water vapor atmosphere. We chose the isothermal and C-C cases because they present upper and lower limits on the rates of hydrodynamic escape, respectively, as described below. Figure 1 provides a conceptual picture of the model.
For a pure water vapor atmosphere around a Sun-like star, an isothermal atmosphere at the effective temperature represents the greatest possible temperature at the top of atmosphere to drive hydrodynamic escape. Water vapor radiates in the IR more efficiently than it absorbs sunlight, so the radiative-convective temperature for a pure water vapor atmosphere will be less (Pierrehumbert 2010;Robinson & Catling 2012). As such, the isothermal atmospheric approximation provides an upper bound on the atmospheric loss rate. In contrast, the lowest possible temperature to drive hydrodynamic escape is the saturated case, where the temperature and pressure at all heights are set by the C-C relation, which is defined by for reference temperature 0 T at reference pressure 0 P , where R is the universal molar gas constant, and v L is the latent heat of vaporization for water (e.g., Pierrehumbert (2010), p.100). The surface temperature is assumed to be in equilibrium with the incoming stellar flux, cooling associated with mass loss via hydrodynamic escape, and latent heat of evaporation. If the temperature were to decrease with altitude faster than the C-C relationship, the water vapor would condense out resulting in a C-C curve that, when extrapolated, would result in a surface temperature no longer in equilibrium with incoming stellar flux and escape. Therefore, the hydrodynamic loss rate of a pure water vapor atmosphere is bounded by the isothermal and saturated cases, which we will now consider in turn.

6
In the blackbody approximation, radiative cooling is given by 4 T  at isothermal temperature T , allowing a straightforward formulation of escape versus radiative cooling. As such, the first order global energy balance for an icy exomoon is between incoming stellar flux versus the energy flux lost to vaporizing the water, lifting molecules out of the gravity well, and radiative cooling, i.e.,   radiative cooling absorbed 4 mass loss flux stellar flux 1 1 4 Here s F is the incoming solar flux available for absorption, A is the Bond albedo,  is the Stefan-Boltzmann constant, s r is the surface radius where the atmospheric density is s  , such that ss u  is the mass flux given an outward radial flow velocity at the surface s u , G is the gravitational constant, and M is the mass of the exomoon.
In addition to the C-C relationship [equation (1)], three equations are needed to derive the steady state, hydrodynamic atmospheric loss in the isothermal approximation (e.g., Catling & Kasting (2017), Ch. 5). The first is steady state mass continuity, given by where r is the radial distance from the planet's center, u is the outward radial flow velocity, and  is the atmospheric density. Steady state momentum conservation is expressed as and pressure p . Finally, the equation for energy balance is given by equation (2). Combining equations (1), (2), (3), and (4) an analytic expression for the isothermal atmospheric mass loss rate in kg s -1 is given by (see Appendix A for the derivation): For an upper limit, we assume the total mass of water present on the exomoon surface, Water M , is 40% of the bulk mass. However, even if 5% water were used [the lower limit for Europa (Schubert, et al. 2004)] from equation (6) we can see that it would translate to a change in Water  by a factor of 8, compared to 40% water. From equation (5) we see that

M , and hence
Water  , has an exponential dependence on mass, so we would anticipate that the difference between 5% and 40% water is not the major factor determining Water  , which is borne out by our results. In addition, if substantial water vapor is lost the bulk density of the moon, m  , may increase over time. However, from equation (5) we see that the exponential term scales like M largely determining the loss rate so the sensitivity to m  is small.
It is important to note that in equation (5) we have assumed the mass loss rate, M , is sufficiently small that energy balance is dominated by radiative loss. This is indeed the case for exomoons of interest in this paper, where the low temperature water vapor atmospheres last for more than 1 Gyr. The surface 8 pressures are well below ~500 Pa until the runaway greenhouse limit is reached. For bodies with rapid hydrodynamic escape the numerical approach defined in Appendix A is appropriate.

Saturated Temperature Profile Case
The saturated case is derived from the same equations as the isothermal case [equations (1), (2), (3), and (4)] but temperature is allowed to change with altitude. The temperature at the critical point at radius c r in Figure 1 (where the isothermal sound speed 0 u equals the radial escape speed) is set such that numerically integrating equations (1), (2), (3), and (4) from the critical point to the surface will result in a surface temperature equivalent to that in equilibrium with incoming solar flux taking into account the evaporative cooling (see Appendix A for details). Once the critical temperature is known the radial outflow velocity is readily calculated and thus the mass loss rate.

Isothermal Case with Greenhouse Effect Considered
In Case (2.1) we let the isothermal atmospheric temperature be set by just the incoming stellar flux and thus be equal to the effective temperature. However, for a thick water vapor atmosphere the surface will be heated by the greenhouse effect of the overlying atmosphere. In this case, we still used an isothermal atmosphere approximation but increased the atmospheric temperature by the total greenhouse warming of the atmosphere at the surface. The larger isothermal atmospheric temperature under this regime will increase the hydrodynamic loss rate compared to Case (2.1).
To account for the atmospheric greenhouse effect, we used a gray, radiative, plane-parallel approximation where the total gray atmospheric optical depth in the thermal infrared at the surface is given by  (7) is appropriate for thick atmospheres, which is the case when the runway greenhouse limit is approached. For thin atmospheres P   is appropriate (Catling & Kasting 2017, p.382). In this study, the surface pressures are in the low-pressure regime (less than ~500 Pa) until the runaway limit is reached. However, the difference between 2 P   and P   in equation (7) is small at such low pressures where the total greenhouse warming is less than a few K until the runaway limit is reached.
Setting P   for such low-pressure moons in equation (7) has negligible impact on the calculated mass loss rate so we approximate the optical depth of all atmospheres in this study with 2 P   . Once the total optical depth of the atmosphere is known from equation (7) (3) and (4) Using equation (10) the time averaged loss rate is calculated and surface water lifetime is then obtained via equation (6).
When the initial surface temperature, and therefore surface pressure, is large (above ~260 K for this model), the optical depth given by equation (7) will be significant. This will cause an increase in surface temperature further increasing the surface pressure and thus the optical depth of the atmosphere. The positive feedback between temperature, pressure, and optical depth will cause equations (1), (7), (8), and (9) to have no valid solution if the initial surface temperature, set by the incoming stellar flux, is large.
The exomoon-star distance where this positive feedback results in no solution is the runaway greenhouse limit, and it is akin the runaway limit found by Ingersoll (1969).
For all three model scenarios, we considered a pure water vapor atmosphere above a surface water reservoir. We looked at icy exomoons with masses ranging from 0.005 to 0.04 Earth masses between 0.9 and 2.0 AU from a Sun-like star. This mass range includes bodies slightly smaller than Europa (0.008MEarth), and slightly larger than Ganymede (0.025MEarth). We set the Bond albedo to 0.2 for each run. We chose a Bond albedo of 0.2 for two reasons, the first is that it approximately represents the lower bound for icy moon Bond albedos in the solar system (Buratti 1991;Howett, Spencer, & Pearl 2010). In addition, a Bond albedo of 0.2 approximates the albedo of open ocean with partial cloud cover (Goldblatt 2015;Leconte et al. 2013). Should an icy moon form surface oceans, the 0.2 Bond albedo gives us the best representation when calculating water longevity.

Figure 2.
For all three plots, the red curve represents the Runaway Flux where the icy moon will be close enough to the star that a runaway greenhouse occurs. The blue curves represent contours of surface water lifetime (plotted in Gyr). The surface temperatures of the moons are shown by the colored background. In plots A and B the surface temperature corresponds to the effective temperature. In plot C the colored background shows the surface temperature, beyond the runaway limit distance, accounting for the effect of the water vapor greenhouse. Comparing the surface temperatures in plot C to those in A and B the water vapor greenhouse is negligible except very close to the runaway limit distance. The rate of hydrodynamic escape depends on both the mass and radius of the exomoon, as such we plot escape velocity vs. distance to incorporate both parameters. Plot A shows the isothermal analytic model (Section 2.1) based on equation (5). Plot B shows the saturated case where the atmosphere was assumed to follow the Clausius-Clapeyron equation and was saturated from the surface to the critical radius for escape (Section 2.2). Plot C shows the isothermal model with the greenhouse effect of water vapor considered (Section 2.3). The results shown in all three figures are dependent on the chosen albedo. If the albedo were to be increased from the chosen value of 0.2, the effect would be a linear decrease in absorbed flux. This would shift the runaway limit and the contours of ocean lifetime closer to the host star. For a Ganymede sized moon with a Bond albedo of 0.2 (shown here), 0.4, and 0.8 the runaway limit occurs at 1.05, 0.91, and 0.52 AU respectively.

RESULTS
For each body in the range of masses and distances considered, we calculated the time averaged mass loss rate, M , using a time step of 10 4 years. With the water content of each world assumed to be 40% of the bulk mass, the water lifetime, Water  , was then calculated via equation (6). The results of these calculations are shown in Figure 2.
The runaway greenhouse star-exomoon distance is shown with red contours on each plot in Figure 2.
From Figure 2A we can see that, in the analytic model, water on a Ganymede-like exomoon (with an escape velocity of ~2.74 km s -1 ) would persist indefinitely at a distance beyond the runaway limit.
However, the ice on a Europa sized moon would only survive for timescales greater than 1 Gyr beyond 13 ~1.5 AU. Given that the isothermal and saturated cases represent the upper and lower bounds on escape rate, the true solution is likely somewhere between the two plots (Figures 2A and 2B).
In Figure 2C, the impact of the water vapor greenhouse effect was considered. Under the radiative model, the greenhouse effect of a pure water vapor atmosphere contributes a few degrees K of warming. However, if the body receives sufficient stellar warming a runaway occurs. With a pure water vapor atmosphere the surface never rises above the freezing point of water without entering a runaway greenhouse. But if clouds were to increase the albedo, a world with a liquid water surface may exist with a marginally stable surface temperature up to 275 K (Goldblatt et al. 2013). However, such a world may be transient and easily swing to either a snowball via the ice-albedo feedback, or a runaway greenhouse state (Goldblatt, et al. 2013).

DISCUSSION
The closely packed lifetime lines in Figure 2 result from a strong dependence on escape velocity and therefore on mass. From equation (5) and (6), with all the constants stripped away, we see that there is an exponential relationship between ocean lifetime and mass, if the mean density of the moon is held constant, given by For constant density,  , the escape velocity from equation (11)  . This strong exponential dependence on 2 esc v can be seen in Figure 2 in both the isothermal and saturated cases. There is a threshold mass region, below which surface water is transient, while moons with masses above this region will last for billions of years.
Ganymede sized moons will persist indefinitely beyond the star-exomoon distance of the runaway limit.
If a gas giant planet possessed rapidly evaporating icy moons future observations may be able to detect them. The escaped H from water would form a torus in the orbit of the moon that may produce detectable scattering in the Lyman-α. However, for young, migrating planets this H torus may be indistinguishable from captured nebular H before it dissipates. This degeneracy could be addressed by observing aging gas giant planets that are just entering the habitable zone as the host star brightens over time. As a Jupiter-like planet enters the habitable zone around an aging star, hydrogen is unlikely to escape from the planet. Indeed, if we assume a Jupiter-like planet at 0.9 AU around a Sun-like star has an exobase temperature of 1500 K then, following Sánchez-Lavega (2011), p. 88, the thermal loss of hydrogen via Jeans' escape from such a planet will be ~10 -37 kg s -1 . This is ~40 orders of magnitude less than the loss rate from icy moons at the same orbital distance so any observed H torus may be an would generally act to lower the rate of escape and water loss. However, a more sophisticated model than presented here is required to study escape from a multicomponent atmosphere.

CONCLUSION
Planetary migration is likely a common phenomenon throughout planetary systems (Tsiganis, et al. 2005). In addition, all the large planets in the solar system have a retinue of icy moons and gas giant exoplanets may have similar icy moons. Inward migration by a gas giant would subject its icy moons to increased stellar heating. Like a comet entering the inner solar system, the moons could evaporate and create atmospheres.
The longevity of such an atmosphere depends strongly on the distance from the host star, and the mass and radius of the exomoon. The smaller the star-exomoon distance, the warmer the icy exomoon will become. As an icy exomoon approaches a distance of ~1.1 AU around a Sun-like star it will enter a runaway greenhouse state when the surface melts. However, this cutoff is dependent on the albedo of the moon, which was set to 0.2 in this paper. Increasing the albedo will allow stable surface conditions at closer orbital distances before the runaway state is achieved. The high temperatures from a runaway state will drive rapid hydrodynamic escape and erode the water from the exomoon on very short timescales.
If the exomoon sits beyond this runaway limit the surface water may persist much longer. Beyond the star-exomoon distance of the runaway limit, there is an exponential relationship between mass and water longevity. For an icy moon of Ganymede's size around a Sun-like star, surface waters will likely persist indefinitely. Large moons of this size will maintain their atmospheres for long periods in the habitable zone and could potentially maintain a liquid surface for timescales greater than 1 Gyr. Thus, such moons could be habitable. However, an icy moon of Europa's size would evaporate rapidly at ~1.1 AU around a Sun-like star, and only beyond ~1.5 AU would surface water (as ice) on a Europa sized moon last for more than 1 Gyr.

A.1. Isothermal Model
The three key equations for hydrodynamic escapecontinuity, momentum, and energycan be written generally (e.g., multiple species, etc. (Koskinen et al. 2013)) but we will use a simplified spherically symmetric model with constant mean molecular mass (see Ch. 5 in (Catling & Kasting 2017) for a more complete discussion of the topic). We assume the atmospheric density and atmospheric flow velocity only change in the radial direction. As such, the derivatives for mass continuity and momentum conservation are complete. Under these assumptions, the time-dependent and steady-state continuity and mass conservation equations are as follows. Continuity is given by: where  is the mass density, r is the radial distance from the planet's center, and u is the atmospheric flow velocity. Momentum conservation is given by:   1 , steady state: u du dp du dp u g u g t dr dr dr dr where p is pressure, and g is gravity.
If we assume an isothermal atmosphere, we can relate pressure and density with the isothermal sound speed  Figure A1. Contours of surface water lifetime comparing the analytic model given by equation (5), shown in red, and the numerical approach where T and s u are solved for simultaneously, shown in dashed blue contours. Plot A shows the analytic model, which does not consider the greenhouse effect, plotted with the numerical model taking into account the greenhouse effect of water vapor as derived in Appendix B. Both models produce identical results until the runaway limit is approached and the numerical model asymptotes along the limit. Plot B shows the analytic and numerical models as well; however, the greenhouse effect is neglected in the numerical model for this plot. In this case, both methods produce identical results, as expected, for slowly evaporating bodies with surface water lasting more than 1 Gyr.
For the slowly evaporating moons of interest in this study, those with surface water lasting more than 1 Gyr, the escape is so slow it does not appreciably cool the moon. Thus an analytic model can be derived by neglecting the mass loss flux cooling term in equation (A9). With the simplified equation (A9) our isothermal temperature is simply calculated from incoming stellar flux. And, from our assumption that the exomoons have an average bulk density of 2 g cm -3 , we can calculate the surface radius temperature set from the solar flux. These two models represent the extremes of atmospheric temperature profiles for a water vapor atmosphere, with the real solution likely somewhere between them.

GREENHOUSE EFFECT AND HYDRODYNAMIC ESCAPE
We would like to calculate the total surface warming due to the greenhouse effect of a water vapor atmosphere considering the energy absorbed to drive atmospheric expansion and escape throughout the atmosphere. We start with the greenhouse effect of a hydrostatic atmosphere, then adapt the equation for a hydrodynamic atmosphere. We assume the atmosphere is transparent to shortwave radiation. From Catling and Kasting (2017), p. 55, for a moon with a gray, radiative, hydrostatic atmosphere the energy balance at the surface is given by In the hydrodynamic atmospheres of interest in this study, the energy flux needed to remove the atmosphere, given by equation (B2), must come from the stellar radiation and the thermal IR flux. That is, it must come from the   1 / 2 net F   energy input term in equation (B1).
As such, the energy balance at the surface will then be given by in the hydrodynamic regime. We also account for the energy required to vaporize the water mass flux at the surface, given by  Equation (B4) is the global energy balance at the surface for an icy moon with the greenhouse effect considered under the hydrodynamic regime. It can be compared with equation (2) in the main text where we assumed an atmosphere that was optically thin in the thermal infrared.