On the Potential Hydrodynamic Loss of Water Vapor During Accretion of the Galilean Satellites
Authors: McKinnon William B. , Zahnle Kevin J. 
Published: September 2022

&lt;p&gt;&lt;strong&gt;Abstract. &lt;/strong&gt;Bierson and Nimmo [1] have proposed that, under certain circumstances, as Galilean-scale satellites accrete they may lose substantial (if not all) of their accreted water ice. This is not, however, a get out of jail free card for satellite accretion models or scenarios. Each satellite accretion model (of which there are many) must be evaluated against the parameters chosen in [1], and the modest ice content of Europa and the apparent anhydrous nature of Io accounted for. Here we evaluate published models in this light, and focus especially on the roles of background protojovian subnebula temperatures and pressures.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Introduction. &lt;/strong&gt;The formation of Jupiter in the core accretion&amp;#8211;gas capture model inevitably ends with a circumplanetary accretion and/or decretion disk (CPD) around the planet. Numerous satellite formation models have been proposed [2-11], with different parameters for disk structure (surface density, temperature, opacity, viscosity). All rely on inferences and assumptions of mass and angular momentum inflow to Jupiter. Numerical gap opening calculations [e.g., 12] provided the basis for the gas-starved model [2-4]. Later 3D radiative-hydrodynamic inflow calculations prompted consideration of heliocentric planetesimal capture as the dominant source of satellite-building solids [13]. Hypotheses for dust and small pebble deposition range from distant infall [e.g., 14] to close-in infall and formation of an outflowing, decretion disk [15]. Multiple generations of satellites may form and be lost [3], Or maybe not, if a magnetospheric cavity opens up close to Jupiter and the innermost satellite, destined to become Io, stalls there [e.g., 16,17].&lt;/p&gt;
&lt;p&gt;The thermal environment of Io and Europa&amp;#8217;s formation could have been hot enough to largely devolatilize any small, accreting satellitesimals, including the ablated fragments of heliocentric planetesimals that encounter the CPD [8]. Some models call for accretion of Io and Europa in the cold outer disk followed by inward type I migration [e.g., 15,18]. Ultimate removal of major amounts of ice is a non-trivial requirement in such scenarios.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Hydrodynamic Wind. &lt;/strong&gt;We consider a water vapor atmosphere at the saturation vapor pressure (SVP) of a water ocean formed during accretion [1]. If the outflow wind speed &lt;em&gt;u&lt;/em&gt;, which increases away from the surface, becomes transonic at the critical radius &lt;em&gt;r&lt;/em&gt;&lt;sub&gt;c&lt;/sub&gt; = &lt;em&gt;GM&lt;/em&gt;/2&lt;em&gt;c&lt;/em&gt;&lt;sup&gt;2&lt;/sup&gt; (where &lt;em&gt;G&lt;/em&gt; is the gravitational constant, &lt;em&gt;M&lt;/em&gt; the satellite mass, and &lt;em&gt;c&lt;/em&gt; the isothermal sound speed), then the outflow can continue unimpeded [19,20]. For example, If Europa accreted with a 300 K water surface, the SVP would be 3.35 kPa, with a &lt;em&gt;c&lt;/em&gt; = 372 m/s. The isothermal atmosphere is a favorable case for hydrodynamic escape, and because it admits simple analytic solutions for &lt;em&gt;u(r)&lt;/em&gt;, it is well-suited to this discussion. The critical distance &lt;em&gt;r&lt;/em&gt;&lt;sub&gt;c&lt;/sub&gt; = 7.4&lt;em&gt;R&lt;/em&gt;&lt;sub&gt;e&lt;/sub&gt;, where &lt;em&gt;R&lt;/em&gt;&lt;sub&gt;e&lt;/sub&gt; is Europa&amp;#8217;s present radius. At the surface &lt;em&gt;r&lt;/em&gt;&lt;sub&gt;s&lt;/sub&gt; [20]:&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; width=&quot;291&quot; height=&quot;50&quot; /&gt;&lt;/p&gt;
&lt;p&gt;For the 300 K Europa case above, &lt;em&gt;u&lt;/em&gt;&lt;sub&gt;s&lt;/sub&gt; &amp;#8776; 0.04 m/s, and for steady-state (mass flux conserved) outflow, both the thermodynamic and ram pressure at &lt;em&gt;r&lt;/em&gt;&lt;sub&gt;c&lt;/sub&gt; are &amp;#8776;6 mPa. Such pressures are low compared with the CPD pressures in even gas-starved models [1,2], even far from Jupiter, which suggests that CPD details as well as accretional mass, radius and surface &lt;em&gt;T, &lt;/em&gt;determine if blowoff is possible (e.g., Fig. 1).&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; width=&quot;845&quot; height=&quot;506&quot; /&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Figure 1. &lt;/strong&gt;Escape of isothermal water vapor atmospheres from Io is suppressed by the back pressure of the CPD if the back pressure exceeds the wind's critical point pressure. E.g., for a nebular pressure of 1 Pa, an escaping wind begins to blow only for &lt;em&gt;T&lt;/em&gt;&gt;400 K. If the back pressure exceeds the critical pressure, the flow reverses sign and Io accretes nebular gas.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Discussion.&lt;/strong&gt; Ram and thermodynamic pressures at the critical, transonic radius are strong functions of satellite radius and accretional surface temperature. Larger values imply free flow to the background CPD, whereas smaller values (if below ambient) imply the wind is inhibited (the pressure boundary condition at &lt;em&gt;r&lt;/em&gt;&lt;sub&gt;c&lt;/sub&gt; cannot be met). Because saturation is assumed at the surface, the actual atmospheric pressure should follow the saturation vapor curve, implying even lower transonic pressures and more stringent limits [19,20]. For cold CPD conditions consistent with the condensation of water ice (&lt;200 K) or its survival in small satellitesimals, any H&lt;sub&gt;2&lt;/sub&gt;O loss will depend critically on accretion timescale and surface temperatures reached [1]. For accretion of icy satellitesimals in warm to hot CPD conditions, such as resulting from inward drift or direct capture from heliocentric orbit, the satellitesimals themselves should actively sublimate prior to accretion. Such comet-style loss would not in itself contribute to isotopic shifts [cf. 1]. Most importantly, however, the hydrodynamic loss model discussed here is an idealization, and implicitly assumes a static background gas subnebula (or lack thereof). The reality of satellite accretion involves a nebular headwind, which will act to sweep any water vapor atmosphere away from the growing satellite. This process is daunting to model, but will both act to accelerate water vapor loss (the ultimate limit being free sublimation) but at the same time increase sublimation cooling.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Acknowledgement. &lt;/strong&gt;WBM thanks NASA&amp;#8217;s &lt;em&gt;Europa Clipper&lt;/em&gt; project.&lt;/p&gt;
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