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16. Thermal Histories of Icy Bodies

Large Icy Satellites The bodies of interest are Ganymede, Callisto, and Titan, perhaps Triton and Pluto, and with Europa as special case (because it is mostly rock). Historically, the earliest ideas for icy bodies ignored convection and postulated a conductive shell over a deep ocean. Typically, you only need to go down 100-200 km on a conductive temperature profile to gain ~200K and thereby reach 273K and melting. Since the late 1970’s, it has been recognized that solid-state convection should be “easy” in these bodies in the sense that the viscosity of water ice near its melting point is low (considerably lower than the viscosity usually attributed to the olivine–rich assemblages in the mantles of terrestrial bodies). The high efficiency of heat transport by ice (since the viscosity of ice at melting ~1014 to 15 cm2/sec) led to the view that any ocean would freeze. On the other hand, the temperature difference available to drive convection is also low. Moreover, water has the very unusual property that it has a negative melting slope for the first two kilobars of pressure (corresponding to ~200 km in bodies such as Ganymede). Of course, there may also be instances where the body is (at least initially) a mixture of ice and rock. There is also the possible presence of low melting point components, especially ammonia, in the ice mixture. Accordingly, the issue of whether the ice will melt is a complicated one. We begin with the issue of what a pure water ice mantle overlying a rock core might do. This is presumably the relevant case for Ganymede, based on the previous discussion of moments of inertia. The central question with these bodies is do they have an ocean? The issue has become less clear with the recognition that the convection cannot involve the near-surface highly viscous ice and is therefore forced to operate with a rather small ΔT in the stagnant lid regime (see last chapter). Meanwhile strong evidence has emerged for the existence of an ocean in Europa and Callisto (from magnetic fields) and Titan (from the eccentricity tide) and some evidence exists for Ganymede also. (See chapter 10; we’ll also talk about this further when we discuss magnetic fields.) We can pose the question more thoroughly as follows: 1. Would you have an ocean even if you had pure water ice and there were

no important effects from first order phase transitions in the ice?

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2. Does the answer depend on initial conditions? 3. What about the effect of phase transitions? 4. What about the effect of salts, ammonia, etc? Answer to Question #1: The answer is equivocal. It depends on the size of the body, the strength of radioactive heating in the rocky component, the (imperfectly known) rheology of ice and the (imperfectly known) scaling of convective heat transport. “Equivocal” means that it is certainly possible (but not assured) for a Ganymede -sized body. Here’s how you go about assessing this. You assume that there is an ocean (as in the next figure) and then calculate the heat flow that convection scaling predicts for a give assumption about the viscosity. The most important parameter (and biggest uncertainty) is the viscosity at the melting point. Of course, you also need to know the temperature dependence of the viscosity but that is less uncertain. Notice that there are two boundary layers, one immediately above the hypothesized ocean and one at the top of the convecting ice I.

Only the ice I rheology matters! The bottom boundary layer is quite thin because the ice viscosity is especially low there. In the pure ice case there must be an exact correspondence between the ice-ocean boundary and the freezing curve. The approximate results of the calculation are shown below.

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The way you read this is by picking off a preferred choice of heat flow and melting point viscosity and then asking whether the predicted temperature at the base of the ice is above or below freezing. (“Temperature” is of course just a proxy for the viscosity, through the very strong dependence of viscosity on T.) As you can see, present day earthlike heat flow for a Ganymede-like body and the high end of viscosity at the freezing point (1015 cm2/sec or more) will put you in the ocean field. Answer to Question #2: Notice that our above analysis assumes there is an ocean! If you assume there is no ocean, then of course you may not need a lower boundary layer, so you may not reach the freezing point. Your T profile would then intercept the ice I-ice III or II boundary along an adiabat. So initial conditions matter... if you start the body off cold then it might conceivably stay in the solid ice regime. In other words, this problem has “hysteresis” and the evolution depends on which regime you start in. The following figure shows two states that might have the same heat flow*. *The heat flows will not be exactly identical. They are only identical for a simple scaling of the stagnant lid convection in which the bottom boundary layer plays no role. In practice, ignoring the bottom of the convecting system is an approximation and numerical simulations are needed to assess the accuracy of this.

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The solution branch with no ocean is unlikely in practice for three reasons: (1) Accretional heating is large and can melt ice. (2) Early radiogenic heating is much larger and tends to push the system into the ocean field from which you must try to freeze out the ocean. (3) Phase transitions matter (answer to next question). Answer to Question #3: The ice I – III transition is endothermic and inhibits flow, much as the 660km discontinuity in earth’s mantle, so this might also favor an ocean. The standard analysis used to assess the effect of 66okm in Earth predicts that ice I-ice II has a bigger effect, other things being equal. However, comparison is tricky (because the systems operate at different Rayleigh number and because stagnant lid operates different from plate tectonics). Answer to Question #4: In all likelihood, the freezing temperature will be lower than 253K (the freezing point of pure water at 2 kilobars). This would guarantee a (thin) ocean at least for Ganymede-like bodies. You can assess this with the results presented above. In the following figure, notice that the ocean-ice interfaces will not (in general) coincide with the thermodynamic boundaries of solidus or liquidus but will be somewhere in between.

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All of the examples discussed here assume that the satellite has fully differentiated (rock core, ice mantle). The conditions for differentiation are not fully understood, but here are some relevant factors:

1. Accretional heating can be enough to melt water ice, provided the body forms quickly and/or is big enough. Callisto might have avoided differentiation by accreting over a period in excess of 105 years. Alternatively, Callisto is differentiated and the gravity data are misleading.

2. Presence of low melting point constituents may aid differentiation. However, this does not guarantee full differentiation as the problem below indicates.

3. Long-lived radiogenic heat sources can cause differentiation if the ice viscosity is high enough. It may matter that the presence of rock mixed in the ice raises the viscosity. In this case, the formation of a rock core may be delayed until hundreds of millions of years (perhaps even 1 Ga) after the body accretes.

4. Rock can also settle in the ice by Stokes flow. This only works well if the rock is almost entirely in the form of large chunks. This is improbable if the rock includes CI chondritic matter (notoriously friable), but possible if the rock has already melted then froze (because of short lived radioactive sources in the planetesimals).

In the case of Titan, elaborate stories have been developed that seek to explain the claimed evidence for cryovolcanism on the surface. See for example Tobie G, Lunine JI, Sotin C Episodic outgassing as the origin of atmospheric methane on Titan, Nature 440 61-64 (2006). These stories necessarily go well beyond the simple considerations provided here.

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In the case of Europa, the ice shell is possibly so thin that you cannot be sure that it is convective (cf problem 15.2). In the case of Pluto or Triton, oceans will only exist if there is antifreeze (ammonia) and there may be additional complications arising from clathrates (water ices containing guest molecules such as methane, carbon monoxide or nitrogen). These ices do not melt but rather decompose into water ice, releasing the trapped volatile. Problem 16.1 In an undifferentiated object consisting of rock and water and ammonia-bearing ices, the body will begin to melt at ~175K and the melt that forms will be of density ~0.92g/cc, coincidentally the same density as pure water ice. But for any amount of rock (other than zero) this means the melt will try to rise, since the coexisting solid will have an average density greater than pure water ice. If this melt does not reach the surface then it may refreeze at some shallower, cooler location. In doing so, it will create a density gradient (i.e., a region of reduced mean rock fraction and increased ammonia fraction) that is resistant to convection. As a consequence, it may turn out that convection is “forbidden” to penetrate into any region where T<175K. (a) What compositional density gradient would be sufficient to shut off thermal convection under typical conditions of interest? (All you need to do for this is to take a typical thermal gradient and the coefficient of thermal expansion α=1 x 10-4. The point of this one line calculation is to convince you that even very small compositional gradients can shut off thermal convection). (b) In the standard formulation of stagnant lid convection that we use for satellites (and terrestrial planets except earth) the convection is assumed to penetrate into a region where the viscosity is up to e9 ~104 higher than the mean viscosity of the deeper warmer regions. If part of this region is unavailable for the circulation because of stable stratification arising from non-uniform rock content then our usual formula for convective transport (eq 15.12) is incorrect. One should instead use a formulation for the possible heat flow that is based on the requirement that

gαΔTδ3 /ν(T-ΔT/2)κ ~103 where T is temperature of the deeper adiabatic region, T-ΔT =175K, and the heat flux F=kΔT/δ. (The assumption here is that the region colder than 175K is stably stratified so convection cannot penetrate to lower T. And the viscosity is evaluated half way through the thermal boundary layer).Under what circumstances does this significantly change the predicted heat flow for a given choice of T? How big is the consequence? Assume ν(T)= 1015exp[25(270/T -1)] cm2/sec, κ=0.01 cm2/sec, g=150cm/s2, α=1 x 10-4, k=2 x 105 erg/K.cm.s

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16.2 Suppose that a satellite forms in time τ. Assuming that it has a single temperature T throughout show that T≈(GM2/4πR3στ)1/4 provided that the increase in thermal energy MCp(T-T0) is small compared to the energy released from gravity. Here, M is final mass, R is radius, σ is Stefan-Boltzmann’s constant and T0 is the initial temperature of the satellite-forming material. It is assumed the satellite radiates into a vacuum. (a) Evaluate T for our Moon assuming the accretion time is 100 years.(Yes, some people think it happens that fast). (b) Evaluate T for Callisto and hence confirm that if Callisto formed fast (e.g. 10,000 years, which is not unreasonable given the short orbital period) then its ice should melt. (c)How is this result for Callisto altered if the satellite forms in a gaseous nebula with temperature T0? (d) In reality the interior must be hotter than the surface during accretion because of burial of impact heat. Could ice convection successful eliminate this heat in this short timescale of accretion without melting the ice? (You don’t need precise numbers in order to assess the likely answer to this.)