The thermal and orbital evolution of Enceladus: observational constraints and models

Francis Nimmo UCSC

Amy C. Barr PSI

Marie Behounkovˇ a´ Charles University

William B. McKinnon Washington University

Enceladus possesses a global subsurface ocean beneath an ice shell a few tens of km thick, and is observed to be losing heat at a rate of ∼10 GW from its south polar region. Two major puzzles are the source of the observed heat, and how the ocean could have been maintained. Tidal dissipation in Enceladus is ultimately controlled by the rate of dissipation within Saturn, parameterized by the factor Qp.A Qp of about 2,000 is indicated by astrometric measurements and generates an equilibrium heating rate at Enceladus sufficient to explain the observed heat and maintain an ocean indefinitely if the ice shell is conductive. If constant, this Qp would indicate an age for Enceladus much less than that of the solar system. An alternative, however, termed the “resonance-locking” scenario, is that the effective Qp is time-variable such that the heating rate is almost constant over geological time. This scenario can explain the long-term survival of the ocean and the present-day heat flux without requiring Enceladus to have formed recently.

1. INTRODUCTION details of how this actually works are not well understood. A particularly difficult aspect of satellite evolution is that Enceladus, despite its limited size, is one of the most their thermal and orbital evolution are often intimately cou- surprising bodies in the solar system with a south polar ter- pled. This is because the amount of tidal heating depends rain (SPT) containing localized active tectonics, remarkable on the satellite’s mechanical properties, many of which geysers connected to four prominent fracture sets (the “tiger are temperature-dependent. There is thus a feedback be- stripes”), and extreme heat flow. Due to these unique char- tween tidal heating and temperature evolution (Hussmann acteristics and a high astrobiological potential, Enceladus and Spohn 2004; Ojakangas and Stevenson 1986, e.g.). As has become one of the primary targets of the Cassini mis- discussed later, these feedbacks can lead to complex, non- sion. However, despite a decade of Cassini observations, monotonic behaviour. Enceladus’s thermal state, long-term evolution and even its The goal of this chapter is to discuss, and attempt to pro- internal structure still remain rather puzzling. vide an answer to, the two puzzles outlined above. In the One major puzzle is the large heat flow measured at the second section, we review the available observational con- SPT. The source of this heat must ultimately be tidal dis- straints on interior structure, heat production and loss, and sipation, arising from the periodic deformation Enceladus surface geology. The third section deals with estimates of experiences in its elliptical path around Saturn. But how the present-day tidal heat production in the ice shell, the this process works in detail, and what it is telling us about ocean and the core. In the fourth section, the tidal evolution the long-term thermal and orbital evolution of Enceladus, is of Enceladus’s orbit, and its implications for the thermal not yet clear. history, are discussed. In sections 5 and 6 we discuss how to As discussed below (and in more detail in chapter HEM- answer the puzzles of the heat output and the ocean survival, INGWAY), there is strong evidence that Enceladus pos- respectively. We conclude with a summary and suggestions sesses a global ocean beneath its ice shell. The second for future work. Other chapters relevant to this one include major puzzle is how Enceladus, a small and thus rapidly- those on the interior of Enceladus (HEMINGWAY), its geo- cooling body, could maintain such an ocean. Tidal heating logical history (PATTERSON), and the nature of the plume is a good way of maintaining such an ocean, but again the

1 source (SPENCER) and ocean (GLEIN). A recent general ice shell could satisfy the observed amplitude. The ampli- review may be found in Spencer and Nimmo (2013). tude of libration moreover suggests that the ice shell is at the thin end (21 − 26 km) of the results derived from the grav- ity/topography data alone (Thomas et al. 2016). Several ways of reconciling the gravity/topography results with the 2. OBSERVATIONAL CONSTRAINTS librations have been proposed, including elastic1 or other non-hydrostatic effects (Cadekˇ et al. 2016; Van Hoolst et al. Most observational constraints, with the exception of 2016), a delayed relaxation to equilibrium (Cadekˇ et al. the surface geology, are relevant to the present-day state 2017) or a modified definition of isostasy (Beuthe et al. of Enceladus. One of the challenges in understanding the 2016). In any event, the libration results are strong evidence long-term evolution of Enceladus is therefore how to best for an ice shell only about 20-25 km thick on average, and use these present-day constraints to infer past behaviour. perhaps as small as a few km beneath the south pole. Fig- ure 1 summarizes the constraints, showing the low-density 2.1 Internal Structure silicate core, the ice shell and ocean, and the reduced shell thickness beneath the SPT. Models of Enceladus’s interior structure rely mainly on One important consequence of the thinned shell at the shape (Thomas et al. 2007), long-wavelength topography SPT is that the ice will tend to flow laterally and fill in the (Nimmo et al. 2011) and gravity field (Iess et al. 2014) mea- hole (Collins and Goodman 2007). The rate of inflow de- surements, and also on the detection of librations - periodic pends mainly on the shell thickness and temperature struc- changes in the body’s rotation rate (Thomas et al. 2016). ture. The fact that inflow is not complete is consistent with These constraints are discussed in detail in Chapter HEM- the shell being locally thin, and might also indicate a rel- INGWAY, so only a brief summary is given here. atively cold ocean and conductive ice shell. Alternatively, Based on its bulk density and observed icy surface, Ence- inflow could be balanced by melting, which would require ladus is thought to consist of a silicate core overlain by there to be a heat source either within the ice shell, or below an H2O layer. Both the global gravity and regional grav- its base (Kamata and Nimmo 2017). ity variations at the SPT are smaller than those expected Another geophysical constraint on Enceladus’s internal from the surface topography (Iess et al. 2014), suggesting structure arises from the fact that the Enceladus plume is that the topography is compensated. One way of achieving modulated on a tidal timescale (Hedman et al. 2013; Nimmo such compensation is to appeal to a subsurface ocean and a et al. 2014). Assuming that the activity is controlled by the reduced ice shell thickness at the SPT (Collins and Good- normal tidal stress along the faults (Hurford et al. 2007), man 2007). Joint analyses of gravity and topography can the exact timing is sensitive to interior properties, most no- separate out the effects of shell thickness variations from tably the viscosity of the ice, the ice shell thickness and the those arising from the deeper internal structure (Iess et al. extent of the internal ocean. A low viscosity (lower than 2014; McKinnon 2015; Beuthe et al. 2016). These models 5·1013 Pa s) at least beneath the SPT and an ice shell thick- suggest the presence of a silicate core which is relatively ness larger than 60 km are required to explain the timing for large (≈190 km radius) and low density (≈ 2500 kg m−3). a global ocean (Behounkovˇ a´ et al. 2015). We note, however, The mean ice shell thickness is probably in the range 25- that this constraint is weaker than those above, because this 60 km, with libration results (see below) supporting the explanation for the timing is not unique (and it contradicts thinner values, and the ocean is at least 10 km thick ev- the other evidence for shell thickness). The timing could in- erywhere. Although these models commonly assume the stead be due to the delayed ascent of water through fractures presence of a global ocean, they cannot, strictly speaking, (Kite and Rubin 2016, e.g.), or perhaps a delayed response be used to prove that such an ocean actually exists. of the ocean beneath. Even prior to the gravity measurements, subsurface liq- uid water was widely assumed to exist on Enceladus. This 2.2 Heat Flow is because of the detection of sodium and potassium salts in ice grains ejected by geysers at the SPT (Postberg et al. Observations of Enceladus by the Cassini CIRS instru- 2009, 2011). Such salts could most naturally be explained ment show that the SPT is a region of locally high heat by the presence of liquid water which had interacted with flow (Spencer et al. 2006; Howett et al. 2011; Spencer et al. silicates. Furthermore, silica-rich nano-particles may be in- 2013). Initial CIRS observations estimated the total power dicative of hydrothermal reactions between water and sili- output to be 5.8 ± 1.9 GW (Spencer et al. 2006), but sub- cates (Hsu et al. 2015); hydrothermal activity is also sug- sequent higher-resolution observations incorporating data gested by the detection of hydrogen in the plume of Ence- from a broader range of infrared wavelengths suggest that ladus (Waite et al. 2017). However, these arguments do not the power output is a factor of ∼ 3 higher, 15.9 ± 3.1 GW distinguish between a regional sea and a global ocean. (Howett et al. 2011). The most recent analysis, using CIRS In contrast, the detection of a large amplitude physical libration (Thomas et al. 2016) is strong evidence for the ex- 1Note that (Cadekˇ et al. 2016) incorrectly mix top-loading and bottom- istence of a global ocean, because only a fully decoupled loading in their calculations; see (Beuthe et al. 2016).

2 Fig. 1.— Sketch of likely internal structure of Enceladus derived from gravity, topography and libration observations (see text). The South Polar Terrain (SPT) has a reduced shell thickness

observations that can spatially resolve the tiger stripes, in- ∆T is the temperature contrast across the ice shell (about dicate an endogenic power of 4.2 GW (Spencer et al. 2013), 190 K) and parameter values assumed for this and subse- with an addition 0.5 GW being liberated as latent heat in the quent equations are given in Table 1. Note that if if the ice plume and possible thermal emission coming from between shell is convecting, or some heat is being advected upwards the tiger stripes. Divided by the ∼ 70, 000 km2 surface area along fractures, the total heat loss will be higher. of the South Polar Terrain (Porco et al. 2006), the 4.7 GW Such a high rate of heat loss suggests that re-freezing of −2 corresponds to a heat flux of Fobs ≈ 70 mW m , compa- the ocean should be rapid. For instance, if a 20 GW cooling rable to terrestrial heat fluxes2. rate is not balanced by any heat production, a 30 km thick What could be the source of such high heat flows? If ocean would completely freeze in about 10 Myr. Numerical all of the radiogenic heating in Enceladus were, for some calculations (Roberts and Nimmo 2008) yield the same re- reason, concentrated in the South Polar Terrain, could Fobs sult. Thus, either we are seeing Enceladus at a special time be provided by radiogenic heating alone? The mean den- (the ocean is a recent phenomenon), or there is some heat sity of Enceladus, ρ¯ = 1607 kg m−3 (Porco et al. 2006), source capable of slowing or halting the re-freezing. We indicates that the satellite has a significant amount of rock. return to this issue in Section 6. Based on the core radius and density inferred from grav- Whether the shell is convecting or conductive is unclear. ity/topography (see Sec 2.1) the mass fraction of rock in Many models have assumed that convection is operating the interior of Enceladus is mr ≈0.7. With a present-day (e.g. Nimmo and Pappalardo 2006; Barr and McKinnon chondritic heating rate H = 4.5 × 10−12 W kg−1 (Spohn 2007; Mitri and Showman 2008; Roberts and Nimmo 2008; and Schubert 2003) and RE (the radius of Enceladus) set Stegman et al. 2009; Han et al. 2012). However, if the mean to 252 km, the total power from radiogenic heat alone is shell thickness is really ≈25 km, then a conductive state is 3 Ptot = [(4/3)πREρ¯]mrH = 0.34 GW, at least an order of more likely (Barr and McKinnon 2007) and is also required magnitude lower than the total power measured by CIRS. to permit incomplete lateral flow (see Sec 2.1). If the ice shell at the SPT is only a few km thick, as sug- gested in Cadekˇ et al. (2016), then the local heat flux there 2.3 Geological constraints (Fobs) could be explained purely by conductive heat trans- fer across the thin shell. A perhaps more likely situation The surface geology of Enceladus records a complex his- is that the shell is somewhat thicker, and that the observed tory (Nahm and Kattenhorn 2015), and provides potentially heat flux is a combination of regional conductive heat, plus important constraints on how Enceladus has evolved (Chap- local heat sources (e.g. warm water being advected from ter PATTERSON). The relative sequence of events can be depth) along the major fractures. established without too much difficulty using a combina- If the mean shell thickness is d¯, then neglecting a small tion of cross-cutting relationships and crater counting. The correction for curvature the global conductive heat loss is big problem is in deriving absolute ages, because these de- pend on inferred impactor fluxes. Even in the conventional k∆T 25 km ˙ 2 picture of (primarily heliocentric) impactors accumulating Econd ≈ 4πRE ¯ = 21 GW ¯ (1) d d since solar system formation, uncertainties in age can ex- where k is the mean ice shell conductivity (3.5 W m−1K−1), ceed a factor of 3 (Zahnle et al. 2003). If the satellites formed more recently from planetocentric debris (Cuk´ et al. 2Le Gall et al. (2017) report even higher heat flows, in excess of 1 W m−2, 2016), the uncertainties are even larger, although the ap- from analysis of Cassini microwave radiometry of the SPT (though outside parent detection of longitudinal crater density variations at the tiger stripe region proper)

3 Rhea and Iapetus (Hirata 2016) suggests that at least for onance,” in which the orbital periods of satellites are in- these satellites, planetocentric debris was not dominant. teger multiples of each other (see Section 4.1.1). Ence- The surface geology of Enceladus may be divided into ladus is presently in a 2:1 mean motion resonance with four regions which exhibit a strong symmetry around the Dione, meaning that Enceladus orbits Saturn twice each present-day tidal and rotational axes (Crow-Willard and time Dione orbits once. In this configuration, the gravi- Pappalardo 2015). The northern cratered plains are the tational perturbations from Dione keep the eccentricity of oldest and do not show evidence of tectonic deformation, Enceladus at a forced value e = 0.0047. though some craters are relaxed, indicating high heat flows The small eccentricity of Enceladus’ orbit means that (see below). The trailing and leading hemisphere areas are its distance to Saturn varies by a small amount during the more deformed, with the former being the older of the two, course of its P = 1.37 day orbit around Saturn. So the while the SPT is most heavily deformed and youngest. The height of the tidal bulge raised on Enceladus varies by a cratered plains are perhaps about 4 Gyr old and the leading small amount, as well. The cyclical raising and lowering hemisphere 1-3 Gyr old, although there are large uncertain- of the tidal bulge results in tidal deformation on a 1.37 day ties (Kirchoff and Schenk 2009). period. This motion is resisted by Enceladus’ own internal Schenk and McKinnon (2009) identified several deep friction. It is this friction which results in tidal heat. basins scattered over Enceladus which have no apparent The rate of energy dissipation in Enceladus can be re- expression in surface geology. The origin of these basins lated to the properties of its orbit and interior (Peale and is unclear, but one possibility is that they arise from com- Cassen 1978), paction of surface pores driven by localized enhanced heat 21 k R5GM 2 n fluxes (Besserer et al. 2013). E˙ = − 2 e P e2, (2) Geology and topography both thus suggest that Ence- 2 Q a6 ladus has experienced several episodes of localized activ- where G is the gravitational constant, MP is the mass of ity, with the present-day locus (the SPT) being the most re- Saturn, n = (2π/P ) is the mean motion, and a is the semi- cent. In some cases, estimates of the heat flux associated major axis of Enceladus. The ratio k2/Q in equation (2) with these episodes can be derived. In an area of the trail- describes how the interior of Enceladus responds to the ap- Giese et al. ing hemisphere, (2008) identified an apparently plied tidal potential, and depends sensitively on Enceladus’ flexural feature and used the derived elastic thickness to in- interior structure. The value k is the Love number describ- mW m−2 Bland et al. 2 fer a heat flux of at least 45 . (2007, ing how the tidal potential of Enceladus responds to the ap- Bland et al. 2012) and (2015) investigated the heat fluxes plied tidal potential, where the subscript 2 indicates the har- required to cause crater relaxation or characteristic defor- monic degree on which the tidal forcing occurs, and Q is mation wavelengths and deduced paleo-heat fluxes of a few the tidal quality factor. hundred mW m−2. Despite the ranges and uncertainties The value of the ratio k2/Q, is uncertain, and depends in these estimates, the heat fluxes derived are similar to the on the details of how the mechanical deformation of Ence- present-day SPT flux measurements, and suggest that dif- ladus’ interior is converted into heat. But one logical ques- ferent regions of Enceladus (including the cratered plains) tion to ask is: what is the value of k2/Q that could explain have experienced high heat fluxes in the past. Unfortu- the observed heat loss, and how does this compare to the nately, it is not yet clear whether there were long intervals of measured values of k2/Q for other solar system bodies? quiescence between these heating episodes, or whether the Evaluating equation (2) for Enceladus yields (Spencer and heating was continuous but simply changed location over Nimmo 2013), time. (k /Q) e 2 E˙ = 15 GW 2 , (3) 0.01 0.0047 3. TIDAL HEATING AT THE PRESENT DAY implying, for the present value of the eccentricity of Ence- 3.9 − 19 3.1. Tidal Heating: The Basics ladus and taking the global heat flow to be GW, that 0.0026 < k2/Q < 0.0127. Note that this calcula- Most of the regular satellites orbiting giant planets are in tion assumes all the measured heat is being produced at a “synchronous state,” in which the satellite’s spin period the present day. An independent constraint arises from is equal to its orbital period. Because the giant planet is the present-day characteristics of the Enceladus-Dione res- so massive, its gravity deforms the satellite, raising a tidal onance, which gives 34(k2p/Qp) < k2/Q < 41(k2p/Qp) bulge that points along the line connecting the center of the (Zhang and Nimmo 2009), where the subscript p indi- satellite to the center of the planet. cates the primary (Saturn). For comparison, fits to lunar If the planet is orbited by multiple satellites, pairs or laser ranging data indicate that k2/Q for Earth’s Moon groups of satellites can enter a resonance, in which the grav- ∼ 6.4 × 10−4 (Williams et al. 2015), a factor of 4 to 20 itational tugs on pairs of satellites occur at regular inter- lower than implied for Enceladus, whereas a similar astro- vals. The most common resonance occurring in multiple metric analysis for Io gave k2/Q = 0.015 ± 0.003 (Lainey satellite systems in the outer planets is a “mean motion res- et al. 2009). These comparisons suggest that Enceladus has

4 a warm interior that can readily deform in response to the 3.3. Ice Shell applied tidal potential, but still retains enough internal fric- 3.3.1. 1-D models tion to result in strong dissipation. Many models of tidal heating in Enceladus assume that 3.2. Where is the Heat Dissipated? the satellite is radially symmetric, meaning that its physical The details of how deformation gives rise to heat in a properties vary as a function of radius, but not location on solid planet, or a body like Enceladus which probably has the satellite. Each layer of the satellite is assumed to have an internal ocean, are not well understood. Enceladus has a viscoelastic rheology; the simplest viscoelastic model is probably fully differentiated over the course of its history that of Maxwell, in which each layer is characterized by (Schubert et al. 2007), and so its interior likely harbors a a viscosity and rigidity. For a satellite with a viscoelastic central rock core and a shell of rock-free ice. Any ocean rheology, k2 is a complex number, and Q is related to the in Enceladus would sit perched between the base of the ice imaginary part of k2 (Zschau 1978; Segatz et al. 1988). shell and the top of the rock core (see Figure 1). Values of k2 and Q can be obtained by representing It not known whether the tidal heat occurs mostly in the the interior of the satellite as a series of layers, each with solid portions of Enceladus (e.g., the ice shell or rock core), its own viscosity and rigidity (Ross and Schubert 1989; the ocean, or potentially at the interfaces between these Roberts and Nimmo 2008; Barr 2008). Because the viscos- layers (e.g., frictional dissipation from the ocean sloshing ity is a strong function of temperature, different assumed against the surface of the rock core). It is also not known at temperature profiles (e.g. conductive vs. convective) can what spatial scale the dissipation occurs. The most common lead to quite different dissipation results. Values of k2 type of tidal heating model assumes that dissipation occurs are calculated using the correspondence principle (Sabadini at the microphysical level: the ice/rock is assumed to have et al. 1982; Tobie et al. 2005; Roberts and Nimmo 2008; a viscoelastic rheology (Findley et al. 1989) , where the de- Wahr et al. 2009). Calculating k2/Q for a convecting ice formation of the material is thought to be accommodated by shell, and substituting into equation 2, yields a tidal heating one of the usual microphysical mechanisms (e.g., volume rate about a factor of 10 lower than observed by CIRS (Barr diffusion, easy slip, dislocation creep, see e.g., McCarthy 2008). and Castillo-Rogez 2013 for discussion). Another possi- An alternative viscoelastic formulation, the Andrade bility is that tidal dissipation is localized along the tiger model, may be more appropriate (McCarthy and Castillo- stripes, either as frictional dissipation along faults (Nimmo Rogez 2013). Fig. 2 shows global tidal heat production et al. 2007) or from turbulent dissipation in water-filled generated with such a model (Behounkovˇ a´ et al. 2015). In cracks (Kite and Rubin 2016). Thus, the tidal heat could Fig. 2a the shell is assumed to have a uniform viscosity, be dissipated in the ice shell of Enceladus, in the ocean, appropriate for a shell which is convecting. For sufficiently in the rock core, or in a combination/all of these locations. low viscosities, the total power output can equal or exceed Each of these alternatives has been explored, with varying that detected at the SPT (shaded region). Heat flow in- degrees of success. creases with decreasing shell thickness because the ampli- If an ocean is not present, the ice shell is then coupled tude of tidal deformation increases with a thinner shell. For to the core, which is nominally less deformable. As a re- a conductive shell, more likely given the thin shell deduced from libration measurements, the total heat production is sult, k2/Q and the amount of tidal dissipation are expected to decrease significantly (Nimmo et al. 2007). It is there- lower (Fig. 2b) and is exceeded by the conductive heat loss fore often argued that the ocean must be a long-lived fea- (dotted line). ture, because if the ocean freezes completely, tidal heating A significant problem with all one-dimensional models is reduced and the ice shell will never re-melt. However, is that, while they predict spatial variations in tidal heat as discussed in Section 4.1.3, this argument is not correct production, they do not explain why only the south pole is in detail, because tidal heating naturally tends towards an currently a hot-spot. All such models result in a symmetri- cal distribution of heat production around the equator. equilibrium value, independent of the satellite k2/Q. The remainder of this section will describe the numerical 3.3.2. 2-D and 3-D models and analytical models that have been constructed to explain the magnitude and distribution of heat coming from Ence- Two types of more sophisticated model have been pro- ladus at present. In 3.3, we describe tidal heating in the posed to explain the distribution and magnitude of tidal heat ice shell of Enceladus, in 3.4 we summarize the results of coming from the south pole of Enceladus. One possibil- calculations of dissipation in the ocean, and conclude by ity is that tidal heating is dissipated on the macro-scale, as discussing the possibility of tidal heating in the rock core frictional heating on faults (Nimmo et al. 2007). In this (section 3.5). The time-evolution of heat generation and model, heating is generated in the cold, near-surface ice transport will be discussed in Section 4. on Enceladus by cyclical strike-slip motion along faults in the centers of the tiger stripes. This model provides a nat- ural explanation for the localization of endogenic heating along the tiger stripes observed by CIRS (Nimmo et al. 2007). However, it is difficult to produce enough heat by

5 Fig. 2.— Present-day tidal heating as a function of ice shell thickness and a viscosity using an Andrade rheology (Behounkovˇ a´ et al. 2015); a) heat production for a constant viscosity (solid line) b) tidal heating (solid lines) and heat loss (dashed line) for a conductive temperature profile and temperature-dependent viscosity (with maximum viscosity 1020 Pa s); gray area the present day heat loss through the SPT (Howett et al. 2011, 2013). this mechanism to explain the 5.8±1.9 GW of power im- and calculate both the local dissipation and the background plied by the Spencer et al. (2006) analysis (Ingersoll and tidal response of the shell. Tidal deformation within the Pankine 2010), let alone enough heat to explain the higher SPT is significantly enhanced by the presence of these weak power output implied by later observations. Furthermore, zones, suggesting an additional way of localizing heating. this model appealed to direct sublimation of water ice to Whether this kind of model is also consistent with the ob- explain the plume, which is inconsistent with the salts de- served modulation of the plume remains to be seen. tected there (Postberg et al. 2009, 2011). A variation on this theme also appeals to dissipation 3.4. Ocean within the tiger stripes, but in the form of turbulent dissi- In principle, tidal heating in the ocean of Enceladus pation in ocean water within the tiger stripes (Kite and Ru- could be a significant source of heat. In particular, if Ence- bin 2016). However, this model appears unable to explain ladus has a non-zero obliquity (tilt), obliquity-driven ocean the higher power outputs implied by some observations, and tides could contribute to the overall heat budget (Tyler even for 5 GW requires a substantially thicker ice shell be- 2009). Unfortunately, however, dissipation drives satellites neath the SPT than is now considered likely. towards a so-called Cassini state, which for Enceladus re- A second set of models build on the idea of calculating sults in an obliquity far too small to generate significant heat k2/Q assuming a Maxwell rheology for the ice and rock, (Chen and Nimmo 2011). Additionally, the resonances re- but relaxing the assumption that the properties of Ence- sponsible for driving ocean dissipation only operate when ladus’ interior are radially symmetric. The basic idea is oceans are thin, typically of order 1 km or less (Tyler 2011; that both the tidal forcing and the temperature structure of Hay and Matsuyama 20017). The results discussed above the ice shell of Enceladus should vary as a function of lat- suggest that the ocean on Enceladus is 10 km or more deep. itude and longitude. If the ice shell is convecting, one ex- Last, the presence of an elastic lid will reduce the ocean pects lateral variations in the temperature structure of the dissipation even further (Beuthe 2016). Thus, ocean cur- shell: warm convective upwellings might experience more rents are unlikely to be a significant source of tidal heat intense tidal dissipation (Wang and Stevenson 2000; Sotin in present-day Enceladus. It is possible that ocean dissi- et al. 2002; Tobie et al. 2003; Mitri and Showman 2008; pation could become more important as the shell thickened, Behounkovˇ a´ et al. 2010; Behounkovˇ a´ et al. 2012). If the thereby preventing total ocean freezing, but even in this case SPT were underlain by a single convective upwelling but the small obliquity limits the total amount of heat produced. the north pole was not (e.g., Han et al. 2012; Rozel et al. Another possible heating mechanism is Joule heating, in 2014), this might explain why all of Enceladus’ tidal heat is which electric currents passing through water pockets of coming out of one location. Alternatively, the shape of the low salinity buried kilometers beneath the surface (Hand core might influence where upwellings and dissipation were et al. 2011). Estimates of the magnitude of this heat source focused (Showman et al. 2013). As a last resort, one can imply that the heat source would be small (something like always appeal to an impact to break the initial symmetry 0.001 to 0.25% of the observed heat flow) (Hand et al. (Roberts 2016, e.g.); the impact site would then reorient to 2011). whichever pole was closer (Nimmo and Pappalardo 2006). The most sophisticated model to date (Soucekˇ et al. 3.5. Rock Core 2016) incorporates aspects of both approaches. These au- At face value, one would expect the rock core of Ence- thors introduce weak discontinuities (faults) within the SPT ladus to be too stiff to experience much tidal heating

6 (Ross and Schubert 1989; Roberts and Nimmo 2008). In at particular frequencies (Ogilvie and Lin 2004; Wu 2005). a Maxwell viscoelastic solid, dissipation is maximized Since a satellite’s orbital frequency depends on its semi- when the period of the cyclical forcing is comparable to major axis, Qp can vary from satellite to satellite, and can the Maxwell time, τM = η/µ, where η is the viscosity of vary in time for a single satellite. the material, and µ is its rigidity. For rock, with η ∼ 1020 The tidal torque exerted by the primary is a strong func- Pa s, and µ = 100 GPa (Barr 2008), τ = 31.7 years, far tion of semi-major axis. It is therefore often the case that longer than the 1.37 day orbital period of Enceladus. (By the inner satellites migrate outwards faster than more dis- comparison, for ice at its melting point, with η ∼ 1014 Pa tant satellites. For example, Fig 3a shows a hypothetical s and µ ∼ 3 GPa, τ = 9 hours.) Thus, it is reasonable to scenario in which Qp is set to a constant value of 16,000 expect that much, or all, of the tidal dissipation would be (dashed lines). The orbits are generally converging, except concentrated in the ice shell of Enceladus. that Tethys moves outwards faster than Enceladus, because Evidence reviewed in Section 2.1 suggests that the sili- Tethys is much more massive and thus raises larger tides on cate core has a low density, most likely due to porosity. The Saturn. This simple calculation suggests that Qp must ex- presence of such (probably water-filled) pores would alter ceed 16,000 unless the age of Mimas is less than that of the the material properties of the core and potentially render it solar system (but see Sec 4.2). more conducive to tidal dissipation (Roberts 2015). Tidal flushing of water through the pores could contribute an ad- ditional heat source, although the only study on this topic 4.1.1. Mean motion resonances to date (Vance et al. 2007) concludes that the contribution is minimal. Alternatively, a core which started warm might An important consequence of the relative outwards mo- be sufficiently dissipative to maintain its temperature over tion of the satellites is that they may have encountered the long term. We note that a warm core is also suggested commensurate locations, in which the inner orbital fre- by the existence of silica nano-particles (Hsu et al. 2015) quency is a simple integer ratio of the outer orbital fre- and hydrogen gas (Waite et al. 2017) in the plume Thus, al- quency. For instance, Enceladus and Dione are currently though little work has been done to date, dissipation in the in a 2:1 eccentricity-type mean motion resonance (MMR). core remains a possibility. Such resonances are extremely important, because they pro- vide a way of exciting orbital eccentricities and thus a po- tential long term heat source (see Sec 4.1.2). In simple 4. TIDAL AND ORBITAL EVOLUTION terms, in an MMR the satellites encounter each other at the same point in succeeding orbits; the mutual orbital pertur- bations thus add coherently, leading to large eccentricities. In detail, several different resonances exist around 4.1. Introduction and general picture a particular commensurate location. For instance, the Enceladus-Dione 2:1 eccentricity commensurability is actu- ally a triplet of resonances: e , e -e and e (the present At a basic level, it is clear that the energy budget of Ence- D D E E resonance) (Meyer and Wisdom 2008a; Zhang and Nimmo ladus must be dominated by tidal heating. Thus, in order 2009). For converging, circular orbits passing through a to understand its long-term thermal evolution, its long-term commensurate location, capture into at least one of these orbital evolution must also be understood. Orbital dynam- resonances is assured as long as the satellites are migrating ics is a complicated subject. Below, we try and summa- sufficiently slowly. Whether or not a particular resonance rize in relatively simple terms the key processes operating. eventually breaks depends on the details of the resonance, More in-depth explanations can be found in e.g., Murray and the extent to which the satellites’ eccentricity is excited and Dermott (1999) and Peale (1999) (higher eccentricities favour resonance breaking). Thus, An orbiting satellite will raise tides on the primary. The for example, a hypothetical ancient Mimas-Enceladus 3:2 resulting tidal bulge will not in general point directly to- e resonance cannot have occurred, because there is no wards the satellite (Fig 3a,inset) ; there will be a lag, with E apparent way to break the resonance (Meyer and Wisdom the magnitude of the lag depending on the rate of energy 2008a). dissipation in the primary. The lag will also cause torques A commensurability of j : j − k has order k and is de- on the satellite; for satellites beyond the co-rotation point, fined by (j−k)n = jn where j is an integer and n and n the torques will cause outwards motion. The Moon is mov- i o i o are the mean motions of the inner and outer satellite. Thus, ing away from the Earth because of tides raised by the 3:2 and 2:1 are both first-order resonances. Capture into Moon on the Earth. higher order resonances has a lower probability than cap- The lag angle (δ/2) may be derived using δ ≈ 1/Q , p ture into first-order resonances (Dermott et al. 1988). On where Q is the so-called dissipation factor of the primary p the other hand, higher-order resonances can lead to higher (Murray and Dermott 1999). A small Q implies a large p equilibrium heating rates (see Sec 4.1.2). lag angle and high dissipation rates. In general, Q is ex- p In contrast to converging orbits, satellites on diverging pected to be frequency-dependent, with peaks in dissipation orbits will pass through commensurate locations without

7 Fig. 3.— a) Outwards satellite migration assuming constant Qp. Dashed lines show evolution when Qp=16,000 everywhere. Solid lines show evolution when Qp is set to a different (constant) value for each satellite, based on astrometric observations of present-day migration rates (Section 4.3). From inner to outer body values for Qp are 3000, 1920, 2600, 2540, 325 and are calculated using equation (9) and the observed migration rate. Inset shows geometry of tidal lag angle δ/2 (see text). b) Outwards evolution based on “resonance locking” scenario (Section 4.3 and see Fuller et al. (2016)). Present-day migration rate is based on the same present-day Qp as in a) and evolution of semi-major axis is calculated using equation (10) for each satellite with tα set to 4, 8, 12, 51 and 49 Gyr, respectively. Example mean-motion resonances are marked, red for diverging orbits and green for converging. getting trapped into resonance. The eccentricities may be heating rate is given by transiently excited (Dermott et al. 1988) but the effect on   the total energy budget will be very small. niTi noTo Ti + To mi mo E˙ eq = + − GM + . p 2 p 2 L + L a a (1 − ei ) (1 − eo) i o i o (4) 4.1.2. Equilibrium eccentricity and heating Here n,e,m,a and L refer to the mean motion,eccentricity, mass, semi-major axis and orbital angular momentum For an isolated satellite, the effect of outwards motion of a satellite, with subscripts referring to the inner i due to torques raised by the primary also results in eccen- and outer o body. The angular momentum is given by p 2 tricity growth. But a satellite in an eccentric orbit experi- m GMpa(1 − e ) with Mp the mass of the primary. T ences diurnal tides which dissipate energy internally (equa- represents the torque from the primary acting on the satel- lite, where tion 2) and cause the orbit to circularize. Because the cir- 2 5 cularization timescale is usually much shorter than the out- 3 Gm Rpk2p T = 6 (5) wards migration timescale, isolated satellites are expected 2 a Qp to have near-zero eccentricity. with Rp and k2p the radius and tidal Love number of the However, the situation changes if two satellites are in an primary, respectively. Inspection of equations (4) and (5) eccentricity-type MMR. In an MMR eccentricity growth is show that the satellite k2/Q is not required and that the ma- much more rapid, and can become comparable to the rate of jor unknown governing the equilibrium heat production rate circularization. One possible result of these competing pro- is Qp. Physically, what is happening is that the energy dissi- cesses is that the eccentricity reaches an equilibrium value, pated in the satellites is ultimately coming from the rotation in which eccentricity excitation due to the MMR is balanced kinetic energy of the primary (a large reservoir); the rate at by the eccentricity damping in the satellite, and the total which this energy can be extracted is controlled by the tidal tidal heat production rate is constant. torque, which depends on Qp. If the satellites are in this equilibrium situation, a very For Enceladus, the equilibrium heat production can thus important result arises: the total rate of energy dissipation be readily calculated (Meyer and Wisdom 2007). If Qp is in the satellites is then independent of the satellite struc- constant, the equilibrium heat production rate is ture. This is extremely useful, because in general the rate   of satellite heating requires its k2/Q, and thus its internal 18, 000 E˙ eq = 1.1 GW (6) structure, to be known (Section 3.1). The partitioning of Qp heat between the two satellites does require their k2/Q to be specified, and so too does calculation of the actual equi- This is an extremely important result, because estimates librium eccentricity; but for thermal evolution calculations, of the current heat output of Enceladus cluster around these are relatively minor disadvantages. 10 GW (see Section 2.2). As discussed in more detail The equilibrium tidal heating rate can be determined by below, there are at least three possible resolutions to this consideration of conservation of mass and angular momen- apparent problem: 1) Enceladus is currently producing heat tum (Meyer and Wisdom 2007; Fuller et al. 2016). The in excess of the equilibrium rate; 2) Enceladus is producing

8 heat at the equilibrium rate, but releases it episodically; 3) the importance of considering orbital and thermal evolution Qp is much smaller than 18,000 at the present day. together.

4.1.3. Periodic Behaviour 4.2. Astrometry

In its current resonance, the eccentricity evolution of The instantaneous outwards evolution of an isolated Enceladus can be described by an equation of the following satellite depends on the torque exerted by the primary form (Ojakangas and Stevenson 1986; Meyer and Wisdom (equation 5) and is given by 2008b):  5 1 da k2p m Rp 1 de 2 2
= 3 n = . (9) = ae 1 − be (7) a dt Q M a t dt p p tide where a and b are constants, with b depending on the k2/Q Accordingly, if da/dt can be measured, then Qp (at that of the satellite. The first term in brackets is the eccentric- particular frequency) can be inferred. ity excitation associated with the MMR, while the second is The situation is slightly more complicated if two satel- the eccentricity damping associated with dissipation√ in the lites are in a MMR. For the commensurability to be main- satellite. The equilibrium eccentricity eeq = 1/ b and is tained (1/ai)dai/dt = (1/ao)dao/dt. If the torque on the inner satellite dominates, the outwards evolution timescale thus dependent on the satellite structure. The satellite struc- √ √ ture, however, is itself evolving at a rate which depends on ttide will be reduced by a factor of [1 + (mo ao/mi ai)] the extent to which temperature changes affect k2/Q. There because the torque is now changing the angular momentum are thus two timescales of interest: the timescale for the of both satellites. For Enceladus-Dione this factor is large eccentricity to equilibrate; and the timescale for k2/Q to (≈13) because Dione is so much more massive. change significantly. Recently, a combination of long-baseline Earth-based Neglecting coefficients of order unity, the eccentricity observations and Cassini spacecraft observations has al- equilibration timescale is (Meyer and Wisdom 2008b) lowed measurements of da/dt to be made for various Sat- urnian satellites (Lainey et al. 2017). These measurements  5   2 1 1 mo ai ai Qp Qp can then be converted to Qp at different frequencies using τe ≈ ≈ 9 Myr 9 ni eeq mi ao Rp k2p 18, 000 equation (9). For Enceladus, two different approaches yield −5 −5 (8) k2p/Qp = 18.1 ± 3.1 × 10 and 27.1 ± 13.5 × 10 . where Ri is the radius of Enceladus, eeq is the equilibrium For the canonical Saturn k2p of 0.34, the implied Qp val- eccentricity (taken to be the present-day value of 0.0047) ues are 1880 ± 280 and 1260(+1240, −420). These results and we have taken k2p=0.34 (Gavrilov and Zharkov 1977). take the Enceladus-Dione MMR into account (R. Jacobson, This timescale is fast compared to the thermal diffusion pers. comm.) and confirm earlier astrometric suggestions timescale of a body the size of Enceladus; it becomes that Qp ≈ 2000 for Saturn (Lainey et al. 2012). Values for shorter if Saturn is more dissipative (lower Qp). Qp derived from Tethys and Dione observations are simi- If the timescales of eccentricity equilibration and ther- lar to those from Enceladus. For Rhea, Qp ≈ 300 with an mal adjustment are similar, an interesting result occurs: uncertainty of about 10%. the orbital evolution can experience periodic behaviour These estimates of present-day Qp are subject to large (Ojakangas and Stevenson 1986; Hussmann and Spohn uncertainties. Nonetheless, it seems very likely that, in 2004; Meyer and Wisdom 2008b). The total heat production contrast to Jupiter, the present-day Qp of Saturn is signif- and the eccentricities of both satellites oscillate around their icantly smaller than the canonical, time-independent value equilibrium values. Thus, Enceladus might currently be in of 18,000 (Section 4.1). a high-heat-production part of the cycle. The long-term average heat production, however, is still fixed at the equi- 4.3. “Resonance Locking” librium value (equation 6). We will return to these issues in more detail below. Another important consequence of the long-term aver- The low Qp values derived from astrometry (see Sec- age heat production being independent of the satellite k2/Q tion 4.2) have two immediate consequences. First, the concerns the fate of the ocean (Section 3.2). If the ocean equilibrium heat production rate in Enceladus (equation 6) freezes completely, then k2/Q of the satellite will be re- could easily be 10 GW or more. This issue is discussed in duced, and so will the instantaneous heat production. How- more detail in Section 5.3 below. Second, if Qp is constant, ever, as equation 7 shows, a reduction in k2/Q will drive the then both Enceladus and Mimas must be much younger than eccentricity to higher values and thus increase the heat flux the age of the solar system (Figure 3a). Recent work, how- back to the equilibrium value. If the equilibrium heating ever, shows that a low present-day Qp does not necessarily is not enough to prevent ocean freezing, it will presumably require young satellites (Fuller et al. 2016). This is an im- not cause re-melting. Nonetheless, this example illustrates portant result, which we discuss in more detail in the rest of this section.

9 Fig. 4.— Cartoon showing how the resonance locking scenario operates. a) Satellite is close to a dissipation peak at absolute synodic frequency f1 and migrates outwards, increasing its frequency. b) At frequency f2 the satellite experiences less dissipation and stops migrating. c) The peak evolves to higher frequency as Saturn evolves, so dissipation in the satellite increases again. The outwards migration rate of the satellite is thus controlled by the timescale tα for Saturn to evolve.

Because the major Saturnian satellites are outside the co- which the rotation rate of Saturn evolves. Because this latter rotation point, from a reference frame rotating with Saturn timescale is expected to be long, we will set tp = ∞ in the the satellites move in a retrograde fashion, with the outer following. satellites having a more negative synodic frequency. In this Equation 10 can be used to derive a simple expression reference frame, outwards satellite migration thus results in for the change in satellite mean motion: an increase in the absolute (synodic) frequency of the tides   t − tnow raised on Saturn. n(t) = Ωp + (nnow − Ωp) exp (11) As discussed above, the dissipation spectrum of Saturn tα is likely composed of discrete peaks. A migrating satel- where the mean motion is set to nnow at time tnow. Of lite may thus sweep through one of these peaks as its orbit course, equation 9 can still be used to derive the effective evolves. Alternatively, a satellite may encounter a peak be- Qp governing a particular rate of outwards evolution. But cause the location of the peaks changes as Saturn’s internal in this approach, Q is not really the fundamental quantity; structure evolves. Fuller et al. (2016) consider this latter p it is a derived quantity, which depends ultimately on tα, the scenario, in the specific case of a peak evolving to higher timescale of Saturn’s evolution. frequencies (Fig 4) The behavior described by equation 11 is very different As the peak approaches the satellite frequency, dissipa- from the conventional picture of satellite orbital migration tion in Saturn increases and the satellite experiences a grow- (Fig 3a). Instead of evolving outwards at a rapidly decreas- ing outwards torque (equation 5), moving the satellite out- ing rate, the pace of orbital migration is almost constant. wards and increasing its absolute synodic frequency. This Fig 3b shows outwards satellite migration calculated using increase, however, moves the satellite away from the dis- equation 11 with the present-day values of tα derived from sipation peak, so that its outwards motion slows down un- astrometry (Lainey et al. 2017). Crucially, the present-day, til the peak catches up again. The rate of outwards evolu- relatively rapid, outwards migration can still be reconciled tion of the satellite is thus governed by the rate at which with the satellites being as old as the solar system. Another the frequency of the Saturnian dissipation peak changes. way of interpreting this result is that the effective Qp that a So the satellite orbit evolution timescale (equation 9) is set given satellite experiences has decreased substantially over t by the timescale α with which Saturn’s interior structure the course of satellite orbital evolution. For instance, Fig 5 t evolves. Note that α can in principle be different at differ- shows that in the case of Enceladus the present-day value ent frequencies, although the physical justification for such of Q is 1,920 (and t =5.7 Gyr) while at 2 Gyr B.P. (be- Fuller et al. p tide an assumption seems weak. (2016) term their fore present) Q was 18,700 (with n = 7.8 × 10−5 s−1 and scenario “resonance locking” because it involves the satel- p ttide=10.7 Gyr) (equations 10 and 11). lite orbital frequency being pinned to a particular resonance A second important consequence of Fig 3b is that the within Saturn. specific resonances that satellites encounter are quite dif- Fuller et al. (2016) give the resulting orbit evolution ferent from those of conventional, constant Q scenar- t p timescale tide as: ios. For Enceladus in particular, the Mimas:Enceladus 3:2     1 1 da 2 Ωp 1 1 1 resonance is never encountered, which is convenient be- = = − − (10) cause this particular resonance appears impossible to break ttide a dt 3 n tα tp tα (Meyer and Wisdom 2008a). Prior to the present day 2:1 where Ωp is the rotation angular frequency of Saturn, n is resonance with Dione, the only major resonance Enceladus the mean motion of the satellite and tp is the timescale over encountered is a 4:3 with Tethys, but since the two bodies

10 are on diverging orbits this resonance is unlikely to have Meyer and Wisdom (2008b) investigated this possibility played a major role. and concluded that Enceladus was unlikely to be experi- A note of caution is required here. First, the astrometric encing periodic behavior, essentially because the eccentric- measurements used to derive tα for the various satellites are ity equilibration timescale (equation 8) was significantly subject to large uncertainties. Even small variations in tα shorter than the thermal adjustment timescale. More re- can lead to substantially different orbital histories. For in- cently, Shoji et al. (2014) used a slightly different approach stance, if tα is set to 50 Gyr for all satellites, then the orbits and found that, for some parameter combinations, periodic are all diverging, prohibiting establishment of resonances. heating could occur (Fig 6a). In principle, such periodicity Second, the evolution shown in Fig 3b is highly simplified could potentially explain the ancient, high-heat flux terrains because it ignores the effect of MMRs on orbital evolution. as well as the present-day heat loss. However, the devia- Thus, for instance, if the Enceladus:Dione 2:1 resonance tion around the long-term average heat production value is were established early, their evolution could have been quite rather small, and at least for a constant Qp of 18,000 even different. Third, the resonance locking theory is heuristic, the peak heat fluxes are still insufficient to explain the ob- in that it uses a very simple description for how the inter- served heat loss. Furthermore, the interval between heat- nal structure evolves. Especially early in Saturn’s history, ing episodes appears short compared to the geologically- evolution was probably quite rapid; thus, the early stages inferred intervals. Although more work could be done, on of orbital evolution and the initial locations of the satellites its own periodicity seems unlikely to be able to explain the (Fig 3b) are highly uncertain. Lastly, the behaviour invoked present-day heat loss. by Fuller et al. (2016) requires that there exists a dissipation peak close to the frequency being excited by the satellite, and that the peak increases in frequency with time. Detailed 5.2. Is Enceladus experiencing episodic heat loss? modelling will be needed to explore how reasonable these assumptions are for Saturn. An alternative possibility is that Enceladus builds up heat in its interior at the equilibrium rate of 1.1 GW, and then releases it episodically. The same duty cycle argument ap- 5. EXPLAINING THE PRESENT-DAY HEAT FLUX plies as in the preceding section: if correct, we are seeing Enceladus in a state it occupies only about 10% of the time. O’Neill and Nimmo (2010) used numerical convection As summarized in Section 2, the present-day measured models to show how this process could work. For a suf- heat output of Enceladus is an order of magnitude larger ficiently low effective yield strength, heat build-up in the than the equilibrium tidal heating value for a conventional interior eventually gives rises to stresses which cause large- scale overturn of the near-surface ice and transiently high Saturn Qp of 18,000 (equation 6). There are at least three possible explanations for this discrepancy. 1) Tidal heat- heat fluxes (Fig 6b). These episodes recur every 0.1-1 Gyr ing on Enceladus is periodic, with the present-day heating and tend to be localized, rather than global. An advantage of rate exceeding the long-term average. 2) The rate of heat this model is that it provides a potential explanation for the loss on Enceladus is episodic, with the present-day loss rate inferred ancient episodes of relaxation and tectonic defor- mation, and the localized present-day heat loss. One draw- exceeding the long-term average. 3) The effective Qp of Saturn is an order of magnitude smaller than the conven- back of this analysis is that it is both 2D and Cartesian, and tional value. Below we discuss each of these possibilities in it is unclear whether the same results would arise in a 3D turn. spherical setting. Additionally, more recent work suggests In this context, the geological observations (Section 2.3) a thinner present-day ice shell (Section 2), which makes are of great importance. In particular, evidence of ancient convection correspondingly more difficult to achieve. And (several Gyr B.P.) heating episodes in different regions of if convection is really operating, rapid lateral flow of ice Enceladus can potentially distinguish between different sce- would make the maintenance of a thin shell over the SPT narios. hard to explain. This explanation seems somewhat more promising than invoking periodic heat production, but it still suffers from 5.1. Is Enceladus experiencing periodic tidal heating? the problem that it requires us to be seeing Enceladus at a special time. More work could undoubtedly be done in this area: for instance, it would be desirable to self-consistently Section 4.1.3 noted the possibility that Enceladus might couple tidal heating into such models, and to explicitly deal be experiencing time-variable heating. If so, we could hap- with the onset or turn-off of convection as the shell thickens pen to be seeing Enceladus at a special time when the tidal or thins. heating rate greatly exceeds the equilibrium value. Since the heat flow is about ten times the conventional equilib- rium value, the probability of this eventuality is about 10%.

11 Fig. 5.— Variation of equilibrium heat production rate for Enceladus and effective Qp as a function of time, based on the resonance locking scenario shown in Fig 3b. Qp is calculated using equations (9) and (10) and heat production using equation (4) with To=0.

Fig. 6.— a) Periodic heat production, modified from Shoji et al. (2014) The dashed line gives the equilibrium heat production rate (1.1 GW). b) Episodic heat loss, modified from O’Neill and Nimmo (2010). Heat flux was converted to heat flow by assuming the area of the South Polar terrain (ten percent of the total area).

12 5.3. Is the effective Q of Saturn small? Global conductive heat losses are given by equation 1. This equation illustrates the difficulty of maintaining a If the astrometry results (Section 4.2) are correct, then global ocean. For instance, to maintain water at 270 K at the Q of Saturn is small, about 2000 at Enceladus frequen- the base of a 65 km thick ice shell (the likely maximum), cies. This in turn implies that the equilibrium heat produc- a power of 8 GW is required. This requirement is only re- tion rate is of order 10 GW, roughly consistent with the ob- duced to 5 GW even if the water is at 200 K due to the served heat flux. The conventional objection to this sce- presence of antifreezes such as salts or ammonia. Surface nario is that such a low Qp implies young satellites. But as porosity could help by reducing the near-surface thermal discussed in Section 4.3, if the “resonance locking” mech- conductivity, but this process is self-limiting (Besserer et al. anism is correct, a low present day Qp does not require 2013) and probably makes only a small difference. The real young satellites (Fig 3b). situation is probably even worse, because any heat gener- As an additional benefit, application of the resonance ated in the shallow ice shell will be conducted out rapidly locking mechanism means that the equilibrium tidal heat and contribute little to the global energy balance. Convec- production varies only slowly with satellite semi-major tive heat transfer, if it is occurring, will also increase the axis - which is not the case with a constant-Q assump- required heat production. These simple calculations make tion. For example, if the present-day ttide of Enceladus it clear that, for an ocean to survive for more than a few tens is 5.7 Gyr, then neglecting the torque on Dione (To=0), of Myr, the heat production rate must be much greater than equation (4) gives a present-day equilibrium heat produc- the conventional equilibrium value of 1.1 GW. tion rate of 22 GW. If the bodies were in resonance 2 Gyr Figure 7 provides an illustrative demonstration of these ago, ttide=10.7 Gyr and the corresponding equilibrium heat results. It models the evolution of a conductive, differ- production rate was 15 GW (Fig 5). Note that these dissipa- entiated Enceladus using the methodology of Nimmo and tion rates do not cause any kind of energy crisis for Saturn: Spencer (2015). The body is initially cold (150 K uniform 20 GW output over 4 Gyr only represents 2 × 10−7 of Sat- temperature) and heated only by radioactive decay in the urn’s rotational energy. core, insufficient to cause melting. At 1 Gyr the internal We again caution readers that the heat production num- heat production in the core is increased by 20 GW to sim- bers are subject to considerable uncertainty. First, the ulate an episode of tidal heating (in reality, tidal heating astrometrically-derived values (e.g. ttide) themselves are will likely also occur in the ice shell, but this is more com- quite uncertain. Second, accounting for the effects of the plicated to model). The heat produced exceeds the shell’s MMR introduces additional uncertainties. Meyer and Wis- ability to transport heat conductively (equation 1) and thus dom (2007) showed that including the torque on the outer the shell begins to melt. Melting continues until the shell body (equation 4) can reduce the total heat production by a reaches an equilibrium thickness of about 20 km, at which factor of around 2. Third, the distribution of heat between the heat produced and the conductive heat loss are balanced. the inner and outer body cannot be derived without knowing If some fraction of the tidal heat were being advected to the the values of the satellite k2/Q. near-surface (e.g. through fluid-filled fractures), the equi- Nonethless, and despite the uncertainties, in our view librium shell thickness would be larger. In any event, these the resonance locking scenario solves many problems. The conditions result in an internal structure similar to that de- present-day observed heat loss does not require special duced for present-day Enceladus (Section 2). Tidal heat pleading, but is simply due to the high equilibrium tidal pro- production is (arbitrarily) stopped at 2 Gyr, at which point duction at Enceladus. This result by itself does not explain the ice shell starts to refreeze, with the freezing process why the heat loss is concentrated in the SPT, or how heat complete in around 100 Myr. A similar calculation using is partitioned between different transport mechanisms. Per- 1.1 GW tidal heating shows that, as expected, melting never haps more importantly, as we will see below, a high heat occurs, even in the presence of antifreeze. flux also helps maintain a liquid ocean for the long term. Of course, this simulation is highly simplified. But it shows that the present-day structure of Enceladus is com- patible with an equilibrium situation, where a thin conduc- 6. LONG TERM THERMAL EVOLUTION tive ice shell is being maintained by a high rate of tidal heat production. When this equilibrium was established is un- The long term thermal evolution of Enceladus depends known: it depends on when Enceladus and Dione entered mainly on the competition between heat production and the 2:1 resonance, and when the orbital frequency became heat loss. The present-day radioactive heat contribution is commensurate with some resonant frequency inside Saturn. about 0.3 GW (Section 2). Primordial stored heat (e.g. from While the geological evidence suggests that tidal heating early 26Al decay) is unlikely to play a role, because the con- has been important over various epochs in Enceladus’s his- duction timescale of the silicate core is of order 100 Myr, tory, it does not reveal whether this heating was episodic much smaller than the age of the solar system (Roberts and in nature, or was continuous but changed its location with Nimmo 2008). The long-term average tidal heat production time. is 1.1 GW for a conventional Saturn Q, but could be up to 22 GW in the resonance-locking case (Section 5.3).

13 Fig. 7.— Thermal evolution of a conductive Enceladus, using the method of Nimmo and Spencer (2015). Heat production in the shell is initially radiogenic, and is then supplemented by a tidal condtribution of 20 GW over the interval from 1 to 2 Gyr. Surface heat flow is plotted in red, shell thickness in black. Ice shell melting commences shortly after tidal heating begins and refreezing is complete within about 0.1 Gyr after the cessation of tidal heating. Model geometry is as in Fig 1, with thermal conductivities set to 3.5 and 3.0 W m−1 K−1 for ice and rock, respectively. Initial temperature is 150 K throughout (no ocean initially).

7. DISCUSSION AND CONCLUSIONS ing transferred mainly by upwards advection of warm ma- terial at the tiger stripes, conductive heat transfer between the tiger stripes, or a combination of the two. We began by posing two, related questions: what is the Several other questions remain to explore. An impor- explanation for the large heat flow observed at Enceladus?; tant question is whether there is any independent evidence and how is the subsurface ocean maintained? At present, for when and how Enceladus (and the other satellites) the most satisfactory explanation appears to be that the ob- formed (see chapter by MCKINNON). The detection of served heat flow is all being produced at the present day by apex-antapex crater asymmetries by Hirata (2016) suggests tidal dissipation, and that the high heating rate is a conse- that at least some satellites experienced mostly heliocentric quence of the low Q of Saturn. An equilibrium tidal heating impacts, and did not reform after a catastrophic disruption rate of 10-20 GW matches the observed heat flow, is suffi- event (which would produce planetocentric debris); more cient to maintain a global ocean over billions of years, and work on this subject is needed. requires a Q consistent with astrometry. Because of the p The resonance locking mechanism itself obviously needs apparently small shell thickness (≈25 km) it appears likely to be explored more deeply, but so too do its consequences that the shell is conductive, not convective. This helps the for the satellites. Equilibrium tidal heating only determines ocean to persist, and is probably required to avoid rapid re- the total rate of heat production, not its location. In or- moval of shell topography by lateral flow. If the “resonance der to understand how the total amount of heating is dis- locking” mechanism of Fuller et al. (2016) is correct, a low tributed between Enceladus and Dione, coupled thermal- Q does not necessarily require young satellites and also p orbital models will need to be developed (cf. Section 5.1). implies a roughly constant rate of equilibrium heat produc- Such models should also be able to investigate whether tion (Fig 5). periodic tidal heating is a possibility, and also the effects The resonance lock model provides a natural way of of ocean freezing on tidal and thermal evolution. For- explaining Enceladus’s thin ice shell and high heat flow ward models, based on plausible initial satellite positions, without requiring either the satellite itself or the subsurface will be required to try and pin down when the current ocean to be recently formed. It thus presents a strongly uni- Enceladus:Dione MMR was established. And the ease or formitarian picture, in contrast to catastrophist ideas for re- difficulty of passing through previously-encountered reso- cent formation of the inner Saturnian satellites (Cuk´ et al. nances provides additional constraints on the system. Of 2016; Asphaug and Reufer 2013). Which of these two pic- course, many of these questions are equally applicable tures is closer to reality will become clear in time, but at to other satellites, such as Tethys, where ancient heating least for Enceladus we favour uniformitarianism. episodes are inferred to have taken place (Giese et al. 2007). It is currently unclear where the bulk of the dissipation Although tidal heating is generally higher at the poles is located. A conductive shell will dissipate mostly towards than the equator, an unresolved issue is why current heat its base (where the ice is warmest). Dissipation in the deep loss is concentrated only at one pole. As mentioned in Sec- ocean seems unlikely, but significant heating in the low- tion 3.3.2 there are several ways in which heat production density silicate core may also be occurring. It is also not could become focused in one region, while an alternative yet clear whether the heat measured at the surface is be-

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15 TABLE 1 LIST OF PARAMETER VALUES ASSUMED

Quantity Symbol Value Units Eqn. Quantity Symbol Value Units Eqn.

−1 −1 Thermal conductivity k 3.5 W m K 1 Enceladus radius RE 252 km 1 26 Temperature drop ∆T 190 K 1 Saturn mass MP 5.68 × 10 kg 2 Mean motion n 5.31 × 10−5 s−1 2 Semi-major axis a 238,000 km 2 Eccentricity e 0.0047 - 2 Saturn radius RP 60,300 km 5 −4 −1 Saturn Love number k2P 0.34 - 5 Saturn spin freq. ΩP 1.65 × 10 s 10

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