Global Subsurface Ocean Research Articles

An inorganic silicate simulant to represent the interior of enceladus

Enceladus, an icy moon of Saturn, consists of an ice shell, global subsurface ocean and a silicate interior. By sampling plume material, the Cassini spacecraft found evidence of ongoing water-rock reactions between the silicate interior and the subsurface ocean. These data showed that these reactions provide a source of bioessential elements to the ocean, making Enceladus one of the leading astrobiological targets in our Solar System. Understanding these water-rock reactions is critical in understanding the potential habitability of Enceladus. To study these reactions experimentally, a chemical simulant to represent the contemporary silicate interior of Enceladus has been designed. Based on the available interpretations of Cassini data about the density, chemical composition, and aqueous alteration of the interior, the chosen starting point for the simulant is a CI chondrite. However, Enceladus is still undergoing active aqueous alteration, thus its silicate mineral assemblage cannot have reached the fully altered assemblage seen in a CI chondrite. To account for this, adaptations have been made to a CI chondrite mineral assemblage, extrapolating back to an assemblage of less aqueously altered minerals whilst maintaining the same chemical composition in terms of major oxide phases. Thus, the chemical and mineralogical composition of this simulant represents a best estimate of the silicate components in the ongoing water rock interactions on Enceladus today.

Laboratory characterization of hydrothermally processed oligopeptides in ice grains emitted by Enceladus and Europa.

The Cassini mission provided evidence for a global subsurface ocean and ongoing hydrothermal activity on Enceladus, based on results from Cassini's mass spectrometers. Laboratory simulations of hydrothermal conditions on icy moons are needed to further constrain the composition of ejected ice grains containing hydrothermally altered organic material. Here, we present results from our newly established facility to simulate the processing of ocean material within the temperature range 80-150°C and the pressure range 80-130 bar, representing conditions suggested for the water-rock interface on Enceladus. With this new facility, we investigate the hydrothermal processing of triglycine (GGG) peptide and, for the first time, analyse the extracted samples using laser-induced liquid beam ion desorption (LILBID) mass spectrometry, a laboratory analogue for impact ionization mass spectrometry of ice grains in space. We outline an approach to elucidate hydrothermally processed GGG in ice grains ejected from icy moons based on characteristic differences between GGG anion and cation mass spectra. These differences are linked to hydrothermal processing and thus provide a fingerprint of hydrothermal activity on extraterrestrial bodies. These results will serve as important guidelines for biosignatures potentially obtained by a future Enceladus mission and the SUrface Dust Analyzer (SUDA) instrument onboard Europa Clipper. This article is part of the theme issue 'Dust in the Solar System and beyond'.

A light scattering analysis of the cryovolcano plumes on enceladus

Observations by NASA's Cassini spacecraft show that Enceladus, the sixth largest moon of Saturn, has cryovolcanoes on its south pole as well as a global subsurface ocean beneath Enceladus's icy crust. Photographs from the Imaging Science Subsystem (ISS) cameras aboard Cassini during flybys of Enceladus show cryovolcano plumes. The images are taken in several filters and with linear polarizers at various angles and orbital positions. Light scattered by the plume particles and observed by Cassini may provide information about the plume composition and potentially the dynamics of the ocean below. Here, we remove the coupling between orbital position and scattered plume intensity as well as analyze ISS data using light scattering models and the Q-space method. We suggest that the plume particles are generally larger than two microns in radius and nonspherical in shape.

Europa’s structural conditions for the existence of subsurface ocean and the absence of metallic core-driven magnetic field

During the Galileo spacecraft’s flyby of Europa, magnetic field measurements detected an inductive signal due to the response of Europa’s interior conductors to temporal fluctuations in the Jovian magnetic field. In contrast, no signatures of intrinsic magnetic field originating from the dynamo motion in the metallic core were acquired. These measurements suggest that a global sub-surface ocean containing electrolytes exists beneath the solid ice shell and that the metallic core lacks convection. Europa’s interior is expected to be divided into the metallic core, rocky mantle and hydrosphere based on the moment of inertia factor estimated from gravity field measurements. Specifically, the thickness of the outermost water layer is 120–170 km, and the radius of the metallic core is 0.12–0.43 times the surface radius. No systematic investigation of Europa’s internal evolution has been conducted to estimate the current state of the subsurface ocean and to explain the absence of a core dynamo field within such uncertainty for internal structure and material properties (especially ice properties). Herein, I performed a numerical simulation of the long-term thermal evolution of Europa’s interior and investigated the temporal changes in the ocean thickness as well as the temperature and heat flow of the metallic core. If the ice reference viscosity is greater than 5 × 1014 Pas, the sub-surface ocean can persist even in the absence of tidal heating. In the case of a tidal heating of 10 mW/m2 and 20 mW/m2, the ice shell thickness is ≤90 km if the ice reference viscosity is ≥1 × 1015 and 1 × 1014 Pas, respectively. Regardless of the ice reference viscosity, if the tidal heating is ≥50 mW/m2, the shell thickness will be ≤40 km. The thermal history of the metallic core is determined by the hydrosphere thickness and the metallic core density, and is unaffected by variations in the ice shell (ocean) thickness. Preferred conditions for the absence of the core dynamo include CI chondritic abundance for the long-lived radioactive isotopes, lower initial core–mantle boundary (CMB) temperature and thicker hydrosphere. The core may be molten without convection if the composition is near the eutectic in a Fe–FeS alloy, or not molten (without convection) if the composition is near the Fe or FeS endmember. Specifically, if the rocky mantle has a CI chondritic radioisotope abundance, any core composition and hydrosphere thickness allow the absence of the core dynamo if the initial temperature at the CMB is lower than 1,250 K. If the rocky mantle has the ordinary chondritic radioisotope abundance, or a higher initial temperature (∼1,500 K) at the CMB, the core density lower than 6,000 kg/m3 is preferred for the absence of the core dynamo. In the case of the core composition near the eutectic one, a hydrosphere thicker than 150 km is required for the lacking core dynamo. The lower pressure of Europa’s rocky mantle due to its thinner hydrosphere compared with that of Ganymede may facilitate heat transfer in the mantle, lowering its temperature and making dynamo motion more challenging.

Discriminating Aromatic Parent Compounds and Their Derivative Isomers in Ice Grains From Enceladus and Europa Using a Laboratory Analogue for Spaceborne Mass Spectrometers

AbstractResults from the Cassini‐Huygens space mission at Enceladus revealed a substantial inventory of organic species embedded in plume and E ring ice grains originating from a global subsurface and putative habitable ocean. Compositional analysis by the Cosmic Dust Analyzer indicated the presence of aromatic species and constrained some structural features, although their exact nature remains unclear. As indicated by many studies, among other organic species, low‐mass aromatics likely played a role in the emergence of life on Earth and may be linked to potential prebiotic or biogenic chemistry on icy moons. Here, we study the behavior of single‐ringed aromatic compounds—benzoic acid and two isomeric derivatives, 2,3‐dihydroxybenzoic acid and 2,5‐dihydroxybenzoic acid—using Laser‐Induced Liquid Beam Ion Desorption (LILBID), an analogue setup to simulate the impact ionization mass spectra of ice grains in space. These compounds share common structural features but also exhibit differences in functional groups and substituent positions. We investigate the fragmentation behavior and spectral appearance of each molecule over three simulated impact velocities, in both positive and negative ion modes. Parent compounds can be distinguished easily from their derivatives due to various spectral differences, including the (de)protonated molecular ion peaks appearing at different m/z values. We conclude that distinction between structural isomers in LILBID is more challenging, but some insights can be revealed by considering intermolecular bonding regimes. This work will guide future investigations into elucidating the composition of isomeric biosignatures in ice grains, relevant for future space missions to Enceladus and Europa.

Crystallization Kinetics of Vitreous Magnesium Sulfate Hydrate and Implications for Europa’s Surface

Brines consistent with that expected for Europa’s global subsurface ocean have been shown to form vitreous salt hydrates when frozen. We report experiments examining the crystallization kinetics of vitreous MgSO4 hydrate in order to better understand the stability of such materials on the surface of Europa. Vitreous MgSO4 hydrates formed from a 2 M parent solution were found to crystallize into MgSO4·11H2O (meridianiite) upon annealing at 195–225 K. Raman spectroscopy was used to monitor the crystallization and reaction rates were determined from the growth of the crystalline fraction as a function of time. Arrhenius analysis yielded an activation energy of 60 ± 9 kJ mol−1 for the vitreous to crystalline transition, implying that such transformation does not occur spontaneously at Europa’s surface temperatures. If emplacement processes favor the formation of vitreous salt hydrates, they are likely to still be stable and could be an important non-ice component on Europa at present day.

A multi-lander New Frontiers mission concept study for Enceladus: SILENUS

Enceladus, with its subsurface ocean, is amongst the top priority targets in the search for life beyond Earth. Following on discoveries from the Cassini mission that Enceladus possesses a global subsurface ocean containing salt and organic compounds, there are many unconstrained properties of the ocean and ice shell that must be investigated to further assess the habitability of Enceladus and begin the search for biosignatures on Enceladus. In this paper, we present a concept study for a New Frontiers class multi-lander and orbiter mission to Enceladus that investigates if there is or ever was a habitable environment on Enceladus. The mission architecture includes an orbiter for detailed chemical analysis of material erupted from Enceladus’ plumes and four impact landers for geophysical measurements. As part of our mission concept study, we explore key trades for orbital and surface science, as well as assess the scientific potential and hazards of candidate landing sites on Enceladus. The novelty of our mission architecture and consideration of both orbital and surface science elements makes this work directly relevant to a broad range of potential future mission architectures under consideration, such as those identified in the 2023–2032 Planetary Science and Astrobiology Decadal Survey.

The Technological, Scientific, and Sociological Revolution of Global Subsurface Ocean Observing

The Technological, Scientific, and Sociological Revolution of Global Subsurface Ocean Observing

Constraining Europa's ice shell thickness with fundamental mode surface wave dispersion

Determining the thickness of Europa's outer ice shell is a key factor for understanding Europa's internal dynamics, evolution, and potential habitability. As such, one of the primary goals of any future lander mission to Europa would be to constrain the thickness of the ice shell and confirm the presence of a global subsurface ocean. Tidally induced ice fracturing events provide a natural source of seismic energy to illuminate the subsurface, thus a seismic instrument onboard a future lander mission could provide a promising means to probe ice shell thickness. A variety of seismic techniques could be used to constrain Europa's interior structure and dynamics, including body wave, surface wave, and normal mode seismology. Here, we use numerical simulations of seismic wave propagation on Europa in order to investigate the potential of using long period dispersion measurements of Rayleigh and flexural waves to constrain the ice shell thickness. Since the sensitivity kernels of group velocity dispersion measurements depend strongly on the structure of the ice shell, inverting for ice shell thickness is a non-linear problem.To address this, we use either a grid search or Markov chain Monte Carlo inversion approach, and test the method on a variety of plausible models of Europa's interior. Additionally, we demonstrate our approach in a “blind” inversion using the 1 week long synthetic catalogs of Europa's seismicity from Panning et al. (2018). We find that under most scenarios, group velocity dispersion measurements between periods of 25–250 s can constrain Europa's ice shell thickness to within several km uncertainty, although the method becomes increasingly inaccurate for thicker ice shells and at large epicentral distances. Our results, which suggest that surface waves from naturally occurring ice fracturing events on Europa can be used to help determine ice shell thickness, may help set instrument requirements for spaceflight capable seismometers aimed at exploring icy ocean worlds.

The In Situ Exploration of a Relict Ocean World: An Assessment of Potential Landing and Sampling Sites for a Future Mission to the Surface of Ceres

Abstract The Dawn orbiter’s exploration of Ceres, the most water-rich body in the inner solar system after Earth, revealed the dwarf planet to be a relict ocean world of great interest to the astrobiology and ocean worlds communities. Evidence for an early global subsurface ocean is preserved in Ceres’ surface minerals. While most of the subsurface ocean froze billions of years ago, its liquid (preserved in a deep brine reservoir) and solid remnants are responsible for spectacular surface features such as the faculae in the Occator crater, Ahuna Mons, and Haulani crater. Therefore, the next step in Ceres exploration is an in situ mission to the surface, with these three features being among the most scientifically compelling targets. Here we demonstrate the process of identifying potential landing/sampling sites. Using the highest resolution data obtained by Dawn (up to ∼5 m/pixel in Occator and up to ∼35 m/pixel elsewhere), we identify safe and scientifically compelling potential landing/sampling sites in these three regions. For future landing/sample return, higher spatial resolution data (a few decimeters per pixel to ∼1 m/pixel) than are currently available are needed to identify hazards on the scale of a ∼1–5 m lander. Such data could be obtained in an orbital reconnaissance phase prior to landing/sampling. Orbital reconnaissance, in combination with terrain relative navigation/hazard avoidance, and the ability to land within a ≦100 m diameter circle (>800,000 of which fit within Occator), makes it highly likely that safe and scientifically compelling landing/sampling sites could be reached by a future in situ mission.

Isostasy with Love – I: elastic equilibrium

SUMMARY Isostasy explains why observed gravity anomalies are generally much weaker than what is expected from topography alone, and why planetary crusts can support high topography without breaking up. On Earth, it is used to subtract from gravity anomalies the contribution of nearly compensated surface topography. On icy moons and dwarf planets, it constrains the compensation depth which is identified with the thickness of the rigid layer above a soft layer or a global subsurface ocean. Classical isostasy, however, is not self-consistent, neglects internal stresses and geoid contributions to topographical support, and yields ambiguous predictions of geoid anomalies. Isostasy should instead be defined either by minimizing deviatoric elastic stresses within the silicate crust or icy shell, or by studying the dynamic response of the body in the long-time limit. In this paper, I implement the first option by formulating Airy isostatic equilibrium as the linear response of an elastic shell to a combination of surface and internal loads. Isostatic ratios are defined in terms of deviatoric Love numbers which quantify deviations with respect to a fluid state. The Love number approach separates the physics of isostasy from the technicalities of elastic-gravitational spherical deformations, and provides flexibility in the choice of the interior structure. Since elastic isostasy is invariant under a global rescaling of the shell shear modulus, it can be defined in the fluid shell limit, which is simpler and reveals the deep connection with the asymptotic state of dynamic isostasy. If the shell is homogeneous, minimum stress isostasy is dual to a variant of elastic isostasy called zero deflection isostasy, which is less physical but simpler to compute. Each isostatic model is combined with general boundary conditions applied at the surface and bottom of the shell, resulting in one-parameter isostatic families. At long wavelength, the thin shell limit is a good approximation, in which case the influence of boundary conditions disappears as all isostatic families members yield the same isostatic ratios. At short wavelength, topography is supported by shallow stresses so that Airy isostasy becomes similar to either pure top loading or pure bottom loading. The isostatic ratios of incompressible bodies with three homogeneous layers are given in analytical form in the text and in complementary software.

Repeated impact-driven plume formation on Enceladus over megayear timescales

Water plumes erupting from the ‘tiger stripe’ features on the south pole of Enceladus are thought to connect to a global subsurface ocean. Proposed origins for the initial stress necessary to form the ‘tiger stripes’ include a giant impact, which would require true polar wander to explain the location of the plumes if the impact did not occur at the South Pole, or tensile stresses, which would require volumetric expansion associated with partial freezing of the subsurface ocean. In addition to the requirement of a fine-tuned impact, true polar wander, or partial freezing of the subsurface ocean, a further issue with these hypotheses is that the ‘tiger stripes’ may be short-lived. We show here that impact resurfacing can seal off plumes and mass loss can lead to their compression and closure over ∼1 Myr. Since plumes are observed at present, a mechanism by which new plumes can be generated every ∼1 Myr and by which such plumes are most likely to form at the south pole is needed. We propose and investigate the possibility that impacts constitute an adequate repeating source for the continual instigation of fractures and plumes. We find that the rate of impacts on Enceladus suggests the formation of ∼103 independent plume systems per Gyr, the majority on the south pole, and is consistent with the Cassini-derived age of the south pole for a lunar-like bombardment history, our estimates of fracture lifetimes, and with the needed parameters for parallel fracture propagation. The model favors a bombardment history similar to that of Triton over one more similar to that of the Galilean satellites.

Heat Production and Tidally Driven Fluid Flow in the Permeable Core of Enceladus

AbstractSaturn's moon Enceladus has a global subsurface ocean and a porous rocky core in which water‐rock reactions likely occur; it is thus regarded as a potentially habitable environment. For icy moons like Enceladus, tidal heating is considered to be the main heating mechanism, which has generally been modeled using viscoelastic solid rheologies in existing studies. Here we provide a new framework for calculating tidal heating based on a poroviscoelastic model in which the porous solid and interstitial fluid deformation are coupled. We show that the total heating rate predicted for a poroviscoelastic core is significantly larger than that predicted using a classical viscoelastic model for intermediate to large (>1014 Pa·s) rock viscosities. The periodic deformation of the porous rock matrix is accompanied by interstitial pore fluid flow, and the combined effects through viscous dissipation result in high heat fluxes particularly at the poles. The heat generated in the rock matrix is also enhanced due to the high compressibility of the porous matrix structure. For a sufficiently compressible core and high permeability, the total heat production can exceed 10 GW—a large fraction of the moon's total heat budget—without requiring unrealistically low solid viscosities. The partitioning of heating between rock and fluid constituents depends most sensitively on the viscosity of the rock matrix. As the core of Enceladus warms and weakens over time, pore fluid motion likely shifts from pressure‐driven local oscillations to buoyancy‐driven global hydrothermal convection, and the core transitions from fluid‐dominated to rock‐dominated heating.

Ammonia-water freezing as a mechanism for recent cryovolcanism on Pluto

NASA's New Horizons flyby of the Pluto-Charon system in 2015 exposed a multitude of geologically active terrains on the surface of Pluto. Possible evidence of cryovolcanism has been reported in association with crustal scale extensional tectonic structures at Virgil Fossae and at two large circular mountains, informally referred to as Wright Mons and Piccard Mons. Our detailed geologic mapping of Wright Mons and the surrounding region reveals additional circular mounds overlying the main edifice and narrow ravines emanating from the summit, possibly produced by liquid flow. We model the overpressure generated during the freezing of a mixed ammonia-water liquid inside an enclosed chamber in Pluto's ice crust and in a global subsurface ocean. We consider scenarios spanning the possible range of Pluto's ammonia abundance and conclude that stresses generated due to freezing-induced pressurization can exceed the tensile strength of ice and create fractures for the liquid to escape. Our results suggest that freezing in a crustal chamber and the subsurface ocean are both viable mechanisms for cryovolcanism on Pluto and may explain the morphological variety of cryovolcanic formations observed by New Horizons.

Experimental Petrology to Understand Europa's Crust

AbstractThe Jovian moon Europa is a prime target for astrobiology. A global subsurface water ocean and a geologically young surface provide evidence of an active planetary body with a potential deep habitable environment. Tectonism and cryomagmatism are both agents of resurfacing, with structures on the surface spatially related to reddish non‐icy materials that could represent crystallized volatile and salt‐rich fluids from the interior, possibly from the ocean or shallower aqueous bodies. Cryomagmatism could therefore be a mechanism for exposing the underlying liquid layers to the surface and could hold paramount importance for understanding the physical and chemical evolution of fluids during their ascent and emplacement and their connection with geological features at the surface. With these premises, we perform a set of laboratory experiments simulating the evolution of different fluids under the conditions in Europa's crust. These experiments allow us to constrain the physico‐chemical and textural changes experienced by the different fluids and solids that are potentially emplaced within the icy crust and determine how they are affected by such secondary processes as reheating, melting, and ultimate recrystallization (e.g., in response to the emplacement of a second diapir close to the first one or tidal reheating). Based on these experimental results, we explore the connection of cryomagmas and their evolution near the surface to geologic features present on Europa's surface, such as pits, uplifts/domes, and microchaos regions, as well as the link with explosive cryovolcanism responsible for putative plumes at Europa.

How Are Microplastics Transported to Polar Regions?

New modeling indicates that global subsurface ocean currents distribute submerged microplastics along very different routes than those traveled by floating plastic debris.

Enceladus's crust as a non-uniform thin shell: II tidal dissipation

Tidal heating is the prime suspect behind Enceladus's south polar heating anomaly and global subsurface ocean. No model of internal tidal dissipation, however, can explain at the same time the total heat budget and the focusing of the energy at the south pole. I study here whether the non-uniform icy shell thickness can cause the north-south heating asymmetry by redistributing tidal heating either in the shell or in the core. Starting from the non-uniform tidal thin shell equations, I compute the volumetric rate, surface flux, and total power generated by tidal dissipation in shell and core. The micro approach is supplemented by a macro approach providing an independent determination of the core-shell partition of the total power. Unless the shell is incompressible, the assumption of a uniform Poisson's ratio implies non-zero bulk dissipation. If the shell is laterally uniform, the thin shell approach predicts shell dissipation with a few percent error while the error on core dissipation is negligible. Variations in shell thickness strongly increase the shell dissipation flux where the shell is thinner. For a hard shell with long-wavelength variations, the shell dissipation flux can be predicted by scaling with the inverse local thickness the flux for a laterally uniform shell. If Enceladus's shell is in conductive thermal equilibrium with isostatic thickness variations, the shell dissipation flux at the south pole is about three times its value for a shell of uniform thickness, which remains negligible compared to the observed flux. Spatial variations of the observed flux can only be reproduced if the ice at the bottom of the shell is ten times more dissipative at tidal frequencies than in standard models of ice rheology. Dissipation in an unconsolidated core can provide the missing power if the rheology of the core is extremely soft, but does not generate any significant heating asymmetry as long as the core is homogeneous. Non-steady state models, though not investigated here, face similar difficulties in explaining the asymmetries of tidal heating and shell thickness.

Expected precision of Europa Clipper gravity measurements

The primary gravity science objective of NASA’s Clipper mission to Europa is to confirm the presence or absence of a global subsurface ocean beneath Europa’s Icy crust. Gravity field measurements obtained with a radio science investigation can reveal much about Europa’s interior structure. Here, we conduct extensive simulations of the radio science measurements with the anticipated spacecraft trajectory and attitude (17F12v2) and assets on the spacecraft and the ground, including antenna orientations and beam patterns, transmitter characteristics, and receiver noise figures. In addition to two-way Doppler measurements, we also include radar altimeter crossover range measurements. We concentrate on ± 2 h intervals centered on the closest approach of each of the 46 flybys. Our covariance analyses reveal the precision with which the tidal Love number k2, second-degree gravity coefficients C¯20 and C¯22, and higher-order gravity coefficients can be determined. The results depend on the Deep Space Network (DSN) assets that are deployed to track the spacecraft. We find that some DSN allocations are sufficient to conclusively confirm the presence or absence of a global ocean. Given adequate crossover range performance, it is also possible to evaluate whether the ice shell is hydrostatic.

Retrieving Temperature Anomaly in the Global Subsurface and Deeper Ocean From Satellite Observations

AbstractRetrieving the subsurface and deeper ocean (SDO) dynamic parameters from satellite observations is crucial for effectively understanding ocean interior anomalies and dynamic processes, but it is challenging to accurately estimate the subsurface thermal structure over the global scale from sea surface parameters. This study proposes a new approach based on Random Forest (RF) machine learning to retrieve subsurface temperature anomaly (STA) in the global ocean from multisource satellite observations including sea surface height anomaly (SSHA), sea surface temperature anomaly (SSTA), sea surface salinity anomaly (SSSA), and sea surface wind anomaly (SSWA) via in situ Argo data for RF training and testing. RF machine‐learning approach can accurately retrieve the STA in the global ocean from satellite observations of sea surface parameters (SSHA, SSTA, SSSA, SSWA). The Argo STA data were used to validate the accuracy and reliability of the results from the RF model. The results indicated that SSHA, SSTA, SSSA, and SSWA together are useful parameters for detecting SDO thermal information and obtaining accurate STA estimations. The proposed method also outperformed support vector regression (SVR) in global STA estimation. It will be a useful technique for studying SDO thermal variability and its role in global climate system from global‐scale satellite observations.

One‐Dimensional Convective Thermal Evolution Calculation Using a Modified Mixing Length Theory: Application to Saturnian Icy Satellites

AbstractSolid‐state thermal convection plays a major role in the thermal evolution of solid planetary bodies. Solving the equation system for thermal evolution considering convection requires 2‐D or 3‐D modeling, resulting in large calculation costs. A 1‐D calculation scheme based on mixing length theory (MLT) requires a much lower calculation cost and is suitable for parameter studies. A major concern for the MLT scheme is its accuracy due to a lack of detailed comparisons with higher dimensional schemes. In this study, I quantify its accuracy via comparisons of thermal profiles obtained by 1‐D MLT and 3‐D numerical schemes. To improve the accuracy, I propose a new definition of the mixing length (l), which is a parameter controlling the efficiency of heat transportation due to convection, for a bottom‐heated convective layer. Adopting this new definition of l, I investigate the thermal evolution of Saturnian icy satellites, Dione and Enceladus, under a wide variety of parameter conditions. Calculation results indicate that each satellite requires several tens of GW of heat to possess a thick global subsurface ocean suggested from geophysical analyses. Dynamical tides may be able to account for such an amount of heat, though the reference viscosity of Dione's ice and the ammonia content of Dione's ocean need to be very high. Otherwise, a thick global ocean in Dione cannot be maintained, implying that its shell is not in a minimum stress state.