RESEARCH LETTER
10.1029/2024GL110409
Key Points:• Measuring physical libration
amplitudes can be used to detectsubsurface liquid water oceans within
the Uranian moons• Combining librations with gravity
measurements can yieldcomprehensive constraints on theinteriors of the Uranian moons• Thick oceans are easier to detect but
finding thin oceans may require
libration amplitudes to be measured towithin better than 10 m

Looking for Subsurface Oceans Within the Moons of Uranus
Using Librations and Gravity
D. J. Hemingway1

and F. Nimmo2

1

Institute for Geophysics, Jackson School of Geosciences, University of Texas at Austin, Austin, TX, USA, 2Department ofEarth and Planetary Sciences, University of California Santa Cruz, Santa Cruz, CA, USA

Abstract Several of the icy moons in the Jupiter and Saturn systems appear to possess internal liquid water

Supporting Information may be found in
the online version of this article.

oceans. Our knowledge of the Uranian moons is more limited but a future tour of the system has the potential todetect subsurface oceans. Planning for this requires an understanding of how the moons' internal structures—with and without oceans—relate to observable quantities. Here, we show that the amplitude of forced physicallibrations could be diagnostic of the presence or absence of subsurface oceans within the Uranian moons. In thepresence of a decoupling global ocean, ice shell libration amplitudes at Miranda, Ariel, and Umbriel will exceed
100 m if the shells are <30 km thick. The presence of oceans could also imply significant tidal heating within thelast few hundred million years. Combining librations with the quadrupole gravity field could provide
comprehensive constraints on the internal structures and histories of the Uranian moons.

Correspondence to:Plain Language Summary Several of the icy moons in the Jupiter and Saturn systems appear to

Supporting Information:

D. J. Hemingway,
douglas.hemingway@utexas.edu

Citation:
Hemingway, D. J., & Nimmo, F. (2024).
Looking for subsurface oceans within the
moons of Uranus using librations andgravity. Geophysical Research Letters, 51,
e2024GL110409. https://doi.org/10.1029/
2024GL110409
Received 23 MAY 2024
Accepted 3 SEP 2024
Author Contributions:Conceptualization: D. J. Hemingway,
F. Nimmo
Formal analysis: D. J. Hemingway
Investigation: D. J. Hemingway
Methodology: D. J. Hemingway
Software: D. J. Hemingway
Visualization: D. J. Hemingway
Writing – original draft:
D. J. Hemingway
Writing – review & editing: F. Nimmo© 2024. The Author(s).
This is an open access article under theterms of the Creative Commons
Attribution‐NonCommercial‐NoDerivs
License, which permits use anddistribution in any medium, provided theoriginal work is properly cited, the use isnon‐commercial and no modifications or
adaptations are made.

HEMINGWAY AND NIMMO

possess internal liquid water oceans. Our knowledge of the Uranian moons is more limited but a future tour of
the Uranus system has the potential to detect subsurface oceans. Planning for this requires an understanding of
how the moons' internal structures—with and without oceans—relate to observable quantities. Here, we showthat certain aspects of their rotational states could be diagnostic of the presence or absence of internal liquidwater oceans within several of the Uranian moons and that combining this with measurements of the gravity
field could provide comprehensive constraints on the internal structures and histories of the Uranian moons.

1. Introduction
One of the most intriguing recent developments in the exploration of our solar system has been the detection ofpotentially habitable subsurface oceans inside several of the icy moons in the Jupiter and Saturn systems(Hemingway et al., 2018; Hussmann et al., 2006, 2015; F. Nimmo & Pappalardo, 2016; Spohn & Schubert, 2003).
Much less is known about the moons of the Uranus system but they offer the possibility of additional ocean worlds
with the potential to raise even more interesting questions. As the planetary science community begins preparation for a Uranus Flagship mission (Simon et al., 2021), it will be important to consider how the Uraniansatellites could have evolved (Bierson & Nimmo, 2022; Castillo‐Rogez et al., 2023; F. Nimmo, 2023; Ćuket al., 2020) and how subsurface oceans could be detected within their interiors. Such results can help guidedecisions about the tour design and the choices of instruments and their capabilities.
A variety of techniques have been used to detect the presence of internal liquid water oceans in the Jupiter andSaturn systems (F. Nimmo & Pappalardo, 2016). In some cases, geologic evidence or the appearance of recent oreven active eruptions (i.e., at Enceladus; Porco et al., 2006) strongly suggest the presence of an internal liquidwater ocean. Where the satellites are exposed to a time‐varying magnetic field, as in the Jupiter system, magnetic
induction signals have provided compelling evidence for electrically conductive layers (e.g., a salty liquid water
ocean) at depth (Biersteker et al., 2023). Although this technique is expected to be effective in the Uranus system
as well (Arridge & Eggington, 2021; Cochrane et al., 2021; Hussmann et al., 2006; Weiss et al., 2021), we will not
focus on magnetic induction here.
Geodetic measurements can also be useful in multiple ways. Measurements of the shapes and gravitational fieldsof the satellites can be used to assess the compensation states of their ice shells, which may be suggestive of a
subsurface ocean (e.g., Beuthe et al., 2016; Hemingway et al., 2018; D. J. Hemingway & Mittal, 2019; Iesset al., 2014; Zannoni et al., 2020; Čadek et al., 2016). Finally, measurements of the time‐varying component of
their shapes and/or gravitational fields (Iess et al., 2012; Moore, 2000), and/or characterization of their rotation
states (i.e., obliquity and forced physical librations) (e.g., Baland et al., 2011; B. Bills & Nimmo, 2008; B. G. Bills
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10.1029/2024GL110409& Nimmo, 2011; Chen et al., 2014; Thomas et al., 2016; Van Hoolst et al., 2013, 2016) can provide key evidence
for internal liquid water oceans because these can vary strongly as a function of internal structure and may bemarkedly different depending on whether or not the ice shell is mechanically decoupled from the deeper interior.
The time variable part of the gravity signal—quantified by the tidal k2 Love number—proved important for
detecting a subsurface ocean at Titan (Goossens et al., 2024; Iess et al., 2012) but the small orbital eccentricities of
the Uranian moons mean that their tidal response amplitudes are likely to be too small to measure without adedicated orbiter. Titan's obliquity (the angle between its spin axis and orbit normal) was found to be a factor of ∼3larger than what would be expected if the ice shell were physically coupled to the deeper interior (B. G. Bills &Nimmo, 2011; Chen et al., 2014), helping confirm the presence of an internal ocean there. Obliquity measurements in the Uranus system, especially at Oberon (Chen et al., 2014), similarly have the potential to reveal
subsurface oceans, though we will not explore this possibility in the present work.
In this study, we will focus primarily on forced physical librations. All of the large satellites in the solar system are
tidally locked with their parent body such that they experience a permanent tidal elongation. The resulting longaxis of the satellite points approximately toward the parent body at all times but, because the satellites haveelliptical orbits, alignment occurs only at pericenter and apocenter and the long axis is otherwise misaligned fromthe line connecting the satellite with the parent body (Murray & Dermott, 1999). This misalignment results ingravitational torques that act toward the east during half of the orbit (between pericenter and apocenter), andtoward the west during the other half (from apocenter back to pericenter). The result is a small oscillation
superimposed on the satellite's otherwise steady rotation rate. This is known as a forced physical libration and its
amplitude can depend sensitively on the satellite's internal structure. For bodies whose ice shells behave rigidly
and are physically decoupled from the deeper interior by a global subsurface ocean, the physical libration
amplitude is a strong inverse function of shell thickness but is less sensitive to the deeper internal structure. We
therefore also discuss the complementary measurement of the satellites' quadrupole gravity field and show that,taken together, these two measurements can provide comprehensive constraints on the internal structures of thelarge Uranian satellites. Though we will not directly model the thermal/orbital evolution of the satellites here (seee.g., Bierson & Nimmo, 2022; Castillo‐Rogez et al., 2023; Ćuk et al., 2020), we will briefly discuss how constraining their present internal structures has implications for their thermal and orbital histories.

2. Methods
The methods necessary for our calculations are all available in the published literature but, for convenience, wesummarize some of the most important equations in Appendix A. Both the libration amplitude and gravitational
field calculations require a model of the satellite's internal structure. We represent each body as a series ofconcentric shells of uniform density, subject to the known radius and bulk density (Table S1 in Supporting Information S1). In our ‘no‐ocean’ models, we specify only the densities of the two layers (icy mantle and rockycore) and we solve Equation A1 for the radius of the core. In our three layer models (ice, ocean, core), we specify
all three layer densities and the thickness of the ice shell and solve Equation A1 for the radius of the core. None of
the Uranian satellites are large enough for their hydrospheres to reach internal pressures at which high pressure ice
phases need to be considered so this simple three‐layer structure is sufficient for our purposes. We assume thebodies have relaxed to hydrostatic equilibrium and we calculate the expected hydrostatic figures for each layer
following the methodology of Tricarico (2014). The resulting set of asymmetric concentric shells are used to
calculate both the corresponding gravitational field asymmetries via Equation A3 and the libration amplitudes via
Equation A11 or Equation A15, depending on whether or not a decoupling ocean is present. We account for thefinite rigidity of the ice shells to allow for the possibility of deformation (we assume a Maxwell rheology) in
response to the dynamical tidal forces, which can reduce libration amplitudes when the ice shells are thin,especially for the larger bodies (see Van Hoolst et al., 2013, and A2.2).

3. Results
For each of the five large Uranian satellites, we calculated the expected amplitude of forced physical librations
under a variety of conditions (Figure 1). As a point of comparison, we also show results for Enceladus, where
librations have been measured (Nadezhdina et al., 2016; Park et al., 2024; Thomas et al., 2016). Libration
amplitude (measured in meters at the equator) increases with the body's radius and orbital eccentricity, anddecreases with the body's bulk density and orbital period. As expected, the libration amplitudes of Titania andOberon are significantly smaller than those of the three interior satellites, due primarily to their much larger semi‐
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Geophysical Research Letters10.1029/2024GL110409

Figure 1. Amplitude of forced physical librations (at the equator) for several possible ocean worlds as a function of ice shell
thickness. Circles represent libration amplitudes expected when no decoupling ocean is present. We assume ρice = 930
kgm 3 , ρocean = 1,030 kgm 3 , and either ρcore = 2,400 kgm 3 (solid curves, filled circles) or ρcore = 3,500 kgm 3 (dashedcurves, open circles). Ice shell rigidity and viscosity are taken to be μ = 4 GPa and η = 1020 Pas (but see also Text S3 and Figure
S6 in Supporting Information S1).

major axes. In spite of their different semi‐major axes, the libration amplitudes of Miranda, Ariel, and Umbriel aresimilar due to opposing effects that approximately cancel: compared to Ariel, Miranda is closer to Uranus but
smaller; compared to Ariel, Umbriel is farther from Uranus but has larger eccentricity (Figure S1 in Supporting
Information S1). Enceladus is similar in size to Miranda but has a much larger eccentricity.
The circles in Figure 1 show the predicted libration amplitude in the absence of a decoupling ocean and are
calculated via Equation A11. Results from Cassini imply core densities for Enceladus, Dione, and Titan in therange 2,300‐2,500 kgm 3 (Durante et al., 2019; D. J. Hemingway & Mittal, 2019; Zannoni et al., 2020), suggestive of porous and/or hydrated silicate rock. Accordingly, we nominally assume ρcore = 2,400 kgm 3 , though
we also show results for ρcore = 3,500 kgm 3 to illustrate the effect of varying core density up to the level of zero‐
porosity, non‐hydrated chondritic rock. When a global subsurface ocean is present, the ice shell is mechanicallydecoupled from the rocky part of the interior and libration amplitude becomes a strong inverse function of shell
thickness (Figure 1). We calculated libration amplitudes for each satellite considering ice shell thicknesses
ranging from as small as 10 km up to the maximum thickness allowable given the bulk density constraints and theassumed density for the rocky core.
Our results for the ‘no‐ocean’ case (circles in Figure 1) are smaller than those obtained by Comstock andBills (2003) primarily because they use satellite eccentricities from Yoder (1995) whereas we use the values from
Jacobson (2014), which are somewhat smaller. Since orbital eccentricities vary over time (e.g., Ćuk et al., 2020),
the values from Jacobson (2014) are merely points of reference that correspond approximately to the present
orbits, but note that varying orbital eccentricity has a straightforward, linear effect on libration amplitude. If weassume an infinitely rigid ice shell and calculate the libration amplitudes using Equation A13, our results (Figure
S6 in Supporting Information S1) are similar to those of Park et al. (2020) (see their Figure 11), though they
considered only the ‘no‐ocean’ case and the case of a 20‐km ocean with an ice shell thickness of either 20 km or

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Geophysical Research Letters10.1029/2024GL110409

100 km. Libration amplitudes are slightly smaller when we allow for the possibility of elastic deformation,
calculating libration amplitudes via Equation A15 (Figure 1).
Figure 1 illustrates that measurements of forced physical librations at the Uranian satellites could provide information about the presence of a decoupling global ocean as well as the thickness of the overlying ice shell.
However, the libration amplitude is not sensitive to other aspects of the internal structure (e.g., ocean thickness,size and density of the rocky core). In contrast, the quadrupole gravity field, if it can be measured (see Castillo‐
Rogez et al., 2023), provides a constraint on the satellite's moment of inertia and thus the total thickness of thehydrosphere (ice shell plus ocean), but cannot easily distinguish between ice and water. To illustrate the utility of
combining the two measurements, we simultaneously calculated the expected libration amplitude and quadrupole
gravity field (we focus here on C2,2 ) across a range of possible internal structures for each satellite. In each case,
we assume a three‐layer body consisting of an icy shell, a liquid water ocean, and a rocky core. We consider awide range of ice shell thicknesses and rocky core densities and solve Equation A1 for the core radius and ocean
thickness given the known total radius and bulk density of each body. The results are summarized in Figure 2,
demonstrating that the combination of gravity plus libration observations simultaneously constrains shell
thickness (teal contours), ocean thickness (dark blue contours), and core density (dashed gray contours); this isenough to also constrain the moment of inertia and core radius (not shown). In addition to the five Uranian moons,
we show results for Enceladus with orange shading representing the observational constraints (Iess et al., 2014;
Tajeddine et al., 2017; Thomas et al., 2016).

4. Discussion
Thomas et al. (2016) estimated the libration amplitude for Enceladus by reconstructing the rotation state basedon optical tracking of surface features in a control point network across 7 years' worth of Cassini imagery. The
resulting estimate was ∼ 530 m at the equator, with a 2‐σ uncertainty of ∼ 60 m (a subsequent study by Parket al., 2024 obtained ∼ 400 ± 40 m, 3‐σ). This strongly implies the presence of a subsurface ocean becauseeven the low ends of these ranges are much larger than the value expected without a decoupling ocean (∼ 140m). The precision of the libration measurement depends primarily on spacecraft/camera positioning andpointing errors, and the design of the tour/flybys (e.g., Park et al., 2020; Thomas et al., 2016). A large numberof images from a variety of different viewing geometries helps to reduce uncertainties in the estimatedlibration amplitudes. In particular, it is important to obtain images that span a range of latitudes because thedisplacement of surface features due to physical librations will be largest at the equator and zero at the poles
whereas displacements due to positioning/pointing errors will not vary with latitude in the same way (e.g.,
Hemingway et al., 2018). It is also essential to sample imagery from a variety of different true anomalies as
the magnitude of displacements varies over the orbital period. Park et al. (2020) estimate that, with a camera
designed for high‐precision pointing, libration amplitudes could be constrained to within ∼ 1 m given enoughmeasurements at sufficiently low altitudes (e.g., for the Enceladus example shown in their Figure 16, ∼ 10
measurements below 100 km altitude or ∼ 100 below 300 km). In that case, subsurface oceans should be
detectable via librations under most conditions. We do not, however, take it for granted that such measurement
precision will be easy to achieve (see Simon et al., 2021) and so we consider a wider range of possible
uncertainties here.
From Figure 1, it is clear that the measurement precision required to detect a subsurface ocean depends on theshell and ocean thicknesses. For example, assuming ρcore = 2,400 kgm 3 , if Ariel has an ice shell thickness of 20
km (and therefore an ocean thickness of ∼ 140 km), the libration amplitude would be ∼ 150 m and 20 m oflibration amplitude uncertainty would translate to just ∼ 2 km of uncertainty in the shell thickness estimate(Figure 1, S2 in Supporting Information S1). However, if the shell thickness is 120 km (ocean thickness is ∼ 30km), the libration amplitude would be ∼ 36 m and 20 m of uncertainty would make a definitive ocean detection
impossible since the ‘no‐ocean’ value is ∼ 24 m. In other words, whereas a thick ocean is readily detectable, thecase of a thin ocean may be indistinguishable from the no‐ocean case unless libration amplitudes can be estimatedwith high precision (Figures S2, S3 in Supporting Information S1). Because the measurement precision required
to distinguish between different internal structure models varies as a function of ice shell and ocean thickness, andis different for different bodies, it is impossible to provide a single number to target for measurement precision.
Nevertheless, as a point of reference, for an ice shell thickness of 40 km, each 10 m of libration amplitude uncertainty translates to ∼ 5 km of ice/ocean thickness uncertainty for bodies like Miranda, Ariel, and Umbriel. For

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Figure 2. Contours show the implied thicknesses of the ice shell (teal) and ocean (dark blue), and the corresponding core density (dashed gray) as a function of libration
amplitude (measured at the equator) and C2,2 gravity (in units of dimensionless potential) for Enceladus and the five large Uranian moons. We assume ρice = 930 kgm 3
and ρocean = 1,030 kgm 3 . Black dash‐dotted lines show expectations without oceans. Ice shell rigidity and viscosity are taken to be μ = 4 GPa and η = 1020 Pas (but see
also Text S3 and Figure S7 in Supporting Information S1). For Enceladus, orange shading marks the observational constraints.

Titania and Oberon, however, the corresponding figures are much larger: ∼ 60 km and ∼ 130 km, respectively.
Hence, if the mission design allows for libration amplitude uncertainties comparable to those obtained by Cassini
at Enceladus (1σ ≈ 20 m at the equator), this method could be suited to detecting subsurface oceans at Miranda,
Ariel, and Umbriel, provided they are not too thin (Text S1, Figures S2, S3 in Supporting Information S1). For

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Titania and Oberon, however, libration amplitudes may be too small to measure with sufficient precision to detectsubsurface oceans.
If the ice shells of the Uranian moons are thin enough to yield large libration amplitudes, significant internal heatproduction may be required. Conservatively assuming that all of the ice shells are conductive, we can compute,
via Equation A17, the total conductive heat loss through the ice shells as a function of their thicknesses (Figure S4
in Supporting Information S1). As a point of reference, Enceladus's ice shell is estimated to be roughly 20 km
thick on average, which corresponds to a total conductive heat loss of ∼ 30 GW (D. J. Hemingway & Mittal, 2019), assuming the ice‐ocean interface is at a temperature of 273 K and that the thermal conductivity varies
with temperature (and therefore depth) as k = c/T, where c = 651 Wm 1 (Petrenko & Whitworth, 1999). In otherwords, roughly 30 GW of internal heat production is required within Enceladus in order to maintain its current ice
shell thickness. The primary sources of internal heat production are tidal heating (most effective for bodies on
short period eccentric orbits) and radiogenic heating (most effective for larger bodies with larger mass fractions ofsilicates). For Enceladus, radiogenic heating is negligible (Figure S5 in Supporting Information S1) so tidal
heating must be the primary source of internal energy dissipation.
Given its similar size to Enceladus, Miranda has a similar relationship between shell thickness and required
internal heat production, the biggest difference being the lower surface temperature of ∼ 60 K (vs. ∼ 75 K for
Enceladus). Given the bulk density constraint and our assumed core density (2400 kgm 3 ), the maximum
permissible ice shell thickness for Miranda is ∼ 100 km (Figure 1), which corresponds to a total conductive heatloss of ∼ 4 GW (Figure S4 in Supporting Information S1) in the presence of a thin ocean. This is far in excess ofradiogenic heat production (Figure S5 in Supporting Information S1) but if tidal heating were sufficiently intense,
total internal heat production could exceed this value, leading to a thinner ice shell and a libration amplitude
greater than ∼ 50 m (Figure 1). However, to yield a libration amplitude comparable to that of Enceladus (∼ 500 m
at the equator), would require Miranda to have an ice shell thickness less than ∼ 10 km. To maintain such a small
shell thickness would require ∼ 66 GW of internal heat production. For Ariel and Umbriel, which have roughly 6
times the surface area of Miranda, correspondingly larger internal heat production would be required to maintainthe same shell thickness. For Titania and Oberon, the surface areas are a factor of ∼ 10 larger than Miranda's,
requiring still greater internal heat production. Their larger volumes of rock yield more radiogenic heat (∼ 6 GW,
Figure S5 in Supporting Information S1), but conductive heat loss is still greater, even for the thickest ice shells
we considered (Figure S4 in Supporting Information S1).
A balanced heat budget may be easier to achieve under slightly different assumptions, however. For example, if
we allow core densities greater than our nominal value (ρcore > 2,400 kgm 3 ), the ice shells could be thicker than
the thickest values shown in Figure S4 in Supporting Information S1, with heat loss being correspondingly
smaller (though this will also reduce libration amplitudes, making them harder to measure). Furthermore, theabove calculations assume pure water (with Tmelt = 273 K) and do not consider porosity. Heat loss could besomewhat more limited if the oceans are very cold. If melting occurs at the ammonia‐water eutectic (176 K), forexample, heat loss could be up to 30% smaller than the numbers calculated above and shown in Figure S4 in
Supporting Information S1. Heat loss could also be reduced if thermal conductivity is less than what we assumed(section A3). For example, Bierson and Nimmo (2022) and Castillo‐Rogez et al. (2023) both showed that some of
the Uranian moons, particularly Titania and Oberon, could maintain long‐lived internal oceans if the ice shells
have very low thermal conductivity (<3 Wm 1 K 1 ), perhaps due to high porosity.
On the other hand, the heat budgets need not be balanced today. The presence of a subsurface ocean may insteadbe the fingerprint of a past epoch of efficient tidal heating, for example, when a larger orbital eccentricity wasmaintained by resonances between the satellites (Peterson et al., 2015; Ćuk et al., 2020). Without efficient tidal
heating at present, any extant oceans may be in the process of freezing today. In that case, the current ice shell
thickness could be used to approximately constrain how recently the ocean began freezing. Roughly speaking,assuming conductive cooling, the freezing timescale is related to thermal diffusion through the ice and can beapproximated as τ ≈ d2/κ, with κ ≈ 10 6 m2 s 1 . This suggests that, in the absence of internal heat production, a
10 km thick ice shell could not have been freezing for more than ∼3 million years while a 100 km thick ice shell
could not have been freezing for more than ∼300 million years (see also Castillo‐Rogez et al., 2023). Hence, if
librations reveal the presence of an internal ocean and yield an estimate of the ice shell's thickness, we can roughly
constrain how recently tidal heating must have been operating—subject to assumptions about ocean chemistry
and the thermal conductivity of the ice. The (past or present) efficacy of tidal heating within the interiors of the

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Uranian satellites is difficult to determine but geologic evidence suggests that Miranda and Ariel experienced
high surface heat flow at some point in their histories (Beddingfield et al., 2015, 2022a, 2022b; Bland et al., 2023;
Pappalardo et al., 1997; Peterson et al., 2015). The energy available for tidal heating is related to the orbital
evolution of the satellites and ultimately depends on the degree of dissipation (Q) within the interior of Uranus (F.
Nimmo, 2023), which is not yet well determined. If the effective Q of Uranus is sufficiently low, a previous
episode of tidal heating could have been enough to support liquid water oceans within the Uranian satellites(Castillo‐Rogez et al., 2023; F. Nimmo, 2023).

5. Conclusions
We have shown that measurement of the forced physical libration amplitudes of the large Uranain satellites has
the potential to reveal the presence of subsurface liquid water oceans within their interiors (Figure 1). We haveshown that combined measurements of the libration amplitude and the quadrupole gravitational field can be used
to simultaneously constrain the ice shell thickness, ocean thickness, and the size and density of the rocky core
(Figure 2). For Miranda, Ariel, and Umbriel, libration amplitudes will exceed 100 m at the equator if the ice shells
are less than 30 km thick. If libration amplitudes can be constrained to within a few tens of meters, subsurface
oceans could be detected as long as they are not too thin (Figures 1 and 2, S2, S3 in Supporting Information S1).
Oceans less than 40 km thick, however, may be difficult to detect unless libration amplitudes can be measured towithin better than 10 m. For Titania and Oberon, libration amplitudes are likely less than 20 m, even if the ice
shells are very thin, and so may be more challenging to measure.
We discussed the importance of constraining ice shell thickness and how it relates to the energy budgets of ocean
worlds. If the ice shells are very thin, significant tidal heating would be required to keep the oceans from freezing(Figure S4 in Supporting Information S1) unless the ice shells have very low thermal conductivity and/or theocean melting temperatures are significantly depressed (e.g., Bierson & Nimmo, 2022; Castillo‐Rogez
et al., 2023). Alternatively, the oceans could be in the process of freezing today. In that case, the current ice
shell thickness could be used to determine roughly how recently tidal heating must have been effective—likely
within the last few hundred million years for the cases we considered.
If flyby geometries and instrumentation can be selected to permit sufficiently precise measurements of libration
amplitudes and the quadrupole gravitational field (Figures 1 and 2), a future tour of the Uranus system has thepotential to reveal the presence of subsurface liquid water oceans and to help constrain the internal structures,
energy budgets, and thermal/orbital evolution of the Uranian moons.

Appendix A: Detailed Methods
A1. Shape, Gravity, Internal Structure
Both the libration amplitude and gravitational field calculations require a model of the satellite's internal structure.
We represent each body as a series of concentric shells of uniform density, subject to the known radius and bulk
density (Table S1 in Supporting Information S1). The bulk density, ρ̄, is related to the layer densities and radii by
R 3ρ̄ = ∑ Δρi ( i )
R
i

(A1)

where R is the known outer radius, Ri is the mean radius of layer i, and Δρi is the density contrast between layer iand the layer above it. The mean moment of inertia, or moment of interia factor, is given byI
2 Δρi Ri 5=∑
( )MR2 5 i ρ̄ R

(A2)

In our ‘no‐ocean’ models, we specify only the densities of the two layers and we use the known total radius andbulk density to solve Equation A1 for the radius of the core. In our three layer models (ice, ocean, core), we
specify all three layer densities and the thickness of the ice shell and again solve Equation A1 for the radius of thecore.

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We assume the bodies have relaxed to hydrostatic equilibrium, and we calculate the expected hydrostatic figures
for each layer using the second order recursive method of Tricarico (2014) (see their Section 2.1.2). The degree‐land order‐m gravitational field asymmetries arising from such a non‐spherical multilayer body, and resolved atradius r, is given by
4πGr
R l+2
∑ Δρi Hilm ( i )r
2l + 1 ibelow

Ulm (r) =(A3)where Hilm represents the shape of the ith layer. The Tricarico (2014) approach to computing hydrostatic equilibrium figures involves approximating the layers as nested triaxial ellipsoids, with shapes represented in terms ofpolar and equatorial eccentricities, ep and eq , or by their semi‐axes whereR
a = √̅̅̅̅̅̅̅̅̅̅̅̅̅ √̅̅̅̅̅̅̅̅̅̅̅̅̅ 1/3( 1 e2p 1 e2q )√̅̅̅̅̅̅̅̅̅̅̅̅̅
b = a 1 e2q√̅̅̅̅̅̅̅̅̅̅̅̅̅
c = a 1 e2p(A4)

abc = R3
which can, in turn, be related to the polar and equatorial flattenings
2c
a+b
b
β=1
a

α=1

(A5)

These triaxial ellipsoids can also be well approximated by degree‐2 spherical harmonic expansions with only two
non‐zero coefficients1
H 2,0 = (2c a b)3
1
H 2,2 = (a b)6(A6)

For simplicity, we have not shown the subscript i in Equations A4–A6, but it should be understood that each layer
has its own R, a, b, c, ep , eq , α, β, and Hlm .
A, B, and C are the principal moments of inertia (in general A < B < C for synchronous satellites) and are given by
A = ∫ ρ y2 + z2 ) dV
B = ∫ ρ x2 + z2 ) dV

(A7)

C = ∫ ρ x2 + y2 ) dV
where ρ is the density of a volume element dV at position x, y, z within the body. From the above, it can be shown
that, for a body consisting of nested triaxial ellipsoids,
B

A
C

HEMINGWAY AND NIMMO=∑i Δρi a2i b2i ) ai bi ci∑i Δρi a2i + b2i ) ai bi ci(A8)

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which is the satellite's dynamical triaxiality (sometimes denoted Σ). Note that while we generally compute the
B Agravity moments via Equation A3, it can also be shown that C2,2 = 4MR
2.
A2. Forced Physical Libration Amplitude
A2.1. Without an Ocean
When no decoupling ocean is present, the ice shell and rocky core rotate together and the libration amplitude isgiven by (Murray & Dermott, 1999, p. 216)γ=

2e(n/ω0 )21

(A9)where e is the orbital eccentricity, n = 2π/P is the satellite's mean motion, where P is the orbital period, and ω0 is
the free libration frequency given by (Comstock & Bills, 2003)√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
B A
ω0 = n 3
C
where the satellite's dynamical triaxiality, (Bγ=Because B

A)/ C, is given by Equation A8. It follows that
2e

1

(A10)

C
3(B A)=6e
B A
3 BCA 1) C

(A11)

A is much smaller than C, the libration amplitude is often approximated as
γ≈

6e

B

A

(A12)

C

A2.2. With a Decoupling Ocean
When the ice shell is mechanically decoupled from the rocky core by a liquid water ocean, the ice shell's response
depends mainly on its thickness and rigidity (Goldreich & Mitchell, 2010; Van Hoolst et al., 2013). Although
there is still gravitational coupling between the layers, this effect is minor compared to the mechanical couplingthat occurs in the absence of an ocean. If the ice shell is sufficiently thick and the body is not too large (i.e., gravity
is relatively weak), elastic forces in the ice shell dominate and the ice shell is effectively infinitely rigid. In thiscase, gravitational torques acting on the ice shell will cause it to oscillate in a way that depends mainly on its own
moment of inertia (primarily a function of its thickness), with the libration amplitude given by (e.g., Thomas
et al., 2016)
γ=

2e[K s K c + 2K int n2 Cc ) + 2K int K c ]Cs Cc n2 ω21 ) n2 ω22 )(A13)

where ω1 and ω2 are the free libration frequencies (Van Hoolst et al., 2008). Cc and Cs are the separate polar
moments of inertia for the rocky core and the icy shell, respectively, and are given by (e.g., Van Hoolst
et al., 2008)
Cc =

8π
2
ρ R5 (1 + αc )15 c c38π
2Cs = ρs [R5s (1 + αs )1532R5o (1 + αo )]

(A14)

3

where we use the subscripts s, o, and c for the shell, ocean, and core, respectively, and where the flattenings, α, aregiven by Equation A5. Finally, K c and K s are the amplitudes of the effective torques applied by the parent body on

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the satellite's rocky core and icy shell, respectively, and K int is the amplitude of the torque applied on the ice shellby the rest of the interior.
Equation A13 is a good approximation for Enceladus, Miranda, Ariel, and Umbriel, since elastic forces will
dominate unless the ice shell is very thin (see Text S3 and Figures S6–S7 in Supporting Information S1). On theother hand, if the ice shell is thin and the body is large (i.e., gravity is strong), elastic stresses within the ice shell
will be overcome by gravitational forces and the ice shell will deform in response to the changing tidal potential.
To allow for this possibility, we in general follow the approach of Van Hoolst et al. (2013), computing the shell's
libration amplitude using their equation (62)γ=4e K 3 K 5 K 2 K 6 n2 K 3 Cc )Cc Cs n2 σ21 ) n2 σ22 )(A15)where the free frequencies σ1 and σ2 are given by Van Hoolst et al. (2013) equation (64). The K 1 ‐K 6 terms, givenby their equations (47)–(52), are related to the various torques that drive the physical librations. These include theexternal torques applied by the parent body on each of the satellite's layers (shell, ocean, and core), as well as theinternal torques of each layer acting on the others.
Although we are assuming hydrostatic equilibrium (see A1), some non‐hydrostatic topography is possible,
especially for the smaller bodies. This can influence the torques by affecting the layers' moment of inertia differences, (B A). However, even in the case of Enceladus, where we have good constraints on the topography
and have reason to believe there are large, non‐hydrostatic lateral shell thickness variations (e.g., D. J. Hemingway & Mittal, 2019), the non‐hydrostatic effects reduce the libration amplitude by just ∼ 6% assuming the mean
ice shell thickness is 21 km; the effects are even smaller for larger shell thicknesses and/or larger bodies.
The Van Hoolst et al. (2013) formulation accounts for elastic deformation of the layers by calculating the
dynamical Love numbers for each layer, given by (see their equation 24)
4πGρs 4Rs ys R4o yo )5R34πGρo 4k2,ot =
Ro yo )
5R3
4πGρok2,ob =
R4c yc )5R3
4πGρc 4k2,c =
Rc yc )
5R3

k2,s =(A16)

which appear in some of the expressions for the torque amplitudes (Van Hoolst et al., 2013 equations 39–41, 54–57), along with the fluid Love numbers for each layer (see their eq. 25). In Equation A16, the effects of the ocean
are divided into their top and bottom parts (ot and ob) because the top of the ocean follows the shape of the base of
the ice shell while the bottom of the ocean follows the shape of the rocky core. Rs , Ro , and Rc are the mean radii ofthe shell, ocean, and rocky core, respectively, and ys , yo , and yc are the radial displacements that arise at each of
those radii in response to the changing tidal potential. We compute these radial displacements following theapproach described by Roberts and Nimmo (2008) (see their eqs. 3–4, and Appendix A; we benchmarked our
calculations against those of Roberts & Nimmo, 2008; Shao & Nimmo, 2022). This approach involves using a
propagator matrix technique (e.g., Sabadini et al., 2016) to solve for the radial functions and approximating theinviscid ocean as a low‐rigidity, low‐viscosity material (we use μ = 106 Pa and η = 1011 Pas). We compute thecomplex rigidity of each layer assuming a Maxwell rheology (e.g., Tobie et al., 2005, equation 13).
A3. Heat Budget
Given an estimate of a satellite's mean ice shell thickness, d, we can estimate the minimum internal heat production required to balance the heat loss through the surface in order to maintain a subsurface liquid water ocean.
We obtain a lower bound on surface heat flux by assuming a conductive ice shell, in which the thermal

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conductivity varies with temperature (and therefore depth) as k = c/T, where c = 651 Wm 1 is an experimentallyderived constant (Petrenko & Whitworth, 1999). Conductive heat flux at the surface is then given by
F = (1d c Ts) ln( )
R d Tb(A17)

where Ts is the mean surface temperature, Tb is the temperature at the base of the ice shell (i.e., the melting
temperature), and R is the mean radius of the body.

Data Availability Statement
All the data used in this research are freely available in the published literature. The estimated properties for the
Uranian satellites (listed in Table S1 in Supporting Information S1) come from Jacobson (2014), P. C.
Thomas (1988), and Widemann et al. (2009). For Enceladus, the shape comes from Tajeddine et al. (2017), thegravitational field comes from Iess et al. (2014), and the libration amplitude comes from Thomas et al. (2016) (butsee also Park et al. (2024)).
Acknowledgments
This work was supported by the Universityof Texas Institute for Geophysics and theJackson School of Geosciences at the
University of Texas at Austin. We thankTushar Mittal, Bob Jacobson, and Matt
Clement for helpful discussions, and threeanonymous reviewers for constructive
suggestions that improved the manuscript.

HEMINGWAY AND NIMMO

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