GEOPHYSICAL RESEARCH LETTERS

123

Supporting Information for “Looking for subsurface
oceans within the moons of Uranus using librations
and gravity”
D. J. Hemingway1 , F. Nimmo2

4

1 Institute for Geophysics, Jackson School of Geosciences, University of Texas at Austin, 10601 Exploration Way, Austin, TX, 78758

52 Department of Earth and Planetary Sciences, University of California Santa Cruz, 1156 High St., Santa Cruz, CA, 95064

6

Contents of this file7

1. Text S1-S3

8

2. Table S19

3. Figures S1-S7

10Text S1: Measurement Precision

11Figure 1 in the main text illustrates how libration amplitude is expected to vary with ice

12shell thickness for each of the bodies we considered. If the ice shells are thin (and therefore

13

the oceans are thick), libration amplitudes are relatively large and easily distinguished14

from the ‘no-ocean’ case. But what measurement precision is required to distinguish15

between the different interior models and to confirm or rule out the presence of a subsurface

16

ocean? This depends on what part of Figure 1 we consider. Because libration amplitude

August 21, 2024, 2:39pm

X - 2 HEMINGWAY & NIMMO: LOOKING FOR URANIAN OCEAN WORLDS USING LIBRATIONS
17

is an inverse function of shell thickness, the curves in Figure 1 (recognizing that this is18a log-log plot) are steep when the ice shell thickness is small and flatter when the ice

19shells are thick (i.e., when the oceans are thin). That is, for small ice shell thicknesses,

20

small changes in ice shell thickness result in large changes in libration amplitude. By

21comparison, for larger ice shell thicknesses, a much larger change in shell thickness is22

required to yield the same change in libration amplitude. The gradients of the curves

23∂din Figure 1 ( ∂γ) can thus be used to quantify the relationship between the precision of24

libration amplitude measurements and the resulting uncertainty in shell thickness (for25

a given thickness, this is very nearly equivalent to the uncertainty in ocean thickness).

26

Considering a range of possible libration amplitude measurement uncertainties (from 1 to27

100 m), we obtain the shell/ocean thickness uncertainties shown in Figure S2. Although28

this information is already implicit in Figure 1, Figure S2 reveals greater detail that can29

further clarify the relationships. Notably, if the thickness uncertainty is larger than the30

ocean thickness, the presence of an ocean cannot be confirmed (region above/right of the31

dash-dotted black line). Whereas Figure S2 assumes ρcore = 2400 kg m−3 , Figure S3 is32

obtained by instead assuming ρcore = 3000 kg m−3 , which may be more applicable to the33

larger moons depending on their thermal evolution (Castillo-Rogez et al., 2023).

34

The white contours in Figures S2 and S3 indicate the precision with which the ice shell

35

thickness can be estimated under various conditions. Estimating the ice shell thickness

36

to within 10%, for example, requires being below the dashed white line; being below the37dash-dotted white line ensures estimates are within 50% of the true value. Because the38∂dgradients ( ∂γ) vary as a function of shell thickess, no single value of libration measurement

August 21, 2024, 2:39pm

HEMINGWAY & NIMMO: LOOKING FOR URANIAN OCEAN WORLDS USING LIBRATIONS X - 339precision can guarantee any given desired precision in shell/ocean thickness determination.

40

Nevertheless, a few broad conclusions can be drawn from Figures S2 and S3. First, using

41

librations to confirm the presence of a subsurface ocean on Titania or Oberon may be42difficult unless the ice shells are thin or the libration measurements can be very precise

43(lower left corners of those subplots). Second, results are similar across Miranda, Ariel,

44and Umbriel, suggesting that one broad set of planning guidelines may be suitable for all

45three. For example, if libration amplitudes can be estimated to within ∼ 10 m, oceans46

can be detected within these bodies even for ice shell thicknesses up to ∼ 100 km. With47

a precision of ∼ 10 m, libration measurements would reveal oceans as thin as ∼ 30 km

48

at Miranda or ∼ 40 km at Ariel and Umbriel. This again assumes ρcore = 2400 kg m−349

but if we assume ρcore = 3000 kg m−3 for Ariel and Umbriel (Castillo-Rogez et al., 2023),50∼ 10 m of libration amplitude precision assures ocean detection only if oceans are at least51∼ 60 km thick. Detection of thinner oceans would require even greater precision in the52libration amplitude estimates, which could be achievable given the right instrumentation53

and tour design (Park et al., 2020).

54

Text S2: Radiogenic Heating

55

Figure S5 illustrates radiogenic heat production for bodies that are a mixture of ice and56

rock. We assume a mixture of water/ice with ρH2O = 1000 kg m−3 and chondritic rock57

with ρchond = 3500 kg m−3 and that the radiogenic heat production rate for this rock is58

H = 4.5 × 10−12 W kg−1 (Spohn and Schubert, 2003). The bodies we consider may consist

59

of a combination of ice, water, and porous and/or hydrated silicates, so their rocky cores60

will generally not consist of purely chondritic rock. We use each body’s bulk density (ρ̄) toAugust 21, 2024, 2:39pm

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HEMINGWAY & NIMMO: LOOKING FOR URANIAN OCEAN WORLDS USING LIBRATIONS

61

H2O
compute the equivalent mass fraction of chondritic rock as f = 1/ρ1/ρ̄−1/ρ
(Hemingwaychond −1/ρH2O

62and Mittal, 2019). The total radiogenic heat production is then simply Hf M where M

63

is the body’s total mass, regardless of the core density. This approach accounts for the64

fact that a body with a lower core density (due to hydration or having ice- or water-filled

65pores) produces proportionally less radiogenic heat.

66

Text S3: Ice Shell Rigidity

67

The magnitude of the radial displacements (the y values discussed in A.2.2) depend

68

mainly on the ice shell’s rigidity and viscosity. If the ice shell has very high rigidity, ys

69

and yo will be small (yc is small in any case) and the libration amplitude given by (A15)70

approaches the value given by (A13). When the ice shells are thin, and especially for71

the larger bodies, libration amplitudes are lower when the finite rigidity is taken into

72

account. For purposes of Figures 1 and 2, we adopt µ=4 GPa and η=1020 Pa s for the73

ice shell but, for reference, Figures S6 and S7 illustrate the maximum possible libration74

amplitudes, obtained by assuming an infinitely rigid shell and computing them instead75

via (A13)—the same result is obtained with (A15) when we adopt very large µ and η.

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HEMINGWAY & NIMMO: LOOKING FOR URANIAN OCEAN WORLDS USING LIBRATIONS

Table S1. Satellite properties and libration amplitudes assuming no ocean. Satellite periods,eccentricities, and masses are from Jacobson (2014). Radii are from Thomas (1988), Widemannet al. (2009), and Park et al. (2024). Moments of inertia are calculated via (A2) and (A8)assuming the bodies consist of a water ice mantle (ρice =930 kg m−3 ) covering a rocky core (withdensities specified in the column headers). γ is calculated via (A11).
ρcore =2400 kg m−3
Satellite

P (days)e20

M (×10 kg) R(km) ρ̄(kg/m )B−ACI
M R2γR(m)B−ACI
M R2γR(m)Enceladus

1.37020.0047

1.08

252.0

1611.1

1.83e-02 0.332 137.47 1.59e-02 0.301 118.52

Miranda1.41350.0014

0.64

235.8

1165.42.53e-02 0.343

52.26

2.47e-02 0.336

50.91

Ariel

2.52040.0012

12.51

578.9

1539.45.49e-03 0.330

23.67

4.87e-03 0.30220.95

Umbriel

4.14420.0039

12.75

584.7

1522.72.04e-03 0.329

28.38

1.82e-03 0.303

25.26

Titania

8.70590.0012

34.00

788.9

1653.24.34e-04 0.334

2.53

3.73e-04 0.300

2.17

Oberon13.46320.0014

30.76

761.4

1663.61.81e-04 0.335

1.16

1.55e-04 0.300

0.99

2100

aTriton

2000

41800

3.5

Titania
Enceladus
Ariel

1600

Oberon1500
Umbriel

1200

3
2.5
2
Miranda
Ariel

11300

b
Umbriel

1.5

1400

-3

Enceladus

4.5

1900

1700

105

Eccentricity

Density (kg/m 3)

ρcore =3500 kg m−33Oberon

Titania

0.50Miranda
24681012140024

Orbital Period (days)

Figure S1.

Triton68101214

Orbital Period (days)

Orbital periods versus bulk densities (a) and orbital eccentricities (b) of the large

Uranian moons plus Enceladus and Triton. Body size is indicated by the size of the circles, which
are shown on an arbitrary but common scale.

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HEMINGWAY & NIMMO: LOOKING FOR URANIAN OCEAN WORLDS USING LIBRATIONS
Ocean Thickness (km)Ocean Thickness (km)103010252010115105

Enceladus
100102030

4080

60

10150
40
30
201080

60

01020

30

20100

Ariel

1090
8070
60

10150
40
30
20
10100204060

80

100

120

140

120

100

80

60

209070
60
15040
30
20
1020

40

60

80

100

120

Ice Shell Thickness (km)Figure S2.

80

60

40

20

1009060

10150
40
30
20
101080

10 0

100

70

0

20160

100

Titania

1080

4060

80

100

120140Ocean Thickness (km)
40140

160

Libration measurement uncertainty (m)

120

70

Ice Shell Thickness (km)Thickness Uncertainty (km)Libration measurement uncertainty (m)101406080

Ocean Thickness (km)16050

Umbriel

10140

2

Ice Shell Thickness (km)240

Ocean Thickness (km)40

Libration measurement uncertainty (m)10090

Ice Shell Thickness (km)Thickness Uncertainty (km)Libration measurement uncertainty (m)

10120

10070

Ocean Thickness (km)1404080

Ice Shell Thickness (km)260

Miranda1050

100

2Thickness Uncertainty (km)20

Thickness Uncertainty (km)

30140

2120

1008060

40

20

Oberon10090807060

10150
40
30

Thickness Uncertainty (km)40

Libration measurement uncertainty (m)50Thickness Uncertainty (km)Libration measurement uncertainty (m)102201010 020406080100120140

Ice Shell Thickness (km)Ice shell/ocean thickness uncertainty as a function of shell/ocean thickness andlibration amplitude uncertainty and assuming ρcore = 2400 kg m−3 . To the right of the dashdotted black line, uncertainties are larger than the ocean thickness, meaning the presence of an
ocean cannot be confirmed. Below the white lines, the ice shell thickness can be constrained towithin 10% (dashed white line), 50% (dash-dotted white line), or 100% (solid white line).
August 21, 2024, 2:39pm

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HEMINGWAY & NIMMO: LOOKING FOR URANIAN OCEAN WORLDS USING LIBRATIONS

Ocean Thickness (km)Ocean Thickness (km)
40

30

20

1070

105040

10

130

20

10

Enceladus
10Thickness Uncertainty (km)60

0102030405060120

260101504030201010 2100Ariel9060

10 15040
30

Thickness Uncertainty (km)

70

20
10050

100

2030

100

100
9060

10 15040302010100150

Figure S3.

200

Thickness Uncertainty (km)70

Ice Shell Thickness (km)

80

10050

10090807060

10 150403020101080

5070

050

100150Ocean Thickness (km)50Titania

10 060Ice Shell Thickness (km)Libration measurement uncertainty (m)Libration measurement uncertainty (m)

10150

50150

Ocean Thickness (km)20040

Umbriel

10150

200

Ice Shell Thickness (km)290

Ocean Thickness (km)
5080

10100

Ice Shell Thickness (km)Libration measurement uncertainty (m)Libration measurement uncertainty (m)

100

4070

Ocean Thickness (km)1506080

Ice Shell Thickness (km)10 280Miranda

10 01070100Thickness Uncertainty (km)50

Thickness Uncertainty (km)60220015010050100

Oberon90807060

10 1504030201010 050100150

Ice Shell Thickness (km)Same as Figure S2 but assuming ρcore = 3000 kg m−3 .

August 21, 2024, 2:39pm

200

Thickness Uncertainty (km)70

Libration measurement uncertainty (m)Libration measurement uncertainty (m)102

HEMINGWAY & NIMMO: LOOKING FOR URANIAN OCEAN WORLDS USING LIBRATIONS

Total Conductive Heat Loss (GW)

X-8

Enceladus
MirandaAriel
UmbrielTitania
Oberon10 3

10 210110 110 2

Ice Shell Thickness (km)Figure S4. Total heat loss as a function of shell thickness for each body assuming a conductive
ice shell with a basal (i.e., melting) temperature of T = 273 K. See section A.3.
1800

106

1

91700

82

1500

7ArielUmbriel

665

11400

4
100.5

1300

48231200

6Miranda
1100
1000

0.5200

400

600

120.5

02

4

1800

1000

Radiogenic Heat Production (GW)

810Enceladus

1600

Bulk Density (kg/m 3)

40.5

OberonTitania

101200Radius (km)Figure S5. Total radiogenic heat production as a function of radius and bulk density assumingthat each body is a mixture of ice and rock (see Text S2 for details).

August 21, 2024, 2:39pm

HEMINGWAY & NIMMO: LOOKING FOR URANIAN OCEAN WORLDS USING LIBRATIONS

Enceladus
MirandaAriel
UmbrielTitania
Oberon10 3

Libration Amplitude (meters)

X-9

10 2

10 1

10 0
10 1

10 2

Ice Shell Thickness (km)Figure S6.

Same as Figure 1 in the main text but assuming an infinitely rigid ice shell andtherefore with libration amplitudes calculated via (A13) rather than (A15).

August 21, 2024, 2:39pm

X - 10HEMINGWAY & NIMMO: LOOKING FOR URANIAN OCEAN WORLDS USING LIBRATIONS

10200

3040

2000

203060

100

40

10200

1630

30500

121490Libration Amplitude (meters)

300

102400110

dicedocean
ρcore

1620300

Miranda
2800

500

2000

142400

50

70
2800

700

Libration Amplitude (meters)

123500

Enceladus
3500

1000

2024002450

C 2,2 Gravity (x106)

2500

100
2000

140

101420200

no ocean25

Figure S7.

10

C 2,2 Gravity (x106)

35

C 2,2 Gravity (x106)

40

200

1045

101214162030640480

32140

no ocean

30

3060
190

1030

140

3180

1080
5170

140

2060

10

160

2800
200

20

30
40

150

60

20

no ocean

140

Oberon

12
14
16

2000

10100

30

30

550

Libration Amplitude (meters)

2400

C 2,2 Gravity (x106)

500

140

3500

Titania

450

200

400

2800

350

120

3500

30

50

30

120

60

2000

1050

100

100

60

30

2400

3060

100

Libration Amplitude (meters)

Umbriel

2350

300

3050

no ocean

2300

2400

2400

14

2250

200
2800

500

10200

2200

1900

100

1800

Libration Amplitude (meters)

1700

C 2,2 Gravity (x106)300

Libration Amplitude (meters)

1600

80

60

2000

200

Ariel1500

1401400

3500

500

1300

2800

1200

no ocean3500

80

no ocean

1213

14

15

16

C 2,2 Gravity (x106)

1718

19

Same as Figure 2 in the main text but assuming an infinitely rigid ice shell andtherefore with libration amplitudes calculated via (A13) rather than (A15).

August 21, 2024, 2:39pm