Electromagnetic induction in the icy satellites of Uranus

6 The discovery of subsurface oceans in the outer solar system has transformed our perspective of ice worlds and has led 7 to consideration of their potential habitability. The detection and detailed characterisation of induced magnetic fields 8 due to these subsurface oceans provides a unique ability to passively sound the conducting interior of such planetary 9 bodies. In this paper we consider the potential detectability of subsurface oceans via induced magnetic fields at the 10 main satellites of Uranus. We construct a simple model for Uranus’ magnetospheric magnetic field and use it to 11 generate synthetic time series which are analysed to determine the significant amplitudes and periods of the inducing 12 field. The spectra not only contain main driving periods at the synodic and orbital periods of the satellites, but also a 13 rich spectrum from the mixing of signals due to asymmetries in the uranian planetary system. We use an induction 14 model to determine the amplitude of the response from subsurface oceans and find weak but potentially-detectable 15 ocean responses at Miranda, Oberon and Titania, but did not explore this in detail for Ariel and Umbriel. Detection of 16 an ocean at Oberon is complicated by intervals that Oberon will spend outside the magnetosphere at equinox but we 17 find that flybys of Titania with a closest approach altitude of 200 km would enable the detection of subsurface oceans. 18 We comment on the implications for future mission and instrument design. 19


Introduction
Evidence of subsurface liquid oceans beneath the frozen ice crusts of the icy satellites of Jupiter and Saturn has 22 changed our perspective of small icy worlds in the outer solar system; these worlds are not inert. Furthermore, the 23 possibility of extant microbial life in their oceans has forced us to reconsider the meaning of planetary 'habitability' 24 throughout the solar system and beyond (e.g., Lazcano and Hand, 2012). Measurements of Europa's gravity field 25 pointed towards a three-shell model consisting of an H 2 O layer overlying a silicate mantle and metallic core. Although 26 the state of the H 2 O layer was unknown, surface geomorphology, including cryovolcanic features, chaos terrain, 27 topography and global tectonics, argued for a liquid subsurface ocean (Anderson et al., 1998;Pappalardo et al., 1999). 28 One of the principal pieces of evidence for the existence of a liquid subsurface ocean was the detection of magnetic 29 field perturbations, in the vicinity of the satellite, that were consistent with an induced field from the interior. As 30 Jupiter rotates, the tilted dipole causes diurnal oscillations in the magnetospheric magnetic field at the satellites, 31 providing an alternating 'primary' or 'inducing field'. Ions dissolved in the subsurface ocean respond to this inducing 32 field and generate eddy currents which produce a secondary or induced field that acts to exclude the varying field from The configuration of the magnetosphere at a given moment can be coarsely parameterised by the solar wind attack 131 angle (e.g., Lepping, 1994) which is the angle, α, that the dipole axis makes to the solar wind (where α=0 represents the 132 dipole pointing into the solar wind). Earth's magnetosphere has an attack angle of ≈ 90 ∘ ± 30 ∘ which means that the 133 dipole is more-or-less perpendicular to the incoming solar wind flow and varies only slightly with Earth's rotation. The 134 combination of large dipole tilt and obliquity at Uranus means that near solstice the attack angle is relatively constant 135 (the magnetosphere essentially rotates around the planet-Sun line) but at equinox the dipole can, at times, be pole-on 136 to the solar wind (Figure 2 bottom right) and so the solar wind attack angle can vary considerably over one Uranus 137 rotation. These significant diurnal and seasonal variabilities are a unique feature of the uranian magnetosphere (e.g., 138 Lepping, 1994; Arridge and Paty, in press). 139 Figure 3 shows the solar wind attack angle for one Uranus year (approximately 84 years). The variation can be thought 140 of a slowly varying mean solar wind attack angle representing seasonal variations with a high frequency diurnal 141 variation forming an 'envelope' around the mean. The figure also shows the angle between the spin axis and the solar 142 wind that informs us about the orientation of the satellite orbital plane relative to the Sun (e.g., the changing 143 orientation in Figure 2). These two angles expose the different characteristics of the dipole-solar wind geometry and 144 satellite orbital plane-solar wind geometry over time. We split this time series into 'zones' to reflect different characteristics at different epochs. The solar wind attack angle is quantised into 45° segments, for example where 0 ≤ 146 < 22.5 ∘ would be represented by 0°, 22.5 ∘ ≤ < 67.5 ∘ by 45 ∘ , and 67.5 ∘ ≤ < 112.5 ∘ by 90 ∘ . Thus we coarsely 147 represent the attack angle as quasi-pole-on, intermediate, and quasi-perpendicular, etc.. The spin axis angle is similarly 148 quantised into 30 ∘ segments. Each unique combination of quantised attack angle and spin axis angle forms a unique 149 'zone' that represents some quantised representation of the geometry of Uranus' magnetosphere and its seasonal and 150 diurnal variation. 151 For example, near the Voyager 2 flyby in 1986, the solar wind attack angle was quasi-perpendicular to the solar wind 152 and the spin axis was approximately parallel to the solar wind implying that the satellites only experienced the 153 magnetospheric flanks and polar regions over an orbit, never encountering the dayside and nightside. Whereas during 154 2015, the magnetosphere transitioned between pole-on and perpendicular configurations over a diurnal rotation and 155 the spin axis was perpendicular to the solar wind implying that the satellites experienced the nightside and dayside of 156 the magnetosphere every orbit. field at the satellites. The distribution and magnitude of currents on the magnetopause change with solar wind attack 165 angles and so affect the primary field by changing the contribution from the magnetopause currents. Furthermore, as 166 can be seen in the insets, the solar wind attack angle variation is also not purely sinusoidal so introduces additional 167 harmonics to the diurnal variation of the magnetopause field. Although the solar wind attack angle is a useful and 168 intuitive tool, the real configuration can be quite difficult to fathom as the field is truly three-dimensional and simple 2D 169 cuts do not fully capture the configuration (Arridge and Paty, in press); 3D tools are essential in understanding the 170 geometry at a given epoch (Arridge and Wiggs, 2019).

172
In this paper we explore the induced response from sub-surface oceans that have been suggested to exist within Titania 173 and Oberon. Hence, we explore the capability of passive magnetic sounding to provide empirical evidence for the 174 presence of an ocean, as carried out at Europa (Khurana et al., 2009), and examine whether such oceans might be 175 detectable by potential future missions. Furthermore, because Uranus' magnetosphere is so diurnally-variable, and this 176 variability changes with season, there may be seasons where an induced response from the oceans is richer and which 177 might provide additional opportunities to learn more about these elusive worlds through multi-frequency passive 178 sounding. Although we focus primarily on Titania, we have learned from the Cassini mission that small icy bodies can be 179 more active than previously thought and so we also include some examination of the inducing field at Uranus' other 180 satellites. We first construct a model for the magnetospheric magnetic field at Uranus and use it to calculate the 181 primary field at each of Uranus' main satellites over a Uranus orbit (section 2). The amplitude of the primary field 182 harmonics are evaluated in section 3 and applied to compute the induced response using the internal structure models 183 of Hussmann, Sohl and Spohn (2006

188
To calculate the primary (inducing) field at the satellites we require a model for the magnetospheric magnetic field. In 189 the terrestrial magnetosphere, the principal contributions to this field are the internal (planetary) magnetic field, the 190 field of the Chapman-Ferraro (magnetopause) currents, the field of the ring current, and the field of the cross-tail 191 current sheet (e.g., Tsyganenko, 2002). At Jupiter and Saturn these must also be augmented by the field of a 192 magnetodisc current sheet which has a dramatic effect on the near-equatorial field (e.g., Smith et al., 1974;Arridge and 193 Martin, 2018). At Uranus we only include the internal planetary field and the Chapman-Ferraro field. Voyager 2 194 observations demonstrated the presence of a well-developed helical tail current sheet at solstice (Behannon et al., 195 1987), however, its geometry for other solar wind attack angles, e.g., at equinox with periods in both pole-on and 196 perpendicular orientations, is much more complex and so we chose not to include a tail current sheet. The implications 197 of this will be considered shortly. In contrast to the tail current sheet, Voyager 2 observations suggested a very weak 198 ring current, relative to the internal field, around an order of magnitude smaller than that at Earth (Connerney et al., 199 1987) and so we chose not to include this in our model. We use the most recent internal field model, which uses 200 magnetometer data and ultraviolet auroral data from Voyager 2 to constrain a 4th degree (hexapole) spherical 201 harmonic model 'AH5' (Herbert, 2009). 202 The final ingredient is a geometrical model for the magnetopause. was fixed for all of our calculations and was not varied with solar wind dynamic pressure (discussed further below), or 209 varied to reflect different seasonal/diurnal configurations. This would be expected to introduce additional variations in 210 the magnetic field at the appropriate periods, but since we have little information to calculate such changing geometry 211 we chose to exclude such effects. 212 The effect of the Chapman-Ferraro currents is to 'confine' or 'shield' the magnetic field inside the magnetopause such 213 that no magnetic flux crosses the boundary. This corresponds to a condition where the magnetic field normal to the 214 magnetopause is zero everywhere on the boundary. In our model we only consider the field of the planet, int , and so 215 we can state this problem as a zero-valued integral over the boundary. For the practical purpose of finding a Chapman-216 (7) 228 where , , and are amplitude coefficients, , , and are scale parameters, Ψ = 2 − is the dipole tilt angle, 230 and , , and are coordinates in the right-handed Uranocentric Solar Magnetospheric (USM) system, where � is the 231 unit vector from Uranus to the Sun, and the X-Z plane contains the dipole vector, , such that � = � × � . Equations 232 (7,8) include 25 harmonics with 64 free parameters to be found through the minimisation of (6). 233 As the quadrupole and higher degree terms are weak at the magnetopause, only the dipole component was included in 234 the minimisation of (6) with a dipole moment of 23000 nT (Connerney et al., 1987). For simplicity the dipole offset was 235 not included in this computation, but only causes a scalar error of less than = 3~0.06 nT in the field strength 236 at the magnetopause, smaller than the quadrupole field at the magnetopause. The minimisation was performed using 237 the Downhill Simplex algorithm (Nelder and Mead, 1965) as implemented in the Python package SciPy (Virtanen et al.,238 in press). The minimiser was iteratively called, each time increasing the number of points that the field was evaluated 239 on, until the root-mean-square normal field and maximum absolute normal field were less than 0.005 nT and 0.01 nT 240 respectively, or until the optimiser had terminated successfully four times in a row. The optimisation of (6) was 241 completed with 37440 points with a root-mean-square normal field of 0.00199 nT and a maximum normal field of 242 0.0104 nT. Figure 4 shows field lines traced for four different dipole tilt angles showing the quality of the 243 magnetopause shielding and the coefficients required for equations (7) and (8)  The entire model was built in Python using NumPy (Oliphant, 2006), SciPy and the SPICE toolkit (Acton, 1996) via 249 SpiceyPy (Annex et al., 2020) packages, and a custom magnetic field modelling package. As the AH5 model is defined in 250 the Uranus Longitude System (ULS) (Connerney et al., 1986) (also known as the U1 coordinate system) which has its 251 pole opposite to that of the IAU defined pole of Uranus, the model was constructed in these coordinates with 252 conversions to/from USM as necessary. USM coordinates can become degenerate for the case where the dipole is 253 exactly parallel or anti-parallel to the solar wind, where the orientation of the X-Z plane around the X axis becomes 254 undefined. This was not encountered during the period of study. Figure 5 shows the diurnal structure in the field for 255 the Voyager 2 epoch, where the configuration remained largely static with the field rotating around the planet-Sun line, 256 and January 2000 where considerable changes in configuration are found over a diurnal cycle. Figure 6 shows a comparison of the residual field, with the AH5 internal field model subtracted, and the model at the Voyager 2 flyby. 258 The model comparison is generally very good on the dayside, with a significant discrepancy near closest approach 259 (possibly reflecting additional unconstrained structure in the internal field) but is worse on the nightside where our 260 neglect of the tail field can be seen. The model has the capability to be varied with the upstream solar wind dynamic pressure through changes in the size of 270 the magnetosphere. We do not include this effect for two reasons: a) to focus on the seasonal and geometrical effects, 271 and b) due to the difficulty in constructing a physically-meaningful timeseries of solar wind pressure over a period of 84 272 years, from a short time series around the Voyager 2 mission. We leave this for future work. However, experiments 273 show that for near solstice conditions using Voyager 2 data the amplitude of such variations amount to ~0.1 nT at both 274 Titania and Oberon. As might be expected, there are also harmonics of the solar rotation period ~500 hours and 275 frequency mixing with orbital periods that amount to a ~0.01 nT amplitude. 276

277
As we were particularly interested in the seasonal effects within the system and were concerned to include effects such 278 as nodal precession (e.g., Jacobson, 2014a), we used SPICE to compute the positions of the satellites, rather than using 279 mean orbital elements as in other studies in the jovian system analyses (e.g., Seufert, Saur and Neubauer, 2011). The 280 positions of Miranda, Ariel, Umbriel, Titania and Oberon were calculated between 01 January 1966 and 01 July 2050, 281 bounding a full Uranus year. The time series was constructed at a cadence of 300 s which produces around 400 data 282 points during one orbit of Miranda and approximately 2500 for Titania. The magnetic field model was used to compute 283 the field at each satellite and the individual sources (dipole, quadrupole, octapole, hexapole, magnetopause) were 284 saved along with the total. Data was stored in HDF5 files for further analysis and are publicly available at 285 doi:10.17635/lancaster/researchdata/411. During analysis a discrepancy was noted between the orbital period of the 286 satellites (from mean elements) and the data from SPICE and so we also computed the orbital period from SPICE for 287 consistency with the analysis. Orbital elements were obtained from state vectors using SPICE using a GM of 288 5.793951322279009 × 10 6 km 3 s −2 from ephemeris URA112 (Jacobson, 2014b). 289 increasing departures from the dipole model (indicated in red) with increasing orbital distance, and increasing 296 differences between seasons, just visible at Umbriel to significant at Titania. Interestingly, the hodogram is relatively 297 constrained near solstice, possibly due to the narrow range of solar wind attack angles at that time and that the orbits 298 are almost in the terminator plane. We do not show the hodogram for Oberon at equinox due to the regular 299 excursions outside the magnetosphere when the satellite is on the dayside. In the jovian system between Ganymede 300 and Callisto, the field becomes almost linearly polarised which is a consequence of the magnetodisc current sheet 301 (Seufert, Saur and Neubauer, 2011). This is absent at in our model results, partly due to the lack of such a current sheet 302 at Uranus, but also due to our neglect of a tail current sheet which would introduce some linear polarisation at 303 particular orbital phases. This may affect Titania but would most strongly affect Oberon. 304 305 306 Figure 7: Hodograms for the variation of the field in the − space for the five satellites in two epochs (near solstice 307 (a-e) and near equinox (f-i). The near equinox hodogram for Oberon is not shown due to periods spent outside of the 308 magnetosphere. Grey backgrounds indicate the envelope of the whole hodogram over an entire Uranus orbit covering 309 the most highly-visited part of the hodogram space; the red envelope is just the hodogram from the dipole component 310 of Uranus' field at the satellite; and the blue envelope shows the total field. 311 312 To explore the seasonal variation in more detail, Figure 8 shows hodograms for Umbriel and Titania over half a Uranus 313 orbit. As expected due to its smaller orbital distance, the hodograms for Umbriel display relatively little variation 314 compared to Titania, however, the hodograms for Umbriel are more elliptically-polarised near solstice, and more 315 circular near equinox. The hodograms for Titania show a large degree of variation with season. Near solstice, in the 316 centre two rows of Figure 8, the hodogram has a relatively narrow elliptical polarisation and occupies a relatively small 317 fraction of the overall envelope of variation over a Uranus year. Near equinox it is broader and almost circular, but it 318 must be highlighted that these intervals are over a wider period of time which may also contribute to the breadth of viewed from above the orbital plane. The curved surface represents the magnetopause. For clarity, the axis scales 336 have been suppressed but they have the same ranges and grid as those in Figure 7. Representative centre-times for 337 each row are approximately 1970,1976,1982,1986,1991,1996 At each satellite the amplitude at the synodic period is approximately constant in time, however, the amplitude at the 361 orbital period exhibits variation over an order of magnitude between different epochs. Near solstice, where the angle 362 between the solar wind and the spin axis is near 0 ∘ or 180 ∘ , the time series shows relatively low amplitudes at the 363 orbital period and its harmonics and the spectrum is relatively simple and uncluttered. Near equinox, where the angle 364 between the solar wind and the spin axis is approximately 90 degrees, there is significant power at the orbital period 365 and its harmonics and also a plethora of other spectral peaks, particularly at periods not consistent with either periods 366 or their harmonics. These additional spectral peaks are more important for Titania, less important for Umbriel and 367

Analysis of the inducing field
Ariel, and essentially absent (relative to the main peaks) at Miranda. As expected, the dipole field component appears 368 at the synodic period, and occasionally at the orbital period if there are effects due to orbital eccentricity of the 369 satellite. The quadrupole field appears at the synodic period in and the 2 nd harmonic in and due to the 370 symmetry of the quadrupole. The magnetopause field always appears at the orbital period of the satellites due to the 371 changing position of the satellite relative to the magnetopause over an orbit. There is also power from the 372 magnetopause field at the synodic period of the satellites due to the variation in dipole tilt over a planetary rotation. 373 The fine spectral structure we see in Figure 9 was determined to be due a set of heterodynes, caused by mixing of 381 different frequency components: ℎ ± = 1 1 ± 1 2 . This was found to be a persistent feature in the magnetopause field 382 due to a mixing of the synodic period with the orbital period and its harmonics, interpreted as the motion of the 383 satellite through an asymmetrical magnetopause cavity thus experiencing a magnetopause field that varies over an 384 orbit. Weaker heterodynes, from mixing of the orbital period with higher harmonics of the synodic period, are 385 interpreted as a consequence of the non-sinusoidal behaviour of the solar wind attack angle over a rotation period. For 386 example, at solstice Titania experiences relatively symmetrical motion relative to the magnetopause as the orbit is 387 roughly in the terminator plane and so the satellite remains near the flanks of the magnetopause. Near equinox, 388 Titania experiences both the dayside and nightside magnetosphere and so experiences an asymmetrical magnetopause 389 field over an orbit. As expected, the heterodyne spectrum is much richer near equinox. There are some hints of such 390 behaviour in similar power spectra for Ganymede presented by Seufert et al. (2011), although the resolution of their 391 spectra is much lower and the amplitudes are smaller, as expected for satellites that are much farther from the 392 magnetopause. 393

Line spectra: Titania and Miranda
394 Figure 10 shows the amplitude spectrum for each field component at Titania near equinox. This spectrum is similar to 395 that near solstice but with larger amplitude heterodynes at equinox. The solstice spectrum for Titania is also similar to 396 Oberon near solstice. The plot highlights the fundamental periods, their harmonics, and heterodynes. Generally the 397 fundamental period and the second harmonics are the strongest lines. For clarity we do not plot heterodynes amongst 398 higher harmonics of both the orbital and synodic periods but note that heterodynes between harmonics generally have 399 a much lower amplitude. As expected for the orientation of the internal field, and have the larger amplitudes 400 amongst the internal field terms. The magnetopause field is somewhat different since the orientation of our spherical 401 coordinate system with respect to the solar wind direction strongly varies with season due to the large obliquity. The 402 magnetopause field presents a great deal of fine structure in the form of heterodynes near the synodic period. 403 Some additional structure associated with the internal field can be seen near the orbital period, for example signals at 404 around 300 and 600 hours, and are not predicted by sets of heterodynes between the orbital and synodic periods. 405 With the assumption that these are additional heterodynes, they require a modulating signal with a period of around 406 600 hours to produce heterodynes between the orbital period and the fundamental and 2 nd harmonic of this 407 modulating period. Without an obvious source within the system, and its restriction to the internal field, made us 408 suspect this was perturbations in the satellite orbits. We computed power spectra of the orbital elements and found a 409 range of spectral peaks in the semi-major axis (computed using SPICE) of Titania matching the periods of low-amplitude 410 peaks in our field data. Similar spectral peaks were also found in the orbit of Oberon and which match similar spectral 411 peaks in the spectrum of Oberon. This also provides an interpretation for the signals near 1000 hours seen in Figure 9.
This was sufficient to for us to conclude a non-magnetospheric effect and conclude that this was to do with the orbital 413 evolution of the satellites and we made no further investigation. For contrast, Figure 11 shows the amplitude spectrum at Miranda. As the orbital and synodic periods are very close 423 together we show the spectrum over a smaller range of periods. Similar to the spectrum at Titania, the heterodynes 424 produce a great deal of fine structure but, in contrast, this fine structure is greatly compressed due to the proximity of 425 the orbital and synodic periods and is compressed into packets near the harmonics of the orbital and synodic periods. ( cos x , y cos y , z cos z ) and = ( cos , cos , cos ) . These harmonics, plus a 440 constant term, were fitted to the synthetic time series using a linear least squares matrix inversion using 441 scipy.optimize.lsq_linear without bounds. For samples and harmonics, the matrix problem for the x 442 component can be written as: 443 The selection of harmonics was performed algorithmically. Prior to the fit, a list of trial periods was constructed from 446 the orbital, synodic and Uranus rotation periods, their higher harmonics, and heterodynes up to a maximum harmonic 447 degree of 8. At each step of the fitting a period was removed from this list and added to a list of periods. The matrix 448 (9) is constructed for each field component and the matrix inverted to provide a list of coefficients. The residuals are 449 calculated and from these the root-mean-square error, the maximum relative (to the peak in that component) error, 450 and the reduced χ 2 . The added period is only retained if the quality of the fit is improved, defined by the root-mean-451 square, reduced χ 2 , or maximum relative error decreasing by a factor of 1-10 -7 , for all three field components. 452 Table 1 contains the amplitudes and driving periods for Titania, at solstice and equinox, and Oberon at solstice. Driving 453 periods were only included that were within a factor of 10 -3 of the maximum amplitude for that field component. The 454 identified source of the driving harmonic is listed in the right-hand column. For some harmonics there are multiple 455 physical sources that contribute to that harmonic and we list all the contributors up to 90% of the amplitude of that 456 harmonic; for example, if the dipole can explain 89% of a harmonic then we also list the next most important 457 contribution to get to at least 90%. 458

A [nT] P [h] A [nT] P [h] A [nT] P [h]
Br 0. Amplitudes are given to 3 s.f. and periods to 4 s.f.. The right hand columns indicate the source of the signal and the 462 Titania. To estimate the effect of random and systematic errors on the measurements we constructed a simple forward 508 model of the Voyager magnetometer. We incorporate the transformation from the sensor to spacecraft frame, 509 sensor−sc , spacecraft to geophysical frame, sc−geo , and incorporated scale factors, , and offsets, , according to 510 equation (10) (after Acuña, 2002) to convert from engineering (measured) units, , to field strength, . 511 = ( − ) sensor−sc sc−geo (10) 512 In a real set of spacecraft observations the transformations sensor−sc and sc−geo contain errors due to twisting and 513 bending motions of the boom away from some calibrated alignment and a finite knowledge and control of the 514 spacecraft attitude. The scale factors and offsets are also subject to uncertainty. These can all be controlled to some 515 degree through calibration but systematic and random errors persist. The data in engineering units are also quantised 516 into a finite number of bits thus generating some quantisation noise. In the case of Voyager the data are quantised into 517 12 bits, although some of these bits are used form 'guard bands' at the upper and lower extrema of each sensitivity 518 range (Behannon et al., 1977;Berdichevsky, 2009) reducing the available bits by five for Voyager. To assess the impact 519 of these uncertainties and quantisation our forward model takes the modelled magnetic field during a flyby, transforms 520 the modelled data into quantised engineering units via equation (10), and then reinverts the data to produce a 521 synthetic timeseries. This is schematically illustrated in Figure 13. 522 The attitude of the spacecraft was specified with some constant axial tilt and a time-varying roll rate to give some 523 constant changing attitude with respect to the ambient field. Both angles were perturbed with normally-distributed 524 angles with a standard deviation of 0.035° to give a maximum RMS error of around 0.05° thus simulating finite 525 knowledge/control of the spacecraft attitude. For simplicity, the error on the alignment of the boom was simulated by 526 twisting the boom around its long axis and we did not consider bending of the boom. This was effected by generating a 527 set of random boom twist angles, equally-spaced in time, that were converted into a continuous boom twist angle via 528 cubic interpolation. The random twist angles were generated from a normal distribution with a standard deviation of 529 0.25° to give a twist amplitude less than 1° (Miller, 1979). Small errors in the offsets and scale factors were introduced 530 by perturbing the offsets by ±6 counts (Berdichevsky, 2009) and the scale factors were scaled by a normally distributed 531 factor ∼ N(1, 0.01) to simulate an error of up to around 4%, e.g., instead of 0.005 nT/count, for example, the scale 532 might be ~0.0048 or ~0.0052 nT/count. No attempt is made to specifically emulate the Voyager magnetometer in great 533 detail -just as a template for a reasonable magnetometer that might measure the fields at Uranus.

562
In this paper we have explored the possibility of detecting subsurface oceans at the uranian icy satellites, focusing on 563 the outer two satellites, Titania and Oberon, as thermal and structural models have identified these as candidates for 564 hosting subsurface oceans. An analytical model for the uranian magnetospheric magnetic field was constructed and 565 used to generate magnetic field time series at the orbits of the five main satellites. These time series were subjected to 566 a spectral analysis to identify the periods of driving signals and their amplitudes were determined via fitting a model 567 harmonic time series. The amplitude of the induced field was calculated at the identified periods to examine the 568 strength of a possible induced response. We found significant periodic signals near the synodic and orbital periods, and 569 their higher harmonics, alongside a rich spectrum of heterodynes particularly associated with the magnetopause field. 570 The heterodynes were found to be a persistent feature in the magnetopause field due to a mixing of the synodic period 571 with the orbital period and its harmonics. This was interpreted as the product of two effects: • Orbital period: A satellite would experience a changing magnetopause field as it orbited within an 573 asymmetrical magnetospheric cavity. 574 • Diurnal period (+harmonics): As Uranus rotates the solar wind attack angle varies in a (generally) non-575 sinusoidal fashion (e.g., Figure 3) and therefore the magnetopause field has a diurnal periodicity plus higher 576 order harmonics due to the non-sinusoidal variation in the attack angle. 577 We found that the identified induced field amplitudes at Titania can vary by a factor of around three between equinox 578 and solstice, due to the variation of both satellite orbit geometry relative to the magnetopause and the solar wind 579 attack angle with season, although this variation in amplitude was mostly restricted to the rich spectrum of 580 heterodynes. The spectrum was found to be generally richer at equinox but contained many closely spaced spectral 581 lines around the synodic period. It remains to be seen if these could be separated and used to constrain a subsurface 582 ocean. It is worthwhile commenting that there is some evidence for similar fine structure at Ganymede, e.g. via an 583 inspection of Figure 4 in Seufert et al., (2011) although the amplitudes are smaller as expected for a satellite much 584 deeper within the magnetosphere (up to around 50% of the magnetopause subsolar distance) than Titania and Oberon. 585 The major seasonal effect is the proximity of Oberon to the magnetopause. For a period of around ±7 years centred on 586 2030 Oberon should remain inside the magnetosphere and the results from Table 1 apply. After that time, e.g., for 587 missions arriving later in the 2030s or in the 2040s, it may be possible to detect an ocean, but only from signals near the 588 synodic period and where the satellite has been inside the magnetosphere for a significant period while the eddy 589 currents establish themselves. This places clear constraints on flyby locations and would require a flyby to be timed for 590 after Oberon had left the vicinity of the dayside magnetosphere (moving towards the nightside) and preferably just 591 before it re-emerges into the dayside from the nightside magnetosphere. 592 These driving periods were combined with a model for the induction response and we showed that induced signatures 593 should be detectable and this was confirmed with a magnetometer forward model and synthetic time series from a 594 Titania flyby. We found that ocean thicknesses of 40 and 52 km should be readily detectable from a flyby with a 200 595 km altitude closest approach, although a 16 km thick ocean was at the limit of detectability. This analysis demonstrated 596 that a 200 km altitude flyby would be acceptable, but would limit ocean depth/conductivity/ice shell ranges and so 597 lower altitude flybys are strongly recommended. Given the weakness of some signals, this also demonstrates that 598 maintaining an AC spacecraft magnetic field below 0.1 nT (preferably below 0.01 nT) at the magnetometer would be 599 advantageous in our ability to resolve less conducting, thinner, and or deeper oceans; although more accurate 600 constraints require further study. It is important to highlight that this work has been guided by the work of Hussmann 601 et al. (2006) and thinner overlying ice shells would result in stronger induced fields and thus would be more readily 602 detectable than the somewhat thicker ice shells predicted by Hussmann. 603 We used an induced dipole model to represent the response of a subsurface ocean. This model is valid when the ocean 604 depth is much greater than the electromagnetic skin depth, = �2/ 0 (e.g., Khurana et al., 1998). For Titania this 605 is almost always satisfied for the 39 and 52 km thick oceans, over a wide range of conductivities above 0.1 S/m. The 606 exception to this is that the 209 hour orbital period signal can only be analysed for conductivities above 1 S/m with this 607 model. The thinner 16 km ocean requires higher conductivities >1 S/m for the main driving periods under this model.