The search for a subsurface ocean in Ganymede with Hubble Space Telescope observations of its auroral ovals

2.1 Scheduling Information and Raw Data Processing

In this work we analyze observations acquired during HST campaign ID 12244, where two visits with five consecutive HST orbits each were dedicated to observing Ganymede at eastern elongation. Thus, HST observed Ganymede's orbitally leading hemisphere, which is the downstream hemisphere with respect to the unperturbed flow of magnetospheric plasma past Ganymede. Visit 1 occurred on 19 November 2010, and visit 2 occurred on 1 October 2011. Details of both visits are summarized in Table 1. With durations of approximately 7 h each, both visits cover more than half of Jupiter's synodic rotation period. The time windows of both visits were designed such that the Jovian magnetic latitudes of Ganymede span the entire possible range, i.e., Ganymede is exposed to the maximum variability of Jupiter's magnetospheric field during each visit. Figure 2 shows that Ganymede's magnetic latitude coverage was very similar for both visits. The magnetic latitudes Θ_M are calculated according to Θ_M=9.5° cos(λ_III−200.8°), where λ_III describes the system III longitude of Ganymede [Dessler, 1983; Connerney et al., 1998]. When Ganymede is at maximum positive latitude, the radial component of Jupiter's magnetospheric field B_r assumes its positive maximum. When Ganymede is at maximum negative latitude, B_r assumes its negative maximum. For the analysis of the rocking of the aurora we use exposures where Ganymede is located near-maximum magnetic latitudes, i.e., , to measure the maximum amplitude of the oscillation of the auroral ovals. The resultant exposures are marked in Table 1 and in Figure 2 with diamonds and crosses.

The observations were performed with the Space Telescope Imaging Spectrograph (HST/STIS) using grating G140L and pseudo aperture 52x2D1. This aperture relocates the position of Ganymede's spectral image away from the region where the Multi-Anode Micro-channel Array (MAMA) detector suffers from large dark currents (i.e., away from the “blotch”). We find the location of the disk of Ganymede on the detector by convolving the Lyman α emission with a theoretical disk of the same size as Ganymede. We assumed that the theoretical disk is uniform due to the lack of appropriate Lyman αmaps of Ganymede. The location of the theoretical disk where the integrated convolved flux attains the maximum corresponds to the center of the Lyman α disk. We also use the long-wavelength trace of the image to independently confirm the vertical location of the disk of Ganymede. The dispersion is predominantly in the horizontal direction on the detector. However, the spectral trace slightly shifts in vertical direction on the detector as a function of its horizontal position in the flat fielded raw data, which we use in this analysis. Here we refer to the vertical direction as the direction parallel to the slit. The horizontal direction is perpendicular to the slit and approximately along the direction of dispersion. We use calibration measurements of the stellar object WD2126+734 to determine the vertical drift of the dispersion when the pseudo aperture 52x2D1 is used in combination with G140L (PI: C. Proffitt, ID: 10040, filename: o8tg02020_flt.fits). In this analysis we investigate the spatial and temporal structure of the oxygen OI 1356 Å emissions due to their superior brightness and low component of solar reflected light compared to other atmospheric emission lines at FUV wavelengths. For comparison, the total averaged flux from Ganymede's atmosphere at 1304 Å is on average weaker by a factor of 1.8 than the fluxes at 1356 Å [Feldman et al., 2000]. About half of the emission at 1304 Å is due to solar radiation reflected from the surface. The solar flux at 1356 Å is less than 10% that of the 1304 Å multiplet, and thus, reflected light contributes only very weakly to the observed emission at 1356 Å. The 1304 Å emission additionally has contributions due to solar resonant scattering, which is negligible for the 1356 Å emission [Hall et al., 1998; Feldman et al., 2000]. For these reasons only the 1356 Å emission is used in this work. The resultant location of the center of the 1356 Å disk calculated from the location of the Lyman α disk and the long-wavelength trace is given by detector pixel i_x= 404 and i_y= 102 for both visits, i.e., in horizontal and vertical direction, respectively.

For a quantitive analysis of the observed auroral fluxes, we first remove the background emission due to dark noise and other sources. For this purpose we average the fluxes in rows above and below the OI 1356 Å image to determine the background fluxes and then remove this flux from the OI 1356 Å image. We use solar spectra measured by the Solar Extreme-Ultraviolet Experiment (SEE) on the Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) mission [Woods et al., 2005] for the particular days of the observations in order to model the solar reflected light from the surface of Ganymede. For determining the albedo, the long-wavelength image of the observations (λ= 1413 Å to 1586 Å) is used. We find a geometric albedo of 0.017 ± 0.003 for visit 1 and 0.019 ± 0.003 for visit 2 on the leading side. The albedo values of this study are similar to values for the albedo on the trailing side of 0.023 ±0.002 derived from fluxes near 1400 Å by Feldman et al. [2000] from HST/STIS observations and an albedo of 0.026 ±0.003 derived by Hall et al. [1998] from HST/GHRS measurements of the reflected C II 1335 Å multiplet. Due to a lack of knowledge of the spatial structure of the FUV albedo, we assumed similar to previous studies that the albedo is spatially constant. We note that modifications of the albedo values have a negligible effect on the derived rocking angles of this analysis.

The modeled reflected flux is then convolved with the point spread function (PSF), which we obtained with the TinyTim simulation software package from the website tinytim.stsci.edu [Krist et al., 2011]. The PSF has a Gaussian structure near the central pixel and power law wings at larger distances from the central pixel similar to the kappa-PSF(κ,γ) [Saur et al., 2011], which is identical to a Lorentzian profile for κ=γ=2. The fluxes of the reflected light are then finally removed from the observed OI 1356 Å flux, and the images are rotated with Jupiter's north pointing upward. The direction from south to north will also be referred to as the y direction. The x direction is perpendicular to it, which is approximately toward Jupiter. After these data processing steps the remaining fluxes are purely due to emission from Ganymede's atmosphere aside from statistical noise.

The resultant images are grainy, i.e., the pixel brightness is somewhat erratic due to the Poisson distributed fluxes. This effect can be reduced by a smoothing filter. We use a filter of the form

(1)

applied 4 times. The filter is referred to as star filter with i_x and i_y referring to pixel coordinates in the rotated images. The effects of the star filter in comparison with other filter applications are discussed in Appendix A.

The resultant images for visit 1 where all exposures are superposed when Ganymede was at sufficiently large latitudes above the current sheet and below the current sheet (Θ_M≲−7°) are shown in Figure 3, respectively. Similar images are shown for the second visit taken roughly one year later in Figure 4.

The apparent emissions above the right limb stem from the weaker emission line (OI 1358.5 Å) of the observed oxygen doublet, whose position is shifted by 5 pixels to the right and up in the rotated images. Since we only use pixels with brightness of 80% and more of the brightest pixels along the ovals, the effect on identifying the auroral ovals is expected to be small.

The pixel size on the STIS FUV-MAMA detector corresponds to 0.0246 arcsec. The spectral resolution (full width at half maximum) for a point source at 1500 Å is 1.5 pixels for all slits [Hernandez et al., 2014]. We use the 2 arcsec slit to obtain spectral images of Ganymede. Ganymede subtends ∼1.75 arcsec in diameter at the time of observations which corresponds to ∼70 pixels, but only 70/1.5 = 47 resolution elements across on a monochromatic image at 1500 Å assuming monochromatic lines. One pixel corresponds to approximately 0.03 R_G (with the Ganymede radius R_G= 2631 km). For Nyquist sampling, we have 70/2 = 35 resolution elements, which corresponds to a spatial resolution of 0.05 arcsec. The resolution of 0.05 arcsec is less than the average width of the oval of 0.18 R_G, which corresponds to 0.16 arcsec or 6.5 pixels. Thus, the ovals are resolved approximately by a factor (0.16 arcsec)/(0.05 arcsec) = 3.2. Note that the applied smoothing procedure broadens the auroral ovals by approximately 1.5 pixels.

2.2 Location of the Ovals

The location of the ovals for each column, i.e., for each i_x, is determined by

(2)

where the summation is over all i_y for which f(i_x,i_y) > f_min(i_x). The minimum is chosen such that f_min(i_x) corresponds to 80% of the brightest pixel in each column i_x. The brightest pixels are searched in each hemisphere within 5° and 60° north and within 5° and 60° south consistent with oval locations from previous observations [McGrath et al., 2013]. The method to define the locations of the ovals in 2 considers the fluxes in the y direction exceeding a cutoff level of 80% of the brightest pixel (in each column i_x, respectively). The method thus considers the emissions in the vicinity of the peak emissions of the ovals, but it discards the effects of other atmospheric UV emissions from Ganymede, i.e., polar excited emissions due to energetic electrons in Jupiter's magnetosphere or equatorial emissions excited from energetic electrons within Ganymede's mini-magnetosphere. The 80% value for the cutoff was deliberately chosen relatively high to clearly separate the emission from the oval compared to other more polar and equatorial emissions. Due to the brightness variation in x direction of the ovals, we choose f_min(x) to vary in x direction. We also investigated other methods to determine the locations of the ovals, which are described and compared against each other in Appendix A.

The auroral images in Figures 3 and 4 show, both, systematic and stochastic variability. The stochastic variability is most likely due to a combination of intermittent reconnection occurring at the open-closed field line boundary [Jia et al., 2010; Eviatar et al., 2001] and statistical effects due to the weakness of the auroral signal. Therefore, the locations of the oval y(i_x) are not a smooth line, and we fit the locations with a polynomial of second degree of the form

(3)

to more easily monitor the systematic temporal evolution of the ovals. Based on the structure of the observed and modeled ovals we find that a polynomial fit of second degree represents the ovals adequately with only a small number of fit parameters. In the polynomial fit the sum of the squared differences between the polynomial and the observation divided by the associated effective width for each i_x is minimized. The effective width is a quantity defined in Appendix B. In Figures 3 and 4, the fitted ovals are shown as dashed and solid green lines when Ganymede was above and below the current sheet, respectively.

To quantitatively determine the change in the locations of the ovals, we introduce the difference

(4)

where y_a and y_b represent the y coordinates (north-south direction) of the polynomial fit when Ganymede is above and below the current sheet, respectively. The differences d(i_x) for visit 1 and visit 2 are shown in Figures 5 and 6, respectively.

The main objective of the data analysis is to determine how strongly the ovals oscillate up and down when Ganymede is above and below the current sheet. The amplitude of the oscillation can be quantitatively determined with a rocking angle α defined by

(5)

The positions x_E and x_W represent the most eastern and western positions on the disk. The northern and southern ovals of visit 1 and visit 2 shown in Figures 3 and 4 represent four independent measurements of the rocking angle α.

4.1 Results of the Rocking Analysis

Figures 3 and 4 show the auroral ovals above and below the current sheet for visit 1 and visit 2, respectively. The figures also include the polynomial fit of second degree to the observations shown as green lines. For comparison we show modeled locations of the auroral ovals with the MHD model from Duling et al. [2014] when an ocean is included (red lines) and when no ocean is present (blue lines). It should be emphasized that the finite width of the observed ovals is larger than the differences in the locations of the modeled ovals with and without ocean in each individual image. Therefore, it is impossible by simple visual inspection of individual images to separate the ocean from the no-ocean hypothesis. However, we will subsequently show that both hypotheses can still be well separated with the statistical analysis techniques discussed in the remainder of this work. The statistical analyses are based on the collective number of resolution elements within each observed image and the availability of measurements of the northern and southern ovals from two different sets of HST data.

To visualize the displacement of the ovals when Ganymede was above and below the current sheet, we show in Figures 5 and 6 the differences d(i_x) from equation 4. The modeled temporal change of the locations is stronger in models without an ocean than in models with an ocean (slopes of blues lines are steeper than slopes of red lines in Figures 5 and 6, respectively). The oscillations of the observed locations of the ovals (green lines) are small and comparable to the expected reduced oscillation of the ovals with an ocean present.

The oscillation can quantitatively be assessed with a rocking angle α defined in 5. Four measurements of the rocking angles are obtained from the observations of the northern and southern ovals during the two visits. The values of these measurements are shown in Table 2. The table also includes formal uncertainties of the rocking angles, which are derived in Appendix B. Note that the individual rocking angles from the four independent measurements are expected to be only approximately similar but not identical mostly due to viewing geometry. All four measurements can be considered independent because emission from the northern hemisphere does not influence the emission from the southern hemisphere and vice versa. Equally the emissions from both visits do not influence each other. As an overall measure for the oscillation properties of Ganymede's aurora we introduce an average rocking angle and an average uncertainty defined by expressions B6 and B7, respectively. From a statistical point of view, each individual rocking angle can be considered a random variable with a certain expectation value and variance. Even though each individual rocking angle does not have exactly the same expectation value, the average rocking angle can be formally introduced as a new random variable with its own statistical properties (such as expectation value and variance). We introduce the average rocking angle for two reasons: (1) It is a single value quantity well suited to test the observations against the ocean and the no-ocean hypotheses and (2) it has an approximately 4 times smaller variance compared to the individual rocking angles. For the average rocking angle we find with a formal uncertainty of ±6.3°. The expected rocking angles are 3.4° with ocean and 8.6° without ocean derived from the MHD model using the same polynomial fits as in the observations (but prior to the consideration of stochastic effects, which will be discussed next).

The above analysis does not address the effects of the stochastic patchiness of the auroral morphology on the rocking angles. The related formal error analysis also strongly overestimates the uncertainty of the observed rocking angles. One reason is that the formal error analysis includes fluxes on a large area of the disk in the calculation of the effective width of the ovals, i.e., within 5° and 60° latitude (see also Appendix B). The effective width includes, in addition to emission directly from the ovals, a global emission contribution, which is excited by globally abundant ‘no-auroral’ electrons. Both are difficult to separate unambiguously. Another reason why the formal error analysis in Appendix B strongly overestimates the real error is that it assumes the emissions along the ovals are fully correlated. The formal error calculation assumes an averaged uncertainty of the ovals positions, i.e., that are fully correlated for each i_x. If the uncertainties along an oval were uncorrelated for each i_x, the total uncertainty would be reduced by approximately the square root of the number of pixels N along the oval in i_x direction, relative to the uncertainty at an individual pixel i_x. This certainly represents a too extreme assumption. Neighboring pixels are correlated to some extent as apparent in the patchy nature of the emissions due to physical correlations and contributions from the data processing such as smoothing and the point spread function. The correlations, however, seem to exist only along a limited fraction of the oval arc. Formally one could calculate the full correlation tensor between each pixel of each exposure and include it in the error analysis. These calculations, however, would get exceedingly complex and will additionally lose transparency.

In cases where the error analysis becomes too complex, a straightforward way to estimate the uncertainty is through a Monte Carlo (MC) method where the errors are estimated from an ensemble of synthetic observations. In the MC test we generate 2048 sets of synthetic HST images based on model runs with and without oceans, respectively, but with randomly generated patchiness and measurement noise comparable to the real observations. Details about how the synthetic images are generated are given in Appendix C. Two examples of synthetic images are shown in Figure 7. These images were created with the MHD model assuming an ocean being present. The underlying model ovals are shown in red. The random patchiness is calculated around these model ovals. For comparison we also show the expected location of the ovals when no ocean was assumed. In Figure 7 we also show the polynomial fits to the patchy synthetic images as green lines. They are calculated in exactly the same way as for the real data.

Subsequently we also apply the same procedure that we apply to the real data to the synthetic images in order to determine the rocking angle and uncertainty . The resultant distribution function for is shown in Figure 8. The expectation values are reduced to 2.2° with ocean and 5.8° without ocean compared to the model results without patchiness given above (see also Table 2). The probability distribution functions of the averaged rocking angles with and without ocean only weakly overlap, which implies that both hypotheses can generally be separated. The associated variances for both the ocean and no ocean models are . The resultant can be considered a measure for the uncertainty of the observed rocking angle. They are smaller than the formal errors calculated with the error propagation method described in Appendix B. The observed α = 2.0° ± 1.3° is consistent with the presence of an ocean, whereas its absence is highly unlikely with a probability smaller than 1%.

As a result of randomly distributed spots along the ovals, the derived rocking angles from the MC test varies for each numerical realization. In some cases the rocking angle can even turn negative depending on how the random patches are distributed (see Table 2). Note that the probability distribution functions of the rocking angles for the individual ovals are roughly twice as wide as the distribution function for the averaged rocking angle from all four ovals shown in Figure 8. This increases the likelihood of an individual α to turn negative.

As stated before, the finite width of the observed ovals is larger than the differences in the locations of the modeled ovals with and without ocean in each individual image. This precludes a simple visual inspection of individual images to separate the ocean from the no-ocean hypothesis. Both hypotheses, however, still can be separated for two reasons. (1) Four independent measurements of northern and southern ovals from two visits are available. This reduces the error approximately by a factor of . (2) Additionally, the pixels along the ovals are not fully correlated, and thus, each vertical column of pixels along the ovals represents partially independent measurements as well. This effect additionally reduces the uncertainty by a factor of 1.3°/6.3° = 1/4.8, based on the ratio of uncertainties from the formal error calculation and the MC test (values of both uncertainties are given in Table 2).

4.2 Model Robustness

An important question of our analysis is whether the amplitude differences of the auroral oscillation with and without ocean are model dependent. Therefore, we investigate both the impact of different upstream conditions of Jupiter's magnetosphere and the impact of different MHD models. For the latter we compare the locations of the open-closed field line boundary (OCFB) calculated with the models by Duling et al. [2014] and with those calculated from the independent model by Jia et al. [2009], introduced in detail further below.

Two examples of the locations of the OCFB are shown in Figure 9 in normal cylindrical projections. Figure 9a shows the OCFB calculated with the model by Duling et al. [2014] with upstream thermal and kinetic plasma pressures 5.7 nPa and 1.8 nPa, respectively. The latitudinal and longitudinal numerical resolution is 1.5°. The jitter/variability of the locations of the ovals is on average even smaller. The uncertainty in tracking the OCFB can be approximated by the numerical resolution of the models used. Figure 9b shows the OCFB from an entirely independent MHD model of Ganymede's magnetic field environment by Jia et al. [2009]. Many physical properties of this model are similar to the ones developed in Duling et al. [2014]. Both models include the internal dynamo field [Kivelson et al., 2002] and can take into account induction in a subsurface ocean. They both reproduce the in situ magnetic field measurements by the Galileo spacecraft remarkably well. The Jia model does not explicitly include an atmosphere and ionosphere. In order to account for the electrically nonconductive nature of Ganymede's surface, the induction equation is solved within Ganymede in the Jia model. For numerical reasons the conductivity jump across the surface of Ganymede needs to be smoothed in the Jia model. The resultant current close to the surface of Ganymede is argued to represent ionospheric currents in the Jia model. In contrast, the ionospheric currents are explicitly included in the Duling model. The upstream magnetic field and velocity values are similar in all model runs used here. The plasma conditions in the Jia model runs used in this work are from Jia et al. [2009] with values of ρ₀=28 amu cm⁻³ and p = 1.9 nPa, i.e., smaller values compared to the other runs shown in this paper (see also Figure 10).

The location of the ovals obtained from the Jia model shows scatter on the order of 2°, similar to the resolution applied to calculate the OCFB with this model. In addition, there is variability due to the nonsmooth magnetic field structure near the reconnection point. However, the amplitude differences in the locations of the ovals with and without ocean is up to 3°–5° in both models and larger than the scatter in the modeled ovals (see Figure 9: runs with ocean as red curves, runs without ocean as blue curves, and Ganymede above and below the current sheet as solid and dashed lines, respectively). An important conclusion from the results shown in Figure 9 is that both models predict measurably less oscillation when an ocean is present compared to when no ocean is present. It thus demonstrates that the predicted differences do not depend on the underlying MHD model.

It needs to be stressed that the differentiation of the models with and without ocean is based on the entire length of the oval's imprint on each hemisphere through a polynomial fit and not single oval data points. Therefore, the ovals only need to be separated on average. To investigate the separability of both hypotheses, we apply the MC test, where we add significantly more patchiness to the data than the jitter in modeled ovals, adjusting the patchiness of model images to the observed level.

In Figure 9 we also compare our modeled locations with the observed average locations of Ganymede's oval from an analysis by McGrath et al. [2013] when Ganymede was at different orbital positions and at different positions with respect to Jupiter's plasma sheet compared to the model runs of this analysis. We nevertheless include the data comparison to demonstrate that our model reproduces key features of the aurora; i.e., the ovals are significantly closer to Ganymede's magnetic equator on the downstream side (around 90° west longitude) compared to the upstream side (around 270°). A detailed comparison of the locations from the Jia model and the previous HST observations is presented in McGrath et al. [2013]. Note that because of the selected position of Ganymede with respect to the plasma sheet, the observations presented in McGrath et al. [2013] are not suited to address the ocean question. The error bars in the extracted locations of the ovals in McGrath et al. [2013] are significant, similar to the finite width of the ovals of the current observations. In this context, it is important to stress again that based on one single set of ovals where the errors bars along the ovals are assumed to be fully correlated, the ocean models cannot be separated from the no-ocean models. It is the combination of having four independent measurements of the movements of the ovals and the partial decorrelation of the uncertainties along the ovals which enables the separations presented here (see also detailed discussion in Appendix C).

In order to investigate the sensitivity of the amplitude differences of the oscillations with and without ocean for different models and model parameters, we performed a series of model runs, where we varied the upstream plasma pressure. During the times of the HST observations no independent measurements of the plasma pressure were available. In our study we investigate the effects of the total upstream pressure defined as sum of ram and thermal pressure. In Figure 10 (top) we show the minimum latitudes of the ovals when no ocean is assumed. The minimum latitudes of the ovals decrease from nearly 25° to approximately 15°, as expected, as a function of increasing upstream plasma pressure. This effect is similar to the variability of Earth's auroral ovals under increasing solar wind pressure. The blue symbols are from the independent model by Jia et al. [2009]. In Figure 10 (bottom) we show the extracted oscillations of the ovals when Ganymede is above and below the current sheet. For all upstream plasma pressures, the expected amplitudes of the oscillation are between 8° and 10° when no ocean is present. With an ocean present the amplitudes of the oscillations are reduced to approximately 4° for all upstream plasma conditions. This holds also for the independent model of Jia et al. [2009]. The plasma pressure controls the absolute latitudinal positions of the ovals, but it has very little effect on the oscillations of the ovals with and without ocean. Thus, both properties of the ovals are independent, and the amplitudes of the oscillations of the ovals in response to externally changing fields are indeed diagnostic of the internal conductivity structure of Ganymede.

As argued by several authors [McGrath et al., 2013; Jia et al., 2009, 2010; Eviatar et al., 2001], the locations where the OCFB intersect with the surface/atmosphere of Ganymede correspond to the location of the auroral ovals. For example, MHD modeling by Jia et al. [2010] shows that shear flow near the OCFB drives strong and localized electric currents along the OCFB, which are likely the root cause for the auroral emission similar to the discrete aurora at Earth. However, if the OCFB only approximately represents the locations of the ovals, then a resultant distance Δybetween the latter and the oval exists. If we reasonably assume that Δy is similar when Ganymede is above the current sheet and below the current sheet, the difference d(x)introduced to measure the oscillation of the ovals can still be obtained from the OCFB with good approximation.

We also applied simpler semianalytical models, where the magnetic field contributions from the plasma interaction have been calculated using a wire model. In the wire model, line currents along the magnetopause (similar to Chapman-Ferraro currents) and line currents continued along the Alfvén wings have been assumed, where the total currents in the wire loops have been calculated with expressions from Neubauer [1998] and Saur [2004]. All other contributors to Ganymede's magnetic field environment have been described similar to the full MHD model of Duling et al. [2014] applied here. These simpler semianalytic models lead to very similar dependences as derived with the fully consistent MHD models: With increasing strength of the plasma interaction, i.e., increasing total electric current, the locations of the ovals move toward the equator on the downstream side and away from the equator on the upstream side. The amplitudes of the oscillations with and without ocean are, however, barely affected by the interaction strength.

We note that Ganymede's atmosphere likely contains deviations from radial symmetry as modeled, for example, by Marconi [2007] and Plainaki et al. [2015]. The atmosphere and its structure influences the plasma interaction. However, even though we did not perform runs with nonsymmetric atmospheres, we generally expect its effects on the properties of the oscillation amplitudes to be small because we generally only see a weak influence of the plasma interaction on the oscillation differences with and without ocean (as discussed earlier in this subsection).

4.3 Properties of the Ocean

The model rocking angle increases when the ocean conductivity is reduced. In our modeling we find that a conductivity σof 0.09 S/m for an ocean assumed to be located between 150 km and 250 km depth provides a lower limit for the model ovals still being consistent with the observations within the uncertainties from the MC test (see Figure 11). This lower limit has been derived from a series of MHD model runs also shown in Figure 11 with variable ocean conductivities σ within 1 × 10⁻³ S/m to 1 S/m. The resultant rocking angle is shown as thin black line marked with crosses. Because the evaluations of the rocking angles from the MC test is numerically expensive, we scaled the rocking angles from the MHD model α_MHD to obtain the rocking angles α_MC according to the MC test based on the rocking angles at zero conductivity (i.e, no ocean) and 0.5 S/m. Values for the rocking angles for these two conductivities are given in Table 2. We applied the linear scaling

(6)

leading to the rocking angle for the MC test for varying conductivities σ (shown as the red curve in Figure 11). The resultant uncertainty is assumed to be 1.3° for every σ similar to the uncertainties of 1.3° derived for 0 S/m and 0.5 S/m (see Table 2). The uncertainty is displayed as red-shaded region in Figure 11. The modeled rocking angles from the MC test within the 1 sigma uncertainty are in agreement with the rocking angle derived from the observations shown as green horizontal line for conductivities larger than 0.09 S/m. These values represent thus a lower limit for the ocean conductivity. For lower conductivities the green curve (observations) no longer overlaps with model expectations (red area). Using the conductivity versus concentration curve for MgSO₄ for Europa from Hand and Chyba [2007], a minimum conductivity of 0.09 S/m would correspond to a minimum salt concentration of 0.9 gram MgSO₄ per kilogram of ocean water.

Note that to a good approximation for small ocean thicknesses H compared to Ganymede's radius R_G and the same average ocean location within Ganymede all modeling runs with the same depth-integrated conductivity Σ=σH are equivalent. Thus, any combination of σ and H(under the H/R ≪ 1 assumption and realistic σ) with values larger than 100 km ×0.09 S/m is consistent with the HST observations (assuming that the average location of the ocean is at a depth of 200 km). Deviations from this law occur when the thickness and the conductivity of the ocean are large. Such deviations are, for example, visible in Figures 11 and 12 of Seufert et al. [2011] near regions where the isolines of Ganymede's induction amplitude are not linear anymore.

The lower limit for the conductivity can be alternatively used to calculate a maximum depth of the ocean. An ocean at depth between 150 km and 250 km with conductivity 0.09 S/m generates an induction amplitude of A = 0.8 (calculated with expressions in Saur et al. [2010]). A similar amplitude would be generated by a perfectly conductive ocean at a depth of 330 km. Any less conductive ocean needs to be located closer to the surface of Ganymede. Note that in this simple equivalence estimate, the effect of the phase shift has been neglected. We also note that our estimation of the maximum depth to the ocean is consistent with theoretical expectations. At depths greater than approximately 150 km in Ganymede, the low-pressure form of ice (ice I_h) is expected to be replaced by high-pressure polymorphs, whose melting temperature increases with pressure, whereas for ice I_h it decreases with pressure. Therefore, the top of an ocean, if it exists, is expected to be most likely at a depth of 150 km or less [e.g., Vance et al., 2014].

4.4 Implication for the Dynamo Field

In the absence of an induced magnetic field a small intrinsic quadrupole component is needed to explain the Galileo magnetic field measurements, yet this component is not required if a conducting ocean is present [Kivelson et al., 2002]. We may therefore consider the ratio of power [Langel and Estes, 1982] of the quadrupole to that of the dipole R₂/R₁= 0.0025 of the oceanless field model as the upper limit for the actual intrinsic field of Ganymede, given that our results demonstrate that an induced component is present. At the top of Ganymede's iron core, whose radius is probably at least 650 km [Sohl et al., 2002], the ratio of quadrupole to dipole power turns out to be <0.04. This is the lowest ratio of all known dynamo fields at the surface of the dynamo region, with the possible exception of Saturn's magnetic field.

This suggests that a unique dynamo is at work. Possible reasons could be that Ganymede's dynamo might operate close to the critical Rayleigh number for dynamo onset due to its small size, that it operates in Jupiter's field [Levy, 1979; Gómez-Pérez and Wicht, 2010] or that it is driven by “iron snow” forming below the core-mantle boundary Hauck et al. [2006]. In particular, the iron snow scenario appears promising. If Ganymede's core contains more than a few percent of sulfur, crystallization of iron would start at the top of the core [Williams, 2009], with iron snow sinking and dissolving at greater depth where the associated iron enrichment of the core alloy drives composition convection. The snow-forming top layer develops a gradient in sulfur concentration and becomes stably stratified. Numerical dynamo simulations show that the presence of the stable layer above the dynamo can strongly reduce the quadrupole-to-dipole ratio [Christensen, 2015].