Analysis of Eddy Current Generation on the Juno Spacecraft in Jupiter's Magnetosphere

The Juno mission to Jupiter, in polar orbit around the gas giant since 4 July 2016, samples the planet's environment with dedicated particle and fields instruments. Juno's magnetometer investigation employs a pair of boom‐mounted vector fluxgate magnetometers colocated with a set of star cameras to map Jupiter's magnetic field with high accuracy. Juno is a spinning spacecraft, rotating at approximately two rotations per minute. In strong magnetic field environments experienced near periapsis, Eddy currents are generated within electrically conductive material near the magnetic sensors. These currents adversely affect measurements of the environment, most evident in the appearance of a spin modulation in the field magnitude. We demonstrate, by finite element modeling and laboratory measurements, that the spin modulation is caused by a physical signal due to Eddy currents generated by the rotation of the conductive spacecraft structure in the presence of a strong magnetic field. We present a finite element model of the induced field and develop a matrix method for removing the Eddy current contribution to the measured field. Juno magnetic field measurements in strong fields are corrected for Eddy current contributions using this model of the interaction.


Introduction
The Juno spacecraft has been in a highly elliptical, polar orbit about Jupiter since 4 July 2016. The mission was designed, in part, to envelope Jupiter in a dense net of observations obtained close to the planet, and spread globally in longitude, by properly phasing periapsis passes. This is achieved by Juno's 53-day orbit period and periapsis passes ("periJoves") during which Juno dips below the most intense radiation, passing just a few thousand kilometers over the cloudtops. Juno's 23 (thus far) orbits sampled Jupiter's environment from~1.05 Jovian radii (R J = 71,492 km) outwards, extending to the distant reaches of the Jovian magnetosphere (~113 R J ). Juno obtains measurements of the magnetic field with two independent identical magnetometer sensor suites, one inboard (IB) and another outboard (OB), located 10 and 12 m from the center of the spacecraft, as illustrated in Figure 1 (adapted from Connerney et al., 2017). Each of the magnetometer sensor suites consists of a triaxial fluxgate magnetometer (FGM) sensor and a pair of colocated star cameras (camera head units) mounted on an ultrastable magnetometer optical bench. Each magnetometer sensor measures the vector magnetic field with 100 parts per million (ppm) absolute vector accuracy over a wide dynamic range (1.6 × 10 6 nT per vector component); see Connerney et al., 2017. The two sensors provide complete hardware redundancy and a means of monitoring spacecraft-generated magnetic fields. The sensor suites are mounted on the magnetometer (MAG) boom, which is fabricated using lightweight aluminum honeycomb panels with carbon composite face sheets. Figure 2 shows an example of the measured field magnitude experienced during Juno's very first periJove pass (PJ1). For about 2 hr, Juno is in a very strong field (above 1 Gauss, with 1 Gauss = 10 5 nT) as the spacecraft transits the north pole, periapsis near the equator, and the south pole. The OB FGM sensor operated in all of its six dynamic ranges in order to capture the wide dynamic range of Jupiter's field with adequate resolution throughout afforded by 16-bit quantization.
Juno is a spinning spacecraft with a nominal spin period of~30 s, that is,~2 rotations per minute (rpm). Juno spins about the symmetry axis of the high-gain antenna (close to the spacecraft payload Z axis). The left panel of Figure 3 shows the magnitude of Jupiter's magnetic field for two consecutive spins of the spacecraft (60 s worth of data) measured during the outbound part of the PJ (note the expanded scale). Apart from the decreasing linear trend, due to the increasing separation between Juno and Jupiter, a superimposed modulation is notable. The slight modulation in the field magnitude is more clearly seen in the rightmost panel of ©2020 The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.  Figure 3 where the decreasing linear trend has been removed. The frequency of the modulation is twice the spin frequency (second harmonic), and its magnitude is about 0.001 Gauss in an ambient field of little more than 3 Gauss, that is, the modulation is a few parts in 10 4 . Further study demonstrates that the modulation scales with the measured ambient field magnitude.

RESEARCH ARTICLE
Invariance of the measured vector field magnitude under rotation is also often employed as a useful tool in calibrating instruments (the "thin shell" method) when magnetic test facilities are unavailable (Merayo, 2000;Risbo et al., 2001) or as an independent and quick alternative to facility calibration (e.g., Connerney et al., 2017). The principle is also widely applied post launch, particularly on spinning spacecraft, to correct for sensor offsets, scale factors, and attitude determination errors (Alconcel et al., 2014;Auster et al., 2002). In particular, sensor offsets manifest as signal in the field magnitude at the spin frequency, whereas scale factor and attitude determination errors manifest as a signal at the second harmonic of the spin frequency. Facility calibration of the Juno magnetometers established rotational invariance of the field magnitude across instrument dynamic ranges 2, 4, and 5 (accessible to field magnitudes monitored by a proton precession magnetometer) to better than 1 part in 10 4 . Appearance of spin-synchronous modulation in the field magnitude at a level of a few parts in 10 4 thus deserves serious attention. Fortunately, Juno carries two identical sensor suites, each operating over several independently calibrated dynamic ranges, facilitating an analysis of the modulation. The Juno experience at Jupiter demonstrates that (a) the modulation is consistent across all Juno periJove passes; (b) it is independent of the instrument's dynamic range and is most readily apparent in a strong magnetic field environment, where magnetospheric fluctuations ("noise") is insignificant; and (c) the same modulation signal is measured by both of Juno's magnetometer (IB and OB) sensors. Both sensors are similarly accommodated on the MAG boom, within a cavity cut in the panel.
A rotating spacecraft is particularly useful in diagnosing spacecraft-generated magnetic fields that might otherwise go unnoticed on a three-axis stabilized platform. This is because a static spacecraft field (fixed in the spacecraft frame) will manifest, given the geometry of Juno's magnetometer optical bench, as a signal in the field magnitude at the spin frequency. Conversely, a spacecraft-generated field that varies with the applied field magnitude and direction will manifest itself in the field magnitude as a signal at the second harmonic of the spin frequency (twice the spin frequency). Rotation of the spacecraft in a strong magnetic field results in the generation of Eddy currents within the spacecraft's conductive structure; these currents produce a field that acts to resist the change in magnetic field on the conductor. While these are expected to be weak currents given the slow spacecraft rotation, Eddy currents generated in close proximity to the magnetometer sensors can contribute significantly to the measured magnetic field and compromise the absolute accuracy of measurement (if not corrected for). For this reason, requirements on structure fabrication in the vicinity of the sensors called for gaps in the conductive (aluminum honeycomb) material near the sensor, much like laminations used in transformer cores. A study of the effects of Eddy currents induced in the electrically conducting parts of a spinning satellite in Earth's magnetic field, although not quite the same as our case, has been made by Smith, 1965. In order to characterize the spacecraft-induced magnetic fields, we develop a finite element model, which is presented in section 2. Section 3 shows the results from the finite element model. Section 4 describes the implementation of the model to remove the spacecraft generated fields from Juno's magnetic field measurements.

Finite Element Model
The finite element analysis requires a problem defined in geometrical space, called the domain, to be partitioned into a set of finite number of smaller regions (elements), called the mesh (Pepper & Heinrich, 2006), over which a set of equations are solved. In our case, the domain is a section of the MAG boom in the vicinity of the OB sensor suite. Figure 4 shows the configuration of the boom and the OB sensor, which is positioned in a rectangular cutout straddled by the optical bench. The FGM sensor elements reside within the bobbins and provide a measure of the magnetic field in the direction of the bobbin axis at the location of the bobbin. The physical configuration and relative dimensions, shown on Figure 4, are used to set up the model domain representing the section of the MAG boom. The modeled section of the boom and the mesh created for the finite element model is shown on Figure 5. The two rectangular plates in the boom cutout represent the XY and Z coils of the FGM sensor. The XY coil measures the horizontal field (the field components in the XY plane), whereas the Z coil measures the vertical field (the field along the Z axis). The accuracy of the finite element model is directly related to the density of the mesh that is used. Our chosen mesh is a combination of 344,969 tetrahedral and 23,184 triangular elements. The average volume size of the tetrahedral mesh elements is < 0.01 cm 3 , whereas the average surface size of the triangular elements is~0.1 cm 2 . Whereas the type and the density of the mesh are important for a reliable solution, the  . Measured field magnitude |B| versus time from Juno's outboard (OB) magnetic field sensor during Juno's first periJove pass (PJ1). The latitude, the longitude at closest approach in System III coordinates (Dessler, 1983), the maximum measured field magnitude, |B| max , the equator crossing, and the closest approach are indicated on the figure.
critical parameter for a realistic solution is the electrical conductivity of the boom structure where the Eddy currents are induced. This is not surprising since the magnitude of the induced currents depend directly on the physical properties of the conductor (the boom). The MAG boom panels were fabricated by Lockheed Martin, and Goddard Space Flight Center acquired samples of the panels that are representative of the configuration as flown on the Juno spacecraft. We have determined the conductivity of a representative boom panel from its inductive response in a laboratory test and found it to be~5 × 10 5 S/m. The measured gross electrical conductivity of the panel is about 1% of the conductivity of bulk aluminum. We run the model with a value of 5.5 × 10 5 S/m, which is consistent with the laboratory measurements on the material sample. The induced field due to Eddy currents on the boom (as a result of the spacecraft spinning in Jupiter's magnetic field) is then estimated using the finite element method (Babuška et al., 2004). This method is used to acquire numerical solutions to differential equations for each element in the mesh. In our case, we are seeking solutions for the 3D time-dependent Maxwell equations, which, in the absence of the boom conductive structure, can be written as (1) where B is the magnetic field, E is the electric field, and μ 0 and ε 0 are the permeability and the dielectric constant of free space. In order to simulate the spacecraft spinning around the Z axis, we apply a time-dependent boundary condition to the domain (boom) for the ambient (inducing) magnetic field, B, such that B x; y; t ð Þ¼B 0 sin ωt ð Þb x þ B 0 cos ωt ð Þb y; where t is time, ω = 2πf J , and f J = 1/30 Hz, the Juno spin frequency. The time-varying ambient magnetic field B induces Eddy currents in the conductive boom structure. The Eddy currents in turn produce an induced field B ind that opposes the change in the applied (inducing) field, dB/dt. Therefore, the Maxwell equations in the presence of the boom can be written as Figure 4. Layout of the MAG boom with the OB magnetic sensor mounted on the optical bench. The FGM sensor coils are also shown. The X/Y bobbin, or XY coil, is responsible for measuring the field in the XY plane, whereas the Z/RY bobbin, or Z coil, is responsible for measuring the field in the Z axis. Dimensions are in inches.

10.1029/2019EA001061
Earth and Space Science KOTSIAROS ET AL.
where J is the induced current density in the boom. Using equations 1 and 4, we find We do not need to take into account the skin effect (Wheeler, 1942) because the spin frequency of the spacecraft, that is, the frequency of the change of the ambient (inducing) field on the boom, is small compared to the dimensions of boom. The skin effect is apparent at higher frequencies and causes the effective resistance of the conductor to increase. Using the finite element method, we solve the system of equations 3, 4, 5, and 6 and estimate the induced current density J and the induced magnetic field B ind in the conductive boom structure.

Results
The finite element method we use to solve the set of 3D time-dependent Maxwell equations is the generalized minimal residual method (Saad, 2003), which is an iterative method for the numerical solution of a nonsymmetric system of linear equations. The selected mesh ( Figure 5) results in a total of 2,304,786 degrees of freedom, which are the unknowns the system of equations is being solved for. The model is run on two Intel® Xeon® CPUs E5-2670 v3 at 2.30 GHz and takes~20 GB out of~258 GB available memory. Convergence is achieved after about 1 hr 15 min. The final solution is stable, and the residual error is below our tolerance limit of 0.1%. Figure 6 shows the path to convergence of the iterative time-dependent solver. Convergence is achieved when the reciprocal of step size reaches a constant asymptotic value. The exponential decrease indicates fast convergence, and the absence of large fluctuations in the reciprocal of step size with increasing time is indicative of a stable solution.

10.1029/2019EA001061
Earth and Space Science KOTSIAROS ET AL. Figure 7 shows the estimated magnitude of the induced magnetic field due to Eddy currents in the boom at six different times. This example uses an ambient field, B, of 4 Gauss, oriented in the boom's XY plane. The maximum field strength experienced by the Juno spacecraft during a periJove is between about 4 and 11 Gauss (thus far). The induced field is color coded and exceeds 200 nT in the vicinity of the XY coil of the FGM sensor. The change of the ambient field, dB/dt, at each time instant is shown with black arrows, whereas the (Eddy) currents induced in the conductive structure of the boom are shown in green. Notice how the model solution yields induced Eddy current loops perpendicular to the change of the ambient field dB/dt in order to create an induced field that opposes that change, consistent with Lenz's law (Griffiths, 1999).
The left panel of Figure 8 shows the field magnitude for an interval of 30 s (one Juno spin) when Juno was between 1.35 R J and 1.34 R J from Jupiter and the magnitude of the ambient field was between~4.12 × 10 5 nT and~4.26 × 10 5 nT (~4.12 and~4.26 Gauss). The corresponding (spin) modulation in the measured field magnitude for that time interval is shown on the right panel of Figure 8. We isolate the spin modulation by subtracting the spin-averaged (moving average with 30-s window) field magnitude from the measured field magnitude at each sample. The variation in field magnitude predicted by the Eddy current model is shown in black. The model captures both the amplitude and phase of the spin modulation remarkably well using a material bulk electrical conductivity (5.5 × 10 5 S/m) remarkably close (within 10%) to that of the material sample measured in the laboratory. The amplitude of the modulation is about 2 to 3 parts in 10 4 with respect to the ambient field, and the frequency is twice the spin frequency, as expected.
Magnetic field data acquired during the first 20 periJove passes (PJ1 to PJ21; science instruments were powered off during PJ2) have been analyzed for second harmonic content in the field magnitude using the fast Fourier transform (FFT) applied to 60-s sample intervals. In Figure 9, left panel, the amplitude of the second harmonic (2f) is plotted with respect to the magnitude of the field, ; in the XY plane. The right panel shows the field, B Z , along the spin axis (Z axis). The amplitude of the spin modulation is linearly correlated with the field in the XY plane and not correlated with the field oriented along the spin axis. This is to be expected, since a field applied along the spin axis ought not induce Eddy currents in the conductive panel. Figure 10 illustrates this with two examples: The first example is drawn from PJ6 when the ambient field was relatively strong (~10 Gauss = 1 × 10 6 nT) and oriented largely in the XY plane (and B z~0 .3 Gauss = 3 × 10 4 nT). The second example is drawn from PJ17 when the ambient field was relatively weak (~0.3 Gauss) in the XY plane and largely oriented along the Z axis (~2 Gauss = 2 × 10 5 nT). The model prediction for the induced field due to Eddy currents is shown in black for both cases. The spin modulation amplitude is large (~220 nT,~220/1 × 10 6 = 2.2 parts in 10 4 ) when the ambient field is oriented along the XY plane and almost zero when the field is oriented along the spin axis. The finite element model predicts the spin modulation remarkably well in both cases, confirming that the spin modulation in the field magnitude is entirely due to Eddy current generation within the boom structure due to the spacecraft rotation in a strong field.

Implementation
The finite element model gives the relation of the induced magnetic field components B ind X and B ind Y to the ambient (inducing) field components B X and B Y . Specifically, from the model, we get with the Eddy current scale factors c 1 , c 2 determined by the finite element model as c 1 = 0.00035 and c 2 = − 0.00058. The scale factors are a few parts in 10 4 , which is consistent with the relative scaling of the spin modulation with the ambient field (see Figures 8 and 10). The scale factors c 1 , c 2 depend primarily Figure 9. FFT amplitude at the second harmonic during Juno's periapsis passes 1 to 21 with respect to the magnitude of the ambient field in the XY plane, |B ρ |, left panel, and the magnitude of the ambient field along the Z (spin) axis, B Z , right panel. Figure 10. Spin modulation in the measured field magnitude for two cases: one when Jupiter's magnetic field is strong in the sensor's XY plane (shown in red) and one when Jupiter's (ambient) magnetic field is strong in the sensor's Z (spin) axis (shown in blue). The finite element model prediction for the magnitude of the induced field as response to Jupiter's field in the two cases is shown in black.
on the electrical conductivity and geometry of the boom structure. The ratio c 1 /c 2 depends on the local symmetry at the point of the model evaluation (in our case, the center of the XY coil). The finite element model indicates a phase difference of about 3 s between B ind X and B Y and −2.5 s between B ind Y and B X . The phase difference between the induced B ind X , B ind Y and the ambient B Y , B X field components respectively is entirely a geometrical effect caused by the geometry of the panel cutout.
The finite element model provides guidance regarding the expected amplitude and phase of the magnetic field due to induced Eddy currents. Magnetic field data acquired during multiple periJoves, and well distributed in spacecraft payload coordinates, can also be used to characterize the field due to eddy currents using an analysis similar to the "thin shell" method described in Connerney et al. (2017). This technique provides a correction matrix, applied to vector observations, that minimizes variation of the field magnitude under rotation. The correction matrix so obtained is necessarily insensitive to a rotation, but this ambiguity can be resolved using the Eddy current finite element modeling as a constraint. The Juno measurements obtained in strong field environments (instrument dynamic ranges 1, 2, 4, 5, and 6) are corrected for induced fields via application of this correction matrix. Figure 11 shows, in a similar fashion to Figure 9, the performance of the correction. Magnetic field data acquired during 10 periJove passes (PJ1, PJ3, PJ4, PJ5, PJ6, PJ10, PJ11, PJ12, PJ19, and PJ20) have been analyzed for second harmonic content in the field magnitude using the FFT applied to 60-s sample intervals before (shown in blue) and after (shown in red) the Eddy currents correction. The amplitude of the second harmonic (2f) is plotted with respect to the magnitude of the field, ; in the XY plane. In the lower ranges (ranges 2 and 4), the amplitude of the spin modulation is significantly reduced after correction, whereas in the higher ranges (ranges 5 and 6), it drops below the quantization step size, that is, below the instrument resolution. The correction performs best for data in range 5. This is not unexpected since range 5 data are well distributed over the sphere and therefore provide a better constraint to the thin shell method for the estimation of the correction matrix consistent with the Eddy current modeling.

Conclusions
Juno's passages through Jupiter's strong magnetic field revealed an unexpected spin modulation in the magnitude of the magnetic field. In the absence of spacecraft-generated magnetic fields, the field magnitude ought to be invariant under rotation. We found that rotation of the spacecraft in very strong magnetic Figure 11. FFT amplitude at the second harmonic with respect to the magnitude of the ambient field in the XY plane, |B ρ |, during ten (1, 3, 4, 5, 6, 10, 11, 12, 19, and 20) Juno periapsis passes where the Eddy current correction has been implemented. The shaded green boxes represent the quantization step size of the magnetometer with 16-bit quantization for each dynamic range (Connerney et al., 2017).
fields results in the generation of Eddy currents within the spacecraft's conductive structure that act to resist the change in magnetic field on the conductor. While the conductivity of the structure near the magnetometers is relatively small (~1% of bulk aluminum) and the spacecraft spins very slowly, the magnetic field due to Eddy currents induced in strong fields is not insignificant. We advocate the use of spin-averaged data whenever that suffices (as in analysis of the internal field, e.g., Connerney et al., 2018). Where higher time resolution is needed, such as with regard to analysis of Birkeland currents (Kotsiaros et al., 2019), the correction applied in strong fields to remove spacecraft fields (due to Eddy currents) will suffice, largely restoring the vector accuracy of measurement provided by the instrument.
The dynamic spacecraft field is well described by the Eddy current model, the predictions of which are in remarkable agreement with the measured signals. This work illustrates that magnetic field measurements obtained by well-calibrated magnetometers can be, and often are, subject to spacecraft-generated fields that may compromise measurement accuracy, if not understood. While it can often be difficult to identify the source of such interference, an understanding of the physical cause of the contribution may provide a means to analytically correct the measurements, thereby restoring measurement accuracy. grant NNN12AA01C to JPL/Caltech. The Juno mission is part of the New Frontiers Program managed at NASA's Marshall Space Flight Center in Huntsville, Alabama. The authors are aware of no real or perceived conflicts of interest with respect to the results of this paper. As agreed with NASA, data supporting the conclusions are released on a schedule via the NASA Planetary Data System at https://pds.nasa.gov. The data analyzed in this paper can be downloaded free of charge at https:// pds-ppi.igpp.ucla.edu/ditdos/ download?id=pds://PPI/JNO-J320 3-FGM-CAL-V1.0/DATA/JUPITER/ PL.