Magnetometer in-flight offset accuracy for the BepiColombo spacecraft

Recently the two-spacecraft mission BepiColombo launched to explore the plasma and magnetic field environment of Mercury. Both spacecraft, the Mercury Planetary Orbiter (MPO) and the Mercury Magnetospheric Orbiter (MMO, also referred to as Mio), are equipped with fluxgate magnetometers, which have proven to be well-suited to measure the magnetic field in space with high precisions. Nevertheless, accurate magnetic field measurements require proper in-flight 5 calibration. In particular the magnetometer offset, which relates relative fluxgate readings into an absolute value, needs to be determined with high accuracy. Usually, the offsets are evaluated from observations of Alfvénic fluctuations in the pristine solar wind, if those are available. An alternative offset determination method, which is based on the observation of highly compressional fluctuations instead of incompressible Alfvénic fluctuations, is the so-called mirror mode technique. To evalu10 ate the method performance in the Hermean environment, we analyze four years of MESSENGER magnetometer data, which are calibrated by the Alfvénic fluctuation method, and compare it with the accuracy and error of the offsets determined by the mirror mode method in different plasma environments around Mercury. We show that the mirror mode method yields the same offset estimates and thereby confirms its applicability. Furthermore, we evaluate the spacecraft observation time within 15 different regions necessary to obtain reliable offset estimates. Although the lowest percentage of strong compressional fluctuations are observed in the solar wind, this region is most suitable for an accurate offset determination with the mirror mode method. 132 hours of solar wind data are sufficient to determine the offset to within 0.5 nT, while thousands of hours are necessary to reach this accuracy in the magnetosheath or within the magnetosphere. We conclude that in the solar wind the 20

mirror mode method might be a good complementary approach to the Alfvénic fluctuation method to determine the (spin-axis) offset of the Mio magnetometer.

Introduction
In October 2018, BepiColombo, a two-spacecraft mission of the European Space Agency (ESA) and the Japan Aerospace Exploration Agency (JAXA), was launched to explore Mercury (Benkhoff 25 et al., 2010). One of the spacecraft is the Mercury Planetary Orbiter (MPO), which is a 3-axis stabilized satellite (quasi nadir pointing) to study the surface and interior of the planet (e.g. Glassmeier et al., 2010). The other is Mio (or Mercury Magnetospheric Orbiter, MMO), a spin-stabilized spacecraft (spin period of about 4 s) to investigate the magnetic field environment of Mercury (e.g. Hayakawa et al., 2004;Baumjohann et al., 2006). During the 7.2 year cruise phase, both orbiters 30 are transported by the Mercury Transfer Module (MTM) as a single composite spacecraft. In late 2025, the composite spacecraft will approach Mercury, where the MTM separates from the other two spacecraft, which are captured into a polar orbit around the planet. As soon as Mio reaches its initial operational orbit of 590 km by 11640 km above the surface, also MPO separates and lowers its altitude to its 480 km by 1500 km orbit. 35 The BepiColombo Mercury Magnetometers (MERMAG) constitute a key experiment of the mission; MERMAG consists of the fluxgate magnetometers onboard both, MPO and Mio. The magnetometers will provide in-situ data for the characterization of the internal field origin as well as its dynamic interaction with the solar wind (see e.g. Wicht and Heyner, 2014, for a discussion). To 40 achieve this goal, accurate magnetic field measurements are thus of crucial importance. Therefore, the components of a linear calibration matrix M and an offset vector O need to be obtained, in order to convert raw instrument outputs B raw to fully calibrated magnetic field measurements (see e.g. Kepko et al., 1996;Plaschke and Narita, 2016): (1) 45 Here, the matrix M transforms B raw into a spacecraft-fixed orthogonal coordinate system. It comprises 9 parameters: 3 scaling (gain) values of the sensor and an orthogonalization matrix, which is defined by the 6 angles that yield the magnetometer sensor directions with respect to the spacecraft reference frame (see e.g. Plaschke and Narita, 2016). The 3D offset vector O, on the other hand, reflects the magnetometer outputs in vanishing ambient fields. These can be attributed to the instru-50 ment and also to the field generated by the spacecraft at the position of the magnetometer sensor.
Frequent in-flight calibration of these offsets is necessary, as they are known to change over time.
To calibrate the magnetometer, all 12 parameters need to be accurately determined. For spinning spacecraft (i.e. Mio) 8 of the calibration parameters can be determined directly by minimizing periodic signatures in the de-spun magnetic field signal at the spin frequency and/or at the second 55 harmonic (Kepko et al., 1996). The remaining 4 parameters i.e. the absolute gain in the spin-plane and along spin-axis, the rotation angles of the sensor around the spin-axis, and the spin-axis offset need to be determined differently. It should be noted that the gains and rotation angle become important at strong fields. In the solar wind and in the Hermean environment they may play a minor role in the magnetic field measurements accuracy in comparison to the offsets.

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The following methods, which can also be applied to non-spinning spacecraft (i.e. MPO), are well established for the offset determination: (1) Cross-calibration of the magnetometer offset with independent magnetic field measurements from other instruments. The Magnetospheric Multiscale mission (MMS, Burch et al., 2016) use independent measurements from the Electron Drift Instruments (EDI, Torbert et al., 2016) to crosscalibrate the spin-axis offset of the magnetometers Plaschke et al., 2014)).
(2) The offset may also be determined in a characteristic region where the magnetic field is known. Goetz et al. (2016) used diamagnetic cavities to determine the magnetometer offsets of the Rosetta spacecraft mission (Glassmeier et al., 2007).

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(3) Spacecraft rotations about the spacecraft principal axes from time to time are also a wellestablished method to obtain magnetometer offsets. This has been done routinely with many missions, most recently including MAVEN (Connerney et al., 2015) at Mars and Parker Solar Probe (Bale et al., 2016).
(4) A common way to calibrate the magnetometer offset in-flight, is to use (a) (nearly-)incompressible 75 or (b) compressible fluctuations of the magnetic field: 4(a) A well-established method is to minimize the variance of the total magnetic field during the passage of (nearly-)incompressible (Alfvénic) variations in the (pristine) solar wind (Belcher, 1973;Hedgecock, 1975). Pure Alfvénic fluctuations are strictly incompressible and circularly polarized.
They are characterized by changes in the magnetic field components while the magnitude of the field 80 stays constant. Particularly in inhomogeneous media such simple classifications are found to be impossible (see Tsurutani et al., 2018, for a review). However, in the solar wind the fluctuations of the magnetic field strength (compressible part) are weak compared to the strong fluctuations of the magnetic field vector direction (Khabibrakhmanov and Summers, 1997). By minimizing the changes of the observed total magnetic field of such fluctuations it is therefore possible to adjust the magne-85 tometer offsets with high precision. In Figure 1(a) a schematic illustration of (nearly-)incompressible Alfénic fluctuations is shown. The changes of the total magnetic field, δ|B|, observed by a virtual spacecraft crossing the fluctuations are minimal (shown in the bottom). Plaschke and Narita (2016) introduced an alternative magnetometer offset calibration technique on the basis of the observation of compressible fluctuation, the so-called mirror mode 90 method, which does not require pristine solar wind measurements. The idea is that for strongly compressible fluctuating fields (e.g. mirror modes) the maximum variance direction of the magnetic field should be nearly parallel to the mean (background) magnetic field (Tsurutani et al., 2011). Mirror mode structures are compressional, non-propagating structures which have been observed in various space plasma environments like in the solar wind (e.g. Winterhalter et al., 1994), planetary magne-95 tosheaths (e.g. Tsurutani et al., 1982) and even near the magnetic pileup boundaries of comets (e.g. Glassmeier et al., 1993). They are typically characterized by the anticorrelation between the magnetic field strength and density fluctuations, with little or no change in the magnetic field direction across the structure (see e.g. Winterhalter et al., 1994). Figure 1(b) shows a sketch of such a mirror mode structure. A virtual spacecraft crossing the structure observes perturbations along the mean 100 magnetic field direction, δB || . Any differences between the maximum variance direction (evaluated from e.g. minimum variance analysis (MVA), Sonnerup and Scheible, 1998) of the magnetic field and the mean (background) magnetic field direction can thus be used for the magnetometer offset determination. Note that although this method is called mirror mode method, observations of highly compressional waves other than mirror mode structures with strong δ|B|/|B| ratios without B di-105 rection changes are sufficient for their application. An advantage of this method is that compressible waves are ubiquitous in the magnetosphere and magnetosheath. Therefore this method can also be applied to calibrate the magnetometers of spacecraft which remain within the magnetosphere (like e.g. MPO).

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In this paper we test the applicability of the mirror mode method in different regions (plasma offset needs to be evaluated) we apply the 1-D mirror mode method (see Plaschke and Narita, 2016) and assume that two offset components are already accurately determined. We test to which degree the 1-D mirror mode method yields vanishing offsets as expected when using calibrated data as input. We also address the question of how much time a spacecraft needs to spend in each individual plasma region until the magnetometer offset can be determined to a specific accuracy from intervals 120 containing compressional magnetic field fluctuations.
The results obtained in this paper enable us to assess whether the mirror mode method would be a useful tool to accurately determine the offset of BepiColombo's magnetometer.

Data and Methodology
We use orbital magnetic field data from the MESSENGER fluxgate magnetometer (FGM, Anderson an initial orbital period of 12 hours, MESSENGER crossed the magnetopause and bow shock four times within 24 hours. We use the 1-Hz calibrated magnetic field data in spacecraft coordinates, where the Y -axis is nominally in the anti-sunward direction (radially away from the Sun), the Z-axis 130 points towards the payload adapter ring at the bottom of the spacecraft, and the X-axis completes the right handed coordinated system. If not noted otherwise, we use the magnetic field components {B x , B y , B z } throughout this paper in these coordinates. MESSENGER was a three-axis-stabilized spacecraft, and its magnetometer offsets were routinely corrected using time intervals of Alfvénic fluctuations in the solar wind. We therefore use the MESSENGER magnetometer data as a calibra-135 tion standard in our magnetometer offset study. To perform a test of the the mirror mode method against the MESSENGER magnetometer data, we determine the 1D offset along the Z-axis, O z , in the same way as introduced by Plaschke and Narita (2016): Within strongly compressible mirror mode structures, the magnetic field variation maximum variance direction, = [l x , l y , l z ] obtained from a principal component analysis (minimum variance 140 analysis, MVA, Sonnerup and Scheible, 1998) should be reasonably aligned with the mean field Under assumption of alignment between the maximum variance direction and the mean field direction Plaschke and Narita (2016) showed that the offset O z can be derived by:

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where θ B = arctan(B z /B xy ) is the elevation angle of magnetic field to the XY -plane and θ = arctan( z / xy ) the elevation angle of the maximum variance direction to that plane. B xy = B 2 x + B 2 y and xy = 2 x + 2 y are the magnetic field and maximum variance within the XYplane.
Since a single offset estimate alone might not be very accurate, a statistically significant offset O zf 150 should be determined by finding the maximum of the probability density function P computed by the kernel density estimator (KDE) method with Gaussian kernel from a sample of individual offset estimations O z : Here N denotes the number of individual offset estimates, O z,n , h is a smoothing parameter which

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A basic condition for the mirror mode method, is the availability of compressional magnetic field fluctuations. As a first step we use the magnetometer data to estimate the occurrence rate of compressional fluctuations in the different plasma environments around Mercury. Then we compare the offset determination by the mirror mode method with the Alfvén wave method in terms of accuracy. Finally, we evaluate the number of offset samples that are sufficient for in-flight calibration 165 with the mirror mode method, given a minimum required accuracy. Therewith, we subsequently determine how many hours the spacecraft (i.e. Mio and/or MPO) need to spend in different plasma environments to obtain reliable offset values.

Occurrence rate of compressional fluctuations
Mercury's plasma environment is highly dynamic and home to a plethora of wave modes and fluctua-170 tions (Russell, 1989;Boardsen et al., 2009Boardsen et al., , 2012Sundberg et al., 2015). We separate the MESSEN-GER orbit segments into solar wind, magnetosheath and magnetospheric parts, based on an extended boundary data set (Winslow et al., 2013, and  given by the difference between the maximum and minimum value of the magnetic field magnitude within the 30-s interval. Subsequently, to determine whether the compressional or the transverse part is dominating, we further define the compressibility index Q ± as which is positive/negative in case of dominating compressional/transverse fluctuations.  Table 1.
The first column of Table 1  Note that we only consider the XY -plane components, because the Z-component are subject to an a-priori unknown offset O z .
φ < 20 • , where φ is the angle between the maximum variance, , and magnetic field, B, 225 directions in the XY -plane. Note that only the XY -plane components is used. Plaschke and Narita (2016) derived this requirement and threshold from Lucek et al. (1999), who identified mirror modes using the angle between maximum variance and magnetic field direction.

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Out of 4-years of MESSENGER data these requirements are met for 3.0 %, 2.1 % and 0.4 % of the intervals within the magnetosphere, magnetosheath and the solar wind, respectively (see also first column of Table 2). It should be noted that these numbers do not have to match the numbers shown in Table 1, since the criteria are different. The numbers in Table 1 reflect the occurrences of strongly compressional fluctuations and here they reflect the occurrences of intervals with mirror mode like 235 characteristics. Interestingly, however, the numbers in the solar wind and the magnetosheath are similar but differ considerably in the magnetosphere.
Using Equation 3 the probability density function P is computed for each region from all determined best-estimate offset O z in that region. Under the assumption that the data are perfectly calibrated and the mirror mode method works accurately, the offset O zf should vanish and P should be highly 240 symmetric with the peak around 0 nT. The average O z,n is found to be quite close to the final offset estimate O zf . Note that, although the standard deviation of the individual offsets O z,n might be large, a larger number of samples or events helps lower the value of the standard deviation of the mean offset O z,n (standard error in Table 2). The dashed lines in Figure 4 mark offset accuracies at 0.5 nT and 1.0 nT. It is visible that the offset accuracy increases with the number of samples following a power law. However, below ∼ 0.5 nT considerably more samples are needed to improve the offset accuracy, which could indicate the lower limit of the offset accuracy of the MESSENGER magnetometer determined by the Alfvénic 270 method. As can been seen in Figure 4, this divergence is only observed in the solar wind, because it is the only region where a significant number of O zf < 0.5 nT are obtained. The solid lines in Figure   4 depict the regression lines of a linear least squares fit of the offset accuracies above 0.5 nT and clearly shows the power law behavior between these offset accuracies and the number of samples. Table 3 shows the fitting parameters of the regression lines of the power laws,  Table 2), finally give the time lengths that a spacecraft needs to spend in the solar wind, magnetosheath or magnetosphere in order to determine the offset with an accuracy better than 0.5 nT or 1.0 nT. Table 4 shows the minimum number N of 285 O z,n estimates with their corresponding time ranges and the necessary spacecraft observation time, required to reach these accuracies.
While 132 hours of solar wind data are sufficient to determine the offset at an accuracy of 0.5 nT, 4241 hours are needed in the magnetosphere and more than 6130 hours in the magnetosheath. This is an interesting result, since the magnetosheath is the region with the highest probability to observe 290 large-amplitude compressional fluctuations and the solar wind with the lowest (see Table 1). A possible explanation might be that the solar wind data are observations in the shock-upstream region in which the shock-reflected backstreaming ions excite large-amplitude Alfvén waves that either pitch the plasma (through ponderomotive force) or develop into magnetosonic-type waves. However, the results suggests that the fluctuations in the solar wind, although observed less often, are better 295 suited to determine the offset with the mirror mode method.

Discussion and Conclusions
We find that the offset determination method proposed by Plaschke and Narita (2016) is well applicable to the data from the Hermean environment. The results reveal that the solar wind is the most suitable region to accurately determine the offset by the mirror mode method, although the lowest 300 percentage of highly compressible fluctuations are observed there (see Table 1). As can been seen in Table 4, offset determination with uncertainties better than 0.5 nT can be achieved with less than 132 hours of solar wind data. However, it is important to note that during this time the instrument offsets need to stay constant to within 0.5 nT, otherwise the intrinsic offset drifts over time would limit the achievable precision, independently from the the amount of solar wind data. Specifically in the 305 magnetosheath and within the magnetosphere this might be a more important limiting factor, since several thousands of hours of data in those regions are needed to ensure that the offset uncertainty is less than 0.5 nT (see Table 4). Figure 4 shows that the offset accuracies down to 0.5 nT diminish with the number of necessary O z estimates and follows a power law. However, below 0.5 nT the power law correlation flattens (spectral index becomes smaller), which might indicate that the lower Furthermore, Mio is a spin-stabilized spacecraft (with a period of about 4 s) and thus two of the three offsets in the de-spun plane can be directly determined by minimization of spin-tone 320 in the data. Since the two spin-plane offsets can be determined very accurate (Kepko et al., 1996), the mirror mode method presented in this study can thus be directly applied to determine the remaining offset (spin-axis offset) very precisely (see, Plaschke and Narita, 2016).
However, the mirror mode method assumes the time independence of the offset properties during the fluctuation measurements. Offsets drifts over time, e.g. due to temperature changes 325 over orbital periods, will be the limiting factors for the achievable accuracies.   Table 3. Fitting parameters of Equation 5 with 95 % confidence intervals, exhibited from linear least squares fit of the offset accuracies above 0.5 nT in Figure 4. a is the 2-σ confidence of the best-estimate O zf determined from only one offset Oz and k is the spectral index of the power law.       is the uncertainty of the determined best-estimate offset O zf with 95 % confidence. The solid lines represent the linear least squares fits of the 2-σ confidence offsets above 0.5 nT.