Spacecraft and interplanetary contributions to the magnetic environment on-board LISA Pathfinder

To understand the coupling of the test mass to magnetic induced forces one can consider the test mass inside the spacecraft as a magnetic dipole in a magnetic field. In such approximation, the test mass will sense a force given by

where $\mathbf{M}$ is the intrinsic moment of the magnetic dipole, $\mathbf{B}$ the surrounding magnetic field and V the test mass volume. Here we use the brackets as an average over the test mass volume. For the sake of simplicity we do not take into account inhomogeneities or anisotropies in the composition of the test mass. A more detailed model of the effect of magnetic field on the test mass would consider also torques applied to the test mass (Diaz-Aguilo et al., 2012). Several considerations need to be taken into account for the free-falling test masses inside the satellite. First, two contributions to the magnetic moment need to be distinguished: a first one from the intrinsic remanent magnetic moment ($\mathbf{M_{r}}$) and a second one coming from induced magnetic moment, which is proportional (through the magnetic susceptibility, $\chi$) to the applied magnetic field ($\mathbf{M_{i}}=\chi\mu_{0}^{-1}\,\mathbf{B}$). Both contributions are determined by the composition of the test mass, which in the LISA Pathfinder case was a Pt (27%) and Au (73%) alloy. Eq. (1) therefore expands to

Since the main objective of the mission is to understand differential acceleration fluctuations down to the mHz, we further need to consider how fluctuations of the magnetic field will impact on the force. The different components to this contribution can be easily visualised if one splits both the magnetic field and the magnetic field gradient into a constant contribution and a term that varies with time,

which leads to several magnetically-induced force contributions

From Eq. (2.1) we clearly observe how the magnetic environment will induce a variety of effects on the test mass that include: constant force contributions ($\mathbf{B_{0}}\cdot\nabla\mathbf{B_{0}}$); terms that will couple the local field at the test mass position to the fluctuations of the gradient of the magnetic field ($\mathbf{B_{0}}\cdot\delta\nabla\mathbf{B}$); couplings of the local magnetic field gradient to fluctuations in the magnetic field ($\delta\mathbf{B}\cdot\nabla\mathbf{B_{0}}$) and coupling of fluctuations in the field and its gradient ($\delta\mathbf{B}\cdot\delta\nabla\mathbf{B}$). Hence, equally important for the experiment are both the monitoring of magnetic field and magnetic field fluctuations since both can couple, through the test mass remanent magnetic moment and susceptibility, into spurious accelerations of the free-falling test masses. We will describe in the following the diagnostics subsystem used on-board to characterise these figures to later describe the results obtained with these sensors.

The LISA Pathfinder magnetic diagnostics subsystem fulfils two related purposes. First, to measure the magnetic field and magnetic field gradient on-board and second, to create magnetic fields and gradients at the position of the test masses to study the contribution of the magnetic forces to the instrument noise budget. The subsystem is therefore composed by four tri-axial magnetometers and two induction coils to fulfil these two purposes, respectively.

The coils are capable of generating a controlled magnetic field at the location of the test masses. The two circular induction coils are located 85.5 mm away from the test mass, each one attached to the external wall of each vacuum enclosure. The coils have a radius of 56.5 mm and are built with 2400 windings of a copper alloy mounted on a titanium support with a dedicated high stability curent driver to ensure that high precision magnetic forces are produced (Diaz-Aguiló et al., 2013). The coils are aligned with the axis joining both test masses so that the generated magnetic field has axial symmetry. Experiments conducted to study magnetically-induced forces on the test masses during flight operations will be reported in a separate publication.

The magnetometers are intended to sense with high precision the evolution of the magnetic field. The magnetometers on-board LISA Pathfinder are fluxgate tri-axial magnetometers, namely TFM100G4-VQS from Billingsley with a sensitivity of $\rm 166.7\mu V/nT$ and a typical root mean square noise figure of $\rm 20\,pT/Hz$ at 1 Hz. In fluxgate magnetometers, each axis consists of a first sensing coil surrounding a second inner drive coil wound around a high permeability magnetic core material. Although providing low-noise measurements, because of the active magnetic core inside the sensor, these magnetometers have to be located far enough from the test mass position so they do not contribute as a source of magnetically-induced force on the test mass. All magnetometers channels are connected to a multiplexer after passing through an analog low-pass filter. The signal is then amplified by means of discrete operational amplifiers (OP471AY/QMLR) in an instrumentation amplifier configuration. Finally, it is digitally converted by a 16 bit analog to digital converter (7809LPRPFH). The sampling frequency is 50 Hz. Hence, since the input is multiplexed by 16 channels, the individual channel sampling frequency is 3.125 Hz. In section 3.2.3 we will come back to the two last components of the read-out chain, instrumentation amplifier and analog to digital converter, which determine the noise performance of the magnetometer read-out.

In terms of data handling, the magnetometers data were downloaded with a dedicated telemetry package, which also included all the diagnostics on-board the satellite. This approach is a characteristic of LISA Pathfinder and differs from most of the scientific space missions where the diagnostics subsystem is considered only for house-keeping purposes. The rationale behind this is that, in gravitational-wave detectors, noise characterisation and noise hunting is crucial for the operation of the instrument. Hence, in our current case, environment sensors entail key information for the achievement of the scientific goal.

In the top panel of Figure 2 we show the time series for the read-out of the four magnetometers on-board LISA Pathfinder since the magnetometers switch-on and until the end of the mission, in July 2017. We provide the magnetic field measurements in the X, Y and Z axes in the LTP reference frame, although the X-component contains most of the interesting features that we will discuss in the following.

The absolute value of the magnetic field at each magnetometer location is dominated by the spacecraft contribution reaching values up to $1\mu T$, far from typical interplanetary magnetic field values which would be of the order of 5-10 nT. Most of the magnetic field measured by the magnetometers corresponds to the contribution of the cold gas micro-propulsion system, in particular to the magnets on the high-pressure latch valves. With an on-ground measured moment of  950 $\rm mA\,m^{2}$, they account for most of the magnetic field measured by the magnetometers (Armano et al., 2015). It is important to mention that since the cold gas thrusters are placed in different locations of the spacecraft the magnetic field they create do not necessarily add in the same direction.

The time series can be divided in six different segments which correspond to the different phases of the mission, namely comissioning, LTP operations, DRS operations, the associated mission extension for both experiments, and decomissioning. The most prominent features in the magnetic field timeline are experiments with the coils to characterise the magnetic contribution to the test mass free-fall. These experiments are listed in Table 1. Other features that affected the magnetic environment and can be identified in Figure 2 are listed in Table 2. In the latter case, these are not associated to the activation of the coils on-board (except for event ‘a’, which was a check of the correct functioning of coil 2) but to the operation of the spacecraft itself, i.e. from changes in configuration to identified anomalies during operations. For the sake of completeness, the mean values of the magnetic field DC for different configurations of the spacecraft can be found in Appendix A.

In terms of the magnetic environment, we notice that DRS operations had an impact in the spacecraft magnetic environment. As seen in the bottom panel of Figure 2, the mean value of the X component shows a steady decrease of  150 nT starting around Aug. 8th and ending around Nov. 13th. for the two magnetometers located near the test masses, i.e. magnetometers PX and MX in the bottom panel of Figure 1. This period is coincident with the start of operations of the DRS instrument. Apart from this long drift, in magnetometers PX and MX we also observe a DC increase of about 40 nT when we turn on the DRS system and the same decrease when we turn it off. This DC change should be related to some units being switched on and off when we change the control system of the satellite.

The reason of this long term trend observed in the magnetic field on-board can be explained by the differences between LTP and DRS micro-propulsion systems. The LTP micro-propulsion system consists of three clusters each featuring four cold gas thrusters (Armano et al., 2019a). The thruster system uses high-pressure Nitrogen propellant to provide very small impulses with a thrust range of 1–500 $\rm\mu N$. Four Nitrogen tanks store the gas at 310 bar with a maximum capacity of 9.6 kilograms of Nitrogen. The DRS micropropulsion system is composed by two Colloid Micro-Newton Thruster (CMNT) clusters (Ziemer et al., 2006). Each cluster includes four complete and independent thruster units. For the CMNT subsystem, thrust is adjustable from 5–30 $\rm\mu N$ by changing the beam voltage and/or propellant flow rate that determines the beam current. In this case, the propellant is stored in four electrically isolated stainless steel bellows compressed by four constant force springs set to supply four micro-valves with propellant at approximately 1 atm of pressure. In terms of the spacecraft magnetic field, this sets a relevant difference between both thrust subsystems since the continuous operation of the CMNT subsystem leads to a steady displacement of the bellows inside the storage tank. Although not having a direct impact on the mission operations in terms of gravitational pull, it is precisely this displacement of the stainless steel bellows occurring during the depletion of the fuel tanks the one that originates the observed magnetic field change. Indeed, given the known geometry and measured magnetic properties of the CMNT we can estimate the impact of the operation of these thrusters on the magnetic environment. The CMNT are aligned in the direction joining both test masses (X direction) as can be seen in the top panel of Fig. 1. They are located at a distance of 28 cm with respect to the two magnetometers that are located along this same axis, the PX and MX magnetometer in the notation of the bottom panel of Fig. 1. The measurement on-ground reported a magnetic moment for these units of 209 $\rm mA\,m^{2}$ (in modulus) which would produce a magnetic field of 1.1 $\rm\mu T$ on the position of the closest magnetometer to each CMNT according to the equation of the magnetic field produced by a magnetic dipole:

with $\mathbf{m}$ being the magnetic moment of the source and $\mathbf{r}$ a vector going from the center of the magnetic dipole to the position where the magnetic field is being measured. If we now assume that the overall variation in the magnetic field measured in the X direction during DRS operations is due to the displacement of the bellow inside the CMNT valves, this would imply a 1 cm displacement which is compatible with the geometry of the valve and the amount of propellant being expelled during this period.

Given that any varying magnetic field or magnetic field gradient will couple into the motion of the test masses, understanding the origin of the fluctuations of the magnetic field measured on-board is an important output of LISA Pathfinder for future gravitational-wave detectors in space. In the previous section we have seen how the electronic units on-board the satellite are the dominant contribution to the magnetometers DC measurement since the interplanetary contribution (typically of the order of 5-10 nT) is at least one order of magnitude below the values reported, for instance, in Table 4. The situation is different when we study the variations of the magnetic field. A wide variety of phenomena can produce a varying magnetic field and, as we show in the following, both fluctuations originated by the spacecraft and by the interplanetary magnetic field are relevant to understand the magnetic field spectra measured by LISA Pathfinder.

We divide the three sections below as follows. First, we provide a characterisation of the different magnitudes describing the fluctuations of the magnetic field on-board to then focus on the physical mechanisms describing these fluctuations for two different frequency regimes that we distinguish in our data.

Until now we have based our analysis on the amplitude spectral density, which entails the information of the magnetic field in the frequency span of interest. The amplitude spectral density effectively describes the fluctuations during a fixed period of time. Hence, it would only be a complete statistical description if the environment on-board LISA Pathfinder were stationary. This is obviously not the case. On the contrary, several situations can induce abrupt changes in the measured magnetic field. In Table 2 we provided a series of events that we identified in the magnetic field timeline. Since these were associated to satellite operations these events are both easily identified and, eventually, removed from the dataset through post-processing. A second, more complex, class of non-stationarity is the one associated to the interplanetary magnetic field. In section 3.2.2 we described the origin of the spectral index of the observed magnetic field fluctuations in the low frequency range and its relation with the solar wind speed. A rich variety of interplanetary structures is superposed to the unperturbed solar wind plasma (Kilpua et al., 2017) &  (Richardson, 2018). Structures such as corotating interaction regions, interplanetary shocks, magnetic clouds or heliospheric current sheet crossings will induce variations in the measured magnetic field on-board the spacecraft.

Since our DC magnetic field is completely dominated by the spacecraft components: around $\rm\sim 1\mu T$, compared with the $\rm\sim 1nT$ coming from the interplanetary media, we can not measure the absolute value of the interplanetary magnetic field in these cases. Nevertheless, if the variations are strong enough (of the order of $\rm\sim 10nT$, for example) and especially if we see them in all four magnetometers, we could indeed assign an interplanetary origin to them. Accurate detection of the magnetic imprint of these events requires the magnetometers to be isolated from spacecraft contributions. This is the case for dedicated space weather mission which place the magnetometers at the end of a deployable boom, such as in the case of ACE (Smith et al., 1998) or WIND (Lepping et al., 1995) missions. Moreover, these missions contain a suite of instruments that allow a complete characterisation of the plasma, tracking parameters such as the solar wind speed or the number density of the plasma that we will refer in the following. Nonetheless, it is worth noticing that, although not designed for that, the radiation monitor on-board LISA Pathfinder (Canizares et al., 2011; Armano et al., 2018a) allowed for the detection of the passage of large scale interplanetary structures such as high-speed solar wind streams and interplanetary counterparts of coronal mass ejections generating recurrent and non recurrent depressions of the galactic cosmic-ray flux (Armano et al., 2018c, 2019c).

As we have previously discussed, in LISA Pathfinder the effect of the interplanetary magnetic field structures can not be easily distinguished in the absolute value of the measured magnetic field because the local magnetic field is largely dominated by the contribution of the spacecraft units. However, as we show in the following, the effect of these structures can have an impact in the spectra of fluctuations in the low frequency band, i.e. below the Hz.

In order to trace the variability of the spectrum we took a closer look at the amplitude of the spectra in two different frequency regions, namely $\rm 20\ \mu Hz$ $<f_{LF}<$ $\rm 50\ \mu Hz$ and $\rm 20\ mHz$ $<f_{HF}<$ $\rm 40\ mHz$. We selected these two frequency regions because, as discussed above, magnetic field fluctuations come from a different physical origin, i.e. interplanetary magnetic field and magnetometer read-out electronics, respectively. For each of these frequency windows we compute the power spectral density. We selected windows of 16 hours to compute each bin. In order to avoid segments containing magnetic experiments or events as the ones reported in Tables 2 & 1, we apply a mask to the data. To do so we take as a figure of merit the amplitude spectral density in the range $\rm 1\ mHz<\ \it f<\rm 10\,mHz$. Those segments where this figure of merit is exceed by five sigma are discarded from our analysis. Following this criterion, we exclude 9 segments out of 300 This analysis allows a generic description of the statistical behaviour of the fluctuations without any previous assumption on its stationarity. Figure 4 shows the results for both frequency ranges. In agreement with our previous analysis, the amplitude of the fluctuations for the low-frequency bin is coherent and follows the same statistical distribution for all four magnetometers. Bins are statistically distributed with similar median values, namely

These values are based on the 16th, 50th and 84th percentiles of the histogram. Also, in this case we can derive a common mean value for the fluctuations in this frequency range of $\rm 207\pm 6\ nT\ Hz^{-1/2}$. On the other hand, the noise power in the higher frequency bins show a different amplitude spectral density for each magnetometer. The median values in this case are:

In the latter we observe that the distribution of the median values are narrower and not overlapping between them. Both behaviours are clearly distinguished in the histograms of Fig. 4, which characterises the variability of the magnetic field fluctuations on-board for the two frequency regimes that we have previously identified.

In the framework of future gravitational-wave detectors in space, the variability in the spectra of magnetic field fluctuations is particularly relevant in the low frequency regime. There are two main reasons for that: first, the mHz band is the main objective of a gravitational-wave detector in space, since these observatories are designed to study the gravitational universe in this frequency band. Second, at the same time, the low frequency fluctuations will be precisely the main contribution to the magnetic induced force noise in the free falling test masses. According to Armano et al. (2016), magnetic induced forces could contribute to a maximum of 3 $\rm fm\ s^{-2}\ Hz^{-1/2}$ of the measured 12 $\rm fm\ s^{-2}\ Hz^{-1/2}$ at 0.1 mHz of the differential acceleration measured on LISA Pathfinder test masses. Although the precise determination of the magnetic noise contribution is still pending, the current estimate is expected to be of the order of other contributions such as the charging noise, with an expected contribution of 1 $\rm fm\ s^{-2}\ Hz^{-1/2}$ (Armano2017) or the actuation noise, which is expected to be the dominant contribution with an expected value of 4.5 $\rm fm\ s^{-2}\ Hz^{-1/2}$ (Armano et al., 2016) at 0.1 mHz. For that reason it is worth characterising further the variability in this frequency regime to provide information for future space-borne observatories. As we show in the following, the fluctuations of the magnetic field in the sub-mHz band are deeply connected to the dynamics of the interplanetary plasma.

Fluctuations in the intensity of the interplanetary magnetic field can be associated with a wide variety of phenomena (Bruno & Carbone, 2013). However, particularly in the inertial range, the amplitude of the magnetic fluctuations is strictly related to the their Alfvénic nature. The solar wind is highly structured in high and low-speed streams, which carry different types of fluctuations. While fast wind is characterised by large-amplitude Alfvénic fluctuations, the slow wind generally advects less Alfvénic fluctuations characterised by a smaller amplitude –with the important exception of the Alfvénic slow wind, see D’Amicis & Bruno (2015). This means that, moving from high- to low-speed regions, the power level of the magnetic fluctuations within the inertial range progressively decreases, though keeping the typical $f^{-5/3}$ Kolmogorov scaling. As a matter of fact, solar wind turbulence may be thought as superposition of a magnetic field background spectrum, common to both fast and slow flows  (Bruno et al., 2017), and a turbulent large-amplitude Alfvénic spectrum, characteristic of the fast solar wind.

In order to study the impact of the solar wind speed in our measurements, we selected three segments representing stable periods of solar wind speed. These periods had to be long enough to allow an estimate of amplitude spectral density down to $\rm 20\,\mu Hz$. The selected time spans correspond to Jul. 7th-14th 2016, May 28th- Jun. 1st 2017 and Feb. 11th-16th 2017 when, according to measurements recorded by ACE, the solar wind had a mean velocity of $\rm 553\pm 47\,km\ s^{-1}$, $\rm 390\pm 53\,km\ s^{-1}$ and $\rm 335\pm 35\,km\ s^{-1}$, respectively. For each of these segments we evaluated the amplitude spectral density of the magnetic field as measured by LISA Pathfinder magnetometers. Although other authors have already studied this phenomena (Bruno et al., 2017), we extended the characterisation to the sub-mHz regime, which is the region of greatest interest for LISA and future gravitational-wave detectors.

Results are shown in Figure 5, where we can distinguish an increase in the power of the low-frequency fluctuations that correlates with the increase of solar wind velocity. Indeed, we observe that fluctuations at $\rm 20\,\mu Hz$ vary from $\rm 170^{+30}_{-10}\,nT\ Hz^{-1/2}$ for a slow wind situation (typical velocities $\rm v\simeq 300\ km\ s^{-1}$) to $\rm 750^{+100}_{-50}\,nT\ Hz^{-1/2}$ when we consider a situation of high-speed wind (typical velocities $\rm v\simeq 500\ km\ s^{-1}$). These two scenarios represent a deviation of 18% and 362%, respectively, with respect to the mean value that we have previously derived for the complete time series, $\rm 207\pm 6\,nT\ Hz^{-1/2}$. Although other phenomena may also contribute to the variability of the sub-mHz fluctuations of the spectra, we consider this correlation with the solar wind as one of the physical mechanisms behind the statistical distribution of the fluctuations in the sub-mHz band that we obtained in Fig. 4. It is thus a dependence that future space-borne gravitational-wave detectors will need to take into account, since as we mentioned in section 2.1, the coupling of the magnetic field fluctuations with the magnetic moment of the test masses could produce spurious forces that a gravitational-wave detector would sense as a change in the acceleration noise, and therefore, possibly mistaking it as a gravitational-wave signal .

In Table 3 we characterise the variation of the shape of the magnetic field fluctuations in terms of the solar wind speed. We fitted the spectra of the magnetic field fluctuations to two power laws, one between 20–100 $\mu$Hz and another one from 100 $\mu$Hz to 10 mHz. We have made this differentiation because, as previously explained, different mechanisms act at different time scales, resulting in different spectral indexes for these power laws. The fits were performed by means of a chi-squared minimization, non-linear fit using derivative-free method. For the sake of completeness, in Table 3 we also show the results of the analysis for the Y and Z axis, which are not shown in the figure.

It is interesting to notice that, according to our parametrisation, the main impact of the wind speed is in the parameter B which accounts for the power of the fluctuations in the frequency band described by the power law. The values for the spectral index C are around 1.5, smaller than the value expected, which is 1.65, for the inertial range of interplanetary magnetic field fluctuations. This small difference may come from the fact that in the LPF case, the read-out noise of the magnetometers starts to flatten the spectrum around 1 mHz, and this may result in a slightly less steep curve when we perform the fit. Moreover, the inertial range of interplanetary magnetic field fluctuations starts between 0.1–1 mHz, while the magnetometers read-out noise starts to be dominant around 1 mHz. Therefore, we can not see a large and clear portion of the inertial range part of the interplanetary magnetic field spectrum. With regard to the parameter A, the comparison between the different wind velocity regimes do not show any significant variation, as we expected given that this parameter describes the noise floor of the instrument, which is dominated by the electronics read-out contribution. When we analyze the parameters corresponding to the lower frequency part, we find that the spectral index E values are around 1, which are in good agreement with the $f^{-1}$ behaviour of the interplanetary magnetic field fluctuations at this frequency range – even though the errors are high because we do not have many points for the fit. Finally, the values obtained for the parameter D are not very well fitted, probably because of the small amount of data points available in this frequency range.

To finalise the study of the non-stationarity of the magnetic field fluctuations, we took a closer look at the impact of this contribution during the noise performance runs. Analogously as it is done with on-ground gravitational-wave detectors during science runs, in LISA Pathfinder these runs were periods where the instrument was configured in its more sensitive configuration and kept in data acquisition mode without introducing any calibration signal. For LISA Pathfinder, these noise performance runs took typically 5 to 10 days although it is worth stressing that future space-borne gravitational wave detectors, such as LISA, aim to spend weeks or months in this scientific acquisition mode.

In Figure 6 we analysed the non-stationarities of the low-frequency part of the magnetic field spectrum during the two LISA Pathfinder noise runs that were published in Armano et al. (2016) (April 2016 noise run) and Armano et al. (2018b) (February 2017 noise run). Following the same approach as we have previously shown, we computed the amplitude spectral density of the magnetic field in the frequency span that goes from 20–50 $\mu$Hz. In the top panels of Figure 6 we show the evolution of this value over the duration of each LPF noise run for LPF and ACE satellites. We compare this with the evolution of the time series of the magnetic field itself $B_{x}$ component in Global Solar Ecliptic (GSE) coordinates (second panel) for ACE. In the same panel, we also show the magnetic field $B_{z}$ component as measured in LISA Pathfinder during the same period. The reason to compare LPF’s $B_{z}$ in LPF reference frame with ACE’s $B_{x}$ in GSE coordinates is that LPF’s Z-axis points always towards the solar panel, which is always pointing towards the Sun. Thus, LPF positive Z-axis is roughly equivalent to positive X-axis in GSE coordinates. To do this comparison, LISA Pathfinder data has been low-pass filtered at 1 mHz in order remove the higher frequency noise coming from electronics and the mean substracted in order to exclude the DC magnetic field coming from the spacecraft components. As we previously showed when analysing the components in the spectra, the main features of the long term evolution are driven by the interplanetary component. As we saw, the phenomena associated with the solar wind requires several parameters that detail the characteristics of the interplanetary plasma. In order to provide a comprehensive view of this phenomena during these two particular noise runs we show as well the solar wind speed (third panel) and the proton density (fourth panel), as measured by ACE.

The results show that, during the February 2017 noise run, there are roughly three clear increases in the ASD of the low frequency bin of the magnetic field. These three peaks seem to be caused by high-speed streams in the solar wind speed, i.e, sudden increases in the solar wind speed, a type of event that also carries a decrease in the proton density, as we can also clearly observe in the lowest panel. Regarding the April 2016 noise run, we can see another high-speed stream around April 13th that causes another small peak in the ASD of the low frequency bin of the magnetic field, associated with the corresponding decrease in proton density. Apart from that, there are two events that cause another increase in the time-series of the first panel. According to Armano et al. (2019c), these events would be associated to a Heliospheric current Sheet Crossing (HCSC) and a Magnetic Barrier (MB). We refer the reader to the aforementioned reference for more details on these events.