Precision measurements of the magnetic parameters of LISA Pathfinder test masses

Precision measurements of the magnetic parameters of LISA Pathfinder test masses M Armano,1H Audley,2J Baird,3P Binetruy,3, ∗M Born,2D Bortoluzzi,4E Castelli,5A Cavalleri,6 A Cesarini,7A M Cruise,8K Danzmann,2M de Deus Silva,9I Diepholz,2G Dixon,8R Dolesi,5L Ferraioli,10 V Ferroni,5E D Fitzsimons,11 M Freschi,9L Gesa,12, 13, ∗D Giardini,10 F Gibert,5, 14 R Giusteri,2 C Grimani,7J Grzymisch,1I Harrison,15 M-S Hartig,2G Heinzel,2M Hewitson,2D Hollington,16 D Hoyland,8 M Hueller,5H Inchausp´e,3, 17 O Jennrich,1P Jetzer,18 N Karnesis,3B Kaune,2N Korsakova,19 C J Killow,20 L Liu,5J A Lobo,12, 13, ∗J P L´opez-Zaragoza,12, 13, †R Maarschalkerweerd,15 D Mance,10 V Mart´ın,12, 13 J Martino,3L Martin-Polo,9F Martin-Porqueras,9N Meshksar,10 P W McNamara,1 J Mendes,15 L Mendes,9M Nofrarias,12, 13, ‡S Paczkowski,2M Perreur-Lloyd,20 A Petiteau,3P Pivato,5 E Plagnol,3J Ramos-Castro,21, 13 J Reiche,2D I Robertson,20 F Rivas,12, 13 G Russano,5L Sala,5 D Serrano,12, 13, §J Slutsky,22 C F Sopuerta,12, 13 T Sumner,16 D Texier,9J I Thorpe,22 D Vetrugno,5 S Vitale,5G Wanner,2H Ward,20 P J Wass,16, 17 W J Weber,5L Wissel,2A Wittchen,2and P Zweifel10 1European Space Technology Centre, European Space Agency, Keplerlaan 1, 2200 AG Noordwijk, The Netherlands 2Albert-Einstein-Institut, Max-Planck-Institut f¨ur Gravitationsphysik und Leibniz Universit¨at Hannover, Callinstraße 38, 30167 Hannover, Germany 3APC, Univ Paris Diderot, CNRS/IN2P3, CEA/lrfu, Obs de Paris, Sorbonne Paris Cit´e, France 4Department of Industrial Engineering, University of Trento, via Sommarive 9, 38123 Trento, and Trento Institute for Fundamental Physics and Application / INFN 5Dipartimento di Fisica, Universit`a di Trento and Trento Institute for Fundamental Physics and Application / INFN, 38123 Povo, Trento, Italy 6Istituto di Fotonica e Nanotecnologie, CNR-Fondazione Bruno Kessler, I-38123 Povo, Trento, Italy 7DISPEA, Universit`a di Urbino “Carlo Bo”, Via S. Chiara, 27 61029 Urbino/INFN, Italy 8The School of Physics and Astronomy, University of Birmingham, Birmingham, UK 9European Space Astronomy Centre, European Space Agency, Villanueva de la Ca˜nada, 28692 Madrid, Spain 10Institut f¨ur Geophysik, ETH Z¨urich, Sonneggstrasse 5, CH-8092, Z¨urich, Switzerland 11The UK Astronomy Technology Centre, Royal Observatory, Edinburgh, Blackford Hill, Edinburgh, EH9 3HJ, UK 12Institut de Ci`encies de l’Espai (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193 Cerdanyola del Vall`es, Spain 13Institut d’Estudis Espacials de Catalunya (IEEC), C/ Gran Capit`a 2-4, 08034 Barcelona, Spain 14isardSAT SL, Marie Curie 8-14, 08042 Barcelona, Catalonia, Spain 15European Space Operations Centre, European Space Agency, 64293 Darmstadt, Germany 16High Energy Physics Group, Physics Department, Imperial College London, Blackett Laboratory, Prince Consort Road, London, SW7 2BW, UK 17Department of Mechanical and Aerospace Engineering, MAE-A, P.O. Box 116250, University of Florida, Gainesville, Florida 32611, USA 18Physik Institut, Universit¨at Z¨urich, Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland 19Observatoire de la Cˆote d’Azur, Boulevard de l’Observatoire CS 34229 - F 06304 NICE, France 20SUPA, Institute for Gravitational Research, School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK 21Department d’Enginyeria Electr`onica, Universitat Polit`ecnica de Catalunya, 08034 Barcelona, Spain 22Gravitational Astrophysics Lab, NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771 USA A precise characterization of the magnetic properties of LISA Pathfinder free falling test-masses is of special interest for future gravitational wave observatory in space. Magnetic forces have an important impact on the instrument sensitivity in the low frequency regime below the millihertz. In this paper we report on the magnetic injection experiments performed throughout LISA Pathfinder operations. We show how these experiments allowed a high precision estimate of the instrument magnetic parameters. The remanent magnetic moment was found to have a modulus of (0.245 ± 0.081) nAm2, the x-component of the background magnetic field within the test masses position was measured to be (414 ±74) nT and its gradient had a value of (−7.4±2.1) µT/m. Finally, we also measured the test mass magnetic susceptibility at 5 mHz to be (−3.3723 ±0.0069)×10−5. All results are in agreement with on-ground estimates. ∗Deceased †jplop[email protected] ‡[email protected] §[email protected] I. INTRODUCTION LISA Pathfinder (LPF) [1, 2] was an ESA mission designed as a technology demonstrator for the future gravitational wave observatory in space, LISA [3]. The main goal of the mission was to demonstrate key technologies required to detect gravitational waves in space. In order to do so, the instrument on-board had to demonarXiv:2407.04431v2 [astro-ph.IM] 5 Nov 2024 2 strate a relative acceleration noise between its two test masses (TMs) in nominal geodesic motion at a level of 3×10−14 ms−2Hz−1/2at 1 mHz, a level of precision impossible to achieve with on ground gravitational wave detectors. LPF launched on December 3rd, 2015 and started its scientific operations on the March 1st, 2016 after reaching the Lagrange point L1 of the Earth-Sun system. The mission was divided into two different experiments onboard, the European Space Agency LISA Technology Package (LTP) and the NASA Disturbance Reduction System (DRS). After seventeen months of scientific operations, the mission successfully demonstrated its main scientific goal, surpassing its requirements and achieving a level of acceleration noise below the LISA requirements in its entire measurement frequency band [4, 5]. As important as achieving this demanding level of geodesic free fall was the development of an understanding all the different contributions that build the noise model of the instrument. Several experiments were planned during the LPF operations in order to isolate and evaluate the most important contributions to the acceleration noise budget. With that objective, LISA Pathfinder carried the Data and Diagnostics Subsystem (DDS), which included a temperature measurement subsystem [6, 7], a magnetic diagnostic subsystem [8, 9] and a radiation monitor [10–13]. In this work we will focus on the results of the magnetic diagnostics and, specifically, on the experiments run to characterize the magnetic parameters of the test masses on-board LPF. Precise knowledge of such values is crucial for the future space-borne gravitational wave observatories, since any magnetic perturbation can have a potential impact on the instrument performance through magnetic parasitic forces. This work is organized as follows. In section II we describe the magnetic diagnostic system on-board designed to study and disentangle the nature of the magnetic forces, introduced in section III, that can perturb the test mass motion. In section IV we describe the in-flight magnetic experiments performed to extract the TMs magnetic parameters and we present our conclusions in section V. II. EXPERIMENTAL SETUP A. The magnetic diagnostics subsystem The magnetic diagnostics subsystem on-board LISA Pathfinder was responsible for monitoring the magnetic environment and creating controlled magnetic fields to perturb the test mass motion in order to properly characterize the contribution of magnetic forces to the total instrument noise budget. To achieve these goals, the subsystem was composed by four triaxial magnetometers and two induction coils. FIG. 1. Coordinate reference system for the coil and the test mass. The convention used for the three angles of rotations along each test mass axis is also shown. The coils—see Fig. 1—were able to produce a controlled magnetic field at the TM locations as well as in the magnetometers closest to them. Both circular induction coils, with an average radius of 56.5 mm, were located 85.5 mm away from the test masses and they were attached to the external wall of each vacuum enclosure. The wire winding used to build up the coils were made of a Titanium alloy (Ti6Al4V) and loop around the structure for a total of 2400 turns. The centers of both coils were aligned with the axis, x, joining both TMs centers so that the induced magnetic field had axial symmetry. The four magnetometers were aligned by pairs in the x−yplane of the TMs in order to be able to measure gradients within the spacecraft in both the xand ydirections. Magnetic field gradients along the zdirection could not be measured. The magnetometers continuously measured the evolution of the on-board magnetic field with a precision of 10 nT Hz−1/2. Each fluxgate magnetometer measurement axis consisted of a sensing coil surrounding a second inner drive coil around a high permeability magnetic core material. This meant that all four magnetometers contained active magnetic sensors that had to be located far enough from the test masses for them not to contribute as a source of magnetic parasitic forces. B. Magnetic environment on-board The background magnetic field measured on-board was completely dominated by the contribution from the electronics of the spacecraft units. Among them, the thruster systems were a major contributor, both the cold gas high pressure latch valves (the ones used by ESA) and the colloidal thrusters (the ones operated by NASA). Cold gas thrusters or, more precisely, some permanent magnets in the cold gas thruster subsystem, contributed with roughly the 80% of the measured magnetic field. Although a strong contribution, this remained constant throughout the mission—partially thanks to the high thermal stability reached on-board [7]. This was 3 not the case for the colloidal thrusters, where a persistent slow drift of around 150 nT in the span of 100 days was observed [9]. The main contribution of the magnetic-induced force noise is below the millihertz [14]. In this frequency regime, magnetic field fluctuations are dominated by the interplanetary magnetic field contribution, which can show an important non-stationary component associated with changes in the interplanetary plasma. For instance, variations in the range of 300−500 km s−1in the solar wind velocity were found to be correlated to variations in the magnetic fields amplitude spectral density in the range 20 −50 µHz of around 170 −750 nT Hz−1/2[9]. In what refers to our analysis in the following, we assume that in all the in-flight experiments where we induced magnetic fields with the coils, the background magnetic field (either generated by the spacecraft or due to the interplanetary contribution) can be safely neglected as it was at least one order of magnitude smaller than the ones induced by the coils. The same is true for the gradients of the magnetic fields. C. Forces on-board LPF We evaluate the induced force in the test mass through the ∆gvariable—the principal scientific output of the mission—nominally defined as the differential acceleration between the two TMs in their nominal position [4]. Since the main objective of the ∆gmeasurement is the evaluation of the free fall of the test masses, those forces arising due to the spacecraft dynamics control loop or other forces caused by spacecraft non-inertial reference frame are subtracted in the definition of this parameter [15]. We will assume that during magnetic injections the dominant forces that TMs will feel will be purely of magnetic origin. ∆g, by construction, is the difference in acceleration between TMs along the axis joining them, the same as the spacecraft xaxis by definition and aligned with the xaxis from Fig. 1, ∆g=F2,x MTM1 −F1,x MTM2 ,(1) where MTM is the mass of each TM and Fis the total force each one feels. When calculating the force measured by each TM during magnetic injections, we assume that the TM furthest from the injecting coil will feel a negligible force when compared to the one nearest to the coil. So, when injecting magnetic signals with the coils, we will estimate the magnetic force that the nearest TM feels as Fx≡MTM∆g. The same is true for the torque such that Nη,ϕ ≡ I∆gη,ϕ, where Iis the moment of inertia of a cube and ∆gη,ϕ are the angular accelerations measured by the interferometric system. Only rotations along the yand z axes, ηand ϕrespectively in Fig 1, are obtainable since the instrument is not sensitive to rotations around the x axis as it is aligned with the laser beam. III. MAGNETIC-INDUCED FORCES AND TORQUES IN A FREE FALLING TEST MASS Magnetic fluctuations can couple into the dynamics of the free falling test masses on-board the satellite. In the following we develop the basic equations needed to describe the experiments carried out with the induction coils in LISA Pathfinder. A. A magnetic dipole in a surrounding magnetic field In a first approximation, the free-falling test masses inside LISA Pathfinder can be considered as a magnetic dipole with total magnetic moment density minside of a surrounding magnetic field B. Parameters in bold refer to vectors. The dipole would therefore feel an associated force and torque given by F=⟨(m·∇)B⟩V,(2a) N=⟨m×B+r×(m·∇)B⟩V,(2b) where rdenotes the distance to the TM with respect to the coil. We use the convention ⟨. . . ⟩ ≡ 1 VRV(. . .)d3xto denote the average of the enclosed quantity over the TM volume, V. The total magnetic moment density mis the sum of two components: the remanent magnetic moment density mrwhich depends on the material and manufacturing process and the induced magnetic moment density, mi. The induced magnetic dipole density of any material is proportional to the applied magnetic field, i.e. mi=χ/µoB,(3) where χis the magnetic susceptibility and µ0is the vacuum permeability. The test mass composition is 73% gold, diamagnetic, and 27% platinum, paramagnetic. Given that the dominant material is gold we should expect the TM to behave like any diamagnetic material, opposing the external magnetic field thus, having a negative magnetic susceptibility. Despite this, we leave the sign undetermined in the following derivation. Also, we have implicitly assumed here an isotropic test mass which allows the use of a scalar susceptibility in the previous equation. Next, we assume the test mass magnetized by a slowly oscillating field B(t) = BAC sin(ω t),(4) where BAC is the amplitude of the oscillating magnetic field and ωits angular frequency, we will obtain a magnetization that varies with time accordingly, mi(t). In diamagnetic, paramagnetic and many ferromagnetic materials, the magnetization also varies sinusoidally and in phase with the applied magnetic field with a constant ratio given by the magnetic susceptibility. However, some ferromagnetic materials show a delayed response that is 4 not in phase with the applied field. This phenomena is typically described by considering the in-phase or real, χr, and the out-of-phase or imaginary, χi, components of the magnetic susceptibility such that χ=χr+iχi, where i=√−1. For the case of LPF experiments the most relevant physical mechanism involved in the imaginary susceptibility are eddy currents since this contribution becomes increasingly important with increasing conductivity of the material. However, for most of the experiments in the low frequency regime this component is expected to be orders of magnitude smaller than the real part thus, its contribution can be neglected. In the following, therefore, we refer to the magnetic susceptibility χas equivalent to its real component χr, unless explicitly stated otherwise. Considering both contributions of the magnetization (the remanent and the induced magnetic moments) we expand Eqs. (2) into F=(mr·∇)B+χ µ0 [(B·∇)B]V,(5a) N=mr×B+r×(mr·∇)B+χ µ0 (B·∇)BV. (5b) In order to describe our experiments in the following sections, we need to develop the equations further. First, we will consider the magnetic field as composed by an applied, oscillating magnetic field BAC , and a stable magnetic field B0, divided into an applied time independent DC magnetic field BDC and some environmental background Bback. B=B0+BAC sin(ω t) =Bback. +BDC +BAC sin(ω t).(6) By substituting in Eq. (5a) and factoring out the components in terms of their frequency response to the input signal, we find that the force can be divided into three components: a constant DC term, a term that oscillates at the same frequency of the induced magnetic field 1 ω and a term oscillating at twice the frequency 2 ω F=FDC +F1ω+F2ω,(7) with FDC ="⟨(Mr·∇)B0⟩ +χV µ0⟨(B0·∇)B0⟩+1 2(BAC ·∇)BAC#, (8a) F1ω="(Mr·∇)BAC +χV µ0(B0·∇)BAC+BAC ·∇B0# ×sin(ω t), (8b) F2ω=−χV 2µ0BAC ·∇BACcos(2 ω t),(8c) where Mr=mrV is the remanent magnetic moment and we have assumed homogeneity and stationarity of the test mass properties. Considering that the relative acceleration measurements in LISA Pathfinder are in the xdirection, the only component of the force from Eq. 7 that will be needed is its xcomponent. Analogously, if we manipulate the torque equations a similar result with the three terms before mentioned should appear. B. Estimate of test mass magnetic parameters The evaluation of both the force and torque expressions, Eqs. (5a) and (5b) respectively, implies the calculation of the average of an external magnetic field and its gradient within the TMs volume as expressed by ⟨. . . ⟩. Making use of the induction coils from Fig. 1 we can control the injected field (BAC,DC and ∇BAC,DC) as it can be calculated by means of Amp`ere’s induction laws under the assumptions of coils with negligible thickness and a wire winding of N turns. Thanks to the symmetry of our system, only the xcomponents of the averaged induced fields are non-zero, BAC, DC x, and their gradients along the yand zaxes are 3 orders of magnitude smaller than ∇xBAC, DC x. Furthermore, the magnetic field in the xdirection and its gradient along xare proportional to one another at any given point in space, that is: BAC, DC x=κ∇xBAC, DC x. This factor constant κonly depends on the coil dimensions and the distance from the coil center. Its value can be found analytically for the simple on-axis magnetic field of a coil but it is harder to obtain for the general off-axis magnetic field formula involving elliptic integrals. Thus, its value was calculated numerically to be κ=−0.04487 m for our particular configuration, with negligible uncertainty originated only due to numerical error. We refer 5 the interested reader to Appendix A for more detail on the calculations involved at the TMs location. Finally, the magnetic force is obtained by applying a heterodyne demodulation at the different frequencies of interest of the on-board measurements of the stray TM force (more detail on this in the upcoming section) resulting in the estimators ˆ FDC,x ,ˆ F1ω,x and ˆ F2ω,x. We now proceed to describe how we will estimate the test mass magnetic parameters from the previous generic expressions. a. Magnetic susceptibility The coupling between an induced magnetic field and its gradient with the magnetic susceptibility of the test mass is responsible for the appearance of a force component at twice the injected modulation frequency in Eq. (8c). Our analysis can take advantage of this by extracting the signal at 2ωfrom the measured test mass force, i.e. χ2ω=−2µ0 V ˆ F2ω,x ⟨BAC x⟩·⟨∇xBAC x⟩.(9) This equation provides a direct estimate of the test mass susceptibility decoupled from any other of the magnetic parameters. The notation in Eq. (9) shows explicitly that the estimate of the susceptibility is obtained at twice the injected frequency by demodulating the encoded information in ∆gand comparing it with the predicted TM average magnetic field and gradient. We notice that, in principle, we could use the signal at 2ωto obtain both real and imaginary contributions to the magnetic susceptibility. To estimate the imaginary contribution we would need to look for a 2ωcontribution with a π/2 phase shift with respect the original injection. We will explore this in the discussion of our results in Section IV. b. Remanent magnetic moment The component of the force at the injection frequency, F1ω,x, depends on all the magnetic parameters of the TM. Taking advantage of the fact that all terms in Eq. (8b) depend on BAC xor ∇xBAC x, we can rewrite the expression as follows ˆ F1ω,x =Mr,x +χV µ0 (B0,x +κ∇xB0,x)∇xBAC x. (10) The term in brackets can be related to an effective magnetic moment such that Mefff,x =Mr,x +χV µ0"Bback.,x +BDC x +κ∇xBback.,x +BDC x#, (11) where we have expanded B0,x as explained in Eq. (6). Bback.,x can be considered negligible compared to the injected magnetic field BDC xas its value is expected to be an order of magnitude smaller than the injected fields through the coils. Thus, we have Meff,x ≃Mr,x +2χV µ0BDC x.(12) If the only variable in Eq. (12) is BDC x, then plotting Eq. (10), we will obtain a straight line with an offset that corresponds to the remanent magnetic moment Mr,x and a slope that is proportional to the magnetic susceptibility χat 1ω. Furthermore, when we induce a magnetic field in the TM position, using the coils, apart from direct forces in the xdirection, we are also generating torques, as described in Eq. (5b). Due to the symmetry of the system the term involving the cross product with rwill integrate to zero accross the TM volume due to the alignment between the coil axis and the TMs center resulting in only two components, see Appendix B for details, where only the 1ωterm will be of interest leading to the following equations ˆ Nϕ,1ω=−Mr,y BAC x;ˆ Nη,1ω=Mr,z BAC x.(13) We can conclude that the 1ωoscillation of the torque in ϕis directly related to the remanent magnetic moment along y,Mr,y, while the 1ωoscillation of the torque in ηis directly related to the remanent magnetic moment along z,Mr,z. Therefore, by demodulating the torque at 1ωfor ηand ϕand considering the values of the injected magnetic field BAC x, we will be able to determine Mr,y and Mr,z. c. Background estimates Similarly to the 1ωterm, in Eq. (8a), the expression of the DC force component of the signal involves again all the unknown parameters. If we group all the terms of Eq. (8a) as a function of BDC x, we can rewrite it as ˆ FDC,x ≃χV µ0κBDC x2 +Mr,x κ+χV µ0∇xBback.,x +Bback.,x κBDC x +(M+∇xBback.,x +χV µ0"3Bback.,x∇xBback.,x +1 2BAC x∇xBAC x#), (14) where M+=Mr,x +Mr,y +Mr,z. We have made the assumption that the background magnetic field is the same in all directions, Bback.,x ≃Bback.,y ≃Bback.,z, because we don’t have any a priori information about its value. We also assume that the three components of ∇Bback.,x are equal: ∇xBback.,x ≃ ∇yBback.,x ≃ ∇zBback.,x, which is a worst case scenario since all components contributing 6 to the background gradient would add up when in reality they could partially cancel each other. If the only variable is BDC x, we can observe that ˆ FDC,x has a quadratic form. IV. IN-FLIGHT EXPERIMENTAL CAMPAIGN Soon after LPF started scientific operations, on March 1st, 2016, magnetic experiments were scheduled to extract the magnetic parameters related to the TMs. The experiments consisted in applying an electric current through the coils to induce a magnetic field in the position of the TMs. The applied current in the coils was a sinusoidal signal I(t) = IDC +IAC sin(ω t), where IDC was a constant offset, IAC the amplitude of the sinusoidal signal and ωits angular frequency. The current induces a magnetic field in the surroundings of the coil of the same type B(t) = BDC +BAC sin(ωt). At the beginning of the commissioning period, all subsystems went through an initial checkout procedure. In this initial phase, coil #2 —the one closest to TM2— showed a malfunction. Due to this fact, the injections performed during the operations period were on coil #1, and the only set of injections performed in coil #2 were done with low currents to prevent any possible current leak to other systems. This resulted in a reduction of the precision achievable with coil #2 experiments. Thus, most of the results that will be shown here will be for TM1 if not specified otherwise. We carried out a total of three sets of magnetics injections. The first set was performed on the April 28th and 29th, 2016. The injections of April 28th consisted of applying a sinusoidal signal through the coil #1 with different DC offsets and different AC amplitudes. The injections of April 29th were exactly the same but through coil #2. The second set of injections were done on June 18th, 2016. They consisted of a series of sinusoidal injections at a wider range of both DC offsets and AC amplitudes than the previous ones but performed exclusively in coil #1. The third, and last, set of injections were performed on March 14th, 15th and 16th , 2017. They consisted of very long-lasting signals at high DC offsets and with small AC amplitudes applied exclusively through coil #1. The complete list of experiments is shown in Appendix C. A typical run of magnetic experiments is shown in Fig. 2, where we show the results from the third set of injections during June 18th. The applied currents through the coil at 5 mHz can be seen in the Appendix within Table VI. The three panels display the main variables of interest in our analysis, these are the magnetic field in the x direction, as measured by the closest magnetometer to each coil, the acceleration produced between the TMs due to the presence of these injections and the torque being induced between both TMs along the yaxis. The estimation of magnetic forces experienced by the test masses during the magnetic injections is performed by demodulating the ∆gmeasurements. Terms ˆ FDC,x, FIG. 2. Experiments with coil #1, June 18th, 2016. Top: Bxas measured in the magnetometer closest to the coil, PX. Middle: ∆g.Bottom: Angular acceleration along the rotation angle η. ˆ F1ω,x and ˆ F2ω,x previously defined in Section III B are estimated by applying a heterodyne demodulation at the corresponding frequencies and rescaling the amplitudes obtained by means of the mass of the TMs, MTM = (1.928 ±0.001) kg. Analogously, the same procedure can be extrapolated to the torque by using the moment of inertia of a cube with the side length of the TMs, lTM = (46.000 ±0.005) mm. Since the magnetic field can not be directly measured in the test mass position we must refer to the magnetometers read-out for calibration. We measured the amplitude in the +xmagnetometer of each of the 20 injections of June 18th, 2016 for coil #1. To do so we demodulate the read-out at the injection frequency. By comparing these amplitudes to the ones predicted by Amp`ere’s law with origin at the coil location we found a systematic discrepancy of 11.85 ±0.45%, with the predicted magnetic field being larger than the measurements from the 7 FIG. 3. ˆ F1ω,x on TM1 as a function of the applied AC magnetic field gradient. The different colors correspond to fixed DC values of the injected signal. magnetometers. This systematic discrepancy can have many origins. For the magnetic field prediction, we are assuming a perfect alignment of both the coil and magnetometer, as well as no tilts. In reality, during both manufacturing and integration of the magnetic diagnostics items (coils and magnetometers) there are unavoidable mechanical tolerances. Furthermore, the launch itself adds more uncertainty on the relative distance and tilts between the diagnostics items, being this a systematic error that is difficult to quantify a priori. All the effects combined can add up to the value reported. Since the magnetometer accuracy is 0.5%, we assume that the mismatch originates in the generation of the magnetic field by the coils and we apply this correction to all the calculated magnetic field values that intervene in all the equations derived in the previous section, redefining them for simplicity, i.e. BAC x≡0.8815 BAC x. A. Remanent magnetic moment The estimate of the remanent magnetic moment is obtained through the dependence with the 1ωcomponent of the force expressed in Eq. (10) together with the approximation of Eq. (11), that we recall here for convenience: ˆ F1ω,x =Mr,x +2χV µ0 BDC x∇xBAC x. In order to evaluate this term, during the June 18th run several injections with different AC field gradients were applied to the TMs. By doing so we can evaluate the term in brackets above at different values of the gradient ∇xBAC x. This is shown in Fig. 3 where we display how the 1ωcomponent of the force changes depending on the intensity of the injected AC magnetic field gradient, for fixed DC offsets. Each result in the plot represents an FIG. 4. Effective remanent magnetic moment plotted as a function of the injected DC magnetic field. The result of the fit, for an equation of the type y = mx + n, is m = (−3.380±0.027) ×10−5and n = (0.140 ±0.138) ×10−9Am2. injection at 5 mHz with AC amplitudes applied to the coil of IAC = 0.5,0.8,1.0,1.5 mA. As explained in Section III B, by running the experiment at different DC levels we can further disentangle the dependencies of the parameters inside the brackets and obtain the estimate for the remanent magnetic moment. Table I gathers the linear fits to the results that we also show in Fig. 4. The offset parameter corresponds to the remanent magnetic moment of test mass #1 in the xdirection, Mr,x = 0.140 ±0.138 nAm2. The slope parameter is directly related to the magnetic susceptibility at 1ω. However, the values that were used for DC offsets of ±0.1,0.2 mA came from the injections of April 28th which were performed at different frequencies (3 mHz), than the rest at 5 mHz. Thus, to obtain the value of the susceptibility at 5 mHz, we will use the fit from Fig. 4, but with only the DC values of ±0.75,1.5 mA giving a result of χ5mHz = (−3.3723 ±0.0069) ×10−5. Analogously, to obtain Mr,y and Mr,z, we demoduTABLE I. Coefficients of the fitted lines of Fig. 3 of the type y=Ax +B. The errors of the fits for IDC =±0.1,0.2 are 0 because there were only two points to fit the line. IDC [mA] A [nAm2] B [fN] 1.50 (−31.24 ±0.25) (−31 ±23) 0.75 (−15.48 ±0.15) (−13 ±13) 0.20 (−4.39 ±0) (−23 ±0) 0.10 (−2.04 ±0) (−14 ±0) -0.10 (2.60 ±0) (−1.3±0) -0.20 (5.00 ±0) (13 ±0) -0.75 (15.58 ±0.21 (−12 ±19) -1.50 (31.09 ±0.23) (−1.3±21) 8 late the amplitudes of the torque measurements around the required angles and apply Eq. (13) for the respective injected magnetic fields. This way, we obtained Mr,y = 0.178 ±0.025 nAm2and Mr,z = 0.095 ±0.010 nAm2which we will need for the background estimations. Note the large uncertainty obtained for Mr,x compared with Mr,y and Mr,z. The difference is due to the different methods used to extract them. The former is calculated indirectly as the offset of the linear fit of the slopes of the force at 1ωwhile the other two components of the remanent magnetic moment are calculated directly from the measured torques along ηand ϕ. Rotations along θwould have allowed a more precise value for Mr,x but the interferometric system is not sensitive along such axis as it is aligned with the laser beam. The results lead to a total remanent magnetic moment of: |Mr|= (0.245 ±0.081) nAm2. B. Background magnetic field The induction of forces in the test mass by means of the controlled injection of magnetic fields allow the determination not only of the test mass magnetic parameters but also of environment parameters that contribute to the magnetic force, which is the case of the background magnetic field in the test mass position. We emphasize here that this is the way to estimate of this parameter since the magnetometers are located too far away from the test masses to guarantee a precise estimate of this variable. In order to do so we evaluate Eqs. (14) using the injections of June 18th, 2016. We express the information provided by these runs by displaying the DC component of the measured force as a function of the DC component of the applied magnetic fields. As predicted, we obtain the parabolas in Fig. 5 for different values of the AC amplitude from where we derive the parabola coefficients of Table II. As derived from Eq. (14), the Acoefficient provides a direct estimate of the magnetic susceptibility A=χV µ0κ(15) which, in contrast with other alternative estimates, is not dependent of the injection modulation frequency. The TABLE II. Coefficients of the fitted parabolas of Fig. 5 of the type y=A x2+B x +C. IAC[mA] A [N/T2](10−2) B [N/T](10−8) C [N](10−12) 1.5 (5.99 ±0.11) (−0.67 ±0.39) (1.047 ±0.025) 1.0 (6.003 ±0.025) (0.265 ±0.089) (0.4948 ±0.0058) 0.8 (5.64 ±0.16) (−1.73 ±0.58) (0.323 ±0.038) 0.5 (5.50 ±0.22) (−2.46 ±0.79) (0.144 ±0.052) FIG. 5. ˆ FDC,x on TM1 as a function of the injected DC magnetic field together with their respective fits to an equation of the type y = Ax2+Bx +C. The different colors correspond to fixed AC values of the injected signal. value obtained using Eq. (15) is χDC = (−3.35 ±0.15) × 10−5. With two remaining terms, Band C, we can build a system of equations, being the two unknowns the parameters that define the background magnetic field in the test mass position, i.e. Bback.,x and ∇xBback.,x C=M+∇xBback.,x +χV µ0"3Bback.,x∇xBback.,x +1 2BAC x∇xBAC x#, B=Mr,x κ+χV µ0∇xBback.,x +Bback.,x κ.(16) Solving this quadratic system of equations, we obtain an expression for the background magnetic field and its gradient in the xdirection at the location of the TMs that we can evaluate for each of the four fit values. From the two mathematically available solutions we select the one closer to the estimates of the magnetic field and field gradient obtained during on-ground characterization of the spacecraft, which were 267 nT and -7575 nT/m, respectively [16]. The values that we obtain for the in-flight estimates are Bback.,x = 414 ±74nT and ∇xBback.,x =−7400 ±2100nT/m. C. Magnetic Susceptibility We have already estimated the test mass magnetic susceptibility as a by-product of the estimate of the remanent magnetic moment and the background magnetic field and field gradient. 9 FIG. 6. Magnetic susceptibility of both test masses at different frequencies together with the model predicted by Eq. (17). The DC value of the susceptibility, that is the frequency independent part, was obtained in section IV B and the value of the susceptibility at 5 mHz was measured in section IV A. The rest of the measurements of the magnetic susceptibility of the TMs have been obtained using the 2ωcomponent of the force and Eq. (9). These values correspond to twice the frequency at which the injection was performed. The results of the susceptibilities can be seen in Table III with the value of the frequencies at which they correspond. Using all the injections, from the three different magnetic experiments runs, the frequencies that could be calculated for the magnetic susceptibility were 2, 5, 6, 10 and 30 mHz. In coil #2, for TM2 results, only the injections from the 29th of April were performed, which were at frequencies 1, 3, 5 mHz meaning that only three susceptibility values could be obtained at twice their frequency. According to [17], the AC magnetic susceptibility of LPF TMs can be approximated at low frequencies as χ(ω)≃χDC +−iωτe 1 + iωτe ,(17) where χDC is the frequency independent term of the susceptibility and with τebeing the magnetic susceptibility cut, i.e., the frequency at which the real and imaginary part of the magnetic susceptibility have the same TABLE III. Susceptibility values of both TMs obtained at 2ω using Eq. (9) for all the injection frequencies. Frequency [mHz] χTM1 (10−5)χTM2 (10−5) DC (−3.35 ±0.15) - 2 (−3.43 ±0.58) (−4.0±2.3) 5 (−3.3723 ±0.0069) - 6 (−2.65 ±0.62) (−2.64 ±0.92) 10 (−3.35 ±0.12) (−3.833 ±0.057) 30 (−4.73 ±0.34) - value. For LPF, this value was measured on-ground to be τe= (2π630)−1Hz−1[18]. If we now plot the measured values of the magnetic susceptibility for TMs 1 and 2 along with the curve in Eq. (17), we obtain the plot shown in Fig. 6. We can see that all magnetic susceptibility results from Table III are compatible within their uncertainty ranges as predicted from Eq. (17), except for the one at 30 mHz. There were only three signal injected at 15 mHz, which had a low SNR and, also, showed an unexpected linear drift that had to be subtracted. We think this explains the systematics affecting the demodulation of its second harmonic at 30 mHz. With respect to the results for TM2, we computed them for completeness but we recall that coil 2 suffered a malfunction at the beginning of operations which could explain the systematic error observed in the derived parameters. The results shown were obtained by heterodyne demodulation of the ∆gsignal with a sinusoidal signal inphase. However, if we made the latter be out of phase by π/2 one would expect the amplitudes measured by this method to be zero only if there were no imaginary component. Thus, the imaginary susceptibility can be obtained demodulating at 2ω, in quadrature. In the frequency regime that we are working in, this value is expected to be orders of magnitude smaller than the real susceptibility. The results that we obtain for the imaginary susceptibility at 10 mHz were consistent with zero, χi= (0.0±1.8) ×10−6. With the limits determined by the force sensitivity of the demodulation around the tenths of femtoNewtons. Thus, we can only confirm that the values of the imaginary susceptibility of the TMs are below |χi|<1.8×10−6at 10 mHz. At the rest of frequencies the result gave a less precise upper bound for the imaginary susceptibility. Finally, we can evaluate the prediction of our force model from Eq. (7) in comparison with the measured acceleration during injections by making use of the extracted magnetic parameters within this article. To do so, we have selected a single injection from all the ones of June 18th, 2016 with a DC offset of 0.75 mA and an AC amplitude of 1.5 mA. We have used these values for the corresponding magnetic field DC and AC calculations together with the magnetic parameters found in the previous sections and substituted into Eqs. (8). The results can be seen in Fig. 7 where, together with the total model force, we can see plotted the different force contributions FDC,F1ωand F2ω. The predicted force and the data match with a residual difference between them of (0.0±1.9) ×10−13 N. Due to the presence of a linear drift within the data originated from the desynchronization between the clocks of the LPF measurement systems, one for the diagnostics and a different one for the optical metrology system, we are able to observe a leftovers signal within the residual data.