Charge-induced force-noise on free-falling test masses: results from LISA Pathfinder

TL;DR

This paper presents detailed measurements of charge-induced electrostatic forces on free-falling test masses in space, demonstrating control of acceleration noise to levels suitable for gravitational wave detection.

Contribution

First in-space measurements of charge-induced forces on test masses relevant for space-based gravitational wave detectors, validating electrostatic models and noise control techniques.

Findings

Charge-induced acceleration noise maintained near 1.0 fm/s^2/√Hz

Charge build-up due to cosmic rays characterized

Electrostatic model predictions agree with measurements

Abstract

We report on electrostatic measurements made on board the European Space Agency mission LISA Pathfinder. Detailed measurements of the charge-induced electrostatic forces exerted on free-falling test masses (TMs) inside the capacitive gravitational reference sensor are the first made in a relevant environment for a space-based gravitational wave detector. Employing a combination of charge control and electric-field compensation, we show that the level of charge-induced acceleration noise on a single TM can be maintained at a level close to 1.0 fm/s^2/sqrt(Hz) across the 0.1-100 mHz frequency band that is crucial to an observatory such as LISA. Using dedicated measurements that detect these effects in the differential acceleration between the two test masses, we resolve the stochastic nature of the TM charge build up due to interplanetary cosmic rays and the TM charge-to-force coupling through stray electric fields in the sensor. All our measurements are in good agreement with predictions based on a relatively simple electrostatic model of the LISA Pathfinder instrument.

Tables

Table 1. Table 1: Test mass charging properties

	TM1	TM2
$\dot{q}$	+22.9	+24.5	e s-1
${\lambda }_{\text{eff}}$	$\mathrm{1060\pm 90}$	$\mathrm{1360\pm 130}$	s-1
${\lambda }_{\text{eff(1+2)}}$333Determined from fit to $\Delta g$ with ${\Delta }_x$=1V	2200$\pm$260		s-1

Table 2. Table 2: Estimates of uncompensated Δ x subscript Δ 𝑥 \Delta_{x}

Date	${\Delta }_{x1}$	${\Delta }_{x2}$
2016-110	$-\mathrm{21.02\pm 0.07}$	$-\mathrm{0.67\pm 0.07}$	mV
2016-155	$-\mathrm{20.93\pm 0.04}$	$+\mathrm{0.66\pm 0.03}$	mV
2017-018	$-\mathrm{18.3\pm 0.9}$	—	mV
2017-030	$-\mathrm{18.7\pm 1.4}$	$-\mathrm{0.1\pm 0.2}$	mV

Equations

$$F_x(q)=-\frac{q}{C_{\text{textsc}t}}\left|\frac{\partial C_x}{\partial x}\right|{\Delta }_x,$$

Full text

Sensitive gravitational experiments employ quasi-free-falling isolated test masses as a reference system for the measurement of the local curvature of space-time. Electrostatic free charge and stray potentials introduce unwanted disturbances that can limit measurement precision. The effect is relevant for gravitational wave observatories both in space Shaul2005 ; Antonucci and on-ground Martynov2016 , tests of the equivalence principal Sumner and measurements of relativistic effects on precessing gyroscopes Buchman2011 .

The LISA Pathfinder spacecraft Mcnamara2008 , a technology-demonstration experiment for a space-based gravitational wave observatory, LISA LisaYB2011 , was launched on December 3 2015. The aim of the mission was to demonstrate the ability to fly free-falling test masses in a single spacecraft with a differential acceleration noise below 30  fm s*-2* Hz*-1/2*. The sensitivity of the instrument has far-exceeded its design specification, achieving a level close to the LISA goal from 0.1-100 mHz and around 5  fm s*-2* Hz*-1/2* in the mHz band Armano2016 . In this paper we describe the measurements and techniques used to minimise charge-related electrostatic forces and evaluate their contribution to the differential acceleration noise of the test masses.

The LISA Pathfinder test masses, identical to those for LISA, are 46-mm cubes of mass 1.928 kg made from a gold-platinum alloy. They sit within a 6 degree-of-freedom capacitive position sensor and actuator, the gravitational reference sensor (GRS) Dolesi2003 ; Weber2003 . The masses are separated from the walls of the sensor by gaps of between 2.9 and 4 mm and have no grounding wire. All test mass and sensor surfaces are gold-coated. The large gaps mitigate the impact of surface forces and the absence of a grounding wire eliminates thermal noise associated with mechanical damping that dominates the low-frequency performance of accelerometers in existing geodesy and fundamental physics missions.

The achieved level of sensitivity to the differential acceleration of the test masses level is made possible by an additional high-precision readout along the $x$-axis provided by a laser interferometer Heinzel2003 ; Heinzel2004 ; Audley2011 with a readout noise of 35 fm Hz*-1/2* measured above 60 mHz Armano2016 .

The GRS consists of a system of 12 electrodes for TM position sensing and actuation, and a further 6 for capacitive biasing of the test mass at 100 kHz. Actuation is achieved with audio-frequency sinusoidal voltages. DC or slowly-varying (f$$\simmHz) voltage signals can be applied to measure TM charge and balance stray electrostatic fields, as will be discussed shortly. Voltages on all electrodes originate from the GRS front-end electronics Mance2012 .

High-energy cosmic rays and solar energetic particles, mostly protons, penetrate the spacecraft and instrument shielding depositing charge on the test mass, either by stopping directly or by secondary emission Jafry1996 ; Araujo2005 ; Wass2005 ; Grimani2005 . Limiting charge accumulation on the electrically isolated TMs is needed to control electrostatic forces, the subject of this paper. In LISA Pathfinder, non-contact discharge is achieved by illuminating the sensor and test-mass surfaces with UV light and transferring charge by photoemission Schulte2006 in a similar way to that already demonstrated on GP-B Buchman1995 . A detailed account of the performance of the LISA Pathfinder UV discharge system will be provided in a subsequent article.

As well as providing desired actuation forces, the GRS is a source of unwanted electrostatic disturbances on the test mass Weber2003 ; Araujo2003 . With the TM centered, the dominant source of electrostatic force noise is the interaction between the TM charge, $q$, and stray electric fields represented by an effective potential difference between opposite sides of the TM, $\Delta_{x}$ 111$\Delta_{x}$ is the equivalent uniform single GRS $x$-electrode potential that would give the same average stray field along the $x$ axis.. The resulting force along the $x$-axis, following the notation of Antonucci , is

[TABLE]

where $\frac{\partial{C_{x}}}{\partial{x}}$ is the derivative of a single sensing electrode capacitance with respect to TM displacement along $x$ and $C_{\textsc{t}}$ is the total capacitance of the test mass with respect to the GRS 222Finite element modelling calculates $C_{\textsc{t}}$= 34.2 pF and $\frac{\partial{C_{x}}}{\partial{x}}$= 291 pF m*-1*..

In LISA Pathfinder, the principle science observable is the differential force per unit mass acting on the two TMs, $\Delta g\equiv\frac{F_{2x}}{m_{2}}-\frac{F_{1x}}{m_{1}}$. The measurement is thus sensitive to the in-band fluctuations of both $\Delta_{x}$ and $q$ for the two TM. Force noise is produced by a non-zero charge $q$, coupling with fluctuations in the average potential difference $\Delta_{x}$ and, likewise, stochastic charge fluctuations mixing with any non-zero potential difference. We measure these effects with a number of dedicated techniques.

The test-mass charge, $q$ can be detected in its effect on the TM potential, $\delta V_{TM}=\frac{\delta q}{C_{T}}$, measured by applying sinusoidally varying voltages with amplitude $V_{\textsc{mod}}$ and frequency, $f_{\textsc{mod}}$ on the $x$-axis electrodes, a technique well-demonstrated in ground-based investigations Wass2006 ; Pollack2010 . The resulting force on the TM is $F_{x}(f_{\textsc{mod}})=-4\left|\frac{\partial{C_{x}}}{\partial{x}}\right|V_{\textsc{mod}}V_{\textsc{tm}}$. A continuous measurement provides an extended time series of $q\left(t\right)$ from which the low-frequency behavior of the charge build up can be studied.

To measure the relevant stray potential-difference $\Delta_{x}$, we introduce a potential $\pm V_{\textsc{comp}}$ to each $x$ electrode. Following the method described in Antonucci it is possible to estimate $\frac{\partial{F_{x}}}{\partial{q}}$ as a function of $V_{\textsc{comp}}$ measuring the change in $\Delta g$ as the charge of one test mass is increased in steps by photoemission under UV illumination. By choosing a value for $V_{\textsc{comp}}$ that provides an equal and opposite potential difference to $\Delta_{x}$, we can cancel $\frac{dF_{x}}{dq}$ to first order.

The LISA Pathfinder sensitivity is sufficient that the effects of in-band fluctuations of $q$ and $\Delta_{x}$, described by their power spectral densities (PSDs), $S_{q}$ and $S_{\Delta_{x}}$ are measurable directly in $\Delta g$ by exaggerating $\Delta_{x}$ or $q$ respectively.

Simulations and ground-based laboratory measurements provide indications of the expected behavior of the test mass charge and stray potentials. High-energy physics simulations Araujo2005 ; Wass2005 predict that a net positive charging rate of 40-70 elementary charges per second (e s*-1*) from galactic cosmic rays (GCR) at the minimum of the 11-year solar activity cycle (20-40 e s*-1* at maximum when GCR flux is suppressed) with a charge-noise equivalent to that produced by a rate of single charges, $\lambda_{\textrm{\scriptsize{eff}}}$, of 200-400 s*-1*. The amplitude spectral density (ASD) of the test-mass charge is expected to have the form $S_{q}^{1/2}$=$\frac{e\sqrt{2\lambda_{\text{eff}}}}{2\pi f}$ with an amplitude of 0.6-0.7 fC Hz*-1/2* at 1 mHz.

$\Delta_{x}$ originates both from surface patch potentials within the sensor and the GRS electronics. Measurements with representative systems in laboratory tests have found static levels of up to  100 mV Carbone2003 ; Weber2007 ; Antonucci ; Pollack2008 . Tests before launch showed $S_{\Delta_{x}}^{1/2}$ coming from voltage fluctuations from the spacecraft electronics to be 30 $\mu$V Hz*-1/2* at 1 mHz Praplan2009 . Torsion pendulum measurements using a representative TM and GRS and similar electronics have placed 2-$\sigma$ upper-limits on the total fluctuations, including patch potentials of 80 $\mu$V Hz*-1/2* at 1 mHz and 290 $\mu$V Hz*-1/2* at 0.1 mHz Antonucci .

A $\sim$3-day measurement of $q$ was made injecting $V_{\textsc{mod}}$=3 V at $f_{\textsc{mod}}$=6 and 9 mHz on TM 1 and TM 2 respectively. The charge was calculated by heterodyne demodulation of $\Delta g$, with the applied $V_{\textsc{mod}}$ as the phase reference.

The average charging rates were +22.9 e s*-1* and +24.5 e s*-1* on TM 1 and 2 respectively. Over 10000-s periods, the charge rate is observed to vary by $\pm$2 e s*-1*, caused by a combination of low frequency noise and drift. The fluctuations around the mean charging rates are shown in the lower-left panel of Figure 1; two $>5\sigma$ glitches in the TM 2 charge fluctuations have been removed. $S_{q}$ was calculated with the Welch method, averaging 11 detrended, 40000-s Blackman-Harris (BH) spectral windows with 50%-overlap. A $f^{-2}$ fit was applied to the PSD, down-sampled by a factor 4 to remove data correlated by spectral windowing. The resulting ASD are shown in the upper left panel of Figure 1 and a summary of the results is given in Table 1.

The $f^{-2}$ dependence of the charge PSD and observed absence of correlation in the two charge time-series are consistent with the model of independent Poissonian processes for the two TMs, at least down to 0.1 mHz. A common drift in the charge rate at very-low frequency (visible in the time series as a quadratic dependence after removal of the linear trends due to the average charging rates) correlates well with measurements in the on-board particle monitor and is therefore likely caused by changes in the incident particle flux. The measured charge-noise levels have roughly 5 times the expected noise power, with effective charge rates between 1000 and 1400 s*-1*. Possible causes for an excess are a larger-than-expected number of high-multiplicity charging events produced by very-high energy ($\sim$TeV) cosmic rays, or a large population of low-energy ($\sim$eV) secondaries emitted from TM and GRS surfaces. These two energy regimes are the source of most uncertainty in the charging predictions Araujo2005 .

In this measurement, made some 3-4 years before solar minimum, we find test-mass charging rates within the expected range but measurably different on the two TMs. The difference in the charge rates may originate in the different Volt-scale AC electrostatic fields used for force actuation in the two GRS. If confirmed, this would favor secondary electrons as the source of excess noise. Further measurements characterizing the charge-rate behavior in detail will be the subject of future work.

The spectral density of the charge noise can also be determined from a measurement of $\Delta g$ with an exaggerated potential difference $\Delta_{x}$. The right panel of Figure 1 shows two measurements of the ASD of $\Delta g$ calculated with the same method described for $S_{q}$. The first lasting $\sim$2.5-days with $\Delta_{x_{1}}$=$\Delta_{x_{2}}$=1 V and calculated by averaging 9 overlapping, 40000-s BH windows. The second $\sim$1-day later with both $\Delta_{x}$ compensated to $\lesssim$3 mV (as described below) using 15 windows covering nearly 4-days. We perform a combined fit to the spectra in the frequency range 0.1\leq$$f$$\leq20 mHz assuming a stationary background and an excess due to random charging proportional to $\Delta_{x}$. We find the excess noise in $\Delta g$ in the presence of the applied electric field is compatible with a total effective charge rate $\lambda_{\text{\scriptsize{eff}}_{1}}+\lambda_{\text{\scriptsize{eff}}_{2}}$=2220$\pm$260 s*-1* in good agreement with the dedicated measurement of the charge fluctuations on each test mass shown in Table 1. The charge noise observed in these two measurements separated by 60-days is stationary to better than 10% and shows no measurable departure from a pure Poissonian behavior.

In order to calculate $\Delta_{x}$ and the required compensation voltages, $\frac{dF}{dq}$ was determined from $\Delta g$ using four charge steps of $\sim$0.6 pC. The charge was measured throughout with $V_{\textsc{mod}}$=50 mV and $f_{\textsc{mod}}$=5 mHz. Figure 2 shows $q$ and $\Delta g$ as a function of time through one of these measurements. The charge steps were repeated with $V_{\textsc{comp}}$ of –20, 0, +20 mV and the dependence of $\frac{dF}{dq}$ on $V_{\textsc{comp}}$ confirms our electrostatic model to better than 2%. Two measurements on each GRS were made 45-days apart, the second with $\Delta_{x}$ on TM 1 compensated within 3 mV. At this level, the contribution to $S_{\Delta g}^{1/2}$ from random charging is 0.2  fm s*-2* Hz*-1/2* at 0.1 mHz. A further three measurements were made 7 months later. The first on TM 1 and a final measurement on each TM after reducing the temperature of the sensor from 22 to 11*∘*C.

The calculated values for $\Delta_{x}$, corrected for applied compensation, are given in Table 2 and plotted in Figure 2 against the system pumping time. We note that the rotational stray-field imbalance, $\Delta_{\phi}$ and $\Delta_{\eta}$, that couple TM charge into torque in analogous fashion to Eqn 1, have been measured in the same experiments to be roughly –32 and +36 mV for TM 1 and +119 and +84 mV for TM 2. When considered with the uncompensated values measured for $\Delta_{x}$, roughly –20 mV and 0 mV for TM 1 and TM 2, the stray fields in the GRS would seem to be similar in magnitude to those observed in various measurements on GRS prototype hardware on ground Carbone2003 ; Weber2007 ; Antonucci . Small but significant changes in $\Delta_{x}$ are observed for the two TMs, consistent with drifts of slightly less than mV/month, roughly an order of magnitude below typical drift values observed, for a limited number of samples, on ground Antonucci ; Pollack2008 . This suggests that only very infrequent repetition of the measurement and compensation scheme will be necessary in LISA to keep acceleration noise from TM charge fluctuations below a tolerable level.

The spectral density of stray voltage fluctuations was measured by increasing the test-mass charge and using a similar method to that used to observe the charge noise effect on $S^{1/2}_{\Delta g}$. Figure 3 shows two measurements of the ASD of $\Delta g$. The first over nearly 2-days (average of 6, overlapping, 40000-s windows) with normal levels of TM potential: $\langle V_{\textsc{tm}1}\rangle$=–16 mV and $\langle V_{\textsc{tm}2}\rangle$=–24 mV. followed within a day by a measurement of just over 2-days (8, 40000-s windows) at $\langle V_{\textsc{tm}1}\rangle$=–1066 mV and $\langle V_{\textsc{tm}2}\rangle$=–1058 mV. Fitting a polynomial to the excess in the PSD of $\Delta g$ proportional to $V_{\textsc{tm}}^{2}$ of the form $S_{\Delta g}$=$\left[A^{2}\left(\frac{1\textrm{mHz}}{f}\right)^{2}+B^{2}\left(\frac{1\textrm{mHz}}{f}\right)\right]\left(\frac{V_{\textsc{tm}}}{\textrm{1V}}\right)^{2}$ we find $A$=3.5$\pm$0.7  fm s*-2* Hz*-1/2* and $B$=6.4$\pm$0.5  fm s*-2* Hz*-1/2* ($\chi^{2}$=1.2). This converts to an ASD of the fluctuations in $\Delta_{x}$ of 34$\pm$2 $\mu$V Hz*-1/2* at 1 mHz and 190$\pm$30 $\mu$V Hz*-1/2* at 0.1 mHz. The measured fluctuations in $\Delta_{x}$ are thus clearly resolved and are consistent with the upper limits placed for a GRS prototype on ground Antonucci . They are also consistent with ground measurements of the low-frequency actuation-circuitry voltage noise, which is indistinguishable from stray surface potential fluctuations in our acceleration noise measurement.

Throughout the majority of the operational phase of the mission, $V_{\textsc{tm}}$ has been controlled within $\pm$80 mV of zero by discharging under UV illumination during weekly or fortnightly interruptions to science measurements for orbital correction. The RMS $V_{\textsc{tm}}$ in a two-week measurement period such as that described in Armano2016 is typically $<$40 mV and the contribution to $S_{\Delta g}^{1/2}$ at 0.1mHz is $1.6\pm 0.2$  fm s*-2* Hz*-1/2*.

We have presented the most sensitive measurements of charge-related electrostatic forces on free-falling test masses relevant for sensitive gravitational experiments in space. Technology and mitigation methods developed for minimizing these forces on test masses in capacitive position sensors have been demonstrated and are directly transferable to LISA. Their contribution to the acceleration noise of individual test masses has been reduced to a level of roughly 1  fm s*-2* Hz*-1/2* at 0.1 mHz, compatible with the budgeted allocation for electrostatic forces contributing to the total acceleration noise in LISA LisaYB2011 .

The optimal method of test-mass charge control in a future space-based GW observatory will minimize the disruption to observations and the total contribution to acceleration noise. A continuous UV discharge scheme trades additional random-charging noise for a reduced coupling between $V_{\textsc{tm}}$ and fluctuating stray potentials. Since the latter dominates the charge-related noise in the periodic discharge scenario described in this paper by a factor of nearly 50 in power, it is likely some improvement can be obtained with a continuous scheme. This will be studied in a future experiment.

The methods and technology described here can enable a new generation of instrumentation for gravity-gradiometry and fundamental physics with improved performance in the milli-Hz regime.

This work has been made possible by the LISA Pathfinder mission, which is part of the space-science program of the European Space Agency.

The French contribution has been supported by CNES (Accord Specific de projet CNES 1316634/CNRS 103747), the CNRS, the Observatoire de Paris and the University Paris-Diderot. E. P. and H. I. would also like to acknowledge the financial support of the UnivEarthS Labex program at Sorbonne Paris Cit (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02).

The Albert-Einstein-Institut acknowledges the support of the German Space Agency, DLR. The work is supported by the Federal Ministry for Economic Affairs and Energy based on a resolution of the German Bundestag (FKZ 50OQ0501 and FKZ 50OQ1601).

The Italian contribution has been supported by Agenzia Spaziale Italiana and Instituto Nazionale di Fisica Nucleare.

The Spanish contribution has been supported by Contracts No. AYA2010-15709 (MICINN), No. ESP2013-47637-P, and No. ESP2015-67234-P (MINECO). M. N. acknowledges support from Fundacion General CSIC Programa ComFuturo). F. R. acknowledges support from a Formaci n de Personal Investigador (MINECO) contract.

The Swiss contribution acknowledges the support of the Swiss Space Office (SSO) via the PRODEX Programme of ESA. L. F. acknowledges the support of the Swiss National Science Foundation.

The UK groups wish to acknowledge support from the United Kingdom Space Agency (UKSA), the University of Glasgow, the University of Birmingham, Imperial College London, and the Scottish Universities Physics Alliance (SUPA).

J. I. T. and J. S. acknowledge the support of the U.S. National Aeronautics and Space Administration (NASA).