††thanks: Deceased

Present address: ]Gravitational Astrophysics Lab, NASA Goddard Space Flight
Center, 8800 Greenbelt Road, Greenbelt, MD 20771 USA ††thanks: Deceased
Present address: ]isardSAT SL, Marie Curie 8-14, 08042 Barcelona, Catalonia,
Spain Present address: ]Institut für Theoretische Physik, Universität
Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany ††thanks: Deceased
Present address: ]Universidad Loyola Andalucía, Department of Quantitative
Methods, Avenida de las Universidades s/n, 41704 Dos Hermanas, Sevilla, Spain.
Present address: ]INAF Osservatorio Astronomico di Capodimonte, I-80131
Napoli, Italy

# Charging of free-falling test masses in orbit due to cosmic rays: results
from LISA Pathfinder

M. Armano  European Space Technology Centre, European Space Agency, Keplerlaan
1, 2200 AG Noordwijk, The Netherlands  H. Audley  Albert-Einstein-Institut,
Max-Planck-Institut für Gravitationsphysik und Leibniz Universität Hannover,
Callinstraße 38, 30167 Hannover, Germany  J. Baird  Université Paris Cité,
CNRS, Astroparticule et Cosmologie, F-75013 Paris, France  P. Binetruy  IRFU,
CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France  M. Born  Albert-
Einstein-Institut, Max-Planck-Institut für Gravitationsphysik und Leibniz
Universität Hannover, Callinstraße 38, 30167 Hannover, Germany  D. Bortoluzzi
Department of Industrial Engineering, University of Trento, via Sommarive 9,
38123 Trento, and Trento Institute for Fundamental Physics and Application /
INFN  E. Castelli  [  Dipartimento di Fisica, Università di Trento and Trento
Institute for Fundamental Physics and Application / INFN, 38123 Povo, Trento,
Italy  A. Cavalleri  Istituto di Fotonica e Nanotecnologie, CNR-Fondazione
Bruno Kessler, I-38123 Povo, Trento, Italy  A. Cesarini  DISPEA, Università di
Urbino “Carlo Bo”, Via S. Chiara, 27 61029 Urbino/INFN, Italy  A. M Cruise
The School of Physics and Astronomy, University of Birmingham, Birmingham, UK
K. Danzmann  Albert-Einstein-Institut, Max-Planck-Institut für
Gravitationsphysik und Leibniz Universität Hannover, Callinstraße 38, 30167
Hannover, Germany  M. de Deus Silva  European Space Astronomy Centre, European
Space Agency, Villanueva de la Cañada, 28692 Madrid, Spain  I. Diepholz
Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik und
Leibniz Universität Hannover, Callinstraße 38, 30167 Hannover, Germany  G.
Dixon  The School of Physics and Astronomy, University of Birmingham,
Birmingham, UK  R. Dolesi  Dipartimento di Fisica, Università di Trento and
Trento Institute for Fundamental Physics and Application / INFN, 38123 Povo,
Trento, Italy  L. Ferraioli  Institut für Geophysik, ETH Zürich,
Sonneggstrasse 5, CH-8092, Zürich, Switzerland  V. Ferroni  Dipartimento di
Fisica, Università di Trento and Trento Institute for Fundamental Physics and
Application / INFN, 38123 Povo, Trento, Italy  E. D. Fitzsimons  The UK
Astronomy Technology Centre, Royal Observatory, Edinburgh, Blackford Hill,
Edinburgh, EH9 3HJ, UK  M. Freschi  European Space Astronomy Centre, European
Space Agency, Villanueva de la Cañada, 28692 Madrid, Spain  L. Gesa  Institut
de Ciències de l’Espai (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n,
08193 Cerdanyola del Vallès, Spain  D. Giardini  Institut für Geophysik, ETH
Zürich, Sonneggstrasse 5, CH-8092, Zürich, Switzerland  F. Gibert  [
Dipartimento di Fisica, Università di Trento and Trento Institute for
Fundamental Physics and Application / INFN, 38123 Povo, Trento, Italy  R.
Giusteri  Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik
und Leibniz Universität Hannover, Callinstraße 38, 30167 Hannover, Germany  C.
Grimani  DISPEA, Università di Urbino “Carlo Bo”, Via S. Chiara, 27 61029
Urbino/INFN, Italy  J. Grzymisch  European Space Technology Centre, European
Space Agency, Keplerlaan 1, 2200 AG Noordwijk, The Netherlands  I. Harrison
European Space Operations Centre, European Space Agency, 64293 Darmstadt,
Germany  M.-S. Hartig  Albert-Einstein-Institut, Max-Planck-Institut für
Gravitationsphysik und Leibniz Universität Hannover, Callinstraße 38, 30167
Hannover, Germany  G. Heinzel  Albert-Einstein-Institut, Max-Planck-Institut
für Gravitationsphysik und Leibniz Universität Hannover, Callinstraße 38,
30167 Hannover, Germany  M. Hewitson  Albert-Einstein-Institut, Max-Planck-
Institut für Gravitationsphysik und Leibniz Universität Hannover, Callinstraße
38, 30167 Hannover, Germany  D. Hollington  High Energy Physics Group, Physics
Department, Imperial College London, Blackett Laboratory, Prince Consort Road,
London, SW7 2BW, UK  D. Hoyland  The School of Physics and Astronomy,
University of Birmingham, Birmingham, UK  M. Hueller  Dipartimento di Fisica,
Università di Trento and Trento Institute for Fundamental Physics and
Application / INFN, 38123 Povo, Trento, Italy  H. Inchauspé  [  Université
Paris Cité, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France  O.
Jennrich  European Space Technology Centre, European Space Agency, Keplerlaan
1, 2200 AG Noordwijk, The Netherlands  P. Jetzer  Physik Institut, Universität
Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland  N. Karnesis
Université Paris Cité, CNRS, Astroparticule et Cosmologie, F-75013 Paris,
France  B. Kaune  Albert-Einstein-Institut, Max-Planck-Institut für
Gravitationsphysik und Leibniz Universität Hannover, Callinstraße 38, 30167
Hannover, Germany  C. J. Killow  SUPA, Institute for Gravitational Research,
School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK
N. Korsakova  Université Paris Cité, CNRS, Astroparticule et Cosmologie,
F-75013 Paris, France  J. A. Lobo  Institut de Ciències de l’Espai (ICE,
CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193 Cerdanyola del Vallès,
Spain  J. P. López-Zaragoza  Institut de Ciències de l’Espai (ICE, CSIC),
Campus UAB, Carrer de Can Magrans s/n, 08193 Cerdanyola del Vallès, Spain  R.
Maarschalkerweerd  European Space Operations Centre, European Space Agency,
64293 Darmstadt, Germany  D. Mance  Institut für Geophysik, ETH Zürich,
Sonneggstrasse 5, CH-8092, Zürich, Switzerland  V. Martín  Institut de
Ciències de l’Espai (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193
Cerdanyola del Vallès, Spain  J. Martino  Université Paris Cité, CNRS,
Astroparticule et Cosmologie, F-75013 Paris, France  L. Martin-Polo  European
Space Astronomy Centre, European Space Agency, Villanueva de la Cañada, 28692
Madrid, Spain  F. Martin-Porqueras  European Space Astronomy Centre, European
Space Agency, Villanueva de la Cañada, 28692 Madrid, Spain  P. W. McNamara
European Space Technology Centre, European Space Agency, Keplerlaan 1, 2200 AG
Noordwijk, The Netherlands  J. Mendes  European Space Operations Centre,
European Space Agency, 64293 Darmstadt, Germany  L. Mendes  European Space
Astronomy Centre, European Space Agency, Villanueva de la Cañada, 28692
Madrid, Spain  N. Meshksar  Institut für Geophysik, ETH Zürich, Sonneggstrasse
5, CH-8092, Zürich, Switzerland  M. Nofrarias  Institut de Ciències de l’Espai
(ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193 Cerdanyola del
Vallès, Spain  S. Paczkowski  Albert-Einstein-Institut, Max-Planck-Institut
für Gravitationsphysik und Leibniz Universität Hannover, Callinstraße 38,
30167 Hannover, Germany  M. Perreur-Lloyd  SUPA, Institute for Gravitational
Research, School of Physics and Astronomy, University of Glasgow, Glasgow, G12
8QQ, UK  A. Petiteau  Université Paris Cité, CNRS, Astroparticule et
Cosmologie, F-75013 Paris, France  IRFU, CEA, Université Paris-Saclay, F-91191
Gif-sur-Yvette, France  E. Plagnol  Université Paris Cité, CNRS,
Astroparticule et Cosmologie, F-75013 Paris, France  J. Ramos-Castro
Department d’Enginyeria Electrònica, Universitat Politècnica de Catalunya,
08034 Barcelona, Spain  J. Reiche  Albert-Einstein-Institut, Max-Planck-
Institut für Gravitationsphysik und Leibniz Universität Hannover, Callinstraße
38, 30167 Hannover, Germany  F. Rivas  [  Institut de Ciències de l’Espai
(ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193 Cerdanyola del
Vallès, Spain  D. I. Robertson  SUPA, Institute for Gravitational Research,
School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK
G. Russano  [  Dipartimento di Fisica, Università di Trento and Trento
Institute for Fundamental Physics and Application / INFN, 38123 Povo, Trento,
Italy  J. Slutsky  Gravitational Astrophysics Lab, NASA Goddard Space Flight
Center, 8800 Greenbelt Road, Greenbelt, MD 20771 USA  C. F. Sopuerta  Institut
de Ciències de l’Espai (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n,
08193 Cerdanyola del Vallès, Spain  T. J. Sumner  High Energy Physics Group,
Physics Department, Imperial College London, Blackett Laboratory, Prince
Consort Road, London, SW7 2BW, UK  Department of Physics, P.O. Box 118440,
University of Florida, Gainesville, Florida 32611, USA  D. Texier  European
Space Astronomy Centre, European Space Agency, Villanueva de la Cañada, 28692
Madrid, Spain  J. I. Thorpe  Gravitational Astrophysics Lab, NASA Goddard
Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771 USA  D. Vetrugno
Dipartimento di Fisica, Università di Trento and Trento Institute for
Fundamental Physics and Application / INFN, 38123 Povo, Trento, Italy  S.
Vitale  Dipartimento di Fisica, Università di Trento and Trento Institute for
Fundamental Physics and Application / INFN, 38123 Povo, Trento, Italy  G.
Wanner  Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik
und Leibniz Universität Hannover, Callinstraße 38, 30167 Hannover, Germany  H.
Ward  SUPA, Institute for Gravitational Research, School of Physics and
Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK  P. J. Wass
<EMAIL_ADDRESS>High Energy Physics Group, Physics Department, Imperial College
London, Blackett Laboratory, Prince Consort Road, London, SW7 2BW, UK
Department of Mechanical and Aerospace Engineering, MAE-A, P.O. Box 116250,
University of Florida, Gainesville, Florida 32611, USA  W. J. Weber
Dipartimento di Fisica, Università di Trento and Trento Institute for
Fundamental Physics and Application / INFN, 38123 Povo, Trento, Italy  L.
Wissel  Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik
und Leibniz Universität Hannover, Callinstraße 38, 30167 Hannover, Germany  A.
Wittchen  Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik
und Leibniz Universität Hannover, Callinstraße 38, 30167 Hannover, Germany  P.
Zweifel  Institut für Geophysik, ETH Zürich, Sonneggstrasse 5, CH-8092,
Zürich, Switzerland

(draft of )

###### Abstract

A comprehensive summary of the measurements made to characterize test mass
charging due to the space environment during the LISA Pathfinder mission is
presented. Measurements of the residual charge of the test mass after release
by the grabbing and positioning mechanism, show that the initial charge of the
test masses was negative after all releases, leaving the test mass with a
potential in the range $-12$ mV to $-512$ mV. Variations in the neutral test
mass charging rate between 21.7 and 30.7 e s-1 were observed over the course
of the 17-month science operations produced by cosmic ray flux changes
including a Forbush decrease associated with a small solar energetic particle
event. A dependence of the cosmic ray charging rate on the test mass potential
between $-30.2$ and $-40.3$ e s-1 V-1 was observed resulting in an equilibrium
test mass potential between 670 and 960 mV, and this is attributed to a
contribution to charging from low-energy electrons emitted from the gold
surfaces of the gravitational reference sensor. Data from the on-board
particle detector show a reliable correlation with the charging rate and with
other environmental monitors of the cosmic ray flux. This correlation is
exploited to extrapolate test mass charging rates to a 20-year period giving
useful insight into the expected range of charging rate that may be observed
in the LISA mission.

Free-falling test masses in precision experiments are susceptible to parasitic
electrostatic forces and torques [1], with significant contributions from the
interaction between stray electric fields and any electrostatic charge [2, 3,
4]. These effects can make up a sizeable fraction of the noise budget for
future gravitational wave observatories such as the Laser Interferometer Space
Antenna (LISA) [5] which will have a measurement bandwidth between 0.1 mHz and
1 Hz. LISA Pathfinder (LPF) [6] was a European Space Agency mission to
demonstrate technology for LISA, especially in measuring the low-frequency
acceleration noise acting on the test masses coming from the spacecraft
environment [7, 8].

In this paper we present measurements of the test-mass charging due to the
space environment over the course of the LISA Pathfinder mission and discuss
the observed variations under a number of different operational conditions.
The paper is organized as follows: in Section I we review the level of
charging due to cosmic rays predicted before the launch of LPF. We describe
the hardware on board relevant to measuring the test-mass charge and
monitoring the space environment responsible for charging. In Section II we
describe the method for evaluating the test mass charge on LPF. We present an
overview of the charge history of the test masses during the mission including
the remnant charge of the test mass after release by the grabbing and
positioning mechanism. In Section III evidence of the variation in test-mass
charging caused by electrostatic effects in the LPF Gravitational Reference
Sensor are presented. Section IV describes the observed dependence of the
charging rate on the cosmic ray flux and correlation with the on board
radiation monitor. Section V presents an extrapolation of the LPF charging
rate across a full solar cycle and finally in Section VI we present our
conclusions.

##  I Introduction

###  I.1 Charge accumulation and discharge

The test mass charge accumulates due to the energetic, charged-particle space
environment, dominated by protons and 4He. These particles which may be
galactic cosmic rays (GCR) or solar energetic particles (SEP) are able to
penetrate the shielding of the spacecraft and deposit charge on the test
masses if they have an energy above $\sim$100 MeV nucl-1 [1].

Prior to launch, the expected charge rate from both cosmic rays and solar
energetic particles had been evaluated using a number of simulations [9, 10,
11, 12, 13] using Geant4 [14] and FLUKA [15] high energy physics simulation
toolkits. Predictions of the mean charging rate were between +38 and +44 e s-1
(where e is the proton charge) in solar minimum galactic cosmic ray conditions
falling to roughly half that during solar maximum. LISA Pathfinder began
science operations in March 2016 as the solar cycle was approaching solar
minimum conditions.

The mean charging rate is the result of both negative and positive charging
with a range of multiplicity events of both polarities. Hence the predicted
stochastic noise behaviour in the charging rate was higher than expected from
the mean rate. The noise is characterised by an effective rate of single
charges, $\lambda_{\mathrm{eff}}$, that would create the observed noise.
Simulations predicted a value for $\lambda_{\mathrm{eff}}\sim 200$-$400$ s-1.

The first in-orbit observations of the LPF test-mass charging rate and its
stochastic noise based on a measurement made on 2016-04-20 were presented in
[16]. Charging rates were found to be 22.9-24.5 e s-1 and
$\lambda_{\mathrm{eff}}=1100$-$1400$ s-1 The observation of a significantly
higher charge noise, together with the charge rate measurements presented here
have motivated a continuing effort to characterize in detail the physics of
the charging process [17, 18, 19, 20, 21]. In particular, reported for the
first time here is experimental evidence that a stable equilibrium for the
test mass charging due to the space environment can be reached. This
equilibrium can be explained by a population of low-energy (eV) electrons
created by secondary emission at the TM and EH surfaces and the electric
fields within the GRS [21].

###  I.2 The Gravitational Reference Sensor

The LPF payload consisted of two Gravitational Reference Sensors (GRS), each
comprised of a $\sim$2 kg gold-plated Au/Pt test mass surrounded by a Au-
plated electrode housing (EH) consisting of 18 electrodes as illustrated in
Fig. 1. Twelve electrodes separated from the test mass by gaps of 2.9–4.0 mm
were used for position sensing and to apply actuation force and torques to the
test mass [16]. Actuation voltages were applied at ac audio frequencies
nominally balanced around the ground potential of the GRS front-end
electronics (FEE). Slowly varying or dc voltages could also be applied as
described below in order to measure test mass charge. A further six electrodes
on the $y$ and $z$ axes of the test mass carried a 4.88 V amplitude 100 kHz ac
voltage which induced a 0.6 V 100 kHz test mass potential bias used for
capacitive position sensing. The test mass was completely isolated from its
surroundings with no electrical connection to ground. In order to reduce
disturbances from electrostatic forces, the test mass charge had to be kept
near zero. In LPF, this was achieved by illuminating the test mass and sensor
with UV light at 253.7 nm from a low-pressure mercury discharge lamp. The
resulting photoelectrons transfer charge to or from the test mass as required.
The detailed behaviour of the discharging system is discussed in [22].

Figure 1: Diagram of LISA Pathfinder GRS showing 46 mm cubic test mass
surrounded by capacitive-sensing bias injection electrodes in blue, $x$/$\phi$
sensing and actuation electrodes in red, $y$/$\theta$ in green and $z$/$\eta$
in yellow.

###  I.3 Electrostatic forces

The test mass charge was evaluated through its interaction with the electric
fields in the GRS. In LPF, the most sensitive axis for force detection is the
$x$-axis connecting the two test masses. The electrostatic force acting on the
test mass (TM) in the $x$ axis is [1]

$F_{x}=\frac{1}{2}\sum\frac{\partial{C_{i}}}{\partial{x}}\left(V_{i}-V_{\textsc{tm}}\right)^{2}\,,$
(1)

where the sum is defined over all $i$ equipotential surface elements of the
sensor with potential $V_{i}$ relative to ground and capacitance with respect
to the test mass, $C_{i}$. All test mass surfaces surfaces are assumed to be
at the same potential, $V_{\textsc{tm}}$, determined by the potential of all
sensor surfaces and the free charge on the test mass $q$, according to

$V_{\textsc{tm}}=\frac{\sum_{i}C_{i}V_{i}}{C_{T}}+\frac{q}{C_{T}}\,,$ (2)

where $C_{T}$ is the total capacitance of the test mass with respect to the
surrounding surfaces (34.2 pF for each test mass on LPF).
Except for dedicated characterization experiments, voltages applied to
electrodes are chosen to give no in-band fluctuation of $V_{\textsc{tm}}$;
however, unavoidable stray fields and in-band fluctuations of the potentials
of the TM and surrounding surfaces give rise to forces that can, if left
unchecked, compromise the acceleration noise performance of the instrument
[3].

In order to measure the test mass potential, a force with frequency
$f_{\textsc{mod}}$ is excited proportional to $V_{\textsc{tm}}$ according to
Eq. (1). Voltages $V_{i}=V_{\textsc{mod}}\sin 2\pi f_{\textsc{mod}}t$ are
applied with opposite phase to the sensing electrodes on either side of the
test mass which have equal and opposite capacitance derivatives
$\frac{\partial{C_{x}}}{\partial{x}}$ (291 pF m-1). To maximize the precision
of the measurement, $f_{\textsc{mod}}$ is chosen to in the sensitive, mHz band
of the instrument. The resulting force along the $x$-axis, assuming all other
potentials are zero, is:

$F_{x}=-4V_{\textsc{mod}}\frac{q}{C_{T}}\left|\frac{\partial{C_{x}}}{\partial{x}}\right|\sin{2\pi
f_{\textsc{mod}}}\,.$ (3)

Equivalently, applying modulating voltages to the diagonally opposite
electrodes produces a torque, $\Gamma$. For example in $\phi$ rotation:

$\Gamma_{\phi}=-4V_{\textsc{mod}}\frac{q}{C_{T}}\left|\frac{\partial{C_{x}}}{\partial{\phi}}\right|\sin{2\pi
f_{\textsc{mod}}}\,.$ (4)

Mixing of the stray potential patches on the electrodes with the charge
measurement modulation voltages will produce an additional force or torque
term modulated at $f_{\textsc{mod}}$. This creates an unknown charge offset
which, to first order is proportional to the average stray potential of the
electrodes used to excite the charge measurement force or torque.

###  I.4 Radiation Monitor

The radiation monitor (RM) on board LPF was specifically designed to monitor
the interplanetary GCR and SEP environment responsible for charge build-up on
the test masses [23]. Two large area (147 mm2), 300 $\mu$m-thick silicon PIN
photodiodes were arranged in a telescopic configuration in a copper enclosure
to give shielding from particles with energy below roughly 100 MeV nucl-1. The
deposited energy resolution was around 15 keV which was sufficient to allow
differentiation between cosmic ray particles—most of which are close to
minimally ionizing in the silicon detectors—and SEPs which result in larger
energy deposits. The intention was to use the RM to measure the dependence of
TM charging on count rate in both GCR and SEP flux conditions. LPF experienced
only one SEP event which did not produce a particle flux enhancement in the
energy range relevant for charging. A Forbush decrease associated with that
event was observed, however [24]. The RM also produced a full history of the
relative fluctuations in the the ambient cosmic ray flux with good counting
statistics. With this data it has been possible to estimate the solar
modulation parameter [24], and observe modulation of the cosmic ray flux due
to solar magnetic field fluctuations [25].

##  II Charge measurement campaign

###  II.1 Experimental details

The test mass charge was measured on-board LPF using both force and torque
excitations as described above. The charge on both test masses could be
measured simultaneously using different modulation frequencies on each mass.
The measurement precision depends on the amplitude of the applied voltages
($V_{\textsc{mod}}$) and the force (torque) sensitivity in the chosen degree
of freedom. The most sensitive measurements were made using the on-board laser
interferometer [26, 27] to measure their differential motion in either $x$ or
$\phi$ (around $z$). The force on the test masses was recovered from their
differential (angular) acceleration, accounting for control forces (torques),
and in the case of the $x$-axis measurements, tilt-to-length and non-inertial
effects due to spacecraft rotations. Typically the sensitivity to the force
(torque) excitation was limited by direct stray forces (torques) on the test
mass. Along the $x$-axis, Brownian force noise on the test masses created a
differential acceleration noise which decreased from around 10 to 3 fm
s${}^{-2}\,\rm{Hz}^{-1/2}$ in the frequency range from 0.5 to 30 mHz over the
course of the mission [22]. In the $\phi$ rotation, the differential angular
acceleration was typically 300 frad s-2 Hz-1/2 for $f_{\textsc{mod}}=5$ mHz,
comparable to the level expected from angular readout sensitivity. The charge
was calculated by applying Eqs. (3) or (4) to the force or torque time series,
demodulating at $f_{\textsc{mod}}$. The residual accelerations mentioned above
correspond to a TM charge (potential) measurement uncertainty of 0.24 and 0.77
fC (7.0 $\mu$V and 23$\mu$V) in a 200 s measurement using
$V_{\textsc{mod}}=50$ mV on the $x$ and $\phi$ degrees of freedom
respectively.

A systematic offset between test mass charge measurements made in the $x$ and
$\phi$ degrees was observed during the mission. In a dedicated experiment to
determine the magnitude of the effect, the test mass charge was measured
sequentially with excitations along the two axes. The resulting potential
difference between the two methods was $20.6\pm 0.3$ mV for TM 1 and $11.6\pm
0.5$ mV for TM 2 while the measurement of an applied polarization of the test
mass was equal for both methods.

The $x$ and $\phi$ charge measurement methods use the same electrodes with the
same average stray potential to excite a force or torque on the test mass and
therefore to first order, the offset in charge measurement should be equal.
The difference observed could originate from the distribution of the patches
over the electrode. In the $\phi$ measurement, patches would have a varying
weighting in their contribution to the measurement offset according to the
lever arm they exert on the test mass rotation. The offset was measured only
once during the LPF mission and therefore it is not possible to make
definitive statement about its stability over time. However, another effect of
voltage patches, charge to force-noise coupling has been found to be stable
over the course of the mission [3].

Extended measurements of the test mass charge allow its time dependence to be
studied. The charging rate can be estimated by taking a linear fit to the time
series. The uncertainty in the charge rate can be improved by increasing
$V_{\textsc{mod}}$ or by measuring for longer. In practice, the latter is
limited by low-frequency, $1/f$ noise arising from the charging shot noise
accumulation and fluctuations in the particle flux responsible for charging.
These have both been measured and reported in [3] and [24], respectively.
Typically, $V_{\textsc{mod}}$ was chosen to be relatively small (10-50 mV) so
as not to excite large signals in the instrument, and measurement durations
were short compared to cosmic ray fluctuations. The dominating uncertainty in
estimating the charge rate was therefore the test-mass acceleration noise
limiting the charge measurement. For a charge measurement using the TM
$x$-axis as described above, the charge-rate noise can be written as
$S^{1/2}_{R}=mS^{1/2}_{a}(f_{\textsc{mod}})C_{T}/4V_{\textsc{mod}}\left|\frac{\partial
C_{x}}{\partial x}\right|2\pi f_{\textsc{mod}}$ where
$S^{1/2}_{a}(f_{\textsc{mod}})$ is the test mass acceleration noise spectral
density at the modulation frequency and $m$ is the mass of the TM. Integrating
over a 1-hr time interval results in a charge rate resolution of 0.34 e s-1.

The LPF RM tracked variations in the particle flux at the spacecraft. For
small changes in flux, the change in charge rate is expected to be
proportional. Assuming a count rate of 9 Counts s-1, the error on the average
count rate in a 1-hr time period is 0.05 Counts s-1. Results presented in
Section IV find a scale factor between charge rate and count rate of around 5
e Count-1 which when applied to the RM count rate resolution gives a
corresponding charge rate estimate of 0.25 e s-1 over 1 hour, comparable to
the charge rate resolution using the $x$-axis with $V_{\textsc{mod}}=50$ mV.
Although the exact values depend on particle flux conditions, the RM is
expected to be a useful diagnostic to confirm the origin of charge rate
fluctuations observed during test mass measurements.

###  II.2 Charge measurement results

Figure 2: LPF test-mass charge as function of time throughout the mission.
Dashed lines indicate the upper and lower limits for the charge during nominal
operations at 3$\times 10^{7}$ e $\approx$ 140 mV. Abrupt decreases in the
test mass charge are the result of charge control operations using the onboard
UV illumination system. Large excursions of the charge occurred at day 152,
425 and 567 during test mass charging and noise investigations.

Charge measurements were made throughout the LPF mission as part of dedicated
science investigations and as part of routine maintenance operations. A
summary plot of measurements is shown in Fig. 2.

In the early phase of the mission: days 80-120, the charge was measured
regularly in order to understand the charging process and make the first
discharging investigations. Over the following 100 days a number of further
charge measurements were made aimed at characterizing the charge-related noise
effects.

As indicated in Figure 2 the test masses were regrabbed six times during the
LISA Pathfinder science mission following the initial release in February
2016. In each case the test mass charge was measured soon afterwards and the
test masses brought back into the operational charge range with the UV
discharge system. In the last two releases, this process was automated on
board LPF and the data necessary to recover the TM charge at release were not
downlinked. In the first five releases however, the charge or potential
$V_{\textsc{tm}}=q/C_{T}$ could be extrapolated to the point of release. Table
1 lists the dates of the test mass releases and the TM potentials where known.
The origin of the residual potential at release is the triboelectric charging
of the test mass due to the release of the adhesion between the TM grabbing
and positioning mechanism fingers and the test mass. A detailed analysis of
this physical process is beyond the scope of this work but the observed
potentials provide useful data for bounding the design parameters of the
control loops for test mass capture during the LISA mission.

Table 1: TM potential after release |  $V_{\textsc{tm}}$ [mV]
---|---
Date | TM 1 | TM 2
2016-02-16 | $-421$ | $-281$
2016-05-11 | $-98$ | $-52$
2016-09-26 | $-98$ | $-80$
2017-10-26 | $-61$ | $-79$
2017-01-16 | $-12$ | $-512$
2017-04-07 | – | –
2017-05-10 | – | –

Later in the mission, the frequency of measurements was reduced but the charge
was measured during regular station-keeping activities in order to maintain
the test mass potential within the desired bounds. The charge was set using
the discharge system to a negative value such that by the time of the next
discharge, the charge would have accrued to the same value with opposite sign
thus maximizing the time spent with the absolute charge below the required
levels. Station-keeping activities were carried out at 1, 2 and 3-week
intervals at different stages of the mission. With the typical charge
accumulation rate of around 25 e s${}^{-1}\approx 10$ mV day-1, the target TM
potentials for this periodic discharge activity varied between $-40$, $-80$
and $-120$ mV.

Other measurements with large TM charge excursions can be seen in the figure
which were used to explore the properties of the TM charging process (as
described here), charge-induced noise [3] and the performance of the UV
discharge system [22].

##  III Charge rate dependency on electrostatics

Early test mass charge modelling work identified the possibility of kinetic
emission from Au surfaces producing a very low-energy (eV) electron population
between the test mass and its surroundings [9]. It was postulated that this
population could play a significant role in TM charging. Altering the test
mass charge by photoelectric discharge changes the potential difference
between the test mass and its surroundings through Eq. (2) providing a
mechanism to probe the low energy electron population. Two experiments were
conducted during the LPF mission dedicated to investigating this effect. In
both cases, the TM charge was increased in a number of steps using the on-
board UV charge control system. The test-mass charge was measured
continuously, allowing an extended period between illuminations in which to
evaluate the charge rate as a function of the average TM potential.

The details for the two measurements are shown in Table 2; during the first,
the test mass potential was varied over a narrower range (around $-0.3$ to
$+0.3$ V) while in the later measurement, a broader range was explored ($-1$
to $+1$ V). In each measurement $V_{\textsc{mod}}=50$ mV and
$f_{{\mathrm{MOD}}}=10$ mHz for TM 1 and 12 mHz for TM 2.

Table 2: Measurement campaigns to investigate the charging rate dependence on test-mass potential | Meas. 1 | Meas. 2
---|---|---
Start date | 2017-01-30 | 2017-06-22
Start time | 19:00:00 UTC | 02:12:00 UTC
Duration | 93600 s | 92550 s
Number steps | 11 | 10
| TM 1 | TM 2 | TM 1 | TM 2
Start potential [V] | $-0.310$ | $-0.236$ | $-1.001$ | $-1.011$
End potential [V] | $0.288$ | $0.046$ | $0.812$ | $0.857$
Slope [e s-1 V-1] | $-37.8\pm 0.8$ | $-40.3\pm 0.9$ | $-31.2\pm 0.6$ | $-30.2\pm 0.6$
$\frac{dq}{dt}$ at $V_{\textsc{tm}}=0$ V [e s-1] | $25.51\pm 0.7$ | $27.5\pm 0.1$ | $28.1\pm 0.3$ | $28.9\pm 0.6$
$V_{\textsc{tm}}$ at $\frac{dq}{dt}=0$ [mV] | $674\pm 16$ | $682\pm 16$ | $901\pm 27$ | $957\pm 39$
Time constant [d] | 65 | 61 | 79 | 82
Figure 3:  LPF charging rate measurement for both test masses with minimum
force and torque actuation authority and $V_{\textsc{mod}}=50$ mV starting at
2017-06-22 at 02:12:00 UTC, the left-hand axis indicates the test mass charge
and the right-hand axis the equivalent test mass potential with $C_{T}=34.1$
pF. The inset shows an expanded plot of the data around
$V_{\textsc{tm}}\approx-808$ mV with linear best fit lines. The charging rates
from the fitted lines are $+53.4\pm 1.2$ e s-1
($\chi^{2}_{\mathrm{red}}=0.99$) and $52.4\pm 1.2$ e s-1
($\chi^{2}_{\mathrm{red}}$=1.2) for TM 1 and 2 respectively.

Figure 3 shows the test-mass charge throughout the second measurement on
2017-06-22. Each data point represents a charge measurement integrated over
500 s. The errorbars shown are calculated from the estimated test mass
acceleration noise and level of $V_{\textsc{mod}}$ using Eq. (3), giving an
error of $\pm$25 fC. The duration of the illumination required to achieve
charge steps increased as the test mass potential increased towards the
equilibrium potential under UV illumination. The inset shows an enlargement of
one of the charge plateaus, chosen when the charges of the two test masses
were approximately equal at $-27.65$ pC. The charge rate is determined with a
relative precision of around 2% in 4500 s and the goodness of fit indicates
that this precision is consistent with the measurement error.

Fig. 4 shows the evaluation of the charging rate at each step of test mass
potential on test masses 1 and 2 during both experiments. In both cases, the
environmental charging rate of both test masses show similar, approximately
linear dependencies on $V_{\textsc{tm}}$. Fitting a line to the data, we
obtain the results shown in Table 2. The steep negative slopes immediately
confirms that low-energy electrons are playing a significant role and the
symmetry either side of $V_{\textsc{tm}}=0$ V implies that these are
originating from both the TMs and from the surrounding surfaces. The data show
that for $V_{\textsc{tm}}\approx+1$ V the negative low-energy electron
charging balances the positive charging from the high-energy cosmic rays and
the overall charging rate goes to zero. This result is important as it would
ensure that the system will reach an equilibrium in the case of a failure of
the charge management system. The time constant of the approach to
equilibrium—inversely proportional to the slope of the charge rate dependence
on test mass potential—can be calculated from the slopes reported above. We
estimate values of 61 to 82 days. Although the existence of an equilibrium is
confirmed, the observed variation indicates that its value is uncertain and is
unlikely to be accurately predictable for LISA.

After the second scan, an additional single charge rate measurement was made
on both test masses. On TM 1 the charge was set so that $q/C_{T}=+0.9$ V, dc
voltages of $-4.8$ V were applied to the $y$ and $z$ electrodes to bias the
mass back to a neutral potential according to Eq. (2). On TM 2, the charge was
set close to zero and no dc voltages were applied. The aim of this measurement
was to probe the energy distribution of low-energy electrons contributing to
charging at a higher energy range—in the regime of potential differences up to
4.8 V—than could be reached by varying the test mass potential alone. In this
measurement, the expectation was that the transfer of kinetic electrons from
TM 1 to the facing $y$ and $z$ electrodes would be suppressed, but not at all
suppressed over the rest of the surface of the test mass. This should lead to
an overall reduction in the charging rate compared to a measurement with the
test mass potential at 0 V but no applied dc electrode voltages. A detailed
analysis of this measurement relies on modeling of electron emission and the
electrostatic fields within the EH [21].

The additional data points near 0 V in Fig. 4 show the results of this
measurement. Under these conditions with the overall TM 1 potential at $-0.03$
V the test mass charging rate was 19.3 e s-1. At the same time, the rate
measured on TM 2 with no applied voltages and $V_{\textsc{tm}}=+0.03$ V was
30.2 e s-1.

Figure 4:  The variation in the test mass charging rates observed with varying
test mass potential for the two measurements described in the text. The upper
panel shows the results from 2017-01-30 and the lower panel form 2017-06-22.

Additional evidence for the influence of low-energy electrons on the charging
rate comes from a charge measurement made during 2016-05-13 as the instrument
was recovering from a test mass release. The charge rate was measured on both
test masses with $V_{\textsc{mod}}=50$ mV from 2016-05-13 00:00:00 UTC.
Initially the test mass actuation system was in a high-authority mode with
higher ac control voltages applied. Each of the 12 actuation electrodes is
applied with a sum of ac voltages which control the one force and one torque
degree of freedom, except the $x$ electrodes of TM 1 which control only
torque. In high-authority mode, the typical peak amplitudes (the sum of two
sinusoids) were around 6 V and 9 V on the $y$ and $z$ electrodes, 2 V on the
TM 1 $x$ electrodes and 5 V on TM 2 $x$. At 2016-05-13 07:30:00 UTC, the
control authority was decreased to optimize force noise performance [8],
resulting in lower ac voltage amplitudes on all electrodes (around 3 V on the
$y$ and $z$ electrodes, 0.5 V on the TM 1 $x$ electrode and 1 V on TM 2 $x$).
Figure 5 shows the test mass charge measurements before and after the
actuation change at $t$=0. A change in slope can be observed in the build up
of charge on both test masses. Fitting to the data, we find the TM 1 rate goes
from 25.3$\pm$0.1 e s-1 to 23.45$\pm$0.05 e s-1 and the TM 2 rate changes from
27.4$\pm$0.1 to 21.59$\pm$0.05 e s-1.

The observed dependence of test-mass charging on ac voltages amplitude are
likely related to the low-energy electron population. The observed behavior
would be consistent with a increase in the net flow of electrons from the EH
to the TM. This may be caused by an asymmetry in the electron emission from
surfaces where ac voltages are applied, or electrostatic effects such as
electric field edge effects which could alter the charge transfer between EH
and TM as the magnitudes of the applied voltages change.

Figure 5:  The test mass charge measured with $V_{\textsc{mod}}=50$ mV during
the change in test mass force and torque authority at 2016-05-13 07:30 UTC
($t=0$). Dashed lines indicate the best fit to the charge trend at $t<0$ in
high-authority mode and dash-dotted lines show the trend at $t>0$ in miminum-
authority mode.

##  IV Charge rate vs GCR flux

###  IV.1 Measurements dispersed through the mission

Over the course of the mission the particle flux as measured by the in-situ
particle detector increased by just less than 40% [24]. Selecting charge rate
measurements done under similar bias conditions, to avoid complications caused
by the behavior of the low-energy electron population, the results can be
plotted as a function of the radiation monitor count rate and compared to the
expectation from the previous section. In Figure 6 we have selected charge
rate measurements in which the magnitude of the test mass charge was less than
2 pC (V${}_{\textsc{tm}}<60$ mV) and the test masses actuation schemes were in
a low force authority mode (as in the second half of the measurement shown in
Figure 5). While the physical processes governing the two measurements are
rather different—the TM charging per particle is energy-dependent while the RM
count rate is not—over the range of variation observed during the LPF mission,
there is a clear correlation between them. For this reason we do not expect a
linear dependence to extend to a zero count rate. The lines on the figure show
weighted linear fits to the data. The resulting slopes are $3.3$ e Count-1 and
$5.6$ e Count-1 for TM 1 and TM 2 respectively, however it is clear that the
error bars are not consistent with a linear dependence in this case. Assuming
there are underlying systematic errors which are not reflected in the
statistics of the individual measurements, we can repeat the fit unweighted
and assume a reduced $\chi^{2}$ of 1. This give slopes of 3.6$\pm$0.5 e
Count-1 and 4.4$\pm$1.1 e Count-1.

Figure 6: LPF test-mass charging rates as a function radiation monitor count
rate for selected measurements with $V_{\textsc{tm}}$ close to 0 V and ac
actuation voltages in minimum-authority mode. Solid lines show the weighted
best fit to the data, dashed lines indicate the unweighted fit.

Not included in Fig. 6 because it was made with a higher test mass force
authority, is the long charge measurement presented in Ref. [16] which began
at 2016-04-20 08:00:00 UTC. The ac actuation voltages in this case were
similar to the high-authority mode described earlier but with reduced voltage
amplitudes on the TM 1 $x$ (1 V) and TM 2 $x$ (2 V) electrodes. This charge
measurement has the highest modulation voltage of any measurements made during
the mission at $V_{\textsc{mod}}=3$ V and therefore the best precision on the
charge rate determination. A measurable change in the radiation monitor count
rate was also observed over the course of the 3-day measurement. Figure 7
shows the charge rate plotted against the radiation count rate for both test
masses. The charge rate data are calculated from the difference between
consecutive charge measurements and averaged over 1-hour intervals, while the
radiation monitor counts are averaged over the same time periods. The slopes
to the best fit lines in this case were $5.2\pm 0.3$ e Count-1 and $5.5\pm
0.3$ e Count-1 for TM 1 and TM 2 respectively.

Figure 7: LPF test-mass charging rates plotted against radiation monitor count
rate over a 3-day period beginning 2016-04-20 08:00:00 UTC. The measurement
was performed with $V_{\textsc{mod}}=3$ V and ac actuation voltages in high-
authority mode but with reduced amplitudes on the $x$-electrodes. Charging and
radiation monitor count rates are evaluated over 1-hour periods. Solid lines
show the best fit to the data.

###  IV.2 A Forbush decrease

Figure 8:  Test mass charge and radiation monitor measurements made during the
Forbush decrease beginning at 2017-07-15 01:00 UTC. The measurement was
performed with $V_{\textsc{mod}}=10$ mV and ac actuation voltages in minimum-
authority mode. Charging and radiation monitor count rates are evaluated over
2-hour periods

On 2017-07-14 at approximately 03:00 UTC, a solar energetic particle event was
recorded by a number of environmental monitors. Following this observation, a
test mass charge measurement on both test masses with $V_{\textsc{mod}}=10$ mV
was commanded to begin onboard LPF, beginning at 2017-07-15 01:00:00 UTC.
During this measurement the ac actuation voltages were in minimum-authority
mode. As reported in Refs. [24, 25] the energy distribution of the solar
protons was not sufficiently hard to register an increase in the onboard
radiation monitor. However, following the passage of the solar particles, a
temporary suppression of the galactic cosmic ray flux associated with a
Forbush decrease was observed. Figure 8 shows the test-mass charging rates and
radiation monitor count rate averaged over 2-hour time intervals during the
event. Over the course of 12 hours from approximately 2017-07-16 08:00 UTC,
the observed count rate decreased by around 17%. A scatter plot of the data is
shown in Fig. 9. The best-fit lines shown on the scatter plot have slopes of
4.5$\pm$0.2 e Count-1 and 4.3$\pm$0.2 e Count-1 for TM 1 and TM 2
respectively.

Measurements presented in this section demonstrate that on timescales of days,
changes in the average charging rate tracked the radiation monitor count rate
with a linear relationship. The two measurements of the charge-count
dependency made at the beginning and end of the mission disagree by several
sigma in this dependency. However, both fall within the errors of the slope
observed over the mission duration as whole shown in Figure 6.

Figure 9:  Scatter plot of the data shown in Figure 8 showing the
correspondence between the test mass charging rate and radiation monitor count
rate during the Forbush decrease. Solid lines show the best fit to the data.

##  V LISA timeline prediction

Ref. [24] established the close correspondence between in-situ measurements of
the cosmic ray flux with the LPF radiation monitor and other solar activity
indicators, especially the Integral Radiation Environment Monitor (IREM),
[28]. Using this correspondence, and the (assumed linear) correlation between
test mass charging and the RM count rate, we can produce timeline of test mass
charging using historical data.

The Level-0 IREM TC2 channel data is sensitive to protons in the same energy
range as that responsible for test mass charging ($E>100$ MeV nucl-1) [29].
The available data time series extends from shortly after the launch of the
INTEGRAL in October 2002 to the present day. The count rate data contain
regular peaks as INTEGRAL traverses the Earth’s radiation belts in its highly-
elliptical, 2.7 day orbit. These peaks are flagged in the data and have been
removed during the analysis shown here. Enhancements in the data can also be
seen due to solar energetic particle events. The correlation between count
rate and charging is not expected to hold during SEP events given the
difference in the energy spectrum of particles responsible for the flux
enhancement. The energy distribution of GCR protons has a peak that varies
from 400 MeV to 800 MeV during the solar cycle. In the range of energies
relevant to test-mass charging however, SEP flux is typically distributed as a
power law in energy. Due to the lower energies of the particles involved, the
charging per incident particle is lower for SEPs. The RM count rate
measurement on the other hand is insensitive to primary particle energy above
around 100 MeV. Despite this caveat, SEP events have not been removed from the
timeline in order to give an impression of the frequency of the events. The
scaled charge rate has, however, been truncated and so does not show the
extrapolated charging rate during SEP events. The charging predictions for a
LPF-like GRS are calculated from these data by scaling to the LISA Pathfinder
count rate and then to TM charging. Over the course of the LPF mission, a fit
to the RM singles count as a function of IREM TC2 count yields a scale factor
of 2.8. From the previous section we adopt a factor of 4.4$\pm$1 e Count-1 to
envelope all of the presented measurements. This charge rate dependence
assumes the low-level of ac actuation voltages used for most of the LPF
science mission.

Figure 10:  Charging rate of the LISA Pathfinder test masses extrapolated over
the 20-year observation range of the INTEGRAL mission by correlation with the
IREM count rate as described in the text. The shaded region shows the duration
of the LISA Pathfinder mission.

Figure 10 shows the resulting estimated test mass charging timeline from
2002-10-20 to 2022-10-10 covering two solar minima and maxima. The charging
rate over the time of the LPF mission, is indicated by the orange band. The
minimum charge rate seen is 5-9 e s-1 during the solar maximum of 2003 and
10-13 e s-1 during the somewhat weaker maximum of 2014. In solar minimum
conditions during 2010 and 2020 the charge rate peaks at 36-39 e s-1. The
variation in the cosmic ray charging rate is within the ranges that have been
assumed in defining the requirements of the LISA mission. The LISA
requirements for the allowable charge on the test mass are stricter than was
implemented on LPF with a maximum absolute charge of 2.4 pC equivalent to 70
mV. Allowing the charge to accumulate from the negative limit, these charge
rates would set the discharge frequency in LISA between roughly once per 70
days and once per 9 days.

Although this charging prediction is based on measurements based on LISA
Pathfinder and not LISA, given the expected similarity between the test mass
and its surrounding electrode housing in the two missions it nonetheless gives
a valuable insight into the range of charging rates that can be expected and
the rate at which they may change over time. Uncertainties in the scaling
between charging and RM count rate introduce an uncertainty of up to 10% as
shown in the rates measured throughout the LPF mission in Figure 6.
Electrostatic contributions to the charging rate could also be different in
the LISA GRS and could introduce a systematic uncertainty at the level of 20%
as seen in Figure 5. A further caveat to apply to these results is that the
assumed linear correspondence between the radiation monitor count rates and TM
charging has not been established in solar maximum conditions when the energy
spectrum of the cosmic ray flux is different from solar minimum. This could
add a systematic error to the lower end of the estimated charge rate range.

The baseline discharging strategy for LISA is to employ a continuous UV
illumination scheme that maintains the test mass potential constant throughout
the mission. In the case this approach introduces excess noise however, the
fallback scheme would be to discharge the test masses intermittently as was
demonstrated successfully in LPF. The estimated range of charging rates over
the course of a solar cycle sets the required test-mass discharge frequency.
Assuming a 1-hour discharging operation for all LISA test masses
simultaneously, and in the pessimistic case that the data during discharging
is unusable for science, the projected worst-case discharge frequency would
cause an observatory down time of 0.5%.

##  VI Discussion

In this paper we have presented a summary of the test mass charging
measurements made on board LISA Pathfinder in order to characterize the
charging process. Upon release the test mass potential was consistently
negative with respect to the 0 V reference of the GRS FEE and its amplitude
varied from $-512$ mV to $-12$ mV. This gives an idea of the triboelectric
charging of the TM upon release by the grabbing, positioning and release
mechanism and the consequent burden on the control system for electrostatic
capture of the TM and the discharge system for its initial neutralization.

We have shown that in both LPF GRS, the test mass charging rate exhibited a
strong dependence on the test mass potential due to the the presence of
electrons with eV-scale energies emitted from the surfaces of the test mass
and surrounding surfaces. The details of the underlying physics of this
process are explored in a number of separate publications [17, 18, 19, 20,
21]. Measurements of the electrostatic charge rate dependence indicated that
an equilibrium test mass potential existed—at which the net environmental rate
of TM charging is zero—between $V_{\textsc{tm}}=600$ mV and 950 mV at various
stages of the mission. The existence and value of this equilibrium potential
means that, if reproduced on LISA, and in the absence of a functioning UV
discharge system due to an instrument failure, the TM potential will at least
remain stable and bounded. The time constants for the test masses to reach
this stable equilibrium are on the scale of 2-3 months. To give an idea of the
cost in TM acceleration noise due to coupling of the charge to actuation
voltage fluctuations on the GRS electrodes, we note that LPF measurements
showed an acceleration noise increase of order 3 fm s-2 Hz-1/2 at 0.1 mHz per
TM with $V_{\textsc{tm}}$ = 1 V, [3] out of a full TM acceleration noise
budget of roughly 10 fm s-2 Hz-1/2 at that frequency.

The impact of SEP events on the equilibrium potential produced by
environmental charging remains to be studied in detail. During such events,
the particle flux can be augmented by several orders of magnitude which will
likely drive the test mass to an equilibrium potential rapidly. However, the
difference in the energy spectrum of particles responsible for charging due to
an SEP event may alter the balance of charging produced by primaries and low-
energy secondary electrons and shift the equilibrium away from its steady-
state value.

We have quantified the variations in the test mass charging rate observed over
the course of the mission as a function of the on-board radiation monitor
record of the galactic cosmic ray flux. The RM count rate provided an accurate
analog to the test mass charge rate—especially during flux variations on the
scale of days—able to track changes at the 10% level.

Due to the short duration of the LPF mission, the energy distribution of the
cosmic ray flux is not expected to have changed significantly and no solar
energetic particle enhancements were seen. The accuracy of extending the
analysis of the count rate dependency outside the conditions of cosmic ray
flux approaching minimum in solar activity is therefore uncertain. For a
longer duration mission such as LISA, the ability to determine the energy
spectrum of the particle flux above 100 MeV nucl-1, combined with detailed
particle tracking simulations would enhance the ability to correlate charging
and particle detection rate. Despite this limitation we have shown that simple
particle count-rate data can be used to track short-term variations in the
charging rate accurately.

We have made use of the correlation between charging and the RM count rate and
the correlation between the LPF RM count rate and that measured by the IREM
TC2 channel to present a 20-year timeline of test mass charging based on
historic data. Using this projection, we extrapolate an estimate for the range
of test mass charging rates that may be seen in a long-duration mission with a
similar GRS such as LISA. The calculated range of 5-39 e s-1 is subject to
considerable uncertainty at the lower bound which could be resolved by
detailed particle tracking simulations with accurate GCR flux models.

## Acknowledgements

This work has been made possible by the LISA Pathfinder mission, which is part
of the space-science programme of the European Space Agency. The French
contribution has been supported by the CNES(Accord Specifique de projet
CNES1316634/CNRS103747), the CNRS, the Observatoire de Paris and the
University Paris-Diderot. E.Plagnol and H.Inchauspé would also like to
acknowledge the financial support of the UnivEarthS Labex program at Sorbonne
Paris Cité (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 (FKZ50OQ0501 and FKZ50OQ1601). The Italian
contribution has been supported by Istituto Nazionale di Fisica Nucleare
(INFN) and Agenzia Spaziale Italiana (ASI), Project No. 2017-29-H.1-2020
“Attività per la fase A della missione LISA”.The Spanish contribution has been
supported by contracts AYA2010-15709(MICINN), ESP2013-47637-P, and
ESP2015-67234-P(MINECO). M.Nofrarias acknowledges support from Fundacion
General CSIC (Programa ComFuturo). F. Rivas acknowledges an FPI contract
(MINECO). The Swiss contribution acknowledges the support of the Swiss Space
Office (SSO) via the PRODEX Programme of ESA. L. Ferraioli is supported by 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). T. J. Sumner also acknowledges support
from the Leverhulme Trust (EM-2019-070\4), J. I. Thorpe and J. Slutsky
acknowledge the support of the U.S. National Aeronautics and Space
Administration (NASA).

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