# MICROSCOPE mission analysis, requirements and expected performance

Pierre Touboul1, Manuel Rodrigues1, Gilles Métris2, Ratana Chhun1, Alain
Robert3, Quentin Baghi2 111Current address: NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA, Emilie Hardy1, Joel Bergé1, Damien Boulanger1, Bruno
Christophe1, Valerio Cipolla3, Bernard Foulon1, Pierre-Yves Guidotti3
222Current address: AIRBUS Defence and Space, F-31402 Toulouse, France,
Phuong-Anh Huynh1, Vincent Lebat1, Françoise Liorzou1, Benjamin Pouilloux3
333Current address: KINEIS, F-31520 Ramonville Saint-Agne, France, Pascal
Prieur3, Serge Reynaud4 1 DPHY, ONERA, Université Paris Saclay, F-92322
Châtillon, France  2 Université Côte d’Azur, Observatoire de la Côte d’Azur,
CNRS, IRD, Géoazur, 250 avenue Albert Einstein, F-06560 Valbonne, France  3
CNES, 18 avenue E Belin, F-31401 Toulouse, France  4 Laboratoire Kastler
Brossel, UPMC-Sorbonne Université, CNRS, ENS-PSL University, Collège de
France, 75252 Paris, France<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The MICROSCOPE mission aimed to test the Weak Equivalence Principle (WEP) to a
precision of $10^{-15}$. The WEP states that two bodies fall at the same rate
on a gravitational field independently of their mass or composition. In
MICROSCOPE, two masses of different compositions (titanium and platinum
alloys) are placed on a quasi-circular trajectory around the Earth. They are
the test-masses of a double accelerometer. The measurement of their
accelerations is used to extract a potential WEP violation that would occur at
a frequency defined by the motion and attitude of the satellite around the
Earth. This paper details the major drivers of the mission leading to the
specification of the major subsystems (satellite, ground segment, instrument,
orbit…). Building upon the measurement equation, we derive the objective of
the test in statistical and systematic error allocation and provide the
mission’s expected error budget.

††: Class. Quantum Grav.

  * Dec. 2020

Keywords: General Relativity, Experimental Gravitation, Equivalence Principle,
Space accelerometers, Microsatellite.

##  1 Introduction

The MICROSCOPE (French acronym: Micro-Satellite à traînée Compensée pour
l’Observation du Principe d’Equivalence) mission was defined in 2000 by the
Centre National d’Etudes Spatiales (CNES), the Observatoire de la Côte d’Azur
(OCA) and the Office National d’Etudes et de Recherches Aérospatiales (ONERA).
Designed to test the WEP in space, the satellite was launched into a low-Earth
sun-synchronous orbit from Kourou on April 25, 2016 at an altitude of 710 km,
and delivered science data for more than two years. In Ref. [1], 7% of the
available data were used to provide first, intermediate results. No evidence
for a violation of the WEP was found at $1.3\times{}10^{-14}$, one order of
magnitude higher than MICROSCOPE’s full mission target accuracy of $10^{-15}$
on the Eötvös parameter.

The MICROSCOPE satellite was designed to provide an environment as stable as
possible. It is finely controlled along its six degrees of freedom thanks to
the Drag-Free and Attitude Control System (DFACS), detailed in Ref. [2]. The
DFACS allows for several modes of operation: inertial pointing or spin mode.
In spin mode, the satellite rotates about the instrument’s $Y$-axis in the
direction opposite to the orbital motion in order to increase the apparent
rate of the Earth gravity field variation. With a frequency rate of rotation
$f_{\rm{spin}_{i}}$, the EP frequency
$f_{\rm{EP}_{i}}=f_{\rm{orb}}+f_{\rm{spin}_{i}}$, where $f_{\rm{orb}}$ is the
satellite’s orbital frequency. Table 1 lists the available frequencies (see
Ref. [3] for details).

Table 1:  Main frequencies of interest.

  * Label | Frequency | Comment
---|---|---
$f_{\rm{orb}}$ |  $1.6818\times{}10^{-4}$ Hz | Mean orbital frequency
$f_{\rm{spin}_{2}}$ |  $\frac{9}{2}f_{\rm{orb}}$=$0.75681\times{}10^{-3}$ Hz | Spin rate frequency 2 (V2 mode)
$f_{\rm{spin}_{3}}$ |  $\frac{35}{2}f_{\rm{orb}}$=$2.94315\times{}10^{-3}$ Hz | Spin rate frequency 3 (V3 mode)
$f_{\rm{EP}_{2}}$ |  $0.92499\times{}10^{-3}$ Hz | EP frequency in V2 mode
$f_{\rm{EP}_{3}}$ |  $3.11133\times{}10^{-3}$ Hz | EP frequency in V3 mode
$f_{cal}$ |  $1.22848\times{}10^{-3}$ Hz | Calibration frequency

The satellite carries the Twin Space Accelerometers for Gravitation Experiment
(T-SAGE) payload. T-SAGE is composed of two sensor units called SUREF and
SUEP. Each sensor unit is a double accelerometer whose test masses are two co-
axial hollow cylinders. SUREF’s test-masses are made of the same material
(PtRh10) while SUEP’s test-masses are made of different material (PtRh10 for
the inner mass, Ti alloys for the outer mass). The principle of the
accelerometers and the description of the instrument are detailed in Ref. [4].
Each Sensor Unit (SU) is associated to a Front End Electronic Unit (FEEU)
which contains the pick-up measurement, the voltage references, the electrode
voltages and the whole instrument temperature acquisition system.

This paper details the per-flight error budget used to establish the
requirement tree for all subsystems of the mission. Based on the measurement
equation (Sect. 2), it is detailed in Sect. 3. After the flight, the
requirements were verified by direct or indirect measurements or by analysis.
An update of the error budget evaluated with flight inputs is presented at the
end of the paper.

##  2 Measurement equation

The acceleration of each test-mass can be given in the satellite moving frame
by:

$\frac{{\rm d}^{2}\vv{O_{\Earth}O_{j}}}{{\rm d}t^{2}}=\frac{{\rm
d}^{2}\vv{O_{\Earth}O_{\rm sat}}}{{\rm
d}t^{2}}+\vv{\dot{\Omega}}\times\vv{O_{\rm
sat}O_{j}}+\vv{\Omega}\times\left(\vv{\Omega}\times\vv{O_{\rm
sat}O_{j}}\right)+2\vv{\Omega}\times\frac{{\rm d}\vv{O_{\rm sat}O_{j}}}{{\rm
d}t}+\frac{{\rm d}^{2}\vv{O_{\rm sat}O_{j}}}{{\rm d}t^{2}},$ (1)

with $O_{\Earth}$ the centre of the Earth as the centre of the Galilean frame,
$O_{\rm sat}$ the satellite centre of gravity and $O_{j}$ the centre of the
$j$th test-mass. $\vv{\dot{\Omega}}$ represents the angular acceleration of
the satellite and $\vv{\Omega}$ its angular velocity. When the test-masses are
servo-controlled, their relative motion to the satellite structure can be
considered as null and thus the terms $\frac{{\rm d}\vv{O_{\rm
sat}O_{j}}}{{\rm d}t}$ and $\frac{{\rm d}^{2}\vv{O_{\rm sat}O_{j}}}{{\rm
d}t^{2}}$ can be neglected in the measurement bandwidth. However, they are
considered when the test-mass position is biased with a sine signal for
calibration [5, 6].

In addition Newton’s second law gives respectively the satellite and the test-
mass acceleration as:

$\frac{{\rm d}^{2}\vv{O_{\Earth}O_{\rm sat}}}{{\rm d}t^{2}}=\frac{\vv{F}_{\rm
ext}}{M_{{\rm I_{sat}}}}+\frac{\vv{F}_{\rm th}}{M_{{\rm
I_{sat}}}}+\frac{M_{{\rm G_{sat}}}}{M_{{\rm I_{sat}}}}\vv{g}(O_{\rm sat}),$
(2) $\frac{d^{2}\vv{O_{\Earth}O_{j}}}{dt^{2}}=\frac{\vv{F}_{{\rm
el},j}}{m_{\rm I}}+\frac{\vv{F}_{{\rm dis},j}}{m_{\rm I}}+\frac{m_{{\rm
G}_{j}}}{m_{{\rm I}_{j}}}\vv{g}(O_{j}),$ (3)

where $M_{{\rm G_{sat}}}$ and $m_{{\rm G}_{j}}$ are the gravitational masses
of the satellite and of the $j^{th}$ test-mass, $M_{I_{\rm sat}}$ and $m_{{\rm
I}_{j}}$ are the inertial masses of the satellite and of the test-mass,
$\vv{g}(O_{\rm sat})$ (resp. $\vv{g}(O_{j})$) is the Earth gravity
acceleration at the centre of mass of the satellite (resp. test mass). The
satellite undergoes non gravitational forces $\vv{F}_{\rm ext}$ such as
atmospheric drag and Solar radiation pressure and thruster forces $\vv{F}_{\rm
th}$. The test mass undergoes electrostatic forces $\vv{F}_{{\rm el},j}$ and
internal disturbing forces $\vv{F}_{{\rm dis},j}$ such as those induced by the
electrostatic stiffness, the gold wire stiffness, radiometer effect, …[7]. The
acceleration measurement is inferred from the measurement of the voltages
applied on the electrodes and from the estimated scale factor [8]. By
combining Eq. (1), Eq. (2) and Eq. (3), the resulting electrostatic
acceleration $\vv{\gamma}_{j}$ of the $j$th mass is defined as the
electrostatic force divided by the mass :

$\vv{\gamma}_{j}=\frac{\vv{F}_{{\rm el},j}}{m_{{\rm
I}_{j}}}=\vv{\gamma}_{\Earth,j}+\vv{\gamma}_{{\rm kin},j}-\frac{\vv{F}_{{\rm
dis},j}}{m_{{\rm I}_{j}}}+\frac{\vv{F}_{\rm ext}}{M_{\rm
I_{s}at}}+\frac{\vv{F}_{\rm th}}{M_{\rm I_{s}at}},$ (4)

with the contribution to the Earth’s gravitational acceleration given by:

$\vv\gamma_{\Earth,j}=\frac{M_{{\rm G_{sat}}}}{M_{{\rm I_{sat}}}}\vv{g}(O_{\rm
sat})-\frac{m_{{\rm G}_{j}}}{m_{{\rm I}_{j}}}\vv{g}(O_{j}),$ (5)

where $\vv{\gamma}_{{\rm kin},j}=\vv{\dot{\Omega}}\times\vv{O_{\rm
sat}O_{j}}+\vv{\Omega}\times\left(\vv{\Omega}\times\vv{O_{\rm
sat}O_{j}}\right)$ is the kinematic acceleration.

Tests of the WEP usually present their result in terms of the Eötvös ratio [9,
10] :

$\eta(2,1)=2\frac{m_{\rm{G_{2}}}/m_{\rm{I_{2}}}-m_{\rm{G_{1}}}/m_{\rm{I_{1}}}}{m_{\rm{G_{2}}}/m_{\rm{I_{2}}}+m_{\rm{G_{1}}}/m_{\rm{I_{1}}}}.$
(6)

Note that the Eötvös ratio depends on the pair of materials used for the test.
In this paper, we use a good first order approximation of the Eötvös
parameter,

$\delta(2,1)\equiv\frac{m_{\rm{G_{2}}}}{m_{\rm{I_{2}}}}-\frac{m_{\rm{G_{1}}}}{m_{\rm{I_{1}}}}.$
(7)

We define the common-mode (resp. differential-mode) of a given instrumental
parameter as the half-sum (resp. half-difference) of this parameter for both
test-masses $o^{(1)}$ and $o^{(2)}$, the inner test-mass being $(1)$ and the
outer being $(2)$:

$\displaystyle o^{(c)}=\frac{1}{2}(o^{(1)}+o^{(2)})$ (8) $\displaystyle
o^{(d)}=\frac{1}{2}(o^{(1)}-o^{(2)}).$ (9)

In what follows, an instrumental parameter can be also defined as a matrix of
sensitivity or alignment and noted $[o]$. However, for the measured
acceleration $\vv{\Gamma}$, applied acceleration $\vv{\gamma}$ or disturbing
accelerations $\vv{b}$ vectors, we use the simple difference instead of the
half-difference when defining their differential-mode:

$\vv{a}^{(c)}=\frac{1}{2}(\vv{a}^{(1)}+\vv{a}^{(2)})$ (10)
$\vv{a}^{(d)}=(\vv{a}^{(1)}-\vv{a}^{(2)}).$ (11)

Finally, the distance between the test-mass ${(1)}$ and ${(2)}$,
$\vv{O_{1}O_{2}}$ (and not $\vv{O_{2}O_{1}}$), is noted $\vv{\Delta}$ and
called “offcentering”.

The instrument frame is defined in Fig. 1: the $X$-axis and the $Z$-axis lie
in the orbital plane, with the $X$-axis along the main axis of the cylindrical
test-masses. The $Y$-axis is normal to the orbital plane, and completes the
triad.

Figure 1: Left: the 4 test-masses orbiting around the Earth (credits CNES /
Virtual-IT 2017) . Right: test-masses and satellite frames; the ($X_{\rm
sat}$, $Y_{\rm sat}$, $Z_{\rm sat}$) triad defines the satellite frame; the
reference frames ($X_{k}$, $Y_{k}$, $Z_{k}$, $k=1,2$) are attached to the
test-masses (black for the inner mass $k=1$, red for the outer mass $k=2$);
the $X_{k}$ axes are the test-mass cylinders’ longitudinal axis and define the
direction of Equivalence Principle test measurement; the $Y_{k}$ axes are
normal to the orbital plane, and define the rotation axis when the satellite
spins; the $Z_{k}$ axes complete the triads. The 7 $\mu$m gold wires
connecting the test-masses to the common Invar sole plate are shown as yellow
lines. The centres of mass have been approximately identified with the origins
of the corresponding sensor-cage-attached reference frames.

The WEP is measured by monitoring the difference in accelerations undergone by
the two test masses along their $X$-axis (the most sensitive). In practice,
this is done by computing the difference of measured electrostatic
acceleration. In the absence of instrumental defects, the electrostatic
acceleration should be in equilibrium with the others actions in Eq. (4) and
lead to a difference of acceleration applied on the test-masses

$\vv\gamma^{(d)}=\delta(2,1)\vv{g}(O_{\rm sat})+([{\rm T}]-[{\rm
In}])\vv{\Delta}+\vv{b_{1}}^{(d)},$ (12)

where $[{\rm T}]$ is the Earth gravity gradient tensor projected in the
instrument’s frame. The kinematic acceleration can be expressed as $-[{\rm
In}]\vv{\Delta}$, where $[{\rm In}]$ is the matrix defined by $[{\rm
In}]=[\dot{\Omega}]+[\Omega][\Omega]$) where $[\Omega]$ is the matrix
representation of the angular velocity of the satellite. At last,
$\vv{b_{1}}^{(d)}$ represents the difference of acceleration due to disturbing
forces acting on the two test masses in Eq. (4). The electrostatic
acceleration applied to each test mass is not perfectly measured, as errors
can come from wrong scale of voltage applied to electrodes or from the
misalignment of the test-mass measurement frames. Instrumental parameters must
therefore be taken into account, such as, for the $j$th test mass:

  * •

the DC bias (offset) due to the measurement pick-up $\vv{b_{0}}^{(j)}$;

  * •

the measurement pick-up noise $\vv{n}^{(j)}$;

  * •

scale factors (close to 1)
$\vv{K_{1}}^{(j)}=\begin{pmatrix}K_{x}^{(j)}&K_{y}^{(j)}&K_{z}^{(j)}&K_{\phi}^{(j)}&K_{\theta}^{(j)}&K_{\psi}^{(j)}\end{pmatrix}^{t}$;

  * •

couplings $\left[\bm{\eta}^{(j)}\right]$ between linear degrees of freedom due
to non orthogonality of the measurement frame; this defect is linked to the
test-mass geometry and is much smaller than 1; note also that the
$\left[\bm{\eta}^{(j)}\right]$ matrix is symmetric;

  * •

couplings between linear degrees of freedom due to electrostatic defects [7]:

$\left[\bm{\beta}^{(j)}\right]=\begin{bmatrix}0&\beta_{xy}^{(j)}&\beta_{xz}^{(j)}\\\
\beta_{yx}^{(j)}&0&\beta_{yz}^{(j)}\\\
\beta_{zx}^{(j)}&\beta_{zy}^{(j)}&0\end{bmatrix}\approx\begin{bmatrix}0&0&0\\\
\beta_{yx}^{(j)}&0&\beta_{yz}^{(j)}\\\
\beta_{zx}^{(j)}&\beta_{zy}^{(j)}&0\end{bmatrix}+O(10^{-3});$ (13)

  * •

couplings of linear acceleration with angular acceleration
$\left[{\mathbf{C}}^{(j)}\right]$;

  * •

rotation from the common instrument frame $\mathcal{R}_{\rm sat}$ linked to
the satellite in which the acceleration $\vv{\gamma}^{(j)}$ is modelled to
each test mass frame $\mathcal{R}_{j}$ (linked to the test mass geometry)
where the acceleration is measured. The alignments of the different frames
have been measured or evaluated on the ground prior to the launch and can be
approximated in the small angle limit ($\rm{<10^{-2}\,rad}$) by

$\left[\bm{\Theta}^{(j)}\right]=\begin{bmatrix}1&\theta_{z}^{(j)}&-\theta_{y}^{(j)}\\\
-\theta_{z}^{(j)}&1&\theta_{x}^{(j)}\\\
\theta_{y}^{(j)}&-\theta_{x}^{(j)}&1\end{bmatrix}.$ (14)

These items are combined to construct the linear acceleration measurement
parameters

$\left[{\mathbf{M}^{(j)}}\right]=\begin{bmatrix}K_{x}^{(j)}&\eta_{z}^{(j)}+\beta_{xy}^{(j)}&\eta_{y}^{(j)}+\beta_{xz}^{(j)}\\\
\eta_{z}^{(j)}+\beta_{yx}^{(j)}&K_{y}^{(j)}&\eta_{x}^{(j)}+\beta_{yz}^{(j)}\\\
\eta_{y}^{(j)}+\beta_{zx}^{(j)}&\eta_{x}^{(j)}+\beta_{zy}^{(j)}&K_{z}^{(j)}\end{bmatrix}$
(15)

and

$\left[{\mathbf{A}^{(j)}}\right]=\left[{\bm{\Theta}^{(j)}}\right]\left[{\mathbf{M}^{(j)}}\right].$
(16)

The models of the actually measured differential acceleration and the common
mode acceleration (up to leading terms) are then

$\vv\Gamma^{(d)}=\vv{b_{0}}^{(d)}+\left[{\mathbf{A}^{(c)}}\right]\vv{\gamma}^{(d)}+2\left[{\mathbf{A}^{(d)}}\right]\vv{\gamma}^{(c)}+2\left[{\mathbf{C}^{(d)}}\right]\accentset{\mbox{\large.}}{\vv\Omega}+\vv
n^{(d)}$ (17)

and

$\vv\Gamma^{(c)}=\vv{b_{0}}^{(c)}+\frac{1}{2}\left[{\mathbf{A}^{(d)}}\right]\vv{\gamma}^{(d)}+\left[{\mathbf{A}^{(c)}}\right]\vv{\gamma}^{(c)}+\left[{\mathbf{C}^{(c)}}\right]\accentset{\mbox{\large.}}{\vv\Omega}+\vv
n^{(c)}.$ (18)

The differential acceleration measurement is used to estimate the Eötvös
parameter $\delta(2,1)$ [5, 6]. Coupling effects are negligible on the
$X$-axis, in particular the two terms $\eta_{z}^{(j)}+\beta_{xy}^{(j)}$ and
$\eta_{y}^{(j)}+\beta_{xz}^{(j)}$ that have been estimated in flight [7]. It
remains that the measurement can be disturbed by the mismatching of scale
factors and the misalignment of the test-masses leading to the projection of
the common mode acceleration on $\vv\Gamma^{(d)}$. In order to correct for
this defect in (17), the applied acceleration $\vv{\gamma}^{(c)}$ is estimated
by using the measurement equation (18). This leads to:

$\vv\Gamma^{(d)}=\vv{B_{0}}^{(d)}+\left[{\mathbf{\tilde{A}}^{(c)}}\right]\vv{\gamma}^{(d)}+2\left[{\mathbf{A}^{(d)}}\right]\left[{\mathbf{A}^{(c)}}\right]^{-1}\vv{\tilde{\Gamma}}^{(c)}+2\left[{\mathbf{C}^{\prime(d)}}\right]\accentset{\mbox{\large.}}{\vv\Omega}+\vv
n^{(d)}$ (19)

where

$\displaystyle\vv{\tilde{\Gamma}}^{(c)}$ $\displaystyle=\vv{\Gamma}^{(c)}-\vv
n^{(c)},$ (20) $\displaystyle\vv{B_{0}}^{(d)}$
$\displaystyle=\vv{b_{0}}^{(d)}-2\left[{\mathbf{A}^{(d)}}\right]\left[{\mathbf{A}^{(c)}}\right]^{-1}\vv{b_{0}}^{(c)},$
(21) $\displaystyle\left[{\mathbf{C}^{\prime(d)}}\right]$
$\displaystyle=(\left[{\mathbf{C}^{(d)}}\right]-\left[{\mathbf{A}^{(d)}}\right]\left[{\mathbf{A}^{(c)}}\right]^{-1}\left[{\mathbf{C}^{(c)}}\right])$
(22) $\displaystyle\left[{\mathbf{\tilde{A}}^{(c)}}\right]$
$\displaystyle=(\left[{\mathbf{A}^{(c)}}\right]-\left[{\mathbf{A}^{(d)}}\right]\left[{\mathbf{A}^{(c)}}\right]^{-1}\left[{\mathbf{A}^{(d)}}\right])\approx\left[{\mathbf{A}^{(c)}}\right]~{}{\rm
at~{}first~{}order}.$ (23)

The DFACS applies a thrust which reduces significantly the mean output
$\vv{\Gamma}^{(c)}$. More precisely, most of the sessions were performed with
the DFACS controlled on the outer test-mass, some on the inner test-mass and
very few on the mean of the two test-masses. In addition, in order to minimise
the gas consumption, the DFACS does not compensate the DC bias of the
accelerometer. The accelerometer bias is regularly estimated as
$\vv{\hat{b}_{0}}^{(c)}$ and injected in the DFACS loop to subtract the DC
bias in the thrust command [2, 3]. As a result, the common mode measurement
(or the test-mass reference for the DFACS loop acceleration measurement) is
not totally nullified and becomes
$\vv{\Gamma}^{(c)}=\vv{\hat{b}_{0}}^{(c)}+\vv{R}_{\rm df}$, where
$\vv{\hat{b}_{0}}^{(c)}$ is the estimation of $\vv{{b}_{0}}^{(c)}$ and
$\vv{R}_{\rm df}$ the residual error control of the DFACS seen by the
accelerometer (i.e. it includes all alignment, coupling and scale factor
errors as they are all compensated by the closed loop).

The WEP is tested along the $X$-axis and the equations are expressed in the
${\mathcal{R}}_{SU}$ frame defined by the test-mass used for the DFACS (or the
mean frame of the two test-masses when using the mean of both). The equations
in this paper are expressed in the mean frame
$\frac{1}{2}({\mathcal{R}}_{(1)}+{\mathcal{R}}_{(2)})$. So the measurement is
the projection of Eq. (19) on the $X$-axis, after substituting Eq. (12) in Eq.
(19):

$\begin{split}\Gamma_{x}^{(d)}&\approx B_{0x}^{(d)}\\\
&+\tilde{a}_{c11}b_{1x}^{(d)}+\tilde{a}_{c12}b_{1y}^{(d)}+\tilde{a}_{c13}b_{1z}^{(d)}\\\
&+\tilde{a}_{c11}\delta g_{x}+\tilde{a}_{c12}\delta
g_{y}+\tilde{a}_{c13}\delta g_{z}\\\
&+\left(T_{xx}-In_{xx}\right)\tilde{a}_{c11}\Delta_{x}+\left(T_{xy}-In_{xy}\right)\tilde{a}_{c11}\Delta_{y}+\left(T_{xz}-In_{xz}\right)\tilde{a}_{c11}\Delta_{z}\\\
&+\left(T_{yx}-In_{yx}\right)\tilde{a}_{c12}\Delta_{x}+\left(T_{yy}-In_{yy}\right)\tilde{a}_{c12}\Delta_{y}+\left(T_{yz}-In_{yz}\right)\tilde{a}_{c12}\Delta_{z}\\\
&+\left(T_{zx}-In_{zx}\right)\tilde{a}_{c13}\Delta_{x}+\left(T_{zy}-In_{zy}\right)\tilde{a}_{c13}\Delta_{y}+\left(T_{zz}-In_{zz}\right)\tilde{a}_{c13}\Delta_{z}\\\
&+2\left(\frac{a_{d11}}{a_{c11}}{\Gamma}_{x}^{(c)}+\frac{a_{d12}}{a_{c22}}{\Gamma}_{y}^{(c)}+\frac{a_{d13}}{a_{c33}}{\Gamma}_{z}^{(c)}\right)\\\
&+2\left(c^{\prime}_{d11}\accentset{\mbox{\large.}}{\Omega}_{x}+c^{\prime}_{d12}\accentset{\mbox{\large.}}{\Omega}_{y}+c^{\prime}_{d13}\accentset{\mbox{\large.}}{\Omega}_{z}\right)\\\
&+n_{x}^{(d)}-2\left(\frac{a_{d11}}{a_{c11}}n_{x}^{(c)}+\frac{a_{d12}}{a_{c22}}n_{y}^{(c)}+\frac{a_{d13}}{a_{c33}}n_{z}^{(c)}\right),\\\
\end{split}$ (24)

where

  * •

$\left[a_{c11},a_{c12},a_{c13}\right]$ is the first line of the common-mode
sensitivity matrix,

  * •

$\left[a_{d11},a_{d12},a_{d13}\right]$ is the first line of the differential-
mode sensitivity matrix $\left[{\mathbf{A}^{(d)}}\right]$,

  * •

$\left[c^{\prime}_{d11},c^{\prime}_{d12},c^{\prime}_{d13}\right]$ is the first
line of the differential-mode angular to linear coupling matrix
$\left[{\mathbf{C}^{\prime(d)}}\right]$,

The first line of Eq. (24) is the effect of the bias in differential and
common mode mostly at DC, which can drift with time. The second line is the
projection onto the sensor frame of the effect of the parasitic differential
accelerations acting on the test-masses with contributions at DC and
$f_{\rm{EP}}$ (systematic errors are detailed in Ref. [5]). The third line is
the projection of the $\delta\vv g$ term on the mean $X$-axis of the two test-
masses. As $a_{c12}$ and $a_{c13}$ are close to $10^{-3}~{}{\rm rad}$, we
consider only $a_{c11}\delta g_{x}$. The 4th, 5th and 6th lines describe the
effect of the test-mass offcentering coupled to the Earth gravity gradient and
the inertia motion projected on the mean $X$-axis of the two test-masses. The
7th line describes the effect of the common mode acceleration due to the
mismatching of scale factors or alignments. The 8th line represents the
coupling of angular motion on the linear acceleration measurement along the
$X$-axis. The last line is the measurement noise contribution.

The calibration process allows us to determine the $a_{c11}\Delta_{x}$,
$a_{c11}\Delta_{y}$, $a_{c11}\Delta_{z}$, $\frac{a_{d11}}{a_{c11}}$,
$\frac{a_{d12}}{a_{c11}}$ and $\frac{a_{d13}}{a_{c11}}$ terms, such that the
calibrated acceleration is given by :

$\begin{split}\Gamma_{x}^{(d,{\rm cal})}=&\left(T_{xx}-In_{xx}\right)^{\rm
mod}[a_{c11}\Delta_{x}]^{\rm cal}+\left(T_{xy}-In_{xy}\right)^{\rm
mod}[a_{c11}\Delta_{y}]^{\rm cal}\\\ &+\left(T_{xz}-In_{xz}\right)^{\rm
mod}[a_{c11}\Delta_{z}]^{\rm cal}\\\
&+2\left(\left[\frac{a_{d11}}{a_{c11}}\right]^{\rm
cal}{\Gamma}_{x}^{(c)}+\left[\frac{a_{d12}}{a_{c22}}\right]^{cal}{\Gamma}_{y}^{(c)}+\left[\frac{a_{d13}}{a_{c33}}\right]^{cal}{\Gamma}_{z}^{(c)}\right)\\\
&-2\left(\left[\frac{a_{d11}}{a_{c11}}\right]^{\rm
cal}n_{x}^{(c)}+\left[\frac{a_{d12}}{a_{c22}}\right]^{\rm
cal}n_{y}^{(c)}+\left[\frac{a_{d13}}{a_{c33}}\right]^{\rm
cal}n_{z}^{(c)}\right),\\\ \end{split}$ (25)

which leads to the corrected measurement equation

$\Gamma_{x}^{(d,{\rm cor})}=\Gamma_{x}^{(d)}-\Gamma_{x}^{(d,{\rm cal})}.$ (26)

##  3 Requirement tree and expected budget performance

Requirements are established by considering all errors that could bias the
measurements (Eqs. 24 and 26). As the looked-for signal is at well known
frequency, aliasing phenomena are also taken into account in the requirement
tree.

We present in this paper two budgets of performance:

  * •

one prior to the launch, with inputs coming from the requirements on all
satellite subsystems for a test in V2 mode;

  * •

an update of this budget established after the flight commissioning phase,
that takes into account new configurations of the satellite (V3 mode) and of
the instrument’s servo-control.

The following subsections focus on the instrument’s $X$-axis. A similar
analysis was performed on the other five degrees of freedom. The derived
requirements are less stringent than on $X$-axis. Ref. [5] gives the
performance obtained with actually measured inputs and should be compared to
the error budget presented here.

###  3.1 Frequencies of interest

Depending on the satellite mode, several frequencies of interest have been
used to check the error budget. Independently of the mode, requirements are
established at the frequency $f_{\rm{EP}}$ of a potential WEP violation signal
and at twice this frequency, $2f_{\rm{EP}}$ corresponding to the main signal
due to the Earth gravity gradient that allows for the estimation of
$a_{c11}\Delta_{x}$ and $a_{c11}\Delta_{z}$. Additionally, $3f_{\rm{EP}}$ has
also been considered to establish the requirements on the system to limit
their projection effect at $f_{\rm{EP}}$ through non linearities. Finally, a
reference signal at $f_{\rm{cal}}$ is used during calibration sessions in
inertial pointing, so that signals at $f_{\rm{cal}}$ and $2f_{\rm{cal}}$ have
also been taken into consideration.

Ref. [11] shows how a signal at any frequency may be projected at
$f_{\rm{EP}}$ because of the finite duration of the observation window, hereby
perturbing the measurement. This leads to the definition of a pattern of the
projection rate at $f_{\rm{EP}}$ of a disturbing signal at a different
frequency. Fig. 2 shows the shape of the pattern determined to reject a signal
at a level of $2\times{}10^{-16}$ m s-2 which is the error allocation of one
error source at $f_{\rm{EP}}$ (see below for error allocation).

Figure 2: Required rejection rate of disturbing signal versus frequency

Frequency | Value
---|---
$f_{0}$ | 0 in inertial pointing
| or ($f_{\rm{EP}}-2f_{\rm{orb}}$) otherwise
$f_{1}$ | $f_{\rm{EP}}-f_{\rm{orb}}/2$
$f_{2}$ | $f_{\rm{EP}}-f_{\rm{orb}}/20$
$f_{3}$ | $f_{\rm{EP}}+f_{\rm{orb}}/20$
$f_{4}$ | $f_{\rm{EP}}+f_{\rm{orb}}/2$
$f_{5}$ | $f_{\rm{EP}}+2f_{\rm{orb}}$

The spin frequencies are fixed so that
$f_{\rm{spin}}=\frac{k}{2}f_{\rm{orb}}$, $k$ being an integer ($k=9$ for
$f_{\rm{spin}_{1}}$ and $k=35$ for $f_{\rm{spin}_{2}})$. Moreover the duration
$T_{ss}$ of the science sessions are fixed to an even number of orbital
periods, $T_{ss}=2nT_{\rm{orb}}$. This ensures that the session contains
integer numbers of orbital periods, of spin periods, and of EP periods:

$T_{ss}=2nT_{\rm orb}=nkT_{\rm spin}=n(k+2)T_{\rm EP}$ (27)

The benefit is a natural decorrelation between periodic signals at frequencies
multiples of $f_{\rm{orb}}$, $f_{\rm{spin}}$ and $f_{\rm{EP}}$. Calibration
sessions are less demanding; in order to ensure an integer number of
calibration periods, their duration has been set to 5.07 orbits (see for
detailed list of sessions in Ref. [3]).

###  3.2 Requirement tree

The specification of all error sources have been determined by considering
four types of errors:

  * •

harmonic errors $S_{\rm{EP}}$ at $f_{\rm{EP}}$ in phase with the gravity
acceleration due to the Earth, source of a possible signal of violation;

  * •

errors $S(f^{\prime})$ at frequencies $f^{\prime}$ around $f_{\rm{EP}}$ that
project to the frequency of interest with a rejection rate
$R^{\prime}(f^{\prime})$;

  * •

harmonic errors $S(nf_{\rm{EP}})$, if $n=1$ in quadrature with the signal we
seek, for $n\neq 1$ at the $n$ multiple $f_{\rm{EP}}$ frequency, the error is
rejected by a factor $R(nf_{\rm{EP}}$);

  * •

stochastic error $S_{\rm{r}}$ around $f_{\rm{EP}}$.

To reach the accuracy of $10^{-15}$, a signal-to-noise ratio of 1 was
considered, implying that the total error of the accelerometric measurement
must be lower than $7.9\times{}10^{-15}$ m s-2 (since the Earth gravity
acceleration amplitude at 710 km is 7.9 m s-2). All errors have been defined
in the frequency domain as a discrete Fourier transform for the errors at the
interest frequencies and as a power spectral density for random errors.
Harmonic errors are added which is a conservative approach since some of the
errors are not correlated and could be added quadratically. The case of
$S(nf_{\rm{EP}})_{n=1}$ has been considered in the same way as $S_{\rm{EP}}$
to establish the requirements. The budget error considered the total error at
$f_{\rm{EP}}$ as $S_{\rm{T}}$ in the following way:

$S_{\rm{T}}^{2}=\left[\sum{S_{\rm{EP}}}+\sum{\frac{S(f^{\prime})}{R(f^{\prime})}}+\sum{\frac{S(nf_{\rm{EP}})}{R(nf_{\rm{EP}})}}\right]^{2}+\sum{\frac{S_{\rm{r}}^{2}}{T_{\rm{ss}}}}.$
(28)

The total allocation error for $S_{\rm{T}}$ is $7.9\times{}10^{-15}$ m s-2
distributed equally between harmonic and random errors to
$(7.9\times{}10^{-15})/\sqrt{2}$ m s-2. This leads to allocate the following
budget over 120 orbits:

$\displaystyle\sqrt{\sum{S_{\rm{r}}^{2}}}$
$\displaystyle<4.7\times{}10^{-12}~{}{\rm m\,s^{-2}Hz^{-1/2}},$ (29)
$\displaystyle\left[\sum{S_{\rm{EP}}}+\sum{\frac{S(f^{\prime})}{R(f^{\prime})}}+\sum{\frac{S(if_{\rm{EP}})}{R(if_{\rm{EP}})}}\right]$
$\displaystyle<5.6\times{}10^{-15}~{}{\rm m\,s^{-2}}.$ (30)

From this global allocation, the number of budget allocations for random error
sources and of systematic errors has been set as follows: 200 allocations for
random sources at a level of $3\times{}10^{-13}$ m s-2 Hz-1/2 each and 28 for
systematic sources at a level of $2\times{}10^{-16}$ m s-2 each. From Eq.
(26), the main error sources have been evaluated in order to distribute
allocations of errors to the subsystems. After several iterative analyses
between the satellite and the instrument team, a final budget is realised for
the main error sources :

  * •

the instrument measurement noise with a particular allocation of
$3.3\times{}10^{-12}$ m s-2 Hz-1/2 in differential mode leaving a budget of
$3.4\times{}10^{-12}$ m s-2 Hz-1/2 for all the other stochastic noises
measured in the difference of acceleration;

  * •

the instrument bias sensitivity to environment (random or systematics). That
includes sensitivity to fluctuations of magnetic field, of local gravity and
of temperature. As the thermal sensitivity is a major constraint in the
design, a larger allocation was fixed to $10^{-15}$ m s-2;

  * •

the instrument parameters $\left[{\mathbf{A}^{(j)}}\right]$ and
$\left[{\mathbf{C}^{(j)}}\right]$ variations (noise or systematics) due to
temperature fluctuations;

  * •

the satellite position knowledge accuracy that could lead to systematic errors
in the evaluation of the gravity field and its gradients;

  * •

the satellite orientation knowledge accuracy that could lead to systematic
errors in the evaluation of the gravity field phase that determines the phase
of the possible WEP violation signal;

  * •

the satellite angular velocity and acceleration noise and systematics;

  * •

the drag-free control of the linear common mode acceleration (noise and
systematics);

  * •

the dating of the measurement data.

###  3.3 Required performance budget

The pre-launch baseline configuration was the spin V2 mode. Table 2 shows the
corresponding expected performance when considering all the error sources.

The sources of error may vary randomly or at the EP frequency. For example, in
science mode, with the DFACS turned on, Eq.(24)’s term
$\frac{a_{d11}}{a_{c11}}{\Gamma}_{x}^{(c)}=\frac{a_{d11}}{a_{c11}}\left({\hat{b}_{0}}_{x}^{(c)}+R_{\rm{df}_{x}}\right)$
can vary with time because the term $\frac{a_{d11}}{a_{c11}}$ varies randomly
(electronic noise) or systematically (temperature variations), and because the
drag-free performance $R_{\rm{df}_{x}}$ varies randomly (gas thruster noise)
or systematically (star sensor harmonic errors). Depending on the origin of
the disturbance, the errors are classified in different topics on the tables:
Earth gravity gradients, instrument gravity, angular motions, temperature
variations, etc. The drag-free performance expressed by $R_{\rm{df}_{x}}$ is
listed in the Drag-free residuals. The accelerometer noise and parasitic
forces comprise the electronic measurement noise, the effect of the
temperature gradients (radiometer effect, radiation pressure), the gold wire
stiffness and damping and the contact potential difference impact.

Table 2:  Expected a priori budget of performance in V2 mode.

  * Error source | Contribution in | Contribution in
---|---|---
SUREF in V2 | Random noise | Harmonic error
| m s-2 Hz-1/2 | m s-2
Earth gravity gradients | $0.0\times{}10^{-13}$ | $0.6\times{}10^{-16}$
Instrument gravity | $0.6\times{}10^{-13}$ | $2.0\times{}10^{-16}$
Satellite gravity gradients | $1.2\times{}10^{-13}$ | $3.1\times{}10^{-16}$
Angular motions | $3.8\times{}10^{-13}$ | $11.0\times{}10^{-16}$
Instrument parameter variations | $6.0\times{}10^{-13}$ | $8.5\times{}10^{-16}$
Accelerometer measurement noise | $14.6\times{}10^{-13}$ |
and parasitic forces |  |
Temperature variations | $2.1\times{}10^{-13}$ | $8.6\times{}10^{-16}$
Drag-Free Residuals | $1.0\times{}10^{-13}$ | $5.0\times{}10^{-16}$
Magnetic sensitivity | $0.8\times{}10^{-13}$ | $4.0\times{}10^{-16}$
Non linearity | $3.7\times{}10^{-13}$ | $8.1\times{}10^{-16}$
Total | $16.9\times{}10^{-13}$ | $55.9\times{}10^{-16}$
EP test budgeted for $\delta:0.7\times{}10^{-15}$ | with $g=7.9$ m/s2 |
| over 120 orbits |

The driving rules to establish each term of the table are detailed below.

###  3.4 Gravity field signal

As they determine of the phase of $\vv{g}$ in Eq. (24), the date, the position
and the orientation of the satellite with respect to the Earth must be
measured. We use the method detailed in Ref. [12] to compute the Earth gravity
acceleration and its gradient tensor projected onto the instrument frame using
the ITSG-Grace2014s gravity potential model [13] expanded up to spherical
harmonic degree and order 50.

The position of the satellite is obtained from the Doppler telemetry
measurements associated to the on-board GPS data [2]. Table 3 gives the
requirements on the satellite’s position to limit the error of estimation on
$T_{xx}$ and $T_{xz}$ at $f_{\rm{EP}}$. The offcentering is specified to
20$\mu$m on $\Delta x$ and $\Delta z$ with a knowledge accuracy of 0.1$\mu$m
and to 20$\mu$m on $\Delta y$ with a knowledge accuracy of 2$\mu$m. The
pointing stability and knowledge has to be also accurate to limit the effect
of the gravity gradients.

The altitude of the satellite determines the intensity of the gravity field
and has been specified to 710 km as a compromise between the maximization of
the signal magnitude and the minimization of the external forces on the
satellite: the Solar radiation pressure and drag forces are of the same order
of magnitude at about 700 km. The final figure was fixed in agreement with the
main passenger of the Soyutz launch. The satellite has to follow a (near
polar) sun-synchronous orbit to maximise the thermal stability. The orbit’s
eccentricity must be lower than $5\times{}10^{-3}$ to limit the amplitude of
the gravity gradient components at $f_{\rm{EP}}$.

Table 3:  Satellite position knowledge and pointing accuracy requirements

  * Frequency | Radial | Tangent | Cross track | Orientation
---|---|---|---|---
DC | 100 m | 100 m | 2 m | 2.5$\times{}10^{-3}$ rad
|  |  |  | known at $10^{-3}$ rad
$f_{\rm{EP}}$ | 7 m | 14 m | 100 m | 10 $\mu$rad
|  |  |  | known at 1 $\mu$rad
2$f_{\rm{EP}}$ | 100 m | 100 m | 2 m | not stringent
3$f_{\rm{EP}}$ | 2 m | 2 m | 100 m | 10 $\mu$rad
|  |  |  | known at 1 $\mu$rad

###  3.5 Instrument and satellite self gravity

The gravity field generated by local moving test-masses could also mimic a
gravity signal at $f_{\rm{EP}}$. As the test-masses cannot be considered as
point masses for local gravity, the difference of gravity exerted on the test-
masses depends on their shape. Refs [14, 15] show that for a cylindrical test
mass, the gravitational potential produced by a moving mass $M_{s}$ at the
distance $R_{s}$ making an angle $\theta$ with the cylinder axis can be
expressed as a sum of Legendre polynomials $P_{2p}$ with form factors $k_{2p}$
depending only on geometry:

$V(R_{s})=-\frac{GM_{s}}{R_{s}}\sum_{p=0}^{\infty}k_{2p}\frac{P_{2p}(cos\theta)}{R_{s}^{2p}},$
(31)

Order 0 is the point source case. When looking for the difference of gravity
field $\Delta g$ exerted on two concentric test-masses by the moving mass,
order 0 disappears and it remains only order 1 depending on the test-mass’s
moments of inertia:

$\Delta g\approx 12\frac{GM_{s}}{R_{s}^{4}}\left(\frac{J}{m}\right)_{\rm
max}\frac{\Delta J}{J},$ (32)

where $\Delta J/J$ represents the mean difference
$(J_{i}-J_{j})_{i=x,y,z;j\neq i}$ of the test-mass’ moments of inertia about
$X$, $Y$ and $Z$. If all moments of inertia are identical about the three
degrees of freedom, then the test-mass can be considered point-like and the
sensitivity to local moving mass is reduced. For a perfect hollow cylinder of
inner radius $R_{1}$ and outer radius $R_{2}$, if the length of the test-mass
is defined as $L=\sqrt{\frac{3}{4}(R_{2}^{2}+R_{1}^{2})}$, then the moments of
inertia are equal along the three axes at second order in $R_{i}^{2}$. As
test-masses have some flat parts and holes, this relation must be adjusted by
computing the moment of inertia of the real shape. The ratio $\Delta J/J$ can
also vary with the geometry accuracy or with the density inhomogeneity. A
specification on each error contributor has been set to $2\times{}10^{-4}$.

Thermal dilation of the satellite and of the instrument changes their mass
distribution and thus the local gravity field, impacting the dynamics of test-
masses. Specifications on the self-gravity at 2$f_{\rm{EP}}$ are driven by the
required precision of the offcentering estimation (performed with the
component of the Earth’s gravity gradient at 2$f_{\rm{EP}}$). The acceleration
obtained with the self-gravity coupled to the offcentering of 20$\mu$m must be
lower than the acceleration residual of the Earth’s gravity gradient effect
once the offcentering is estimated at a precision of 0.1$\mu$m. The main
specifications are presented in Table 4.

Table 4:  Main requirements on self gravity induced by the satellite and the
instrument.

  * Component | Specification | Constraining source
---|---|---
DC gravity |  $5\times{}10^{-9}$ m s-2 | Mass distribution
Differential acceleration at $f_{\rm{EP}}$ | 10-16 m s-2 | Test-mass shape
Common mode acceleration at $f_{\rm{EP}}$ | 10-12 m s-2 | Thermal expansion
Common mode acceleration at 2$f_{\rm{EP}}$ |  $2\times{}10^{-11}$ m s-2 | Thermal expansion
Gravity gradients at $f_{\rm{EP}}$ |  $5\times{}10^{-12}$ s-2 | Mass distribution
|  | with thermal expansion
Gravity gradients at 2$f_{\rm{EP}}$ |  $5\times{}10^{-10}$ s-2 | Mass distribution
|  | with thermal expansion

###  3.6 Angular motion

Angular motion is dominated by centrifugal and Coriolis effects defined by the
matrix $[{\rm In}]$. The 20$\mu$m offcentering specification is considered as
baseline. At first order the first line of $\left[{\mathbf{A}^{(c)}}\right]$
can be approximated by $\left[1+O(10^{-2}),O(10^{-3}),O(10^{-3})\right]$. The
differential linear acceleration induced by the angular velocity and the
angular acceleration thus reads:

$(-\Omega_{y}^{2}-\Omega_{z}^{2})\Delta_{x}+(\Omega_{x}\Omega_{y}-\dot{\Omega}_{z})\Delta_{y}+(\Omega_{x}\Omega_{z}+\dot{\Omega}_{y})\Delta_{z}.$
(33)

A precise estimation of the angular motion is estimated a posteriori [2] which
helps to estimate this error contribution. It can be also corrected in the
data science process [16].

To derive the requirements from Eq. (33), we note that $[\Omega]$ and
$[\dot{\Omega}]$ are distributed over several frequency components:
$[\Omega]=[\Omega]^{0}+[\Omega]^{f_{\rm{EP}}}+[\Omega]^{2f_{\rm{EP}}}+[\Omega]^{3f_{\rm{EP}}}$,
since $[\Omega]$ is squared in the equation. When spinning the satellite,
$\Omega_{y}=2\pi f_{\rm{spin}}$ is much larger than the angular velocity
residual in inertial pointing. This last term determines the specification at
$f_{\rm{EP}}$, then the specification at $2f_{\rm{EP}}$ and $3f_{\rm{EP}}$ are
derived.

Requirements have been established for all satellite modes. Table 5 summarises
the main specifications in V2 rotating mode. During calibration sessions, the
specifications at $f_{\rm{cal}}$, 2$f_{\rm{cal}}$ and 3$f_{\rm{cal}}$ are
relaxed by two to three orders of magnitude compared to the ones at
$f_{\rm{EP}}$, 2$f_{\rm{EP}}$ and 3$f_{\rm{EP}}$ as the sensitivity needed in
the differential acceleration measurement is about a few 10-12 m s-2, three
orders of magnitudes higher than the WEP test sensitivity target. Pointing is
also specified to prevent a projection of transverse component of the gravity
gradients at $f_{\rm{EP}}$ and 2$f_{\rm{EP}}$.

Table 5:  Main requirements on satellite angular motion in V2 mode.

  * Component | Specification | In instrument frame
---|---|---
Sat. pointing |  |
at $f_{\rm{EP}}$ and $3f_{\rm{EP}}$ |  $10\mu$rad | all axes
A posteriori knowledge |  $1\mu$rad | all axes
Angular velocity |  |
at $f_{\rm{EP}}$ |  $5\times{}10^{-9}$ rad s-1 |  $\Omega_{x}$ and $\Omega_{y}$
at $f_{\rm{EP}}$ |  $3.5\times{}10^{-7}$ rad s-1 | $\Omega_{z}$
at 2$f_{\rm{EP}}$ and 3$f_{\rm{EP}}$ |  $3.5\times{}10^{-7}$ rad s-1 | all axes
Angular acceleration |  |
at $f_{\rm{EP}}$ |  $5\times{}10^{-12}$ rad s-2 | all axes
at 2$f_{\rm{EP}}$ and 3$f_{\rm{EP}}$ |  $10^{-9}$ rad s-2 | all axes

###  3.7 Instrumental parameters

This subsection applies to the effect of temporal variations of instrumental
parameters $[A^{(c)}]$ and $[A^{(d)}]$. The temperature stability of the
instrument parameters is discussed in section 3.9. The expression of the
disturbance can be summarised as
$[nA^{(c)}]\gamma^{(d)}+[nA^{(d)}]\gamma^{(c)}$, where $[nA^{(j)}]$ is the
stochastic noise of the matrix $[A^{(j)}]$. Table 6 lists the main
requirements on the instrument’s parameters stability under some hypotheses on
$\gamma^{(d)}$ and $\gamma^{(c)}$. For $\gamma^{(c)}$, a mean acceleration of
$3\times{}10^{-8}$ m s-2 has been specified. This acceleration is the residual
acceleration applied to the satellite, the DC bias of the accelerometer being
subtracted from the thruster command. For the mean differential acceleration
$\gamma^{(d)}$, the DC gravity gradient or angular motion effects have a
negligible effect coupled to the stability of alignments. For the scale factor
stability, $\gamma^{(d)}$ is dominated by the part of the differential bias
sensitive to the scale factor variation on $X$: $2.5\times{}10^{-8}$ m s-2.

Table 6:  Main requirements on instrument’s parameters.

  * Component | Common | Differential | SU | FEEU
---|---|---|---|---
| noise | noise | Temp. sensitivity
Scale factor |  $10^{-5}$ Hz-1/2 |  $3\times{}10^{-6}$ Hz-1/2 |  $8\times{}10^{-6}$ K-1 |  $2\times{}10^{-6}$ K-1
Alignment |  $10^{-5}$ rad Hz-1/2 |  $10^{-7}$ rad Hz-1/2 |  $10^{-7}$ K-1 |  $2\times{}10^{-8}$ K-1
Lin. Coupling |  $10^{-7}$ rad Hz-1/2 |  $5\times{}10^{-8}$ K-1 |  $6\times{}10^{-9}$ K-1
Ang. Coupling |  $10^{-8}$ (m s-2)/( rad s-2) Hz-1/2 |  $5\times{}10^{-9}$ K-1 |  $6\times{}10^{-10}$ K-1

###  3.8 Accelerometer noise

This subsection is devoted to the stochastic measurement noise of the
accelerometer. Systematic errors are discussed in subsections 3.5, 3.9 and
3.11. The measurement noise of the accelerometer is driven by several sources
[8, 7]:

  * •

Electrostatic stiffness coupled to the servo loop position noise;

  * •

Actuation noise of the electronics that applies the voltage to the electrodes
in order to create electrostatic forces; this noise includes the digital
conversion noise;

  * •

Contact Potential Difference (CPD) at the surface of electrodes and of test-
masses that disturbs the applied voltages, evaluated to less than $10\mu$V
Hz-1/2;

  * •

The stiffness and mechanical damping of the 7 $\mu$m gold wire that applies
the reference voltage to the test-mass.

Random or systematic temperature variations that have an effect on scale
factors, alignments, electronics biases and parasitic forces (radiation
pressure, radiometer effect, outgassing) are discussed in subsection 3.9.

The dominant term is the gold wire damping on the $X$-axis and the
electrostatic stiffness effect and actuation noise for the $Y$ and $Z$ axes.
Prior to the launch, two measurement channels were foreseen for $X$: one
science channel with a specification of $2\times{}10^{-12}$ m s-2 Hz-1/2 and
one DFACS channel with a specification of $10^{-11}$ m s-2 Hz-1/2. On the less
demanding axes, $Y$ and $Z$, the specification was set to $10^{-10}$ m s-2
Hz-1/2.

After the launch, the science channel on one SUEP test-mass was saturated,
which led to a new measurement strategy: the DFACS channels are used for the
science, the three remaining science channels being used only for instrument
characterisation. The performance presented here is realised on the DFACS
utilisation baseline.

###  3.9 Thermal sensitivity

The acceleration measurement is sensitive to the temperature of the SU and of
the FEEU. The sensitivities of $[A^{(c)}]$, $[A^{(d)}]$ and $[C^{(d)}]$ to the
SU and to the FEEU temperature interface have been specified. The driving
terms are the same as in subsection 3.7. In addition to these terms, parasitic
forces depending on the temperature must be taken into account.

Residual gas molecules hit the test-mass and generate a radiometer effect
force that can be expressed as a disturbing acceleration

$\frac{1}{2}P_{r}\frac{S_{\rm TM}}{m_{i}T_{\rm SU}}\alpha(f)\nabla T_{\rm
SU}L_{\rm TM},$ (34)

with the SU internal pressure specified to $P_{\rm{r}}=10^{-5}$ Pa, with
$S_{\rm TM}$ the area along $X$ of the test-mass of mass $m_{i}$ and length
$L_{\rm TM}$, with $\alpha(f)$ the thermal filtering depending on the
frequency of measurement and $\nabla T_{\rm SU}$ the temperature gradient
along the $X$-axis.

Similarly, photons hit the test mass with a radiation pressure which
accelerates the test-mass by

$\frac{16}{3}\sigma_{b}T_{\rm SU}^{3}\frac{S_{TM}}{m_{i}c}\alpha(f)\nabla
T_{\rm SU}L_{\rm TM},$ (35)

with $\sigma_{b}$ the Boltzmann constant and $c$ the light velocity.

The thermal filtering is considered between the SU interface (where
temperature probes are) and the electrode set surrounding the test-mass from
which thermalised molecules or photons leave the surface to hit the test-mass.

The bias can vary because of the Contact Potential Difference sensitivity to
temperature which is specified to 15 $\mu$V K-1. It can also vary because of
the variation of the electronics voltages (the AC and DC test-mass voltages
–$V_{p}$ and $V_{d}$–, the actuation voltages, the secondary power voltage).
For instance, the requirement on $V_{p}$ has been set to $8\mu$V K-1, the bias
stability actuation voltage is specified to better than $2\mu$V K-1 and the
actuation gains specified to $2\times{}10^{-4}$ K-1 for the most constraining
specifications. A particular care has also been taken to limit the variations
of the power supply at $f_{EP}$ in order to minimise the variation of the
power dissipation inside the FEEU [5].

The bias due to the electrostatic stiffness is proportional to [7]

$\frac{S_{\rm TM}}{e_{j}m_{k}}(V_{p}^{2}+V_{d}^{2}),$ (36)

with $e_{x}$ the gap between the $j$th test-mass and any grounded surface.
This term is sensitive to the SU temperature because of the thermal expansion
of the electrode set or of the test-mass and sensitive to the FEEU temperature
because of the test-mass voltage sensitivity.

Finally, the bias due to the gold wire stiffness varies with the SU
temperature through the thermal expansion of the gold wire and its Young
modulus sensitivity to temperature.

The budget of the temperature sensitivity is summarised in Table 7.

Table 7:  Thermal sensitivity requirements of the instrument on $X$-axis.

  * Component | SU | FEEU
---|---|---
| Temp. sensitivity
Scale factor |  $8\times{}10^{-6}$ K-1 |  $2\times{}10^{-6}$ K-1
Alignment |  $10^{-7}$ K-1 |  $2\times{}10^{-8}$ K-1
Lin. Coupling |  $5\times{}10^{-8}$ K-1 |  $6\times{}10^{-9}$ K-1
Ang. Coupling |  $5\times{}10^{-9}$ K-1 |  $6\times{}10^{-10}$ K-1
Bias |  $10^{-12}$ m s-2K-1 |  $10^{-13}$ m s-2K-1
Bias sensitivity to |  |
temperature gradients |  $2\times{}10^{-12}$ m s-2(K/m)-1

###  3.10 Drag-Free

The specification on the linear acceleration level applied to the
accelerometer is driven by the DFACS control residuals. The leading component
is similar to the instrument parameter $[A^{(d)}]\gamma^{(c)}$. But here we
focus on the stochastic and systematic variations of $\gamma^{(c)}$ detailed
in Table 8. The scale factor matching and the alignments calibrated to about
$10^{-4}$ accuracy allow us to approximate $[A^{d}]$.

Table 8:  Drag free requirements along $X$, $Y$ and $Z$.

  * Frequency | Specification | Applicability
---|---|---
Random about $f_{\rm{EP}}$ |  $3\times{}10^{-10}$ m s-2 Hz-1/2 | EP test and calibration
At $f_{\rm{EP}}$, $2f_{\rm{EP}}$ and $3f_{\rm{EP}}$ |  $10^{-12}$ m s-2 | EP test in inertial pointing
At $f_{\rm{EP}}$ |  $10^{-12}$ m s-2 | EP test in rotating mode
At $2f_{\rm{EP}}$ and $3f_{\rm{EP}}$ |  $10^{-11}$ m s-2 |
At $f_{\rm{cal}}$, $2f_{\rm{cal}}$ and $3f_{\rm{cal}}$ |  $5\times{}10^{-12}$ m s-2 | Calibration in inertial pointing

###  3.11 Magnetic field variations

The analysis of the magnetic effects on the acceleration of the test-mass is
detailed in Ref. [5]. An allocation of $4\times{}10^{-16}$ m s-2 at $f_{\rm
EP}$ is considered for the difference of acceleration, and $8\times{}10^{-15}$
m s-2 at $2f_{\rm EP}$.

###  3.12 Non linearities

On the $X$-axis, a non-linearity can be modelled by an adding the term

$4K_{2x}^{(c)}{\gamma_{x}^{(c)}}{\gamma_{x}^{(d)}}+K_{2x}^{(d)}\left(4{\gamma_{x}^{(c)}}^{2}+{\gamma_{x}^{(d)}}^{2}\right)$
(37)

to Eq. (17), where $K_{2x}^{(c)}$ and $K_{2x}^{(c)}$ are respectively the
common and differential measurement quadratic term. By considering the DFACS
performance specification above, the resulting specification on non-linear
term has been established with an allocation of 4 times
$2\times{}10^{-16}~{}{\rm m\,s}^{-2}$ at $f_{\rm EP}$ for the two main terms
in $\gamma_{x}^{(c)}$. This leads to specify $K_{2x}^{(c)}$ and $K_{2x}^{(d)}$
to be lower than $14000~{}{\rm s^{2}m}^{-1}$.

##  4 A posteriori budget from in-flight measurement

Table 2 has been updated after the commissioning phase. Several inputs to the
budget of error had to be modified. First, the instrument bias was greater
than expected due to the higher gold wire stiffness. That led to use the DFACS
channel for the $X$-axis science measurement instead of the particular science
channel. To cope with the necessity of a higher ratio measurement range over
bias, the electrode voltages were biased by a constant in order to change the
scale factor [8]. That also implied updating the instrument’s control laws.
Secondly, this stiffness also induces a higher damping of the gold wire and
thus a higher noise. By increasing the frequency rate of the satellite
rotation (V3 mode), it is possible to take advantage of the decreasing effect
of the noise with frequency and of a better temperature filtering. The thermal
sensitivity is also higher than expected but fortunately, the variation of
temperature is much lower than considered in Table 2 because of the higher
rotation rate of the satellite: this topic is detailed in Ref. [5]. Table 9
takes into account the new flight operation configuration for the satellite
and the instrument and their associated environment. The random noise is
directly deduced from accelerometer in flight measurement and implicitly
includes all error sources. The DFACS performance results from the analysis of
the accelerometer and the star sensor outputs combined to a fine
trajectography in Ref. [2]. It corresponds to the budget actually estimated in
orbit, with the real instrument configuration and noise performances in the
case of the SUEP instrument.

Table 9:  Expected budget of performance updated after commissioning phase,
with updated data from instrument and DFACS in V3 mode.

  * Error source | Contribution in | Contribution in
---|---|---
SUEP in mode V3 | Random noise | Harmonic error
| m s-2 Hz-1/2 | m s-2
Earth gravity gradients |  | $2.6\times{}10^{-16}$
Instrument gravity |  | $2.0\times{}10^{-16}$
Satellite gravity gradients |  | $1.7\times{}10^{-16}$
Angular motions |  | $3.8\times{}10^{-16}$
Instrument parameter variations |  | $18.6\times{}10^{-16}$
Accelerometer measurement noise | $404\times{}10^{-13}$ |
and parasitic forces |  |
Temperature variations |  | $41.2\times{}10^{-16}$
Drag-Free Residues |  | $1.7\times{}10^{-16}$
Magnetic sensitivity |  | $4.0\times{}10^{-16}$
Non linearity |  | $3.8\times{}10^{-16}$
Total | $404\times{}10^{-13}$ | $79.5\times{}10^{-16}$
EP test budgeted for $\delta:6.1\times{}10^{-15}$ | over 120 orbits |
and for $\delta:2.0\times{}10^{-15}$ | over 1400 orbits |
with $g=7.9$ m/s2 |  |

##  5 Conclusion

The mission MICROSCOPE has been designed to fulfil the objective of a test at
$10^{-15}$ accuracy level. All requirements on satellite subsystems have been
defined to achieve this objective. As shown, a detailed analysis of all error
sources leads to establish a budget of error for the mission. This analysis
was updated during the whole mission development duration and served as a
guideline to make compromises when it was necessary. This tool was shared
within the mission team and in particular within the satellite and instrument
team in the framework of the MICROSCOPE performance group which has been
meeting monthly for more than 15 years. The last version of the mission error
budget was established with the analysis of the inflight performance. As some
subsystem could not be fully tested on ground in their flight configuration,
reaching the objective was a very challenging task. Also some unexpected
events and breakdowns led the team to reconsider the mission scenario [3] in
order to preserve firstly the integrity of all systems and secondly to get the
best possible performance despite difficulties. Ref. [8, 2, 7, 5] detail the
analysis of the major subsystem errors that finally leads to reach an accuracy
test of about $2\times{}10^{-15}$ by considering 1400 orbits of measurement.
This figure is not far from the obtained result when each session performance
is considered and cumulated [17]. It is almost an improvement by a factor 6
compared to the first published results [1, 18]. The long path to this success
gives precious lessons that should help to design a new mission with much
better accuracy.

The authors express their gratitude to all the different services involved in
the mission partners and in particular CNES, the French space agency in charge
of the satellite. This work is based on observations made with the T-SAGE
instrument, installed on the CNES-ESA-ONERA-CNRS-OCA-DLR-ZARM MICROSCOPE
mission. ONERA authors’ work is financially supported by CNES and ONERA
fundings. Authors from OCA, Observatoire de la Côte d’Azur, have been
supported by OCA, CNRS, the French National Center for Scientific Research,
and CNES. ZARM authors’ work is supported by the DLR, German Space Agency,
with funds of the BMWi (FKZ 50 OY 1305) and by the Deutsche
Forschungsgemeinschaft DFG (LA 905/12-1). The authors would like to thank the
Physikalisch-Technische Bundesanstalt institute in Braunschweig, Germany, for
their contribution to the development of the test-masses with funds of CNES
and DLR.

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