MICROSCOPE's constraint on a short-range fifth force
Joel Bergé, Martin Pernot-Borr?s, Jean-Philippe Uzan, Philippe Brax, Ratana Chhun, Gilles Métris, Manuel Rodrigues, Pierre Touboul
aa r X i v : . [ g r- q c ] J a n MICROSCOPE’s constraint on a short-range fifthforce
Joel Berg´e , Martin Pernot-Borr`as , Jean-Philippe Uzan ,Philippe Brax , Ratana Chhun , Gilles M´etris , ManuelRodrigues , Pierre Touboul DPHY, ONERA, Universit´e Paris Saclay, F-92322 Chˆatillon, France Sorbonne Universit´e, CNRS, Institut d’Astrophysique de Paris, IAP, F-75014 Paris,France CNRS, Institut d’Astrophysique de Paris, IAP, F-75014 Paris, France Institut Lagrange de Paris, 98 bis, Bd Arago, 75014 Paris, France Institut de Physique Th´eorique, Universit´e Paris-Saclay, CEA, CNRS, F-91191Gif-sur-Yvette Cedex, France Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS, IRD, G´eoazur, 250avenue Albert Einstein, F-06560 Valbonne, FranceE-mail: [email protected]
July 2020
Abstract.
The MICROSCOPE experiment was designed to test the weakequivalence principle in space, by comparing the low-frequency dynamics of cylindrical“free-falling” test masses controlled by electrostatic forces. We use data taken duringtechnical sessions aimed at estimating the electrostatic stiffness of MICROSCOPE’ssensors to constrain a short-range Yukawa deviation from Newtonian gravity. Wetake advantage of the fact that in the limit of small displacements, the gravitationalinteraction (both Newtonian and Yukawa-like) between nested cylinders is linear, andthus simply characterised by a stiffness. By measuring the total stiffness of the forcesacting on a test mass as it moves, and comparing it with the theoretical electrostaticstiffness (expected to dominate), it is a priori possible to infer constraints on theYukawa potential parameters. However, we find that measurement uncertainties aredominated by the gold wires used to control the electric charge of the test masses,though their related stiffness is indeed smaller than the expected electrostatic stiffness.Moreover, we find a non-zero unaccounted for stiffness that depends on the instrument’selectric configuration, hinting at the presence of patch-field effects. Added to significantuncertainties on the electrostatic model, they only allow for poor constraints on theYukawa potential. This is not surprising, as MICROSCOPE was not designed forthis measurement, but this analysis is the first step to new experimental searches fornon-Newtonian gravity in space.
Keywords : Experimental Gravitation, Modified Gravity, Yukawa potential, Electrostaticaccelerometer
Submitted to:
Class. Quantum Grav. ifth force stiffness
1. Introduction
A hundred years after its invention, Einstein’s theory of General Relativity (GR) stillpasses all experimental tests [1], from early tests (the Mercury perihelion puzzle andthe measurement of the gravitational deflection of stars’ light passing near the Sunby Eddington) to current tests (gravitational lensing [2, 3], gravitational redshift [4, 5],gravitational waves direct detection [6]). However, in order to withstand (not so recent)astrophysical and cosmological observations, GR must be supplemented by dark matterand dark energy. The former explains the flat rotation curve of galaxies and theirdynamics in clusters [7, 8], while the latter explains the acceleration of the cosmicexpansion [9, 10]. Whether our theory of gravitation must be revised or the contentof our Universe better understood is still an open discussion [11, 12]. In this article, weadopt the former possibility.Theories beyond the standard model propose the existence of new fields andparticles. For instance, string-inspired theories introduce a spin-0 dilaton-like particle(see e.g. Refs. [13, 14]), while scalar-tensor models modify GR’s equations via theintroduction of a new scalar field (see e.g. Refs. [11, 15, 16]). Although a new verylight scalar field should entail the appearance of a new long-range force incompatiblewith current Solar System tests, its existence can be made compatible with experimentalconstraints by virtue of a screening mechanism that makes the field’s mass environment-dependent, thereby hiding it from local experimental tests [13,17–24]. Those models cannevertheless have measurable effects, such as an apparent violation of the equivalenceprinciple (see e.g. Refs. [20, 25]) or a variation of fundamental constants [26, 27].Looking for short-range deviations from Newtonian gravity is essential to testlow-energy limits of high-energy alternative theories (such as string theory or extradimensions) and is the goal of several experimental efforts (see Refs. [28–30] for reviewsand references therein, and Refs. [31, 32] for recent results). While most of them arehighly optimised to look for specific minute signals, we aim, in this article, to searchfor a short-range deviation from Newtonian gravity as a byproduct of MICROSCOPEdata.The MICROSCOPE space experiment tested the weak equivalence principle (WEP)to a record accuracy [33, 34] via the comparison of the acceleration of two test massesfreely falling while orbiting the Earth. If the WEP is violated, a signal is expectedat a frequency defined as the sum of the satellite’s orbital and spinning frequencies.Since MICROSCOPE orbits the Earth at a 700 km altitude, the experiment is thenmostly sensitive to long-ranged (more than a few hundred kilometers) modifications ofgravitation. Its first results thus allowed us to set new limits on beyond-GR modelsinvolving long-range deviations from Newtonian gravity parametrised by a Yukawapotential, a light dilaton [35] and a U-boson [36, 37]. Updates of those works are underway following the final MICROSCOPE results [38, 39].In this article, we use MICROSCOPE sessions dedicated to the in-flightcharacterisation of its instrument to look for short-range deviations of Newtonian ifth force stiffness
2. Yukawa gravity
We parametrise a deviation from Newtonian gravity with a Yukawa potential, which issimply added to the Newtonian potential. The total gravitational potential created bya point-mass of mass M at distance r is then V p r q “ ´ GMr ” ` α exp ´ ´ rλ ¯ı , (1)where G is Newton’s gravitational constant, α is the strength of the Yukawa deviationcompared to Newtonian gravity and λ is the range of the corresponding fifth force.Despite its simplicity, the Yukawa parametrisation is useful as it describes thefifth force created by a massive scalar field in the Newtonian regime (see e.g. theSupplemental material of Ref. [35] and references therein). The range λ correspondsto the Compton wavelength of the scalar field, and α is linked to its scalar charge.Phenomenologically, this charge can depend on the composition of the interacting bodies ifth force stiffness α as the parameter to constrain.Many experiments have already provided tight constraints on its range and strength,from sub-millimeter to Solar System scales (e.g. Refs. [28,29] and references therein, andRefs. [35, 41–44] for more recent works). In this article, we are concerned with rangesbetween λ « ´ m and λ « ´ m, corresponding to the scale of MICROSCOPE’sinstrument. The best constraints on the strength of a Yukawa potential for such rangesare | α | ď ´ [31, 45, 46].
3. MICROSCOPE experiment concept and test masses dynamics
MICROSCOPE was designed as a test of the universality of free-fall, relying on an easyrecipe: drop two test bodies and compare their fall. However, instead of letting twotest bodies freely orbit the Earth and monitoring their relative drift, MICROSCOPEuses electrostatic forces to maintain two test masses centered with respect to each other.This is done with a differential ultrasensitive electrostatic accelerometer, consisting oftwo coaxial and concentric cylinders made of different materials. The difference ofelectric potentials applied to keep the cylinders in equilibrium is a direct measure of thedifference in their motion.This section provides a primer about the MICROSCOPE experimental concept. Westart with a description of the capacitive detection and electrostatic control principle,driving us to a short presentation of the instrument. We then present a simplifiedequation of the dynamics of a test mass. Details can be found in Refs [47–49].
The electrostatic control of the test masses relies on two nested control loops. Thefirst one is inside the payload: each test mass is placed between pairs of electrodes andits motion with respect to its cage is monitored by capacitive sensors. It can be keptmotionless by applying the electrostatic force required to compensate all other forces,such that the knowledge of the applied electrostatic potential provides a measurementof the acceleration which would affect the test mass with respect to the satellite in theabsence of the control loop. Note that even if there is no net motion with respectto the satellite, it is common and convenient to call electrostatic acceleration theelectrostatic force divided by the mass; this definition will be used all along the paper.The second control loop is included in the satellite’s Drag-Free and Attitude ControlSystem (DFACS), which aims to counteract external disturbances via the action of coldgas thrusters. This system also ensures a very accurate control of the pointing and ofthe attitude of the satellite from the measurements of angular position delivered by thestellar sensors and of the angular acceleration delivered by the instrument itself. ifth force stiffness C and C . The motion of the test mass induces avariation in the capacitances; the detector senses it and outputs a related voltage V det “ V d p C ´ C q{ C eq , where V d is the potential of the test mass and C eq is thecapacitance of the capacitor formed by the test mass and the electrodes when the testmass is at the centre of the cage. The capacitances C i depend on the geometry ofthe sensor, and therefore their form differ along the longitudinal and radial axes of theinstrument; nevertheless, it can be shown that at first order, V det is proportional to thedisplacement δ of the test mass about the center of the cage along all axes [48].The control loop digitises the detector output voltage V det and computes theactuation voltage to apply to the electrodes in order to compensate for the displacementof the test mass, and recentre it in the cage. The (restoring) electrostatic force appliedby an electrode i is then F el ,i “ p V i ´ V p q ∇ C, (2)where ∇ C is the spatial gradient of the capacitance. The (polarisation) potential of thetest-mass V p is maintained constant and the potential V i of the electrode is tuned bythe servo-control loop. This “action” takes the general form F el ,i « ´ m G act V e ` ω p « ` ˆ V e V p ˙ ff δ + , (3)where the sensitivity factor G act and the stiffness coefficient ω p depend on the geometryof the sensor, V e is the voltage output from the control loop and applied to the electrodes,and m is the mass of the test mass.If G act is known well enough, the acceleration of the test-mass can be measuredthrough the voltage V e required to apply the restoring force. This measurement isperturbed by the electrostatic stiffness, which introduces a bias if the test-mass is notservo-controlled to the equilibrium point. Nevertheless, the asymmetry in the designof the electrostatic configuration and the displacement are sufficiently small to ignoreit during nominal WEP test operations. Instead, in this paper, we use measurementsessions where the displacement δ is not negligible, allowing for the estimation of theelectrostatic stiffness. The core of MICROSCOPE’s instrument consists of two differential accelerometers (orSensor Units – SU), the test masses of which are co-axial cylinders kept in equilibrium ifth force stiffness Figure 1.
Cutaway view of a MICROSCOPE sensor, with its two test masses, theirsurrounding electrodes-bearing cylinders, cylindrical invar shield, base plate, upperclamp and vacuum system. The volume of a sensor is 348 ˆ ˆ
180 mm ; the testmasses have length ranging from 43 mm to 80 mm and outer radii ranging from 39 mmto 69 mm, and are separated from the electrode-bearing cylinders by 600 µ m gaps [48].The reference system is shown on the left of the figure. Figure from Ref. [34, 48]. with electrostatic actuation [48]. The test masses’ materials were chosen carefully soas to maximize a potential violation of the WEP from a light dilaton [13, 50, 51] andto optimise their industrial machining: the SUEP (Equivalence Principle test SensorUnit) test masses are made of alloys of platinum-rhodium (PtRh10 – 90% Pt, 10% Rh)and titanium-aluminium-vanadium (TA6V – 90% Ti, 6% Al, 4% V), while the SUREF(Reference Sensor Unit) test masses are made of the same PtRh10 alloy. For eachsensor, we call “IS1” the internal test mass and “IS2” the external one. For instance,the internal mass of SUREF is named IS1-SUREF, and the external mass of SUEP iscalled IS2-SUEP.As shown above, the test masses of each SU are controlled electrostatically with nomechanical contact but a thin 7 µ m-diameter gold wire used to fix the masses’ electricalpotential to the electronics reference voltage. Two Front-End Electronics Unit (FEEU)boxes (one per SU) include the capacitive sensing of masses, the reference voltage sourcesand the analog electronics to generate the electrical voltages applied to the electrodes;an Interface Control Unit (ICU) includes the digital electronics associated with theservo-loop digital control laws, as well as the interfaces to the satellite’s data bus.Fig. 1 shows a cutaway view of one SU, with its two test masses, their surroundingelectrodes-bearing cylinders, cylindrical invar shield, base plate, upper clamp andvacuum system. ifth force stiffness The dynamics of MICROSCOPE’s test masses, as required for testing the WEP, isdiscussed at length in Ref. [47]. In particular, Ref. [47] focusses on the differentialmotion of two test masses and discriminates between inertial and gravitational masses.In this section, we summarise the equations pertaining to the dynamics of a given testmass, and since this article is not concerned with the WEP, we identify the inertial andgravitational masses (thence some simplification with respect to Ref. [47]).Up to electrostatic parasitic forces (see below), the electrostatic force (3)corresponds to a “control” acceleration responding to the contribution of the variouscontributors to the dynamics of the test mass, ~ Γ cont “ ~F el m “ ∆Γ C ` ~ Γ kin ´ ~F loc m ´ ~F pa m ` ~F ext M ` ~F th M (4)where m and M are the masses of the test mass and of the satellite, ~F ext are non-gravitational forces affecting the satellite (atmospheric drag, Solar radiation pressure), ~F th are forces applied by the thrusters (to compensate for external forces) and ~F loc and ~F pa are local forces (inside the sensor) that we can consider individually (e.g.electrostatic stiffness, gold wire stiffness, self-gravity) or as collective contributions (e.g.electrostatic parasitic forces), respectively. We denote as ∆Γ C the difference betweenthe Earth gravitational acceleration at the center of the satellite and that at the centerof the test mass. We assume that the test masses are homogeneous. Moreover, sincewe are concerned with short-range Yukawa deviations only, we assume that the Yukawacontribution to the Earth’s gravity acceleration acting on the test-masses is negligible.Finally, the second term of the r.h.s. of Eq. (4) contains the contribution from thesatellite’s inertia and from the motion of the test-mass, Γ kin “ r In s P ` r Ω s P ` : P , (5)where P is the position of the test mass with respect to the center of the satellite, r In s ” r Ω s ` r Ω sr Ω s is the gradient of inertia matrix of the satellite and r Ω s its angularvelocity.Noting that since the applied electrostatic force of Eq. (4) is the sum of themeasured “action” force (3) summed over all electrodes and parasitic electrostatic forces, ~F el “ ~F el , meas ` ~F elec , par , (6)we show in Appendix A that the measured acceleration of a test mass, expressed in theinstrument frame, is Γ meas | instr “ B ` ∆Γ C | sat ` Γ kin | sat ´ ~F loc | instr m ` n , (7)where n is the measurement noise and B is an overall bias defined from the localparasitic forces and measurement bias. ifth force stiffness
4. Stiffness: experimental measurement and contributors
A short-range Yukawa fifth force may hide in the local component ~F loc of the force. Asdescribed in the remainder of the paper, our analysis is based on Eq. (7) and consistsin: ‚ measuring the overall stiffness using dedicated sessions; ‚ estimating (when possible) or modelling all possible contributors to ~F loc , thensubtracting them (but a Yukawa interaction) from the measured overall stiffness; ‚ extracting constraints on a Yukawa interaction from the residuals.In this section, we first describe the experimental approach to measure the overallstiffness (subsection 4.1), before listing (subsection 4.2) and discussing one by one thecontributions to ~F loc , including the Yukawa interaction, in the remaining subsections.The second item of our programme is completed in this section (contributions thatcannot be estimated and are thus modelled –electrostatic stiffness, thermal effects andNewtonian gravitational interaction) and in Sect. 5, where contributions that can beextracted from the data (gold wire and Yukawa interaction) are estimated. The lastitem is the subject of Sect. 6. The stiffness is the derivative of force with respect to the position. Measurement sessionswere dedicated to measure MICROSCOPE’s instrument stiffness [52], the stiffness beingexpectedly dominated by an electrostatic stiffness (see Sect. 4.3 and Ref. [48]). Theprinciple of the measurement is to impart a f “ {
300 Hz sinusoidal excitation ofamplitude x “ µ m to the test mass through the electronics control loop. The positionof the test mass is thus forced to be x p t q “ x sin p ωt ` ψ q , (8)where x p t q is any axis ( x, y, z ) of the instrument (along which we aim to estimate theelectrostatic stiffness), ω “ πf and ψ a given phase. The acceleration (7) measured asthe test mass motion is forced in position is Γ meas “ Γ exc ` B ` ∆Γ C ` Γ kin ´ ~F loc m ` n , (9)where we dropped the subscripts “ | instr” and “ | sat” for simplicity, and where Γ exc “ x ω sin p ωt ` ψ q is the excitation acceleration imparted to the test massfrom the electronics control loop.Two measurements lasting 1750 s were performed for each axis of each testmass (one measurement per available electrical configuration –subsection 4.3 andAppendix B). Figure 2 shows the process to measure the stiffness of SUREF’s internalmass along its Y axis: the acceleration measured by the sensor (red) is compared to itsinput position (blue). ifth force stiffness Figure 2.
Experimental process to measure a MICROSCOPE’s sensor’s stiffness(here, the stiffness of SUREF’s internal mass is estimated along its Y axis): the testmass is excited in position with a known amplitude of 5 µ m, and we measure itsresponse in acceleration (left panel); the acceleration is mostly due to the electrostaticstiffness. Right panel: test mass’ acceleration as a function of its position. In the remainder of this paper, we subtract the Earth gravity modeled as describedin Refs. [34, 53], as well as the kinematic acceleration inferred from satellite’s attitudemeasurements, from the measured acceleration. We thus deal with the acceleration Γ ” Γ meas ´ ∆Γ C ´ Γ kin (10) “ Γ exc ` B ´ ~F loc m ` n . (11)This acceleration is dominated by the local one ~F loc { m , while the excitationacceleration Γ exc is negligible. In Ref. [52], it was assumed that only the electrostaticstiffness k ǫ played a significant role, such that ~F loc “ ´ k ǫ ~x , where ~x is the displacementof the test mass with respect to its equilibrium position, and Eq. (11) became (ignoringthe quadratic factor) Γ “ Γ exc ` B ` k ǫ m~x ` n . (12)Under these assumptions, the electrostatic stiffness is simply the slope of the Γ ´ ~x relation (up to the factor m ), as shown in the right panel of Fig. 2. Chhun et al. [52]used this simple technique to estimate the electrostatic stiffness on the three axes of eachMICROSCOPE’s test mass. They found significant disagreements with expectationsfrom the theoretical model summarised in Sect. 4.3, which they explained by modelinaccuracies and contribution from the gold wire aimed to control the charge of the testmasses (Sect. 4.4). We discuss their results in Appendix C. In what follows, we go beyond the simple assumptions of Ref. [52] and explore how thesame measurement could shed light on short-range non-Newtonian gravity. ifth force stiffness F el (Sect. 4.3); force due to the gold wire F w (Sect. 4.4); radiation pressure F p (Sect.4.5); radiometric effect F r (Sect. 4.6); Newtonian F N and non-Newtonian (Yukawa) F Y gravity (Sect. 4.8), such that F loc “ F el ` F w ` F p ` F r ` F N ` F Y . The electrostatic force used to control the test mass is discussed at length in Ref. [48].Here, we shall only state that it consists of a bias b ǫ and a stiffness k ǫ , F el “ b ǫ ´ k ǫ x . (13)Those factors depend on the geometry of the test mass and of the electrodes, andon the electric configuration (voltages applied to the different parts of the sensor). Inparticular, the electrostatic stiffness along the X -axis is expected to be zero for allsensors. For completeness, and since this paper particularly focuses on the stiffness, weprovide below the electrostatic force imparted by the full set of electrodes on the radialaxes when the test mass moves along the Y -axis [48]: F el p y q « ´ ǫ S y e i sin p α y { q α y { p V p y ´ V p q v y ` ǫ Se i ˆ ` sin α y α y ˙ rp V p y ´ V p q ` V d s y ` ǫ Se i ˆ ´ sin α z α z ˙ rp V p z ´ V p q ` V d s y ` πǫ L x R x e e rp V p x ´ V p q ` V d s y ` πǫ R φ L φ e e rp V p φ ´ V p q ` V d s y, (14)where ǫ is the vacuum permittivity, e i ( e e ) is the gap between the inner (outer) electrodecylinder and the test-mass, and where we assumed that all control voltages listed inRef. [48] are small compared to the V p and V d voltages. Those two voltages describethe electric configuration. Two configurations are available: high-resolution mode(HRM) and full-range mode (FRM). They are detailed in Ref. [48] and summarisedin Appendix B.The first term of the r.h.s. of Eq. (14) defines the gain of the detector (the forcebeing proportional to the control voltage v y ); the other terms define the stiffness createdby the Y , Z , X and φ electrodes. In this equation, S is the surface of the Y and Z electrodes, R x and R φ are the inner radius of the X and φ electrodes, and L x and L φ aretheir length. The angles α y and α z are defined as the angle between the displacement ifth force stiffness Y and Z axes, respectively. Appendix D proves the form ofthe stiffness created by the Y electrodes (second term of the r.h.s. of the equation).We assessed the accuracy of the stiffness terms of the model (14) with finiteelements simulations. We found it to be biased high: finite elements models providean electrostatic stiffness 7% to 10% lower than the model (14). Nevertheless, in theremainder of this paper, instead of relying on finite elements simulations, we useEq. (14) corrected by a 8.5% bias. This allows us to easily propagate metrology andvoltage uncertainties in the electrostatic stiffness model, without the need to run atime-consuming simulation for each allowed set of parameters. We then add an extra3% statistical error to those uncertainties to reflect the uncertainty on the bias of themodel. The 7th column of Table 2 (denoted k ǫ, th ) lists the electrostatic stiffness expectedfor each test mass of MICROSCOPE. The electric charge on test masses is controlled via a gold wire linking them to thespacecraft body. The wire can be modelled as a spring acting on the test mass with theforce F w “ ´ k w r ` i φ p f qs x ´ λ w x , (15)where λ w describes the viscous damping of the wire, k w is the wire stiffness and φ p f q describes the internal damping; note that φ can depend on the frequency f . The wirequality factor Q “ { φ .For a sinusoidal motion of the test mass (along the j th axis) x j p t q “ x j sin p ωt ` ψ q ,the force exerted by the gold wire is the sum of an out-of-phase sinusoidal signal [54]and a (velocity-proportional) quadrature signal F w,j p t q “ ´ k w,j x j sin p ωt ` ψ ´ φ q ` λ w x j ω cos p ωt ` ψ q . (16)Thermal dissipation in the wire is the origin of the f ´ { low-frequency noise thatlimits MICROSCOPE’s test of the WEP [33]. Using the dissipation-fluctuation theorem,it can be shown that this acceleration noise is [54, 55]Γ n,w p f q “ m d k B T π k w Q p f q f ´ { ms ´ {? Hz , (17)where m is the mass of the test mass, T is the temperature and k B is the Boltzmannconstant. This allows for an estimation of the k w { Q ratio from the spectral density oflong measurement sessions (see Sect. 5.2). The electrode-bearing cylinders, being at temperature T , emit thermal radiation throughphotons that eventually hit the test mass and transfer their momentum to it, thuscreating a pressure. A gradient of temperature and a difference of temperature ∆ T ifth force stiffness F p “ c Sσ ∆ T T e , (18)where T is the average temperature, c the speed of light, σ the Stefan-Boltzmannconstant, S the surface of the test mass, and e is the vector directed from the hottestto the coldest region.The temperature and its gradient did not evolve in time during the measurementsessions used in this paper (six temperature probes are positioned on each sensor in sucha way that we can monitor the temperature and have a glimpse at its gradient [58]; in theworst case, we could note a 0.003K evolution of the temperature during the measurement–while its mean is about 280K–, with all probes affected by the same evolution, entailingan unmeasurably small variation of the temperature gradients). Therefore, as far as weare concerned, we can consider the radiation pressure-induced force as a simple bias.Given the measured temperatures, an order of magnitude estimation allows us to expectthe corresponding acceleration to be at a level of a p À ´ ms ´ . Taking its name from Crookes’ radiometer, originally thought to prove the photonpressure, the radiometer effect is actually a residual gas effect affecting test massesin rarefied atmospheres whose mean free path exceed the size of the container. In thiscase, equilibrium conditions do not happen when pressure is uniform, but when theratios of pressure to square root of temperature equal one another [56, 57].This entails a force on the test mass proportional to temperature gradient aboutits faces ∆ T , F r “ P S ∆ TT e , (19)where P is the pressure in the container, S the surface of the test mass orthogonal tothe temperature gradient, T the average temperature in the container and, as before, e is the vector directed from the hottest to the coldest regions.Even when stationary, a non-linear temperature profile can cause a position-dependent radiometric effect and potentially a stiffness. However, the sparsetemperature measurements in MICROSCOPE sensors do not allow us to go beyond thelinear temperature profile hypothesis, thereby limiting our modelling of the radiometriceffect to a constant acceleration. Orders of magnitude estimates provide a level ofacceleration of the same order as the radiation pressure, a r À ´ ms ´ . The test mass moves in an imperfect vacuum, so thatdrag may be expected. Orders of magnitude estimates provide a related acceleration « ´ ms ´ , well below our capacity to detect it [48]. ifth force stiffness Gas molecules are released from the materials of the instrument’sparts (in particular the electrode-bearing cylinders) and can impact the test mass andmodify the pressure inside the instrument [56]. However, the vacuum system wasdesigned, and the materials chosen, such that outgassing can be safely ignored [48].
Test masses have a non-zero magnetic moment, and can thereforebe affected by Lorentz forces, either from the Earth magnetic field or local magneticfields. The former applies a periodic signal at the orbital frequency and therefore doesnot affect the stiffness measurements (besides the fact that the Earth magnetic field islargely suppressed by MICROSCOPE’s instrument magnetic shield). Local magneticfields are more difficult to assess. However, noting that their effect on the test of theWEP is subdominant [58], we ignore them in this paper.
Inhomogeneous distributions ofsurface potentials create a force between charged surfaces. MICROSCOPE’s instrumentcan be affected by such patch effects, which act as an additional stiffness dependent onthe test masses’ voltages, thus on the electric configuration [59]. It goes beyond thescope of this article to develop a model of patch effects in MICROSCOPE, and wewill not try to quantify them. Note that they may affect MICROSCOPE only in thestiffness measurement sessions used in this paper, where test masses are set in motion;in MICROSCOPE’s test of the WEP, test masses are kept motionless, and thus lesssensitive to patch effects (which act as a bias).
Very small misalignments betweenMICROSCOPE’s cylinders can be estimated [58]. As they break the cylindricalsymmetry, they can introduce additional terms in the electrostatic stiffness [60].However, as we show below, the error budget in stiffness measurement sessions is largelydominated by the gold wire, so that we can safely ignore them for the purpose of thispaper (thereby justifying our r θ j s “ Id assumption in Appendix A).
The local gravity force applied to a MICROSCOPE test mass is the sum of the forcesbetween that test mass and the parts making the corresponding sensor (Fig. 1): ‚ seven co-axial cylinders: two silica electrode-bearing cylinders surrounding thetest mass, the second test mass and two other silica electrode-bearing cylinderssurrounding it and two cylindrical invar shields, ‚ and four plain cylinders: a silica base plate, an invar base plate, an invar upperclamp, and a vacuum system.The characteristics of those elements can be found in Ref. [48]. As we show below, thegravity force is dominated by the closest elements, so that we can safely neglect thecontribution from the other sensor and from the satellite itself. ifth force stiffness O and O is ÝÑ F “ ´ ż V d V ż V d V B V pÝÑ r ´ ÝÑ r qB r ÝÝÝÑ O O , (20)where the 3-dimensional integrals are taken over the volume of the two bodies, and r i “ p x i , y i , z i q is the coordinate vector of an infinitesimal volume element of the i thbody. Noting ρ i the i th body’s density, the Newtonian potential between infinitesimalvolumes V N pÝÑ r ´ ÝÑ r q “ ´ Gρ ρ d V d V |ÝÑ r ´ ÝÑ r | , (21)and the Yukawa potential of strength α and range λ between infinitesimal volumes V Y pÝÑ r ´ ÝÑ r q “ ´ α Gρ ρ |ÝÑ r ´ ÝÑ r | exp ˆ ´ |ÝÑ r ´ ÝÑ r | λ ˙ d V d V . (22)In the present case, as shown in Fig. 1, all contributions are interactions betweencylinders, either empty (test masses, electrode-bearing cylinders, shield) or full (baseplate, upper clamp). For simplicity, we also assume that the vacuum system is a fullcylinder. Computing the gravitational force applied to the test mass then boils down tocomputing the interaction between perfectly aligned cylinders (as we assumed in Sect.3), and therefore computing the 6-dimensional integral (20).Appendix E shows that, in the limit of small displacements with which we areconcerned in that article, the 6D integral (20) can be reduced to a 1D integral dependingon the geometry of the pair of cylinders. In this case, the gravitational force can beTaylor-expanded, and is dominated by a stiffness term K . The expressions given belowapply both to the Newtonian ( α “ λ Ñ 8 ) and Yukawa forces. They give the forceexerted by any one of MICROSCOPE’s cylinders on a test mass.
In the limit of small displacements δ of the test mass alongthe cylinders’ axis, the force is given by F x p x , δ q « ´ π Gρρ α ÿ i K i p x q δ i , (23)where x is the distance between the center of the test mass and the center of the sourcecylinder along their longitudinal axis ( x “ | x | ą x subscript corresponds to MICROSCOPE’s (longitudinal) X -axis but is referred to as z in the more conventional cylindrical coordinate systemused in Appendix E. The K i coefficients depend on the geometry of the test mass –source pair as follows. If a and b are the inner and outer radii of the cylinder source,2 ℓ its height and ρ its density; and if a and b are the inner and outer radii of the testmass, 2 L its height and ρ its density, then: ifth force stiffness $’’’’’’’’&’’’’’’’’% K p x q “ K p x q “ ż W p k ; a , b q W p k ; a, b q κk e ´ κℓ sinh p κL q d kK p x q “ K p x q “ ż κ W p k ; a , b q W p k ; a, b q k e ´ κℓ sinh p κL q d k, (24a)(24b)(24c)(24d)where we introduced the parameter κ “ a k ` { λ (25)and the function W p k ; a, b q “ bJ p kb q ´ aJ p ka q , (26)where J i are Bessel functions of the first kind.(ii) if the test mass is longer than the source and they are concentric (which is the caseof the pair made of the internal test mass as the source and the external test mass):the force is formally identical to that of the previous case, with ℓ and L switchingtheir roles.(iii) if the test mass and the source are above each other, $’’’’’’’’’’’’&’’’’’’’’’’’’% K p x q “ x | x | ż W p k ; a , b q W p k ; a, b q κ k e ´ κ | x | sinh p κℓ q sinh p κL q d kK p x q “ ´ ż W p k ; a , b q W p k ; a, b q κk e ´ κ | x | sinh p κℓ q sinh p κL q d kK p x q “ x | x | ż W p k ; a , b q W p k ; a, b q k e ´ κ | x | sinh p κℓ q sinh p κL q d kK p x q “ ´ ż κ k W p k ; a , b q W p k ; a, b q e ´ κ | x | sinh p κℓ q sinh p κL q d k. (27a)(27b)(27c)(27d) Similarly, at third order in δ { a , where δ is the displacement of thetest mass along a radial axis ( Y or Z ), the radial force created by any one of the othercylinders is F r p x , δ q « ´ π Gρρ α p K p x q δ ` K p x q δ q , (28)where the K i coefficients depend on the geometry of the test mass – source pair:(i) if the test mass is shorter than the source and they are nested (which is the case ifth force stiffness $’’’&’’’% K p x q “ ż kW p k ; a , b q W p k ; a, b q κ „ L ´ e ´ κℓ κ sinh p κL q cosh p κ | x |q d kK p x q “ ´ ż k W p k ; a , b q W p k ; a, b q κ „ L ´ e ´ κℓ κ sinh p κL q cosh p κ | x |q d k, (29a)(29b)with x « ℓ and L switchingroles.(iii) if the test mass and the source are above each other, $’’’&’’’% K p x q “ ż kW p k ; a , b q W p k ; a, b q κ e ´ κ | x | κ sinh p κℓ q sinh p κL q d kK p x q “ ´ ż k W p k ; a , b q W p k ; a, b q κ e ´ κ | x | κ sinh p κℓ q sinh p κL q d k (30a)(30b) The gravity force applied to a MICROSCOPE testmass is just the sum of the Newton and Yukawa forces created by the aforementionedinstruments’ parts, F g “ p F N ,x ` F Y ,x q e x ` p F N ,r ` F Y ,r q e r (31) “ ÿ j p F N ,x,j ` F Y ,x,j q e x ` ÿ j p F N ,r,j ` F Y ,r,j q e r , (32)where the r subscript stands for the Y and Z axes, and where the forces created by the j th part of the instrument F N ,x,j and F Y ,x,j are given by Eq. (23) and F N ,r,j and F Y ,r,j by Eq. (28).As shown in Appendix E, a first-order Taylor expansion of Eqs. (23) and (28) isenough to precisely account for the gravitational interactions in the present article, wheredisplacements are limited to 5 µ m. This means that the local gravitation effectively actsas a stiffness on the test masses. We thus define the Newtonian and Yukawa, radial andlongitudinal stiffnesses by F N ,r “ ´ k N ,r r (33) F N ,x “ ´ k N ,x x (34) F Y ,r “ ´ k Y ,r r (35) F Y ,x “ ´ k Y ,x x, (36) ifth force stiffness Figure 3.
Newtonian (plain bars) and Yukawa (hashed bars) radial stiffnesses acted bythe SUEP’s parts (the external test mass –TM–, the four electrode-bearing cylinders–IS1-int, IS1-ext, IS2-int, IS2-ext–, and the shielding cylinders) on its internal testmass. The Yukawa potential is set such that p α, λ q “ p , .
01 m q . The contributionof the base plates, upper clamp and vacuum system to the Yukawa interaction is toosmall to appear on the plot. where x and r are the displacement of the test mass along the longitudinal and anyradial axes of the instrument. Newtonian gravity
The plain bars of Fig. 3 show the Newtonian stiffnesses from allcylinders on SUEP’s internal test mass along its radial axis. The force between nestedcylinders is destabilising (negative stiffness), whereas the force from the base plates,upper clamp and vacuum system stabilises the test mass, with the total radial forcebeing destabilising. It can also be shown that the Newtonian gravitational interactionalong the X (longitudinal) axis acts as a stabilising stiffness.Finally, it can be seen from the figure that the contribution from the outer shield issubdominant. Thence, those from the other differential sensor and from the other partsof the satellite are even more subdominant, and we ignore them.The next-to-last column of Table 2 lists the Newtonian gravity stiffness of the fourMICROSCOPE test masses along their radial and longitudinal axes. Yukawa gravity
The hashed bars of Fig. 3 show the Yukawa stiffnesses from allcylinders on SUEP’s internal test mass along its radial axis, for p α, λ q “ p , .
01 m q . Itcan be noted that only co-axial cylinders contribute, since the base, upper clamp andvacuum system are more distant than 0.01 m from the test mass. Similarly, the closest ifth force stiffness Figure 4.
Yukawa stiffness (normalised by α ) for the four MICROSCOPE test masses,on the radial (left) and longitudinal (right) axes, as a function of Yukawa’s range. cylinders provide most of the signal. It can be noted that the Yukawa stiffness of theclosest cylinders is larger than their Newtonian stiffness. This difference comes fromthe fact that with λ “ λ for allMICROSCOPE test masses, along their radial (left panel) and longitudinal (right panel)axes. Starting from the smaller λ reachable (linked to the distance between a test massand its closest cylinder), the radial stiffness increases steadily as more and more co-axialcylinders are within reach of λ and contribute to the gravity signal. The stiffness peaksaround λ « .
01 m, where the base and upper cylinders start to contribute but withan opposite sign stiffness, thereby decreasing it until the Newtonian regime is reachedwhen λ becomes larger than the sensor’s largest scale. The longitudinal stiffness showsa similar behaviour, though it changes sign while more and more cylinders contributeto the signal.Comparing Fig. 4 with Table 2, it is clear that the gravity stiffness (and therefore,signal) is largely subdominant. We put it to test in Sect. 5. Taking all the forces above into account, the acceleration (11) of a test mass measuredalong the i th axis during a stiffness characterisation session isΓ i p t q “ b ǫ,i ` a p,i ` a r,i ` mω ` k ǫ,i ` k N,i m x i sin p ωt ` ψ q` k w,i m x i sin p ωt ` ψ ´ φ q ` λ w,i m ωx i cos p ωt ` ψ q ` k Y,i p α, λ q m j x i sin p ωt ` ψ q , (37)where we singled out the Yukawa gravity contribution and made its stiffness’ dependenceon p α, λ q explicit, since it is this very dependence that we aim to constrain in the ifth force stiffness
5. Data analysis
This section presents least-squares estimates of the parameters introduced in theprevious section. We use the acceleration data introduced in subsection 4.1 to estimatetwo components of the stiffness (that of the gold wire k w and a linear combination k of the electrostatic and gravitational stiffness), the quality factor of the gold wire, anda velocity-dependent coefficient for each axis of each sensor (Eq. 38 below). Then,in subsection 5.3.3, we subtract theoretical models of the electrostatic and Newtoniangravity stiffness from the estimated k ; any residual should come either from a Yukawastiffness or from unaccounted for contributors. We perform the exercice in the twoelectrical configurations (HRM and FRM) summarised in Appendix B. The measurement equation (37) could be used in its original form to extract the unknownparameters from the data and simultaneously constrain the Yukawa interaction’sparameters. However, since the Yukawa contribution is expected to be at most of theorder of the Newtonian contribution, which is itself largely less than the electrostaticstiffness, its parameters have a small constraining power on the data, and we find moresuited to first estimate an overall stiffness, from which we can eventually extract the p α, λ q parameters.Moreover, Eq. (37) requires the estimation of two phases. The first one, ψ , is thatof the excitation signal and can be estimated a priori by fitting the position data, thenused as a known parameter in the following analysis. The second, φ , is the phase-offsetinduced by the gold wire’s internal damping. Instead of trying to estimate it from thedata (which may be difficult given the 4 Hz sampling of data, when assuming that thequality factor of the wire is in the range Q « ´ p t q “ b ` p κ ` κ w cos φ q sin p ωt ` ψ q ´ p κ w sin φ ´ κ λ q cos p ωt ` ψ q , (38)where b ” b ǫ ` a p ` a r , κ ” x p mω ` k ǫ ` k N ` k Y q{ m , κ w ” x k w { m and κ λ ” x λ w ω { m ,and we dropped the i subscript for simplicity.Five parameters are left for estimation: b , κ , κ w , κ λ and φ “ { Q . It is howeverclear that fitting Eq. (38) will provide only three independent constraints. If estimating b will be easy, the other parameters will remain degenerate unless we can use some priorknowledge. We show in Sect. 5.2 that we can obtain an independent estimate of the κ w { Q “ κ w φ combination. ifth force stiffness Table 1.
Gold wire stiffness [10 ´ Nm ´ ] and orientation of the force [deg] estimatedfrom long measurement sessions’ acceleration noise spectral density. k w,x { Q k w,y { Q k w,z { Q k w { Q ϕ θ
IS1-SUEP 1 . ˘ .
20 0 . ˘ .
05 1 . ˘ .
20 1 . ˘ .
10 48 ˘ ˘ . ˘ .
01 0 . ˘ .
10 0 . ˘ .
06 0 . ˘ .
08 84 ˘ ˘ k w { Q ratio As shown in Sect. 4.4, fitting the low-frequency part of the spectral density of theacceleration measured along a given axis can provide an estimate of the ratio k w { Q forthis axis (from Eq. 17) once temperature data are available (which is the case for allmeasurement sessions). Performing this task for the three linear axes, we can get anestimate of the gold wire stiffness along each axis, and the orientation of the force due tothe wire. This force is presumably collinear with the wire, although the glue clampingprocess may complexify it. Noting ϕ the angle between the force and the test masslongitudinal axis ( X -axis), and θ the angle between the Y -axis and the projection ofthe wire on the ( y, z ) plane (Fig. 5), the three stiffnesses that can be measured are k w,x “ | k w | cos ϕ (39) k w,y “ | k w | sin ϕ cos θ (40) k w,z “ | k w | sin ϕ sin θ (41)from which we can recover the modulus of the stiffness and the orientation of the wire.Fig. 6 shows the fit corresponding to IS1-SUEP’s X -axis from the session used toestimate the WEP in Ref. [33]. Values obtained for the internal sensor of both SU aregiven in Table 1. We checked that estimates from different sessions are consistent. Notethat rigorously, since the drag-free system is controlled by the external sensor, fitting theinternal sensor’s spectral density only provides information about the sum of the k w { mQ ratios of both sensors (where m is their mass). Nevertheless, under the assumptions thattheir masses are similar (which is enough given the goals of this article), that their wireshave similar k w { Q ratios, and that their spectral density are uncorrelated, fitting theinternal sensor’s spectral density indeed provides a constraint on each sensor’s wire’s k w { Q ratio.Willemenot & Touboul [55] used a torsion pendulum to characterise a goldwire similar to those used by MICROSCOPE. Assuming that the wire is deformedperpendicular to its principal axis (i.e. in flexion), they give a convenient scaling toquantify the wire’s stiffness k w “ . ˆ ´ ˆ r w . µ m ˙ ˆ . l w ˙ ˆ E . ˆ Nm ˙ , (42)where r w is the radius of the wire, l w its length and E its Young modulus. UsingMICROSCOPE’s gold wires’ characteristics ( r w “ . µ m, l w “ . ifth force stiffness Figure 5.
Gold wire and test mass geometry.
Figure 6.
Typical (square-root) spectral density of the acceleration measured alongthe X -axis of IS1-SUEP. The f ´ { low-frequency part originates from thermaldissipation in the wire (Eq. 17), and the high-frequency f increase is due to thecapacitive detector’s noise [61]. The orange line is the best fit of the low-frequencypart. ifth force stiffness E “ . ˆ Nm ), we expect k w « ˆ ´ . Combined with a quality factor Q «
100 as measured in Ref. [55], this scaling provides k w { Q « ´ N { m, in flagrantcontradiction with the values estimated from flight data (Table 1).Two explanations can be proposed: (i) the wire does not behave as shown inRef. [55] or (ii) its quality factor is much lower than expected. In the former explanation,the wire may work in compression (i.e. it is deformed along its principal axis), whichpotentially increases its stiffness. In the latter, the mounting process (wires being gluedto the test masses) may decrease the overall quality factor; differences between the gluepoints in MICROSCOPE and in Ref. [55] may explain a significant difference of qualityfactor.Assuming that the electrostatic model of the instrument is correct and that themeasured stiffness is dominated by the electrostatic stiffness hints at a low quality factor Q «
1. Note however that, even if the quality factor is really that low, MICROSCOPE’smain results (the test of the WEP) depend on the k w { Q ratio, and are thus unaffectedby the current analysis. We now come back to Eq. (38), with the aim to estimate the model parameters for thefour sensors, starting with radial axes. Y and Z ) The following assumptions allow us to break thedegeneracy between the parameters mentioned in Sect. 5.1: ‚ for a given sensor and a given axis j , the gold wire’s ratio k w,j { Q is independent ofthe electrical configuration (HRM or FRM), and can be estimated as shown in Sect.5.2 for the internal sensors. We further assume that the mounting of gold wiresis general enough to assume that the external sensors’ k w,j { Q ratio is the same asthat of the internal sensor [48]. ‚ the ratio k w,j { Q varies from one axis to another, but the quality factor Q is a trueconstant for a given sensor. In other words, since k w,j and Q are degenerate, weassume that only the stiffness depends on the direction. ‚ by cylindrical symmetry, the total stiffness of the radial axis j ( j “ y, z ) k ,j “ m ω ` k ǫ,j ` k N ` k Y is independent of the axis, and depends on theelectrical configuration only through the electrostatic stiffness k ǫ . This assumptionis reasonable given the estimated metrology uncertainties [48].Denoting ˆ χ y “ k w,y { Q and ˆ χ z “ k w,z { Q the radial gold wire’s ratios estimated inTable 1, and combining constraints from the model (38), where we add the subscripts‘F’ and ‘H’ for measurements in FRM and HRM modes, we obtain the following system ifth force stiffness $’’’’’’’’’’’’’’’’’’’’’’’&’’’’’’’’’’’’’’’’’’’’’’’% ˆ a yF “ κ rF ` κ wy cos φ ˆ a yH “ κ rH ` κ wy cos φ ˆ a zF “ κ rF ` κ wz cos φ ˆ a zH “ κ rH ` κ wz cos φ ˆ a wyF “ ´ κ wy sin φ ` κ λyF ˆ a wyH “ ´ κ wy sin φ ` κ λyH ˆ a wzF “ ´ κ wz sin φ ` κ λzF ˆ a wzH “ ´ κ wz sin φ ` κ λzH ˆ χ y “ k wy Q ˆ χ z “ k wz Q , (43a)(43b)(43c)(43d)(43e)(43f)(43g)(43h)(43i)(43j)where we recall that φ “ { Q , κ wj “ x k wj { m , and similarly for κ r (with the subscript r “ y, z ), κ λ and the ˆ a and ˆ a w coefficients are the estimates of the sine and cosinecoefficients of Eq. (38).On the one hand, Eqs. (43e-43h) trivially give the velocity-dependent terms asa functions of the unknown Q and estimated χ j and a wj . On the other hand, Eqs.(43a-43d), (43i-43j) can be combined to give2 p ˆ χ y ´ ˆ χ z q x m Q cos ˆ Q ˙ “ ˆ a yF ` ˆ a yH ´ ˆ a wzF ´ ˆ a wzH , (44)thus providing the equation x cos ˆ x ˙ ´ ξ “ Q is a root, where ξ is defined through parameters estimated from Table 1 andfitting Eq. (38) for the sensor’s two radial axes in each electrical configuration.Once Q is estimated, Eqs. (43a-43d) readily provide κ rF and κ rH . Actually, theygive two estimates of each, which we checked to be consistent. X Under the same assumptions, it is then straightforwardto estimate the X -axis stiffness from Eq. (38) and Table 1, for a given electricalconfiguration (that we do not make explicit in the equations below for simplicity): $’’’&’’’% κ x “ ˆ a x ´ x m ˆ χ x Q cos ˆ Q ˙ κ λx “ ˆ a wx ` x m ˆ χ x Q sin ˆ Q ˙ . (46a)(46b) ifth force stiffness Figure 7.
Difference between theoretical electrostatic stiffness and measured totalin-phase stiffnesses corrected for the excitation and Newtonian gravity stiffnesses,∆ k “ ˆ k ´ k N ´ mω ´ k ǫ, th , for all axes (longitudinal and radial) of each sensor,in the HRM (diamonds) and FRM (squares) electric configurations. Our results are listed in Table 2 for each sensor, in their two electricalconfigurations. The left group’s four columns show the instrumental parameters: totalin-phase stiffness k , gold wire stiffness k w and quality factor Q , and velocity-dependentcoefficient λ w . The next two columns give the theoretical electrostatic stiffness k ǫ, th and the Newtonian gravity stiffness. The last column lists the difference between thetheoretical and the estimated electrostatic stiffness ∆ k “ ˆ k ´ k N ´ mω ´ k ǫ, th . Errorbars give 1 σ uncertainties.The electrostatic stiffness estimated in the HRM electrical configuration isconsistent with the theoretical one for most sensors and axes, with at most a « σ discrepancy. However, we note a significant difference in the FRM configuration (Fig.7). Being dependent on the electrical configuration, this discrepancy hints at theexistence of an electric potential-dependent additional stiffness completely degeneratewith the electrostatic one. Patch effects could be at its origin: the discrepancy beingsignificant for higher voltages is indeed consistent with the voltage-dependence of patcheffects’ stiffness. Disentangling this puzzle would require modeling patch effects inMICROSCOPE’s sensors. As this goes far beyond the scope of this paper, and sinceour experiment is not competitive with other short-ranged forces searches (as we discussbelow), we let the question of this discrepancy open. In the remainder of this paper, wethence use HRM measurements only.The gold wire’s quality factor is lower than could be expected from Ref. [55].Nevertheless, Q being close to 1 is consistent with our discussion in Sect. 5.2. ifth force stiffness k and k w are degenerate in the amplitude of Eq. (38)’s sine, meaningthat ˆ k errors actually come from those on k w { Q ). if t h f o r ce s t i ff n e ss Table 2.
Estimated model parameters. The left group’s four columns show the instrumental parameters: total in-phase stiffness k ,gold wire stiffness k w and quality factor Q , and velocity-dependent coefficient λ w . The next two columns give the theoretical electrostaticstiffness k ǫ, th and the Newtonian gravity stiffness. The last column lists the difference between the theoretical and the estimated electrostaticstiffness ∆ k “ ˆ k ´ k N ´ mω ´ k ǫ, th . Error bars give 1 σ uncertainties. Sensor Axis (mode) ˆ k ˆ k w ˆ Q ˆ λ w k ǫ, th k N ∆ k [ ˆ ´ N { m] [ ˆ ´ N { m] [ ˆ ´ Ns { m] [ ˆ ´ N { m] [ ˆ ´ N/m] [ ˆ ´ N { m]SUEP X (HRM) -0.00 ˘ ˘ ˘ ˘ ˘ ˘ Y (HRM) -1.55 ˘ ˘ ˘ ˘ ˘ ˘ Z (HRM) -1.65 ˘ ˘ ˘ ˘ ˘ ˘ X (FRM) 0.04 ˘ ˘ ˘ ˘ ˘ ˘ Y (FRM) -18.85 ˘ ˘ ˘ ˘ ˘ ˘ Z (FRM) -18.87 ˘ ˘ ˘ ˘ ˘ ˘ X (HRM) -1.35 ˘ ˘ ˘ ˘ ˘ ˘ Y (HRM) -6.96 ˘ ˘ ˘ ˘ ˘ ˘ Z (HRM) -8.20 ˘ ˘ ˘ ˘ ˘ ˘ X (FRM) -1.32 ˘ ˘ ˘ ˘ ˘ ˘ Y (FRM) -78.47 ˘ ˘ ˘ ˘ ˘ ˘ Z (FRM) -78.56 ˘ ˘ ˘ ˘ ˘ ˘ X (HRM) 0.06 ˘ ˘ ˘ ˘ ˘ ˘ Y (HRM) -1.58 ˘ ˘ ˘ ˘ ˘ ˘ Z (HRM) -1.71 ˘ ˘ ˘ ˘ ˘ ˘ X (FRM) 0.06 ˘ ˘ ˘ ˘ ˘ ˘ Y (FRM) -19.25 ˘ ˘ ˘ ˘ ˘ ˘ Z (FRM) -19.16 ˘ ˘ ˘ ˘ ˘ ˘ X (HRM) 0.20 ˘ ˘ ˘ ˘ ˘ ˘ Y (HRM) -8.91 ˘ ˘ ˘ ˘ ˘ ˘ Z (HRM) -9.56 ˘ ˘ ˘ ˘ ˘ ˘ X (FRM) 0.15 ˘ ˘ ˘ ˘ ˘ ˘ Y (FRM) -80.24 ˘ ˘ ˘ ˘ ˘ ˘ Z (FRM) -80.09 ˘ ˘ ˘ ˘ ˘ ˘ ifth force stiffness Figure 8.
Difference between theoretical electrostatic stiffnesses and measuredtotal in-phase stiffness corrected for the excitation and Newtonian gravity stiffnesses,∆ k “ ˆ k ´ k N ´ mω ´ k ǫ, th , for the radial axes of each sensor, in the HRM electricconfiguration. The dashed line is the ∆ k weighted average and the grey area shows its1 σ weighted uncertainty.
6. Constraints on short-ranged Yukawa deviation
In the previous section, we invoked patch effects to account for the non-zero differencebetween theoretical electrostatic stiffness and measured total in-phase stiffness correctedfor the excitation and Newtonian gravity stiffnesses, ∆ k “ ˆ k ´ k N ´ mω ´ k ǫ, th . Actually,∆ k also contains the putative Yukawa potential that we aim to constrain in this paper.Given the obvious dependence of ∆ k on the electric configuration, which cannotbe explained by a Yukawa-like gravity interaction, we exclude the obviously biasedFRM measurements from our analysis below. Furthermore, as shown in Fig. 4, aYukawa potential has a stronger signature on the radial axes than on the longitudinalone. Therefore, we use only the stiffness estimated on the radial axes in the HRMconfiguration to infer constraints on the Yukawa interaction. Fig. 8 shows thecorresponding ∆ k estimate, together with their weighted average and 1 σ uncertainty(dashed line and grey area), x ∆ k y “ p . ˘ . q ˆ ´ N { m. It can be noted that SUEPand SUREF have similar behaviours. This is expected since they are identical –up totheir external test mass and small machining errors.The marginal offset from 0 is surely due to unaccounted for patch effects and apossible suboptimal calibration of our electrostatic model. However, as error bars arelargely dominated by gold wires, and are significantly larger than the remaining bias, weuse this estimation of x ∆ k y to infer the 95% (2 σ ) upper bound on the Yukawa potential ifth force stiffness Figure 9.
95% confidence contour for a Yukawa potential. The light yellow areashows the excluded region by various experiments: Irvine [62], E¨ot-Wash 2007 [41],HUST 2012 [45], HUST 2020 [31], and the yellow area shows the region excluded bythe current work. in Fig. 9, noting that a positive x ∆ k y corresponds to a negative α . Note that since ourestimated ∆ k is consistent with 0, we merely consider that a Yukawa interaction can bepresent within the error bars; we do not claim that it explains ∆ k ’s slight offset from 0.The curves in the lower part of Fig. 9 show the current best upper bounds on aYukawa potential, inferred from dedicated torsion balance experiments [31, 41, 45, 62].Note that the E¨ot-Wash group recently updated its constraints [32]; however, since theyhave been improved below the ranges of λ considered here, we do not show them inFig. 9. Our constraints are clearly poor compared to the state of the art. It would havebeen surprising otherwise, since MICROSCOPE was not designed to look for short-range deviations from Newtonian gravity. However, our results suggest that thanks toits non-trivial geometry, an experiment looking like MICROSCOPE, if highly optimised,may allow for new constraints of gravity through the measurement of the interactionbetween several bodies.
7. Conclusion
We used in-flight technical measurements aimed to characterise MICROSCOPE’sinstrument to search for short-ranged Yukawa deviations from Newtonian gravity.MICROSCOPE not being designed for this task, this article serves as a proposal fora new experimental concept in the search of small-scale modifications of gravitation, aswell as a first proof of concept. The analysis is based on the estimation of the stiffness ifth force stiffness α , λ ) are not competitive with the published ones, obtained with dedicated laboratorytests. We find | α | ă ´ for 10 ´ m ď λ ď ´ m ď λ ď α ă ´ , corresponding to a stiffnessseven orders of magnitude lower than the electrostatic stiffness. Since MICROSCOPE’scapacitive control and measurement prevents us from using an electrostatic shield similarto that used by torsion pendulum experiments, a competitive experimental constrain willthus require a control of the instrument’s theoretical model of one part in 10 millions.Whether this endeavour is possible remains an open question. Nevertheless, it couldbe circumvented by performing the measurement with several (more than two) voltagesswitching on and off different sets of electrodes to empirically determine the geometrydependence of the electrostatic stiffness.In the meantime, we use the measurements presented in this paper to provide newconstraints on the chameleon model in a companion paper [65] based on Refs. [66, 67]. Acknowledgments
We acknowledge useful discussions with Bruno Christophe and Bernard Foulon, andthank Vincent Lebat for comments on this article. We acknowledge the financial supportof CNES through the APR program (“GMscope+” and “Microscope 2” projects). MPBis supported by a CNES/ONERA PhD grant. This work uses technical details of theT-SAGE instrument, installed on the CNES-ESA-ONERA-CNRS-OCA-DLR-ZARMMICROSCOPE mission. This work is supported in part by the EU Horizon 2020research and innovation programme under the Marie-Sklodowska grant No. 690575. ifth force stiffness
Appendix A. Test mass dynamics
Equation (4) is an idealised version of the more realistic description of Ref. [47]. First,the sensor is not perfectly aligned with the satellite’s frame, as described by the r θ s matrix ~ Γ cont | instr “ r θ s ˜ ∆Γ C | sat ` ~ Γ kin | sat ` ~F ext | sat M ` ~F th | sat M ¸ ´ ~F loc | instr m ´ ~F pa | instr m , (A.1)where the subscripts “ | instr” and “ | sat” mean that forces and accelerations are expressedin the instrument or satellite frame, respectively.Moreover, the measured acceleration is given by the control acceleration (A.1)affected by the matrix r A s containing the instrument’s scale factors, by electrostaticparasitic forces (since the applied electrostatic forces are the sum of the measured andparasitic electrostatic forces m~ Γ cont | instr “ ~F el “ ~F el , meas ` ~F elec , par ), by the measurementbias b due to the read-out circuit and by noise n : Γ meas | instr “ b ` r A s ˜ Γ cont | instr ´ F elec , par | instr m ¸ ` K ” Γ cont | sat ı ` n . (A.2)We can then wrap up and write the measured acceleration explicitly: Γ meas | instr “ B ` r A sr θ s ˜ ∆Γ C | sat ` Γ kin | sat ` F ext | sat M ` F th | sat M ¸ ´ r A s ~F loc | instr m ` K ” Γ cont | sat ı ` n , (A.3)where B ” b ´ r A s ˜ ~F pa | instr m ` ~F elec , par | instr m ¸ (A.4)is the overall bias and K is the quadratic factor accounting for non-linearities in theelectronics.In this article, following the measurements of Ref. [34], we assume that r A s “r θ s “ Id (Identity matrix), that the drag-free perfectly cancels the external forces andwe ignore the quadratic factor (see Refs. [33, 49, 52]), so that our main measurementequation is Γ meas | instr “ B ` ∆Γ C | sat ` Γ kin | sat ´ ~F loc | instr m ` n . (A.5) ifth force stiffness Table B1.
High-resolution mode (HRM) electric configuration. All voltages are in V. V d V p V px V py { z V pφ IS1-SUEP 5 5 -5 2.5 -10IS2-SUEP 5 5 0 2.5 -10IS1-SUREF 5 5 -5 2.5 -10IS2-SUREF 5 5 -10 0 -10
Table B2.
Full-range mode (FRM) electric configuration. All voltages are in V. V d V p V px V py { z V pφ IS1-SUEP 1 42 0 0 0IS2-SUEP 1 42 0 0 0IS1-SUREF 1 42 0 0 0IS2-SUREF 1 42 0 0 0
Appendix B. Electric configurations
MICROSCOPE can be used with two electric configurations: in the full-range mode(FRM), voltages are high enough to be able to acquire the test masses, while the high-resolution mode (HRM), with lower voltages, allows for an optimal control of the testmasses. Tables B1 and B2 summarise the corresponding voltages (which appear in Eq.14). See Ref. [48] for details.
Appendix C. Discussion of Chhun et al. [52] analysis
In Ref. [52], Chhun et al. compute the electrostatic stiffness in HRM, using the samemeasurement sessions as those used here, with a simple ratio of sines amplitudes. Theyneglect the local gravity stiffness and assume a negligible gold wire’s stiffness k w « λ w “ x p t q “ x sin p ωt ` ψ x q (C.1)Γ p t q “ Γ sin p ωt ` ψ Γ q , (C.2)and infer k ǫ “ m Γ { x ´ mω , with the implicit assumption that ψ x “ ψ Γ . Table C1sums up their results.Two important points need to be highlighted. First, the stiffnesses estimatedunder the very restrictive assumptions of Ref. [52] are close to (yet inconsistent with)the expected electrostatic stiffnesses (with an accuracy ranging from a few to a dozenpercent, especially on the radial axes, see Table 2). Second, the stiffness estimated onthe radial axes are consistent with each other, thus showing a good degree of cylindrical ifth force stiffness Table C1.
Total stiffness (identified as the electrostatic stiffness) measured in Ref.[52]. The expected values can be found in Table 2. k ǫ,x rˆ ´ N { m s k ǫ,y rˆ ´ N { m s k ǫ,z rˆ ´ N { m s IS1-SUEP 1 . ˘ . ´ . ˘ . ´ . ˘ . . ˘ . ´ . ˘ . ´ . ˘ . . ˘ . ´ . ˘ . ´ . ˘ . . ˘ . ´ . ˘ . ´ . ˘ . Q " k w { Q enters the measurement). Thus assuming φ Ñ
0, we re-writeEq. (38) as (Taylor expanding the sine and cosine at first order in φ )Γ p t q “ ˘ b κ ` κ κ w ` κ w p ` φ q sin ˆ ωt ` ψ ´ arctan κ w φκ κ w ˙ , (C.3)which tends to lim φ Ñ Γ p t q “ ˘| κ ` κ w | sin p ωt ` ψ q . It is thus clear that using Eq. (C.1),Ref. [52] estimates the total stiffness. Nevertheless, a subtlety remains. Rigorously,although the phase in Eq. (C.3) should be that of the excitation, ψ “ ψ x , which may(and does) differ from the phase of the acceleration ψ Γ , Ref. [52] assumes ψ x “ ψ Γ (whichis consistent with the assumption that the gold wire has zero stiffness). Unfortunately,the experiment contradicts this assumption (at least on the radial axes).Relaxing the ψ x “ ψ Γ hypothesis of Eq. (C.1), we find almost unchanged totalstiffnesses (with percent-level modifications), but a small residual with a π { k w { Q ratio of the gold wires (when assuming λ w “ k w or Q . Appendix D. Radial electrostatic stiffness due to the Y electrodes
In this appendix, we give a detailed computation of the electrostatic stiffness created byMICROSCOPE’s Y electrodes on a given test mass as the test mass moves along the Y -axis (but remains at z “ ifth force stiffness Y -axis by two pairs of diametrically-opposed Y electrodes (at potential V e ` and V e ´ ), completed by two pairs of Z -electrodes, as shownin Fig. D1. Appendix D.1. Electrostatic force between the plates of a capacitor
At constant potential, the electrostatic force between conductors reads F elec “ ∇ U ,where U is the electrostatic energy. For a capacitor, U “ CV , (D.1)where C is its capacitance and V the potential difference between its plates. Theelectrostatic force created along the y -axis is then F p y q “ B C B y V . (D.2) Appendix D.2. Capacitance of one Y electrode – test mass pair Assuming electrodes are on an infinite cylinder (this assumption is reasonable sinceelectrodes are far enough from the edges of the cylinder) and using the Gauss theorem,it is easy to show that the electric field of an electrode (of surface charge σ ) at a distance r from the axis of the cylinder is E p r q “ σR ey ǫ r . (D.3)The electric potential of the electrode is thus V p r q “ R ey σǫ ln r. (D.4)Finally, the capacitance of the electrode-test mass pair C “ Q ∆ V “ ˆ π ´ d R ey ˙ L y ǫ ln R mi R ey , (D.5)where the charge Q “ σS “ σ ˆ π ´ d R ey ˙ R ey L y , (D.6)where S is the surface of an electrode (of length L y ).Denoting e ” R mi ´ R ey the gap between the cylinder and the test mass, in thelimit e ! R ey „ R mi , Eq. (D.5) reads C “ ˆ π ´ d R ey ˙ L y ǫ R mi ` R ey e . (D.7) ifth force stiffness Figure D1.
Test mass’ Y - and Z -axes control geometry. Upper panel: test mass(light brown) and inner electrode-bearing silica cylinder, with its two rings of pairsof electrodes to control the Y -axis (red) and the Z ´ axis (blue). The outer electrode-bearing silica cylinder controls the X -axis and is not shown here (see Ref. [48]). Lowerpanel: Radial cut of a ring of Y and Z electrodes geometry, when the test mass isoffset by y along the Y -axis, with e being the gap between the electrodes and thetest mass in equilibrium. The inner cylinder carries the electrodes ( Y and Z alongthe corresponding axes – Y electrodes are shown at potential V e ` and V e ´ ) of externalradius R ey ; electrodes are separated by dips of width d . The test mass (of inner radius R mi and potential V TM ) surrounds this inner cylinder, and can move around it. ifth force stiffness Appendix D.3. Y electrodes electrostatic stiffness When moving the test mass by an amount y along the Y -axis, the electrostatic forcebetween the electrodes and the test mass is the sum of the forces between the test massand the V e ` and V e ´ electrodes, F “ F ` ` F ´ (so far we consider only one pair ofelectrodes).Those forces are, from Eq. (D.2), F ` “ B C ` B y p V TM ´ V e ` q , (D.8) F ´ “ B C ´ B y p V TM ` V e ´ q , (D.9)with C ˘ “ ˆ π ´ d R ey ˙ L y ǫ R mi ` R ey p e ˘ y q . (D.10)The total force is thus F “ k „ ´ p V TM ´ V e ` q p e ` y q ` p V TM ´ V e ´ q p e ´ y q , (D.11)where k ” ˆ π ´ d R ey ˙ L y ǫ p R mi ` R ey q . (D.12)Assuming y ! e , the force reads, at first order in y { e , F “ k e ” p V TM ´ V e ´ q ´ ` ye ¯ ´ p V TM ´ V e ` q ´ ´ ye ¯ı . (D.13)Keeping only the (stiffness) terms proportional to the displacement y and expandingthe square sums, we get F “ k e “ ´ p V e ` ` V e ´ q V TM ` V e ´ ` V e ` ` V ‰ y, (D.14)with [48] $’&’% V e ´ “ V p ´ v y V e ` “ V p ` v y V TM “ V p ` ? V d sin ω d t, (D.15)of which we take the mean value x V TM y “ V p and x V y “ V p ` V d (and omit the x . . . y symbol hereafter), such that the stiffness contribution to the force is F “ k e “ p V p ´ V p q ` V d ‰ y. (D.16)Considering now the two pairs of electrodes, and substituting Eq. (D.12) to k , F “ ˆ π ´ d R ey ˙ L y ǫ R mi ` R ey e “ p V p ´ V p q ` V d ‰ y. (D.17)Since R mi « R ey , using the expression for the surface of an electrode (Eq. D.6), wefind the expression given in Eq. (14), with α y “ ifth force stiffness Appendix E. Gravitational force between hollow cylinders
Ref. [68] derives the longitudinal F z p r, z q and axial F r p r, z q forces between two hollowcylinders by a Yukawa gravitation. In this appendix, we use those results to complementthem with the cases at hand in this paper. Note that contrary to the MICROSCOPEreference frame used in the main text, we use a more intuitive coordinate frame, wherethe z -axis is along the main axis of the cylinders, so that the natural cylindrical system p r, ϕ, z q holds. This is the convention of Ref. [68].The gravitational force created along the z -axis on a unit mass at p r, θ, z q by ahollow cylinder of inner and outer radii a and b , height 2 ℓ and density ρ is [68] F z p r, z q “ ´ πGαρ ż J p kr q d kκ r bJ p kb q ´ aJ p ka qs ˆ $’&’% h p z ; k q if ´ ℓ ď z ď ℓh p z ; k q if z ą ℓh p z ; k q if z ă ´ ℓ. (E.1)where κ is defined in Eq. (25), with λ the Yukawa interaction range, J i are Besselfunctions of the first kind and the h i functions depend on the altitude of the unit massand are defined below ; . The Newtonian interaction is straightforward to recover bysetting λ Ñ 8 (and α “ F r p r, z q “ ´ πGαρ ż kJ p kr q d kκ r bJ p kb q ´ aJ p ka qs ˆ $’&’% h p z ; k q if ´ ℓ ď z ď ℓh p z ; k q if z ą ℓ ´ h p z ; k q if z ă ´ ℓ (E.2)The h i functions are defined as h p z ; k q “ exp r´ κ p z ´ ℓ qs ´ exp r´ κ p ℓ ` z qs h p z ; k q “ exp r´ κ p ℓ ´ z qs ´ exp r´ κ p ℓ ` z qs h p z ; k q “ exp r κ p z ´ ℓ qs ´ exp r κ p ℓ ` z qs h p z ; k q “ ´ exp r´ κ p ℓ ´ z qs ´ exp r´ κ p ℓ ` z qs . (E.3) Appendix E.1. Forces on a full cylinder
The forces exerted by the previous cylinder (called the “source”, centered on p x, y, z q “p , , q ) on another full cylinder (called the “target”, centered on p x s , , z s q ) of radius a ,height 2 L and density ρ , is obtained by integrating Eqs. (E.1) and (E.2) on the volumeof the target (at this point in the computation, we do not care whether the geometry isphysically sound –i.e. cylinders may overlap; this will be done below): F z p x s , z s q “ ρ ij d x d y ż z max z min d zF z p r, z q , (E.4) ; Note that h and h are confused in Ref. [68] ifth force stiffness F r p x s , z s q , where, for convenience, we express the volume in Cartesiancoordinates (though we will quickly return to cylindrical coordinates below), with r “ a x ` y . The z -integral is taken from the base z min to the top z max of thetarget cylinder, and the p x, y q -integral is taken over the disk section of the cylinder. Weexplicit them below. Appendix E.1.1. z -integral F z p r, z q and F r p r, z q depend on z only through the h i functions, so it is enough to compute H i p k q “ ş z max z min h i p z ; k q d z . Several cases dependingon the position of the target with respect to the source must be considered:(i) Target’s z -extension fully contained in source’s z -extension: in this case, z s ´ L ą ´ ℓ and z s ` L ă ℓ , and only h and h are defined. Their integrals are straightforwardto compute, with z min “ z s ´ L and z max “ z s ` L : H p z s , k q “ ´ κℓ κ sinh p κL q sinh p κz s q (E.5)and H p z s , k q “ L ´ ´ κℓ κ sinh p κL q cosh p κz s q . (E.6)(ii) Target’s z -extension fully covering source’s z -extension ( z s ´ L ă ´ ℓ and z s ` L ą ℓ ):in this case, all h i are defined, and H p z s , k q “ ´ κ ` e ´ κ p z s ` L q ´ e ´ κℓ ˘ sinh p κℓ q , (E.7) H p z s , k q “ ´ κ ` e ´ κℓ ´ e κ p z s ´ L q ˘ sinh p κℓ q , (E.8) H p z s , k q “ ℓ ´ ´ κℓ κ sinh p κℓ q (E.9)and H p z s , k q “ z s ´ L ą ℓ ): in this case, only h is defined and H p z s , k q “ ´ κz s κ sinh p κℓ q sinh p κL q . (E.10)(iv) Target fully below source ( z s ` L ă ´ ℓ ): in this case, only h is defined and H p z s , k q “ ´ κz s κ sinh p κℓ q sinh p κL q . (E.11)(v) Other cases correspond to the target’s and the source’s z -extension overlapping,with none completely covering the other. Since they are not of use inMICROSCOPE, we do not consider them here. ifth force stiffness Figure E1. p x, y q -integration geometry. Appendix E.1.2. p x, y q -integral With no loss of generality, we can set the target cylinderon p x s , y s q “ p x s , q in the p x, y q -plane ( y s ‰ f , ij d x d yf p x, y q “ ż θ ` θ ´ d θ ż R ` p θ q R ´ p θ q f p r, θ q r d r, (E.12)where the integration boundaries depend on the geometry of the problem. Let us assumethat the disk over which we take the integral is centered on p x s , y s q “ p x s , q and has aradius a (not to be confused with the radius of the source –which is of no use here).(i) | x s | ą a This case is illustrated by the left panel of Fig. E1. It is easy to show that the θ integral runs from θ ´ “ ´ arcsin p a {| x s |q to θ ` “ arcsin p a {| x s |q . For a given θ inthat domain, the r -integration then runs from R ´ p θ q to R ` p θ q which are solutionsof the quadratic equation R ´ x s R cos θ ` x s ´ a “ , (E.13)and are given by R ˘ p θ q “ x s cos θ ˘ b a ´ x s sin θ. (E.14)(ii) | x s | ď a In this case, shown in the right panel of Fig. E1, the θ boundaries are trivially θ ´ “ θ ` “ π . It is also trivial that for a given θ , R ´ p θ q “
0. Finally, itcan be shown that the upper r -boundary is the same as that of the previous case, R ` p θ q “ x s cos θ ` a a ´ x s sin θ . ifth force stiffness Appendix E.1.3. Longitudinal and radial forces
Noting that the r -dependence of the F z p x s , z s q force appears only in the J Bessel function, and using Eq. (E.1) we re-writeEq. (E.4) as F z p x s , z s q “ ´ πGρρ α ż K z p k q κ r bJ p kb q ´ aJ p ka qs r H p z s , k q ` H p z s , k q ` H p z s , k qs d k, (E.15)where we abusively sum the H i functions (setting them to 0 outside their definitionrange).Since all cylinders of a given MICROSCOPE’s sensor unit are co-axial, we consideronly the | x s | ď a case in this paper, so that the p x, y q -integration is K z p k q “ ż π d θ ż R ` p θ q J p kr q d r, (E.16)and we note that I z p k, θ q ” ż R ` p θ q rJ p kr q d r “ R ` p θ q J r kR ` p θ qs k . (E.17)Similarly, the radial force F r p x s , z s q “ ´ πGρρ α ż K r p k q κ r bJ p kb q ´ aJ p ka qs r H p z s , k q ` H p z s , k q ´ H p z s , k qs d k, (E.18)with K r p k q “ ż π cos θ d θ ż R ` p θ q krJ p kr q d r, (E.19)and I r p k, θ q ” ş R ` p θ q krJ p kr q d r “ π t R ` p θ q J r kR ` p θ qs H r kR ` p θ qs ´ R ` p θ q J r kR ` p θ qs H r kR ` p θ qsu , (E.20)where H and H are Struve functions (not to be confused with the previous H i functions).Without any further assumptions, we cannot integrate Eqs. (E.16) and (E.19)over θ analytically, and we end up with a 2D integral for the force between the twocylinders. We show below that in the limit of small displacements, we can integrate themanalytically. Nevertheless, in the general case, the θ integrations are easily performednumerically. Appendix E.1.4. Small displacements limit: longitudinal force
We assume that thetarget cylinder (of radius a ) moves about a “reference” position (¯ x, , ¯ z ), with a smalldisplacement δ along the z -axis. Assuming that ¯ x ! a , at first order in ¯ x { a , I z p k, θ q « a J p ka q k ` a J p ka q cos θ ¯ x, (E.21) ifth force stiffness K z p k q « π a J p ka q k . (E.22)Denoting z s “ ¯ z ` δ the altitude of the target’s center, and expanding the H i functions in the limit of small δ , taking care of their definition ranges (Appendix E.1.1),we find that the longitudinal force created on the cylinder of radius a is, at third order:(i) if ¯ z ! p ℓ, L q and ℓ ą L (target’s z -extension fully covered by that of the source): F z p ¯ z, δ q « ´ π Gρρ α p K δ ` K δ q , (E.23)where K “ ż a J p ka qr bJ p kb q ´ aJ p ka qs κk e ´ κℓ sinh p κL q d k (E.24)and K “ ż κ a J p ka qr bJ p kb q ´ aJ p ka qs k e ´ κℓ sinh p κL q d k. (E.25)(ii) if ¯ z ! p ℓ, L q and ℓ ă L (source’s z -extension fully covered by that of the target):the force is formally identical to that of the previous case, with ℓ and L switchingtheir roles.(iii) if | ¯ z | ą ℓ ` L (cylinders above each other): F z p ¯ z, δ q « π Gρρ α p K ` K δ ` K δ ` K δ q , (E.26)with K “ ´ ¯ z | ¯ z | ż a J p ka qr bJ p kb q ´ aJ p ka qs κ k e ´ κ | ¯ z | sinh p κℓ q sinh p κL q d k, (E.27) K “ ż a J p ka qr bJ p kb q ´ aJ p ka qs κk e ´ κ | ¯ z | sinh p κℓ q sinh p κL q d k, (E.28) K “ ´ ¯ z | ¯ z | ż a J p ka qr bJ p kb q ´ aJ p ka qs k e ´ κ | ¯ z | sinh p κℓ q sinh p κL q d k, (E.29)and K “ ż κ k a J p ka qr bJ p kb q ´ aJ p ka qs e ´ κ | ¯ z | sinh p κℓ q sinh p κL q d k. (E.30) Appendix E.1.5. Small displacements limit: radial force
We assume that the targetcylinder (of radius a ) moves about a “reference” position (¯ x, , ¯ z ), with a smalldisplacement δ along the X -axis. Assuming that ¯ x ! a , at third order in δ { a , I r p k, θ q « πa r J p ka q H p ka q ´ J p ka q H p ka qs ` r ka J p ka q cos θ s δ ` k r ka J p ka q cos θ ´ J p ka q sin θ s δ ` k r J p ka q cos p θ q ´ ka J p ka q cos θ s δ , (E.31) ifth force stiffness K r p k q « πka J p ka q δ ´ πk ka J p ka q δ . (E.32)The radial force created on the cylinder of radius a is thus, at third order F r p ¯ z, δ q « ´ π Gρρ α p K δ ` K δ q , (E.33)where, in the definition ranges (Appendix E.1.1):(i) if ¯ z ! p ℓ, L q and ℓ ą L (target’s z -extension fully covered by that of the source): K “ ż ka J p ka qr bJ p kb q ´ aJ p ka qs κ „ L ´ e ´ κℓ κ sinh p κL q cosh p κ ¯ z q d k (E.34)and K “ ´ ż k a J p ka qr bJ p kb q ´ aJ p ka qs κ „ L ´ e ´ κℓ κ sinh p κL q cosh p κ ¯ z q d k. (E.35)(ii) if ¯ z ! p ℓ, L q and ℓ ă L (source’s z -extension fully covered by that of the target):the force is formally identical to that of the previous case, with ℓ and L switchingtheir roles.(iii) if | ¯ z | ą ℓ ` L (cylinders above each other): K “ ż ka J p ka qr bJ p kb q ´ aJ p ka qs κ e ´ κ | ¯ z | κ sinh p κℓ q sinh p κL q d k (E.36) K “ ´ ż k a J p ka qr bJ p kb q ´ aJ p ka qs κ e ´ κ | ¯ z | κ sinh p κℓ q sinh p κL q d k. (E.37) Appendix E.2. Forces between hollow cylinders
We finally come back to the problem at hand: the gravitational force between the twohollow cylinders defined at the beginning of this appendix. By virtue of the superpositionprinciple, it is given by subtracting the force between the hollow source cylinder andtwo full target cylinders of radii a and b . Thus, in the limit of small displacements, thelongitudinal and radial forces are formally given by Eqs. (E.23), (E.26) and (E.33), withthe K i coefficients given below (they are obviously identical to those given in the maintext in the MICROSCOPE coordinates system, where the x and z -axes are inverted). Appendix E.2.1. Longitudinal force (i) if ¯ z ! p ℓ, L q and ℓ ą L (target’s z -extension fully covered by that of the source): F z p ¯ z, δ q « ´ π Gρρ α p K δ ` K δ q , (E.38)where K “ ż r b J p kb q ´ a J p ka qsr bJ p kb q ´ aJ p ka qs κk e ´ κℓ sinh p κL q d k (E.39) K “ ż κ r b J p kb q ´ a J p ka qsr bJ p kb q ´ aJ p ka qs k e ´ κℓ sinh p κL q d k. (E.40) ifth force stiffness z ! p ℓ, L q and ℓ ă L (source’s z -extension fully covered by that of the target):the force is formally identical to that of the previous case, with ℓ and L switchingtheir roles.(iii) if | ¯ z | ą ℓ ` L (cylinders above each other): F z p ¯ z, δ q « π Gρρ α p K ` K δ ` K δ ` K δ q , (E.41)with K “ ´ ¯ z | ¯ z | ż r b J p kb q ´ a J p ka qsr bJ p kb q ´ aJ p ka qs κ k e ´ κ | ¯ z | sinh p κℓ q sinh p κL q d k (E.42) K “ ż r b J p kb q ´ a J p ka qsr bJ p kb q ´ aJ p ka qs κk e ´ κ | ¯ z | sinh p κℓ q sinh p κL q d k (E.43) K “ ´ ¯ z | ¯ z | ż r b J p kb q ´ a J p ka qsr bJ p kb q ´ aJ p ka qs k e ´ κ | ¯ z | sinh p κℓ q sinh p κL q d k (E.44) K “ ż κ k r b J p kb q ´ a J p ka qsr bJ p kb q ´ aJ p ka qs e ´ κ | ¯ z | sinh p κℓ q sinh p κL q d k, (E.45) Appendix E.2.2. Radial force F r p ¯ z, δ q « ´ π Gρρ α p K δ ` K δ q , (E.46)(i) if ¯ z ! p ℓ, L q and ℓ ą L (target’s z -extension fully covered by that of the source): K “ ż k r b J p kb q ´ a J p ka qsr bJ p kb q ´ aJ p ka qs κ „ L ´ e ´ κℓ κ sinh p κL q cosh p κ ¯ z q d k (E.47) K “ ´ ż k r b J p kb q ´ a J p ka qsr bJ p kb q ´ aJ p ka qs κ „ L ´ e ´ κℓ κ sinh p κL q cosh p κ ¯ z q d k. (E.48)(ii) if ¯ z ! p ℓ, L q and ℓ ă L (source’s z -extension fully covered by that of the target):the force is formally identical to that of the previous case, with ℓ and L switchingtheir roles.(iii) if | ¯ z | ą ℓ ` L (cylinders above each other): K “ ż k r b J p kb q ´ a J p ka qsr bJ p kb q ´ aJ p ka qs κ e ´ κ | ¯ z | κ sinh p κℓ q sinh p κL q d k (E.49) K “ ´ ż k r b J p kb q ´ a J p ka qsr bJ p kb q ´ aJ p ka qs κ e ´ κ | ¯ z | κ sinh p κℓ q sinh p κL q d k. (E.50) ifth force stiffness Figure E2.
Relative difference between the exact expression (E.2) and its first orderTaylor expansion (first term of Eq. E.46) for the radial force created by the parts ofMICROSCOPE’s SUEP on the inner test mass, as a function of the displacement of thetest mass. Left: Newtonian force. Right: Yukawa force, for p α, λ q “ p , .
01 m q ; onlythose cylinders which create a non-negligible Yukawa force allowing for a well-behaved F stiffness { F exact ratio are shown. Appendix E.3. MICROSCOPE gravitational stiffness
Fig. E2 shows the relative difference between the exact expression (E.2) and its first-order Taylor expansion (first term of Eq. E.46) for the radial force created by the partsof MICROSCOPE’s SUEP on the inner test mass, when the test mass moves withinthe range used to estimate the stiffness in flight. A first-order approximation providesa 10 ´ accuracy on the gravitational forces, and can thus be safely used.Finally, note that we compared the analytic results of this appendix with numericalsimulations and with the (different) analytic expressions of Ref. [69], and found a goodagreement between all methods. References [1] Will C M 2014
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