Charge Induced Acceleration Noise in the LISA Gravitational Reference Sensor
Timothy J Sumner, Guido Mueller, John W Conklin, Peter J Wass, Daniel Hollington
CCharge Induced Acceleration Noise in the LISAGravitational Reference Sensor
Timothy J Sumner , , Guido Mueller , John W Conklin , PeterJ Wass and Daniel Hollington Department of Physics, University of Florida, 2001 Museum Road, Gainesville,Florida 32603, USA Imperial College London, Prince Consort Road, London. SW7 2AZ, UK Department of Mechanical & Aerospace Engineering, University of Florida, 231MAE-A, P.O. Box 116250, Gainesville, Florida 32611, USAE-mail: [email protected], [email protected]
September 2019
Abstract.
The presence of free charge on isolated proof-masses, such as those within space-borne gravitational reference sensors, causes a number of spurious forces which will giverise to associated acceleration noise. A complete discusssion of each charge inducedforce and its linear acceleration noise is presented. The resulting charge accelerationnoise contributions to the LISA mission are evaluated using the LISA Pathfinderperformance and design. It is shown that one term is largely dominant but that a fullbudget should be maintained for LISA and future missions due to the large number ofpossible contributions and their dependence on different sensor parameters.
1. Introduction
The build-up of charge on isolated free-flying proof-masses fully enclosed withinspacecraft is inevitable due to cosmic-rays and solar energetic particles which are sohighly penetrating that they can reach the enclosed proof-mass and interact with itand its surrounding environment [1]. Once a proof-mass is charged there will be forcesbetween it and its surrounding conducting enclosure which include an electrostatic forcefrom its own mirror charges, an interplay with any applied or stray electric potentialsand a Lorentz force from motion through ambient magnetic fields. Associated with thoseforces will be induced acceleration noise which could be a limiting noise component forweak-force experiments, and the charge has then to be controlled [2, 3, 4, 5, 6, 7]. Thelevel of control needed depends on the sensitivity of the sensor to charge.This study documents all the charge induced noise terms for gravitational referencesensors of the type used for LISA Pathfinder [8] and proposed for LISA [9]. In section 2a sensor model geometry is defined and its charge sensitivity is derived. In section 3 a r X i v : . [ phy s i c s . i n s - d e t ] S e p harge noise in LISA
2. The Sensor Model
The generic model developed here is based on the gravitational reference sensor(GRS) [10] within the LISA Technology Package (LTP) [11] flown on LISA Pathfinder.The basic sensor design [12] is a metallic cube surrounded by a number of electrodepairs enclosed within a metallic housing. The electrode pairs perform different functions(signal injection, sensing and actuation in all six degrees of freedom) and are arrangedsymmetrically on either side of the proof-mass in all three axes. One such pair is shownschematically in Figure 1.
Figure 1.
Schematic representation of the proof-mass with one electrode pair.
In total the GRS has 9 electrode pairs providing only a partial coverage of theproof-mass surface area. The electrodes are close-mounted into a metallic housing whichthen provides an effective ground plane coverage of the rest of the proof-mass. For thepurposes of the electrostatic model the housing exposed directly to the proof-mass canbe thought of as 3 additional grounded electrode pairs, one for each axis. The LTP usedgold-coated cubic proof-masses with a side length 46 mm. The nominally symmetricgaps, d , ranged from 2.9 mm to 4.0 mm depending on the function of the electrode pair.Relevant to the charge sensitivity of this design will be the individual capacitancesand, more importantly, the individual capacitance gradients. Finite element modellingis the most accurate way to find these with this type of complicated geometry [13] harge noise in LISA C T , is given by C T = (cid:88) i =1 C i where C i = (cid:15)A i d i (1)where (cid:15) is the electrical permittivity within the gap, A is the cross-sectional area,and values of i from 1 to 18 refer to the electrodes proper and 19 to 24 refer to thehousing ground planes. The ground planes are designed to provide some degree of cross-coupling isolation and constitute some 61% of the total capacitance. They also help tosuppress fringing fields from around the edges of the individual electrodes coupling tothe proof-mass making the parallel plate approximation more accurate.Figure 1 shows the proof-mass with a matched, and nominally identical, pair ofelectrodes along the x − axis. Identifying these two electrodes from left to right with i = 1 and i = 2 the contribution to C T is simply C + C . The capacitance gradientalong x , as experienced by the proof-mass moving along x , is ∂C T ∂x = ∂C ∂x + ∂C ∂x = − C d + C d (2)Therefore, if the two electrodes are identical and d = d = d and C = C = C x , thegradient will be zero. If the proof-mass is not at the centre, but is displaced by a smalloffset x o the capacitance gradient due to the matched pair of electrodes becomes ∂C T ∂x = C x d x o d (3)The first factor is the single-sided gradient and the second factor is << ∼ .
01 forthe LTP experiment). The total single-sided capacitance gradient in the most sensitiveaxis for the LTP design is 1 . A = A − A , (through machiningtolerances) then there will be an gradient even if the proof-mass is centred geometrically.Equation 3 becomes [15] ∂C T ∂x = C x d (cid:18) x o d + ∆ AA (cid:19) (4)In practice the co-ordinate system can be redefined to pass through the electrostaticcentre to take account of any mismatch in the electrode (and housing) symmetry. Henceequation 3 will be used. harge noise in LISA k will come from any voltages present on the electrodesaccording to:- F k = 12 (cid:88) i ∂C i ∂k V i (5)There will be two types of deliberately applied voltage; the signal injection voltageat high frequency (needed to perform the position measurement) and somewhat lowerfrequency voltages to apply actuation forces to control the proof-mass in all six degreesof freedom. The signal injection is applied in the two axes orthogonal to the sensitivedirection. The control voltages are applied on matched pairs of electrodes such that (cid:80) V i = 0.The presence of a free charge, Q , on the proof-mass will result in two newelectrostatic force terms to give an overall charge related force, F Qk F QEk = Q C T ∂C T ∂k − QC T (cid:88) i V i ∂C i ∂k (6)The first term is the interaction between Q and its mirror charges induced in theelectrodes. The second term is the interaction between Q and any voltages present onthe electrodes, both deliberately applied and stray [5].In addition to the electrostatic forces there will also be Lorentz forces due to theproof-mass (charge) motion through any magnetic fields present, both internal andexternal. These will have the form F QLk = Q (cid:16) (cid:126)v SC × (cid:126)B ext + (cid:126)v P M × (cid:126)B ext + (cid:126)v P M × (cid:126)B int (cid:17) k (7)where (cid:126)v SC is the velocity of the spacecraft through the interplanetary magneticfield, (cid:126)B ext , (cid:126)v P M is the velocity of the proof-mass relative to the spacecraft and (cid:126)B int isany magnetic field generated within, and locked to, the spacecraft. As first pointed outby J-P Blaser [16] the metallic enclosure around the proof-mass will act as a shield forit against the first term in equation 7 through the generation of an effective Hall voltageacross the enclosure. For an ideal, completely closed and perfectly conducting enclosurethe effect should be total. However any apertures in the enclosure will result in magneticfield leakage and there will be some residual small efficiency, η , to be applied to the firstterm. F QLk = Q (cid:16) η(cid:126)v SC × (cid:126)B ext + (cid:126)v P M × (cid:16) (cid:126)B ext + (cid:126)B int (cid:17)(cid:17) k (8)A first estimate of the efficiency was evaluated at a very early LISA design phase [17]giving η < .
03. Since then the GRS design became that of a very much more enclosedproof-mass. harge noise in LISA
3. Acceleration Noise from forces involving Q The forces arising due to a free-charge on the proof-mass are those in equations 6 and 8.Any parameter within those force terms which exhibits a noisy behaviour will give riseto acceleration noise of the proof-mass. Noise formulae for each term will be evaluatedby considering which parameters in each one will exhibit noise [18].
4. Electrostatic acceleration noise
The first term in equation 6 can be combined with equation 3 to give the accelerationas a Q Ek = (cid:32) Q M C T C x d (cid:33) x o (9)where M is the mass of the proof-mass. The acceleration is directly proportionalto the displacement and the factor in parenthesis is thus an effective negative springconstant. Note that C x and d are nominal design values and are fixed. C T on theother hand is the total capacitance and will depend on displacements in all three axesand angles through combinations of offsets in them [17]. However the dominant noisein C T resulting in acceleration noise along the x -axis is directly from displacementnoise of the proof-mass relative to the housing along the x -axis, charaterised by itsamplitude spectral noise density, S / x . This displacement noise also affect x o directly.Finally the charge, Q , arises due to stochastic charge deposits from cosmic-rays andsolar energetic particles [19] and so there are three acceleration noise terms, S / a j arisingout of equation 9.From S / x , via C x , there is S / a = Q M C T (cid:18) C x d (cid:19) S / x (10)From S / x , via C T , there is S / a = Q M C T (cid:18) C x d x o (cid:19) S / x (11)From S / Q , via Q , there is S / a = QM C T (cid:18) C x d (cid:19) x o S / Q (12)The second force term in equation 6 involves the interaction between Q and voltagespresent on each electrode. Due to the nominal symmetry of the sensor it is convenientto split this term into two parts. The first due to common-mode voltages and the seconddue to differential mode voltages.Forces due to common-mode voltages for a perfectly-centred proof-mass should sumup to zero as each electrode pair will combine with equal and opposite capacitancegradient. However as already noted any displacement of the proof-mass from the harge noise in LISA V c there will be an acceleration a Q Ek = (cid:18) QM C T (cid:19) (cid:32) V c C (cid:48) x d x o (cid:33) (13)where the result of the summation of the all relevant voltages and capacitances hasbeen combined into the final factor and C (cid:48) x is the total capacitance from the electrodesproper; i.e. not including the ground plane. This acceleration term involves Q , C T , V c and x o all of which have noise associated with them. Hence there are four accelerationnoise components.From S / Q there is S / a = 1 M C T (cid:32) V c C (cid:48) x d (cid:33) x o S / Q (14)From S / x , via C T , there is S / a = QM C T V c (cid:32) C x C (cid:48) x (cid:33) (cid:32) C (cid:48) x d x o (cid:33) S / x (15)From S / V c there is S / a = QM C T (cid:32) C (cid:48) x d x o (cid:33) S / V c (16)From S / x there is S / a = QV c M C T (cid:32) C (cid:48) x d (cid:33) S / x (17)Forces due to differential-mode voltages for a perfectly-centred proof-mass will notsum up to zero. Instead they will act through the individual capacitance gradient(s) ofwhichever electrode(s) they are present on, including the ground planes of the housing.Differential mode voltages could arise either through mismatch in applied voltagesotherwise intended to balance (due to scale factors or incoherent noise), or throughuncontrolled stray potentials associated with individual surfaces [20, 21, 22, 23, 24, 25].For simplicity it will be assumed here that only one surface, labelled simply as i , isimplicated with a stray voltage, V i . Then a Q Ek = (cid:18) QM C T (cid:19) V i ∂C i ∂k (18)This acceleration term involves Q , C T , ∂C i /∂k and V i all of which have noiseassociated with them. Hence there are four more acceleration noise components.From S / Q there is S / a = 1 M C T V i ∂C i ∂k S / Q (19)From S / x , via C T , there is S / a = QM C T V i ∂C i ∂k (cid:18) C x d x o (cid:19) S / x (20) harge noise in LISA S / x , via ∂C i /∂k , there is S / a = QM C T V i ∂ C i ∂k S / x (21)From S / V i there is S / a = QM C T ∂C i ∂k S / V i (22) From equation 8 the maximum acceleration experienced by the charged proof-mass dueto magnetic field interactions is a QLk = QM ( ηv SC B ext + v P M ( B ext + B int )) k (23)This acceleration involves Q , v SC , B ext , v P M and B int all of which have noiseassociated with them. Indeed even η could have a time dependent, and hence noisybehaviour, due to pitch angle variation in the external B field, but this will not beaddressed here. Hence there are five more acceleration noise components.From S / Q there is S / a = 1 M ( ηv SC B ext + v P M ( B ext + B int )) S / Q (24)From S / v SC there is S / a = QM ( ηB ext ) S / v SC (25)From S / B ext there is S / a = QM ( ηv SC + v P M ) S / B ext (26)From S / v PM there is S / a = QM ( B ext + B int ) S / v PM (27)From S / B int there is S / a = QM ( v P M ) S / B int (28)There are thus sixteen individual contributions to the overall charge-inducedacceleration noise and these will be in the next section to find the overall noise budget.
5. Overall Acceleration Noise Budget Evaluation
The linear acceleration noise will be evaluated along the sensitive x -direction for eachof the contributions. harge noise in LISA In the sensitive direction the nominal amplitude spectral density of the position noiseto be used in equations 10, 11, 15, 17, 20 and 21 is taken directly from the requirementsdetailed in the LISA proposal [9] for the interferometric ranging, assuming equalcontributions from the two proof-masses in a link. S / x ≤ . × − (cid:118)(cid:117)(cid:117)(cid:116) (cid:32) × − f (cid:33) m / √ Hz (29)where f is the frequency. However, this is a requirement on the differential noisebetween two widely separated proof-masses forming a link, and it does not directlygive the relative spatial noise behaviour of one proof-mass within its own spacecraftenclosure which will depend on the inherent platform stability, the local displacementmeasurement and the closed-loop drag-free performance used to control the spacecraftmotion. This has been studied in detail for LISA Pathfinder [26] and we use resultsfrom that study to provide representative performance data. Figure 2 in [26] shows anamplitude spectral density between 0.05mHz and 30mHz which we approximate to S / x ≈ × − f m / √ Hz (30)Although equations 29 and 30 have very different forms their values at 1mHzare almost identical. In the less-sensitive directions, relevant to equation 27, thedisplacement noise requirement is relaxed to 5 nm/ √ Hz and the LISA Pathfinder data(Figure 5 in [26]) show an in-flight performance of S / y,z in the range 0 . − × − m / √ Hzbetween .01 and 30 mHz ‡ .The charge noise, S / Q to be used in equations 12, 14, 19 and 24 was firstderived using Monte-Carlo simulations [19] but has since been modified by in-orbitmeasurements made by LISA Pathfinder [14]. The charging process is due to theinteraction of cosmic-rays within the proof-mass and surrounding structures. Thecosmic-rays have random arrival times and each cosmic-ray charges the proof-massby an amount drawn at random from a broad distribution [19]. Hence the chargingrate will exhibit shot noise, but at a level exceeding that expected for single chargecontributions; i.e. S / I = e (cid:113) λ eff where λ eff >> ˙ Q/e is the effective single-chargenoise rate. The average charging rate seen on the two proof-masses in LISA Pathfinderwas +23 . λ eff = 1100 charges/s. The charge, Q is then the result of integrating the current. Consequently the charge spectral noisedensity relevant to equations 12, 14, 19 and 24 is: S / Q = 12 πf S / I = e πf (cid:113) λ eff C / √ Hz (31)This charge spectral noise density is relevant to equations 12, 14, 19 and 24.The appropriate form for the amplitude spectral density of the common-modevoltage, S / V c , depends on how the voltages are produced and applied. The lowest ‡ Using the ‘State reproduction from the model’ as a true indicator of this noise harge noise in LISA √ n higher whereas if the noise sources are correlated (i.e derivedfrom a common reference) the overall noise scales as n . Typically high-precision voltagereferences have noise at the ppm level, which will be assumed for equation 16.In equation 22 the spectral noise density for differential mode (stray) voltages isrequired. Stray voltages on gold surfaces have been measured up to 100 mV levels. Earlymeasurements of the noise associated with such stray voltages gave a white spectraldensity of 30 µ V / √ Hz above 0.1 mHz but rising to lower frequencies [23]. More recentlyLISA Pathfinder has made a new more representative measurement in space, whichincluded both stray voltages and low-frequency applied voltage noise [14]. The resultwas no longer white above 0.1mHz but could be well fitted by a form S / V i = (cid:115) . × − f + 2 . × − f V / √ Hz (32)It should be noted that this mathematically assigns the effect of all the differentialstray voltages to just one of the two sensing electrodes along the x -direction, followingthe convention adopted in [24]. If this assumption is relaxed and the stray voltages,for example, are uniformly distributed between both sensing electrodes and the groundplane the overall result would be the same as obtained with equation 32 as the voltagefluctuations would be reduced in proportion to the electrode areas and the gaps are thesame for each. The spacecraft velocity noise within equation 25 will be due to solar wind forces.To generate an acceleration noise along the x -axis the spacecraft velocity noise mustbe in a perpendicular direction. In that direction drag-free satellite control willsuppress the actual movement of the spacecraft using the external micro-thrusters usingmeasurements from capacitance sensing of the proof-mass. The resulting residual motionof the spacecraft relative to the local gravitational frame will be from a combination ofuncertainty (noise) in the capacitance measurement and noise in the thrusters givingimperfect drag-free control. The requirement specification for measurement of relativeproof-mass displacement within the spacecraft calls for 5nm/ √ Hz. The in-flight LISAPathfinder platform stability performance, including the closed-loop drag-free, is shownin Figure 6 of [26]. The spacecraft acceleration curves recovered from the simulation havea characteristic ’V’ shape which, for the relevant transverse axes, can be approximatedto within a factor of two, to a functional form S / a SC = 3 × − f + 2 × − f ms − / √ Hz (33)From this the amplitude spectral density of the velocity noise is S / v SC = 3 × − πf + 2 × − fπ ms − / √ Hz (34) harge noise in LISA /f n fit to the noisiest componentin their data over the relevant frequency range gives S / B ext = 0 . × − f . T / √ Hz (35)Equation 27 requires the velocity noise spectrum for the proof-mass relative to itshousing. It does not matter whether the proof-mass is actually moving but rather thatthere is relative motion between the proof-mass and the magnetic field source. Therequirement specification for measurement of relative proof-mass displacement withinthe spacecraft calls for 5nm/ √ Hz. The in-flight LISA Pathfinder platform stabilityperformance, including the closed-loop drag-free, gave a resultant maximum amplitudespectral density of 20nm/ √ Hz. Conservatively treating this as frequency independentimplies, S / v PM = 4 πf × − ms − / √ Hz (36)The magnetic field noise was measured in situ by magnetometers on boardLISA Pathfinder. At low frequencies ( < < / √ Hz [30]. The magneticfield noise measured by LISA Pathfinder would have included spatial variations in thefield converted into time dependent variations due to the spacecraft motion. Theseshould be fairly representative of that expected for LISA given the L1 orbit around theSun is close to 1AU.
Table 1 lists the nominal values of all the parameters used in estimating the accelerationnoise from each of the 16 terms identified.
Figure 2 shows the contributions from the electrostatic terms, S / a through S / a , to theacceleration noise spectral density. Also shown is the performance requirement for theoverall LISA GRS acceleration noise for each proof-mass [9]. S / a ≤ × − (cid:118)(cid:117)(cid:117)(cid:116) (cid:32) . f (cid:33) (cid:118)(cid:117)(cid:117)(cid:116) (cid:32) f (cid:33) ms − / √ Hz (37)The ‘Sum’ curve is the quadrature sum of all terms. In principle terms driven bythe same noisy parameter should be added linearly but in practice they also containfactors whose absolute sign is uncertain. The values of all the parameters have been set harge noise in LISA Table 1.
Relevant GRS parameters used for charge-induced noise evaluation
Parameter Value Units EquationsM 1.928 kg [10 – 28] C T C x d x o µ m [11 – 16], 20 V c V i a mV [19 – 21] ∂C i /∂k b pF/m 19, 20, 22 ∂ C i /∂k c pF/m η .01 d - [24 – 26] v SC × m/s 24, 26 B ext e nT 24, 25, 27 v P M πf (3 / f nm/s 24, 26, 28 B int g § µ T 24, 27 a Assuming dc compensation following [22] as demonstrated on LISA Pathfinder [14] b Expressed as an effective single-sided gradient from the two sensing electrodes asassumed for LISA Pathfinder [14] c ( ∂C i /∂k ) /d d Taking the very conservative early estimate [17] reduced by a further factor of 3 forthe newer more closed designs e See [27] f Estimated from S / y,z . Note f in Hz g See [30]to those appropriate for continuous discharge during science mode operations, as shownin table 1, with a charge level, Q , set to 5 × elementary charges. S / a is the most dominant term, and is almost coincident with the ‘Sum’ curveon the plot. It arises due to the interplay between Q and the spectral noise densityfrom the differential (stray+ applied) voltages. S / a and S / a are next most dominantterms, and on the plot there are coincidentally almost equal. S / a comes from theinteraction between differential (stray) voltages and the charge noise spectral density. S / a comes from the interaction between common mode (injection) voltages and thecharge noise spectral density. The total spectral noise density from all electrostaticterms is comfortably below the performance requirements, partly, it should be noted,due to the stray voltage compensation scheme [22], now successfully demonstrated inflight with LISA Pathfinder [14].Figure 3 shows a similar plot but with Q = 1 . × elementary charges, which harge noise in LISA Figure 2.
Acceleration noise density associated with electrostatic contributions, S / a to S / a , in continuous discharge mode assuming Q = 5 × charges. is the maximum charge expected in science mode if an intermittent discharge schemeis adopted. S / a has increased proportionately to Q . The ‘Sum’ remains below therequirement specification of the GRS. The electrostatic noise approaches the requirementspecification closest at low frequencies, below a mHz.Figure 4 shows the fraction of the GRS acceleration noise budget absorbed by thecharge induced noise at 0.1 mHz as a function of Q . Below Q = 10 charges the fractionbecomes charge independent, but non-zero. This is due to S / a and S / a which donot depend on absolute charge, but do depend on the charge noise, and between themcontribute ∼ × − ms − / √ Hz at 0.1mHz. Q can increase up to 5 × charges beforethe GRS budget is fully absorbed by charge induced noise. Also shown in figure 4 isthe overall contribution from the Lorentz force noise terms, S / a through S / a . It canbe seen that these are insignificant compared to the electrostatic terms, although thedominant term within the Lorentz noise benefits from the (fortuitous) shielding factor, η and, without that, the Lorentz force noise would dominate above Q ∼ charges.Figure 5 shows the individual Lorentz force noise components with a relatively high Q of 5 × charges. The dominant term is S / a which is coincident with the ’Sum’curve and is the interaction between Q and the external magnetic field fluctuations,after allowing for an assumed 99% effective shielding factor ( η = 0 .
01) from the Halleffect in the metallic housing. harge noise in LISA Figure 3.
Maximum acceleration noise density associated with electrostaticcontributions, S / a to S / a , during science operations using the intermittent rapiddischarge mode with Q max = 1 . × charges. Figure 4.
Fraction of the LISA GRS acceleration noise budget absorbed by thecharge induced noise as a function of Q at 0.1 mHz. harge noise in LISA Figure 5.
Acceleration noise density associated with Lorentz contributions, S / a to S / a . Q = 5 × is used to enhance these terms.
6. Discussion
A comprehensive evaluation of the linear acceleration spectral noise density terms arisingfrom free charge residing on the proof-mass has been carried out. Sixteen first-orderterms have been identified including both electrostatic interactions and magnetic fieldinteractions. All terms have been informed by the recent in flight experience with LISAPathfinder, during which some of the experiments were specifically designed to helpconsolidate the charge induced noise understanding.
The dominant charge related terms contributing to the GRS acceleration noise areelectrostatic terms. The most dominant involves the differential voltage fluctuations inequation 32 while the next two involve the noisy charging process via equation 31.For both continuous discharging and intermittent discharging modes of operationthe charge related noise remains within the overall budget, for the parameters adoptedin table 1. Most of the key parameters in that table are based on LISA Pathfinder whichwas ultimately able to show compliance with the LISA budget [31].From figure 4 it can be seen that there is no advantage to reducing the charge muchbelow 10 charges from the point of view of acceleration noise. Hence the continuousdischarge mode only needs to ensure Q < × charges. harge noise in LISA Forces will build up as charge is accumulating on the proof-mass. Thesize of these forces could in principle cause bias effects. The three basic forces are givenin table 2 for various values of Q . Note there is a coincidental equality between the sizesof the second two forces given the parameter values used in Table 1 Table 2.
Charge induced forces
F orce
Q Value Units Q C T ∂C T ∂k . × − N1.5E7 2 . × − N5E8 3 . × − N QC T (cid:80) V i ∂C i ∂k . × − N1.5E7 1 . × − N5E8 3 . × − N QC T V com C k d δx . × − N1.5E7 1 . × − N5E8 3 . × − N The presence of free charge on the proof-mass will introducenew terms into the spring constant and these must be small enough not to unduly affectthe dynamics of the system. Typical spring constants observed on LISA Pathfinderwere ∼ − N/m [8] and from Table 3 it can be seen that charge induced electrostaticspring terms will not become problematic until Q approaches 10 charges. During solar-quiet times the charging rate will follow the ambientcosmic-ray environment [32]. The cosmic-ray rate will be fairly steady but with quasi-periodic modulations at the few percent level with the occasional larger short-termForbush depressions [33]. The overall trend in the charge on the proof-mass will belinear with time if the charge is allowed to build-up. This linear charge build-up willgive rise to forces which grow both linearly and quadratically with time and this will behappening quasi-independently on the six proof-masses within the LISA constellation.In the case of an intermittent discharge scheme, with a periodic discharge sequence, thiscould produce fourier components in the data which could become problematic [34]. harge noise in LISA Table 3.
Charge induced spring constants
F orce/ ( SpringConstant ) Q Value Units Q C T ∂C T ∂k . × − N/m (cid:18) Q C T C x d + Q C T (cid:16) C x d δx (cid:17) (cid:19) . × − N/m5E8 3 . × − N/m QC T (cid:80) V i ∂C i ∂k . × − N/m (cid:18) QV i MC T (cid:16) C x d δx + ∂ C i ∂x (cid:17)(cid:19) . × − N/m5E8 8 . × − N/m QC T V com C k d δx . × − N/m (cid:18) QC T V com C x d + QC T V com (cid:16) C x d δx (cid:17) (cid:19) . × − N/m5E8 3 . × − N/m
Acknowledgments
TJS acknowledges support from the Leverhulme Trust (EM-2019-070 \ References [1] Jafry Y, Sumner T J and Buchman S 1996 Electrostatic charging of space-borne test bodies usedin precision experiments
Class. Quantum Grav. A97–A106[2] Buchman S, Quinn T, Keiser M, Gill D and Sumner T J 1995 Charge management and controlfor the Gravity Probe B gyroscopes
Rev. Sci. Instrum. et al Phys. Rev. D et al Adv. Space Res. et al Class.Quantum Grav. S597– S602[6] Sun K-X, Allard B, Buchman S, Williams S and Byer R L 2006 LED deep UV source for chargemanagement of gravitational reference sensors
Class. Quantum Grav. S141–S150[7] Pollack S E, Turner M D, Schlamminger S, Hagedorn C A and Gundlach J H 2010 Chargemanagement for gravitationa-wave observatories using UV LEDs
Phys. Rev. D et al Phys. Rev. Lett. et al arXiv:1702.00786 [10] Dolesi R et al
Class.Quantum Grav. S99–S108[11] Anza S et al
Class. Quantum Grav. S125–S138 harge noise in LISA [12] Weber W J et al Proc. SPIE
Commun. Num. Meths. Eng. et al Phys. Rev. Lett.
Proc. STEP Symp. in Pisa
ESAWPP-115
Final Report to ESA: AO/1-2787/94/NL/JG - Drag-freesatellite control [18] Ara´ujo H M, Howard A, Davidge D and Sumner T J 2003 Charging of isolated proof masses insatellite experiments such as LISA
Proc. SPIE
Astropart. Phys. Class. Quantum Grav. A291–A297[21] Carbone L et al
Phys. Rev. Lett. et al Adv. Space Res. Phys. Rev. Lett. et al
Phys. Rev. Lett. et al
Phys. Rev. D et al Phys. Rev. D Space Sci. Rev. et al Astron. Astrophys.
J. Phys.: Conf. Ser. et al µ Hz Phys. Rev. Lett. et al
Astropart. Phys. et al Astrophys. J.
113 (12pp)[34] Shaul D N A, Ara´ujo H M, Rochester G K, Sumner T J and Wass P J 2005 Evaluation ofdisturbances due to test mass charging for LISA
Class. Quantum. Grav.22