Measuring the absolute non-gravitational acceleration of a spacecraft: goals, devices, methods, performances
aa r X i v : . [ phy s i c s . i n s - d e t ] D ec SF2A 2011
G. Alecian, K. Belkacem, S. Collin, R. Samadi and D. Valls-Gabaud (eds)
MEASURING THE ABSOLUTE NON-GRAVITATIONAL ACCELERATION OF ASPACECRAFT: GOALS, DEVICES, METHODS, PERFORMANCES
B. Lenoir , B. Christophe and S. Reynaud Abstract.
Space provides unique opportunities to test gravitation. By using an interplanetary spacecraftas a test mass, it is possible to test General Relativity at the Solar System distance scale. This requires tocompute accurately the trajectory of the spacecraft, a process which relies on radio tracking and is limitedby the uncertainty on the spacecraft non-gravitational acceleration.The Gravity Advanced Package (GAP) is designed to measure the non-gravitational acceleration withoutbias. It is composed of an electrostatic accelerometer supplemented by a rotating stage. This article presentsthe instrument and its performances, and describes the method to make unbiased measurements. Finally, itaddresses briefly the improvement brought by the instrument as far as orbit reconstruction is concerned.Keywords: Electrostatic accelerometer, Bias rejection, Colored noise, Allan variance, Non-gravitationalacceleration, General Relativity, Orbit reconstruction
With the ever-increasing precision of measurements, space has become a privileged place to test the two funda-mental theories which emerged during the 20 th century: General Relativity and Quantum Theory. In additionto providing a very clean environment, it opened new ways of testing these theories: as an example of interest forthis article, precise navigation of interplanetary spacecrafts allows probing the scale dependence of gravitationat the Solar System distance scale (Jaekel & Reynaud 2006).Even if most experimental tests support General Relativity (Will 2006), there are still open windows fordeviations. Indeed, the fact that these two fundamental theories are difficult to reconcile suggest that GeneralRelativity may not be the final description of gravitation. The reason is that gravitation is the only interactionnot having a quantum description. The validity of the Newton potential has been extensively tested for distancesbetween the millimeter and the characteristic size of planetary orbits (Fischbach & Talmadge 1999). But thereremain open windows outside this distance range for violations of the inverse square law (Adelberger et al. 2003,Fig. 4): below the millimeter or for distances of the order or larger than the Solar System characteristic size.Long range tests are performed using the motion of planets and interplanetary probes. Monitoring ofthe Moon and Mars delivers high precision tests of the validity of General Relativity at these distances (e.g.Kolosnitsyn & Melnikov 2004; Williams et al. 1996). However, the navigation data of the Pioneer probes show adiscrepancy with respect to the predictions of General Relativity (Anderson et al. 1998, 2002a; L´evy et al. 2009).This discrepancy can be described as an anomalous acceleration directed toward the Sun with a roughly constantamplitude of approximately 8 × − m.s − . The origin of this anomaly is yet unexplained despite a huge effort ofthe scientific community (Turyshev & Toth 2010, and references therein): it may be an experimental artifact aswell as a hint of considerable importance for fundamental physics (Brownstein & Moffat 2006; Jaekel & Reynaud2005). At larger scales, the rotation curves of galaxies and the relation between redshift and luminosities ofsupernovae are accounted for by introducing respectively “dark matter” and “dark energy”, which represent25 % and 70 % of the energy content of the Universe (Frieman et al. 2008). Since these dark components havebeen introduced on the basis of gravitational observations solely, the hypothesis that General Relativity is nota correct description of gravitation at these scales needs to be considered (Carroll et al. 2004). Onera – The French Aerospace Lab, 29 avenue de la Division Leclerc, F-92322 Chˆatillon, France. Laboratoire Kastler Brossel (LKB), ENS, UPMC, CNRS, Campus Jussieu, F-75252 Paris Cedex 05, France.
SF2A 2011It is therefore essential to test gravitation at all distance scales. To this extend, several mission con-cepts have been proposed to improve the experiment made by the Pioneer probes (Anderson et al. 2002b;Bertolami & Paramos 2007; Johann et al. 2008; Christophe et al. 2009, 2011; Wolf et al. 2009). In many pro-posals, the addition of an accelerometer being able to measure without bias the non-gravitational accelerationof the spacecraft is central. ESA included this idea in the roadmap for fundamental physics in space (ESA 2010)and recommended the development of accelerometer compatible with spacecraft tracking at the 10 pm.s − level.This article presents such an instrument, called the Gravity Advanced Package (Lenoir et al. 2011b). First, adescription of the instrument and its performances is given. Then, the method used to make absolute measure-ment is described (Lenoir et al. 2011a). Finally, the expected improvements of the orbit reconstruction processusing the instrument are briefly discussed. The Gravity Advanced Package is an important technological upgrade for future fundamental physics missions inspace. It is composed of two subsystems: MicroSTAR is a three-axis electrostatic accelerometer (Josselin et al.1999) based on Onera’s experience (Touboul et al. 1999; Touboul & Rodrigues 2001), and the Bias RejectionSystem is a rotating stage with piezo-electric actuator used to rotate MicroSTAR around its x axis. Theaccelerometer aims at measuring the non-gravitational acceleration of the spacecraft but other quantities are alsomeasured. In fact, MicroSTAR measures the components of the vector a on its three orthogonal measurementaxis called x , y and z : a = 1 m S F NG ext → S + ˙Ω ∧ l + Ω ∧ ( Ω ∧ l ) − (cid:18) m A + 1 m S (cid:19) F G S → A + (cid:18) m S F G ext → S − m A F G ext → A (cid:19) (2.1)where m S and m A are the masses of the satellite and the proof mass respectively, F NG ext → S is the non-gravitationalforce acting on the spacecraft, Ω is the rotation vector of the instrument with respect to a Galilean referenceframe, l is the vector between the center of mass of the satellite and the instrument (lever arm), F G S → A is thegravity of the spacecraft, and the last term in parenthesis is the gravity gradient expressed in term of acceleration, F G ext → S and F G ext → A being the gravitational forces acting on the satellite and the proof mass respectively. Allthese additional terms can be removed (Lenoir et al. 2011b).Of course, the measurement is plagued by scale ( δk ν ) and quadratic factors ( k ν ), by bias ( b ν ) and by noise( n ν ), such that the actual measurement on the axis ν ∈ { x ; y ; z } is : m ν = (1 + δk ν ) a ν + k ν a ν + b ν + n ν (2.2)where a ν is the projection of a on the axis ν . −5 −4 −3 −2 −1 −10 −9 Frequency (Hz) A cc e l e r o m e t e r no i s e ( m . s − . H z − / ) −2 −12 −10 −8 −6 Temps d’integration (s) A cc e l e r o m e t e r no i s e ( m . s − ) f c = 0.1 Hzf c = 1 Hzf c = 10 Hzf c = 100 Hz Fig. 1. Left:
Square-root of the power spectrum density of the accelerometer noise.
Right:
Square-root of the Allanvariance of the accelerometer noise. The curves are plotted for different cut-off frequencies. The oscillations for 2 πτ f c < The measurement noise is characterized by the following power spectrum density (Lenoir et al. 2011b) for aeasuring the absolute non-gravitational acceleration of a spacecraft 3measurement range of 1 . × − m.s − (cf. fig. 1): S n ( f ) = (cid:16) . × − m . s − . Hz − / (cid:17) × " (cid:18) f . (cid:19) − + (cid:18) f .
27 Hz (cid:19) (2.3)The characterization of the noise can also be given in term of Allan variance A ( τ, f c ), where f c is the cut-offfrequency. A simplified expression of the Allan variance (AVAR) (Allan 1966) for 2 πτ f c ≫ A n ( τ, f c ) = (cid:16) . × − m . s − . Hz − / (cid:17) × (cid:20) τ + 2 ln(2) × . f c π τ × (0 .
27 Hz) − (cid:21) (2.4)Figure 1 shows the dependence of Allan variance with respect to integration time and cut-off frequency withoutthe 2 πτ f c ≫ x axis of a monitored angle called θ .Assuming that the quadratic factors are equal to zero and that the scale factors are perfectly known ∗ , thequantities measured along the axis y and z are : (cid:26) m y = [cos( θ ) a Y + sin( θ ) a Z ] + b y + n y (2.5a) m z = [ − sin( θ ) a Y + cos( θ ) a Z ] + b z + n z (2.5b)with a µ ( µ ∈ { Y ; Z } ) being the components of the acceleration in the reference frame of the spacecraft. The method for removing the bias of the instrument consists in flipping MicroSTAR. The underlying idea isthat when θ = 0 rad, the accelerometer measures the quantities m y = a Y + b y and m z = a Z + b z , and when θ = π rad, it measures m y = − a Y + b y and m z = − a Z + b z . Subtracting these measurements allows recoveringthe external signal without bias, under the assumption that they are constants. The complete method, whichcan handle time variations of the external signal and of the bias, is described in (Lenoir et al. 2011a). It isshown in particular that the modulation signal, i.e. the time variation of θ , must fulfill some conditions in orderto correctly remove the bias from the measurement (cf. eq. (3.1)).The modulation signal is supposed to be periodic, τ being the period. Moreover, the measurements usedfor data post-processing are the ones made when the angle θ is constant with time and only two positions areconsidered: 0 rad and π rad (so that assuming k ν = 0 is not restrictive). On the contrary, the measurementsmade when the accelerometer is rotating are not used because they may be spoiled by unwanted signal (vibration,fictitious acceleration). The duration of the rotation per period is called T M and will be referred to as the maskingduration.Assuming that the signal to measure and the bias are affine functions of time for each modulation period(which is correct if τ is small compared to the characteristic variation time of the signal and the bias), theconditions for completely removing the bias from the measurements read Z τ/ − τ/ m ( t ) cos( θ ( t )) dt = Z τ/ − τ/ t m ( t ) cos( θ ( t )) dt = Z τ/ − τ/ t m ( t ) cos( θ ( t )) dt = 0 (3.1)where m ( t ) is equal to 0 when the accelerometer is rotating and 1 when it is not. These conditions allow derivingthe time variation of the angle θ , which is shown in figure 2 (left). The pattern depends on the ratio of themasking duration T M and the period τ , as shown by figure 2 (right).With such a signal and after post-processing, it is possible to recover the mean of the external signal withoutbias over a modulation period τ . It is possible to characterize these unbiased measurements in term of noise.The level of noise depends on the modulation period τ and the masking time T M but is approximately whiteas shown by figure 3 (left). For τ = 600 s and T M = 200 s, the uncertainty on the unbiased measurements is4 . × − m.s − (this value is the integral of the power spectrum density shown in fig. 3 (left)). This allowsreaching a precision of 1 pm.s − for an integration time of three hours. ∗ These assumptions are made for simplicity purpose. The complete treatment of the problem is presented in (Lenoir et al.2011a).
SF2A 2011 A ng l e θ ( deg r e s ) Time (arbitrary unit) π rad 0 rad π rad 0 rad0 → π π → → π π → P e r c en t age o f m a sk i ng t i m e Time (arbitrary unit)
Fig. 2. Left:
Modulation signal (-) for a masking duration representing 33.33 % of the modulation period τ = 1 arbitraryunit. Two periods are represented, separated by circles ( ◦ ) Right:
The figure describes how the modulation signal changeswhen the ratio T M /τ changes (one period is represented). The abscisse is time and the ordinate is T M /τ expressed inpercentage. The curves indicate the values of the angle θ : between two curves of the same style θ changes, between thecurves (-) and ( · ) and between the curves (- -) and (- · ) θ = π rad, and θ = 0 rad elsewhere. As mentioned in the introduction, the Gravity Advances Package (GAP) aims at improving the orbit reconstruc-tion of interplanetary probes so as to test the theories of gravitation. So far, models have been used to correctfor the non-gravitational acceleration of the spacecrafts. But the computed orbit is then subject to errors dueto uncertainties or inaccuracies in the models. By providing unbiased measurements of the non-gravitationalacceleration in the orbit plane, the GAP enhances orbit reconstruction: it removes parameters to be fitted, itmeasures the fluctuation of the non-gravitational acceleration, and it removes correlations. −7 −6 −5 −4 −3 −10 Frequency (Hz) U nb i a s ed m ea s u r e m en t no i s e ( m . s − . H z − / ) −8 −6 −4 −2 −10 −8 −6 −4 −2 Frequency (Hz) R ad i a t i on no i s e ( m . s − . H z − / ) Fig. 3. Left:
Square root of the power spectrum density of the unbiased measurements obtained after post-processing fora modulation period τ = 600 s and a masking duration T M = 200 s. Right:
Square root of the power spectrum densityof the solar radiation pressure noise (Fr¨ohlich & Lean 2004) in term of acceleration for a spacecraft with a ballisticcoefficient of 0.1 m .kg − at 10 AU of the Sun (Biesbroek 2008). The bump at 3 mHz corresponds to the 5-minuteoscillations of the Sun. There are several sources for the non-gravitational acceleration, the main ones being the direct solar radiationpressure and the anisotropic thermal radiation of the spacecraft. Figure 3 illustrates the interest of the GAP asfar as the fluctuations of the non-gravitational acceleration are concerned. The power spectrum density of theunbiased measurement noise is compared to the noise on the radiation pressure expressed in term of acceleration.Whereas these fluctuations are not taken into account in the models, it shows that the GAP allows measuringthem with a very high precision.easuring the absolute non-gravitational acceleration of a spacecraft 5
The Gravity Advances Package, designed to improve the tests of gravitation at the Solar System distance scale,displays performances which will allow improving the accuracy of orbit reconstruction significantly. Indeed, witha carefully designed calibration signal for the rotating stage, it is possible to remove completely the bias of theelectrostatic accelerometer and to obtain the mean non-gravitational acceleration of the spacecraft with no biasand a precision of 1 pm.s − for an integration time of three hours. This expected precision will allow reachingthe 10 pm.s − level for spacecraft tracking recommended by ESA. To do so, the accelerometer measurementswill have to be taken into account during the orbit reconstruction process. The authors are grateful to CNES for its financial support.
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