Phys revlett.106.221101

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Phys revlett.106.221101

  1. 1. Selected for a Viewpoint in Physics week endingPRL 106, 221101 (2011) PHYSICAL REVIEW LETTERS 3 JUNE 2011 Gravity Probe B: Final Results of a Space Experiment to Test General Relativity C. W. F. Everitt,1,* D. B. DeBra,1 B. W. Parkinson,1 J. P. Turneaure,1 J. W. Conklin,1 M. I. Heifetz,1 G. M. Keiser,1 A. S. Silbergleit,1 T. Holmes,1 J. Kolodziejczak,2 M. Al-Meshari,3 J. C. Mester,1 B. Muhlfelder,1 V. G. Solomonik,1K. Stahl,1 P. W. Worden, Jr.,1 W. Bencze,1 S. Buchman,1 B. Clarke,1 A. Al-Jadaan,3 H. Al-Jibreen,3 J. Li,1 J. A. Lipa,1 J. M. Lockhart,1 B. Al-Suwaidan,3 M. Taber,1 and S. Wang1 1 HEPL, Stanford University, Stanford, California 94305-4085, USA 2 George C. Marshall Space Flight Center, Huntsville, Alabama 35808, USA 3 King Abdulaziz City for Science and Technology, Riyadh, Saudi Arabia (Received 1 April 2011; published 31 May 2011) Gravity Probe B, launched 20 April 2004, is a space experiment testing two fundamental predictions of Einstein’s theory of general relativity (GR), the geodetic and frame-dragging effects, by means of cryogenic gyroscopes in Earth orbit. Data collection started 28 August 2004 and ended 14 August 2005. Analysis of the data from all four gyroscopes results in a geodetic drift rate of À6601:8 Æ 18:3 mas=yr and a frame-dragging drift rate of À37:2 Æ 7:2 mas=yr, to be compared with the GR predictions of À6606:1 mas=yr and À39:2 mas=yr, respectively (‘‘mas’’ is milliarcsecond; 1 mas ¼ 4:848 Â 10À9 rad). DOI: 10.1103/PhysRevLett.106.221101 PACS numbers: 04.80.Cc Introduction.—In 1960, Schiff [1] showed that an ideal at the equator, eliminates ‘‘seeing’’ in the measurement togyroscope in orbit around Earth would undergo two the guide star, and vastly reduces torques from suspendingrelativistic precessions with respect to a distant inertial the gyroscope against 1g gravity. Cryogenics brings newframe: (1) a geodetic drift in the orbit plane due to motion levels of magnetic shielding, thermal isolation, ultrahighthrough the space-time curved by Earth’s mass and (2) a vacuum operation, and a uniquely effective gyro readoutframe-dragging due to Earth’s rotation. The geodetic term based on the London moment in a spinning superconduc-matches the curvature precession of the Earth-Moon sys- tor. The two together give the instrument ultimate me-tem around the Sun given by de Sitter in 1916 [2]. The chanical stability: in zero g, there is no sag; at zero K,Schiff frame dragging is related to the dragging of the orbit there is no thermal distortion.plane of a satellite around a rotating planet computed by Essential to GP-B as a controlled physics experimentLense and Thirring in 1918 [3]. Frame dragging has im- was the calibration phase, a 46-day period following theportant implications for astrophysics; it has been invoked main science phase designed to set limits on a range ofas a mechanism to drive relativistic jets emanating from potential disturbing effects and quantify any that mightgalactic nuclei [4]. prove larger than expected. The measurement requires one or more gyroscopes ref- Experiment description.—The heart of the instrumenterenced to a remote star by an onboard telescope. In the was a 0.92 m long fused silica structure containing four642 km polar orbit of Gravity Probe B (GP-B), the two gyroscopes and a star-tracking telescope mounted in aeffects are at right angles, as in Fig. 1. The predicted 2440 l superfluid helium Dewar operating at 1.8 K. Eachgeodetic drift rate is À6606:1 mas=yr; the frame-draggingdrift rate with the chosen star IM Pegasi is À39:2 mas=yr. GP-B was conceived as a controlled physics experimenthaving submas=yr stability (106 times better than the bestmodeled navigation gyroscopes) with numerous built-inchecks and methods of treating systematics. Three prin-ciples guided the design: (1) make Newtonian gyro driftsmuch less than the predicted GR effects; (2) add sensors sothat modeling hinges on physical understanding as againstthe ad hoc observational modeling used in navigationgyroscopes; (3) exploit natural effects such as stellar aber-ration in calibrating the instrument. Meeting the manymechanical, optical, and electrical requirements rested ona conjunction of two technologies, space and cryogenics. Operation in space separates the two effects, increases FIG. 1 (color). Predicted drift rates of GP-B gyroscopes. Seethe geodetic effect 12.4 times as compared to a gyroscope [17] for definitions of WE and NS inertial directions.0031-9007=11=106(22)=221101(5) 221101-1 Ó 2011 American Physical Society
  2. 2. week endingPRL 106, 221101 (2011) PHYSICAL REVIEW LETTERS 3 JUNE 2011gyroscope comprised a 38 mm diameter niobium-coated TABLE I. The seven near zeros.fused quartz sphere suspended electrically, spun up by Property Requirement Achievedhelium gas, and read out magnetically. The four were setin line, two spinning clockwise and two counterclockwise, Rotor propertieswith axes initially aligned to the boresight of the telescope. Mass unbalance, r=r 6 Â 10À7 2 À 5 Â 10À7 Asphericity (nm) 55 33The space vehicle rolled with 77.5 s period about the line of Patch dipole (C Á m) 10À15 see textsight to IM Pegasi, which was, however, occulted by Earth Environmentfor almost half each orbit (Fig. 1). During occulted periods, Cross-track acceleration (g) 10À11 10À11pointing was referenced to star trackers and rate gyros on the Gas pressure (torr) 10À11 10À14outside of the spacecraft. Drag compensation, originated by Rotor trapped field (G) 9 0.2–3Pugh in an independent proposal for an orbiting gyroscope Mixedexperiment [5], two months prior to Schiff’s Letter, was by a Rotor electric charge (electrons) 108 108control system referred to one of the gyroscopes as a proofmass. Attitude, translational, and roll control authority wasprovided by the helium boil-off gas from the Dewar vented neity and asphericity for the rotor, residual accelerationthrough proportional thrusters [6]. and magnetic field for the environment. The others are gyro A gyro readout that did not disturb the spin orientation electric charge, gas pressure, and patch effect. Six metwas vital. Superconductivity supplied three essentials. The requirements; the complication discovered during thespinning rotor generated a London moment equivalent at calibration phase was the patch effect. Even so, the idea80 Hz to a uniform 5 Â 10À5 G field aligned with the spin of seeing relativity in the ‘‘raw’’ data was preserved.axis. A SQUID magnetometer coupled to a superconduct- Figure 2 shows the north-south (NS) drifts of the fouring loop on the gyro housing provided a readout capable of gyroscopes with no torque modeling. The geodetic driftdetecting a 1 mas change of spin direction in 10 h. Finally, is visible in all four.a combination of high permeability and ultralow field The term ‘‘patch effect’’ [7,8] refers to contact potentialsuperconducting shields around the instrument with local differences between crystalline or contamination regionsshields for each gyroscope achieved (1) 240 dB isolation on a metal surface. Prelaunch studies focused on eddyfrom external magnetic disturbances, (2) a limit on trapped current losses and torques from interaction of the rotor’sfields in the rotors $1% of the London moment, and patch-induced electric dipole moment with support volt-(3) virtual elimination of magnetic torques. ages and the housing dipole moment. Far more important The gyro readout scale factor Cg ðtÞ was calibrated on was the on-orbit discovery of forces and torques caused byorbit against the aberration of starlight. During each half- interacting patches on the rotor and housing. Put simply,orbit when IM Pegasi was visible, the telescope pointed rotor and housing were spherical mechanically; electri-at its apparent position, and the 20.495 86 arcsec annual cally, they were not.and 5.18560 arcsec orbital aberrations, derived, respec- Three unforeseen effects emerged, differing in detail fortively, from the JPL Earth ephemeris and GPS orbit data, each gyroscope: a changing polhode path and two patchappeared in the gyro readout. effect torques. The changing polhode, originating in a An elementary calculation captures the stringencies of 1 pW dissipation of the rotor’s kinetic energy, compli-the experiment. Consider a spherical, not quite homoge- cated the Cg determination. Beginning with their body axesneous rotor under transverse acceleration f. Let r sepa- arbitrarily oriented and a 2–6 h period, the rotors transi-rate the centers of mass and support, vs $ 9 m=s be the tioned to a final 1–4 h period, each spinning nearly aboutrotor’s peripheral velocity, 0 the maximum alloweddrift rate 0:1 mas=yr (1:5 Â 10À17 rad=s). Then, we haver=r 2vs 0 =5f, and mass unbalance requirementsr=r: 6 Â 10À18 on Earth, 6 Â 10À10 in a typical 642 kmaltitude satellite, and 6 Â 10À7 for GP-B in 10À11 g drag-free mode. Without drag compensation, the experimentwould have been impossible. Likewise for an asphericalhomogeneous rotor, the electrical support torques, whilegreatly reduced on orbit, only reached the desired levelthrough a further symmetrizing factor, spacecraft roll. Thenet allowed asphericity was 55 nm. Aiming for a $0:5 mas=yr mission, we created an errortree setting limits on 133 disturbing terms, and verified inadvance that all were negligible. Seven ‘‘near zeros’’ FIG. 2 (color). North-south gyro orientation histories with no(Table I) were central. Four noted already are inhomoge- modeling of torque or scale factor. 221101-2
  3. 3. week endingPRL 106, 221101 (2011) PHYSICAL REVIEW LETTERS 3 JUNE 2011its maximum inertia axis. The two torques were (1) Then the ‘‘step over’’ in the NS–WE plane follows, toa spin-to-roll misalignment torque 200–500 larger than lowest order, a Cornu spiral winding out from its initialpredicted from mechanical asphericity, and (2) a ‘‘roll- direction a day or so before the resonance, moving across,polhode resonance’’ torque, where the roll averaging men- then winding back into the new direction, its angular pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitioned above would temporarily fail, making a particular magnitude proportional to a2 þ b2 [9]. m mgyroscope axis realign, or step over, by as much as 100 mas Science data. Determining rNS and rWE requiresin 1–2 days when a high harmonic of its polhode rate came (1) data from gyroscope, telescope, and roll referenceinto resonance with spacecraft roll. The patch effect mis- signals (preprocessed and synchronized at 2 s intervals),alignment torque was discovered and quantified during the with supporting TFM analyses based on separatecalibration phase by commanding the spacecraft to point to 2200 sample=s gyro readout signals, and (2) the measure-a series of positions at known angles to the guide star. ment model of Eq. (2) below. The data came from 11 Remarkably, a key to modeling all three effects was months of science operations, where spacecraft anomaliesmagnetic asphericity: two patterns, magnetic fluxons and from events such as solar-flare-induced CPU reboots re-voltage patches, remained locked together in the rotor. sulted in 10 distinct but connectable data segments. MinorA process of trapped flux mapping (TFM) allowed exact interruptions came from calibrations, transient electronicstracking of the evolving polhode phase and angle needed noise, and passages through the South Atlantic Anomaly.for computing both Cg ðtÞ and torques. Measurement model. With s, , r , and Cg ðtÞ mean- ^ ^ Further extensive and diversified postscience calibra- ings already defined, we write SQUID signal zðtÞ astions showed no significant disturbing effects other thanthe three just discussed. zðtÞ ¼ Cg ðtÞ½ðNS À sNS Þ cosðr þ Þ þ ðWE À sWE Þ Data analysis.—Two data analysis methods were used to  sinðr þ ފ þ bias þ noise; (2)determine and cross-check the relativity results. One,called ‘‘algebraic,’’ was based on a parameter estimator (NS À sNS ) and (WE À sWE ) are the NS and WE projec-utilizing the gyro dynamics and measurement models tions of the misalignment ¼ À s, and is a constant ~ ^ ^detailed below. It produced the primary science results roll phase offset.(see Tables II and III, and Fig. 3). The other method, called Trapped flux mapping. The information gained from‘‘geometric,’’ though currently less precise, neatly elimi- trapped flux mapping was central to the three main model-nated the need to model the misalignment torque. This ing challenges [determining Cg ðtÞ and handling the mis-torque, being Newtonian, causes drift rates perpendicular alignment and resonance torques]. The magnetic fluxto the misalignment vector . The drift in that direction is ~ trapped in each rotor on cooling through its transitionthus a combination of relativistic and Newtonian contribu- temperature formed a unique, stable fluxon pattern. Bytions, while the drift rate parallel to —when there is no ~ fitting to the spin harmonics of the 2200 sample=s dataroll-polhode resonance—is pure relativity. The annual stream, we constructed detailed maps from which thevariation in misalignment direction allows determination three-dimensional orientation of each gyroscope could beof both relativity effects. tracked continuously over $2  109 turns. Rotor spin Gyro dynamics. The rNS (geodetic) and rWE (frame- speed and spin-down rate were determined to 10 nHz anddragging) drift rates and two patch effect torques [9] 1 pHz= sec , respectively [10]. Crucial for data analysisconnect to the unit vector sðtÞ along the gyro spin axis by ^ were (1) polhode phase p ðtÞ and angle p ðtÞ both goodthe equations of motion: to 1 over the 353 days of science, and (2) detailed knowl- X edge of the trapped flux contribution to Cg ðtÞ. s NS ¼ rNS þ kðtÞWE þ ðam cosÁm À bm sinÁm Þ; _ m TFM and Cg ðtÞ. The scale factor Cg ðtÞ to be calibrated X against aberration comes from the London moment MLsWE_ ¼ rWE À kðtÞNS þ ðam sinÁm þ bm cosÁm Þ: m aligned with the rotor’s spin axis, plus a complex pattern of trapped flux $10À2 ML fixed in its body, i.e., CLM þ g (1) CTF ðtÞ. To meet the necessary 10À4 accuracy for Cg ðtÞ gThe kðtÞWE , kðtÞNS terms give the misalignment drift required connecting data from $5200 successive guide starrates, WE , NS being the components of the misalignment valid half-orbits enabled by TFM.vector ¼ À s, and kðtÞ the polhode-dependent torque ~ ^ ^ TFM and resonance torques. The role of TFM incoefficient, is the unit vector along the spacecraft roll ^ computing the resonance torque was even more remark-axis. The resonance torques are governed by the third term able. Solving Eq. (1) required precise knowledge of p ðtÞof Eq. (1) where Ám ðtÞ ¼ r ðtÞ À mp ðtÞ, r and p and p ðtÞ. It is curious to reflect that if GP-B had had anbeing the known roll and polhode phases, with exact ideal London moment with no trapped flux, the computa-resonance occurring when the corresponding frequencies tion would have been impossible.coincide, !r ¼ m!p . To clarify the picture of a resonance, Spacecraft pointing. To model the misalignmentwe keep only the resonance term on the right of Eq. (1). torque, the accurate value of ðtÞ is needed continuously. ^ 221101-3
  4. 4. week endingPRL 106, 221101 (2011) PHYSICAL REVIEW LETTERS 3 JUNE 2011During the half-orbit periods when the guide star is visible, baseline and nine ‘‘sensitivity run’’ estimates for eachit is the sum of annual and orbital aberrations and minor gyroscope, whereby in each run the number of terms in aterms including parallax, bending of starlight by the Sun, single submodel is increased by 1 above the baselineand pointing error from the telescope signal. For the oc- value. The postfit residuals were white with 2 % 1 forculted periods, ðtÞ was determined from the four science ^ all gyroscopes.gyros’ SQUID signals. The connection process required The joint four-gyro result is a weighted average of thefast computation of the gyro spin vector sðtÞ through the ^ four individual drift rates using their 2 Â 2 rNS , rWE sub-entire 11 months of science data. blocks of the full covariance matrix for each gyroscope. In Parameter estimator. The science data set was pro- the NS direction the scatter of the individual estimates iscessed by a nonlinear batch weighted least-squares (WLS) 28% larger than the individual uncertainties indicate, whileestimator implemented in iterative sequential information in the WE direction it is 37% smaller. The individual andfilter form [11]. The cumulative information matrix for the combined statistical uncertainties are corrected for theirfull mission ($ 1:4 Â 107 data points per gyroscope) was ‘‘over’’ and ‘‘under’’ dispersion using the 2 of the indi-calculated sequentially based on one-orbit data batches. To vidual estimates in the NS and WE directions. The ‘‘pa-ensure robust convergence of numerous runs with different rameter sensitivity’’ covariance matrix is the scatter of allnumbers of estimated parameters, we used the sigma-point 104 combinations of drift rate estimates obtained in 10filter technique [12]. The lack of convergence observed in sensitivity runs for each of the four gyroscopes. The totalsome runs of a standard WLS estimator was thus elimi- covariance matrix is a sum of three matrices: the correctednated. Computations were critically facilitated by the re- statistical one, the sensitivity one, and the covarianceplacement of differential equations (1) with their analytical matrix of unmodeled systematic effects. The 1 uncertain-solution sNS ðtÞ, sWE ðtÞ for an arbitrary kðtÞ. ties of the joint result shown in Table II are the square roots Results and conclusions.—The four gyroscope signals of the diagonal elements of the total covariance matrix.were analyzed independently and the results combined and Table III summarizes uncertainties of the joint four-gyrocross-checked in various ways. Table II lists the four rela- result. The three main disturbances (evolving scale factortivity estimates and final joint result with 1 errors, also and the two Newtonian torques) are modeled and thusplotted as 95% confidence ellipses in Fig. 3. The values contribute to the statistical uncertainty. The systematicare referenced to inertial space by correcting for the uncertainty in Table III includes (1) uncertainty from thesolar geodetic contribution of 7:3 Æ 0:3 mas=yr NS sensitivity analysis to the number of model parameters andand À16:2 Æ 0:6 mas=yr WE, and for the guide star (2) small effects not incorporated in the model.proper motion, 27:3 Æ 0:1 mas=yr NS and À20:0 Æ We performed several important data analysis cross-0:1 mas=yr WE (the latter numbers are based on the decli- checks. First, the gyro drift rate results of Table II arenation and right ascension rates of 27:27 Æ 0:11 mas=yr confirmed by separate analyses of the segmented data.and À20:83 Æ 0:10 mas=yr, respectively, as determined The 24 independent results from six data segments forin [13]). each gyroscope are all consistent with the joint result The result for each gyroscope is the average of 10 drift within their confidence limits, demonstrating the internalrate estimates based on 10 distinct parameter sets deter-mined by the following uniform procedure. The submodelsfor scale factor Cg ðtÞ, misalignment torque coefficient kðtÞ,polhode phase p ðtÞ, etc., are linear combinations of cer-tain basis functions with coefficients to be estimated. Thenumber of terms in each submodel is increased from zerountil the change in the relativistic drift rate estimatesbecomes less than 0:5. This defines the baseline set fora given gyroscope. Table II gives the weighted mean of the TABLE II. Results.Source rNS (mas=yr) rWE (mas=yr)Gyroscope 1 À6588:6 Æ 31:7 À41:3 Æ 24:6Gyroscope 2 À6707:0 Æ 64:1 À16:1 Æ 29:7Gyroscope 3 À6610:5 Æ 43:2 À25:0 Æ 12:1Gyroscope 4 À6588:7 Æ 33:2 À49:3 Æ 11:4Joint (see text) À6601.8 Æ 18.3 À37.2 Æ 7.2 FIG. 3 (color). North-south and west-east relativistic drift rateGR prediction À6606:1 À39:2 estimates (95% confidence) for the four individual gyroscopes (colored ellipses) and the joint result (black ellipse). 221101-4
  5. 5. week endingPRL 106, 221101 (2011) PHYSICAL REVIEW LETTERS 3 JUNE 2011 TABLE III. Contributions to experiment uncertainty. Arlene Coffman with matching funds by KACST wasContribution NS (mas=yr) WE (mas=yr) crucial to finalizing this Letter. Support from ICRA/ ICRANet is also gratefully acknowledged. GP-B wouldStatistical (modeled) 16.8 5.9 never have succeeded without the work of hundreds ofSystematic (unmodeled) people at Stanford, NASA, Lockheed Martin, and else-Parameter sensitivity 7.1 4.0 where. Some 500 graduate, undergraduate and high schoolGuide star motion 0.1 0.1 students have contributed much to the project. Our specialSolar geodetic effect 0.3 0.6 thanks go to members of the NASA-appointed ScienceTelescope readout 0.5 0.5 Advisory Committee, chaired by Clifford Will, for theirOther readout uncertainties 1 1 support and advice, and to Charbel Farhat for all his help.Other classical torques 0:3 0:4Total stat + syst 18.3 7.2consistency of the model. Second, the misalignment torque *francis1@stanford.educoefficients k determined during the calibration phase [1] L. I. Schiff, Phys. Rev. Lett. 4, 215 (1960).proved to be in excellent agreement with the end-of- [2] W. de Sitter, Mon. Not. R. Astron. Soc. 77, 155 (1916).mission values estimated by both algebraic and geometric [3] J. Lense and H. Thirring, Phys. Z. 19, 156 (1918).analysis methods. No less impressive was the agreement [4] K. S. Thorne, in Near Zero: New Frontiers of Physics,between the time history kðtÞ throughout the mission ob- edited by J. D. Fairbank et al. (W.H. Freeman and Co.,tained by the two methods. As for the roll-polhode reso- New York, 1988), pp. 573–586. [5] G. E. Pugh, Research memorandum 11, Weapons Systemnance torques, the gyro dynamics model predicts that Evaluation Group, the Pentagon, Washington, DC, 1959,during a resonance the gyroscope orientation axis approxi- reprinted in Nonlinear Gravitodynamics. The Lense-mately follows a Cornu spiral. Indeed, that is typically Thirring Effect., edited by R. J. Ruffini and C.observed in orbit-by-orbit gyro orientations determined Sigismondi (World Scientific, Singapore, 2003),by both data analysis methods. pp. 414–426. Further cross-checks from the geometric method in- [6] D. B. DeBra, in Near Zero: New Frontiers of Physics,cluded relativity estimates for 2 of the 4 gyroscopes, in edited by J. D. Fairbank et al. (W.H. Freeman and Co.,both NS and WE directions, in statistical agreement with New York, 1988), pp. 691–699.the algebraic results, and an estimate of the gravitational [7] J. B.Camp, T. W. Darling, and R. E. Brown, J. Appl. Phys.deflection of light by the Sun. IM Pegasi’s closest approach 69, 7126 (1991).to the Sun occurs on March 11th, with ecliptic latitude [8] C. C. Speake, Classical Quantum Gravity 13, A291 (1996).22.1 as compared with the grazing incidence value 0.265 [9] G. M. Keiser, J. Kolodziejczak, and A. S. Silbergleit,in classical light deflection tests. The maximum deflection Space Sci. Rev. 148, 383 (2009).predicted by GR is 21.7 mas; the observed 21 Æ 7 mas [10] A. Silbergleit et al., Space Sci. Rev. 148, 397 (2009).serves as a neat structural confirmation of Gravity Probe B. [11] G. J. Bierman, Factorization Methods for Discrete Lunar laser ranging has reported a measurement of the Sequential Estimation (Dover, New York, 2006), 2nd ed.de Sitter solar geodetic effect to 0.7% [14]. Analyses of [12] R. van der Merwe, E. Wan, and S. J. Julier, in Proceedingslaser ranging to the LAGEOS and LAGEOS II spacecraft of the AIAA Guidance, Navigation Control Conferencereport a 10%–30% measurement of the frame-dragging (AIAA, Providence, RI, 2004), pp. 16–19.effect, assuming the GR value for the geodetic precession [13] I. I. Shapiro et al. (to be published).[15,16]. GP-B provides independent measurements of the [14] J. G. Williams, X. X. Newhall, and J. O. Dickey, Phys.geodetic and frame-dragging effects at an accuracy of Rev. D 53, 6730 (1996). [15] I. Ciufolini and E. C. Pavlis, Nature (London) 431, 9580.28% and 19%, respectively. (2004). This work was supported by NASA Contract NAS8- [16] L. Iorio, Space Sci. Rev. 148, 363 (2008).39225 through Marshall Space Flight Center and by King [17] The inertial axes used in the Letter are west-east (WE)—Abdulaziz City for Science and Technology (KACST). along the cross-product of the unit vector to the guide starFunding was also made available through a personal gift and the unit vector z of the inertial JE2000 frame, and ^by Richard Fairbank with matching funds from NASA and north-south (NS)—along the cross-product of the WE unitStanford University. A personal gift from Vance and vector and the unit vector to the guide star. 221101-5

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