1. The Gravity Probe B Experiment: Data Analysis Journey Michael Heifetz On Behalf of GP-B Data Analysis Team

2. 2 MG12 Paris 12-18 July, 2009 Gravity Probe B Concept

3. 3 MG12 Paris 12-18 July, 2009 Science Signal aberration Guide Star - spin axis direction - roll axis direction Apparent Guide Star Measurement: Spin-Roll Misalignment

4. 4 MG12 Paris 12-18 July, 2009 Science Signal Science Signal Spectral Shift (Roll Frequency) SQUID Readout System SQUID Pick-up Loop Rolls with S/C aberration Guide Star - spin axis direction - roll axis direction Apparent Guide Star ☼

5. 5 MG12 Paris 12-18 July, 2009 3 Sep 04 30 Mar 05 Science Signal Apparent Guide Star Aberration Aberration -- Nature's calibrating signal for gyro scale factor C g Guide Star Annual aberration Readout Output Orbital motion: –Varying apparent position of star (v orbit /c + special relativity) Spacecraft around Earth: - 5.1856 arcsec (97.5 min period) Earth around Sun: –20.4958 acrsec (1 yr period)

6. 6 MG12 Paris 12-18 July, 2009 Pre-Flight Data Analysis Strategy Constant - calibrated based on orbital and annual aberration Surprise A: variations Gyro orientation trajectory and - straight lines Surprise B: Patch Effect Torque Scale Factor

7. 7 MG12 Paris 12-18 July, 2009 Expected Gyroscope Behavior Geodetic effect (-6571 marcsec/yr) Frame-dragging effect (-75 marcsec/yr) Newton’s universe Includes Solar Geodetic and Guide Star Proper Motion

8. 8 MG12 Paris 12-18 July, 2009 Flight Data (Gyro 2)

9. 9 MG12 Paris 12-18 July, 2009 Three Pillars of GPB Data Analysis Information Theory Filter Implementation: Numerical Estimation Techniques Understanding of Gyroscope Motion: Trapped Flux Mapping (TFM) Torque Models Underlying Physics Readout Science Signal Structure: Measurement Models Underlying Physics, Engineering

10. 10 MG12 Paris 12-18 July, 2009 Data Analysis Structure: ‘Two-Floor’ Processing Torque Modeling Gyro Orientation Time History Data Analysis Building SQUID Readout Processing First Floor Second Floor Relativity Measurement Full Information Matrix

11. 11 MG12 Paris 12-18 July, 2009 Structure of Two-Floor Analysis SQUID Science Signal (2 sec sampling rate) 1 st Floor One Orbit Estimator 1 st Floor No Torque Modeling Gyro Orientation Profiles (NS, EW) (1 point per orbit) Gyro Scale Factor Estimates Kalman Filter (Smoother) Torque Model 2 nd Floor Relativity Estimate Torque coefficients Estimates Gyro Orientation Profiles (NS, EW) (1 point per orbit) Data Reduction

12. 12 MG12 Paris 12-18 July, 2009 1 st Floor Challenges: How to Pull Out Gyro Orientation from SQUID Data complete Readout scale factor time-variations (“C g Polhode modeling”) Pointing error compensation (“Gyro/Telescope scale factor matching”) Data Grading (quality of inputs) Bias modeling (e.g. polhode variations, bias jumps) Electronic Control Unit noise elimination Most Difficult Problems

13. 13 MG12 Paris 12-18 July, 2009 Trapped Flux & Readout Scale Factor Trapped magnetic potential (V) I2I2 I3I3 I1I1 6 Sept 2004 26 June 20054 Oct 2004 14 Nov 200420 Feb 200520 Dec 2004 Gyro 1 body frame polhode ˆ ˆ ˆ ^ s ^ s ^ s ^ s ^ s ^ s

14. 14 MG12 Paris 12-18 July, 2009 Successes of Trapped Flux Mapping ParameterError Angular velocity,  10 nHz ~ 10 -10 Polhode phase,  p ~ 1  Rotor orientation ~ 2  Trapped magnetic potential~ 1% Gyroscope scale factor, C g ~ 10 -4 I3I3 I2I2 Path of spin axis in gyro body I1I1 I3I3 I2I2 I1I1 Trapped magnetic potential ^ s Relative C g variations

15. 15 MG12 Paris 12-18 July, 2009 Scale Factor Model blue – a n (t) and b n (t) red - fit to ε(t) 0 0 0 0 Harmonic expansion in polhode phase with coefficients that depend on polhode angle Trapped Flux Mapping - Polhode phase - Polhode angle Gyro principle axes of inertia and instant spin axis position pp I3I3 I1I1 I2I2

16. 16 MG12 Paris 12-18 July, 2009 Gyroscope-Telescope Scale Factor Matching Reduces coupling of vehicle motion to science signal from 20 to 0.1 marc-sec SQ1 Signal PSD - Unmatched Frequency (Hz) 1 2 3 456 Roll ± Orbital 1 – Roll 2 – 2xRoll 3 – dither 1 4 – dither 2 5 – 3xRoll 6 – 4xRoll SQ1 Signal PSD - Matched Matched Gyroscope (SQUID) Data Telescope Data Spectrum of SQUID Signal: before and after matching Pointing error compensation

17. 17 MG12 Paris 12-18 July, 2009 SQUID Data SQUID No-bias Signal Nonlinear Least-Squares Estimator (No Torque Modeling) Roll Phase Data Aberration Data Grading τ μ Batch length: 1orbit Bias Estimator C g (t k *) C T (t k *) δ φ(t k *) Residuals Pointing/Misalignment Computation Telescope Data Roll Phase Data Aberration Data OUTPUT: Pointing GSV/GSI Polhode Phase Data Trapped Flux Mapping Polhode Angle Data Full Information Matrix Gyro Orientation (1 point/orbit) Full State Vector Estimates Gyro Scale Factor Model Let’s look at the gyro orientation profiles … G/T Matching First Floor: SQUID Readout Data Processing

18. 18 MG12 Paris 12-18 July, 2009 1 st Floor Output: Gyro Orientation (NS direction) Seeing Strong Geodetic in ‘Raw’ Data

19. 19 MG12 Paris 12-18 July, 2009 1 st Floor Output: Gyro Orientation (EW direction) The Name of the Game – Frame-Dragging!

20. 20 MG12 Paris 12-18 July, 2009 Patch Effect & Pre-launch Investigations rotor surface housing surface SEM image of rotor Nb film The patch effect surface layer with variable electric dipole moment density Pre-launch investigation Rotor electric dipole moment + field gradient from suspension Kelvin probe measurements: Contact potential differences ~ 100 mV Mitigated / eliminated by grain size, < 1 μm << 30 μm gap

21. 21 MG12 Paris 12-18 July, 2009 Evidence for Patch Effect Exhibit A: Gyroscope spin-down GyroSpin-down period (yr) 115,800 213,400 37,000 425,700 Polhode period vs elapsed time since January 1, 2004 Gyro 1, T p (hr) Gyro 4, T p (hr) Time (days) Blue: Worden Red: Santiago & Salomon Exhibit B: Changing polhode period

22. 22 MG12 Paris 12-18 July, 2009 More Evidence for Patch Effect Exhibit C: Orbit determination Anomalous z-axis acceleration ~ 10 -8 N, modulated at polhode frequency Exhibit D: Large misalignment torques Mean East-West misalignment Mean North-South misalignment Mean rate (marcsec/day) vs. mean misalignment (arcsec) 1000 2000 3000 4000 30 210 60 240 90 270 120 300 150 330 1800 05001000150020002500300035004000 0 0.5 1 1.5 2 2.5 3 3.5 4 Drift rate magnitude (arcsec/day) Mean misalignment (arcsec) k = 2.5 arc sec/day/degree

23. 23 MG12 Paris 12-18 July, 2009 Misalignment Torque (Roll Averaged) Guide Star  ^ s ^   NS vs. EW misalignment,  -20 -10 0 10 20 EW misalignment (arcsec) µ NS misalignment (arcsec) -15 -10 -5 0 5 10 15 Torque   Drift  Torque coefficient: k(  p ) Relativity fixed in inertial frame  Aberration spectrally shifts misalignment torque   2 nd Floor Torque Model (2006- 2007)

24. 24 MG12 Paris 12-18 July, 2009 Relativity Estimates (Misalignment Torque Modeling) 2007 Gyro 3 Gyro 4 Gyro 1 Gyros 1, 3, 4 combined GR prediction Gyro 3 Gyro 4 Gyro 1 Gyros 1, 3, 4 combined

25. 25 MG12 Paris 12-18 July, 2009 Discovery of Roll-resonance Torque (non roll-average) Exhibit E: Roll-polhode resonance ‘jumps’ –‘Jumps’ occur when high harmonic of changing polhode rate, m polh, is coincident with roll rate,  roll Date (2005) Or Even More Evidence for Patch Effect 142139140141145144143138146 s EW res. m EW orientation, s EW (arcsec) Gyro 2 flight data

26. 26 MG12 Paris 12-18 July, 2009 Discovery of Roll-Polhode Resonance Torques Resonance

27. 27 MG12 Paris 12-18 July, 2009 Full Torque Model -Unknown (estimated) parameters Resonances: - S/C roll axis direction Trapped Flux Mapping Polhode Phase ( ), Polhode Angle ( ) Roll-resonance torque Relativity Misalignment torque

28. 28 MG12 Paris 12-18 July, 2009 2 nd Floor Kalman Filter Output: Torque related variables: - torque coefficients - modeled torque contributions - Reconstructed “relativistic” trajectory Kalman Filter / Smoother Torque Contribution Subtraction Relativity Estimates Gyro Orientation Profiles State vector: Propagation Model: Measurement Model: “Measurements”

29. 29 MG12 Paris 12-18 July, 2009 Measured & Reconstructed Orientations

30. 30 MG12 Paris 12-18 July, 2009 Measured & Reconstructed Orientations (G4) 1 st Floor Output 2 nd Floor Output

31. 31 MG12 Paris 12-18 July, 2009 Measured & Reconstructed Orientations (G2)

32. 32 MG12 Paris 12-18 July, 2009 Current Results Einstein’s prediction NS: -65711 marcsec/yr EW: -751 marcsec/yr (includes solar GR effects and guide star proper motion) Relativity estimates from 155-day analysis For the first time GR estimates agree among gyros Statistical uncertainty: < 0.5% of geodetic effect ~ 14% of frame-dragging 4-gyro combined result NS: -656512.3 marcsec/yr EW: -80.45.4 marcsec/yr (50% probability)

33. 33 MG12 Paris 12-18 July, 2009 Locking in the Final Results Current (statistical) limit: ~14% of frame- dragging Fundamental limit from covariance analysis: ~ 5% of frame-dragging Reaching this fundamental limit requires: 1.Expanding analysis to full year of science data 2.Once-per-orbit averaging  2-sec processing Enabled by parallel computing A definitive result requires completing critical and detailed treatment of systematic effects

34. 34 MG12 Paris 12-18 July, 2009 One OrbitGyro Motion Why 2-sec Filter?

35. 35 MG12 Paris 12-18 July, 2009 Serial 2-sec processing (160 days) Aug ’09 Complete transition to parallel processing Oct ’09 Extension to full mission (353 days) Dec ’09 Complete treatment of systematics Feb ’10 Grand synthesis ~ 2 marcs/yr Jun ’10 4-gyro limit  Final results to be announced at Fairbank Workshop on Fundamental Physics & Innovative Engineering in Space Aug ’10 Path to Completion