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

2 MG12 Paris July, 2009 Gravity Probe B Concept

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

4 MG12 Paris 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 MG12 Paris July, Sep 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: arcsec (97.5 min period) Earth around Sun: – acrsec (1 yr period)

6 MG12 Paris 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 MG12 Paris 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 MG12 Paris July, 2009 Flight Data (Gyro 2)

9 MG12 Paris 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 MG12 Paris 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 MG12 Paris 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 MG12 Paris July, 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 MG12 Paris July, 2009 Trapped Flux & Readout Scale Factor Trapped magnetic potential (V) I2I2 I3I3 I1I1 6 Sept June Oct Nov Feb Dec 2004 Gyro 1 body frame polhode ˆ ˆ ˆ ^ s ^ s ^ s ^ s ^ s ^ s

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

15 MG12 Paris July, 2009 Scale Factor Model blue – a n (t) and b n (t) red - fit to ε(t) 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 MG12 Paris 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) 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 MG12 Paris 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 MG12 Paris July, st Floor Output: Gyro Orientation (NS direction) Seeing Strong Geodetic in ‘Raw’ Data

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

20 MG12 Paris 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 MG12 Paris July, 2009 Evidence for Patch Effect Exhibit A: Gyroscope spin-down GyroSpin-down period (yr) 115, ,400 37, ,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 MG12 Paris July, 2009 More Evidence for Patch Effect Exhibit C: Orbit determination Anomalous z-axis acceleration ~ 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) Drift rate magnitude (arcsec/day) Mean misalignment (arcsec) k = 2.5 arc sec/day/degree

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

24 MG12 Paris 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 MG12 Paris 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 s EW res. m EW orientation, s EW (arcsec) Gyro 2 flight data

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

27 MG12 Paris 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 MG12 Paris July, 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 MG12 Paris July, 2009 Measured & Reconstructed Orientations

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

31 MG12 Paris July, 2009 Measured & Reconstructed Orientations (G2)

32 MG12 Paris 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 MG12 Paris 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 MG12 Paris July, 2009 One OrbitGyro Motion Why 2-sec Filter?

35 MG12 Paris 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