1. GP-B/Aero-Astro 1 Data Analysis September 30, 2008 Stanford The Gravity Probe B Experiment: “Testing Einstein’s Universe” (Data Analysis Challenges) Dr. Michael Heifetz (Hansen Experimental Physics Laboratory)

2. GP-B/ Aero-Astro 2 October 21, 2008 Stanford Data Analysis What is Gravity Probe B? Gravity Probe B (GP-B) is a NASA physics mission to experimentally investigate Albert Einstein’s 1916 general theory of relativity – his theory of gravity. GP-B directly measures in a new way, and with unprecedented accuracy, two extraordinary effects predicted by the general theory of relativity: 1.The geodetic effect – the amount by which the Earth warps the local spacetime in which it resides 2.The frame-dragging effect – the amount by which the rotating Earth drags its local spacetime around with it. The frame-dragging effect has never before been directly measured!

3. GP-B/ Aero-Astro 3 October 21, 2008 Stanford Data Analysis The Enigma of Gravity Sir Isaac Newton: Space and time are absolute or fixed entities. Gravity is a force that acts instantaneously between objects at a distance, causing them to attract one another. Albert Einstein: Space and time are relative entities, interwoven into a spacetime fabric whose curvature we call gravity. Spacetime tells matter how to move, and matter tells spacetime how to curve.

4. GP-B/ Aero-Astro 4 October 21, 2008 Stanford Data Analysis Geodetic Effect –Space-time curvature ("the missing inch") Frame-dragging Effect –Rotating matter drags space-time ("space-time as a viscous fluid") The Relativity Mission Concept Leonard Schiff

5. GP-B/ Aero-Astro 5 October 21, 2008 Stanford Data Analysis A “Simple” Experiment GP-B Co-Founder, Bill Fairbank, once remarked: “No mission could be simpler than GP-B; it’s just a star, a telescope and a spinning sphere.” However, it took over four decades to develop all the cutting-edge technologies necessary to carry out this “simple” experiment.

6. GP-B/ Aero-Astro 6 October 21, 2008 Stanford Data Analysis Brief History of Gravity Probe B 1957 Sputnik – Dawn of the space age 1958 Stanford Aero-Astro Department created 1959 L. Schiff conceives of orbiting gyro experiment as a test of General Relativity 1961 L. Schiff & W. Fairbank propose gyro experiment to NASA 1972 1 st drag-free spacecraft: TRIAD/DISCOS 1975 SQUID readout system developed 1980 Rotor machining techniques perfected 1998 Science instrument assembled 2002 Spacecraft & payload integrated 2004 Launch and vehicle operations 2005 End of data collection Start of Data Analysis 2007 Preliminary results presented at April APS meeting 2008 -2009 Final results 84 doctorates (29 Phys; 54 AA, EE, ME; 1 Math) 15 Master’s degrees, 5 Engineer’s degrees 13 doctorates completed at other universities Stanford Student Participation

7. GP-B/ Aero-Astro 7 October 21, 2008 Stanford Data Analysis Spacecraft gyros (3x10 -3 deg/hr) 10 2 10 1 0.1 0.01 39 Frame dragging <0.3% accuracy 10 3 6606 Geodetic effect <0.002% accuracy marcsec/yr 0.5 GP-B requirement 10 4 10 5 10 6 10 7 10 8 10 9 Best laser gyros (1x10 -3 deg/hr) Best mechanical gyros on Earth (10 -2 deg/hr) Electrostatically suspended gyroscope (ESG) on Earth with torque modeling (10 -5 deg/hr) Why a Space-Based Experiment? marcsec/yr Best terrestrial gyroscopes 10,000,000 times worse than GP-B 1 marcsec/yr = 3.2x10 -11 deg/hr

8. GP-B/ Aero-Astro 8 October 21, 2008 Stanford Data Analysis GP-B Instrument Concept Gyros 4 & 3 Gyros 2 & 1 Fused quartz block (metrology bench) Star tracking telescope Guide star IM Pegasi Operates at ~ 2 K with liquid He Rolls about line of sight to Guide Star –Inertial pointing signal at roll frequency –Averages body-fixed classical disturbance torques toward zero –Reduces effect of body-fixed pointing biases

9. GP-B/ Aero-Astro 9 October 21, 2008 Stanford Data Analysis Ultra-Precise Gyroscopes To measure the minuscule angles predicted by Einstein's theory, it was necessary to build near-perfect gyroscopes ~10 million times more precise than the best navigational gyroscopes. The GP-B gyro rotors are listed in the Guinness Database of World Records as the most spherical man-made objects.

10. GP-B/ Aero-Astro 10 October 21, 2008 Stanford Data Analysis SQUID Magnetometers How can one monitor the spin-axis orientation of a near-perfect spherical gyroscope without any physical marker showing the location of the spin axis on the gyro rotor? The answer lies in superconductivity. Predicted by physicist Fritz London in 1948, and most fortunate for GP-B, a spinning superconductor develops a magnetic moment exactly aligned with its spin axis.

11. GP-B/ Aero-Astro 11 October 21, 2008 Stanford Data Analysis Dewar & Probe GP-B’s 650-gallon dewar, kept the science instrument inside the probe at a cryogenic temperature (2.3K) for 17.3 months and also provided the thruster propellant for precision attitude and translation control.

12. GP-B/ Aero-Astro 12 October 21, 2008 Stanford Data Analysis Pointing Telescope A telescope mounted along the central axis of the dewar and spacecraft provided the experiment’s pointing reference to a “guide star.” The telescope’s image divider precisely split the star’s beam into x-axis and y-axis components whose brightness could be compared.

13. GP-B/ Aero-Astro 13 October 21, 2008 Stanford Data Analysis Integrated Payload & Spacecraft Built around the dewar, the GP- B spacecraft was a total- integrated system, comprising both the space vehicle and payload, dedicated as a single entity to experimentally testing predictions of Einstein’s theory.

14. GP-B/ Aero-Astro 14 October 21, 2008 Stanford Data Analysis

15. GP-B/ Aero-Astro 15 October 21, 2008 Stanford Data Analysis Challenges of Data Analysis…

16. GP-B/ Aero-Astro 16 October 21, 2008 Stanford Data Analysis θ Apparent Guide Star aberration Guide Star - gyro spin axis orientation - vehicle roll axis orientation - gyroscope misalignment Relativity: slopes of (Geodetic) and (Frame- dragging) (significantly more complex problem) SQUID Readout Data Roll Phase Data Telescope Data, Orbital and Annual Aberrations Scale Factor Gyro orientation trajectory and - straight lines Surprise B: Patch Effect Torque - calibrated based on orbital and annual aberration Surprise A: variations ‘Simple’ GP-B Data Analysis Pointing Error via Telescope

17. GP-B/ Aero-Astro 17 October 21, 2008 Stanford Data Analysis Three Cornerstones of Dynamic Estimation (Filtering) Information Theory Filter Implementation: Numerical Techniques Gyroscope Motion: Torque Models Underlying Physics SQUID Readout Signal Structure: Measurement Models Underlying Physics, Engineering

18. GP-B/ Aero-Astro 18 October 21, 2008 Stanford Data Analysis Data Analysis Structure: ‘Two-Floor’ Processing Torque Modeling SQUID Readout Processing Gyro Orientation Time History Data Analysis Building First Floor Second Floor Relativity Measurement Full Information Matrix Patch Effect Torque Theory (mathematical physics) Scale Factor Modeling Trapped Flux Mapping

19. GP-B/ Aero-Astro 19 October 21, 2008 Stanford Data Analysis Polhode Motion, Trapped Flux & C g Actual ‘London moment’ readout Body-axis Path Trapped magnetic fields London magnetic field at 80 Hz: 57.2 μG Gyro 1: 3.0 μG Gyro 2: 1.3 μG Gyro 3: 0.8 μG Gyro 4: 0.2 μG Scale factor Cg modulated at polhode frequency by trapped magnetic flux Two methods of determining Cg history - Fit polhode harmonics to LF SQUID signal - Direct computation by Trapped Flux Mapping

20. GP-B/ Aero-Astro 20 October 21, 2008 Stanford Data Analysis Polhode Motion and Readout Scale Factor: C g Model pp I3I3 I1I1 I2I2  Gyro principle axes of inertia and instant spin axis position Harmonic expansion in polhode phase with coefficients that depend on polhode angle Trapped Flux Mapping (TFM) - Polhode phase - Polhode angle Unknowns Guide Star Trapped Flux John Conklin

21. GP-B/ Aero-Astro 21 October 21, 2008 Stanford Data Analysis First Floor: SQUID Readout Data Processing 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/Misalign. 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) State Vector Estimates Gyro Scale Factor Model Let’s look at the gyro orientation profiles … G/T Matching

22. GP-B/ Aero-Astro 22 October 21, 2008 Stanford Data Analysis Inertial Orientation Time-history: Gyro 1 NS Direction De-trended m=42m=41 EW Direction time milliarcsec m=42 m=41 resonance NS Direction time milliarcsec Strong Geodetic Effect

23. GP-B/ Aero-Astro 23 October 21, 2008 Stanford Data Analysis NS Direction EW Direction Inertial Orientation Time-history: Gyro 2 NS direction de-trended m=214 m=142m=214 74 resonances! m=142 time milliarcsec EW Direction Resonance Schedule Resonances:

24. GP-B/ Aero-Astro 24 October 21, 2008 Stanford Data Analysis Torque Modeling θ Apparent Guide Star aberration Guide Star - gyro spin axis orientation - vehicle roll axis orientation - gyroscope misalignment Misalignment torque Roll-Resonance torque k(t), c + (t), c - (t) are modulated by harmonics of polhode frequency – roll/polhode resonance: relativity 2006-20072008

25. GP-B/ Aero-Astro 25 October 21, 2008 Stanford Data Analysis Torque Coefficients: Polhode Variation Roll-resonance torque coefficients c +, c - : Misalignment torque coefficient k: and have the same structure as and The same polhode structure as in Readout Scale Factor Model (1 st Floor) Trapped Flux Mapping - polhode phase - polhode angle

26. GP-B/ Aero-Astro 26 October 21, 2008 Stanford Data Analysis 2 nd Floor Roll-Resonance Torque Dynamic Estimator Orientations Profiles Roll Phase Misalignment Polhode Phase/Angle State vector: Propagation Model: Measurement Model: Estimator (separate for each segment) Output: - Torque related variables: - torque coefficients - modeled torque contributions - Reconstructed “relativistic” trajectory (Orientation profile minus torque contributions) Combine reconstructed trajectories for all segments Fit to a straight line Relativity: Slope estimate Full 1 st Floor Information is not yet used

27. GP-B/ Aero-Astro 27 October 21, 2008 Stanford Data Analysis Measured Inertial Orientation Modeled Inertial Orientation Gyro 2: Estimation Results (Modeled Orientation vs Measured Orientation) Subtracting the torque contributions… 74 Resonances!

28. GP-B/ Aero-Astro 28 October 21, 2008 Stanford Data Analysis NS Gyro 2: Reconstructed “Relativistic” Trajectory Reconstructed Trajectory +1σ -1σ Weigted LS fit based on input noise Frame-dragging effect!

29. GP-B/ Aero-Astro 29 October 21, 2008 Stanford Data Analysis Current Relativity Estimates for Gyros 1,2,3, and 4 GR prediction Gyro 3 (2007) Gyro 1,3,4 combined (2007) Gyro 1 (2007) Gyro 4 (2007) Gyro 1,2,3,4 combined G1 G3 G4G2

30. GP-B/Aero-Astro 30 Data Analysis September 30, 2008 Stanford Where we stand now Roll-Resonance Torque Modeling: reduced large part of systematic errors: previously unmodeled torque-related errors are now modeled properly dramatically enhanced the agreement between the gyroscopes The same torque model works for all 4 gyros over entire mission Developed estimator is not good enough : Orientation time step, currently 1-orbit (97min) should be made much less than 1 roll period (77 sec) Final improvement of Algebraic Method: “2-sec Filter”: That is where we need your help!

31. GP-B/ Aero-Astro 31 October 21, 2008 Stanford Data Analysis Two-Second Filter: Nonlinear Stochastic Optimization Problem New Filter is formulated as a Dynamic Nonlinear Estimation Problem: θ Apparent Guide Star aberration Guide Star (!) SQUID Data 307 days = 4605 orbits x 97 min x 30 (2-sec data points) Nonlinear Model Nonlinear Dynamic Gyro Motion Model Requires multiple cost-function minimum search iterations going through millions of data points For 1 Gyro

32. GP-B/ Aero-Astro 32 October 21, 2008 Stanford Data Analysis Main Equations T r = 97 sec Geodetic Frame-dragging

33. GP-B/ Aero-Astro 33 October 21, 2008 Stanford Data Analysis Main Equaitions -cont

34. GP-B/ Aero-Astro 34 October 21, 2008 Stanford Data Analysis Challenges of 2-sec Filter Dealing with several millions of ‘measurement’ equations requires new assessment of numerical techniques and computational capabilities Analyzing gyroscopes together and the nonlinear structure of the estimation problem probably will require parallel processing (in which we have no experience) Evaluation of the analysis results, given the complexity of 2-sec filter, will probably require the development of new “truth model” simulations