The Gravity Probe B Experiment: “Testing Einstein’s Universe” (Data Analysis Challenges)

1. The Gravity Probe B Experiment: “Testing Einstein’s Universe” (Data Analysis Challenges) Dr. Michael Heifetz (Hansen Experimental Physics Laboratory)

2. 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: • The geodetic effect – the amount by which the Earth warps the local spacetime in which it resides • 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. 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. The Relativity Mission Concept • Geodetic Effect • Space-time curvature ("the missing inch") • Frame-dragging Effect • Rotating matter drags space-time ("space-time as a viscous fluid") Leonard Schiff

5. 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. 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 1st 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 • 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. 6606 Geodetic effect <0.002% accuracy Frame dragging <0.3% accuracy 39 0.5 GP-B requirement Why a Space-Based Experiment? Best mechanical gyros on Earth (10-2 deg/hr) 109 103 108 Spacecraft gyros (3x10-3 deg/hr) 102 marcsec/yr Best laser gyros (1x10-3 deg/hr) 107 10 Electrostatically suspended gyroscope (ESG) on Earth with torque modeling (10-5 deg/hr) marcsec/yr 106 1 105 0.1 Best terrestrial gyroscopes 10,000,000 times worse than GP-B 104 0.01 1 marcsec/yr = 3.2x10-11 deg/hr

8. GP-B Instrument Concept • 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-fixedpointing biases Guide star IM Pegasi Gyros 2 & 1 Star tracking telescope Fused quartz block(metrology bench) Gyros 4 & 3

9. 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. 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. 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. 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. 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.

15. Challenges of Data Analysis…

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

17. Gyroscope Motion: Torque Models Underlying Physics Three Cornerstones of Dynamic Estimation (Filtering) SQUID Readout Signal Structure: Measurement Models Underlying Physics, Engineering Filter Implementation: Numerical Techniques Information Theory

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

19. Body-axis Path Polhode Motion, Trapped Flux & Cg • Actual ‘London moment’ readout 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 Trapped magneticfields • 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. Guide Star I3 g - Polhode phase I2 I1 Fp - Polhode angle Polhode Motion and Readout Scale Factor: Cg Model Trapped Flux Gyro principle axes of inertia and instant spin axis position John Conklin Trapped Flux Mapping (TFM) Harmonic expansion in polhode phase with coefficients that depend on polhode angle Unknowns

21. τ μ Telescope Data Gyro Scale Factor Model Trapped Flux Mapping Gyro Orientation (1 point/orbit) State Vector Estimates First Floor: SQUID Readout Data Processing OUTPUT: Pointing SQUID Data GSV/GSI Batch length: 1orbit Aberration Data Roll Phase Data Bias Estimator SQUID No-bias Signal Pointing/Misalign. Computation Roll Phase Data Aberration Data Cg (tk*) CT (tk*) δφ(tk*) G/T Matching NonlinearLeast-Squares Estimator (No Torque Modeling) Polhode Phase Data Data Grading Polhode Angle Data Residuals Full Information Matrix Let’s look at the gyro orientation profiles…

22. resonance EW Direction m=42 m=41 Inertial Orientation Time-history: Gyro 1 NS Direction milliarcsec Strong Geodetic Effect m=42 milliarcsec m=41 time NS Direction De-trended time

23. Resonance Schedule EW Direction NS Direction time m=142 m=214 m=142 m=214 74 resonances! Resonances: Inertial Orientation Time-history: Gyro 2 NS direction de-trended EW Direction milliarcsec

24. Guide Star Apparent Guide Star Misalignment torque Roll-Resonance torque aberration relativity θ - gyro spin axis orientation - vehicle roll axis orientation - gyroscope misalignment Torque Modeling 2006-2007 2008 k(t), c+(t), c-(t) are modulated by harmonics of polhode frequency – roll/polhode resonance:

25. Trapped Flux Mapping - polhode phase Misalignment torque coefficient k: • polhode • angle and have the samestructure as and Torque Coefficients: Polhode Variation Roll-resonance torque coefficients c+, c-: The same polhode structure as in Readout Scale Factor Model (1st Floor)

26. Orientations Profiles Propagation Model: Roll Phase Misalignment Measurement Model: Polhode Phase/Angle State vector: 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 2nd Floor Roll-Resonance Torque Dynamic Estimator Full 1st Floor Information is not yet used

27. Measured Inertial Orientation Modeled Inertial Orientation Gyro 2: Estimation Results(Modeled Orientation vs Measured Orientation) 74 Resonances! Subtracting the torque contributions…

28. NS Weigted LS fit based on input noise Reconstructed Trajectory +1σ -1σ Gyro 2: Reconstructed “Relativistic” Trajectory Frame-dragging effect!

29. G2 G4 Gyro 4 (2007) Gyro 1,3,4 combined (2007) Gyro 1 (2007) Gyro 3 (2007) Current Relativity Estimates for Gyros 1,2,3, and 4 Gyro 1,2,3,4 combined GR prediction G1 G3

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

32. Main Equations Geodetic Tr = 97 sec Frame-dragging

33. Main Equaitions -cont

34. 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