Misalignment and Resonance Torques and Their Treatment in GP-B Data Analysis

1. Misalignment and Resonance Torques and Their Treatment in GP-B Data Analysis Mac Keiser and Alex Silbergleit

2. Outline • Misalignment Torques • Observations • Explanation and Calculation of Torque • Data Analysis • Resonance Torques • Observations • Explanation and Calculation of Torque • Data Analysis • Summary

3. Gravity Probe B Mission Timeline Initialization Phase Science Data Collection Phase Calibration Phase Launch April 20, 2004 Gyroscopes Spun Up and Aligned August 29, 2004 Aug. 15, 2005 Liquid Helium Depeleted Sept. 29, 2005 Proton Flux, Jan. 20-22, 2005, Measured by GOES Satellite Gyro 3 West-East Spin Axis Orientation Particles/(cm2 sec sr) Arc Sec Time (days) from Jan. 1, 2005 Time (days) Misalignment Torque - Observations

4. Additional Evidence for Torques:Gyroscope Orientation History

5. Gravity Probe B Mission Timeline NhS1 (acquired) Initialization Phase Science Data Collection Phase Calibration Phase HR Peg (acquired) Launch April 20, 2004 Gyroscopes Spun Up and Aligned August 29, 2004 Aug. 15, 2005 Liquid Helium Depleted Sept. 29, 2005 20 • Calibration Phase Spacecraft Maneuvers • Increased the Misalignment Between the Satellite Roll Axis and the Gyroscope Spin Axes • 19 Maneuvers to Nearby Stars or “Virtual” Stars • Operating Conditions Changed • DC or AC Suspension Voltages • Spacecraft Attitude Control IM Peg Guide Star 20 Calibration Phase ObservationsMisalignment Torques

6. Observations – Gyroscope 3 Gyroscope 3, Mean Rate (mas/day) vs. Mean Misalignment (as) Mean North-South Misalignment Mean West-East Misalignment

7. Gyroscope 1 Gyroscope 2 90 4000 120 60 90 4000 120 60 3000 3000 150 30 2000 2000 150 30 1000 1000 Mean North-South Misalignment 180 0 180 0 210 330 210 330 240 300 240 300 270 270 Gyroscope 4 Gyroscope 3 90 4000 120 90 4000 3000 120 60 2000 150 30 150 30 Mean North-South Misalignment 1000 180 0 180 0 210 330 210 330 240 300 240 300 270 270 Mean West-East Misalignment Mean West-East Misalignment Observations – All Gyroscopes

8. Observations–Change of Electrode PotentialGyroscope Drift Rates, DC Preload, Misalignment 10

9. Summary: Calibration Phase Measurements

10. E Dipole Layer Non-uniform potential Calculation of Torque due to Patch Effect Fields Electric Field at a Metallic Surface E Uniform Potential No Patch Effect Field Torques due to Patch Effect Potential on Rotor and Housing • Expand Potential on Each Surface in Terms of Spherical Harmonics • Use Rotation Matrices to Transform to a Common Reference Frame • Solve Laplace’s equation, find energy stored in electric field • Find the torque by differentiating the energy with respect to the angles which determine the mutual orientation of the conductors

11. Calculated Misalignment Torque Torque roll spin housing rotor

12. Analytical Expression for Torque Torque Coefficient • Proportional to Misalignment • Perpendicular to Misalignment Direction • Modulated at Polhode Frequency • Depends of Polhode Path • Depends of Patch Effect on Rotor and Housing Calculated Misalignment Torque Averaged over spin of gyroscope and roll of housing  Torque roll spin

14. Misalignment Torques - Data Analysis Is it possible to separate the gyroscope drift rate due to misalignment torques from the drift rate due to relativistic effects? Characteristics of Misalignment and Uniform Drift Simulated Data • Radial Component of Drift Rate Contains NO Contribution from Misalignment Drift • Magnitude and Direction of Uniform (Relativistic) Drift Rate May Be Determined From Variation of Radial Component with Misalignment Phase

15. Two Data Analysis Methods • Explicitly Include Misalignment Torques in Analysis of Data • Only Use Information on Radial Rate • Precision of Drift Rate Estimates ~ 1/T3/2 • Initial Application of This Method In N Batches ~ N/T3/2 • New Data Analysis Approach Recovers Full Precision • Explicit Use of Sequential Correlated Noise in Rate Estimates

16. Resonance Torques Observation*: Offsets in Orientation of Gyroscope Axis Tend to Occur when a harmonic of the gyroscope polhode frequency is equal to the satellite roll frequency Roll Frequency = 143 * Polhode Frequency * J. Kolodziejczak, MSFC

17. Observations of Resonance Torques Start Roll Frequency = 143 * Polhode Frequency End

18. Resonance Torques – Gyroscope 4

19. Resonance Torques – Gyroscope 4

20. Analytical Expression for Torque Torque Components Calculation of Patch Effect Resonance Torque:Harmonic of Polhode Frequency Equal to Roll Frequency Torque spin roll • Properties of Resonance Torques • Resonance Condition, nfp = fr • Independent of Misalignment • Direction Depends on Relative Phase and Distribution of Patches • Depends on Polhode Path

21. Resonance Torques – Predicted Cornu Spiral Fresnel Integrals: Integration of Equations of Motion With Linearly Varying Polhode Frequency, Constant Polhode Angle

22. Resonance Torques: Data Analysis • Exclude data in vicinity of resonances • Explicitly include resonances in data analysis • Two Parameters Uniquely determine each resonance

23. Example: Analysis of Data for Gyroscope 4 Misalignment Torques: Use only radial rate information (along the misalignment vector) Resonance Torques: Exclude Data in Vicinity of Resonance Formal Statistical Rate Errors: NS = 16 mas/yr WE = 14 mas/yr

24. Summary • Patch Effect Torques are dominant classical torques acting on the gyroscopes • Motion of gyroscope spin axis due to patch effect torques can be separated from the relativistic motion of the gyroscopes. • Misalignment Torque: • Acts in Direction Perpendicular to Misalignment • Resonance Torque • Displacement Occurs in Finite Time • Unique Time Signature