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

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

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

4. Additional Evidence for Torques: Gyroscope Orientation History

5. Calibration Phase Observations Misalignment Torques
Initialization
Phase
Science Data Collection
Calibration
Launch
April 20, 2004
Gyroscopes Spun Up and Aligned
August 29, 2004
Liquid Helium Depleted
Sept. 29, 2005
Aug. 15, 2005
Gravity Probe B Mission Timeline
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
HR Peg
(acquired)
NhS1 20

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

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

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

9. Summary: Calibration Phase Measurements
Torque Direction
Perpendicular to Misalignment
Torque Dependence on Misalignment
Proportional to Misalignment < 10
Torque Magnitude
= k,
k ~ 1 arcsec/(deg day)
= 3 × 10-9/sec
Dependence on Electrode Voltages
Independent with 20 Hz modulation.
k changes with dc voltage
Stability
Evidence for long term changes in k

10. Calculation of Torque due to Patch Effect Fields
Electric Field at a Metallic Surface
Dipole
Layer
Non-uniform potential E 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
roll
spin
housing
rotor

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

13. Measurements
Calculation
Torque Direction
Perpendicular to Misalignment
Torque Dependence on Misalignment
Proportional to Misalignment < 10
Proportional to misalignment,  << 1
Torque Magnitude
= k,
k ~ 1 arcsec/(deg day)
= 3 × 10-9/sec
Depends on rotor and housing potential
Increases with increasing l
Consistent with 50 mV patches, l = 30
Dependence on Electrode Voltages
Independent with 20 Hz modulation.
k changes with dc voltage
Indep. of voltage with 20 Hz modulation
Electrode dc voltage changes k
Stability
Evidence for long term changes in k
k depends on angle between spin axis and maximum inertia axis
Modulation of torque at harmonics of polhode period
Torque is modulated at harmonics of polhode period
Est. orientation change < 1 mas.

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. Calculation of Patch Effect Resonance Torque: Harmonic of Polhode Frequency Equal to Roll Frequency
Analytical Expression for Torque
Torque
spin
roll
Torque Components
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