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

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

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. 20-22, 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

Additional Evidence for Torques: Gyroscope Orientation History

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

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

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

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

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

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

Calculated Misalignment Torque roll spin housing rotor

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

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.

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

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

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

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

Resonance Torques – Gyroscope 4

Resonance Torques – Gyroscope 4

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

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

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

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

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

End of Presentation