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

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

• Mac Keiser and Alex Silbergleit

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Outline

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

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Misalignment Torque - Observations
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Additional Evidence for TorquesGyroscope
Orientation History
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Calibration Phase ObservationsMisalignment
Torques
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Observations Gyroscope 3
Gyroscope 3, Mean Rate (mas/day) vs. Mean
Misalignment (as)
Mean North-South Misalignment
Mean West-East Misalignment
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Observations All Gyroscopes
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ObservationsChange of Electrode
PotentialGyroscope Drift Rates, DC Preload,
Misalignment 10
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Summary Calibration Phase Measurements
Measurements
Torque Direction Perpendicular to Misalignment
Torque Dependence on Misalignment Proportional to Misalignment lt 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
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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

1. Expand Potential on Each Surface in Terms of
   Spherical Harmonics

1. Use Rotation Matrices to Transform to a Common
   Reference Frame

1. Solve Laplaces equation, find energy stored in
   electric field

1. Find the torque by differentiating the energy
   with respect to the angles which determine the
   mutual orientation of the conductors

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Calculated Misalignment Torque
Torque
roll
spin
housing
rotor
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Calculated Misalignment Torque Averaged over
spin of gyroscope and roll of housing
?
Torque
roll
spin
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Measurements Calculation
Torque Direction Perpendicular to Misalignment Perpendicular to Misalignment
Torque Dependence on Misalignment Proportional to Misalignment lt 10 Proportional to misalignment, ? ltlt 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 lt 1 mas.
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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

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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

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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
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Observations of Resonance Torques
Start
Roll Frequency 143 Polhode Frequency
End
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Resonance Torques Gyroscope 4
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Resonance Torques Gyroscope 4
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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

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

• Exclude data in vicinity of resonances
• Explicitly include resonances in data analysis
• Two Parameters Uniquely determine each resonance

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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
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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