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HEPL Seminar

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July 1, 2009

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HEPL SeminarJuly 1, 2009

2

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

Analysis

Mac Keiser and Alex Silbergleit

HEPL Seminar

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July 1, 2009

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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July 1, 2009

Misalignment Torque - Observations

InitializationPhase

InitializationPhase

Science Data Collection Phase


CalibrationPhase

CalibrationPhase

LaunchApril 20, 2004LaunchApril 20, 2004

Gyroscopes Spun Up and AlignedAugust 29, 2004Gyroscopes Spun Up and AlignedAugust 29, 2004

Liquid Helium DepeletedSept. 29, 2005Liquid Helium DepeletedSept. 29, 2005Aug. 15, 2005Aug. 15, 2005

Gravity Probe B Mission TimelineGravity Probe B Mission Timeline

Proton Flux, Jan. 20-22, 2005, Measured by GOES Satellite

Par

ticl

es/(

cm2

sec

sr)

Time (days)

0 10 20 30 40 50 60-2.45

-2.4

-2.35

-2.3

-2.25

-2.2

-2.15

-2.1

-2.05

-2

-1.95Gyroscope 3 WE Orientation, Jan. 2 - Feb. 24, 2005

Arc

Se

c

Time, Days from Jan. 1, 2005

Gyro 3 West-East Spin Axis Orientation

Arc

Sec

Time (days) from Jan. 1, 2005

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July 1, 2009

Additional Evidence for Torques:Gyroscope Orientation History

250 300 350 400 450 500 550 600-4

-3

-2

-1

0

1

2

3WE Orientation

WE

Ori

enta

tio

n (

Arc

sec)

250 300 350 400 450 500 550 6000

10

20

Elapsed Days since Jan 1, 2004

Err

or

(mas

)

G1

G4

G3

G2

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July 1, 2009

Calibration Phase ObservationsMisalignment Torques

InitializationPhase

InitializationPhase



CalibrationPhase

CalibrationPhase

LaunchApril 20, 2004LaunchApril 20, 2004

Gyroscopes Spun Up and AlignedAugust 29, 2004Gyroscopes Spun Up and AlignedAugust 29, 2004

Liquid Helium DepletedSept. 29, 2005Liquid Helium DepletedSept. 29, 2005Aug. 15, 2005Aug. 15, 2005

Gravity Probe B Mission TimelineGravity 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

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 PegGuide Star

HR Peg(acquired)

NhS1(acquired)

2020

2020

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Observations – Gyroscope 3

1000

2000

3000

4000

30

210

60

240

90

270

120

300

150

330

180 0

0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1

1.5

2

2.5

3

3.5

4Gyro 3, Drift Rate Magnitude vs. Mean Misalignment

Drift

Rate

Magn

itude

(arc

sec/d

ay)

Mean Misalignment (arc sec)

k = 2.5 arc sec/day/degree

Gyroscope 3, Mean Rate (mas/day) vs. Mean Misalignment (as)

Mean West-East Misalignment

Mea

n N

ort

h-S

outh

Mis

alig

nmen

t

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July 1, 2009

Observations – All Gyroscopes

1000

2000

3000

4000

30

210

240

90

270

120

300

150

330

180 0

Gyroscope 4


1000

2000

3000

4000

30

210

60

240

90

270

120

300

150

330

180 0

Gyroscope 2Gyroscope 1

30

210

60

240

90

270

120

300

150

330

180 0

Gyroscope 3 4000


30

210

60

240

90

270

120

300

150

330

180 0

1000

2000

3000

4000

Mea

n N

ort

h-S

outh

M

isa

lignm

ent

Mea

n N

ort

h-S

outh

M

isa

lignm

ent

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

10

20

30

30

210

60

240

90

270

120

300

150

330

180 0

Gyroscope 3 Drift Rate (as/day), DC preload

South

East

DC Preload, c-axis at -14vDC Preload a-axis at -14vAC PreloadDir:Cal Star to GS

1

2

3

4

30

210

60

240

90

270

120

300

150

330

180 0

Gyroscope 1 Drift Rate, DC preload

South

East


5

10

15

20

30

210

60

240

90

270

120

300

150

330

180 0


South

East


5

10

15

30

210

60

240

90

270

120

300

150

330

180 0


South

East


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July 1, 2009

Summary: Calibration Phase Measurements

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/secDependence 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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July 1, 2009

Calculation of Torque due to Patch Effect Fields

Electric Field at a Metallic SurfaceElectric Field at a Metallic Surface

Uniform PotentialNo Patch Effect Field

Uniform PotentialNo Patch Effect Field

EE

Dipole LayerDipole Layer

Non-uniform potentialNon-uniform potential

EE

Torques due to Patch Effect Potential on Rotor and HousingTorques due to Patch Effect Potential on Rotor and Housing

1. Expand Potential on Each Surface in Terms of Spherical Harmonics1. Expand Potential on Each Surface in Terms of Spherical Harmonics

2. Use Rotation Matrices to Transform to a Common Reference Frame2. Use Rotation Matrices to Transform to a Common Reference Frame

3. Solve Laplace’s equation, find energy stored in electric field3. Solve Laplace’s equation, find energy stored in electric field

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

4. 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

Torqueroll

spin

rotorrotor

housinghousing

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Calculated Misalignment Torque Averaged over spin of gyroscope and roll of housing

Torqueroll

spin

1 ,ˆ k

Analytical Expression for TorqueAnalytical Expression for Torque

• Proportional to Misalignment• Proportional to Misalignment

• Perpendicular to Misalignment Direction• Perpendicular to Misalignment Direction

• Depends of Patch Effect on Rotor and Housing• Depends of Patch Effect on Rotor and Housing

• Depends of Polhode Path• Depends of Polhode Path

1

0

20 )0,(2 l

llmim

plm HReYd

ak p

Torque CoefficientTorque Coefficient

• Modulated at Polhode Frequency• Modulated at Polhode Frequency

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Measurements Calculation

Torque Direction Perpendicular to Misalignment

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.

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Misalignment Torques - Data Analysis

1

2

30

210

60

240

90

270

120

300

150

330

180 0

Misalignment Drift

-100 0 100-1.5

-1

-0.5

0

0.5

1

1.5Radial component of Misalignment Drift

Misalignment Angle (degrees)

Dri

ft R

ate

1

2

30

210

60

240

90

270

120

300

150

330

180 0

Uniform Drift

-100 0 100-1.5

-1

-0.5

0

0.5

1

1.5Radial Component of Uniform Drift

Misalignment Angle (degrees)

Dri

ft R

ate

Characteristics of Misalignment and Uniform Drift

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

Is it possible to separate the gyroscope drift rate due to misalignment torques from the drift rate due to relativistic effects?Is it possible to separate the gyroscope drift rate due to misalignment torques from the drift rate due to relativistic effects?

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July 1, 2009

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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July 1, 2009

Resonance Torques

5660 5680 5700 5720 5740 5760-2

-1

0

1

2Cosine of Roll After Removing DC and Linear Drift, Gyro 2, Res 277

mV

5660 5680 5700 5720 5740 5760-2

-1

0

1

2Sine of Roll

mV

Orbit Number

Gyroscope 2 From: May 10, 2005, 2amTo: May 15, 2005, 5 pm

1 day

50 mas

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

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

* J. Kolodziejczak, MSFC* J. Kolodziejczak, MSFC

Roll Frequency = 143 * Polhode Frequency


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July 1, 2009

Observations of Resonance Torques

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5Sine and Cosine at Roll Frequency, Gyro 2, Res 277

Co

sin

e o

f R

oll

Fre

qu

ency

(m

V)

Sine of Roll Frequency (mV)

North

West

Approximate Scale:

50 mas

StartStart

EndEnd



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Resonance Torques – Gyroscope 4

-25 -20 -15 -10 -5 0 5 10 15 202.8

3

3.2

3.4

3.6

3.8NS Orientation, Common Cg and Dphi, Gyro 4

Arc

Sec

-25 -20 -15 -10 -5 0 5 10 15 20-3.5

-3.48

-3.46

-3.44

-3.42

-3.4WE Orientation, Common Cg and Dphi, Gyro 4

Arc

Sec

Time (days) from 2005/001-00:00:00.0

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July 1, 2009

Resonance Torques – Gyroscope 4

-25 -20 -15 -10 -5 0 5 10 15 202.8

3

3.2

3.4

3.6

3.8NS Orientation, Common Cg and Dphi, Gyro 4

Arc

Sec

-25 -20 -15 -10 -5 0 5 10 15 20-3.5

-3.48

-3.46

-3.44

-3.42

-3.4WE Orientation, Common Cg and Dphi, Gyro 4

Arc

Sec

Time (days) from 2005/001-00:00:00.0

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July 1, 2009

Calculation of Patch Effect Resonance Torque: Harmonic of Polhode Frequency Equal to Roll Frequency

roll

spin Torque

cossin

,sincos

BAy

rpBAx nΔφ

Analytical Expression for TorqueAnalytical Expression for Torque

Torque ComponentsTorque Components

nllp

lnb

nllp

lnA

HRdlld

a

HRdlld

a

*1,ln0

20

*1,ln0

20

Im)()1(

Re)()1(

Properties of Resonance Torques

• Resonance Condition, nfp = fr

• Independent of Misalignment

• Direction Depends on Relative Phase and Distribution of Patches

• Depends on Polhode Path

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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July 1, 2009

Resonance Torques – Predicted Cornu Spiral

-1 0 1 2 3 4 5 6 7

-3

-2

-1

0

1

2

3

Orientation of Gyroscope Spin AxisN

ort

h-S

ou

th O

rien

tati

on

West-East Orientation

Fresnel Integrals: Integration of Equations of Motion With Linearly Varying Polhode Frequency, Constant Polhode AngleFresnel Integrals: Integration of Equations of Motion With Linearly Varying Polhode Frequency, Constant Polhode Angle

dtntL

dtntL

ts

dtntL

dtntL

ts

ntn

tB

tA

WE

tB

tA

NS

rp

2'

2'

2'

2'

2

cossin)'(

sincos)'(

HEPL Seminar

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July 1, 2009

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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July 1, 2009

Example: Analysis of Data for Gyroscope 4

-4 -3 -2 -1 0 1 2 3 4-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02Radial Rate vs. Misalignment Phase, Gyro 4

Ra

dia

l Ra

te (

arc

se

c/d

ay)

Misalignment Phase (rad)

GeometricSuperGeometric

Misalignment Torques: Use only radial rate information (along the misalignment vector)

Resonance Torques: Exclude Data in Vicinity of Resonance

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

Formal Statistical Rate Errors:

NS = 16 mas/yr

WE = 14 mas/yr

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July 1, 2009

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

HEPL SeminarJuly 1, 2009

26

End of Presentation

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