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HEPL Seminar
1
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
3
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
HEPL Seminar
4
July 1, 2009
Misalignment Torque - Observations
InitializationPhase
InitializationPhase
Science Data Collection Phase
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
HEPL Seminar
5
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
HEPL Seminar
6
July 1, 2009
Calibration Phase ObservationsMisalignment Torques
InitializationPhase
InitializationPhase
Science Data Collection Phase
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 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
HEPL Seminar
7
July 1, 2009
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
HEPL Seminar
8
July 1, 2009
Observations – All Gyroscopes
1000
2000
3000
4000
30
210
240
90
270
120
300
150
330
180 0
Gyroscope 4
Mean West-East Misalignment
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
Mean West-East Misalignment
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
HEPL Seminar
9
July 1, 2009
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
DC Preload, c-axis at -14vDC Preload a-axis at -14vAC PreloadDir:Cal Star to GS
5
10
15
20
30
210
60
240
90
270
120
300
150
330
180 0
Gyroscope 2 Drift Rate (as/day), DC preload
South
East
DC Preload, c-axis at -14vDC Preload a-axis at -14vAC PreloadDir:Cal Star to GS
5
10
15
30
210
60
240
90
270
120
300
150
330
180 0
Gyroscope 4 Drift Rate (as/day), DC preload
South
East
DC Preload, c-axis at -14vDC Preload a-axis at -14vAC PreloadDir:Cal Star to GS
HEPL Seminar
10
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
HEPL Seminar
11
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
HEPL Seminar
12
July 1, 2009
Calculated Misalignment Torque
Torqueroll
spin
rotorrotor
housinghousing
HEPL Seminar
13
July 1, 2009
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
HEPL Seminar
14
July 1, 2009
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.
HEPL Seminar
15
July 1, 2009
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?
HEPL Seminar
16
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
HEPL Seminar
17
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
Roll Frequency = 143 * Polhode Frequency
HEPL Seminar
18
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
Roll Frequency = 143 * Polhode Frequency
Roll Frequency = 143 * Polhode Frequency
HEPL Seminar
19
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
HEPL Seminar
20
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
HEPL Seminar
21
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
HEPL Seminar
22
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
23
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
HEPL Seminar
24
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
HEPL Seminar
25
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