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The Gravity Probe B Experiment: Data Analysis Journey
Michael Heifetz
On Behalf of GP-B Data Analysis Team
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MG12 Paris 12-18 July, 2009
Gravity Probe B Concept
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MG12 Paris 12-18 July, 2009
Science Signal
s
aberration
GuideStar
s
- spin axis direction
- roll axis direction
Apparent Guide Star
s
EWe NSe
GSe Measurement:
Spin-Roll Misalignment
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MG12 Paris 12-18 July, 2009
Science Signal
)sin()(
)cos()(
rEWEW
rNSNS
s
s
Science Signal Spectral Shift (Roll Frequency)
SQUID Readout System
SQUID Pick-up Loop Rolls with S/C
aberration
GuideStar
s
- spin axis direction
- roll axis direction
Apparent Guide Star
s
EWe NSe
GSe
☼
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MG12 Paris 12-18 July, 2009
3 Sep 04
30 Mar 05
Science Signal
noisebias
s
sCZ
rEWEW
rNSNSgSQUID
)]sin()(
)cos()([
Apparent Guide Star
Aberration
Aberration -- Nature's calibrating signal for
gyro scale factor Cg
s
EWe NSe
GSe
GuideStar
Annual aberration
Readout Output
• Orbital motion:– Varying apparent position of star
(vorbit/c + special relativity)• Spacecraft around Earth:
- 5.1856 arcsec (97.5 min period)• Earth around Sun:
– 20.4958 acrsec (1 yr period)
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MG12 Paris 12-18 July, 2009
Pre-Flight Data Analysis Strategy
Constant - calibrated based on orbital and annual aberration
,gCSurprise A: variationsSurprise A: variationsgC
Gyro orientation trajectory and - straight lines )(tsNS
)(tsEW
Surprise B: Patch Effect TorqueSurprise B: Patch Effect Torque
noisebias
s
sCZ
rEWEW
rNSNSgSQUID
)]sin()(
)cos()([
Scale FactorScale Factor
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MG12 Paris 12-18 July, 2009
Expected Gyroscope Behavior
Geodetic effect(-6571 marcsec/yr)
Frame-dragging effect(-75 marcsec/yr)
Newton’s universe
Newton’s universe Includes Solar Geodetic and Guide Star Proper Motion
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MG12 Paris 12-18 July, 2009
Flight Data (Gyro 2)
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MG12 Paris 12-18 July, 2009
Three Pillars of GPB Data Analysis
InformationTheory
Filter Implementation: Numerical Estimation Techniques
Understanding of Gyroscope Motion: Trapped Flux Mapping (TFM)
Torque Models
UnderlyingPhysics
Readout Science Signal Structure: Measurement Models
UnderlyingPhysics,
Engineering
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MG12 Paris 12-18 July, 2009
Data Analysis Structure: ‘Two-Floor’ Processing
Torque Modeling
Gyro Orientation Time History
Data Analysis Building
SQUID Readout Processing First Floor
Second Floor
RelativityMeasurement
Full Information Matrix
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MG12 Paris 12-18 July, 2009
Structure of Two-Floor Analysis
SQUID Science Signal
(2 sec sampling rate)
1st Floor One Orbit Estimator
1st Floor
No Torque Modeling
Gyro Orientation Profiles (NS, EW) (1 point per orbit)
Gyro Scale Factor Estimates
Kalman Filter (Smoother)
Torque Model2nd Floor
Relativity Estimate
Torque coefficients Estimates
Gyro Orientation Profiles (NS, EW) (1 point per orbit)
Data Reduction
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MG12 Paris 12-18 July, 2009
1st Floor Challenges: How to Pull Out Gyro Orientation from SQUID Data
complete
Readout scale factor time-variations(“Cg Polhode modeling”)
Pointing error compensation (“Gyro/Telescope scale factor matching”)
Data Grading (quality of inputs)
Bias modeling (e.g. polhode variations, bias jumps)
Electronic Control Unit noise elimination
Most Difficult ProblemsMost Difficult Problems
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MG12 Paris 12-18 July, 2009
Trapped Flux & Readout Scale Factor
Tra
pp
ed
mag
neti
c p
ote
nti
al (V
)
I2
I3
I1
6 Sept 200426 June 20054 Oct 200414 Nov 200420 Feb 200520 Dec 2004
Gyro 1body frame
polhode
ˆ
ˆ ˆ
s
s
sss
s
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MG12 Paris 12-18 July, 2009
Successes of Trapped Flux Mapping Parameter Error
Angular velocity, 10 nHz~ 10-10
Polhode phase, p ~ 1
Rotor orientation ~ 2
Trapped magnetic potential ~ 1%
Gyroscope scale factor, Cg ~ 10-4
I3
I2
Path of spin axis in gyro body
I1
I3
I2I1
Trapped magnetic potential
s
Rela
tive C
g vari
ati
ons
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MG12 Paris 12-18 July, 2009
Scale Factor Model
N
ngg CtC
0
10blue – an(t) and bn(t)
red - fit to ε(t)
0000
00
00
Harmonic expansion in polhode phase with coefficients that depend on polhode angleHarmonic expansion in polhode phase with coefficients that depend on polhode angle
Trapped Flux Mapping
Trapped Flux Mapping
- Polhode phase - Polhode phasep
- Polhode angle - Polhode angle
.2
)(tan,)()(,)()(
,))(sin()())(cos()(1)(
00
00
ttbtbtata
tnbtnag
Ctg
C
K
k
knk
nn
K
k
knk
nn
N
npnpn
Gyro principle axes of inertia and instant spin axis position
p
I3
I1
I2
p
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MG12 Paris 12-18 July, 2009
Gyroscope-Telescope Scale Factor Matching
Reduces coupling of vehicle motion to science signal from 20 to 0.1 marc-sec
SQ1 Signal PSD - Unmatched
Frequency (Hz)
1 2 3 4 5 6Roll ±Orbital
1 – Roll2 – 2xRoll3 – dither 14 – dither 25 – 3xRoll6 – 4xRoll
SQ1 Signal PSD - Matched
Matched Gyroscope (SQUID) DataMatched Gyroscope (SQUID) Data
Telescope DataTelescope Data
Spectrum of SQUID Signal:before and after matchingPointing error compensation
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MG12 Paris 12-18 July, 2009
SQUID DataSQUID Data
SQUID No-bias Signal
SQUID No-bias Signal
Nonlinear Least-Squares Estimator
(No Torque Modeling)
Nonlinear Least-Squares Estimator
(No Torque Modeling)
Roll PhaseData
Roll PhaseData
AberrationData
AberrationData
Data Grading
Data Grading
ττ
μμ
Batch length: 1orbit Batch length: 1orbit Bias
EstimatorBias
Estimator
Cg (tk*)CT (tk*) δφ(tk*)
Cg (tk*)CT (tk*) δφ(tk*)
ResidualsResiduals
Pointing/Misalignment Computation
Pointing/Misalignment Computation
TelescopeData
TelescopeData
Roll PhaseData
Roll PhaseData
AberrationData
AberrationData
OUTPUT:Pointing
OUTPUT:PointingGSV/GSIGSV/GSI
Polhode PhaseData
Polhode PhaseDataTrapped
Flux Mapping
Trapped Flux
Mapping Polhode AngleData
Polhode AngleData
Full Information Matrix
Full Information Matrix
Gyro Orientation(1 point/orbit)
Gyro Orientation(1 point/orbit)
Full State Vector Estimates
Full State Vector Estimates
gC Gyro Scale Factor ModelGyro Scale Factor Model
Let’s look at the gyro
orientation profiles…
G/T MatchingG/T Matching
First Floor: SQUID Readout Data Processing
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MG12 Paris 12-18 July, 2009
1st Floor Output: Gyro Orientation (NS direction)
Seeing Strong Geodetic in ‘Raw’ Data
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MG12 Paris 12-18 July, 2009
1st Floor Output: Gyro Orientation (EW direction)
The Name of the Game – Frame-Dragging!
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MG12 Paris 12-18 July, 2009
Patch Effect & Pre-launch Investigations
rotor surface
housing surface
SEM image of rotor Nb film
• The patch effect• surface layer with variable electric dipole moment density
• Pre-launch investigation• Rotor electric dipole moment + field gradient from suspension• Kelvin probe measurements:
• Contact potentialdifferences ~ 100 mV
• Mitigated / eliminated by grainsize, < 1 μm << 30 μm gap
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MG12 Paris 12-18 July, 2009
Evidence for Patch Effect• Exhibit A: Gyroscope spin-down
Gyro Spin-down period (yr)
1 15,800
2 13,400
3 7,000
4 25,700
Polhode period vs elapsed time since January 1, 2004
Gyro
1,
Tp (
hr)
Gyro
4,
Tp (
hr)
Time (days) Time (days)
Blue: Worden Red: Santiago & Salomon
• Exhibit B: Changing polhode period
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MG12 Paris 12-18 July, 2009
More Evidence for Patch Effect• Exhibit C: Orbit determination
• Anomalous z-axis acceleration ~ 10-8 N, modulated at polhode frequency
• Exhibit D: Large misalignment torques
Mean East-West misalignment
Mean N
ort
h-S
outh
mis
alig
nm
ent Mean rate (marcsec/day) vs. mean misalignment (arcsec)
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
4
Dri
ft r
ate
mag
nit
ud
e (
arc
sec/
day)
Mean misalignment (arcsec)
k = 2.5 arc sec/day/degree
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MG12 Paris 12-18 July, 2009
Misalignment Torque (Roll Averaged)
GuideStar
s
NS vs. EW misalignment,
-20 -10 0 10 20EW misalignment (arcsec)
Misalign-ment angle
Misalignment Phasey
Uniform Radial
Precession(Relativity)
Torque-induceddrift
RN
E
µ
NS
mis
alig
nm
ent
(arc
sec)
-15
-1
0 -5
0
5
1
0 1
5
NSNSEWEW
EWEWNSNS
stkrdt
ds
stkrdt
ds
)(
,)(
• Torque • Drift
• Torque coefficient: k(p)• Relativity fixed in inertial frame
Aberration spectrally shifts misalignment torque
2nd Floor Torque Model
(2006- 2007)
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MG12 Paris 12-18 July, 2009
Relativity Estimates (Misalignment Torque Modeling) 2007
Gyro 3
Gyro 4
Gyro 1
Gyros 1, 3, 4 combined
GR predictionGR prediction
Gyro 3
Gyro 4
Gyro 1
Gyros 1, 3, 4 combined
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MG12 Paris 12-18 July, 2009
Discovery of Roll-resonance Torque (non roll-average)
• Exhibit E: Roll-polhode resonance ‘jumps’– ‘Jumps’ occur when high harmonic of changing polhode
rate, mpolh , is coincident with roll rate, roll
Date (2005)
Or Even More Evidence for Patch Effect
142 139140141145 144 143 138146
sEW
res. m
EW
ori
enta
tion,
s EW (
arc
sec) Gyro 2 flight data
)(tm pr
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MG12 Paris 12-18 July, 2009
Discovery of Roll-Polhode Resonance Torques
Resonance
Resonance
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MG12 Paris 12-18 July, 2009
Full Torque Model -Unknown (estimated) parameters-Unknown (estimated) parameters
)()()(
);()()(
0
0
tmtt
tmtt
rm
rm
Resonances: )(tm pr
- S/C roll axis direction),(EWNS
)](sin)()(cos)([)(0020
001
tmktmktkm
kM
mm
k
mnkN
nmn
mn
m
m Mmtk
k
k
k,...,1,0),(01
00
2
1
2
1
Trapped Flux
Mapping
Polhode Phase ( ),Polhode Angle ( )
00
)2/tan( 00
)](sin)(cos)(sin)(cos[))((
)](sin)(cos)(sin)(cos[))((
0 10
0 10
tatbtatbstkrdt
ds
tbtatbtastkrdt
ds
mmnmmnmmnmmn
Mc
m
Nc
n
n
NSNSEW
EW
mmnmmnmmnmmn
Mc
m
Nc
n
n
EWEWNS
NS
Roll-resonance torqueRelativity Misalignment torque
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MG12 Paris 12-18 July, 2009
2nd Floor Kalman Filter
Output:• Torque related variables:
- torque coefficients - modeled torque contributions
- Reconstructed “relativistic” trajectory
Kalman Filter / Smoother
Torque Contribution Subtraction
Relativity
Estimates
Gyro Orientation
Profiles
State vector: }{},{,,),(),( ckrrtstsxEWNSEWNS
)(),()( 11 kkkk txttFtx
kkk tHxtz )()( 1
Propagation Model:
Measurement Model:
“Measurements”
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MG12 Paris 12-18 July, 2009
Measured & Reconstructed Orientations
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MG12 Paris 12-18 July, 2009
Measured & Reconstructed Orientations (G4)
1st Floor Output
2nd Floor Output
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MG12 Paris 12-18 July, 2009
Measured & Reconstructed Orientations (G2)
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MG12 Paris 12-18 July, 2009
Current Results
Einstein’s prediction
NS: -65711 marcsec/yr
EW: -751 marcsec/yr
(includes solar GR effects and guide star proper motion)
Relativity estimates from 155-day analysis
• For the first time GR estimates agree among gyros• Statistical uncertainty: < 0.5% of geodetic effect
~ 14% of frame-dragging
4-gyro combined result
NS: -656512.3 marcsec/yr
EW: -80.45.4 marcsec/yr
(50% probability)
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MG12 Paris 12-18 July, 2009
Locking in the Final Results• Current (statistical) limit: ~14% of frame-
dragging• Fundamental limit from covariance analysis:
~ 5% of frame-dragging
• Reaching this fundamental limit requires:
1. Expanding analysis to full year of science data
2. Once-per-orbit averaging 2-sec processing Enabled by parallel computing
• A definitive result requires completing critical and detailed treatment of systematic effects
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MG12 Paris 12-18 July, 2009
One OrbitGyro Motion
Why 2-sec Filter?
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MG12 Paris 12-18 July, 2009
• Serial 2-sec processing (160 days) Aug ’09
• Complete transition to parallel processing Oct ’09
• Extension to full mission (353 days) Dec ’09
• Complete treatment of systematics Feb ’10
• Grand synthesis ~ 2 marcs/yr Jun ’10 4-gyro limit Final results to be announced at Fairbank Workshop on
Fundamental Physics & Innovative Engineering in SpaceAug ’10
Path to Completion
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