1. Improved Accuracy of the Gravity Probe B Science Results
John W. Conklin for the GP-B Data Analysis Team
Stanford University

2. Gravity Probe B Concept

3. Expected Gyroscope Behavior
Geodetic effect* (-6571 marcsec/yr)
Newton’s universe
Frame-dragging effect* (-75 marcsec/yr)
Newton’s universe
*includes solar GR effects and guide star motion

4. Processed Flight Data (Gyro 2)

5. Three Complications (due to Patch Effect)
Polhode damping  complicates Cg determination
Blue: Worden Red: Santiago & Salomon
Gyro 1, Tp (hr)
Gyro 4, Tp (hr)
Time (days)
Time (days)
Misalignment Torques

6. Three Complications (due to Patch Effect)
Roll-polhode Resonance Torque
‘Jumps’ occur when harmonic of polhode rate coincident with roll rate
Gyro 2 per orbit orientation
sEW res. m
EW orientation, sEW (arcsec)
146
145
144
143
142
141
140
139
138
Date (2005)

7. Two Champions of the Data Analysis
Spectral separation
Rotor spin ~ 60 Hz - 80 Hz (changing with time)
Spacecraft roll = 13 mHz (from on-board star trackers)
Spacecraft orbit = 0.17 mHz (from on-board GPS)
Rotor polhode ~ 0.1 mHz (changing with time)
Earth’s orbit = 32 nHz (from JPL Earth ephemeris)
General Relativity acts at zero-frequency
Trapped magnetic flux  Enables determination of a) & d)
model of patch effect & gyro dynamics allow separation of relativistic & Newtonian effects

8. Why is the data analysis taking so long?
Pre-launch data analysis completely rewritten (almost)
Data Analysis timeline
Launch
20 April 2004
Post science calibration
15 August 2005
Helium depletion
September 2005
Geometric separation of misalignment torque
August 2006
Trapped Flux Mapping
August 2007
Complete Torque Modeling
August 2008
Complete (2-second) data processing
August 2009
Release of Science Results (NASA HQ)
13 October 2010

9.   s s  Science Signals   r ^ ^ ^
Telescope measures S/C pointing:
SQUID measures misalignment: = –    ^ s ^ s ^  
North (inertial)  ^
S/C x-axis r
Gyros 2 & 1
Guide Star
(IM Pegasi)
Science telescope
Gyros 4 & 3

10. Trapped Flux & Readout Scale Factor
Gyro 1 ˆ I3
polhode
Trapped Magnetic Potential (V)
6 Sept 2004 s 
~ 1% ˆ I1 ˆ I2

11. Trapped Flux & Readout Scale Factor
Gyro 1 ˆ I3
Trapped Magnetic Potential (V)
4 Oct 2004 s  ˆ I1 ˆ I2

12. Trapped Flux & Readout Scale Factor
Gyro 1 ˆ I3 s 
Trapped Magnetic Potential (V)
14 Nov 2004 ˆ I1 ˆ I2

13. Trapped Flux & Readout Scale Factor
Gyro 1 ˆ I3 s 
Trapped Magnetic Potential (V)
20 Dec 2004 ˆ I1 ˆ I2

14. Trapped Flux & Readout Scale Factor
Gyro 1 s  ˆ I3
Trapped Magnetic Potential (V)
20 Feb 2005 ˆ I1 ˆ I2

15. Trapped Flux & Readout Scale Factor
Gyro 1 s  ˆ I3
Trapped Magnetic Potential (V)
26 June 2005 ˆ I1 ˆ I2

16. Trapped Flux Mapping (August 2007)
Parameter
Error
Angular velocity, 
10 nHz ~ 10-10
Polhode phase, p
~ 1
Rotor orientation
~ 2
Trapped magnetic potential
~ 1%
Gyroscope scale factor, CgTF
~ 10-4 s ^ I1 I2
Path of spin axis in gyro body I3
Gyro 1
Relative Cg variations I1 I2
Trapped magnetic potential

17. Modeling Patch Effect Torques
Arbitrary rotor & housing potentials (spherical harmonics)
Rotate to common frame, spin average
Solve Laplace’s equation between rotor & housing  Energy stored in E-field  Patch effect torque: (a) roll averaged torque and (b) torque at 1st harmonic of roll significant s ^ s ^   s p  ^
Guide Star p
Spacecraft dynamics
Rotor dynamics

18. s  Misalignment Torque (Roll Averaged)   ^ ^ Torque   Drift s ^  
Guide Star  ^
Torque  
Drift 
Proportionality coefficient: k(p, p)
Relativity fixed in inertial frame
 Aberration spectrally separates misalignment torque
NS vs. EW misalignment,   
NS misalignment (arcsec)
EW misalignment (arcsec)

19. Roll-polhode Resonance Torque
Torque at 1st roll harmonic significant during “resonance”
Electrostatic model predicts:
Defines Cornu spiral
Requires accurate p(t)
Provided by TFM
North-South Orientation (arbitrary units)
East-West Orientation (arbitrary units)

20. Relativity + Newtonian Torques
Gyro 2 per orbit orientations
Misalignment torque
Roll-polhode resonance torque
Relativity

21. Newtonian Effects Removed (Aug. 2008)
Gyro 2, orientations – Newtonian torques
2.0
1.5
1.0
NS orientation (arcsec)
0.5
0.0
NS uniform drift
–0.5
Jan Jan Feb Mar Mar Apr May 8
+1
1.74
1.72
1.70
EW orientation (arcsec)
1.68
EW uniform drift
1.66
1.64
–1
Jan Jan Feb Mar Mar Apr May date (2004)

22. The 2-second Filter (August 2009)
Nonlinear simultaneous estimation of
Uniform drift, scale factor, torque coeffs., telescope params., …
LF SQUID data sampled every 2 sec over ~1 yr (4)
Batch least-squares fit
Iterative linearization & linear least squares fit
Sigma point algorithm for Jacobian computation
Robust convergence & unbiased uniform drift estimates

23. Correct Model  Consistent Results Gyro-to-gyro consistency
Segment-to-segment consistency
Spinup & Alignment Complete Gyros 1, 2, 3
Gyro 4
2004
2005
Aug
Sept
Oct
Nov
Dec
Jan
Feb
Segment 2
Seg. 3
Segment 5
Segment 6
1 - Gyro 3 Analog Backup
2 - SRE Safemode
3 - bad GPS config
4 - roll notch filter
5 - Jan 20 Solar Flare
6, 7,8 - computer reboots
9 - roll notch filter
2005
Mar
Apr
May
Jun
Jul
Aug
Sept
Segment 9
Segment 10
Calibration

24. Gyro 1 Uniform Draft Rate Estimates
95% confidence ellipses
Seg. 10
Seg. 2-3
Seg. 5,6,9,10
Seg. 5-6
Seg. 9
 NS / EW observability varies due to Annual Aberration

25. Four-gyroscope Results

26. Experiment Control Unit
A Few Remaining Tasks
-4.6
ECU noise
SQUID readout electronics variations (~10-3)
Data segments 2 & 3
Sensitivity Analysis
-4.7
-4.8
-4.9
-5.0
Gyro 1 SQUID Signal (V)
-5.1
-5.2
-5.3
-5.4
-5.5
time (sec)
Experiment Control Unit
SRE Variations – Gyro 4

27. Expected Experiment Accuracy
Primarily depends on treatment of 2 systematic effects
Covariance analysis (including nearly all systematics)
Parameter optimization + additional data = additional 10%
Analysis excludes ~15% of potentially valuable data (i.e. outliers)
Misalignment Torque Coeff.
 rNS (mas/yr) [% GR]
 rEW (mas/yr) [% GR]
Upper Bound
21 [0.3%]
12 [30%]
Complete Treatment of ECU Noise
13 [0.2%]
7 [18%]
Complete Treatment of ECU Noise & SRE variations
6 [0.1%]
4 [10%]
All credible results to date consistent with GR (within 3), but not yet at final experiment accuracy

28. Announcement & Fairbank Workshop
Science Results Announcement:
13 Oct. NASA HQ
Fairbank Workshop
Summer 2011
NASA MSFC Huntsville, AL
Summer 2011

29. backup slides …

30. Low & High Frequency SQUID Data
LF SQUID channel (780 Hz LP filter → 4 Hz LP filter → gain)
5 Hz continuous
HF SQUID channel (780 Hz LP filter)
106 “snapshots” over 1 year (2200 Hz, ~ 2 sec duration)
HF channel
Gyro 1 HF “snapshot”, 10 Nov. 2004
LF channel

31. Rotor body-fixed frame
Trapped Flux Mapping
Trapped magnetic potential in spherical harmonics
TF fixed in body  Euler rotation to inertial frame
About 3-axis by p (polhode phase)
About 2-axis by p (polhode angle)
About 3-axis by –s (“spin” phase)
Trapped Flux Mapping:
Nonlinear estimation of rotor dynamics, p(t), p(t), s(t)
Linear fit for spherical harmonic coefficients, Alm, l odd s p p
Rotor body-fixed frame

32. Geometric Method (from SAC 15)
Misalignment Phase (deg)
Data simulated for illustration purposes
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