The Gravity Probe B Experiment: Data Analysis Journey Michael Heifetz On Behalf of GP-B Data Analysis Team

The Gravity Probe B Experiment: Data Analysis Journey

Michael Heifetz

On Behalf of GP-B Data Analysis Team

2

MG12 Paris 12-18 July, 2009

Gravity Probe B Concept

3


Science Signal

s

aberration

GuideStar

s

- spin axis direction

- roll axis direction

Apparent Guide Star

s

EWe NSe

GSe Measurement:

Spin-Roll Misalignment

4


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

☼

5


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)

6


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

7


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

8


Flight Data (Gyro 2)

9


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

10


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

11


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

12


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

13


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

14


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

15


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

16


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

17


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



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

18


1st Floor Output: Gyro Orientation (NS direction)

Seeing Strong Geodetic in ‘Raw’ Data

19


1st Floor Output: Gyro Orientation (EW direction)

The Name of the Game – Frame-Dragging!

20


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

21


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

22


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

23


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)

24


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

25


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

26


Discovery of Roll-Polhode Resonance Torques

Resonance

Resonance

27


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

28


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”

29


Measured & Reconstructed Orientations

30


Measured & Reconstructed Orientations (G4)

1st Floor Output

2nd Floor Output

31


Measured & Reconstructed Orientations (G2)

32


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)

33


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

34


One OrbitGyro Motion

Why 2-sec Filter?

35


• 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

Documents

The Gravity Probe B Experiment: Data Analysis Journey Michael Heifetz On Behalf of GP-B Data Analysis Team