gp-b/aero-astro 1 data analysis september 30, 2008 stanford the gravity probe b experiment:...

GP-B/Aero-Astro

1Data AnalysisData Analysis

September 30, 2008 • Stanford

The Gravity Probe B Experiment: “Testing Einstein’s Universe”

(Data Analysis Challenges)

Dr. Michael Heifetz(Hansen Experimental Physics Laboratory)

GP-B/ Aero-Astro

2

October 21, 2008 • Stanford

Data AnalysisData Analysis

What is Gravity Probe B?

• Gravity Probe B (GP-B) is a NASA physics mission to experimentally investigate Albert Einstein’s 1916 general theory of relativity – his theory of gravity.

• GP-B directly measures in a new way, and with unprecedented accuracy, two extraordinary effects predicted by the general theory of relativity:

1. The geodetic effect – the amount by which the Earth warps the local spacetime in which it resides

2. The frame-dragging effect – the amount by which the rotating Earth drags its local spacetime around with it.

The frame-dragging effect has never before been directly measured!The frame-dragging effect has never before been directly measured!

GP-B/ Aero-Astro

3



The Enigma of Gravity

Sir Isaac Newton:Space and time are absolute or fixed entities. Gravity is a force that acts instantaneously between objects at a distance, causing them to attract one another.

Albert Einstein:Space and time are relative entities, interwoven into a spacetime fabric whose curvature we call gravity. Spacetime tells matter how to move, and matter tells spacetime how to curve.

GP-B/ Aero-Astro

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• Geodetic Effect– Space-time curvature ("the missing inch")

• Frame-dragging Effect– Rotating matter drags space-time ("space-time as a viscous fluid")

The Relativity Mission Concept

ωRωR

vR23232

3

2

3

RRc

GI

Rc

GMΩ

Leonard SchiffLeonard Schiff

GP-B/ Aero-Astro

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A “Simple” Experiment

GP-B Co-Founder, Bill Fairbank, once remarked: “No mission could be simpler than GP-B; it’s just a star, a telescope and a spinning sphere.” However, it took over four decades to develop all the cutting-edge technologies necessary to carry out this “simple” experiment.

GP-B/ Aero-Astro

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Brief History of Gravity Probe B1957 Sputnik – Dawn of the space age

1958 Stanford Aero-Astro Department created

1959 L. Schiff conceives of orbiting gyro experiment as a test of

General Relativity

1961 L. Schiff & W. Fairbank propose gyro experiment to NASA

1972 1st drag-free spacecraft: TRIAD/DISCOS

1975 SQUID readout system developed

1980 Rotor machining techniques perfected

1998 Science instrument assembled

2002 Spacecraft & payload integrated

2004 Launch and vehicle operations

2005 End of data collection

Start of Data Analysis

2007 Preliminary results presented at April APS meeting

2008 -2009 Final results

•84 doctorates (29 Phys; 54 AA, EE, ME; 1 Math)•15 Master’s degrees, 5 Engineer’s degrees•13 doctorates completed at other universities

Stanford Student Participation

GP-B/ Aero-Astro

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Spacecraft gyros(3x10-3 deg/hr) 102

10

1

0.1

0.01

39Frame dragging<0.3% accuracy

103

6606

Geodetic effect <0.002% accuracy

mar

csec

/yr

0.5 GP-B requirement

104

105

106

107

108

109

Best laser gyros (1x10-3 deg/hr)

Best mechanical gyros on Earth(10-2 deg/hr)

Electrostatically suspended gyroscope (ESG) on Earth with torque modeling(10-5 deg/hr)

Why a Space-Based Experiment?

mar

csec

/yr

Best terrestrial gyroscopes 10,000,000 times worse than GP-B

1 marcsec/yr = 3.2x10-11 deg/hr

GP-B/ Aero-Astro

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GP-B Instrument Concept

Gyros 4 & 3

Gyros 2 & 1

Fusedquartz block

(metrology bench)

Star tracking telescope

Guide star

IM Pegasi

• Operates at ~ 2 K with liquid He

• Rolls about line of sight to Guide Star– Inertial pointing signal at roll frequency– Averages body-fixed classical disturbance torques toward zero– Reduces effect of body-fixed

pointing biases

GP-B/ Aero-Astro

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Ultra-Precise Gyroscopes

To measure the minuscule angles predicted by Einstein's theory, it was necessary to build near-perfect gyroscopes ~10 million times more precise than the best navigational gyroscopes. The GP-B gyro rotors are listed in the Guinness Database of World Records as the most spherical man-made objects.

GP-B/ Aero-Astro

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SQUID Magnetometers

How can one monitor the spin-axis orientation of a near-perfect spherical gyroscope without any physical marker showing the location of the spin axis on the gyro rotor? The answer lies in superconductivity.

Predicted by physicist Fritz London in 1948, and most fortunate for GP-B, a spinning superconductor develops a magnetic moment exactly aligned with its spin axis.

GP-B/ Aero-Astro

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Dewar & Probe

GP-B’s 650-gallon dewar, kept the science instrument inside the probe at a cryogenic temperature (2.3K) for 17.3 months and also provided the thruster propellant for precision attitude and translation control.

GP-B/ Aero-Astro

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Pointing Telescope

A telescope mounted along the central axis of the dewar and spacecraft provided the experiment’s pointing reference to a “guide star.” The telescope’s image divider precisely split the star’s beam into x-axis and y-axis components whose brightness

could be compared.

GP-B/ Aero-Astro

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Integrated Payload & Spacecraft

Built around the dewar, the GP-B spacecraft was a total-integrated system, comprising both the space vehicle and payload, dedicated as a single entity to experimentally testing predictions of Einstein’s theory.

GP-B/ Aero-Astro

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Data AnalysisData Analysis 19

Redundant spacecraft processors, transponders.

16 Helium gas thrusters, 0-10 mNea, for fine 6 DOF control.

Roll star sensors for fine pointing.

Magnetometers for coarse attitude determination.

Tertiary sun sensors for very coarse attitude determination.

Magnetic torque rods for coarse orientation control.

Mass trim to tune moments of inertia.

Dual transponders for TDRSS and ground station communications.

Stanford-modified GPS receiver for precise orbit information.

70 A-Hr batteries, solar arrays operating perfectly.

GP-B Spacecraft

6.4 m 3240 kg

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Challenges of Data Analysis…

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θ

Apparent

Guide Star

aberration

Guide Star

GSe

EWeNSe

s

- gyro spin axis orientation

- vehicle roll axis orientation

- gyroscope misalignment

s

Relativity: slopes of (Geodetic) and (Frame- dragging)

(significantly more complex problem))(ts

NS)(ts

EW

noisebias

rEWEW

rNSNSgSQUID

s

sCtZ

]

[

)sin()(

)cos()()(

SQUID Readout Data

Roll Phase Data

Telescope Data, Orbital and Annual

Aberrations

Scale FactorScale Factor

Gyro orientation trajectory and - straight lines )(tsNS

)(tsEW

Surprise B: Patch Effect TorqueSurprise B: Patch Effect Torque

- calibrated based on orbital and annual aberration

,g

CSurprise A: variationsSurprise A: variationsgC

‘Simple’ GP-B Data Analysis

Pointing Error via

Telescope

GP-B/ Aero-Astro

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Three Cornerstones of Dynamic Estimation (Filtering)

InformationTheory

Filter Implementation: Numerical Techniques

Gyroscope Motion: Torque Models

UnderlyingPhysics

SQUID Readout Signal Structure: Measurement Models

UnderlyingPhysics,

Engineering

GP-B/ Aero-Astro

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Data Analysis Structure: ‘Two-Floor’ Processing

Torque Modeling

SQUID Readout Processing

Gyro Orientation Time History

Data Analysis Building

First Floor

Second Floor

RelativityMeasurement

Full Information Matrix

Patch Effect Torque Theory

(mathematical physics)

Scale Factor Modeling

Trapped Flux Mapping

GP-B/ Aero-Astro

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Polhode Motion, Trapped Flux & Cg• Actual ‘London moment’ readout

Body-axis Path Trapped magnetic

fields

London magnetic field at 80 Hz: 57.2 μG

Gyro 1: 3.0 μG

Gyro 2: 1.3 μG

Gyro 3: 0.8 μG

Gyro 4: 0.2 μG

• Scale factor Cg modulated at polhode frequency by trapped magnetic flux•Two methods of determining Cg history

- Fit polhode harmonics to LF SQUID signal- Direct computation by Trapped Flux Mapping

• Scale factor Cg modulated at polhode frequency by trapped magnetic flux•Two methods of determining Cg history

- Fit polhode harmonics to LF SQUID signal- Direct computation by Trapped Flux Mapping

GP-B/ Aero-Astro

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Polhode Motion and Readout Scale Factor: Cg Model

p

I3

I1

I2

Gyro principle axes of inertiaand instant spin axis position

00

2 2

0 0

( ) 1 ( )cos( ( )) ( )sin( ( )) ,

, , ( ) tan( / 2).

N

g g n p n pn

K Kn k n k

n nk n nkk k

C t C a n t b n t

a a b b t

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 (TFM)Trapped Flux Mapping (TFM)

- Polhode phase - Polhode phase

p

- Polhode angle - Polhode angle

Unknowns

3I

1I

2I

Guide StarGuide Star

Trapped

Flux

Trapped

Flux

John ConklinJohn Conklin

GP-B/ Aero-Astro

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First Floor: SQUID Readout Data Processing

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/Misalign. Computation

Pointing/Misalign. 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)

State Vector Estimates

State Vector Estimates

gC Gyro Scale Factor ModelGyro Scale Factor Model

Let’s look at the gyro

orientation profiles…

G/T MatchingG/T Matching

GP-B/ Aero-Astro

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Inertial Orientation Time-history: Gyro 1

NS Direction De-trended

m=42 m=41

EW Direction

timetime

mill

iarc

secm=42

m=41

resonance

NS Direction

)(tmpr

timetime

mill

iarc

sec

Strong Geodetic EffectStrong Geodetic Effect

GP-B/ Aero-Astro

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NS Direction EW Direction

Inertial Orientation Time-history: Gyro 2NS direction de-trended

m=214 m=142m=214 74 resonances! m=142 timetime

mill

iarc

sec

mill

iarc

sec

EW Direction

Resonance Schedule

Resonances: )(tm proll

GP-B/ Aero-Astro

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Torque Modeling

)(tRTorqMis

θ

Apparent

Guide Star

aberration

Guide Star

GSe

EWeNSe

s

- gyro spin axis orientation

- vehicle roll axis orientation

- gyroscope misalignment

s

sˆ

)]cos()()sin()([))((

)]sin()()cos()([))((

rrNSNSEW

EW

rrEWEWNS

NS

tctcstkrdt

ds

tctcstkrdt

ds

Misalignment torque

Roll-Resonance torque

k(t), c+(t), c-(t) are modulated by harmonics of polhode frequency – roll/polhode resonance: k(t), c+(t), c-(t) are modulated by harmonics of polhode frequency – roll/polhode resonance:

)(tm proll

relativity

2006-2007 2008

GP-B/ Aero-Astro

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Torque Coefficients: Polhode Variation

Roll-resonance torque coefficients c+, c-:

,00

1010

nN

nn

c

cc

,0

02

1

0

,..2,12

1 ncN

nmn

mn

cMmm

m

c

c

c

c

2

)(tan 0

0

t

Misalignment torque coefficient k:

)sincos()( 0

2

0

01 pmp

M

mm mkmktk

k

and have the same structure as and

mk

1 mk

2

mc

1

mc

2

)sincos()()( 21

1010 pmp

M

mm mcmcctc

c

The same polhode structure as in Readout Scale Factor Model (1st Floor)The same polhode structure as in Readout Scale Factor Model (1st Floor)



)(tp - polhode phase

)(0t - polhode

angle

GP-B/ Aero-Astro

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2nd Floor Roll-Resonance Torque Dynamic Estimator

Orientations

Profiles

Roll Phase

Misalignment

Polhode Phase/Angle

State vector: }{},{,,),(),( ckrrtstsxEWNSEWNS

)(),()( 11 kkkk txttFtx

kkk tHxtz )()( 1

Propagation Model:

Measurement Model:

Estimator (separate for each segment)

Output: - Torque related variables:

- torque coefficients - modeled torque contributions

- Reconstructed “relativistic” trajectory (Orientation profile minus torque contributions)

Combine reconstructed trajectories for all segments

Fit to a straight line

Relativity:

Slope estimate

Full 1st Floor Information is not yet usedFull 1st Floor Information is not yet used

GP-B/ Aero-Astro

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Measured Inertial OrientationMeasured Inertial Orientation Modeled Inertial OrientationModeled Inertial Orientation

Gyro 2: Estimation Results(Modeled Orientation vs Measured Orientation)

Subtracting the torque contributions…

74 Resonances!74 Resonances!

GP-B/ Aero-Astro

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NSNS

Gyro 2: Reconstructed “Relativistic” Trajectory

Reconstructed TrajectoryReconstructed Trajectory +1σ+1σ

-1σ-1σ

Weigted LS fit based on input noiseWeigted LS fit based on input noise

Frame-dragging effect!

GP-B/ Aero-Astro

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Current Relativity Estimates for Gyros 1,2,3, and 4

GR prediction

Gyro 3 (2007)

Gyro 1,3,4 combined

(2007)

Gyro 1 (2007)

Gyro 4 (2007)

Gyro 1,2,3,4 combined

G1G1 G3G3

G4G4G2G2

GP-B/Aero-Astro

30Data AnalysisData Analysis

September 30, 2008 • Stanford

Where we stand now

Roll-Resonance Torque Modeling:

• reduced large part of systematic errors: previously unmodeled torque-related errors are now modeled properly

• dramatically enhanced the agreement between the gyroscopes

Roll-Resonance Torque Modeling:

• reduced large part of systematic errors: previously unmodeled torque-related errors are now modeled properly

• dramatically enhanced the agreement between the gyroscopes

The same torque model works for all 4 gyros over entire mission

The same torque model works for all 4 gyros over entire mission

Developed estimator is not good enough: • Orientation time step, currently 1-orbit (97min) should be made much less than 1 roll period (77 sec)

Developed estimator is not good enough: • Orientation time step, currently 1-orbit (97min) should be made much less than 1 roll period (77 sec)

Final improvement of Algebraic Method: “2-sec Filter”: That is where we need your help!

Final improvement of Algebraic Method: “2-sec Filter”: That is where we need your help!

GP-B/ Aero-Astro

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Two-Second Filter: Nonlinear Stochastic Optimization Problem

• New Filter is formulated as a Dynamic Nonlinear Estimation Problem:

θ

Apparent

Guide Star

aberration

Guide Star

GSe

EWeNSe

s(!)(!)

noisetbaCshtZnknkgkk

))...,,(,,()(

SQUID Data

6108.11,...2,1 Nk307 days = 4605 orbits x 97 min x 30 (2-sec data points)307 days = 4605 orbits x 97 min x 30 (2-sec data points)

Nonlinear Model

sec21

kk

ttt

• Nonlinear Dynamic Gyro Motion Model• Nonlinear Dynamic Gyro Motion Model

)},{},{,,( tcksfdt

sd

Requires multiple cost-function minimum search iterations going through millions of data points

Requires multiple cost-function minimum search iterations going through millions of data points

For 1 GyroFor 1 Gyro

GP-B/ Aero-Astro

32



Main Equations

noisebias

rEWEW

rNSNSgSQUID

s

sCtZ

]

[

)sin()(

)cos()()(

00

2 2

0 0

( ) 1 ( )cos( ( )) ( )sin( ( )) ,

, , ( ) tan( / 2).

N

g g n p n pn

K Kn k n k

n nk n nkk k

C t C a n t b n t

a a b b t

)sincos()()(2

11010 pmp

cM

mm

mcmcctc

Tr = 97 sec

)]cos()()sin()([))((

)]sin()()cos()([))((

rrNSNSEW

EW

rrEWEWNS

NS

tctcstkrdt

ds

tctcstkrdt

ds

GeodeticGeodetic

Frame-draggingFrame-dragging

GP-B/ Aero-Astro

33



Main Equaitions -cont

)sincos()(2

01 pmp

kM

mm

mkmktk

GP-B/ Aero-Astro

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Challenges of 2-sec Filter

• Dealing with several millions of ‘measurement’ equations requires new assessment of numerical techniques and computational capabilities

• Analyzing gyroscopes together and the nonlinear structure of the estimation problem probably will require parallel processing (in which we have no experience)

• Evaluation of the analysis results, given the complexity of 2-sec filter, will probably require the development of new “truth model” simulations

gp-b/aero-astro 1 data analysis september 30, 2008 stanford the gravity probe b experiment:...

Documents

engineeringgpb aeroastro

floorgpb aeroastro

kggpb aeroastro

polhode structure

coarse orientation control

readout scale factor

magnetic torque rods

g gyro