The Gravity Probe B Experiment: “Testing Einstein’s Universe” (Data Analysis Challenges)

GP-B/Aero-Astro

1Data 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 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 resides2. 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!

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Data Analysis

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.

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Data Analysis

• 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

ωRωRvR 23232

323

RRcGI

RcGMΩ

Leonard Schiff

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Data Analysis

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.

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Data Analysis

Brief History of Gravity Probe B1957 Sputnik – Dawn of the space age1958 Stanford Aero-Astro Department created1959 L. Schiff conceives of orbiting gyro experiment as a test of General Relativity1961 L. Schiff & W. Fairbank propose gyro experiment to NASA1972 1st drag-free spacecraft: TRIAD/DISCOS1975 SQUID readout system developed1980 Rotor machining techniques perfected1998 Science instrument assembled2002 Spacecraft & payload integrated2004 Launch and vehicle operations2005 End of data collection

Start of Data Analysis2007 Preliminary results presented at April APS meeting2008 -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

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Data Analysis

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

10

1

0.1

0.01

39 Frame 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

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Data Analysis

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

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Data Analysis

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.

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Data Analysis

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.

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Data Analysis

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.

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Data Analysis

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.

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Data Analysis

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.

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Data 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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Data Analysis

Challenges of Data Analysis…

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Data Analysis

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)

)(tsNS

)(tsEW

noisebiasrEWEW

rNSNSgSQUID

s

sCtZ

][

)sin()(

)cos()()(

SQUID Readout Data

Roll Phase Data

Telescope Data, Orbital and Annual

Aberrations

Scale Factor

Gyro orientation trajectory and - straight lines )(tsNS

)(tsEW

Surprise B: Patch Effect Torque

- calibrated based on orbital and annual aberration

,g

CSurprise A: variationsgC

‘Simple’ GP-B Data Analysis

Pointing Error via

Telescope

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Data Analysis

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

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

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Data Analysis

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

Body-axis Path Trapped magnetic

fields

London magnetic field at 80 Hz: 57.2 μGGyro 1: 3.0 μG

Gyro 2: 1.3 μG Gyro 3: 0.8 μGGyro 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

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Data Analysis

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 angle

Trapped Flux Mapping (TFM)

- Polhode phase

p

- Polhode angle

Unknowns

3I

1I

2I

Guide Star

Trapped Flux

John Conklin

GP-B/ Aero-Astro

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Data Analysis

First Floor: SQUID Readout Data Processing

SQUID Data

SQUID No-bias Signal

Nonlinear Least-Squares Estimator

(No Torque Modeling)

Roll PhaseData

AberrationData

Data Grading

τ

μ

Batch length: 1orbit Bias

Estimator

Cg (tk*)CT (tk*) δφ(tk*)

Residuals

Pointing/Misalign. Computation

TelescopeData

Roll PhaseData

AberrationData

OUTPUT:PointingGSV/GSI

Polhode PhaseDataTrapped

Flux Mapping Polhode Angle

Data

Full Information Matrix

Gyro Orientation(1 point/orbit)

State Vector Estimates

gC Gyro Scale Factor Model

Let’s look at the gyro orientation profiles…

G/T Matching

GP-B/ Aero-Astro

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Data Analysis

Inertial Orientation Time-history: Gyro 1

NS Direction De-trended

m=42 m=41

EW Direction

time

mill

iarc

secm=42

m=41

resonance

NS Direction

)(tmpr

time

mill

iarc

sec

Strong Geodetic Effect

GP-B/ Aero-Astro

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Data Analysis

NS Direction EW Direction

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

m=214 m=142m=214 74 resonances! m=142 time

mill

iarc

sec

EW Direction

Resonance Schedule

Resonances: )(tm proll

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Data Analysis

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

tctcstkrdtds

tctcstkrdtds

Misalignment torque

Roll-Resonance torque

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

)(tm proll

relativity

2006-2007 2008

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Data Analysis

Torque Coefficients: Polhode Variation

Roll-resonance torque coefficients c+, c-:

,00

1010n

N

nn

c

cc

,0

02

1

0

,..2,12

1 ncN

nmn

mn

cMmm

m

cc

cc

2)(tan 0

0

t

Misalignment torque coefficient k:

)sincos()( 02

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)

Trapped Flux Mapping

)(tp - polhode phase

)(0t - polhode

angle

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Data Analysis

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 used

GP-B/ Aero-Astro

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Data Analysis

Measured Inertial Orientation Modeled Inertial Orientation

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

Subtracting the torque contributions…

74 Resonances!

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Data Analysis

NS

Gyro 2: Reconstructed “Relativistic” Trajectory

Reconstructed Trajectory +1σ

-1σ

Weigted LS fit based on input noise

Frame-dragging effect!

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Data Analysis

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

G1 G3

G4G2

GP-B/Aero-Astro

30Data 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

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)

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

GP-B/ Aero-Astro

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Data Analysis

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)

Nonlinear Model

sec21

kk

ttt

• Nonlinear Dynamic Gyro Motion Model

)},{},{,,( tcksfdtsd

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

For 1 Gyro

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Data Analysis

Main Equations

noisebiasrEWEW

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

tctcstkrdtds

tctcstkrdtds

Geodetic

Frame-dragging

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Data Analysis

Main Equaitions -cont

)sincos()(2

01 pmp

kM

mm

mkmktk

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Data Analysis

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

The Gravity Probe B Experiment: “Testing Einstein’s Universe” (Data Analysis Challenges)

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