Absolute Gravity Meter Accuracy vs Time1817: Pendulum accuracy from 10-4 (Kater)1960: Pendulum Potsdam: 10-5 (10 mGal)1967 :1967 : Faller & Dicke : White light fringes; accuracyof 10-5 (1 mGal)1972 :1972 : Hammond and Faller : laser light fringes; accuracy of 10-6 (100 µGal)1980 :1980 : Zumberge and Faller/Sakuma; accuracy of10-8 (10 µGal)1985 :1985 : Niebauer and Faller : accuracy of 5 µGal; 10 µGal in field conditions, 6 « JILAg » built.1993 :1993 : FG5 accuracy of 10-9 (1 µGal)

Historical Progress

Pendulum-Absolute Gravimeters

Simple Pendulum

( )( )

( )

( ) ( )

! V.

sinsin IV.

III.

sin II.

cos1 I.

2

2

ImgL

ImgL

I

mgLUU

mgLmghU

=

−=−=

=Γ

−=∂∂

−=Γ

−==

ω

θωθθ

θ

θ

θ

&&

&&m

L

θ

h=L [1-cos(θ) ]

Simple Pendulum (Is it absolute?)

Measuring g requires measurement of:1. Pendulum period: Easy2. Mass of pendulum: Medium difficulty3. Position of center of mass: Difficult4. Distance between support point and

center of mass: Medium difficulty5. Moment of inertia of the pendulum: Very

difficult

Not really absolute because it is difficult to use absolute standards to measure moment of inertia.

Kater’s Pendulum:Absolute Gravity Meter

Reversible Pendulum Gravity MeterCaptain Henry Kater(1777-1835)

Reversible PendulumPendulum Elements

• Two Support Points on opposite sides of the pendulum

•Weights can be adjusted to move the center of gravity (CG)

• Fixed Support Platform

• Device adjusted until period of swing is the same for both pivot points

CG

Support Point 1

Knife EdgeContacts

Adjustable weights

L1

L2

Stopwatch

Support Point 2

Support Platform

Reversible Pendulum Equations

( ) ( )( ) ( )( )

( )( )

( ) ( )212

2

212

21

22

22

1

1

2121

21212

22

1

22

21

22

22

1

121

2

4g XI.

then, if X.

IX.

VIII.

VII.

VI.

LLLL

LLLLLLg

LLLLmLLm

mLImLII

LLmgIII

mgLI

cmcm

+Τ

=+=

==

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−=

+−=−=

+−+=Δ

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−=Δ

=

πω

ωωωωω

ωω

ω

L1

L2

Support point 1

Support point 2

Kater’s Reversible PendulumMeasurement requires only two measurements

Distance between pivot pointsPeriod (time)

Does not depend upon massDoes not depend upon shapeDoes not depend upon location of center of massDoes not depend upon moment of inertia

This is an example of a “good” absolute gravity measurement because a simple time and distance measurement tied to absolute standards gives a “calibrated” value for gravity. No reference is needed to any other gravity measurement.

Reversible Pendulum Gravimeter(Royal Observatory: Madrid)

Two Absolute Gravimeters(Royal Observatory: Madrid)

Free-Fall Laser Measurement

Drop an object and time how fast it falls.Record the time & distance pairs.Solve the equation of motion

Test Object

D1, T1

D2, T2

D3, T3

Fixed Reference

LASER

Beam Splitter

Raw Fringe

Photodiode

D1, T1 D2, T2 D3, T3

200 2

1iii gttvxx ++=

Faller/Dicke White Light Interferometer Gravity Meter(Princeton 1963)

•First interferometer based gravimeter

•Fixed Path could be calibrated with monochromatic light

• Calibrated time and distance measurements gives absolute gravity

Detector

g

Free Falling Cat’s Eye

Fixed Mirror

White light source

Three Equal Paths for White Light Fringes (Faller/Dicke 1963)

PD

Object

FixedMirror

PD

Object

FixedMirror

PD

Object

FixedMirror

h

hh

Case I Case II Case III

JILAg Absolute Gravimeter

Laser

g

First absolute gravimeter with multiple copies (6 units produced ca. 1985)

FG5 Gravimeter

State of the art absolute gravimeter: 2005

FG-5 Principle of OperationVacuum Chamber

Interferometer

Stationary LowerMirror

Upper Mirror

DetectorA freely falling reflective test mass is dropped in a vacuum. This causes optical fringes to be detected at the output of an interferometer. This signal is used to determine the local gravitational acceleration.

Freefalling

InterferenceLaser

FG5 Specifications

Accuracy: 2 μGal (observed agreement between FG5 instruments) Precision: at a quiet site, 10s drop interval, 15-30μGal/√Hz

~1 μGal in 3.75 minutes or 0.1 μGal in 6.25 hoursOperating dynamic range: World-Wide Operating temperature range: 15°C to 30°C

FG5 Schematic Laser is frequency-stabilized He-Ne laser (red light @ 633 nm)

Interferometer splits beam into test and reference beams

The test beam bounces off falling corner cube then off stationary spring corner cube

The reference beam travels straight through interferometer.

Beams are recombined and interference signal (fringes) is used to track falling test mass

The time intervals between the occurrence of each fringe are measured by a Rubidium oscillator

FG5 SubsystemsDropping ChamberSuperspringInterferometerLaserElectronicsSoftware

Real-Time Data AcquisitionPost-Processing Data Analysis

FG5 Dropping ChamberDrag Free Cart Mechanical DriveVacuum system (Ion Pump 10-6 Torr)Test Object (ball&vee contacts)

Corner CubeLock Mechanism

Drag-free chamber

Mach-Zenderinterferometer

Ion pump (always connected)

Dropping chamber(Vacuum ~ 10-4 Pa⇔ 10-9 atm. )

Drag-free Dropping Chamber

Reduces drag due to residual gas molecules

Follows the dropped corner cube, gently arrest and lift it

Shields the corner cube from external electrostatic forces

FG5 Superspring60s PeriodTwo Stage nested spring systemSphere DetectorCoil transducerLock MechanismTemperature compensationSpring height adjustmentBubble level adjustments

Delta rodsZeroing the sphere position (S-shaped response)

1 km

Superspring Isolation

F

Practical RealizationNested Spring Systemfree period ~ 60 sHeight: 30cm

x

Ground

1

2

Passive Spring (60s ⇒1km)

Active Spring Concept

The SuperspringPivots

Aneroid

CornerCube

CoilMagnets

Sphere

1st StageSpring2nd StageSpring

The superspring: long period isolation spring that provides the inertial reference frame

Inertial reference corner cube

The Superspring

… and with the superspring

Without the superspring ...

Measurement Scatter

Interferometry MAX

Laser

PhotodiodeB.S.

Fringes

Michelson’s interferometer

min

time recorded (w.r.t. rubidium oscillator) at each minimum creating (t,d) pairs at every λ/2

0 1 2 3 4 5

4

2

0

2

4

5

5−

sin 2πx2( )

50 x

fringe signal sweeps in frequency as test mass falls under influence of gravity

λ/2

FG5 InterferometerMach-Zender typeInsensitive to rotations and translationsThree optical outputs

Main signal interferometer (APD)Telescope (verticality and/or beam alignment)Viewing port

Two Electronic SignalsAnalog (Alignment)TTL (Timing)

WEO Iodine Stabilized LaserPrimary Standard (BIPM Certified)Stabilized to rotational states ( hyperfine splitting) of iodineAccuracy at 1 part in 1011

Automatic peak lockingFiber launching system

Faraday Isolator (prevents feedback into laser)5-axis stagePolarized fiberOutput collimation (~6mm)

Operating Temperature: 15 – 25 °C

Fringe = l/2 xiFor each xi , a measured time ti, The following function is fitted to the data xi , ti :

γ is the vertical gravity gradient (~3 µGal/cm),c the speed of lightx0 the initial positionv0 the initial velocityg0 the initial acceleration

cxxtt i

i)(~ 0−

−=xi , ti , i = 1, …,700

g Determination

2

~~ 20

00i

iitgtvxx ++=

2

~20 itxγ

+ 40

30

~241~

61

ii tgtv γγ ++

Windows BasedGraphics packageGravity correctionsEarth Tide ModelsOcean Load CorrectionStatistical analysisReal time data acquisitionPost processing

g Gravity Acquisition and Processing Software

g Software control

Site SpecificationInstrument ParametersData Acquisition ParametersGravity CorrectionsGraphicsReports

Site SpecificationLatitudeLongitudeElevation (std pressure)Gradient (-3.1 μGal/cm)Polar Motion

Data Acquisition ParametersNumber of drops/setNumber of setsInterval between drops

(normally 1s)Start time of data acquisitionProjects (sets of sets)

g Input Parameters

Gravity Corrections & Error Sources

Gravity CorrectionsEarth TidesOcean LoadingBarometerPolar motionGradientSpeed of Light

Common Error SourcesVerticality: 9 arcsec = 1μGal“1 spot” = 4μGal

Environmental ErrorsWater Table: 2.5 cm = 1μGalAir Pressure: 1mBar = 0.3-0.4μGal

Common Gravity Corrections

Earth tides (+/Earth tides (+/-- 150 150 µµGals)Gals)

Ocean loading (+/Ocean loading (+/-- 3 3 µµgal in Brussels)gal in Brussels)

Barometric variations (Barometric variations (-- 0.3 0.3 µµgal / hPa, i.e. 9 gal / hPa, i.e. 9 µµGal Gal for a typical 30 hPa variation )for a typical 30 hPa variation )

Polar motion (+/Polar motion (+/-- 5 5 µµgal)gal)

Residuals

MeasurementsBest FitResiduals

Note: vertical scale exaggerated, normal residuals are approximately 1nm.

0 2 4 6 8 100

20

40

60

80

100100

0

t2 runif 1 4−, 4,( )0+( )− 100+

t2− 100+

t2− t2 runif 4 4−, 1,( )0+( )+ 50+

100 t

Simple Statistics: “How much data should I take?”

dropsstat N/σδ =

•First, some definitions:• σ = drop scatter (standard deviation of measurements)• δstat = statistical uncertainty • δsys = systematic uncertainty (“built in” system uncertainty and model uncertainties)• δtotal = sum, in quadrature, of statistical and systematic uncertainties

22statsystotal δδδ +=

•Measure drop scatter, σ•Pick your desired statistical uncertainty, δstat •This determines Ndrops•Spread this Ndrops over a convenient number of sets.

Remember the balls & vees: only run as long as you need to!

Simple Statistics (cont)FG5: ~2μGal Systematic Uncertainty

Example: •Drop scatter = 15μGals•2μGals statistical uncertainty => ~100 drops

•For 100μGal scatter (noisy site!) => 2500 drops total•Lifetime ~250,000 drops => 100 site occupations

“Typical” Good Data Set

Overnight Run

•100 Drops/set

• New Microg Site

•HzGal/ 23

min 17Gal/ 74.0

μ

μ

=