past-present & future of absolute and relative gravimetry
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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
=
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