THE FREE-FALL DETERMINATION OF THE
UNIVERSAL CONSTANT OF GRAVITYbyJOSHUA PETER SCHWARZ
B. A., University of Massachusetts,1993
M. S., University of Colorado, 1996

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial ful llmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
1998

This thesis entitled:
The Free-Fall Determination of the Universal Constant of Gravity
written by Joshua Peter Schwarz
has been approved for the Department of Physics

James Faller

Douglas RobertsonDavid Bartlett

John HallDate
The nal copy of this thesis has been examined by the signatories, and we ndthat both the content and the form meet acceptable presentation standards ofscholarly work in the above mentioned discipline.

iii
Schwarz, Joshua Peter (Ph. D.,Physics)
The Free-Fall Determination of the Universal Constant of Gravity
Thesis directed by Professor Adjoint James Faller
The Newtonian constant of gravity, G, is the constant of proportionalitythat scales the magnitude of the gravitational force between masses. G is the least
precisely known of all fundamental constants. Recent high-precision experiments
to measure G have produced highly discordant results, with values spread over arange of 0.7%. The situation is almost incredible; the second digit of the value isin contention and experiments disagree by as much as 50 of their individual error
estimates.
We have developed a new determination that uses an absolute free-fallgravimeter to sense the gravitational attraction between a known source mass
and a falling test mass. We use a one-half metric ton source mass whose position is alternated from above to below the dropping region, either decreasing or
increasing the observed acceleration of the test mass. The source mass generates
a di erential signal of 8 parts in 107 of the local acceleration due to the Earth.
This method is independent of many of the systematic errors associated with thetraditional torsion balance experiments upon which the accepted value of G isbased.
Using this \free-fall" method in two data runs of approximately two weeks,
we have found a value of G at the level of 1100 ppm. Our value is: G = (6.6873 
0.0076) X10,11 m3/kg-s2. This value lies in the approximate center of the rangeof recent results

ACKNOWLEDGEMENTS

Many people have o ered assistance with my experiment. Tim Niebauer
originally suggested the method of using the gravimeter to measure the acceleration signal of a source mass to determine G. But there is no doubt in my mind
that the person most responsible for the free fall experiment was Jim Faller. Jim
provided the energy and drive to get the experiment ying, and in many respects
drew me along for the ride. His was the insight that lead to our source mass
design, and also provided an example of the type of thought that I most enjoy
attempting.
Doug Robertson was like a second advisor to me. While Jim discussed (andargued) di erent points of apparatus with me, Doug and I worked closely on thesoftware framework that was crucial for the experiment. Doug o ered liberalconstructive criticism on my rough drafts, and good ideas about the experiment.
In more practical ways the instrument makers John Andru, Dave Alchenberger, Jim Csotty, Blaine Horner, Cole Briggs, Seth Wieman, Hans Rohner, and(especially!) Hans Green helped me (or did I help them?) with the design of thelifting assembly. I certainly was no help to Hans Green or Blaine when they did
the actual work.
I didn't understand the purpose of the supply oce until it was too late.
That didn't stop Ed Holliness, Maryly Dole, Brian Lynch and David Tegart fromdefying all the laws of physics I learned about to get me everything I desperately

v
needed (UPS RED!) on time. Ed and Brian also helped me work on my ping ponggame and throwing arm. In this capacity they boosted my self con dence to theheights required to submit a dissertation.
The last two paragraphs have mentioned many people, yet it only covers
the rst oor of JILA. The Computing and Electronics Department is on thesecond oor. Terry Brown, Joel Frahm, Alan Dunwell, James Fung-A-Fat, Paul
Beckingham, Mike Whitmore, Ralph Mitchell, Nick Metro, Peter Ruprecht, andChela Kunasz have helped me with all my electronics and computer needs. Terry
Brown, especially, worked hard to ll my electronics-empty skull with ideas about
the discriminator circuit of the gravimeter (Chapter 3). He also helped me withmy arm, and kindly ran patterns for a rookie QB. James kept me honest by toying
with me on the raquetball court. He would have ruined my self con dence, if I
had been so foolish as to challenge him while working on this thesis.
The Scienti c Reports Oce is also on the second oor. It's in a corner inthe building, where the Jewels are kept. Laurie Kovalenko and Marilee DeGoede
waded through so many sentences like this one that I can't help su ering fromfeelings of deep remorse | although I also experience feelings of thanks andadmiration because they were both such good writers, knowledgeable like I am
not, and feelings of regret because I'm leaving and won't be able to depend on
their help again, never ever, ever any more | maybe.
Outside of JILA the kind folks at Micro-g Solutions, Inc. were a great help
to us. They worked hard to help us get the experiment up and running quickly
so that we could capitalize on the gravimeter time were allocated. Tim Niebauer,
Fred Klopping, Kip Buxton, Jess Valentine, Jerry Gschwind and Echo never let
us down. I'd especially like to thank Fred and Jess, who were always very nice tome even though I was needy.
From NOAA and the Table Mountain Gravity Observatory J.D. Williams,

vi
Tonie van Dam, Dan Winister and Glenn Sasagawa helped us wade through redtape on our way to borrowing the gravimeters we used.
I've thanked the people who have helped with the experiment, and now
I'd like to give a hearty handshake1 to all the people who have helped me tomake it through | not by pointing out systematic errors in our experiment, or bysuggesting alternate data analysis techniques, but by helping with \the life part".
Many people I've already mentioned fall into this class. Jim Faller, for example,
supported my interests in subjects apart from physics. We had many enjoyable
conversations about Music, Sport, and Fine Dining that helped me keep proper
perspective. Every one in the instrument shop, too, always made me feel welcome
to shoot the breeze, have some popcorn and join up for a ride up Flagsta .
Lisa Perry, LaReina Romero, Cheryl Glenn, Fran Haas, and even Leslie Haas
were almost always available for a quick chat whenever the mood arose. Cherylhas su ered through hummed melodies from my singing classes, while Lisa wasalways quick to demonstrate proper technique (not just in singing).
Before there was JILA there was the Department of Physics, and classes.
Both Susan Thompson and Linda of \The Department" made me feel like morethan a cog in the machine. Somehow they also made sure that I ful lled thebureaucratic requirements of the grad school | two tough jobs.
Without Harold Volkie Parks, Darren Link, and Greg Rakness I never would
have scraped by. I needed three full-time tutors, and luckily that's what I got.
These three even helped feed me, by working out a system where I was taken to
lunch by everyone who got better exam scores than mine. Such generosity of time,knowledge and food is too rare. Nevertheless I found similar generosity (well, twoof three. . . ) in all my classmates. I think that we were all lucky in our peers.
That leaves all the special people. I hope they know who they are.
1

I had \hug" here, but the rst person I started to write about was Jim. . .

CONTENTS

CHAPTER
1 INTRODUCTION
1.1 G-Whiz: The Current Situation . . . . . . . . . . . . . . . . . . .
1.2 The Free-Fall Method . . . . . . . . . . . . . . . . . . . . . . . . .

2 THEORY
2.1 Mutual Attraction of the Free-Falling and Source Masses . . . . .
2.2 Theoretical Determination of the Trajectory . . . . . . . . . . . .
2.3 The Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 EXPERIMENTAL APPARATUS
3.1 The Measurement System . . . . . . . . . . . . . . . . . . . . . .
3.1.1 The Dropping System . . . . . . . . . . . . . . . . . . . .
3.1.2 The Interferometer System . . . . . . . . . . . . . . . . . .
3.1.3 The FABIO system . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Phase Shifts at the Discriminator Input . . . . . . . . . . .
3.1.5 The Position Reference System . . . . . . . . . . . . . . .
3.2 The Source Mass . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 The Tungsten Masses . . . . . . . . . . . . . . . . . . . . .
3.2.2 The Drive/Support Structure . . . . . . . . . . . . . . . .
3.3 Superconducting Relative Gravimeter . . . . . . . . . . . . . . . .

1
6
10
15
16
20
22
26
26
27
31
33
34
36
38
43
44
47

viii

4 ANALYSIS
4.1 A Big Least Squares Solution . . . . . . . . . . . . . . . . . . . .
4.2 The Method of Parallels . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Di erencing Techniques . . . . . . . . . . . . . . . . . . .
4.3 Dependence of go on First Fringe Fit . . . . . . . . . . . . . . . .

5 ERROR ANALYSIS
5.1 General Error Analysis . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Common mode errors . . . . . . . . . . . . . . . . . . . . .
5.1.2 Positioning Errors . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Modeling Errors . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 Numerical Techniques . . . . . . . . . . . . . . . . . . . .
5.1.5 Spurious Signals . . . . . . . . . . . . . . . . . . . . . . . .
5.1.6 Contaminated Data . . . . . . . . . . . . . . . . . . . . . .
5.2 Sources of Systematic Error . . . . . . . . . . . . . . . . . . . . .
5.2.1 Air Gap Modulation . . . . . . . . . . . . . . . . . . . . .
5.2.2 Thermal Signals . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Systematic Laser Excitation . . . . . . . . . . . . . . . . .
5.2.4 Laser Verticality and Floor Tilts . . . . . . . . . . . . . . .
5.2.5 Ball Wear . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.6 Phase Coherent Excitation of the Super Spring . . . . . .
5.2.7 Magnetic E ects - Static Fields . . . . . . . . . . . . . . .
5.2.8 Magnetic E ects { Dynamic Fields . . . . . . . . . . . . .
5.2.9 Contaminated Fringes . . . . . . . . . . . . . . . . . . . .
5.2.10 Unquanti ed Error Sources . . . . . . . . . . . . . . . . .
5.3 Evidence of other Systematic Errors . . . . . . . . . . . . . . . . .
5.3.1 Residual Analysis . . . . . . . . . . . . . . . . . . . . . . .

51
51
54
55
57
62
62
62
64
67
68
69
69
70
73
82
88
88
91
92
94
96
101
104
108
110

ix

6 DATA AND RESULTS
6.1 Data Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 The Bronze Mass Experiment . . . . . . . . . . . . . . . .
6.1.2 May 1997 Experiment . . . . . . . . . . . . . . . . . . . .
6.1.3 May 1998 Experiment . . . . . . . . . . . . . . . . . . . .
6.2 Combined Results . . . . . . . . . . . . . . . . . . . . . . . . . . .

113
113
113
118
127
132

7 CONCLUSION

136
7.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.2 Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

BIBLIOGRAPHY141

APPENDIX
A MASS METROLOGY146
A.1 Metrology of the Source Mass . . . . . . . . . . . . . . . . . . . . 147
A.2 Modeling the Test Mass . . . . . . . . . . . . . . . . . . . . . . . 159

B ERROR CALCULATIONS
B.1 Positioning Errors . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.1.1 Source Mass Positioning Errors . . . . . . . . . . . . . . .
B.1.2 Proof Mass Positioning errors . . . . . . . . . . . . . . . .
B.1.3 Extracted Positions . . . . . . . . . . . . . . . . . . . . . .
B.2 Spurious Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2.1 Di erential Magnetic Forces. . . . . . . . . . . . . . . . . .
B.3 Modeling Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3.1 Source Mass Modeling Errors . . . . . . . . . . . . . . . .

166
166
166
167
167
168
168
174
174

x
B.3.2 Proof Mass Modeling errors . . . . . . . . . . . . . . . . .
B.4 Numerical System . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.4.1 Absolute Accuracy of Quadrature: The Integration Overthe Source Mass . . . . . . . . . . . . . . . . . . . . . . . .
B.4.2 Assumption of Cylindrical Symmetry . . . . . . . . . . . .
B.4.3 Di erential Precision of the System . . . . . . . . . . . . .

177
181
181
183
185

FIGURES

Figure
1.1
1.2
1.3
1.4
1.5

Schematic of a torsion balance . . . . . . . . . . . . . . . . . . . .
Schematic of the time-of-swing method. . . . . . . . . . . . . . . .
Recent G results . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of the experimental apparatus . . . . . . . . . . . . . .
Photograph of the experimental apparatus . . . . . . . . . . . . .

4
5
7
13
14

2.1 Acceleration eld of the source mass . . . . . . . . . . . . . . . .
2.2 Perturbation to the free-fall solution. . . . . . . . . . . . . . . . .

24
25

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8

Schematic of the dropping chamber . . . . . . . . . . . . . . . . .
Interferometer schematic. . . . . . . . . . . . . . . . . . . . . . . .
Schematic of the super spring . . . . . . . . . . . . . . . . . . . .
Schematic of the source mass . . . . . . . . . . . . . . . . . . . .
Photo of the source mass . . . . . . . . . . . . . . . . . . . . . . .
View of pivot-balance system . . . . . . . . . . . . . . . . . . . .
Photo of screw drive . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of superconducting relative gravimeter . . . . . . . . .

28
32
37
41
42
45
47
49

4.1 Di erencing methods . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Quality of t for position of source mass . . . . . . . . . . . . . .
4.3 Comparison of di erent SOD results . . . . . . . . . . . . . . . .

56
58
60

xii
4.4 Extracted values of the start-of-drop position . . . . . . . . . . . .

61

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8

66
71
74
75
77
79
81

Change of proof mass acceleration with radial position . . . . . .
Commentary on data exclusion . . . . . . . . . . . . . . . . . . .
Vertical motion of the airgap . . . . . . . . . . . . . . . . . . . . .
Amplitude spectrum of the airgap motion . . . . . . . . . . . . .
Comparison of airgap theory to measurement . . . . . . . . . . .
Weighted motion of airgap interface . . . . . . . . . . . . . . . . .
Scaling of airgap theory to measurement . . . . . . . . . . . . . .
Comparison of airgap motion with source mass in upper or lower
position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Temperature signal in a test for heating errors . . . . . . . . . . .
5.10 Time-binned di erences from the 1998 experiment . . . . . . . . .
5.11 Residuals corrected for laser-bauble signal . . . . . . . . . . . . .
5.12 Pointwise-averaged sets, 1997 and 1998 data runs . . . . . . . . .
5.13 Setup to measure AC magnetic forces . . . . . . . . . . . . . . . .
5.14 Plot of data with missed fringes . . . . . . . . . . . . . . . . . . .
5.15 A test for corrupt data: variation of g with rst t fringe . . . . .
5.16 Plot of suspicious data, set averaged . . . . . . . . . . . . . . . .
5.17 Di erence signal: theory and measurement . . . . . . . . . . . . .
5.18 Residuals-to-drop averaged over 150,000 drops . . . . . . . . . . .

83
85
87
89
94
100
102
104
107
109
111

6.1
6.2
6.3
6.4
6.5
6.6

115
117
117
120
121
122

Photo of bronze mass experiment . . . . . . . . . . . . . . . . . .
Bronze mass experiment, data sample . . . . . . . . . . . . . . . .
Bronze mass experiment, results . . . . . . . . . . . . . . . . . . .
May, 1997 experiment. Individual di erences . . . . . . . . . . . .
May, 1997 experiment, results . . . . . . . . . . . . . . . . . . . .
May, 1997 experiment. Power spectral analysis. . . . . . . . . . .

xiii
6.7 1997 experiment, G as a function of rst t fringe . . . . . . . . .
6.8 1997 experiment, Uncertainty in G related to G values . . . . . .
6.9 Daily results of the 1998 experiment . . . . . . . . . . . . . . . .
6.10 Individual di erences from the 1998 experiment . . . . . . . . . .
6.11 Power spectral analysis of 1998 data. . . . . . . . . . . . . . . . .
6.12 1998 experiment, G as function of rst fringe t . . . . . . . . . .
6.13 Final results of the Free Fall experiment . . . . . . . . . . . . . .

124
125
129
130
131
133
135

A.1 Cylinder arrangement in source mass . . . . . . . . . . . . . . . . 152
A.2 Photograph of the proof mass. . . . . . . . . . . . . . . . . . . . . 164
A.3 Drawing of the proof mass model. . . . . . . . . . . . . . . . . . . 165
B.1 Vertical magnetic eld and gradient, theory . . . . . . . . . . . . 169
B.2 Study of error introduced by proof mass model vertical granularity 178
B.3 Study of error introduced by proof mass model radial granularity 179
B.4 Study of error arising from vertical interpolation grid density . . . 182
B.5 Study of error arising from vertical interpolation grid density on
the radial dimension . . . . . . . . . . . . . . . . . . . . . . . . . 183
B.6 Study of error arising from radial interpolation grid density . . . . 184
B.7 Study of error arising from radial interpolation grid density . . . . 184
B.8 Vertical acceleration eld with the assumption of cylindrically symmetric source mass . . . . . . . . . . . . . . . . . . . . . . . . . . 186
B.9 Errors arising from assumption of cylindrical source mass symmetry 186

TABLESTable
5.1 Summary of major error sources. . . . . . . . . . . . . . . . . . .

72

6.1 Main moments of the 1998 experiment distribution . . . . . . . . 128
A.1 Characteristics of the tungsten cylinders . . . . . . . . . . . . . .
A.2 Density variations in the tungsten cylinders . . . . . . . . . . . .
A.3 Model of the lower support plate, 1997 experiment . . . . . . . . .
A.4 Model of the upper plate, 1997 experiment . . . . . . . . . . . . .
A.5 Main model of the source mass, 1997 experiment . . . . . . . . . .
A.6 Model of the support plates, 1998 experiment . . . . . . . . . . .
A.7 Main model of the 1998 experiment . . . . . . . . . . . . . . . . .
A.8 Model of the Top Hat cap of the proof mass. All dimensions (position, height, rinner , routter , o set) are in centimeters. The masses
are in grams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.9 Model of proof mass: Tungsten balls, glass sphere, top hat cap. All
dimensions (position, height, rinner , routter , o set) are in centimeters. The masses are in grams. . . . . . . . . . . . . . . . . . . . .
A.10 Model of proof mass: Corner cube holder and counter weight. All
dimensions (position, height, rinner , routter , o set) are in centimeters. The masses are in grams. . . . . . . . . . . . . . . . . . . . .

150
150
154
155
156
157
158

160

160

161

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A.11 Model of proof mass: Test mass body, testmass nut and legs. All dimensions (position, height, rinner , routter , o set) are in centimeters.
The masses are in grams. . . . . . . . . . . . . . . . . . . . . . . . 162
A.12 Model of proof mass: Corner Cube. All dimensions (position,height, rinner , routter , o set) are in centimeters. The masses are
in grams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
B.1 Uncertainties in proof mass position and orientation . . . . . . . .
B.2 A simple proof mass model . . . . . . . . . . . . . . . . . . . . . .
B.3 Magnetic Field of the drive motor. . . . . . . . . . . . . . . . . .
B.4 Calibration of the voice coil. . . . . . . . . . . . . . . . . . . . . .
B.5 Calibration of the voice coil. . . . . . . . . . . . . . . . . . . . . .
B.6 Errors due to magnetic forces . . . . . . . . . . . . . . . . . . . .
B.7 Uncertainties in proof mass position and orientation . . . . . . . .
B.8 Uncertainties in proof mass position and orientation . . . . . . . .
B.9 Errors due to the numerical system. . . . . . . . . . . . . . . . . .

167
170
172
173
174
175
177
181
185

CHAPTER 1
INTRODUCTION

The world we live in is largely de ned by gravity. When we see the wavesof the ocean below or the sky above we are conscious of the e ects of this far
reaching force. The atmosphere we breathe, the light of the sun, the fall of rain,
the shape of a ame | all are consequences of the attraction of matter.
For my ally is the Force. And a powerful ally it is. . . Its energy
surrounds us and binds us. . . You must feel the Force around you.
Here, between you. . . me. . . the tree. . . the rock. . . everywhere!
Yes, even between this land and that ship!
| Yoda, The Empire Strikes Back, George Lucas
The force that de nes the arrangement of the stars above us is a subject
well deserving of appreciation, contemplation, and study. It is no wonder that for
thousands of years questions about gravity have occupied the minds of the curious
and contemplative.
Sir Isaac Newton provided the rst model that we of the modern world citewhen considering the e ects of this powerful attraction, in a description that allphysicists learn in the formative years of their studies:
F = GMmr2
Just this snippet of information is very valuable, for it forms the basis of muchknowledge of how things interact { how the forces of gravity vary with mass anddistance.

2
Without the universal gravitational constant, G, it is not possible to scale
the force { to calculate the gravitational attractions between objects, even if theirmasses and relative positions are known. Although the practical bene ts of performing this calculation are far and few between, the value of G is of real interest
in some areas of physical research. Geophysicists, geologists, and seismologists
use the value of G to deduce information about the structure and makeup of
the Earth. Cosmologists and astronomers, on the other hand, base information
concerning astronomical structure and the early universe on G[1]. Theorists who
work on uni cation the four known forces need to know G to gauge the success of
their theories.
It is the search for this constant that forms the basis of this dissertation, andhas motivated many experiments since the late eighteenth century. Experiments
to measure G have all had to deal with the same set of tough issues. Most
important is the requirement that the experimental system be very sensitive tothe gravitational eld of a well-known mass, yet simultaneously be very insensitive
to all other forces. This is a dicult demand to meet because the force of gravity
is so weak in comparison to other forces. The following calculation illustrates thispoint. The accepted value for G, as of 1998, was

G = (6:6726  0:00085)  10,11 m3=kg , s2

(1.1)

based on the 1986 adjustment of standards. We can use this value to estimate
the gravitational attraction between two average sized people (Jack and Jill) in aclose hug:Jack )(MassJill )F = G  (MassDistance2

which may be approximately,11 3
 s2 )(65kg)  (50kg) = 2:4  10,7 N
F = (6:7X 10 m =(0kg:3m)2

3This is equivalent to the weight of three tenths of one billionth of a gram (luckily
gravity is not the only attractive force!).
We don't generally think of gravity as weak because we associate it with thepull of the entire Earth. Yet other forces can easily swamp gravity on many lengthscales. An example is the electrostatic force; the minute charge picked up whenyour feet scu a rug is enough to cause slips of paper to stick to your hair, defying
the gravitational attraction of the whole world. A more quantitative comparison
can be made between the magnitude of the electric attraction of a proton for an
orbiting electron and the gravitational force between them. The electric force is1040 times larger.
Another hard fact of G-measurements is that gravity can't be shielded. If I
wished to produce a region that was free of electric elds, all I would have to do isconstruct a conducting shell around the region. If I wanted to exclude magnetic
elds, I could make the shell super conducting. If I need to exclude gravitationalelds, then I had better think of a di erent experiment!
For this reason G experiments are designed so that it is not necessary tohave a gravity-free environment for their success. Most experimenters, however,would like to circumvent one major component of the gravity eld that exists on
Earth { namely the attraction due to the planet itself. Experimentalists havecommonly attempted to do this by using a torsion ber to support a dumbbellshaped beam. This allows horizontal forces to be expressed in twists of the ber
support without strong dependence on the local acceleration.
Fibers have been most commonly used in torsion balances, rst conceived by
the Minister John Price and used by Henry Cavendish in his famous determinationof G [2]. The torsion balance is used by measuring the shift of the equilibrium
position of the beam due to the gravitational attraction of external masses (Figure1.1). Because bers twist very easily even small gravitational forces produce a

4

 . . .

.
...
.
.
x. . . . . . . . 
x Beam...
w . . .
..

Fiber





.

..
. . .,

..
...

..

x 


..

Figure 1.1: Schematic of a torsion balance. A dumbbell-shaped beam is hungfrom a torsion ber. External source masses (hollow circles) de ect the beamfrom its equilibrium position (dotted line). The de ection changes depending on
where the source masses are placed.
signi cant shift in the equilibrium position.
To relate the change in equilibrium position to the value of G it is necessary
to calibrate the torque produced by the twisted ber. This is typically doneby measuring both the oscillation frequency of the system without the source
masses present and the moment of inertial of the beam. These two measurements
can be combined to give a value for the torsion constant of the ber. The torsion
balance is sensitive to more than just the gravitational signal of the source masses.
Changes in the ber can a ect both the equilibrium position of the beam and thenatural frequency of the isolated system, skewing the results of the experiment.
A very similar experimental setup can be used in a di erent type of measurement called a \time-of-swing," \torsion pendulum," or \Heyl-type" experiment[3].
This method requires that the frequency of oscillation of the beam be recorded.
Source masses add a gravitational torque that acts on the beam either to enhanceor decrease the e ective torsion constant of the ber (Fig. 1.2). By quantifying
the e ect of the masses on the period, the value of the gravitational constant canbe found.

5
The vast majority of precision G experiments done since Cavendish haveused either torsion balances or torsion pendulums. Over the ninety years before
the 1986 setting of the accepted value, more than three quarters of all G measurements used one of these two systems. Of the three most accurate experiments
before the reassesment of the accepted value in 1986, two (Luther and Towler[4] and Sagitov et al.[5]) used torsion pendulums, while the third (Karagioz andIzmailov ([6])) used a torsion balance. The accepted value of the constant, today,
is based on the Luther and Towler number. More complete discussion of the measurement of G in the last 200 years may be found in Gillies[7], Bagley [8], andKoldewyn [9].

}


}




} }


B

A
Figure 1.2: A schematic of the time-of-swing method. A torsion pendulum (in its
equilibrium position) is represented by the dumbbell-shaped rod. represents thetorsion support. The hollow circles represent source masses. A) The source masses
gravitationally augment the restoring torque of the ber, increasing the frequencyof oscillation. B) The source masses decrease the frequency of oscillation.
Recently ber-based torsion experiments have come under attack. They
are subject to a systematic e ect that was rst recognized in 1995 by Kazuki
Kuroda[10]. The error arises because the anelastic relaxation e ects in twisted
bers change the torsion constant of the bers; real bers diverge slightly fromthe simpler model of twisted members that had been used until Kuroda released

6
his ndings. This systematic error a ects both torsion balance and pendulum
experiments. Continuing research on this characteristic of bers is ongoing [11, 12].
There are also plans to operate a torsion pendulum at cryogenic temperatures, toreduce anelastic e ects [13].

1.1

G-Whiz: The Current Situation

The dominance of torsion ber based determinations of G, coupled withuncertainty about ber anelasticity, lead to four new experiments that have produced results in the last three years (Figure 1.3). The results vary over such alarge range (0.7%) that the probability that the disagreement is due to a statisticaluctuation is approximately one in 518 .
The rst measurement was conducted by Walech, Meyer, Piel and Schurr,
at the University of Wuppertal, Germany [18, 19]. The group used two simple
pendulums to support mirrors de ning a microwave Fabry-Perot resonator. By
placing a source mass system on the line between the pendulums, the distance between the two bobs was changed. The resonator measured the separation change,
thereby scaling G. Their initial result[18], released in 1995, was:G = 6:6719  0:0006  10,11m3=kg  s2
This value was the only recent result consistent with the accepted value. In 1998
large systematic errors associated with the positioning of their source masses cameto light. When they correct their previous result to re ect these errors, their valuechanges to:
G = 6:6637  0:0004  0:0044  10,11 m3=kg  s2
This value lies 2 standard errors below the accepted value [16].
Fitzgerald and Armstrong of the New Zealand Standards used a nulled torsion balance [15]. This is a normal torsion balance experiment with one important

7

6.73

6.72
PTB

G X 1011 m3/kg−s 2

6.71

6.7

6.69

6.68

6.67

Luther, Towler

Wuppertal

Swiss

New Zealand
6.66

6.65
ResultFigure 1.3: Four recent results of G experiments. The 1982 result of Luther andTowler at the NBS [4] is the basis of the accepted value. The results of the PTB(German Standards, 42 high [14]), the New Zealand Standards (10 low [15]),
and a group at the University of Wuppertal (2 low [16]) were all released in 1995.
A group in Switzerland released a preliminary result in 1998 (2 high [17])8di erence | the torque of the source masses was not matched by the twist in theber, but rather by a torque produced by electrostatic forcers. The forcers wereservoed to maintain the orientation of the beam. This method has the advantagethat the orientation of the beam doesn't change, so spatially variant horizontal
gravity gradients do not in uence the results. Also, this type of system is not
subject to ber anelasticity problems. The major hurdle with this method is in
the calibration of the torque produced by the forcers. Fitzgerald and Armstrong
released a result in 1995:
G = 6:6656  0:00063  10,11m3=kg  s2
This result is 11 standard errors ( about 0.1% ) below the accepted value.
At the Physikalisch-Technische Bundesanstalt (PTB)1 de Boer, Haars andMichaelis have carried out a similar experiment for which planning began in 1976
[14, 20]. They used a compensated torsion balance with a mercury bearing, rather
than a ber, to support the beam. This type of bearing is nearly free of static
friction. The experimenters made measurements of G at di erent distances andbetween di erent materials. These measurements gave consistent results. In 1995
the nal result of their measurement was released:
G = 6:7154  0:00055  10,11m3=kg  s2
This represents a disagreement with the accepted value of more than 40  (0.6%high).
A group at the University of Zurich, consisting of J. Schurr, F. Nolting and
W. Kundig [17, 21] is currently measuring the di erence in the weight of two
masses2 . They modulate this di erence by moving two source masses around the
The PTB is the German Bureau of Standards
This method is similar, in some respects, to our own \free fall" method, and shares manyof its bene ts. The di erence, for example, is measured in the vertical direction, so this determination is made with the perturbing gravitational attraction in-line with the attraction of the
Earth.
12

9
two weights. The group uses two large asks containing 13.5 tons of mercury as
their source masses. They released a preliminary result (using water instead of
mercury in the asks) in February of 1998:
G = 6:6754  0:0015  10,11m3=kg  s2
This result is 2 above the accepted value.
Let us summarize the situation. Many experiments have been e ected recently, yet at the level of a quarter percent there is no clear value for G. The newexperiments use a variety of methods, most moving away from twisted ber ideology, but their results fail to converge. The large spread in results compared tosmall error estimates, as shown in Fig.1.3, indicates that there are large systematic
errors in at least two of the three high precision results.
A need for experiments that are independent of the torsion/twisted support
methodology exists. Consider that the two values with the largest discrepancies
with the accepted value are based on similar experiments using di erent torsion
supports (PTB and New Zealand). Also recall that the great majority of G
experiments relied on a twisted ber torsion approach to the measurement, amethod subject to systematic errors that are only now becoming understood.
The \Free Fall Measurement of G" is presented in this thesis. The free
fall measurement is unique in that it uses an unsupported test mass to sense thegravitational force. It senses this force in a direction parallel to the acceleration
due to the Earth (no torque). This method, therefore, has a very di erent setof systematic errors than other G-experiments. We hope that our results will
add useful information in the current situation of nearly one-half a percent of
uncertainty in G.

10

1.2

The Free-Fall Method

The free-fall method depends on our ability to measure the amount that an
external source mass changes the acceleration of a freely falling object. First asource mass is placed above the region in which the \test mass" falls. Here thegravitational pull of the source mass acts in the opposite direction to the attractionof the Earth, decreasing the downward acceleration of the test mass. Second
the source mass is placed below the drop region, where it augments the Earth's
attraction and the acceleration of the falling mass. If the change in acceleration
can be measured accurately, and if the geometry of the source and test masses arewell known, then we can determine G.
This is conceptually simple, yet in practice the subtleties of the measurement
are plentiful. These subtleties arise from the lack of an ideal mechanism to measure
the acceleration of the test mass. The acceleration signal produced by our source
mass is only 1 part in 107 of the signal of the Earth. Thus we need to precisely
measure a small acceleration signal on top of a huge o set. To achieve an accuracy
in our measurement of 0.01%, we must have a precision of 1 part in 1011 in themeasurement of the absolute acceleration of the falling object. This is not an easytask.
Fortunately free-fall gravimeters have been developed to precisely measure
the local acceleration of gravity, g. The gravimeter drops an object and records its
position as a function of time. By tting the position/time curve with a parabola
the acceleration of the test mass is extracted.
Working at JILA we were perfectly poised to capitalize on the utility of
free-fall gravimeters for the G measurement. The gravimeter we used, a Micro-g
Solutions, Inc. model FG-5, grew out of free-fall gravimeter systems that weredeveloped in JILA by Jim Faller[22], James Hammond, Robert Rinker[23], Mark

11
Zumbergezumberge, and Tim Niebauer[24]. Tim originally suggested the idea for
the free-fall measurement of G, and is the head of the Micro-g Solutions, Inc.
Thus we had, at our elbows, a group of people with tremendous expertise withthe gravimeter system. Also the University of Colorado, which is associated with
JILA, owns an FG-5 that we were able to borrow for our measurements.
The experiment was carried out at the Table Mountain Gravity Observatory (TMGO), a center for the research and development of absolute and relative gravimeters, 10 miles north of Boulder. TMGO is operated by the National
Oceanic and Atmospheric Administration (NOAA). Running the experiment at
TMGO was crucial to the success of the free-fall measurement because of the exceptional quality (low vibration noise) of the site. Also TMGO is equipped withsecondary measurement systems that were extremely helpful in our experiment.
A free fall gravimeter uses a laser interferometer system to track the motion of the falling test mass. The test mass contains a corner-cube retrore ector
that de nes one arm of a Michaelson-type interferometer. Thus the interferometer
fringe crossings correspond to changes of the test mass's position. By incorporating a precise time standard, a stable position reference, and all the electronics andhardware required to drop the test mass in vacuum, it is possible to record theposition as a function of time of the falling mass. The absolute accuracy of thesystem is 1Gal, or approximately one part in 109 of the acceleration due to the
Earth3 . It is limited by machine-dependent uncertainties arising from (among
other things) residual air pressure in the vacuum can, electrostatic forces, and theattraction of the gravimeter itself.
In the free-fall G experiment we were more interested in the change of theacceleration of the test mass than its absolute acceleration. This meant that wewere not limited by the accuracy of the gravimeter so much as by its precision. We
3

A Gal is a cm/s2 , and a Gal is 1e-6 Gal.

12
operated the gravimeter in a di erential mode that eliminated \common mode"errors in the measurement | errors that were independent of the source mass
position. This allowed us to achieve a precision in our result that is more thantwenty times greater than the absolute accuracy of the gravimeter.
The perturbing gravity eld that changed the acceleration of the falling mass
was produced by a ring-shaped source mass made of 500 kg of tungsten alloy. The
use of a ring-shape gives a tremendous advantage over traditional spherical or solid
cylindrical source masses because the axial acceleration eld produced by a ringhas two extrema. This means that there are two positions at which the eld of
the masses doesn't change with a movement from that position; there are two
positions for the source mass that cause our results to be nearly insensitive topositioning errors. The source mass was located alternately above and below thedrop zone in these two places, as shown in Fig.1.4. The actual experimental setup
is shown in Fig.1.5 with the source mass in its upper position.
Conceptually the experiment was a variant of a classical orbit determination
problem in which our free-falling test mass was the orbiting body [25]. In e ect, we
did a satellite laser ranging experiment in a laboratory. But in contrast to classicalspacecraft tracking problems, where one determines the quantity GM (G times
the mass of the primary planet or star, M), we were able to extract G because
we measured M in the laboratory. The close analogy to conventional spacecraft
tracking problems allowed us to use a number of classical orbit determination
techniques.

13

Figure 1.4: A schematic cross-section of the experimental apparatus. The source
mass was alternately placed at positions A and B at twenty minute (100 drop)intervals.

14

Figure 1.5: A photograph of the experimental apparatus. The source mass isin its upper optimal position around the dropping chamber. The interferometer
looks like a black box located below the dropping chamber. The super spring
hangs below the interferometer. The large aluminum structure was built for theG-experiment, and supported the source mass.

CHAPTER 2
THEORYThe underlying physics of the free-fall determination have been well understood for hundreds of years. Two equations are sucient to classically describe thegravitational interaction between masses. Sir Isaac Newton rst recognized therelationship between the mass of two point particles (M and m), their separation(d), and the magnitude of the gravitational force (F ) between them:

F = GmMd2(2.1)

F = ma(2.2)

Newton's second law,describes the acceleration, a, of a mass m, arising from the application of a vector
force, F. The second law, although only valid in a classical regime of low velocity,
is suciently accurate to describe the action of the free-fall experiment.
These equations allowed us to calculate both the attraction between thetest and source masses, and the theoretical path that the falling test mass was
expected to follow. Although these calculations were developed from a straightforward model, they were nontrivial. The test mass and the source mass wereboth complicated objects, composed of many di erent pieces with a variety ofdensities. Also, the attraction between them depended on their relative positions,which was time dependent.

16
The problem of determining the motion of a falling object is often treatedin spacecraft tracking experiments. We used a number of numerical methods thatwere developed for tracking applications. These included the use of the numericalquadrature of Eqn. 2.1, the numerical integration of the equation of motion,and \Encke's Method" for dealing with di erential equations that include small
perturbative terms[26].
This chapter deals with the mathematical and computational work necessary
to solve for the motion of a test mass falling in the presence of a xed mass. Firstthe calculation of the mutual attraction of the two masses is presented. Second
the determination of the expected path of the falling object is laid to view. These
two steps fully de ne our expectations of the physical behavior of the system. The
manner in which these \great expectations" and the observed data are brought
into agreement is dealt with in Chapter 4.

2.1

Mutual Attraction of the Free-Falling and Source Masses

To calculate the instantaneous force between the source and proof masses
we integrated the gravitational attraction of each di erential element of one mass
to each bit of the other mass. This involved a three-dimensional integral over eachvolume:mp ddt~r2p =Gsp dr~0 dr~0
(2.3)
s p
Vp Vs jr~s0 , r~p0 j2
where the subscripts p and s refer to the proof and source mass respectively, V
is the volume,  is the density, mp is the mass of the proof mass, and ~r is the
position vector. For this theoretical work we used an assumed value (the 1986
CODATA number) for the gravitational constant. The vertical component of theforce is given by:
ZZZ ZZZ
2Gspz0 dr~0 dr~0
mp ddtz2p =
(2.4)
s p
Vp Vs jr~s0 , r~p0 j3
2

ZZZ

ZZZ

17
in which z is the vertical component of ~r. This integral cannot be analyticallysolved for an arbitrary mass con guration, so it was numerically solved. However,by limiting the description of the source and proof masses to collections of rightcylinders, the integral could be partially reduced analytically. This greatly simpli ed the computation of the mutual attraction, and also eased the modeling of
the two masses. Working now in cylindrical coordinates with z representing thevertical position, and ~r the polar vector, the integral for the attraction between
each pair of cylinders was reduced to four dimensions. The vertical integration
over each cylinder of the source and test mass was analytically solved:
X

ZZ ZZ

Cp Cs Ap Ashd2~rs d2~rp Gps ln(zp2 ,zs2 +p(r~p ,1r~s+~)2 +(zp2 ,zs2 )2 )(2.5)

, ln(zp ,zs +p(r~p,1r~s+~) +(zp ,zs ) )+ ln(z ,z +p(r~ ,1r~ +~) +(z ,z ) )
p s
p
s
p
s
i
, ln(z ,z +p(r~ ,1r~ +~) +(z ,z ) )
2

1

2

2

1 2

1

2

2

1

2 2

p1

s1

2

p1

s1 2

p

s

where the sum is over the set of cylinders comprising the proof (Cp) and source(Cs) mass models.  is the horizontal o set between the integration cylinders. The
integrals are over only the ends, A, of the cylinders because the vertical integral
has been analytically solved.
We used Romberg quadrature to execute the integrations. In Romberg
quadrature simple trapezoidal evaluations of the integrations are computed witha variety of step sizes. The set of results is then extrapolated to nd the value
corresponding to an in nitesimal step size. By increasing the number of step values
used in the extrapolation, and recording the progression of the extrapolated value,
limits can be placed on the error of the extrapolation. This method is an example
of \Richardson's deferred approach to the limit." The actual code that we used
for each dimension of the integral is presented in Refs. [27, 28]
The model of the test mass is made of nearly 100 cylinders, that of the18source mass of approximately 60. Thus the simpli ed integral of Equation 2.5
must be computed for approximately 60  100 = 6000 cylinder pairs (for a single
test mass position relative to the source mass). This was an extremely large task
because the integrations are layered; if a single numerical integration requires 20
evaluations of the integrand to converge, then a single four-dimensional integral
would require 204 evaluations of the integrand and the computing cycles required
to execute the integration routine 203 times. To complete our problem an average
workstation1 would take approximately eight months.
To speed up the integration I split the integral into two portions, each performing a set of two dimensional integrals. This allowed the use of an interpolation
grid to break the layering of the integrals. With an interpolation grid the valueof the inner two integrations could be \looked up" for whatever range of variables
the grid covers, avoiding the inner layer integrations2 . Thus, for the outer two
integrals, the integrand doesn't contain a double integral, but only a lookup of
the value for the inner integrals at the appropriate point. If a four dimensional
layered integral is split into two two-dimensional integrations in this way, it would
only require 2X202 evaluations of the integrand, a factor of 200 decrease over a
straight evaluation.
The rst set of integrals were evaluated over the cylinders of the source mass,
returning the magnitude of the vertical acceleration eld produced by them. The
integrals were evaluated along a two-dimensional grid covering the two dimensionsof vertical and radial position with respect to the source mass. Only two dimensions were required (instead of three) because I made the assumption that theHere \average" refers to an HP 730/9000 series workstation. This probably was average inthe mid-1990s.
2 \Lookup," as used here, isn't a rigorous term. It refers to the operation of interpolating
the values associated with the grid points to an intermediate point. I have made the assumptionthat such a lookup takes negligible time | compared to the time required for a double integral
this is very reasonable.
119source mass eld was cylindrically symmetric. This was a reasonable assumption
given the symmetry of both the test and source mass, and was highly accurate.
The completed interpolation grid was, essentially, a description of the source mass
gravity eld.
The second set of integrations were calculated over the volume of the proof
mass, on the interpolation grid. Each di erential bit of mass, m, was subject toa force proportional to the value of the gravity eld, a(z; r) at its position:

F = m a(z; r)Thus the net force3 on the proof mass was calculated.
Alternatively, this set of integrations may be thought of as the averaging of
the acceleration eld over the volume of the proof mass, weighted by the densityof each cylinder of the model. The result of the second set of integrations is asingle number | the acceleration of the entire test mass due to the entire source
mass. The use of the interpolation eld decreased computation time by a factorof 500.
It was necessary to know the in uence of the source mass on the proof mass
throughout the length of the drop. This was found by evaluating the second setof integrations at a sequence of (vertical) test mass positions. The sequence wasused to create a second interpolation array, containing the acceleration of theproof mass due to the source mass. Thus the source mass perturbation, at any
vertical position, was readily available by interpolated from this one-dimensionalgrid.
It was the acceleration of the test mass, not the force acting on it, that interested us. Thus
the force was divided by the net mass of the proof mass, returning the acceleration. Note thatthe acceleration of a falling mass is independent of its mass, but not its mass distribution
3

20

2.2

Theoretical Determination of the Trajectory

Once the gravitational attraction between the source and proof mass wasquanti ed we needed to calculate the theoretical path that the proof mass would
follow. This was done by integrating the acceleration of the proof mass twice withrespect to time. For an object in a constant acceleration eld, ,gz^, the equationof motion is:
d2z = ,g:(2.6)
dt2
When integrated twice with respect to time, the equation gives the path of theobject,(2.7)z(t) = , 12 gt2 + V0t + Z0
with the boundary conditions z(t = 0) = Z0 and z0 (t = 0) = V0.
The actual acceleration eld the proof mass fell through was not constanteither in time nor space. A linear gradient term is the rst order correction to a
constant eld. With such a term the equation of motion becomes:
d2z = ,g + z
(2.8)
dt2
where is the magnitude of the linear gradient | typically on the order of 3 partsin 109 of the local acceleration per centimeter for the gravity eld of the Earth.
Integration yieldsppz(t) = g , 2g (e, t + e t)
(2.9)
p
p
, 2Vp0 (e, t , e t )pp, Z20 (e, t + e t )Alternatively we can substitute Eqn. 2.7 into Eqn. 2.8 to get an approximate
polynomial solution:
43(2.10)
z(t) = 21 g(t2 + 12t ) + V0(t + 6t ) + X0(1 + 21 t2 )

21This approximation is clearly very good over the length scale of this experiment.
It is merely the rst few terms of the Taylor expansion of the hyperbolic functions.
By expanding Equation 2.9 to sixth order in p t, with X0 and V0 set to zero forclarity we get Equation 2.10 with a single correction term, C (t):2 6
C (t) = , 6!gt + O(t8 )Over the length of the drop the di erence between the two parabolas is negligible.
This completes the description of the motion of a particle falling in a uniform
gravitational eld with a linear gradient, but we still must include the e ect of thesource mass distribution. The perturbing acceleration, P (z; G), was numericallycalculated from Eqn.2.3. Adding this term to the equation of motion gives:

d2z = ,g + z + P (z; G)
dt2(2.11)

Clearly P (z; G) depends on both the proof mass position relative to the source
mass and the actual value of the constant of gravity. Eqn. 2.11 must be integrated
numerically because P (z; G) was not analytically described. We used the BulirschStoer algorithm [27], which relies on the Richardson extrapolation technique muchas Romberg quadrature does4 .
To further increase the accuracy of the integration, we used a techniqueknown as \Encke's Method"[26]. The di erential equation 2.11 contains a large
simple di erential equation, Equation 2.6, that is perturbed by a small complicated term, z + P (z; G). Encke's method is simply a change of variables, to allow
separate integrations of the constant eld and the perturbation terms. We de ne as the solution of the unperturbed equation:d2 = ,gdt2

(2.12)

Here we integrate a di erential equation, not a simple integrand, so the intermediate points
used in the extrapolation come from modi ed-midpoint integrations.
4

22
thus  is the second degree polynomial of Equation 2.7. If z is the exact solution
to the whole di erential equation, and z is the di erence between the z and ,then:d2z = d2z , d2 
dt2
dt2 dt2= P (z; G)
Encke's method was used to avoid round-o error introduced by the large position
and acceleration values of the \total solution."

2.3

The Solutions

The numerical integration of the equation of motion allowed us to calculate
the position versus time path of the falling test mass as a function of: 1) The
relative position of the start of the drop relative to the source mass. 2) The
source mass con guration. 3) The proof mass con guration. 4) The value of G.
5) The value of the local gradient, . 6) The initial velocity of the proof mass.
To illustrate the abilities this conferred upon us, I include some plots describing the (theoretical) proof mass motion. Figure 2.1 displays the acceleration
experienced by the test mass due to the source mass. Note that the acceleration
eld is highly symmetric around the center of the source mass (approximately aposition of 10 cm) because of the vertical symmetry of the source mass. There are
two optimal source mass positions in the experiment, corresponding to the extremaof the acceleration curve. The two positions maximize the acceleration signal of
the source mass; this is how the they are de ned. These positions also minimize
sensitivity of the signal to the relative position of the drop and the source mass5 .
Additionally, they minimize signal sensitivity to source mass inhomogeneities andtest mass modeling errors.
5

The signal is insensitive to changes of both the vertical and horizontal position of the drop.

23
Figure 2.2 shows the perturbation to the parabolic path of the test mass
when dropped with the source mass either in its upper or lower position. The
symmetry of these perturbations is somewhat unexpected. It arises because thesource mass was positioned to take advantage of the extrema of the acceleration
curve; the source mass was placed to produce an e ect most like that of a constant eld (this de nes the positional invariance characteristic of the two optimal
positions).
The experiment consisted of quantifying the source mass's perturbation to
the path. This measurement is described in the next chapter, on \Experimental
Apparatus." The task of scaling the theory to experimental data is discussed in
Section 4.

24

40

Vertical acceleration, Gal

30
20
10
U
0

L

<−−−−−−−−−−−−−−−−−−−<−−−−−−−−−−−−−−−−−−−

−10
−20
−30
−40
−40

−30

−20

−10

0

10

Position, cm

20

30

40

Figure 2.1: Vertical acceleration of a test mass due to the source mass. Plottedagainst the test mass position above the bottom of the source mass. A point
mass on axis (  ), 2 cm o axis (|), and the acceleration of the actual proof
mass of the experiment ({{). The acceleration of the test mass was less than thatexperienced by a point mass because of its vertical extent. The lines marked by\L" and\U" indicate the region of the eld that the proof mass fell through when
the source mass was in its lower or upper optimal position, respectively. Note thatthe eld magnitude increased with distance from the axis. The eld is assumed
to be cylindrically symmetric.

25
8

6

Residual, nm

4

2

0

−2

−4

−6

−8

0

0.02

0.04

0.06

0.08

0.1

0.12

Time, seconds
A

0.14

0.16

0.18

0.2

0.03

Residual, nm

0.02

0.01

0

−0.01

−0.02

−0.03
0.02

Time, seconds
B
Figure 2.2: Source mass perturbation (position as a function of time) to thetheoretical trajectory for the freely falling test mass. With the source mass in
its lower (|{) and upper(  ) positions. A) The total perturbation. B) The
perturbation with parabolic components removed.
0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

CHAPTER 3
EXPERIMENTAL APPARATUS

This chapter will present the basic design of the apparatus and some ofits subtleties. We have used di erent incarnations of the experimental apparatus
described here in three data runs. The rst was a proof-of-concept experimentthat used a relatively light source mass. This \bronze mass data run" is describedin Section 6.1.1. Our experience with this preliminary set up inspired the main
design, described in this chapter, and used for a data run in 1997. The data
recorded during this run motivated several modi cations that were made to thegravimeter system and source mass for a nal run in 1998. The nal experimentis discussed in Section 6.1.3.

3.1

The Measurement System

All the utility we needed to drop an object and measure its position as afunction of time was provided by a+
Micro-g Solutions FG-5 Free Fall Absolute Gravimeter. Free fall gravimeters
work by measuring trajectory of a falling object. By tting a parabola to thepath of the dropped object, a value for g is obtained. The FG-5 extends these
simple ideas to high levels of accuracy and complexity. When used in a traditional
measurement of local gravity, the FG-5 is accurate enough to sense the gravity

27change resulting from a vertical movement of a third of a centimeter1 . This isroughly equivalent to a 100 picosecond change in the time necessary to fall twenty
centimeters (the length of the drop). At this level of accuracy there are many
pitfalls in all phases of the measurement.
Many subtleties of the g measurement exist, and most in uence the G results. It is important to note that the gravimeter was designed to measure thelocal acceleration of gravity, not perturbations to it. This fact allowed the Gmeasurement to exploit the precision of the gravimeter, as opposed to its accuracy. We extended the utility of the gravimeter to a regime never before explored,
sharpening our need to understand the system fully.
The description of the gravimeter is spread over three sections covering
the dropping system, the interferometer, and the position reference used by theinterferometer. These are not physical, so much as logical, divisions.

3.1.1

The Dropping System

The dropping system of the gravimeter (Fig.3.1) was designed to drop the
proof mass from a well known and constant position (the \start-of-drop" or \SOD"
position), and allow it to fall without any rotational velocity while shielded from
non-gravitational forces. The system's success is dependent on its ability tominimize the e ects of residual gas in the vacuum, magnetic elds, electrostatic
charges, and thermal gradients. The dropping system is very important to theexperiment. The repeatability of the start-of-drop position, the angular motion
imparted to the test mass, and the alignment of the drop with the chamber in uence the accuracy we can achieve with the free-fall method.
The gravity gradient is approximately 3Gal/cm. The acceleration is falling o because
the drop occurs at a greater distance from the Earth. 1Gal is the advertised uncertainty in theaccuracy of the FG-5.
1

28

Figure 3.1: A schematic of the dropping system. The vacuum can forms a continuous conducting shell around the drop region, reducing electrostatic forces. The
viewing port at the top of the chamber is made of glass with a thin conductive
coating. This drawing is courtesy Micro-g Solutions, Inc.

3.1.1.1

The Vacuum System

The rst level of shielding from gas, thermal, and electrical signals is provided by the vacuum chamber. The chamber is made of aluminum which, besides

29
its conductive properties, is non-magnetic and has high thermal conductivity. It
forms an unbroken conductive shell around the drop region, reducing the magnitudes of stray electric elds. The can's thermal properties helps minimize temperature gradients (a discussion of thermal e ects and their magnitudes may befound in Section 5.2.2). An ion pump at the bottom of the chamber maintains
a vacuum of approximately 10,6 torr, reducing the e ects of air to a level of 0.1
Gal. The vacuum chamber is supported on a thick aluminum \party tray" withthree legs. The party tray clamps onto the chamber to x its orientation.
The dropping can of the FG-5 incorporates a few \bonus features". The
can is equipped with an optically at window at its top. The window provides aport for optical determinations of the position of the proof mass inside the can, or
for measurement of the proof mass rotation during a drop. The window is coatedwith a conductive layer and is grounded to the rest of the chamber.
In the G experiment we used a custom chamber dimensioned so that thesource mass could be positioned as near to the drop as possible. During the data
runs the vacuum can was covered with a layer of nylon mesh and aluminum foil,
decreasing the sensitivity of the dropping system to thermal signals.

3.1.1.2

The Co-falling Chamber

The structure that executes the drop and catch of the test mass is locatedwithin the vacuum can. Three stainless steel ground rods guide an \elevator" that
supports the test mass. This elevator, or \co-falling chamber," is servoed to follow
the test mass during the drop, providing a zero-g environment that minimizes thee ects of residual gas in the chamber. The co-falling chamber provides a secondary
level of thermal, electrical, and magnetic shielding. Because it surrounds the test
mass it does not have a strong e ect on the test mass acceleration ( 1Gal).
The quality of the servo is so poor ( 0.01 mm) that there is no concern that30
there could be a systematic di erential bias in our results from this source.
A simple mechanical system is used to drive the co-falling chamber. A DC
servo motor drives a belt (via a pulley) that moves the lift along its guides. A shaft
encoder registers the angular position of the pulley, and thus a relative vertical
position of the co-falling cart. Before each drop the co-falling chamber rests on
a stop at the bottom of the vacuum can. At this position the shaft encoder isreset. This allows referencing of the vertical position of the cart both at the stop
and at the SOD position2 . The SOD position is an important parameter in the
G experiment, which requires that it be constant3 . Because the shaft encoder isreset at the known position of the stop, there can be no cumulative slip between
the belt and the drive pulley. It is not possible that there be slip between thetime that the shaft encoder is reset and the time that the cart is lifted to the SOD
position. This is because the cart could not consistently track the falling mass if
the belt was signi cantly loose. If the drop is not completely botched, then there
will be no problem with the relatively gentle lift of the cart to the SOD position.
The drop begins with the initialization of the time standard and data storageboards of the gravimeter computer system. The co-falling cart motion is controlled
by several analog servos, each controlling a di erent phase of the movement.
In a 20-30 ms \lift o " phase the co-falling cart is accelerated downwards
at 2g, resulting in a test mass { cart separation of 3 mm. As soon as the drop
starts, the whole dropping chamber rebounds upwards due to the weight change
arising from the cart's acceleration. This rebound motion is discussed in depth inSec. 5.2.1.
The drop continues for 20 cm with the cart tracking the position of the proof
The shaft encoder/motor system is servoed to match a voltage output by the encoder witha reference voltage. The reference voltage de nes the distance from the reference stop to the
SOD position.
3 The SOD position is also important in absolute gravity measurements, but at a lower level
than in the G measurement.
2

31
mass, and ends with a \soft-catch, " in which the cart gently rises to meet theproof mass. The cart and test mass are decelerated and set against the referencestop where the shaft encoder is reset. The cart is then servoed to the SOD position
where vibrations die down until the next drop sequence starts.

3.1.2

The Interferometer System

The measurement of the test mass position during the drop is made withan interferometer. The test mass acts as one arm of the interferometer, resulting
in a correlation between its position and the number of fringes observed at theinterference output. Figures 1.4 and 3.2 contain schematic views of the interferometer.
The light source for the interferometer is a He-Ne laser locked to an optical
hyper ne absorption peak in I2 (red light,  633 nm). The iodine peak de nes afrequency standard so the wavelength of the laser can be used as a length standard
without calibration.
The interferometer output during a drop is a chirped sinusoid in light intensity superimposed with a constant light intensity. The sinusoid arises fromthe motion of the test mass 4 . This signal is monitored with an avalanche photo
diode that converts the intensity signal into a voltage signal, allowing a high-speed
discriminator to identify fringe crossings.
The photo diode voltage is high-pass ltered (with a 700Hz corner) to remove
any DC voltage o set5 . Thus, at frequencies above 700 Hz each transition of the
The time for the test mass to travel a distance corresponding to a fringe decreases is
proportional to its velocity. The velocity increases linearly with time. This is why the sinusoidal
signal is chirped. The constant light intensity occurs because there is a slight di erence in thelight power sent down each arm of the interferometer. Thus only some fraction of the lightinterferes.
5 Voltage o sets are removed to help ensure that the discriminator is triggered when the slew
rate of the intensity signal is maximum. This reduces the sensitivity of the trigger to noise on
the signal.
4

32

Figure 3.2: Schematic of the standard FG-5 interferometer. The drawing lies in
the horizontal plane. Thus the laser beam extends out of the page to illuminate
the proof mass, and into the page to strike the reference mirror. This drawing iscourtesy Micro-g Solutions, Inc.
voltage from positive to negative represents a light fringe. Each fringe representsa shift in length between the two arms of the interferometer of z = 2 . The
number of fringes (actually zero-crossings) that occur are counted, and a scaling
factor, n, is applied to the fringes; the time of each nth fringe is recorded. Becauseof software limitations, the minimum scaling factor is 500. This corresponds toa time/position datum at every 1/3 mm of travel, providing approximately 650
data points for each 20 cm drop. The fact that the data are equally spaced inposition implies that they are chirped in time { the data are more dense at theend of the drop than at its start (in time).

33
The stabilization scheme used to lock the laser requires that its frequency
be modulated over 6MHz at 1.2 kHz [29]. This dither is seen as a high frequency
sinusoidal position signal, but by tting for a sinusoid at the appropriate frequency,
the e ect of the dither on the data is minimized. Ideally the laser light would be
provided without any dither, to avoid the additional processing and tting errors
associated with removing this signal.
As an aside, note that the errors in the measurement arising from the lightpressure of the laser beam on the test mass, time delays due to the nite speedof light, and Doppler shifting of the laser frequency are all common mode errorsthat don't a ect the experiment.

3.1.3

The FABIO system

In a data run in 1998 the interferometer of the gravimeter was replaced witha ber-optic system, called the Fiber Absolute Interferometer (FABIO). FABIO
uses a ber-optic link to pipe laser light to the interferometer arms (the laseris placed a few meters from the dropping chamber). FABIO reduces the number of mirrors used in the interferometer, thereby improving the stability of theverticality of the laser beam.
When we used FABIO a polarization-maintaining (PM) ber was used. To
our surprise, the set-to-set scatter acceleration values doubled. This was especially
unexpected, because the short-term drop-to-drop scatter did not increase. At
the time we suspected the FABIO system, and speci cally the PM bers. Tim
Niebauer suggested a mechanism to explain the increase in scatter at the longer
time scale of sets that involved drifts of the polarization carried by the ber.
Testing of the gravimeter and FABIO system after the G measurement wasover showed that the set-to-set scatter was normal and that there was no dependence of g on the polarization carried by the ber (at levels interesting in absolute34
determinations of g). Thus the choice to use FABIO appears to be unrelated tothe increased scatter. Section 6.1.3 includes discussion about possible sources of
the noise.

3.1.4

Phase Shifts at the Discriminator Input

Frequency dependent phase errors introduced in the fringe signal by theelectronics of the discriminator/APD system result in position/time signals that
may mimic accelerations of the proof mass. Phase shifts could be introduced
by electronic lters or by the limited bandwidth of the APD. In the FG-5 the
APD has a bandwidth of 50 MHz so the phase errors introduced at 6 MHz (themaximum frequency of the interferometer signal) are very small (corresponding
to a error in g of approximately 0.1 Gal).
The phase error problem doesn't a ect the di erential signal unless the error
is dependent on the source mass position. We don't expect any such dependency
because the APD/ lter/discriminator circuit was positioned within the interferometer, far from the source mass and doubly shielded from its direct thermal
signature.
The fact that phase errors don't a ect the G measurement (unless they are
truly gigantic) raised an interesting possibility. By limiting the bandwidth of the
APD circuit we could reduce its statistical noise in the measurement of the fringe
crossings. This would reduce the drop-to-drop scatter in g-values. Truncationof the bandwidth necessarily results in phase bumps in the circuit, but in thedi erential mode, this was not a concern. The lter would have the same e ect
as increasing the data sampling rate6 .
I wrote a software simulation that showed that a four-fold decrease in the
We could not simply increase the data density because of software limitations in the gravimeter system. In the future experimenters would be well served to put in the work required toconvert the software system to a form that would support storage of arbitrarily dense data.
6

35
bandwidth (achieved by placing a low-pass lter with a corner of 12.5 MHz between the APD output and the discriminator input) would cause only a 10 Gal
shift in absolute gravity and negligible bias in our results. If the noise introducedby the large bandwidth of the APD were the main source of drop-to-drop scatterthen this lter would win a factor of close to 2 in scatter, and thus the same
factor in the integration time required to reach a given precision in G. This wasa reasonable expectation because the residuals in least squares ts to drop data
correspond well with the actual scatter observed.
We tested a low-pass lter at 12.5 MHz for use in a data run in 1998. The
scatter did not decrease as quickly as expected. There was no signi cant decreasein the size of the residuals until we used a lter corner frequency of only 8 MHz.
Worried that the lters were acting as antennas and injecting additional noise into
the system, we decided to forgo the use of the lter altogether.

3.1.4.1

The Test Mass

The test mass is a complicated object designed to ful ll several criteria. It
incorporates a mirror, a spherical lens, and a counterweight. The test mass is 9
cm long, approximately 2 cm wide, and is made of aluminum, tungsten, vespel,glass and beryllium copper. The test mass used in the G-experiment weighed
101.2 grams. Di erent test masses generally weigh within a gram of 101 grams.
The test mass includes a corner cube retrore ector. Corner cubes have theproperty that incoming light rays are re ected anti-parallel and o set from theirincoming paths. The re ected ray is insensitive to rotations of the corner cube solong as the rotations occur around a point known as the \optical center" of the
cube. To minimize path length errors (and the associated acceleration bias) due
to the inevitable rotations of the freely falling test mass, the optical center of thecorner cube must as close as possible to the center of mass of the entire falling36
object. Therefore the test mass incorporates a counterweight that is used to alignthe center-of-mass to within 4 microns of the optical center of the re ector.
To allow the co-falling chamber to track the test mass as it falls, the test
mass contains a small optical glass sphere at its upper end. The sphere acts as alens that focuses the light of an LED onto a linear position sensor. Both the LED
and the detector are hard mounted to the co-falling chamber. The output of thisposition sensor is used in the servo loop that controls the drive motor.
Three tungsten balls support the test mass on matching tungsten vees on
the co-falling chamber. Tungsten is used to reduce wear in the balls resulting
from the catch phase of the drop. The balls and vees are placed and oriented to
impart minimal rotational and horizontal velocity to the test mass at the start ofthe drop. The fact that the balls do wear and change the mass distribution within
the test mass is discussed in Sec. 5.2.5.
The framework that holds these components is made of aluminum. The aluminum aids in reducing electrostatic forces, but is subject to eddy-current forces.
Mark Zumberge suggested constructing the proof mass from non-conductive material, but was concerned about increasing the e ects of thermal signals [30]. Theory
and experiment have shown that the eddy-current e ects are negligible in normal
conditions. Because aluminum has low magnetic susceptibility the test mass is
not strongly in uenced by direct attraction to magnetic elds.

3.1.5

The Position Reference System

A stable reference is required to make any position measurement. In the
FG-5 this reference is provided by a \super spring" one-dimensional active isolation unit. The super spring was originally built by Robert Rinker as a dissertation
project with James Faller [23]. Since its conception it has undergone many re nements, nally resulting in the super spring used in the G-measurement. Figure 3.3

37

<
>
<
>
<
>
m1
forcer <
>
<
>
<
>
<
>
<
>
<
>
<
>
m2

       

?
6

Earth

6

D
?

Figure 3.3: Schematic of the super spring one-dimensional isolation system.
shows a schematic of the design of this system.
The super spring may be considered as normal spring that has been fooled
into acting like it is very long ( 1 km). A servo loop applies a force to m1 ,trying to keep D constant (to some level of gain). Thus m1 tracks m2, e ectively
servoing the portion of the spring between the two masses as though it was thelast few coils of a much longer spring. Because the period of a spring depends on
the amount it is stretched (S ) in the following way:
mg = kSm = S
kgr
T = 2 mks
T = 2 Sg ;m2 is isolated from much lower frequency vibration signals than would be lteredby a simple spring system of the same physical length as the super spring.
The super spring supports the reference arm of the interferometer system,and therefore has direct in uence on the level of scatter observed in the acceleration measurements. The e ective period of the super spring is nearly 60 seconds,
so the amplitude of oscillation must be kept under 5 nm to insure that the peak

38
acceleration signal introduced by the reference is less than 5 Gal. The super
spring system succeeds in this goal, an impressive achievement considering that
this corresponds to a 200-fold decrease in the vibratory acceleration signal of theearth at the fundamental frequency of the spring. For information about how thesuper spring reacts to the source mass position see Section 5.2.6.
The super spring is supported below the dropping system by a tripod (thatalso holds the interferometer). There is no contact between the dropping systemand this tripod lest the vibrations triggered by the drop excite the super spring.
At the Table Mountain Gravity Observatory, where our data was taken, there wasadditional isolation of the spring system. The interferometer tripod was placedon a pier of concrete embedded7 in the oor. This reduced coupling of vibrations
through the oor.

3.2

The Source Mass

A hollow cylindrical ring-shaped source mass was rst used in a G experiment by Michael Hulett, working on a senior thesis project under Jim Faller. He
used two bronze rings fashioned from the bushings for a ship's screw propeller.
Later, in the mid-seventies, William Koldewyn8 [31] used this same pair of cylinders in a G experiment that used a magnetically supported torsion pendulum. We
used the identical masses in a preliminary version of our experiment, described in
Section 6.1.1.
Hollow cylinders allow much laxer positioning requirements for G experiments because they produce a gravitational eld with extremum values, as discussed in Sec. 2.3. Figure 2.1 shows the axial acceleration eld for a hollow
cylinder. At a certain distance from the center of the cylinder the eld on axis
The piers extended approximately 5 feet below the oor, and were surrounded with Styrofoam.
8 Koldewyn also worked on his Ph.D. with Jim Faller, at Wesleyan University, 1976
7

39
reaches a maximal value that doesn't change with position. Thus the extrema represent areas in which the eld is changing much more slowly than the traditional
1/r2 roll-o of the eld produced by a spherical mass.
The axial force is also unchanging with respect to radial translations (fromthe axis of the cylinder). This is expected from symmetry considerations | theforce must get either larger or smaller (than it is on axis) as you move o axis inany direction. The axial force increases closer to the inner walls of the cylinder,
so the axis represents a minimum in axial force as a function of radius.
Thus a ring-shaped mass generates a eld insensitive to translations of themass in any direction (at two points). The G-experiment, however, integrates
the eld of the source mass over a nite distance | the length of the drop.
Although the absolute position insensitivity of the eld exists at only two points,
the extrema result in two SOD positions that maximize the perturbation to thetest mass trajectory. These positions correspond to the upper and lower source
mass positions, the one decreasing g, the other increasing it.
These properties of the source mass were essential to the success of theexperiment. Let's examine the situation if we were forced to use a spherical source
mass, and contrived to place it above and below the drop path. The di erential
acceleration of a point test mass due to a sphere placed in two reasonable positions,above and below the dropping region9 is strongly a ected by positioning errors.
A 1.0 mm error in positioning would result in a signal error of 1.5%. The same
positioning error in our ring-mass system (correctly set up) would result in a signal
change of less than 0.003%, 500 times less signal error. The sphere design is noteven an option for the free-fall method.
The inner and outer radius and the height of the cylindrical source mass
For this example, I assume that a sphere of the same mass and density of our source mass
could be placed within 4cm of the top or bottom of the drop.
940
were determined by our requirements for the experiment. If the cylinder were tootall, for example, it would be physically impossible to place the source mass in its
lower optimal point without very drastic modi cations to the dropping system.
On the other hand, if the mass were shorter than the greatest possible length, wewould loose signal-to-noise.
The inner radius of the source mass is roughly inversely proportional to thesignal magnitude | a smaller inner radius increases the signal. This parameteris limited by the size of the vacuum can and the clearance required between boththe vacuum can and the co-falling chamber, and the can and the source mass. To
minimize the inner radius we constructed a thin-walled chamber with a minimumof clearance.
The ring's outer radius a ects both the signal size and the position of theacceleration extrema. This choice of particular parameter was based on boththe ability of a single robust person to setup the mass, and the importance of theadditional signal achieved by increasing the radius. As the ring increases in radius
the signal gain is only linear, but the mass increases with the radius squared.
The approximate dimensions of the cylinder were 50 cm diameter, 27 cm
height, 16 cm inner radius. Our source mass weighed approximately 521 kg. Thisgeometry produced an acceleration signal that peaked at a value a bit larger than
43 Gal. The total signal sensed by the gravimeter was approximately 82 Gal.
This is smaller than twice the peak signal of the source mass because of the extent
of the test mass and the averaging action of measuring the acceleration over thelength of the drop.
Once the approximate dimensions for the ring were xed we concerned ourselves with limiting possible density variations throughout the mass. The moststraightforward manner of determining the density structure within an uncharacterized mass is to chop it into small pieces and measure the density of each41
small bit. The amount of density information extracted depends on the size ofthe bits (minced tells you more than chopped). We decided to split the mass into
12 cylindrical pieces, as in Figs. 3.4 and 3.5.

Figure 3.4: Schematic of the source mass. The source mass is constructed of nonmagnetic materials including tungsten, aluminum, titanium, and bronze. Somestainless steel xtures were also used.
Cylinders were chosen (instead of cubes, or irregular geometries) becauseof their rotational symmetry and amenability to form a ring shape. Cylinders
can be characterized by more than just their density. Linear and angular density
variations can be measured (as discussed in the next Section, 3.2.1).
The 12 cylindrical tungsten billets were mounted in two levels. Cylinders
can be oriented in many ways so we were able to arrange them to best mimic ahomogeneous set; any linear or quadratic gradients in the density over the wholeheight of the source mass were minimized. The gaps between the cylinders werelled with small tungsten rods, increasing the net density of the source mass.
Aluminum plates were used to sandwich the cylinders together. The plates

42

Figure 3.5: Photo of the source mass. The locating ring of the lower plate isclearly visible. Note how little clearance there was between the source mass andthe vacuum can.
had rims with radii three times that of the cylinders (plus  200 m clearance),
insuring that the masses were precisely located in the radial direction. The upper
plate provided additional strength, reducing ex of the bottom plate and deformation of the source mass.
The plates were bound together with tensioning rods and bolts tapped into
the small tungsten rods. Throughout the design of the source mass we tried topreserve as much exibility in mass positioning and support as possible, while
maintaining symmetry to ease modeling tasks. The source mass was originallydesigned so that the top and bottom plate would be interchangeable, preserving
its vertical symmetry. As the experiments progressed and modi cations weremade this symmetry was broken.

43

3.2.1

The Tungsten Masses

The tungsten cylinders were custom made for us by Schwarzkopft in Austria.
They are made of Densimet 18k | a sintered metal of 95% tungsten with 5% copper and nickel as binding material. This material has good magnetic properties;
it has volume susceptibility of 0.00066 (MKS) and is paramagnetic [32, 33]. We
ordered the cylinders slightly oversized and had them nished in house. Blaine
Horner, of the JILA instrument shop, insured that the cylinders would be as
regular as possible and dimensionally consistent from piece to piece.
The cylinders were machined to 4.080" (10.363 cm) in height and 6.530"(16.586 cm) in diameter. Their diameters and heights are all consistent to 0.001"(0.002 cm), with masses of approximately 39700 grams. The mean density of thecylinders is 17.724 g/cm3 with a standard deviation of 0.008 g/cm3 ( 0.04% ). The
peak to peak density variation was 0.02 g/cm3 ( 0.12% ). Details of the cylindermasses and volumes are provided in Appendix A.
To determine whether the cylinders were cylindrically symmetric they wereplaced on an air bearing. Any inhomogeneity with angular dependence would
result in a torque along the axis of rotation. The `light' side of the cylinder wasmarked and then the frequency of oscillation due to the un-balancing torque wasmeasured. It is possible to model an angular density variation as a bubble in thematerial. With the cylinder axis oriented along the z^ directionmgr
I
where ! is the angular frequency of oscillation due to a bubble of mass m at radius
r. I is the moment-of-inertia of the cylinder.
The results of this test showed that 11 of the 12 masses had an angulardensity variation equivalent to a 25 gram surplus at the outer radius. This suggeststhat the variation is not due to bubbles (a bubble larger than 1 cc is very unlikely
! =

r

44
to occur) but rather a density gradient. Such a gradient was most probably a
result of cooling e ects. The billets were cut from 3 long cylinders that had
been cooled on their sides. It is possible to nd out more about the angular
distribution of mass in the cylinders, but this test was sucient to bound possible
angular density variations.
Density variations along the length of the cylinders were also measured, as
in Fig. 3.6. Given this information I was able to arrange the cylinders so that
there was only a very small net linear density change from the bottom to the topof each layer of tungsten. This measurement was sucient to limit linear andquadratic density dependence along the height of the whole source mass.
Using the information about density variations from one cylinder to thenext, from one side of each cylinder to the other, and from their tops to theirbottoms, I was able to make an error estimate for the assumption (made in ourmodels) that the masses were homogeneous. The error from this source forms
approximately a quarter of our net uncertainty. In a new experiment I would
suggest constructing the source mass out of much smaller cylinders, spheres, or
cubes than we used this time around. By decreasing the size of the individualcomponents density variations could be better controlled.

3.2.2

The Drive/Support Structure

The support structure for the source mass consisted of a large aluminum
tripod that supported three screws synchronously rotated by a stepper motor.
The tripod was designed to allow easy disassembly for transportation purposes.
Each leg was bolted into a thick \main plate" that formed the acme of the tripod.
The legs had adjustable leveling feet. Tensioning wires ran from the plate to thelegs increasing the rigidity of the structure and preventing twisting motions.
On top of the main plate of the tripod rested the \drive plate" that sup-

45

Figure 3.6: Photo and schematic of the pivot and balance system used to measuredensity variations along the length of the tungsten cylinders. Two 3/4" ballbearings were used to de ne the pivot line.
ported the screws and drive structure. The drive plate could be moved 2 cm inthe horizontal plane to position the source mass concentrically with the dropping

46
chamber (clearance for the screws was cut into the main plate). The drive plate
was also equipped with a central retractable pivot that could be used to produce
a pure rotation of the plate and screws in the vertical direction. This degree of
freedom was incorporated in the design to insure that the screws could be positioned about the FG-5 with maximum clearance. After position adjustments thedrive plate was clamped to the main plate to prevent creep.
The screws were supported by bolts riding on cylindrical roller bearings
located in aluminum towers bolted to the drive plate (Fig.3.7). The towers provide
space for a secondary set of cylindrical roller bearings that increased the rigidity
with which the screws were held. Attached to each screw was a plastic sprocket
keyed to prevent slippage. The three screw/sprocket assemblies were linked toeach other with a stainless steel chain that was annealed to reduce its magnetic
susceptibility. This chain cycle was driven by a sprocket rotated by a separate
chain loop connected to a stepper motor. This constrained the screws to rotate
synchronously (in the limit that there was no slack). The worst case phase shift
between screws is equivalent to a single link misplacement because the sprockets
provide, in some sense, a quantized positioning mechanism. Even this worst caseshift was negligible.
The stepper motor was bolted to the main plate of the tripod as far from theFG-5 as possible. It was held with vibration isolating mounts The drive chain was
turned through a rubber shaft attached to the motor. These isolating measures
were taken to minimize the vibration of the ground due to the drive system,
which could disturb the gravimeter. Although the vibrations were of too high afrequency to excite the super spring much, no vibration is good vibration.

47

Figure 3.7: Photo of the screw drive/support system. A) The drive plate. B) The
main support plate. C) A tower supporting one of the screws. D) One of threeclamps.

3.3

Superconducting Relative Gravimeter

Local acceleration may be a ected through many mechanisms10 . Atmospheric e ects, for example, can change in gravity either by changing the mass of
the air above the observation site, or by compressing the ground11 . This signal
has been calculated to be approximately -0.40 Gal /mbar at Boulder ([24, 35]).
Tides, ground water changes, continental uplift and ocean loading can change
gravity as well. Tides, by far the largest source of gravity signals ( 150 Gal),occur on a much longer time scale than the modulation of our source mass. Tides
People have thought of many ways gravity measurements may be a ected. It has beensuggested that atmospheric pressure associated with solar eclipses could a ect gravity[34]. The
umbra moves with supersonic speed across the surface of the earth, creating a transient pressure
disturbance due to the decrease in air temperature within the sunspot. The pressure change
would then change the value of the local acceleration. Luckily solar eclipses don't happen every
few minutes!
11 The corresponding change in the distance to the center of the Earth increase the local
acceleration.
10

48
are approximately biurnal (6 hour period), while our source mass position wasalternated every 20 minutes, at the slowest. Thus tides did not correlate strongly
with our signal. Some faster environmental signals, however, might have randomly
correlated with the source mass position and increased the scatter in our results.
These signals would not have systematically biased our nal value.
To avoid unnecessary scatter we used a superconducting relative gravimeter(SRG) to record real time varying gravity signals concurrently with the G experiment. The SRG cannot provide an absolute value for the local acceleration(something like 9.8 m/s2) but can only track relative changes in g. For example,
between time t and T , the relative meter might give a measurement equivalent to:g(T ) = g(t) + 12:345 Gal. We subtracted the gravity signal (as recorded with theSRG) from the acceleration values extracted from the absolute drop data. This
reduced the scatter introduced into our experiment by real time varying gravity
signals due to environmental sources.
The SRG works by measuring displacements of a mass on a spring. As
the local gravity decreases, the mass is lifted by the spring. As the local gravity
increases, the spring is stretched, lowering the mass. The change in equilibrium
position can be related to the change in the weight (due to the change in gravity)of the mass through Hooke's law. A schematic of the system is seen in Figure
3.8. The mass in the SRG system is a superconducting niobium sphere and thespring is actually provided by a magnetic eld generated by persistent currents
running in two superconducting coils. The motion of the equilibrium position of
the sphere is recorded with an AC capacitance bridge surrounding the sphere.
The spring constant of the SRG system must be calibrated before the position of the spring/mass equilibrium can be related to changes in gravity. The
output can be calibrated in a variety of ways, but the SRG we used was calibrated

49

Figure 3.8: Schematic of the superconducting relative gravimeter used to record
real time varying gravity signals. This drawing is courtesy GWR Instruments.
from another SRG12 . The two systems were placed side by side and allowed tomeasure environmental signals concurrently. Comparison of the results completed
the calibration. Any errors in the calibration would lead only to residual environmental signals in our data. It is important to note that the SRG system could notremove time varying signals arising from drifting errors in the free-fall gravimeter
This second SRG was previously calibrated with a mechanical system that physically accelerated the entire SRG apparatus at known rates. Both calibrations were done by NOAA
personnel independently of the G-measurement. The quality of the SRG raises the question\Why not do the free fall measurement with out the free fall? Just use a SRG." This is by
no means an unreasonable question, and in fact the idea is good. There is no reason why thisexperiment should not be carried out. One issue that limits the utility of the SRG for a G
measurement, however, is the diculty of calibrating the output. Currently SRGs can only becalibrated to several parts in 104.
12

50
system that actually recorded the acceleration signal of the source mass.
The SRG is subject to drifts and steps in its output. Some drifts result frommass changes of the niobium sphere (it adsorbs helium). Throughout our data
run we performed linear least squares tting for the drift and determined that it
was less than 0.2 Gal per day | having negligible impact on our values for theG.

CHAPTER 4
ANALYSIS

A central task that we faced with the free fall method was implementing thesoftware system used to extract G from gravimeter data. This chapter describes
the algorithms we used to this end. Preliminary work was discussed in Chapter
2. Section 2.1 included the calculation of mutual attraction of the test mass andsource mass over the large range of relative positions in which they are placed. In
Sec. 2.2 the method of numerically integrating the equation of motion of the test
mass was presented.

4.1

A Big Least Squares Solution

A statistically correct way to extract G from observed data is to calculate
a least-squares solution in which G is a tting parameter of the problem.
One might imagine tting the parabolic path of the dropped object to afunction, z(t) composed of a standard solution for a falling object (as in Section2.2), and a term explicitly dependent on G:
43z(t) = 21 g(t2 + 12t ) + V0(t + 6t ) + X0(1 + 21 t2 ) + G f(t)where f(t) is a basis function describing the e ect of the gravitational interaction between the source and proof masses. Unfortunately this was not reasonablebecause f(t) looked very much like a parabola; the correlation between the perturbation we were trying to measure and the e ect of the local acceleration was

52strong (as is evident in Fig.2.2). For this reason the problem had to be posed withthe di erential nature of the data evident. We did this by folding all the information from drops with the source mass in both positions into one \big least-squares
solution" (LSS).
The LSS had to convert the information of the observations (the position/time points for each drop), the superconducting relative gravimeter (SRG)
data (essentially an acceleration value), and the gravitational signal of the source
mass into a single form that would allow them to be compared. Least-squares
tting minimizes the total chi-squared error between individual data points anda tting function. In our case the individual data points were position/time pairs
from drop data. The tting function was a fourth-degree polynomial describing
the motion of a freely falling object in a simple gravity eld (Eqn.2.10).
It is not immediately clear how all this information can be combined togenerate a chi-squared value that depends on the choice of the tting parameters.
The problem was complicated by the many di erent relationships between theobservations and the various tting parameters. For example, Xo; Vo, and g are
local to the data within a single drop. On the other hand, the variable G, the o set
from the SRG from the absolute local acceleration (O), and a drift parameterfor the SRG, D1 , were global to the whole problem. An individual datum isassociated with all the other data of its drop in one way, in a di erent way withdata sharing the source mass position, all the remaining data in yet another way.
Even data within a single set is separated in time | environmental gravity signals
change appreciably on a time scale of minutes.
The LSS began with the removal of the laser dither signal from the drop
To remove environmental signals, data from a superconducting relative gravimeter (SRG)
is included in the procedure. Unfortunately relative meters can only give information aboutchanges in local gravity { an o set between the relative value and the absolute value must be
included in the t. Relative meters also su er from small drifts, which must be t for as well.
See Sec. 3.3.
1

53
data. The dither of the laser signal (Sec. 3.1.2) correlated very weakly with G,
which is why it was dealt with before applying LSS. Fitting the data of each dropfor:43z(t) = 21 g0(t2 + 12t ) + V0 (t + 6t ) + Z0(1 + 21 t2 ) + A sin !d t + B cos !dt

where !d is the dither angular frequency ( 1200 Hz), allowed only the dither
signal to be removed. Note that this \pre-cleaning" was only possible because thedither basis function was of such high frequency that it was nearly orthogonal tothe other basis functions.
The t for the dither provided starting values for g', V0, and Z0. These
values were used to set up the design matrix of the problem. g0 was correctedwith the acceleration value from the super conductor, gsc, appropriate for thetime of the drop. g0 was also corrected with the global values of the o set anddrift:

ge ective = g0 + O + Dt + gsc
The design matrix also must include information about our gravity signal.
This was included, for each datum, by integrating the equation of motion (Eqn.
2.11) with the global value of G. It was the position value returned by this integration that was used in the design matrix for the G parameter.
Since the base unit of information dealt with in the LSS was a single positiontime datum and because there were more than 650 such data per drop, and morethan 150,000 drops in our data sets, the least squares matrix was necessarily large.
The matrix resulting from our set up had more than 16,000,000 observations andrequires nearly 200,000 adjusted parameters. Dealing with such a large matrix
requires greater addressing capacities than possessed by desktop computers |extra cleverness is required.

54
Because none of the parameters local to a drop were correlated with theequivalent parameters of other drops, the design matrix was sparse2 . The matrix
was therefore largely empty; it had a banded-block structure, in which the only
non-zero elements of the matrix were in bands at its edges and in blocks along its
diagonal. Algorithms developed to deal with sparse-matrices made the problem
tractable { it could be solved in about two weeks with a desktop workstation[36, 37].

4.2

The Method of Parallels

A more intuitive method of extracting the value of G from the observed data
exists. The method of parallels (MOP) was faster than the LSS, but included lessinformation about correlations between the initial conditions of drops and thevalue of G. In MOP a theoretical prediction of the change in observed gravity due
to the source mass was compared to a measured change in g.
MOP required that an assumed value of the constant of gravity, Gt , beadopted so that the magnitude of the source mass signal on the measured local
acceleration could be predicted. The prediction, gt , was found by tting parabolas to theoretical position-time curves3 generated modeling the source mass in its
two positions. The magnitude of gt was linear in Gt so it was easily related toan experimentally observed change in gravity, go:

gt = go :
Gt G

(4.1)

Of course the tting functions used to generate the prediction of the signal had
to be the same functions used to t the actual drop data.
It was only because G correlated so weakly with the other parameters of the
Only the value of the constant, the value of the o set to the relative meter, and its drift
were global to the whole problem.
3 Describing the motion of the test mass, as generated in Sec. 2.2.
2

55
free-fall problem that MOP was successful. As we saw from the correlation matrices from the LSS4 , the correlation between G and any other tting parameter(used in the LSS) was less than 0.1. MOP didn't juggle all the information we hadin a single least squares solution, but the information it dealt with was suciently
independent to allow the solution to be partitioned. First the position/time pairsof each drop were t for an acceleration value. Second an analysis of the acceleration values produced a go value. Third go and gt were compared, producing
a value for G. Because LSS and MOP produced highly consistent results, MOP
was used in the nal analysis.

4.2.1

Di erencing Techniques

We used a variety of methods to extract the experimental go value from thedata. The simplest way to calculate the di erence was to subtract the average thevalue of the local acceleration measured with the source in its lower position fromthe average with the mass in the upper position. This method is not optimal;
it does not recognize underlying time varying signals in the local acceleration
resulting in increased scatter.
A second method that we used involved taking individual di erences, as
shown in Fig.4.1. This was essentially the same as the rst method, but allowed
identi cation of outlying di erence values, so that they could be removed. With
this method the di erences we measured were not completely uncorrelated | each
set of data was used to calculate two di erence values. Therefore we had to becareful in our analysis of statistical uncertainty.
We used a third method to avoid the e ects of linear drifts on the scatter ofthe di erences (Fig.4.1). It involved estimating the drift of the local acceleration
across three sets of data by averaging two consecutive sets that shared a common
4

And qualitatively from self-consistency tests with MOP.

Acceleration

6

56

Method 3

Method 2::::::::::::::::::

ppppppppp:::::::::

p p :::::::::p p p p p p p p p p p p p p p p :::::::::

XX
XX

XX
X

:::::::::
Time
Figure 4.1: Two methods of calculating go given acceleration data. (:::::) represents a set of acceleration data with the source mass in either its upper or lower
position. The vertical lines represent the measured di erence value associatedwith the pair or trio of sets.
X
X

source mass position. By taking the di erence of this average with respect tothe intervening set (with the source mass in its other position), the e ect oflinear drifts on the di erences was minimized. Method three had even stronger
correlations between the di erences than method two, for three di erence values
shared information from one set.
A statistically correct implementation of the third method was used in our
nal analysis. First uncertainty values were associated with the time and acceleration of each drop, allowing the importance of the information of each drop tobe appropriately weighted. The uncertainty value was based on the chi-squared
value of the parabolic t to that particular drop. A line was t, with linear leastsquares, to each pair of subsequent sets sharing a common source mass position.
Then the weighted mean of the intervening set was removed from the value of theline at that time5 . This system produced a statistically correct estimate for theerror in each di erence value. The nal go was formed by the weighted mean
There is no reason that the time of each drop cannot be weighted according to the uncertainty of the drop. Thus it was possible to associate a \weighted mean time," a single time best
representing the data in the set.
5

57
value of these di erence results, with an error estimate taken from the weighted
mean standard deviation of the individual di erences.

4.3

Dependence of go on First Fringe Fit

The methods of MOP and LSS destroyed a great deal of information. MOP
reduced the approximately 96 billion position/time pairs of the experiment into a
series of less than 2000 measurements of the experimental go. These di erenceswere then averaged to return a signal best estimate for go. Yet it was possible
to extract other information about the free-fall system. This section outlines anapproach (using MOP) that I used to extract information about the position of
the source mass with respect to the SOD position. In Secs. 3.1.1.2 and 5.1.2 analternate \direct" measurement of the SOD is presented.
The gravimeter system recorded approximately 650 usable position/time forthe test mass during each drop. By varying the set of data within each drop thatwas t the SOD could be found by analysis of the dependence of go on the choiceof the sub set. A reasonable manner of altering the subset was to change the rst
or last fringe that was t. It was more useful to modify the rst t fringe (FFF)
than the last t fringe, because the e ect of the acceleration parameter is mostobvious at the start of the drop.
The go value extracted from the data had a very similar dependency on
the FFF as the acceleration eld of the source mass had on position. Imagine
an acceleration eld that contains a step increase occurring in the middle of thedrop region (in the spatial middle of recorded data). As the FFF approaches thisstep, the extracted value of g will increase. As soon as the rst fringe is beyond
the step, the value of g will no longer change with the fringe number. Clearly the
position of the step can be determined by analysis of the gravimeter data.
In a very similar manner the changing (in space) acceleration eld of the58source mass left a signature dependence of go on the FFF. By comparing theoretical curves, generated assuming di erent SOD positions, to the observed dependence, a best-agreement SOD position was found6 . Fig. 4.2 shows the qualityof agreement for di erent SOD positions on di erent days of the experiment. By
scaling the best- t theoretical curve, generated at the extracted SOD, to the data(with a single multiplicative factor), a scaling of theoretical value of G used in ourmodels could also be made.
2

1.8

Fit quality

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0
−0.4

Start-of-drop position, cm
Figure 4.2: Here the quality of a t of theoretical data to experimental datais plotted. The curves that were t represented the dependence of g on the
FFF in the position/time data recorded by the gravimeter. Each line plottedhere represents the quality of the agreement between theory and experiment,
for a particular day of the 1998 experiment. The weighting/quality assessment
was based on the chi-squared value in a linear least squares routine that t thetheoretical curve (at a given start-of-drop position) to the experimentally derived
curve. The t was in one-dimensional (for a scaling factor).
−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Noise in the data results in greater and greater deterioration of the go value
extracted with less and less data t 7 . This constrained the region over which the
In the free-fall measurement the translation of the source mass between its upper and lower
positions was very well known compared to the mass's position relative to the SOD.
7 There are two reasons for this deterioration. The rst is that the statistical uncertainty
6

59
FFF was varied to the fringes between the earliest usable data in each drop andthe fringe where the uncertainty in go became large. We were fortunate that thissub-set encompassed the region where the second derivative of go with respect to
the FFF was greatest. If the second derivative were small the correlation between
the scaling factor for the theoretical G and the SOD position would increase, as
would the uncertainty in the best- t SOD value.
This method to nd the SOD position was applied to two independent
data runs. In the rst run the extracted SOD position was consistent with the\direct" measurement (Sec. 5.1.2). In the second run, however, the two values
were inconsistent, disagreeing by nearly 2 mm | about 4 standard deviations. I
attribute this discrepancy to the fact that in the second data run changes in theshaft-encoder voltage were monitored. Conceivably, as in quantum physics, theobserver a ected the observed, and changed the SOD position. Thus the extracted
SOD position was used for the nal determinations of G in both data runs.
The extracted SOD value is more appropriate than the direct value. Examination of the values of G obtained for a set of FFF values for the two di erent
SOD positions results in Figure 4.3. There is no physical dependence of G on
FFF, but the directly reference value suggested that there is | because the directreference was incorrect. Note that the G values extracted for FFF about fringe 18
agree well between the two SOD positions. This agreement is due to our lack ofsensitivity to positioning errors, and is a consequence of the extrema of the source
mass eld. In some sense the positioning error doesn't matter if a reasonable FFF
is chosen for analysis.
To check for a systematic drift or signal in the SOD position during a datarun (conceivably due to ball wear, Sec. 5.2.5) the best-agreement SOD positionp(inversely proportional to N where N is the number of data points t) increases. The second
is that the correlation between g and the other tting parameters increases as the beginning of
the drop is excluded from the t.

60

6.72

6.71

6.7

6.68

G X 1011

3

m /kg−s

2

6.69

6.67

6.66

6.65

6.64

6.63

0

20

40

60

80
First fit fringe

100

120

140

160

Figure 4.3: The value of G extracted from the data as a function of the rst fringet. The points marked with asterisks are generated using the SOD value extracted
from drop data. The points marked with crosses use the directly referenced SOD
position. The small points bounding each set of error bars show the G value using
the appropriate SOD 1.
for each day is plotted in Fig.4.4. There is no signi cant drift. There is also nocorrelation between the value of G produced by a day of data, and its best- t
SOD.

61

5.4

Extracted drop position, cm

5.3

5.2

5.1

5

4.9

4.8

4.7

4.6

140

145

150

155

Day of experiment

Figure 4.4: The start-of-drop position extracted from each day of the 1998 experiment. The error bars are based on the chi-squared agreement of the least-squarest of the data to theory. The error bars are used only for weighting purposes.
The horizontal line shows the weighted mean position.

CHAPTER 5
ERROR ANALYSIS

All mechanisms that introduce errors to the G experiment lead to one result| the breakdown of the connection between the theoretical model of the experiment and its physical reality. The careful calculation, determination, and, in somecases, estimation of the magnitudes of the errors associated with our experimentis of the utmost importance in describing our results. Because the standard used
to evaluate the level of agreement between experimental results is the number ofstandard errors between their values (the t-test), the uncertainty associated with
our results will a ect the way we perceive agreement levels.
A discussion and summary of the main sources of error in the free-fall experiment is in presented in Section 5.1. I have chosen to present some systematic
error sources in depth in section, Sec. 5.2. Some techniques that we have used tosearch for indicators of systematic errors are presented in the last section of thischapter. The \hard" presentation of signi cant errors, including the values usedin determining their magnitudes is provided in Appendix B.

5.1

General Error Analysis

5.1.1

Common mode errors

Many error signals fall into the class called \common mode". These signals
a ect the absolute accuracy of the FG-5 system, but have little impact on its

63
di erential precision. Mechanisms that in uence the absolute accuracy of thegravimeter are easy to imagine|and become even more so when you start trying
to think of errors capable of acting on the gravimeter di erentially.
Consider the e ect of the building that housed the experiment. This building
certainly a ected the acceleration of the test mass, probably at the level of 5 Galin the common mode. Yet we don't expect it to introduce a change our measured
source mass signal on a level greater than 5 parts in 109 because it was on thislevel that the di erential equation describing the motion of the test mass wasinvalidated. A general list of common mode e ects that are not important in thedi erential mode includes:

 gravitational attraction of the co-falling chamber drag e ects of residual gas in the vacuum
 light pressure of the laser sensing the test mass position
 forces due to magnetic eld of the earth gravitational attraction of stationary apparatus
 slowly varying gravitational e ects due to:|Lunar Tides|Earth Tides|Ocean Loading|Atmospheric pressure changes|Water table changes or snow fall
|Temperature drift e ects
|Continental uplift slowly varying machine dependent drifts:|Laser verticality|Laser frequency or time standards drifts

64|Laser intensity changes|Vacuum pressure changes
Although common mode errors don't bias our results, they could strongly
in uence the scatter in the free fall measurement if they modulated the g-signal
on a time scale close to the source mass modulation. Two data runs with di erent
modulation rates were made. The rst, in 1997, modulated the source mass
every twenty minutes. During the rst 17 minutes of each \set", 100 drops were
measured with 10 seconds between each drop. In a second data run in 1998,
the source mass was modulated at 11 minutes, with 90 drops per set (5 seconds
between drops). Scatter in this second data run would only be strongly sensitive
to common mode signals changing at periods less than about 30 minutes.

5.1.2

Positioning Errors

Errors made in positioning the source and proof masses resulted in systematic errors that were not ameliorated by the di erential nature of the experiment.
The analysis of these positioning errors is divided into a treatment of the sourceand proof masses separately.
Errors in the source mass vertical position with respect to the start-of-drop(SOD) biased the calculation of the acceleration experienced by the test mass andour value of G. This type of errors was introduced by wear in the support nuts ofthe source mass, by inaccuracy in referencing the mass to the vacuum chamber,and uncertainty in the SOD position within the chamber. Because we operated
at the extrema positions of the source mass eld, however, positioning errors were
not a major source of error in the experiment. The uncertainties associated withthe source mass positioning issues are fully presented in Sec. B.1.1, in Appendix

65
B. The net e ect of vertical positioning errors was only 10 ppm.
Error positioning the tungsten cylinders within the source mass was a much
more important source of uncertainty. Because the radial position of the large
tungsten cylinders was determined by the locating lip of the sandwiching plates(which allowed some clearance) the cylinders' radii from the drop were not absolutely de ned. The plate that supported the tungsten mass distorted under theirweight producing additional uncertainty. Density inhomogeneities, so carefully
dealt with in Sec. 3.2.1, had the same e ect as misplacing elements of the source
mass. These sources of error were responsible for nearly 100 ppms of uncertainty.
The complicated proof mass, which underwent free fall motion, was a greater
source of positioning errors than the source mass. Most important was the radial
position of the proof mass, which depended on horizontal velocities (that could
never be made identically zero), and the centering of the proof mass within the
dropping chamber. Estimates of radial positioning errors were made with a simpli ed model of the proof mass that was increasingly miss-positioned horizontally.
The variation in acceleration signal with the radial o set of the proof mass fromthe symmetry axis is roughly quadratic. Figure 5.1 displays the e ect on thevertical acceleration of the proof mass by moving it o the axis of symmetry of
the source mass.
The proof mass optical center was \directly" referenced to the source mass
only through a long chain of measurements:

 Optical center of corner cube to tungsten balls (within test mass) Tungsten balls to tungsten vees (test mass {> co-falling chamber).
 Tungsten vees to bottom of co-falling chamber.
(within co-falling chamber).
 Co-falling chamber to stop at bottom of dropping chamber.

66

0.025

Acceleration di erence, Gal

0.02
0.015
0.01
0.005
0

−0.005−0.01−0.015−0.02−0.025

−20

−10

0

Position, cm
10

20

30

40

Figure 5.1: Di erence in vertical acceleration (due to the source mass) sensed bya simple proof mass (8 cm long hollow cylinder with 1 cm diameter and 1 mmwall) falling on axis and (|) 0.1 cm, ({) 0.2 cm, (  ) 0.4 cm o axis. Positionis with respect to the source mass. Note that radial position errors have precisely
the same e ect as a scaling of G.
assembly (via shaft encoder servo).
 Gravimeter bottom stop to bottom of gravimeter vacuum.
(within drop chamber) chamber.
 Bottom of dropping chamber assembly to top of lower ange on
vacuum can.
 Top of ange to bottom of the source massin its lower position.
These measurements were made directly1 , with the vacuum chamber open to air.
This trail of references added error at every step. Alternatively information about
For the most part. The shaft encoder travel to the SOD was interpolated from the diameterof the drive pulley and calibrations of the shaft encoder output.
1

67
the position of the test mass could be extracted from the data (as in Section 4.3).
In a data run in 1997 experiment the two independent determinations (extracted and directly measured) of the position agreed well, but in another run
in 1998 they were not consistent. The set-to-set variation in the position, as
measured by the shaft-encoder, was the same magnitude as this disagreement (inthe 1998 experiment). Possibly the system that was used to monitor the shaftencoder voltage throughout the 1998 measurement may have a ected the system.
For these reasons we adopted the position value extracted from the data (Sec.
4.3 includes a discussion of why this was reasonable). Using the value given by
the \direct" measurement changes the nal value by less than 1/2 sigma.

5.1.3

Modeling Errors

Errors in the measurement of masses contributed weakly to our nal error
estimates. The direct measurement of the masses of the tungsten cylinders andthe two aluminum plates was the largest source of error. Buoyancy corrections
were applied for these objects (Sec. B.3.1). The smaller components of thesource mass (nuts and bolts) could be weighed with higher precision because of
their small mass. The total uncertainty associated with errors of this type wasabout 40 ppm.
Although the total mass of the test mass had no bearing on the acceleration
di erence we measured, its mass distribution did. Thus it was important that therelative densities and positions of the various components of the test mass werecorrectly modeled. Although there was essentially no error introduced during themodeling of the source mass2 , modeling the test mass was very dicult because of
the complexity of its design. Errors in the relative masses ( 0.05 g) and positions
2 Certain approximations were made during the modeling of screw threads, bolt heads, andgroups of bolts, as discussed in Appendix A. These approximations were very good.

68( 0.01 cm) of the components of the proof mass contribute about 90 ppm to our
uncertainty.
Over the course of the free fall experiment the tungsten balls that supported
the test mass experienced some wear. This introduced three errors: Firstly achange in the relative mass of the balls to the rest of the test mass, secondly achange in the mass distribution within the balls themselves, and thirdly a change
in the SOD position. These are all very small errors, and are discussed in Sec.
5.2.5.

5.1.4

Numerical Techniques

A primary assumption of the free-fall method of measuring G was that thetheoretical framework we built was accurate enough to describe the experiment tobetter accuracy than it could be carried out. Position versus time data for the test
mass lacked meaning if we could not relate them and their variations to the source
mass and G. There were many levels at which our numerical approximations may
have di ered from reality. I think it reasonable to start the discussion of possible
errors with the integration over the source mass.
The Romberg quadrature algorithms we used to evaluate the integrations
over the source and proof mass incorporated an error estimating functionality (Sec.
2.1). We speci ed that the individual integrations return a value good to better
than 1 part in 108. The individual integrations agreed at this level with analytic
test cases. The error introduced by this limit of accuracy was overshadowed by
the approximation we made in integrating over an interpolation eld (Sec. 2.1).
By specifying a suciently dense grid, however, this source of error could becontrolled to arbitrary precision. The nal grid precision we used corresponded
to an uncertainty of 20 ppm.
To reduce the interpolation eld to a two-dimensional grid we assumed that69
the gravitational eld produced by the source mass was cylindrically symmetric.
This was very good approximation to a level of 20 of ppm.

5.1.5

Spurious Signals

Thermal and magnetic e ects were among the hardest sources of uncertainty
to quantify. These sources of discussed in depth later in this chapter, within Sec.
5.2. Here a very brief discussion is included.
We had to perform secondary experiments to get the magnitude of signals
arising from these two areas. A test for thermal e ects was made by placing
modulated heat sources around the gravimeter, recording the power incident on
the gravimeter and the resulting g signal. The modulation frequency matched
the modulation rate in the 1997 experiment, 20 minutes per set. This would
overestimate the thermal signal in a 1998 data run in which only 11 minutes wereallocated per set.
We bounded the magnitudes of the magnetic signals both by calculating thee ect (for eddy current damping during the drop) and by experimentation (forthe force of the magnetic AC signals). The theoretical calculations were basedon measurements of the DC magnetic eld in the region of the dropping chamberand on the susceptibility and (simpli ed) geometry of the source and test masses.
The experiment is presented in Sec. 5.2.8 on page 96.

5.1.6

Contaminated Data

There were several ways data could be ruined. Earthquakes resulted inhuge scatter in the measurement of acceleration values. When the scatter waslarge enough (several hundred Gal ) the source mass signal could be completely
swamped out. Occasionally fringes were miscounted, ruining individual drops(Sec. 5.2.9, pg. 101). Rarely the laser lost its frequency lock, resulting in a

70position measurement made with a uncalibrated \ruler".
Drops contaminated by missed fringes were easily identi ed and removed
from the set of drops used in the di erencing. Some unreasonable data was not
so obvious. The criterion that we chose to remove this data was simple: Drops
that produced g values more than 3 standard deviations from the mean g-value
for the particular set were removed.
Our treatment of earthquake contaminated data was even more casual.
Earthquakes were removed by eye. Generally just the beginning of the earthquake
was removed, leaving several sets with large scatter due to the motion of the super spring. The di erence values produced from these sets were \automatically"
associated with greater uncertainty with the di erencing algorithm discussed inSec. 4.2.1. Entire earthquakes were left in the data to test the method, andno signi cant change (> 0:2) in any days' G result was observed. Figure 5.2
summarizes our feelings about earthquake contaminated data.

Totaled Non-statistical Error Estimates
Table 5.1 presents a summary of the major systematic error sources of thefree-fall experiment. The table is split into two sections for the two data runs
because some errors were better controlled in the 1998 run.

5.2

Sources of Systematic Error

There are many sources of real gravity changes that are sensed by free
fall and super conducting gravimeters (tides, atmospheric signals, water tablechanges. . . ). These sources do not complete the set of mechanisms that in uence
measured gravity; instrumental errors are sources of false signals that can haveequal bearing on the results of the acceleration measurement as real gravitationale ects. All the systematic non-gravitational signals we have discovered in the

71

Figure 5.2: A Hallmark postcard { commentary on our data analysis technique?
Because our results don't change by more than a tenth of a standard uncertainty
whether we take or leave the little fellow, we don't think it much matters.

Source of Uncertainty
Positioning Errors

Vertical positioning
Radial positioning
Internal positioning
RSS TOTAL:
Modeling Errors

1997 Error 1998 Error
10 ppm85 ppm85 ppm120 ppm

10 ppm60 ppm45 ppm75 ppmGranularity of mass models
Symmetry Assumption
Interpolation Density
Modeling
Source mass density variations:
Point mass
LinearQuadratic
AngularRSS TOTAL:

52 ppm20 ppm20 ppm90 ppm52 ppm20 ppm20 ppm90 ppm60 ppm50 ppm60 ppm25 ppm145 ppm

60 ppm50 ppm60 ppm25 ppm145 ppm

Thermal Signals
Airgap60 ppm75 ppm60 ppm75 ppm72

Table 5.1: A table of the magnitudes of the important error sources. Error estimates are separated between two data runs in 1997 and 1998. Generally the error
estimates are the same or better in the 1998 run because this run incorporated
more robust support plates for the source mass and more precise alignment of thesource mass and dropping system.
free fall G-measurement were consequences of di erential modulation of thesemechanisms.
This section contains discussion of some sources of error in the measurementof the local acceleration, and the manner that the G experiment might modulate
the errors to create systematic bias in our results. The subsections are arrangedby the magnitude of the uncertainty due to the mechanism. One or two of thesubsections discuss broader subjects than a single error source (Secs. 5.2.9 and5.2.10 ).
More complete discussion of all the error sources (including common mode)

73
in the free fall gravimeter may be found in [30, 24, 38]. Only errors springing
from the FG-5 system are discussed here because we don't expect (or see evidenceof) any systematic connection between the free fall measurement and the super
conducting system.

5.2.1

Air Gap Modulation

Because the dropping chamber is in high vacuum whilst the rest of thegravimeter is in air, an air-vacuum interface exists between the arms of the interferometer. The index of refraction of air is slightly larger than the index of
vacuum, so motions of the interface (the \airgap") result in phase shifts of thefringe signal[39, 38]. If the interface moved upwards by an amount  the laser
beam would pass through a distance  where vacuum had been replaced by air.
This would result in a phase shift mimicking a motion of the test mass upwards,
away from the interface. If the interface moved downwards, there would be acorresponding false motion of the proof mass towards the interface.
At the start of the drop the co-falling chamber is accelerated quickly fromrest jarring of the dropping chamber and introducing a vibration in the airgap that
couples into the g measurement in just this way | these vibrations necessitate theseparation of the interferometer/spring system from the dropper least the super
spring be excited. Figure 5.3 is a plot of the vertical position of the interface as afunction of time averaged over 90 drops.
This noise signal is systematically linked to the drop | it is triggered by thedrop motion. Power spectral analysis of the motion (Figure 5.4) shows that theairgap motion might be modeled as a sinusoid with frequency 63 Hz (a very bad
approximation). A sinusoidal motion at this frequency has a small but non-zerocorrelation with the acceleration parameter of the linear t for gravity.
The e ect of the airgap modulation was not completely common mode be-

74

−5

8

x 10

6

Vertical position, cm

4

2

0

−2

−4

−6

−8

0

0.05

0.1
Time, seconds

0.15

0.2

Figure 5.3: Vertical motion of the air-vacuum interface between the dropping
chamber and the interferometer. This trace was averaged over 90 drops of the test
mass. The two vertical dotted lines represent the typical start and stop positions
for the t for local acceleration. Note that the mean position of the interface is
non-zero during the drop because the test mass and its co-falling chamber became
weightless during each drop causing the rest of the tripod/dropping assembly tojump up on its spring-like legs. Because the test mass is in free fall at this point
other e ects of the weight change are unimportant.
cause the motion of the interface was a ected by a damping force dependent on
the source mass position. In its lower position the source mass was placed close to
the lower ange of the dropping chamber, resulting in an increase in air-dampingof the motion. In this position the source mass was near to the ion pump and its
quickly-changing magnetic eld. The ion pump magnets (which were rigidly connected to the dropping can) induced eddy currents within the conducting source
mass. This magnetically damped the airgap motion di erentially.

75

−8

2.5

x 10

Amplitude, cm

2

1.5

1

0.5

0

0

50

100

150

200

250
300
Frequency, Hz

350

400

450

500

Figure 5.4: The amplitude spectrum of the airgap motion { the square-root of
the power spectrum. The data analyzed was averaged over approximately 11,000
drops. The analysis included 0.24 seconds of data following the drop trigger. Thus
it incorporates data that is not seen in the t for gravity.

76
We measured the motion of the interface throughout a data run in 1998.
A split-photo diode and high-intensity LED were supported by a rigid aluminum
tube bolted to the lifting assembly. A knife edge bolted to the top of the dropping
chamber blocked some of the light (from the LED) from one half of the photodiode. The whole arrangement was shielded from stray light by a telescoping
cover, insuring that the source mass position would not directly in uence themeasurement.
We attempted to model the e ect of the airgap by tting a parabola tothe apparent test mass motion that would have been introduced by the interface
vibrations:za = 2(n , 1)zag(5.1)

where za is the apparent motion of the test mass, zag is the actual motion of
the air/vacuum interface, and n is the index of refraction of air in Colorado(1.00026). The factor of two is included because the laser beam passed throughthe air/vacuum interface twice, once in each direction. Einstein said \Everythingshould be made as simple as possible, but not more so." Unfortunately Equation
5.1 crosses the line of proper simplicity.
We can compare the \expected apparent" proof mass motion, the za of
Eqn. 5.1, to the residuals of the t to drop data, which are roughly equivalent
to a \measured apparent" motion3 . The agreement between the expected andactual motion would be a good measure of the correctness of the simple model.
Figure 5.5 contains such a comparison. Although the simple model agrees well(except for a scaling factor) at the beginning of the drop, the model does not
successfully describe the apparent motion throughout the drop.
The scaling factor might be due to our assumption that when the air/vacuum
The measured apparent motion, as recorded by the residuals to the t, would naturally
have all parabolic components removed.
3

77

−8

x 10

6

Residual position, cm

4

2

0

−2

−4

0.04

0.06

0.08

0.1

0.12
Time, sec

0.14

0.16

0.18

0.2

0.12
Time, sec

0.14

0.16

0.18

0.2

A

−8

x 10

6

Residual position, cm

4

2

0

−2

−4

0.04

0.06

0.08

0.1

B
Figure 5.5: Comparing the expected false motion of the test mass arising fromairgap motion to residuals from the t for g. The solid curve is the expected signal,
calculated as in Equation 5.1 from the measured motion of the airgap averaged
over 90 drops. The dotted line is the residuals to the t of the same 90 drops. A)
The modeled motion is scaled according to the model. B) the modeled motion isscaled for best agreement with the residuals.
interface moved the displaced air was not compressed but only displaced from thelaser path. This assumption might not be valid. The index of refraction of air isdirectly proportional its density, so unless air was moved out of the path of the78
laser there would be no phase error. This might not have occurred, for example,
in a sealed column of air, where one end was a vibrating plunger. The degree
that the air was being compressed rather than displaced would be indicated by
the measured signal that is seen in the residuals.
I cannot explain why the model appears reasonable at the beginning of thedrop, up to about 0.08 seconds, but not for other portions of the drop. Thisdisagreement was signi cant. Fitting the expected apparent motion, za , witha parabola to extract an acceleration correction for each set of data gave very
unexpected results. The corrections were very close to sinusoids and consistently(day to day) had the same period (about an hour and a half) and amplitude(about 1.5Gal ). Examination of the airgap data showed that phase shifts in theairgap motion occurring around the middle of the drop were responsible for thissignal. We have no idea how the phase shifts were modulated in such a way as toproduce the sinusoidal time dependence. Clearly, however, the actual acceleration
data recorded by the gravimeter did not exhibit any such underlying signal.
To nd out if any useful information about the magnitude of the di erential
airgap error source could be extracted from the data, I added a weighting function
to the model to decrease sensitivity to the less reasonable information. I t thisweighted model (Fig. 5.6) with a parabola in the same way as before. The
correction values extracted from the weighted model were much more reasonable.
When I removed the acceleration corrections for each set from the actual measured
values of g the set-to-set scatter in the g-values decreased slightly. This showedthat there was some correspondence between the information in the airgap motion
and the data recorded by the gravimeter.
At this point I processed the airgap corrections for a g correction, by using
di erence method two (Sec. 4.2.1). This introduced a new problem, however.
The magnitude of the computed corrections to g and g was highly dependent

79

−8

x 10

4

Airgap position error, cm

2

0

−2

−4

−6

0.04

0.06

0.08

0.1
0.12
Time, seconds

0.14

0.16

0.18

0.2

Figure 5.6: The weighted expected false position signal of the proof mass due tothe airgap vibration. The dotted lines represent an exponential weighting of theairgap data. This weighting served to decrease the in uence of the airgap data at
the end of the drop.
on the weighting function used to curb the disagreement between the theoreticaland observed airgap signal. How was the correct damping function to be found?
Because the airgap signal accounted for a very small portion of the scatter in gvalues, it would very dicult to test di erent damping coecients to nd a \bestt".
To avoid this dependence on the weighting function I assumed a single damping coecient and calculated the correction to g as a function of the rst fringet (FFF). This is exactly analogous to the work in Sec. 4.3. One reason this was80
a good idea is that parameter we modi ed (the rst t fringe) only varied in arange where the agreement between the theoretical airgap motion agreed well withthat observed in the residuals. This gave an expected correction to our data as afunction of the FFF: ga(FFF). This curve, which looked roughly like a sinusoidwith a constant o set from zero, was, in turn, compared to the residuals of thego as a function of FFF (obtained from the gravimeter data with the theoreticaldependence of go on FFF removed). Figure 5.7 shows the two curves. One isderived from the free fall data (we know our models of the free-fall motion are verygood). The other comes from airgap data (with a questionable model), scaled tomatch up.
Now the trouble was that the residual to the observed dependence of goon FFF would have no information about a net bias | any mean value or o set
would have been removed with the theoretical curve. Our best estimate for thebias was the o set as calculated with the airgap information, because our result
was averaged over the g values for each start fringe (Discussed in Sec. 6.1.2).
To summarize up to this point: We have a model for the airgap's e ect on
gravity as measured with the FG-5. This model unfortunately incorporates a (tothis point) arbitrary weighting function that strongly in uenced the calculationsof the magnitude of the airgap signal. We have found a way to compare changes inthe theoretical airgap signal as a function of FFF (looks like a sinusoid in FFF) to
actual observed anomalous changes in g as a function of FFF (which also lookedsomething like a sinusoid in FFF). Since we can only compare changes in these
two signals as a function of FFF, there can be no direct comparison of the biasof the airgap signal. Of course if we could extract the bias from the free fall data
directly, we would never have to measure the airgap motion independently. Thus
the bias must be taken from the airgap motion and model.
By calculating di erent ga(FFF) curves using di erent damping functions,

81

0.03

0.02

0.01

δg signal, µGal

0

−0.01

−0.02

−0.03

−0.04

−0.05

−0.06

−0.07
10

15

20

25

30

35
40
First fit fringe

45

50

55

60

Figure 5.7: Scaling the theoretical g correction derived from airgap motion data.
({{) g as a function of start fringe from free-fall data, with theoretical value
removed. (|) g as a function of start fringe from airgap data, and scaled forbest agreement with the free-fall data. An o set has been removed from bothcurves. An additional o set has been removed from the airgap derived curve aspart of the t.
I was able to show that the bias in ga was linearly dependent on the amplitudeof the sinusoid. We were able to extract from free-fall data the amplitude of thesinusoid, and thereby x the bias value. The important point is that I was able
to show that the calculated airgap systematic bias, calculated in this way, wasindependent of the choice of damping function used in the model.
In this convoluted way the e ect of the airgap on g was seen to be 0.006
Gal with an uncertainty (derived from the t) of 0.0005Gal. The whole airgap
motion caused approximately a 1 Gal shift in the absolute value of g extracted

82
from the drop data. This means that the source mass modulates the magnitudeof the vibrations by about 1%. An additional measurement strengthens these
conclusions. By integrating the power spectra density calculated from the airgap
motion an estimate for the power of the motion is made. By averaging the airgap
motion for the mass in upper and lower position, and then taking the di erence,the change in power between the two positions could be calculated. This measurement showed that the di erential motion was 1% of the power of the net motion,
as expected from the model. Figure 5.8 is a plot of the relative motions of theairgap.
Because of the weakness of the model the entire 0.006 Gal e ect was
adopted as the error estimate arising from di erential airgap damping (also, acorrection of this magnitude is made). In the 1997 experiment the airgap motion
was not measured, so we can't use this method for that data. We do know thatthe damping was smaller in the 1997 experiment because the source mass was notplaced as close to the gravimeter as it was in the 1998 experiment. Thus an error
estimate of 0.006 Gal (75 ppm) was adopted for the airgap e ect in that run,
too.

5.2.2

Thermal Signals

The source mass had a large thermal mass so it did not readily change
temperature. Therefore it acted as heat reservoir and could introduce a thermal
signature into the signal we were trying to measure. The source mass position
could thermally couple into the gravimeter in two ways. The mass could either
carry a thermal signature (due to the temperature gradient of the air in the room)and heat or cool di erent portions of the vacuum can or ion pump, or it could
block radiation incident on the gravimeter and thereby perform a cooling function.
The second mechanism is unimportant in the free fall measurement because no

83

Apparent proof mass motion, cm

−7

2.5

x 10

2

1.5

1

0.5

0

−0.5

−1

−1.5

A)

−2

−2.5

0

0.05

1.0

Time, seconds
0.1

0.15

0.2

0.25

0.9

Relative Power

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

B)

0

0

50

100

150

200

250

300

350

400

Frequency, Hz
Figure 5.8: Comparing the net airgap motion to the di erence in motion between
the two source mass positions. The motion has been averaged over all the airgap
data of a two week data run in 1998 (approximately 170,000 drops). A) Airgap
motion: ({{) Net airgap motion. (|) Di erence signal, multiplied by a factor of
10. B) Amplitude spectral density of the airgap motion. ({{) for the net motion.
(|) for the di erential motion.

84
signi cant localized sources of infrared radiation were located close to the system.
Thermal gradients can cause large shifts in acceleration measured with theFG-5 gravimeter. A thermal gradient creates a pressure gradient (in the residualgas of the vacuum system) across the proof mass. This gradient results in abuoyant force on the test mass. Niebauer et al. [38] cite a coecient of 0.40
Gal/oC over the length of the proof mass. The temperature at the surface ofthe proof mass is also related to the \kick" that a gas molecule bouncing o thesurface is given. If the upper and lower portions of the proof mass are at di erent
temperatures there would be a greater transfer of momentum and energy in one
direction than in the other, causing a net force on the test mass.
It is dicult to place a theoretical limit on thermal signals, so I did a simple
experiment to test the response of the gravimeter. A modulated heat source madeof heating tape was hung in a ring about the upper half of the vacuum can. A crude
parabolic aluminum re ector focused its power (80 watts) on the upper half the
dropping chamber. The tape was turned on and o at the same frequency as thesource mass modulation (in the 1997 experiment), thereby mimicking the actual
thermal signal of the source mass on a single portion of the vacuum chamber4 .
The vacuum can was covered with a layer of nylon mesh and aluminum foil, justas it was in the G measurement. The temperature of the can was measured witha small thermistor taped under the nylon/aluminum shield.
The temperature signal recorded with the thermistor had less than a 0.25o
C amplitude, as may be seen in Figure 5.9. The set-averaged temperature (theaverage temperature registered by the thermistor for a 20 minute set of 100 drops)
varied from set-to-set by less than 0.03 oC. The resulting di erential signal in g
was less than 0.7 Gal.
A second test was done with the heating tape arranged about the lower
4

The heat was on for twenty minutes, then o for twenty minutes, and so on.

85

29

28

o

Temperature, C

27

26

25

24

23

0

5

10

15

20

25

Time, hours

Figure 5.9: Temperature recorded by a thermistor taped to the dropping chamber
during two test of thermal sensitivity. (  ) - Upper portion of the can heated.
The upper dotted curve is the curve of the average temperature for each 20 minuteset, with an o set included for clarity. (|) - Lower portion of the can heated.
Once again the upper solid line is the temperatures averaged over 100 drops, toprovide the average temperature for each set.
portion of the vacuum can, to simulate the lower source mass position. In thisrun (the temperature signal is again seen in Figure 5.9) the temperature signal
had less than a 0.2 oC amplitude, with the set averages varying by less than 0.05oC
due to the modulation of the tape. The resulting di erential signal in g was lessthan 0.15 Gal.
We can calculate the worst case power incident on the dropping chamberof the gravimeter due to black body radiation emitted by the source mass. The

86
Stefan-Boltzmann law gives

P = eA(Ts4 , Tc4)
where P is the power,  is the Stefan-Boltzmann constant (5.6e-8 W m,2 K,4 ), e
is the emissivity (the worst case value is 1), and A is the surface area between thesource mass and the vacuum chamber (about 0.12 m2). T is the temperature. I
assume a (generous) 0.5o C temperature di erence between the source mass andthe vacuum can in either of its two positions. This is based on a measured 1.5oC
temperature gradient from the oor to the ceiling of the building, coupled withthe assumption that the source mass will carry an average temperature between
its upper and lower positions. These values lead to a value of 0.4 watts for the
power radiatively transfered to the can.
Convective transfer of heat is also a concern. I assume a worst case scenario| that the body of air between the vacuum can and the source mass is held at
one position, maximizing heat ow. Then the heat transfer to the can would be
the ow from the source mass through the air to the can:Q = KA(Ts , Tc)tw
where K is the thermal conductivity of air (0.026 W/m-oC), and w is the widthof the layer of air (about 0.2 cm). The 0.5oC temperature di erence results in aow of 0.01 W by convection/conduction.
Assuming a linear dependence of g on incident power, we can immediately
see that the source mass thermal signal must be less than::01W) = 0:004 Gal
0:7Gal (:4 +80W
in its upper position, and less than

:01W) = 0:001 Gal
gThermal = 0:15Gal (:4 +80W

87
in its lower position. Thus the total thermal e ect of the source mass is less than
0.005 Gal or 60 ppm.
It was possible that there was a correlation between the go values and adiurnal thermal signal. We didn't expect any such dependence, but to check it
the di erence values were averaged in time. Every di erence that was measured
between 2 and 3 pm (Rocky Mountain Time) was averaged together to give atime-binned picture of the di erences at around 2:30. This picture is in Figure
5.10. The larger error bars around 8 and 9 in the morning are due to the small
amount of data at those times. Traditionally I would reset the gravimeter every
morning at about 8:30.

83.4

Average difference signal, µGal

83.2

83

82.8

82.6

82.4

82.2

0

5

10
15
Rocky Mountain Time, hours

20

Figure 5.10: Time binned di erences from the 1998 experiment. The times are forRocky Mountain time zone. I reset the gravimeter at approximately 8:30 everyday. This explains the larger error bars at those times.

88

5.2.3

Systematic Laser Excitation

High-precision lasers do not like to be bumped. When the laser of the FG5 is tapped, it undergoes an oscillation in frequency at  100 Hz (the cornerfrequency of the piezo servo). The vibration of the drop and the associated soundimpulse wave provide a tap that excites this response in the laser. This frequency
modulation signal is triggered by the start of the drop and therefore systematicallyin uences the t for g.
The sound wave that strikes the laser was a ected by the position of thesource mass. Fig. 5.11 is a plot of 6300 averaged, corrected and di erenced
residuals | the di erence in the average residuals with the mass up and the mass
down. Di erence residuals only show signals arising from systematic errors. These
residuals clearly showed that one-tenth of the laser signal was either damped by
the source mass in its lower position or ampli ed by it in its upper position. We t
for the \laser bauble" signal in the di erenced residuals, and extracted information
about the amplitude and phase of the oscillation. In a second data run the laser
was located far away from the dropping chamber and its light was introduced tothe interferometer through a ber-optic (Sec. 3.1.3). In this system the powerin the signal was observed to decrease by 50%.
Modeling this signal in a simulation of the system shows that the di erential
e ect in our signal was 0.0015 Gal . This corresponds to 20 ppm, about 1/30 of
our statistical uncertainty.

5.2.4

Laser Verticality and Floor Tilts

For the interferometer fringes to represent vertical position measurements
the laser beam must be parallel to the local vertical. If this condition is notsatis ed and the beam deviates from vertical by a small angle , a cosine scaling

89

0.08

0.06

Residuals, nanometers

0.04

0.02

0

−0.02

−0.04

−0.06

−0.08
0.02

0.04

0.06

0.08

0.1

0.12
Time, sec

0.14

0.16

0.18

0.2

0.22

Figure 5.11: Corrected and di erenced residuals from 64 sets of the 1997 data.
Residuals from drops with the source mass in its lower position were averaged andthen subtracted from the averaged residuals of the drops from the upper position.
This \di erenced residual" was then corrected by removing a theoretical curve
representing the di erence in residuals due to the source mass perturbation. The
sine t has a frequency of 100.3 Hz and an amplitude of 12 picometers.
error of magnitude

g = 2
g 2(5.2)

occurs.
Verticality shifts are caused by tilts of the components of the interferometer,of the laser, or of the interferometer supports. These tilts can be introduced byground tilt or thermal expansion signals. Although no di erential signal is introduced by a steady common mode error the verticality may change synchronouslywith the G measurement due to random correlation of environmental thermal effects. If the verticality could, in fact, respond to temperature signals at the high90frequency of the source mass modulation, it would in uence the scatter of ourresults. It is possible that there is signi cant power in the temperature signal atthis high frequency but because the environment was not phase locked with ourexperiment only the day to day scatter of our results would have been in uenced.
The verticality of the gravimeter system can not be measured while thegravimeter is running5 . This means that we could not directly measure changes
verticality at scales that were important to our results, nor could we e ectively
monitor the verticality throughout our data runs. Therefore it was not possible
to do more than conjecture about the random e ects of verticality on scatter.
The likelihood that the thermal signal of the source mass a ected verticality
is small. The interferometer optics were shielded from the source mass by a 1/4"
thick aluminum box and the thick aluminum party tray of the gravimeter. Theinterferometer was approximately 60 cm below the lower source mass position.
The verticality could have been systematically modulated by oor tilts coupled with the free fall measurement. The source mass was moved between its
two optimal positions with as little horizontal movement as possible, minimizing
systematic oor tilts. The weight of the source mass was sucient to constrain
the support screws to hang within 0.6 mRad of vertical. This was measured byexamining the separation of both the screws to a long rod suspended from a thin
wire, and the source mass to the dropping chamber6 . An angle of 0.6 mRad in
the screws corresponded to a horizontal source mass motion of 0.02 cm between
its two positions.
To bound the magnitude of systematic oor tilt I measured the tilt of thepier7 as I shifted weights on the oor. I used a prototype opto-electric bubble
The verticality can only be measured to a precision of 0.2 Gal (about 0.02 mRad) [40]Di erential thermal expansion of the lifter legs is a negligible in uence on the verticalmotion of the source mass
7 The interferometer was supported on a 5 foot pier of concrete embedded into the oor.
56

91
level ([41]) to measure the pier tilt, which was measured in two directions to avoid
undo sensitivity to reinforcing bars in the concrete oor. The tilt was checked ata period matching the possible oor loading signal from the G-experiment.
To connect the tilt of the screws to a oor loading, I assumed that the tiltof the oor was linearly proportional to the horizontal displacement of the centerof mass of the source mass. Thus the measurements of oor tilts from di erent
weights placed at di erent distances from the pier showed that the pier tilted
0.5 radians for a 1 cm horizontal movement of the source mass. This gives anestimate of 10 nanoradians systematic oor tilting synchronously linked to theG-measurement.
The experimental error due to a tilt of d depends on the mean value of theverticality of the laser,  | the di erential error we are concerned with resultsfrom the change in the absolute error of the verticality:

(g) = d
g

(5.3)

Assigning a base angular verticality error corresponding to 1 Gal , the worst caseerror arising from systematic oor tilt is only 15 ppm. This is negligible.

5.2.5

Ball Wear

The tungsten balls that support the proof mass on the co-falling chamber
wear with time. This introduced two possible diculties for the G measurement.
Firstly, the wear changed the relative position of the start-of-drop and the source
mass. Secondly, as the tungsten wore the mass distribution in the proof masschanged, weakening the connection between our model and the actual test mass.
Both mechanisms introduced little error in the G measurement. Their combined
contribution was 12 ppm.

92

5.2.6

Phase Coherent Excitation of the Super Spring

It is possible to view the super spring as a very long ( 1 km) simple
spring; the temperature response of the equilibrium position of the spring/mass
system follows this model. We were concerned that such a `soft' spring would
respond to the shift in gravitational potential resulting from the translation of thesource mass. When the source mass was moved to its lower position closer to thespring, the proof mass would `suddenly' be subject to an increase in the upwardgravitational attraction of the source mass. This would cause the equilibrium
position to rise. When the source mass moved to the other location, the opposite
would occur, and the super spring would `fall' back to its old equilibrium. A
calculation incorporating the source mass eld and information about the spring's
fundamental frequency shows that the acceleration amplitude resulting from these
excitations would be about 1Gal.
The acceleration signal of the excited super spring is a damped sine wave.
The super spring would oscillate for approximately three periods before the amplitude of the harmonic motion would become small enough to loose phase coherency[42]. Although the drop to drop scatter of the FG-5 was too large for this underlying error signal to be seen in a single data set, the di erential signal would havebeen preserved over the whole experiment. Because the integral of a damped sine
wave over 3 oscillations is non-zero, this super spring excitation would bias thego measurement.
To decrease this coupling the source mass was moved slowly compared tothe period of the super spring. This allowed the equilibrium position of the spring
to change slowly while keeping the amplitude of oscillation as small as possible.
Approximately 30-40 seconds of time were allowed after each move for the super
spring to damp down from whatever excitations it may have experienced.

93
As a quick test for this coupling we examined drop data from the beginningof each set of data to its end, looking for a damped sinusoid. Figure 5.12 shows theresults for data runs in 1997 and 1998. This analysis, although too noisy to placea bound on the super spring excitation, is interesting because it would re ect any
trends in the data. Also, within smaller averaging groups (only 20 sets instead of
all the data of the experiment) some low frequency trends were observed. These,
discussed in more depth in Sec. 5.2.10, could be the result of a higher frequency
mode of the super spring beating with the drop measurements.
To bound this error compare the theoretical amplitude of a spring excitedinstantaneously and one excited (roughly) adiabatically, following the discussion
in Faller[22]. The ratio of the Fourier components at the resonant frequency of thespring when released in these two ways will be equal to the amplitude of motion
at that frequency, so:
R0e,i!ot dt
ASudden =
,1
1
R0
R
ASlow
e,i!otdt + e,kt e,i!ot dt,10

where !o is the fundamental frequency of the super spring (about 0.1 rad/sec), andk is the decay rate for the equilibrium position (approximately 0.03 rad/sec). Plugging in the numbers we nd that the amplitude of the slowly-release spring would
be approximately one twentieth of amplitude of the suddenly released spring.
Thus an approximation of the signal amplitude would be 1Gal/20 = 0.05Gal.
A worst case scenario would be that a single drop would consistently be skewed
by this whole amount in each set. This would introduce a bias of 0.05/100 =0.0005Gal in each set (100 drops per set), and a di erence bias of 10 ppm.

94

1.5

1

0.5

Acceleration, µGal

0

−0.5

−1

−1.5

−2

−2.5

−3

A)

0

10

20

30

40

50
Drop number

60

70

80

90

100

1

0.5

Acceleration, µGal

0

−0.5

−1

−1.5

−2

−2.5

B)
Figure 5.12: Plots of drop-wise-averaged sets. (  ) Source mass in its upper
position, with mean value removed. (|) | source mass in its lower position withmean value and 1.5 Gal removed. A) Sets averaged for the 1997 data run. B)
Sets averaged for the 1998 data run.

5.2.7

0

10

20

30

40
50
Drop number

60

70

80

90

Magnetic E ects - Static Fields

An external magnetic eld will induce magnetization within any object with
non-zero magnetic susceptibility, resulting in a force on the object. This force isthe \direct attraction" of the object to the external eld. The magnetization will

95
also result in a perturbation to the original eld that depends on the susceptibility,
geometry, and position of the object.
In the free fall measurement a static magnetic eld, Bext, was generated by
the Earth and the permanent magnets of the ion pump and drive motor. The
source mass had a susceptibility of  = 0:00066 in MKS units. Thus the source
mass underwent some magnetization | it looked like a magnet that produced aeld of approximate magnitude Bext . This static magnetic eld could di erentially in uence the acceleration of the proof mass.
If a conducting object, such as the proof mass, is in motion through aspatially variant magnetic eld an eddy current damping e ect will occur. Eddy
current damping is a frictional loss resulting from the dissipation of energy by theaction of an electric current induced in a current loop by a time varying magnetic
ux. From Faraday's law we see that an electro-motive force (emf), " is generated:
jBj
(5.4)
" = ddt = A ddt
for a loop of area A within a time changing magnetic eld, B, if the magnetic ux,
, penetrating the loop is AB. Thus a loop moving through temporally static yet
spatially dynamic eld with velocity v will experience an emf related to the rateof change of the eld. If that the loop is moving in the z^ direction:dB = dB dzdt
dz dt= v dBdz ; so :" = vA dB
dz
For a loop of metal with resistance R the emf will result in a current I, and apower dissipation, P, of:vA dB
I = R" = Rdz
2 2 dB 2
2
P = "R = v A R( dz ) :96The power dissipated by the current is equal to the force acting on the object
times its velocity, so the eddy current force must be proportional to the velocity
and the gradient squared:vA2( dB )2
F = Pv = Rdz
The actual magnitude of the elds associated with the source mass, andtheir spatial dependence, were directly measured without the gravimeter present.
These measurements and the detailed calculation of the di erential eddy current
e ect are presented in Sec. B.2.1. Di erential eddy current damping arisingfrom all static elds associated with the source mass introduce error into the G
measurement only at the levels below pGal.

5.2.8

Magnetic E ects { Dynamic Fields

Time varying magnetic elds also exert forces on conducting loops. Although the magnetic susceptibility of the source mass is very moderate, as discussed in the previous section, its conductive characteristics cause it to shield theproof mass from most AC magnetic elds | there is no ameliorating factor of
0.0006 (the susceptibility) to reduce the signal. The entire force contribution of
any AC magnetic eld modulated by the source mass would be represented as adi erential error. After considerable work (described here) we found that errors
arising from AC elds were thousands of times smaller than our uncertainties.
The source mass only shields the test mass from elds propagating from thedirection of the source mass towards the test mass. Magnetic elds were produced
by AC currents owing in anything from a fan to the power conduits around theexperiment. The most important source of an AC magnetic eld was the servo
motor used to drive the co-falling chamber. This motor was located below, yetclose to, the region of free fall. Its eld would be largely shielded only when the97source mass was in its lower position.
Alternating magnetic elds acted on the test mass in three ways. They
caused a sinusoidal force that sprang from the sinusoidal variation in the ux intersecting the cross section of the proof mass, a repulsive force that was dependent
on the inductance of the proof mass, and a damping component that arose fromthe eddy current damping associated with the necessary spatial dependence of theA.C eld. The last was discussed in the previous section.
The sinusoidal and constant forces can be calculated in the following way,following [43, 44, 45]. A time variant magnetic eld, B sin (wt), impinges on acurrent loop with cross sectional area A, resistance R, and impedance L. Using
Lenz's, Faraday's and Kircho 's laws we see that the current, I in the circuit
satis es:

Bw cos wt = IR + L dI
dt
A solution to this di erential equation is:

I = R2BAw(R cos wt + wL sin wt)+ w 2 L2

(5.5)

the vertical force Fv on the loop is thenFv = (IXB )v= IBr
where Br is the radial eld. Assuming that the AC B- eld is generated by an
oscillating magnetic dipole, it is reasonable to use (for a small current loop) thisexpression:!t
Br = Bo sin
3
h

(5.6)

where h is the height of the loop above the source of the magnetic eld. Combining
Equations 5.6 and 5.5, we get an expression for the force on a current loop falling

98
through the magnetic eld produced by an oscillating dipole:!t B0 A! (R cos !t + !L sin !t)
F = Bo sin
h3 R2 + !2L22
2 L sin2 !t B 2 A!R cos(!t) sin(!t)
o= Bho3(A!
R2 + !2L2) + h3 (R2 + !2L2 )(5.7)

The second term of this equation represents an oscillating signal, while the timeaverage of the rst term is:
2 2L
Fr = 21 h3 (BRo2A!
:+ ! 2 L2 )Thus Fr is a repulsive force acting to push the loop away from the oscillating
dipole that generates the eld. This derivation is only accurate for a small rangeof motions. If life were simple we could express the eld produced by the motor
as an expansion of oscillating dipoles and then calculate, in theory, the resultant
force on the test mass. Unfortunately our system was so complicated that it wasnigh well impossible to perform the calculation.
To calculate the force it is necessary to know the time varying amplitude of
the radial and vertical magnetic elds at the position of the test mass. To do this
we must rst create a model for the eld produced by the drive motor. Direct
measurement showed that the motor creates a eld containing two frequencies.
The lower frequency is the same frequency as the rotation of the spindle. The
other is 13 times higher. These elds are caused by permanent magnetizations
within the motor (which has 13-fold symmetry). Thus the motor is easily modeled
as a series of oscillating dipoles. The amplitude of the elds depend only on the
speed of the motor.
This information only lets us model the elds outside of the vacuum can.
Inside the can, and inside the co-falling chamber where the test mass is truly
located, the situation is more complex. The skin depth, or thickness at which a

99
conductor will damp out approximately 65% of the amplitude of the eld is:s2 = !
where  and  are the permittivity and conductivity of the shielding material.
The attenuation of the eld depends on both the thickness of the shield andthe frequency of the signal. Because the motor signal is chirped the frequency
and amplitude of the original signal changes. The vacuum can is cylindrical so
the amount of shielding varies across its length. Because the proof mass falls,
the angle at which the signal intercepts the vacuum can changes, changing the
thickness of the shield even as the eld amplitude increases. Of course the relative
proportion of the vertical and radial components of the eld that make it throughthe shield change as well.
We could (and did) directly measure the eld at each point in the can,for each position/velocity of the proof mass (if we ignored the e ect of the cofalling chamber). This doesn't help with a theoretical solution because of thecomplexity of the proof mass, which is a far cry from a ring with a well de ned
area, impedance, or inductance. It contains aluminum, beryllium copper, tungsten
and glass. Portions of it are threaded. Portions are bored, bent, burnished, slitted,
slotted, slanted. So much for theory.
Presented with such a dicult theoretical situation it was necessary to experimentally determine the magnitude of the acceleration signal of the AC magnetic elds. Figure 5.13 is a picture and schematic of the apparatus used to do
this.
The apparatus consisted of a high precision balance8 that supported a Plexiglas beam. The beam was coated with a ne carbon power to reduced electro-statice ects. It was enclosed within a conductive shell, shielding it from air currents
This balance was previously used in an experiment to check whether gravity diminishes
above a spinning disk, [46].
8

100

A

B
Figure 5.13: The apparatus used to measure the force of AC magnetic elds acting
on the test mass. A) A schematic of the system. B) A photo of the apparatus. A
balance (center) with precision 0.0001 gram, or 1 part in 106 of the mass of thetest mass. The balance supported a beam that extended past its sides, into thevacuum can (upside down, on the right) and a cardboard box lined with aluminum
foil (on the left). In the box the beam supported a non-conducting counter weight.
In the vacuum can the beam supported an actual test mass from an FG-5 system(also upside down). Two coils (positioned around the vacuum can) provided AC
magnetic elds at a variety of frequencies. The change in the balance reading
represents the magnetic force on the proof mass. The balance, alone, was notsensitive to the elds.

101and electric elds. On one side of the beam was an actual FG-5 proof mass, on
the other side was a non-conducting counterweight. The proof mass was placedon a pedestal on the beam so that it was positioned within the same vacuum can(open to air) used in the G experiment.
Magnetic elds were produced by voice coils positioned around the vacuum
can and measured without the test mass in position. The elds were measured atthree positions inside the vacuum can at frequencies of 5, 10, 20, 40, 80, 160, 300
and 600 Hz. The low frequency component of the drive motor eld chirps from 0
to 40 Hz during a drop. The eld amplitude generated by the coils was as large as
could be produced with the ampli er we used. The balance showed no responseto these elds.
After this \ eld calibration" the proof mass was placed on the beam at
the same three positions. The resulting forces, measured by the balance, wereused to place limits on the drive motor's e ect on the test mass during the G
measurement. The largest force generated by this system was approximately 1.5
mg. At very low frequency (where the force is small) and at frequencies higher
than 300 Hz (where our ampli er had trouble driving the voice coils) the balancegave null results. Ground stainless rods were placed in the vacuum chamber tosimulate the e ect of the guide rods in the dropping system. The entire signalof the servo motor was 4 orders of magnitude smaller than the precision reachedwith the G experiment.

5.2.9

Contaminated Fringes

Generally neither the points at the very beginning nor those at the very endof the drop are used in the t for local acceleration. Data at the beginning is very
noisy due to airgap noise (Section 5.2.1) and missed fringes. When real fringes

102
were missed by the discriminator9 , the fringes recorded by the gravimeter areincorrectly scaled. Missed fringes are very easily identi ed | either by the acceleration value they produce, or by the chi-squared of the t. Figure 5.14 shows theacceleration parameter from ts to drop data, some of which was contaminated.
Only about 10% of the data has a missed fringe beyond fringe 15. That data wasalways removed for the nal analysis. At the other end of the drop the data could
be contaminated by the servoing of the co-falling chamber to catch the test mass.
1000

Acceleration, Gal

800

600

400

200

0

1000

2000

Drop number

6000

7000

8000

Figure 5.14: Plot of data with missed fringes. Banding due to the quantized
nature of missed fringes is easily seen. A single missed fringe results in a bias ofapproximately 170 Gal. The bias varies by 5 Gal depending on which fringe
was missed. The occasional lack of contamination (e.g. at drop 6000) occurs when
the amplitude of the interferometer output increases. Note that if the ts for g
were started just a few fringes later in the drop no mis-scaled fringes would be
included and no banding would occur.
Fringes were only missed before fringe #18. By processing data with an FFF beyond
that, the problem could be completely avoided. Of course that would mean giving up some
information. . .
9

103
The measurement of the absolute acceleration is much more sensitive toerrors arising from the inclusion of contaminated data than was the free fall G
measurement. In the di erential mode there is much greater freedom to choose
the data to t. A typical \start fringe" is (scaled) fringe #20, while a typical \endfringe" is #640.
At some choice of start and stop fringes all the data too poor to use (evenin the di erential mode) lay beyond the boundary fringes. It is not necessarily
clear, however, that the entire region of reasonable data should have been t. The
quality of the data can vary from the beginning of the drop to the end. As earlier
and earlier fringes are t, more and more drops become contaminated by missed
fringes. Although the statistical uncertainty in the t is inversely proportional to
the square root of the number of data points used in the analysis, more points do
not necessarily better results make.
One way to identify the fringes bounding good data was to evaluate thedependence of the local acceleration on the set of data t | a good start or stop
fringe lies in a region where the value of g is independent of the choice of fringe.
This is a technique commonly used in absolute gravity measurements. Figure5.15 contains a plot such as might used in this method. A similar system involves
examining the residuals to the t and directly identifying the point at which thecatch, for example, begins.
Because it is not the absolute value of the g value that interested us we useda very simple method to decide which data to use. The choice of start fringe thatcaused the uncertainty in the value of G (as opposed to g) to increase sharply
de ned the edge of \forbidden territory." The boundary fringes were #10 { #150
for the start fringe and #580 { #670 for the end fringe.

104

0.15

0.1

acceleration signal, µGal

0.05

0

−0.05

−0.1

−0.15

−0.2

−0.25
20

25

30

35

40

45
first fringe fit

50

55

60

65

70

Figure 5.15: Dependence of g and go on the rst fringe t. ({{) The absolute
acceleration, g (source mass in its upper position), as a function of the rst fringet (an o set value has been removed). (|) The acceleration di erence signal fromthe free-fall experiment as a function of rst fringe t (with o set removed). In
both curves the last fringe t was No: 640. Note that the curvature of the solidline and some drift of the dot-dashed line, is a signal generated by the source mass
{ and is expected.

5.2.10Unquanti ed Error Sources

We have thought of several poorly de ned mechanisms that could have inuenced our results but that we couldn't (and/or didn't) test. When the results of
the 1998 run displayed high set-to-set scatter (but not drop-to-drop scatter), we
tried to think of sources of systematic error that could change our measured value
for G on the time scale of a day. In this section a few of the possible mechanisms
we thought of are presented.
One of the rst mechanisms that came to our minds was a grounding prob-

105
lem between the experimental apparatus of the free-fall experiment and the FG-5.
Errors in the measurement of g as large as 10Gal have been attributed by Microg Solutions Inc. to unspeci ed grounding errors, generally involving the super
spring system. The mechanism that would result in a mismeasurement of theabsolute acceleration is not clear.
There certainly were opportunities for ground loops to occur in our lastmeasurement. The drop initialization trigger was used in the airgap measuringapparatus, in the FG-5 system, and in the shaft encoder output recording system. The shaft encoder voltage was actually read by the FG-5 in place of thesuper spring position. Additionally, the aluminum layer surrounding the dropping chamber was held at 1.5 volts over a ground referenced to the chassis of apower supply itself ground to the wall.
During the experiment, worried about such ground loops, I disconnected
the shaft encoder recording system for a quarter day, but didn't see any change
in drop-to-drop scatter. Since it was our set-to-set scatter that was surprising,
this really doesn't tell us much. Unfortunately the question of how ground loops
might a ect the scatter is left unanswered. It is very dicult to imagine a nonpathological mechanism that would link any grounding problems to the source
mass position.
A second mechanism that could in uence low-frequency drifts in our experiment was a random coupling of a higher order mode of the super spring to themeasurement. The super spring has a mode in which the spring that supports
the reference mirror twists. This mode has a frequency of approximately 0.3 Hz.
An amplitude of 7 Gal would correspond to drop-to-drop scatter of 5 Gal. The
decay constant of this mode is approximately 9 minutes. The twist mode is responsible for the majority of the g signal injected to the system by energy in the
super spring [47].

106
It is possible that the movements of the source mass maintained energy in thetwist mode without strongly changing its phase. Because a small vertical torqueis sensed by the oor when the source mass was moved, the excitation would bysystematic. Also, this would explain why the scatter in our G-values increased in
the data run of 1998, where there source mass was moved more frequently.
This mechanism is not very likely, however, because the amplitude of thee ect would be very small. The period of the torsion mode was observed to be
3.93  0.1 sec. At this frequency the worst case bias to the di erence signal would
be 0.17 Gal. The signal we saw in the data had a much larger amplitude than
this. The fact that no increased scatter was observed in the beginning of each set
as opposed to the end of the set (Section 5.3) seems to indicate that this wasn't animportant e ect. Finally, the gravimeter has been run continuously throughout
the movement of the source mass. As soon as the source mass motion ended, thescatter immediately returned to its value before the move.
Let us, nevertheless consider a pathological case in which the beat frequency
between the measurement rate (1/5 Hz) and the spring (the pendulum mode of
the spring was close to 1 Hz) were very low (less than 0.02 Hz). Then the bias
introduced by the spring would not be averaged out over many cycles in each set.
In the case that the beat was 0.015 Hz, the bias could easily be 0.4 Gal , in one
set. Assuming the phase and amplitude were maintained, the next set would be
biased by -0.4Gal .
Thus, in the special case that the frequency of measurement was very close
to a multiple of the spring frequency, the underlying signal could very well havebeen caused by phase drifts in the spring. It would not require a phase modulationon the time scale of the sets, but only on the scale of the drifts. The fact that thecoherence did not clearly extend between di erent days of data10 provides some10

Each morning I suspended the measurements and down loaded the previous day's data

107
support for this mechanism. Because the coherence was not completely continuouswithin a day, however, this is not a strong statement.
Examination of the Figure 5.16 shows that a low (mHz) signal can be seen
in some averaged sets. The sets were averaged in a region where the coherent
signal in the di erence data was clearly evident. This is consistent with the ideathat the spring mode was randomly phase locked with the experiment.
6

4

2

Acceleration, µGal

0

−2

−4

−6

−8

−10

−12

0

10

20

30

40
50
Drop number

60

70

80

90

Figure 5.16: Plot of acceleration value by drop averaged over 26 sets (13 withthe source mass down). Notice that there is an underlying signal in the datathat suggest the presence of systematic drifts linked to the start of each set. This
underlying systematic is consistent with the a random phase locking of the super
spring torsion mode and the measurement cycle of the gravimeter.
from the gravimeter computer. Sometimes I allowed the source mass to continue its motions,because this seemed to reduce wear in the support nuts. Generally it took about an hour todown load the data and reinitialize the gravimeter.

108

5.3

Evidence of other Systematic Errors

Systematic errors that do not perfectly mimic parabolic motion would leave
telltale signatures in the record of the test mass motion. For this reason I hoped
to unearth evidence of systematics by looking for anomalous signals in the drop
data. I used a variety of methods, which are presented here.
Any dependence of G on the portion of the drop we analyzed is evidenceof a systematic error. This is related to the work in Secs. 4.3 and 5.2.1, where
the examination of the dependence of G on the rst t fringe (FFF) was used to
extract information about the position of the start of drop and the airgap signal.
Plots of the dependence of G on FFF for two data runs are shown in Figures 6.7
and 6.1.3. It is also possible to look for the dependence on the last fringe t, as
in Figure 5.17, but it was more reasonable to analyze dependence on start fringe
than on end fringe because the start of the drop in uenced the t for accelerationmuch more strongly than the end. In fact the rst half of the data accounts for
90% of the information about the acceleration. Also the source mass signal wasmuch more clear at the beginning of the drop than at the end.
Other signs of corrupting signals may be found by analyzing the g values
measured with the mass up, as opposed to the mass down. A di erent amount ofdrop-to-drop or set-to-set scatter could indicate a systematic e ect. No signi cantdi erences were found. The drop-to-drop scatter was also compared between the
beginnings and ends of sets of data, and was identical, suggesting that there was
no appreciable excitation of the super spring during the movements of the source
mass. In Section 5.2.6 sets of data representing the average set with the source
mass either up or down were computed. The acceleration values of the nth dropof every mass-up set were averaged together to give the value of the nth dropof the average mass-up set. This was also done for the source mass in its lower

109

δg signal, µGal

84.5

84

83.5

83

560

580

600
Last fringe fit

620

640

660

Figure 5.17: A comparison of delta g as a function of last fringe t to theory. The
rst fringe t was fringe 20. This data is extracted from a single day of data in
the 1998 data run. The plot with error bars and circles is the experimental curve,
while the smooth line of stars represents theory. Note the small sinusoidal signal
in the data. I attribute this to the oscillating airgap signal.
position. The averaged sets are shown in Figure 5.12. These sets would re ect
the presence of any systematic error parameter that would shift from one value toanother over the time of a single set. Thermal signals coupled to the source mass
position, for example, could slowly change the temperature of the ion pump, inturn changing its eciency, the vacuum pressure, and the residual gas forces on
the test mass. No evidence of these drifts is seen in the averaged results. Figure
5.16 shows the set-average for a short segment of suspicious data. Less than 20
sets were averaged for this plot which does display drifts.
Another test for systematics involves using the only half the data: Computing the di erences using just the second half of each set and comparing them tothe di erences calculated using just the rst half of each set would allow linear
drifts (in g, as a function of the time from the start of the sets) to be identi ed.
Processing all the 1998 data in this way resulted in two values for G (one value

110
using the rst half of set data, one value with the second half of set data). The
two values were consistent. The di erence between them was (0.007  0.01) x
10,11 m3/kg-s2 .

5.3.1

Residual Analysis

Valuable information remains in the residuals to each t of drop data. Eacht produces values for the parameters of the parabolic function Eqn. 2.10. By
subtracting the curve speci ed by these parameters from the data, a set of residual
values is found, as in Fig. 5.18.
Information about various systematic errors may be extracted from theseresiduals. In the method of Klopping [39] the set of decaying sinusoids is assumed
to describe many systematic errors in the measurement of g. Decaying sinusoidal
signals might arise from vibrations excited by the drop, as in Secs. 5.2.3 and5.2.1. A damped sine wave apparent in the test mass motion would clearly not
result from any reasonable gravitational source.
Klopping developed a system that t and removed decaying sinusoids fromthe data. The frequencies of the sinusoids were found by spectral analysis of theresiduals to the drops, then the damping rate and amplitude were found with nonlinear tting techniques. His method is cited by Micro-g Solutions as an accepted
provider of corrections for the airgap e ect and is used to correct for di erent oor
responses at absolute gravity sites.
We hoped to use Klopping's software to remove the airgap e ect, and anyother unknown signals. I wrote a program that read in the frequencies and decay
constants11 of the four largest decaying sinusoids for each set of data and then used
The frequencies and decay constants were provided by a modi ed version of the TREX.EXE
software that is standard to the FG-5 system. TREX.EXE is only con gured for use with low
data density data (180 data points per drop). The modi cation allowed permitted use with our
high-density data (700 data points per drop)
11

111

−8

3

x 10

2

Residuals, cm

1

0

−1

−2

−3

−4
0.02

0.04

0.06

0.08

0.1

0.12
0.14
Time, seconds

0.16

0.18

0.2

0.22

Figure 5.18: The residuals-to-drop averaged over 150,000 drops. The upper curveis the average residual, including data from both source mass positions. The
choppy middle curve is the di erence between the averaged residuals with thesource mass in its upper and lower positions (with o set included). Note thatthe slight \M" shape of the di erenced residuals is due to the higher order e ectsof the source mass (as in Fig.2.2). The theoretical expectation for the di erence
in residuals is shown with the smooth curve overlying the di erenced residuals.
The lowest curve is the di erenced residuals with the model removed (and o set
added). Any signal visible in the lowest curve represents the trace of a systematic
error.
them to provide set corrections for the data. The set corrections would remove
the airgap e ect from all the data and the airgap problem would be solved.
This method of attack was unsuccessful. The set corrections did not reduce
set-to-set scatter of g-values signi cantly, suggesting that the corrections calculated by this analysis were not valid. Also the uncertainty in go grew very large

112
due to the inclusion of many tting parameters (a damped cosine and sine function for each decaying sinusoid; a total of 8 extra parameters) in the least-squares
t. The lowest frequency of the sinusoids was 10 Hz | low enough to correlatestrongly with the cubic and quartic components of the perturbing signal (as inFig. 2.2, part B). This meant that the sinusoids would remove part of the source
mass signal as well as any airgap errors.

CHAPTER 6DATA AND RESULTS

6.1

Data Runs

We used the free fall method in three data runs, presented here in chronological order. The rst was a proof-of-concept experiment that I call the \bronze
mass run." The second acquired 9 days of data over a 13 day period in early May
1997. The third run incorporated several improvements to the apparatus and was
conducted in late May of 1998.

6.1.1

The Bronze Mass Experiment

In the summer of 1995 David Newell and Paul Keyser, both graduates of
JILA in physics, developed a system that allowed a bronze ring-shaped mass to beplaced above and far above an FG-5 free fall gravimeter. In this way a di erential
measurement of the ring's gravity eld could be made, but with only (at best) one
half the optimal signal-to-noise. The system was built in JILA and was sucient
to show that the acceleration of gravity, as measured with a free fall gravimeter,
could be perturbed with an external source mass.
I modi ed the apparatus in order to allow the source mass to be placed bothabove and below the region of the drop. The modi ed apparatus was portable,
and was setup at the Table Mountain Gravity Observatory (TMGO), a muchbetter (vibrationally quiet) site for gravity measurements than JILA. Figure 6.1

114
is a picture of the setup. This arrangement did not make provisions for highaccuracy positioning of the source mass.
The source mass of this run was made of two bronze rings that had beenused in two previous G experiments. The rst experiment was that of Michael
Hulett, who in 1969 measured the de ection of a horizontal pendulum due thegravitational attraction of the rings. His work was performed for his senior honors
thesis with James Faller.
The bronze masses were reused in a G experiment that used a magnetic
support system to escape the relatively sti torsion constants of a ber. This
experiment was conducted by William Koldewyn in the mid 1970s, again withadvisor Jim Faller [9]. Koldewyn had the masses precision machined and carefullymeasured their volumes. We used his density values in our theoretical modelsof the system. The two rings weighed approximately 100 Kg (combined), andproduced an acceleration signal of approximately  10Gal. We were very luckythat the bronze rings were just large enough to t over the vacuum chamber ofthe gravimeter, but we were not able to get the mass low enough to reach its lower
optimal position | the rings were too tall.
The lifting arrangement relied upon a counterweight to balance the weightof the source mass. The translation of the chain and counterweight meant that the
movement of the source mass could not be treated simply as a translation of theperturbing acceleration eld about the start-of-drop position. This complication,
coupled with an ugly system of bearings, hangers, and plates, made the massarrangement dicult to model accurately.
The FG-5 gravimeter we used had a standard (at the time) data acquisition
system that was only capable of recording 180 position-time pairs for each drop of
the proof mass. Nevertheless the drop-to-drop scatter in the data was not muchworse than the best data we would ever take (only about 20% greater scatter). A

115

Figure 6.1: A photo of the source mass and a portion of the lifting assembly of
the bronze proof-of-concept experiment. Note that the mass is made only two
cylinders. It was supported by a chain, and was guided by three rods.

116
sample of the data is plotted in Figure 6.2. This data acquisition system had someother disagreeable properties. There was a delay of approximately 30 milliseconds
between the start of the drop and the start of data acquisition. This was dueto the time required for the system to initialize after receiving the trigger signal
indicating the start of the drop. Thus the fringes recorded by the system were notwell connected to the position of the start of the drop. The system could not keep
up with the fringe data from the interferometer, and would miss fringes towards
the end of the drop.
Because our software implementation of the theoretical system was not completed at the time of the bronze mass experiment the source mass was not correctly
placed in its optimal positions. The data from this run were taken with a variety
of upper and lower source mass positions, including some scans where the source
mass was slowly stepped through its range.
All in all the bronze mass run was a valuable learning experience but notmuch more. Figure 6.3 shows a plot of the results of this rst data run. The
results are clearly \in the ball park" of the correct value of G, though they arealso only of proof-of-concept quality.
The bronze mass experiment illustrated the need for a robust system withimproved signal-to-noise and a simpler mass arrangement (for modeling). During
1996 we worked on the completion of the software system as well as the designand construction of the apparatus to be used in the actual measurement.

117
40

Acceleration, Gal

30

20

10

0

−10

−20

−30

−40

0

2

4

6

8

10

12

Time, hours
Figure 6.2: A 13 hour sample of acceleration data from the bronze proof-of-concept
experiment. Note that the signal of the bronze ring source mass has an amplitudeof only 10 Gal.
7

6.9

G X 1011 , m3/kg−s 2

6.8

6.7

6.6

6.5

6.4

0

2

4

6

8

10
Day number

12

14

16

18

20

Figure 6.3: Results of the bronze mass proof-of-concept experiment. The value
marked with an asterisk represents the nal value of the data run. For comparison,
the accepted value of G is 6.6726  0.0009 X 10,11 m3/kg-s2 .

118

6.1.2

May 1997 Experiment

The May 1997 experiment used the lifting assembly and apparatus describedin Chapter 3. By this point we had developed a sophisticated software system tocalculate the proper source mass positions and to analyze data. The gravimeter
was equipped with a new data acquisition system that allowed four times denser
data than could be recorded previously. This was expected to reduce the drop todrop scatter of acceleration measurements by close to a factor of two (sqrt(4) = 2).
The high-speed data acquisition system solved the problems with missed
fringes late in the drop and the uncertainty in the delay between the start of drop
and the start of data acquisition. There was still a delay for the fringe signal
frequency to rise past 700 Hz (where it could trigger the discriminator) but thiswas a negligible error source (a fringe signal above 700 Hz is achieved in just a
few microns of vertical displacement).
The nine days of data that we took (over a 13 day period) gave the results
shown in Figs. 6.4 and 6.5. In order to compare the internal and external scatterof the data I averaged individual di erences in a variety of bin sizes. With a bin
size of 20, 30, 50, and 81 the external scatter was 1.5, 1.3, 1.6 and 1.3 times aslarge as the internal. Averaging these results indicates that a reasonable ratio is
1.4; there is more variation in the G value from bin to bin than would be expectedfrom the scatter within each bin.
The di erence values (Fig.6.6) produce a power spectrum that is at, however, suggesting that there was no unusual power at low frequency (that could
strongly bias our result). The lack of a clear drift over the whole data set supported this assumption. Fitting the individual di erence with a line gave a driftin go equivalent to 0.20  0.13 Gal from the start of the data run to the end(13 days). The drift estimate is suggestive, but not clearly indicative of a drift.

119
The reduced chi-squared comparison of the daily results to a Gaussian (aspresented in [48, 49]) returned ~2 = 0:8 (40% probability). This indicates good
agreement with a Gaussian. Calculating the reduced chi-squared from the individual di erences, however, returns a value of ~2  3 (10% probability).
These results do not damn the 1997 data run, but they do suggest thepresence of either a random coupling of environmental signals to the G-experiment,or the random modulation of an unknown systematic. Because the experiment
was run during a hot spell in Colorado we attributed the low-frequency scatter to
air conditioning cycles correlating with the experiment. The air conditioning at
the site was observed to switch on and o at slightly higher frequencies than thesource mass was modulated.
An estimate of the bias introduced by the underlying signal could be madeby modeling it as a sinusoid in the data and calculating the o set introduced
by a fractional period. Alternately an estimate of the e ect could be made by
selectively removing bins of data, and recording the e ect on the mean. These
two methods returned estimates of the bias that agreed to 10%. Using the second
method, we adopted an estimate of 450 ppm for the random drift bias error in
the 1997 result.
The nal result of the 1997 was calculated in a more complicated way thanby simply averaging di erences. Instead G was calculated as a function of the rst
fringe t (FFF) from the go curve found in Section 4.3. Figure 6.7 is a plot of
this curve. The value of the constant produced by the data depends on the choiceof the rst fringe t. I characterize this dependence as weak because the standard
deviation of the values computed with di erent FFFs is 1/3 of the uncertainty
of each result with a single choice of FFF. A component of the \noise" between
the di erent start fringes was due to the fact that we extract G from a slightly
di erent sets of data for each value.

120
85

84

Difference signal, µGal

83

82

81

80

A

79

0

50

100

150

200

250
300
Difference number

350

400

450

500

550

45

40

35

30

Count

25

20

15

10

5

0
79

δ g value, µGal
B
Figure 6.4: Data from the 1997 experiment. A) Di erences as measured in Sec
4.2.1. The dotted line represents the individual di erences, while the solid line isthe di erence value averaged over blocks of 20 individual values. B) Histogram ofthe di erences. The solid line is the best t Gaussian curve.
80

81

82

83

84

85

121

6.72

6.71

6.69

G X 1011

3

m /kg−s

2

6.7

6.68

6.67

6.66

130

132

134

136
Day

138

140

142

Figure 6.5: Results of the May 1997 experiment. Daily results. The value marked
with an asterisk represents the nal value of the data run. The error bars represent
a single standard error, only. The lowest horizontal line represents the 1995 New
Zealand result [15], the middle line is the accepted value [4], and the upper lineis the 1995 German result [14].

122

10

Power Spectrum Magnitude (dB)

5

0

−5

−10

−15

−20

A)

0

0.1

0.2

0.3

0.4

0.5
Frequency

0.6

0.7

0.8

0.9

1

450

500

550

82.4

82.3

82.2

Acceleration, µGal

82.1

82

81.9

81.8

81.7

81.6

B)

81.5

0

50

100

150

200

250
300
Difference number

350

400

−5

Power Spectrum Magnitude (dB)

−10

−15

−20

−25

−30

−35

C)
Figure 6.6: Power spectral analysis of individual di erences from the 1997 experiment. A)The power spectrum from di erences calculated using method two(of Sec 4.2.1). Only half of the di erences were used in the calculation to avoid
any correlation between the individual values. B)The di erence values averaged
in groups of 20. This corresponds to periods of approximately 7 hours. C) The
power spectrum of the binned data. The power spectrum with bins of 3.5 hours
looks much same, with just a bit more de nition.
0

0.1

0.2

0.3

0.4

0.5
Frequency

0.6

0.7

0.8

0.9

1

123
The variation in the value raises the question of which value is appropriate
to summarize the result. Because we don't expect that G be physically dependent
on the FFF, a t of all this data to a at line gives a reasonable indicator (theweighted mean value) for our best estimate for G.
Some portion of the dependence of the measured G on the FFF was systematic. The data points in Fig. 6.7 are not completely independent, as is especiallyobvious between fringes 24{30 and beyond fringe 60. An estimate of the sensitivityof our \best value" to possible systematics in the data can be made by calculating
the change in the value when di erent portions of the curve are excluded from thecalculation of the mean (this assumes that there is no signi cant DC systematicbias). Figure 6.8 is a plot of the error estimate in G as a function of start fringe.
The two \especially suspicious" regions between fringes 24{30 and 60 onwards aremarked. The mean value is very insensitive to the G-values in these regions, inpart because of their larger uncertainties. The weighted mean value shifts only
1/10th of a standard uncertainty when these data points are excluded. The pointthat the exclusion of this data may be considered arbitrary is moot because ourresults are insensitive to this manipulation.
The value of G extracted from the 1997 run is the weighted mean of thedata shown in Figure 6.7 with a statistical uncertainty of 450 ppm. The statisticaluncertainty was based on the complete data set evaluated from fringe 20 to 640 (aregion chosen for its compatibility with absolute gravity measurements). The error
bar includes estimates of error sources as discussed in Section 5.2 and Appendix
B. The 1997 result is:
G = (6:6863  0:0015  0:0042  10,11m3=kg , s2
where the rst uncertainty estimate is the root sum square of the systematic
errors of Table 5.1 and the second uncertainty is the random error associated with

124
6.7

G X1011 m3/kg−s2

6.695

6.69

6.685

6.68

6.675

A.

20

40

60

80
Start fringe

100

120

140

160

20

40

60

80

100

120

140

160

6.69

6.689

6.688

G X 1011 m3/kg−s2

6.687

6.686

6.685

6.684

6.683

6.682

6.681

Start fringe
B.
Figure 6.7: G as a function of rst fringe t to a xed nal fringe (#640). A)
With error bars associated with the analysis of the complete 1997 data set witheach particular choice of rst t fringe. B) Without error bars.

125
−3

8

x 10

7.5

Error esimate (m3/kg−s2 X 1011)

7

6.5

6

5.5

5

4.5

4

3.5

A.

0

20

40

60

80
Start fringe

100

120

140

160

6.691

6.69

6.689

6.687

2

11

G (m /kg−s X 10 )

6.688

3

6.686

6.685

6.684

6.683

6.682

6.681

Start fringe
B.
Figure 6.8: Statistical uncertainty in G as a function of rst fringe t to a xed
nal fringe (#640). A) The error estimates plotted against start fringe. The
vertical lines are at fringes 24, 30, and 60. B) Error (|) scaled for comparison to
the actual G values ({{) error.
0

20

40

60

80

100

120

140

160

126
a bias in the data from the underlying signal and the statistical uncertainty. The
combined statistical and systematic error estimate is 0.0045 x 10,11 m3/kg-s2 (670
ppm).

127

6.1.3

May 1998 Experiment

We made several improvements to the apparatus for the 1998 data run.
Concerned about thermal signals we added a layer of insulation (Cryotex) aroundthe whole lifting assembly and a separate loose thermal shield for the laser andelectronics system of the gravimeter. We increased the frequency of modulationof the source mass position by nearly a factor of two, decreasing the correlationof random environmental signals and machine drifts to the G measurement.
One way temperature signals could in uence the scatter in the free fall experiment was by causing tilts of the laser verticality (see Section 5.2.4). For
this reason we used a ber-optic interferometer system that had shown improved
verticality stability over the standard FG-5 interferometer(see Sec. 3.1.3 for discussion).
We directly measured the motion of the air-vacuum interface throughout
the experiment. We hoped that this auxiliary data would help us to place bounds
on the di erential airgap signal (Section 5.2.1). In addition to airgap data we
logged the shaft-encoder voltage (Sec. 3.1.1.2) and ion-pump voltage1 for eachdrop.
We attempted to increase the precision of the experiment by using a super
spring that had demonstrated superior performance, and by doubling the dropping
rate from a drop every 10 seconds to one every 5 seconds.
We made several \miscellaneous" changes. To remove the magnetic signal ofthe drive motor (Section 5.2.8) and reduce magnetic damping of the airgap motion(Section 5.2.1) magnetic shielding was placed around the motor and ion pump.
To improve the performance of the lifting assembly and decrease deformation of
the source mass the support plates (Section 3.2) were replaced with thicker plates
The ion pump voltage gave a crude estimate of the vacuum pressure in the dropping
chamber.
1

128with extra support rods.
Figures 6.9 and 6.10 show the results of these improvements. The scatter in
set-to-set g values increased by a factor of 1.9. The the scatter in go increased bya factor of 1.5. The increase in scatter was not seen at the high frequency of drop
measurements, but only in set-to-set scatter. The ratio of internal to external
scatter increased to 1.9. The value of G extracted from this data, however, wasconsistent with the 1997 value. We don't know what caused this unexpected
increase in scatter.
There was some coherence between the individual g di erences, as in Figure 6.10. The histogram of the whole set of data, however, displays a normal
distribution (Table 6.1 gives the major moments of the distribution, with \typical" values for the kurtosis and skew), and the power spectrum (Fig. 6.11) isreasonably at. This suggests that the correlations are random. A t of a line tothe data gives a drift value corresponding to a change of 0.065  0.10Gal fromthe beginning to the end of the data set (15 days). Assuming that the signal was
random, a bias error estimate of 0.004 x 10,11 m3/kg-s2 (600 ppm) was adopted.
This was calculated in the same way as in the 1997 data run.
MomentMean
Average Deviation
Standard Deviation
Variance
Skewness
Kurtosis

Value
Typical82.8633 Gal
0.90 Gal
1.16 Gal
1.34 Gal2
0.12
0.1
0.34
0.24

Table 6.1: Main moments of the 1998 experiment distribution of g observations.
The values for the kurtosis and skew are close enough to \typical" values to beconsidered normal.
The 1998 data displays much the same scatter in G as a function of rst
fringe t as the 1997 data (Figure 6.1.3). It, too, displays coherence between

129

6.76

6.74

G X 1011, m3/Kg−s2

6.72

6.7

6.68

6.66

6.64

6.62
138

A.

140

142

144

146

148

150

152

154

156

87

88

Day
250

200

Count

150

100

50

0
78

79

80

81

82

83

84

85

86

δ g value, µGal
B.
Figure 6.9: Results of the 1998 data run. A) Daily results. The horizontal lines
represent values for G from other recent experiments: The upper line is the PTB
result, the middle line is the accepted NBS result, and the low line is the New
Zealand Standards result. The daily results re ect the coherence obvious in Fig.
6.10. B) A histogram of the individual 1998 g di erences. The smooth curves
represent the normal probability for the mean and variance of this data. The
larger curve consists of 1660 points in 30 bins, while the lower curve is the samedata in 70 bins.

130
88

Difference signal, µGal

86

84

82

80

78

0

A.

200

400

600
800
Difference number

1000

1200

1400

84.5

84

Difference signal, µGal

83.5

83

82.5

82

81.5

0

200

400

600

800

1000

1200

1400

1600

1800

Difference number
B.
Figure 6.10: Individual di erences from the 1998 data run. A) The di erences,calculated using method 3 of Sec. 4.2.1. B) The di erences averaged in bins ofwidth 40. The dotted line is a running average, while the solid line is simply
binned data.

131
12

10

Power Spectrum Magnitude (dB)

8

6

4

2

0

−2

−4

−6

A)

0

0.1

0.2

0.3

0.4

0.5
Frequency

0.6

0.7

0.8

0.9

1

1.5

1

Acceleration, µGal

0.5

0

−0.5

−1

−1.5

B)

−2

0

200

400

600
800
1000
Binned difference values

1200

1400

1600

5

Power Spectrum Magnitude (dB)

0

−5

−10

−15

−20

−25

C)
Figure 6.11: Power spectral analysis of the individual di erences from the 1998
experiment. A) The power spectrum from di erences calculated using method 2of Sec. 4.2.1, but only every other value was used in the power spectrum to avoid
correlated data at the higher frequencies. B) The di erence data was placed into
bins representing 3.6 hour periods. C) The power spectrum of the binned data.
0

0.1

0.2

0.3

0.4

0.5
Frequency

0.6

0.7

0.8

0.9

1

132
the values. Once again the weighted mean value of G given by these results isinsensitive to exclusion of suspicious data, changing by less than 1/10th sigma
when the data is removed. The weighted mean of this data (associated with 700
ppm statistical uncertainty and adjusted with the airgap correction) forms the1998 data run result:G = (6:6896  0:0013  0:0062)  10,11m3=kg , s2
where the rst uncertainty is root sum square of the systematic errors of Table 5.1
and the second uncertainty is the RMS error associated with a bias in the datafrom the underlying signal and the statistical variation of the data. The combined
statistical and systematic error estimate is 0.0063 x 10,11 m3/kg-s2 (940 ppm).

6.2

Combined Results

The large underlying signal in the 1998 data could represent a randomly
modulated systematic error. If this was the case there could be a systematic biasthat we would not be able to identify in the data. Consider the hypotheticalsituation that our assessment of the thermal coupling between the source mass
and gravimeter was vastly underestimated. Clearly the coupling would not exist if
all components of the gravimeter and the source mass were the same temperature.
If there was a xed o set in the source mass temperature, however, a constant bias
to the g signal would occur. And if the source mass temperature had a constanto set with a random time varying component data such as we've seen might haveresulted. It is not clear that there was such an o set, but to be properly cautious,
we must deal with this possibility.
The only information that we have that there is no low frequency or DC
bias in our results is the agreement between our two data runs. The fact thatthey were not completely independent is unimportant because the scale of known

133

6.7

G X 1011 m3/kg−s2

6.695

6.69

6.685

6.68

6.675

20

40

60

80
Start fringe

100

120

140

160

80
Start fringe

100

120

140

160

A

6.695

G X 1011 m3/kg−s2

6.69

6.685

6.68

20

40

60

B
Figure 6.12: G as a function of rst fringe t to a xed nal fringe (No: 640).
A) With error bars associated with the analysis of the complete '98 data for eachparticular value of the rst fringe t. B) Without error bars.

134
systematic errors is very small compared to statistical and possible systematic
uncertainties. We can make an estimate of our uncertainty arising from such a
low-frequency bias by taking the di erence of the two runs. This di erence is only
400 ppm (the two results don't di er in a statistically signi cant way), but theuncertainty in the di erence is 1100 ppm (0.11%).
For the nal result of the free fall experiment we combine the 1997 and1998 data runs. The two experiments were not fully independent; they used thesame tungsten masses arranged in roughly the same orientations. Therefore thecorrect combined error of the two data runs is slightly larger than the uncertainty
in the mean of the combination. Adopting the uncertainty in the di erence as
an estimate of a possible underlying systematic o set increases the uncertainty inthe mean to a level were these distinctions are unimportant.
The combined nal result for the free fall experiment is:
G = (6:6873  0:0076)  10,11 m3=kg , sec2 :The uncertainty corresponds to 1150 ppm, 0.12%. This result is shown with all
the daily results and some other recent results in Figure 6.13. This value for theconstant lies 1.5 standard errors above the accepted value, and 3 standard errors
below the German PTB result. Although this result does not uphold the PTB
value at all, it does lend some support to the region above the accepted value.

135

6.76

6.74

6.72

6.7

G X 1011

3

m /kg−s

2

3

6.68
5

1
2

6.66

4

6.64

6.62

1997 Results1998 Results

Recent Results

Figure 6.13: A plot of G results. At the right side of the gure, recent results: 1.
NBS. 2. New Zealand Standards. 3. PTB. 4. Wuppertal. 5. Zurich. The nal free
fall result is the value marked with a circle, and includes all error estimates. The
other results marked with crosses are the daily results of the free fall experiments.
The error bars of the daily results represent statistical uncertainty, only. The
weighted mean result of each data run is marked with an asterisk. These values
include the error estimates of Table 5.1, statistical uncertainty, and the estimateof bias errors from the underlying signals.

CHAPTER 7
CONCLUSION

The free fall determination of G is conceptually simple. An idealized model
describing the motion of a falling mass has been well understood for literally
hundreds of years. The study of objects moving in gravitational elds begins at
the start of an education in physics.
Yet somehow G measurements are full of complexity, subtlety, and beauty| and the free fall experiment is no exception. The complexity of the experimentarises from the fact that our measurement pushes on the boundary between theidealized model and the hard reality of measurement. The calculation of themutual attraction of the source mass and the proof mass involved over 6000 sixdimensional integrals calculated at 1000 positions along the drop range of the testmass.
The subtleties of the experiment are found mostly in the search for systematic errors. The source mass in uences the resonances of the lifting assembly indi erent ways, depending on its position. These resonances, which extend to below 70 Hz, a ect the response of the oor to the changing weight of the dropping
system. The weight changes because portions of the dropping chamber (the cofalling chamber and the proof mass) become weightless during the drop. Thus thechange in source mass position a ects the airgap measurement (Section 5.2.1).
The beauty of the free fall method lies in the hints of simplicity that remain

137
in the experiment. The fact that the sensing mass of the experiment is unsupported means that dependencies on material properties are greatly reduced if notcompletely removed. The fact that the measurement is di erential means that
common mode errors of gigantic magnitudes can be tolerated. The modulation
rate of the source mass pushes the gravity signature of the source masses beyond
the frequencies of most environmental signals.

7.1

Results
Our nal value for G is
G = (6:6873  0:0076)  10,11 m3=kg , sec2 :This result incorporates error estimates of random drift bias errors and an estimateof a DC shift between the results of the two data runs. This was not the ideal
situation, as made clear by Churchill Eisenhart:
I realize that some of the subjectively estimated uncertainties tabulated by experimenters are intended, when combined \in quadrature" with the statistical standard error of the mean, to provide an
in ated \standard error" of the mean that can be used to \provide
an indication of how closely the result of this experiment is likely
to agree with other results that might be obtained if the entire
experiment were carried out afresh `from the beginning,' " butexperience shows that this is a delusion: there is no substitutefor comprehensive realistic repetitions of the entire experiment!
|Churchill Eisenhart [50]
It is unfortunate that the gravimeter system was not available for in depth
testing with the source mass system, or for additional experimental runs. In somesense it was this lack of time with the gravimeter that was our greatest nemesis.
It was for this reason that we had to adopt the di erence in the results of the1997 and 1998 data runs as an estimate of systematic bias errors. This estimate

138
is not unreasonable:
The uncertainty assigned to a precision measurement ideally should
not be solely determined by random measurement errors, butshould partly be the result of subjective judgments by the experimenter on the accuracy of his work." | Abstract by Peter
Franken [51]7.2

FutureAt the time that we did our experiments it was not quite clear to what
precision we would be able to extend the performance of the FG-5. Now, withthe lifting assembly packed away and the tungsten masses waiting for their next
leading role, I believe I can clearly see the future of the free fall method.
There are only two ways in which the free fall experiment will be continued. In a series of small improvements, or in a single big step. The combined
uncertainty of our results is 1300 ppm, the systematic errors arising from mass
homogeneities, modeling errors, and positioning (internal to the source mass) only
rear their ugly heads at the level of 100 ppms. Clearly our results can be improved
dramatically if the sources of the underlying drifts could be identi ed and controlled.
If the random signal were removed from our data and questions of lowfrequency biases were laid to rest then a result at the level of 300 ppm could beachieved with the present system. Clearly our data would be close to this level if
not for the underlying signals.
I have thought of several lines of attack that might be used to track down
the drift source. For example, \double moving" the source mass in between setsmight give information about spring e ects. After each set of data the source mass
would be moved half way to its alternate position, and then immediately moved
back to its original position. Increased scatter in this situation would indicate

139that the move was linked to the noise source.
Once the source of the drift was identi ed, however, and a measurement at
the level of 300 ppm completed, further improvement of more than a factor of
two would be very dicult to achieve. In this context many of the improvement
ideas that I entertained are simply not worth the e ort. The \return" on small
improvements to the system would not be worth the investment. A dramaticscale-up of the system would be more interesting.
A large scale-up of the experiment would involve a large increase in thetime, money and energy spent on the determination. The experiment would be
performed in its ideal incarnation, incorporating many ideas independently of
the constraints of weight and portability that in uenced the current gravimetersystem.
The entire dropping assembly would be custom designed to operate at asingle location. It would include a simpli ed proof mass less sensitive to eddy
currents, and counterweighted to reduce vibration. The position of the test mass
would be recorded at every fringe crossing, not each 500th. The interferometersystem would be used to reference the start-of-drop position to the source mass.
External magnetic elds would be removed either with shielding or active control.
The vacuum pump could be moved far away from the rest of the system. The
laser would be, for example, a frequency doubled Nd:YAG green laser. This typeof laser can be stabilized to better stability than the HeNe, without the large
frequency dither of the current system.
The source mass would be located in the same evacuated room1 as thetest mass. The source mass would be made not of just twelve tungsten billets,
but of hundreds of smaller precision machined objects placed as close to the dropas possible. They would be supported by a system capable of switching their
1

Including thermal isolation, naturally. A Big Step!

140
positions after every drop.
A custom super spring would be made for the experiment. Perhaps an multidimensional active isolation unit as have been developed for use in gravity wave
detectors could be incorporated into its design [52]. The spring could be located
far from the dropping system (perhaps in the same vacuum) to reduce couplingof oor motions.
It is certainly reasonable to wonder whether such a costly experiment using
the free fall method will ever be carried out. The free fall measurement currently
su ers from lower signal-to-noise ratio than some other comparable experiments.
It also also has exhibited unexplained drifts. But the great characteristic of themethod is its use of an unsupported mass in a system that, in theory, is extremely
simple.
There are several G-experiments that are expected to produce results by theend of this year. If their results are consistent, and the value of G can be statedwith con dence and precision, then I don't believe there is a strong future for thefree fall method. If the results continue to show great deviation in G, however,then I believe that the conceptual simplicity of the free fall method, coupled with
its lack of dependence on material properties, will provide reasonable impetus for
an improved experiment.
I know what must be done to solve the problems inherent in the measurement of G. Intelligent thought, unrelenting e ort, big money, and a dose of luck
must be pooled for one carefully planned, constructed, and executed measurement. I hope that the work presented in this thesis will help future experimenters(as it helped me) to grasp the challenge of the measurement of G, and to recognize
some of the possible pitfalls in that measurement.

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APPENDIX A
MASS METROLOGY

In an experiment in which we measured the mutual gravitational attractionof two complex masses it was important to describe the mass distributions as wellas possible. If the size or mass of one (or both) of the objects were assigned anincorrect value, then our theoretical model would misrepresent the actual physicalsituation. Correctly describing the objects may sound like a simple task, but it
was not. Density variations internal to the masses correspond to errors if they
were not accounted for in the models. Unmodeled density variations could arise
from voids in the material and from density gradients that occur due to coolinge ects. Not to mention that both the source and test masses (our two objects)
were made of many of eensy-weensy bits and pieces.
The source and proof mass were each subject to errors that di erently a ect
the experiment. The source mass created an acceleration eld that was sensitive
to the radial placement of the 12 large tungsten cylinders. The large volumes of
the cylinders lent themselves to density variations. The proof mass sampled theeld of the source mass over an extended volume. The mass of each component
acted as a weighting-value for the acceleration contribution of that component. If
the components were mislocated they sampled a slightly di erent eld than wasmodeled.
This appendix contains the details of the metrology of the tungsten cylinders

147
and the modeling of both the source and proof mass. First I discuss the source
mass and then the proof mass.

A.1

Metrology of the Source Mass

The most important contribution to the mass of the source object is fromthe 12 tungsten alloy cylinders that form the bulk of its volume. The cylinders
were custom made for the experiment by Schwarzkopf Technologies Corporation,a well known producer of tungsten products. Because tungsten has an extremely
high melting temperature most tungsten products are sintered rather than cast.
The sintering process glues small tungsten particles together with a binding
agent; in our case the binding agent was a mixture of copper and nickel. A wellmixed powder of the tungsten and binding metals was rst pressed into forms,
then heated. The heat melted the binder, which seeped through the pores of thepowder. This process reduces the volume of the mass by about 20%. The binding
agent made up 4.5% of the total mass of the alloy. This particular alloy is called"Densimat 18K", a paramagnetic material with low susceptibility; it was tested
for magnetic susceptibility, and has a volume susceptibility of 0.00066  5% in
MKS units[32, 53].
The tungsten was cast in three long rods that were each cut into smaller
lengths to produce the cylinders. When the cylinders arrived in JILA they weremachined to matching heights and diameters (to about 0.001 cm). At this point
one end of the cylinders was marked as the \R" side, or reference end. This surface
was cut at the factory, while the opposite side was cut at JILA. I arbitrarily decided
that the \R" sides are the bottom ends of the cylinders, and use the two terms
interchangeably.
We tested for density variations between and within the cylinders in three
ways. First we weighed each cylinder and calculated its density from its dimen-

148
sions. The densities of the billets were treated as a statistical ensemble and indicate a mean density of 17.724 gr/cc with a standard deviation of 0.008 gr/cc,
representing an variation in density from one piece to another of only 0.04%. The
peak to peak variation in density was 0.02 gr/cc, or 0.12% [54]. This measurement
gave a good idea of the magnitude of density variations over the volume scale ofa single cylinder ( 2000 cm3). We measured possible variations over smaller
volumes using more sophisticated techniques.
To characterize angularly dependent inhomogeneities we measured the period of oscillation of the cylinders on an air bearing. The cylinders were placed on
the air bearing so that they could rotate about their axes of cylindrical symmetry
with low friction. The bearing was placed on a slight incline and the cylinders were
kept from slipping o lengthwise with a pin placed close to the axis of rotation.
The frequency of oscillation is related to the torque of the o set center ofmass:dmg = ,I00(A.1)

where I is the moment of inertia of the cylinder, d is the displacement of thecenter of mass from the geometric center, m is the mass of the cylinder, and  isthe angle from equilibrium. Thus d is given by:2
d = I!
mg(A.2)

with ! being the angular frequency of the oscillation. The direction of the o set d
was also measured, simply by letting the cylinder ring down to equilibrium. Aftereach measurement the cylinder was swapped end-for-end and both measurements
were repeated. The equilibrium position varied slightly depending on whether
the bottom or top of the cylinder was facing the pin. When the cylinders werepositioned on the support plates the direction of the density gradient was set to149
be tangential to the radius from the drop axis. This helped limit possible signal
errors that arose from angularly dependent density variations.
To obtain some information about linear density variations along the lengthof the cylinders we compared the weight of each end of the cylinders. Figure 3.6
includes a schematic of the apparatus. Each cylinder was placed such that its
weight was supported between a pivot point and a electronic balance. A spacer
was used to put the geometric center of the approximately halfway midway thepivots. The balance was zeroed, and the cylinder was then swapped end forend while insuring that the geometric center was not moved. The change in thereading of the balance gave an indication of the linear density variation within eachcylinder. The balance used was a Setra brand scale accurate to 1 gram (I checked
its calibration through the local NIST chapter). This scale uses a stressed quartz
element to determine the weight placed on its pan; there is very little \give" in thepan so it maintains its vertical position even while supporting di erent weights.
Each cylinder was numbered at the time it was machined. Table A.1 is alisting of the height, diameter, weight, density of each cylinder. Table A.2 lists
the results of the air bearing and linear weight di erence tests. It includes theperiod of oscillation on the air bearing for each orientation of the cylinders, andthe linear weight di erence. The di erence (in arc) of the equilibrium position of
the cylinders in their two orientations on the air bearing was typically 1 cm.
Only the large cylinders were tested in these ways. All the other componentsof the source mass were assumed to be uniform in density. Once the cylinders werecharacterized as carefully as possible they were arranged between the support
plates so as to minimize possible linear and radial density variations. Figure A.1
shows how the cylinders were actually placed for the May experiment. The top
layer of masses includes cylinders 4, 6, 8, 9, 10, and 11, for a total mass of 238110
grams. Cylinders 1, 2, 3, 5, 7, and 12 formed the lower layer, with a total weight

150
CylinderHeight, cm Diameter, cm Weight, g Density, g/cm3
110.3637
16.586239682
17.712
2
10.3637
16.585739709
17.7343
10.3627
16.5837
39669
17.723
4
10.3632
16.586239664
17.7145
10.3632
16.5862
39662
17.713
6
10.3637
16.585439700
17.731
7
10.3632
16.586239678
17.720810.3632
16.586239679
17.721910.3627
16.586239701
17.732
1010.3627
16.5857
39662
17.71511
10.3637
16.586739704
17.7301210.3632
16.587039704
17.730AVERAGE 10.36324 16.58543
39684.5 17.7229
Table A.1: Dimensions, weight and density of the 12 Densimet 18K tungsten
cylinders. All length measurements are in centimeters  0.0005 cm. The weightmeasurements are grams  1 gram. The density values are g/cm  0.001 g/cc.

CylinderPeriod Top, s Period Bottom, s T, s Linear Di erence, g
1
23.25
22.5
1.0
-9
2
29.0
28.8
0.9
-4
3
29.5
28.0
0.6
3
4
37.0
39.5
1.9
2
5
31.5
31.0
0.9
-2
6
25.0
24.5
0.3
1
7
27.0
27.5
0.6
-3
8
28.8
29.0
1.3
-3
9
24.5
24.5
1.0
17
10
26.5
26.5
0.9
-4
11
25.8
25.0
0.3
-12
12
23.0
23.2
0.8
3
AVERAGE 27.6
27.5
0.9
-1.8
Table A.2: The results of testing the angular and linear density variations of the
12 cylinders. The period is in seconds,  0.25 second. The quality of the equilibria
positions is indicated by the di erence in period between the two orientations, T.
The linear weight di erence is in grams,  2 grams.

151
of 238104 grams. Thus the top and bottom layers only di er in weight by 6 grams.
In the top row cylinders 8, 9, 10 and 11 were placed \right-side up" (R-sidedown), while cylinders 4 and 6 were placed upside-down (R-side up). Thus there
was a net di erence in weight from the top of the cylinders to the bottom of -1
gram. In the bottom row, cylinders 1, 3, 5, and 12 were right side up and 2 and 7
were upside down. This resulted in a total gradient of +2 grams over the heightof the bottom row. The net linear gradient of the whole mass assembly was thus
,1 + 2 + 6 = 7 grams over the height of both layers.
All the components of the source mass were modeled as cylinders characterized by a vertical position with respect to the bottom of the lower support
plate, a height, an inner and outer radius, an o set from the symmetry axis of
the source mass, and a mass. Because many of the cylinders occur in symmetricpolar arrays about the central symmetry axis, the model also includes an \array
number" which is assigned the number of cylinders in the array. Thus a single
entry in the model might represent more than one cylinder. The entry1 : position= 1, height = ,1, inner radius= 0:2, outer radius = 0:5, mass = 1, o set = 10 andnumber = 6 represents an array of hollow cylinders (with the appropriate innerand outer radii) placed 10 cm from the axis of source mass symmetry. Each islocated such that the bottom of the cylinder is 1 cm above the bottom of the lower
support plate, and there is a 60 ( 360
6 ) degree polar angle between each cylinder.
The emphasis in the model was on preserving the net mass of each elementof the source mass, as well as its radial and vertical position. Some arrays ofscrews were modeled not as individual cylinders, but as a single ring of mass.
This approximation is good for small objects. Because the eld of the source
mass is very close to cylindrically symmetric the only error in this method is anincorrect density as a function of radius | a tiny error.
1

All positions are given in centimeters. The mass is given in grams.

152

Figure A.1: Arrangement of the 12 tungsten cylinders within the source mass.
\U" represents cylinders placed right side up, while \D" marks those that wereplaced up side down. The arrow points only approximately to the North.

153
The threads of bolts and the threaded portions of holes were not modeled
precisely. In the source mass model I occasionally modeled a width, x, of threads
on the inner or surface of a hole as an extra thickness of the surface of x2 . Alternatively I modeled the threads as a layer with one-half the density of the solidmaterial. The mass of the whole object was constrained to be the measured mass,
so this corresponds to a slight density and positioning error, also a tiny error.
Often the objects that make up the source mass were modeled as a seriesof cylinders. The \inner tungsten rods" are a good example. They were small(1" diameter) tungsten rods with holes tapped in both ends, placed in the gaps
around the main cylinders of the source mass. Each rod was modeled as threecylinders: First a solid cylinder of the same outer radius, o set and height of therods was modeled. This cylinder was assigned a mass value that was the sum of
the actual mass of the rod and the additional mass removed from the bolt holes.
Second a small solid cylinder located at the lower end of the rod, of dimensions
matching the bolt holes and a negative mass to exactly cancel the \extra" massmodeled in the main rod. Third, another \hole" with negative mass was modeledat the top of the rod.
The complete source mass model used for the '97 experiment is shown inthree tables: Table A.3 includes the model of the lower support plate and nut
anges. Table A.4 describes the upper plate, with bushings. Table A.5 embraces
the main tungsten cylinders, the tungsten rods, and the support rods.
The 1998 experiment incorporated several improvements. Thicker support
plates were used to reduce exing, and thus cocking of the support nuts. Additional support rods, made of aluminum, also reduced exing. Table A.6 contains
the '98 model of the support plates, while Table A.7 contains the rest of thesystem.

154

Description
Position Height rinner routter Mass O set #Lower Plate
01.283 10.658 25.96 6651.5 0
1lower ring
1.283 0.993 24.892 25.963 467.3 0
1holes for tears
01.283 00.337 -1.26 11.366 6
CB for tear bolts
00.759 .337 0.517 -1.01 11.366 6
machining holes
01.283 00.700 -5.34 15.24 4Counter Bore foroutter W rods
00.635 01.045 -6.00 19.06 3
Holes for O.W rods
0.635 0.648 00.52 -1.48 19.06 3
C.B. for anges
00.635 03.203 -56.28 20.32 3
Holes for Bronze Nuts 0.635 0.648 01.93 -20.86 20.32 3
Extra ange bolt holes 0.635 0.648 1.905 2.15 -2.855 20.32 3
Necessary F.B. holes
0.635 0.648 2.288 2.776 -5.71 20.32 3
Flanges, Whole-.629 1.264 1.700 3.187 78.92 20.32 3
Slots in Flanges-.629 1.264 2.195 2.883 -32.19 20.32 3
Threaded portion of nuts -.629 1.264 .857 1.7 73.6 20.32 3
Clean portion of nuts
0.635 2.542 .857 1.911 202.1 20.32 3
Bolts,washers of anges -1.30 2.54 2.19 2.88 28.4 20.32 3
Table A.3: Model of the lower support plate of the source mass, '97 experiment.
The masses are corrected bor buoyancy e ects. All dimensions (position, height,
rinner , routter , o set) are in centimeters. The masses are in grams.

155

Description
Position Height rinner routter Mass O set #Holes for support rods
22.01 1.01 0.337 -1.26 11.366 6
Top Plate
22.01 1.01 8.3 25.96 5344 0
1
Inner W holes
22.01 .375 0.55 -.98 9.55 6
Inner W clearance
22.38 .635 0
1.04 -6 9.55 6bolt heads
22.38 .384 0.55 2.54 9.475 6
Top Inner W Bolts
20.9
1.68 9.08 9.88 4.9 0
1Outter W clearance
22.38 .635 0
1.04 -6 19.06 3
bolt heads
22.38 .384 0.55 2.54 19.252 3
screws part
21.115 1.27 0.33.46 19.252 3
Holes for bushings
22.01 .375 01.27 -5.22 20.32 3
counterbore for bushings
22.38 .635 02.54 -35.4 20.32 3
holes for bolts for bushings 22.01 .375 1.64 1.67 -1.2 20.32 3
holes for machining
22.01 1.02 0.69 -4.2 15.25 4
lip of plate
21.74 .27 24.89 25.96 126.9 0
1
Long part of bushing
17.3
5.08 .960 1.24 82.09 20.32 3
Flange of bushing
22.38 .635 .960 2.54 86.6 20.32 3
holes for screws, IN bushings 22.38 .635 1.57 2.21 -.53 20.32 3bolt heads for bushings
23.02 .62 1.34 2.3 8.8 20.32 3
screws body for bushing
22.09 .93 1.52 2.1 8.84 20.32 3
Table A.4: Model of the Upper Plate of the source mass, '97 experiment. The
masses are corrected bor buoyancy e ects. All dimensions (position, height, rinner ,routter , o set) are in centimeters. The masses are in grams.

156

Description
Position Height rinner routter Mass O set #Bottom Row W cyls 1.283 10.3632 08.2931 39684.0 16.586 6
Top Row W cyls
11.6462 10.3632 08.2931 39685.0 16.586 6
Inner W rods *1.556 20.45 0
1.27 1920.91 9.475 6
O W rods *1.283 20.45 0
1.27 1920.91 19.252 3
L holes I rod
1.556 1.27 0.32 -7.2
9.475 6
L holes O rod
1.283 1.27 0.32 -7.2
19.252 3
H holes I rod
20.73 1.27 0.32 -7.2
9.475 6
H holes O rod
20.46 1.27 0.32 -7.2
19.252 3
Washers of O W rods 21.73 .27.32 1.27 3.52 19.252 3
SR *1.556 20.45 0.476 116.05 11.366 6
L holes of SR
1.556 1.52 0.285 -3.2
11.366 6
H holes of SR
20.73 1.27 0.285 -3.2
11.366 6
Teardrops(as ring) 1.283 .273 10.658 13.477 24.8 0
1
L. inner W rod bolts 1.283 1.78 9.16 9.79 14
0
1
L Bolts for SR
0.07
2.54 11.05 11.68 37.5 0
1
H Bolts for SR
20.88 2.54 11.05 11.68 37.5 0
1
Table A.5: Model of the Main Mass and Sandwich stuctures of the source mass,'97 experiment. * | Mass includes extra mass correction for holes. L(H) | atthe lower (upper) portion of the source mass. I(O) | at the inner (outter) rod
position. SR | support rods. The masses are corrected bor buoyancy e ects. All
dimensions (position, height, rinner , routter , o set) are in centimeters. The masses
are in grams.

157

Description
Position Height rinner routter Mass O set #LP base*02.2454 10.541 24.8857 10168.19 0
1
LP ring03.7770 24.8857 26.6675 3092.13 0
1
Inner SR holes
02.2454 0.3454 -2.388 11.3665 6
Inner CB0.6680 .3454 .5359 -.99965 11.3665 6
2nd SR holes
02.2454 0.3454 -2.388 17.3812 3
2nd CB
0.6680 .3454 .5359 -.99965 17.3812 3
3rd SR holes
02.2454 0.3454 -2.388 19.252 3
3rd CB
0.6680 .3454 .5359 -.99965 19.252 3
4th SR holes
02.2454 0.3454 -2.388 22.86 3
4th CB
0.6680 .3454 .5359 -.99965 22.86 3
Nut hole
02.2454 0
1.9177 -73.596 20.32 3
Nut CB01.2725 1.9177 3.1979 -74.271 20.32 3
threaded holes
1.2725 .970 2.3813 2.699 -7.501 20.32 3
Nut anges*
0
1.264 1.700 3.187 78.92 20.32 3
Slots in Flanges 0
1.264 2.195 2.883 -32.19 20.32 3
Threaded part nuts 01.264 .857 1.7
74.6
20.32 3
Clean part nuts
1.264 2.542 .857 1.911 202.1 20.32 3
Flange bolts-.671 2.54 2.19 2.88 28.7
20.32 3TP top22.972 2.21 10.541 24.884 10001.45 0
1
TP ring
21.420 3.762 24.884 26.671 3087.33 0
1
TP scratch
22.972 .1956 13.202 14.1745 -46.39 0
1
TP in holes
22.972 2.21 0.3454 -2.349 11.3665 6
2ndTP in holes
22.972 2.21 0.3454 -2.349 17.3812 3
3rdTP in holes
22.972 2.21 0.3454 -2.349 19.252 3
4th inner holes
22.972 2.21 0.3454 -2.349 22.86 3
Bushing holes
22.972 2.21 0
1.270 -31.75 20.32 3
TP bushing bolts 23.1
2
1.7
2.1
17.7
20.32 34 threaded holes 22.972 2.21 1.588 2.222 -6.5073 20.32 3
Table A.6: Model of the support plates, '98 experiment. All length measurments
are in centimeters, and all mass values are in grams. LP stands for \lower plate",
CB stands for \counterbore", SR stands for \support rods", and TP stands for\top plate". \Inner", \2nd", \3rd" and \4th" refer to rings of bolt holes, counted
from the center of the plates. The masses are corrected bor buoyancy e ects.
Objects marked with \*" include extra mass to make up for holes.

158
Description
Position Height rinner routter Mass O set #Upper W cyls
2.2454 10.3632 08.2931 39678.6 16.5887 6
Lower W cyls
12.6086 10.3632 08.2931 39679.6 16.5887 6IW*
2.5184 20.45 0
1.27 1920.74 9.475 6OW*2.2454 20.45 0
1.27 1920.74 19.252 3
washers for OW 22.695 .273 .3175 1.25 3.5
19.252 3
l-corr IW
2.5184 1.27 0.32 -7.2
9.475 6l-corr OW
2.2454 1.27 0.32 -7.2
19.252 3
h-corr IW
21.6984 1.27 0.32 -7.2
9.475 6h-corr OW
21.4254 1.27 0.32 -7.2
19.252 3
inner SRs*
2.5184 20.44 0.476 40.07 11.3665 6
l-corr
2.5184 2
0.32 -1.395 11.3665 6
h-corr
20.958 20.32 -1.395 11.3665 6
long middle SR* 2.2454 20.71 0.476 40.61 17.3812 3
l-corr middle SR 2.2454 2
0.32 -1.395 17.3812 3
h-corr for MSR 21.2
2
0
.32 -1.395 17.3812 3
H: i-l SR bolts 0.635 .254 .4763 1.925 11.3665 6
S: i-l SR bolts .635
3.175 0.3175 3.00 11.3665 6
H: 2nd-l SR bolts 0.635 .254 .4763 1.925 17.3812 3
S: 2nd-l SR bolts .635
3.175 0.3175 3.00 17.3812 3
H: 4th-l SR bolts 0.635 .254 .4763 1.925 22.860 3
S: i-l SR bolts .635
3.175 0.3175 3.00 22.860 3
H: o-l TR bolts 0.635 .254 .4763 1.925 19.252 3
S: o-l TR bolts .635
2.0574 0.3175 2.308 19.252 3
H: i-u SR bolt 25.5120 .635 .254 .4763 1.925 11.3665 6
S: i-u SR bolts 21.702 3.81 0.3175 4.12 11.3665 6
H: 2nd-u SR bolt 25.1820 .635 .254 .4763 1.925 17.3812 3
S: 2nd-u SR bolts 21.372 3.81 0.3175 4.12 17.3812 3
H: o-u SR bolt 25.1820 .635 .254 .4763 1.925 22.860 3
S: o-u SR bolts 21.372 3.81 0.3175 4.12 22.860 3
H: o-u TR bolt 25.1820 .635 .254 .4763 1.925 19.252 3
S: o-u TR bolts 22.007 3.175 0.3175 3.0
19.252 3
H: i-u TR bolt 25.5120 .635 .254 .4763 1.925 9.475 6
S: i-u TR bolts 21.797 3.175 0.3175 3.00 9.475 6Teardrops
2.2454 .273 10.658 13.477 24.8 0
1l-bolts IW
2.2454 1.78 9.16 9.79 14.1 0
1
IW locator body 22.972 2.548 .3454 .890 14.0358 9.475 6
IW \""" ange 25.182 .330 .3454 .890 1.82 11.3665 6
Table A.7: '98 model of tungsten elements, support rods, and bolts. All length
measurments are in centimeters, and all mass values are in grams. \Inner", \2nd",\3rd" and \4th" refer to rings of bolt holes, counted from the center of the plates.
IW stands for \inner tungsten rod", OW stands for \outter tungsten rod", SRstands for the aluminium support rods, \MSR" refers to SRs in the middle ringe.
S: stands for \screw" and H: for \head" The masses are corrected bor buoyancy
e ects. Objects marked with \*" include extra mass to make up for holes.

159

A.2

Modeling the Test Mass

The test mass was an extremely important part of the absolute gravimeter.
It was designed to be non-magnetic, conducting, and rugged (see Sec. 3.1.4.1). It
is a complex object that was dicult to model, especially at the level of accuracy
required by the experiment (as discussed in Appendix Sec. B.3.2).
The proof mass model consists of a collection of cylinders, just as that of thesource mass. I was more careful in the modeling of threads, however. They wereconsistently modeled as regions of one half the density of the rest of the material.
The proof mass is \blessed" with more components that resist casting as cylinders
than the source mass | the corner cube is essentially a pyramid, and the opticallens is a sphere. The corner cube was modeled as ten stacked cylindrical slabs,
while the sphere was modeled as a tiny cylinder closely mimicking a point mass.
Because in a source free region a sphere acts as if all it's mass were concentrated
at the center of the sphere, this is a reasonable approximation. It is not quite
perfect because the interpolation grid of the source mass eld is imperfect, andthe tiny cylinder isn't truly a point mass.
The entire proof mass model is contained in the following tables. Figure
A.2 is a photo of the proof mass. Figure A.3 is a drawing of the proof mass model
elements. Table A.8 models the top hat of the proof mass. Table A.9 includes thetungsten balls, the top hat cap and optical sphere. Table A.10 encompasses thecorner cube holder and the counter weight. Table A.11 embraces the test mass
body, nut and legs. Table A.12 only holds the corner cube model.

160
Description
Position Height rinner routter Mass O set #lowest part2.607 .201.025 1.7680 0
1main base top hat
2.807 .401.60 8.6179 0
1
thin part central hole 2.607 .6
0.075 -.0289 0
1
thick part central hole 3.207 .131 0.15 -.0244 0
1
air hole for W ball
3.169 .169 0.1-.0146 .4
3hole for W balls
2.607 .562 0.198 -.1850 .4
3holes in TM body legs 2.807 .4
0.15 -.0759 1.27 3
top of top hat base3.207 .131 0.975 1.0481 0
1bottom tophat can
3.3376 .449 .812 .975 1.1008 0
1
lowest part OVAL hole 3.787 .118 .812 .975 0.1754 0
1
2nd lowest
3.905 .1
.812 .975 0.1318 0
1center of oval hole
4.005 .2
.812 .975 0.2385 0
1
1st above center
4.205 .1
.812 .975 0.1318 0
1
top of oval hold4.305 .118 .812 .975 0.1754 0
1
Can above oval
4.423 .134 .812 .975 0.3284 01solid part of can top 4.556 .65 .860 .975 1.1566 0
1threaded part can top 4.707 .5.815 .86 0.1397 0
1
Table A.8: Model of the Top Hat cap of the proof mass. All dimensions (position,height, rinner , routter , o set) are in centimeters. The masses are in grams.

Description
Position Height rinner routter Mass O set #tungsten ball stock2.569 .62 0.1985 1.447 .4
3
1st part ball end W ball 2.469 .08 0.198 0.148 .4
3
top W ball
2.369 .05 0.11 0.039 .4
3
Glass sphere4.104 .001 0.01 4.65 01main part top hat cap 4.785 .35 .45 .8244 1.314 0
1threads
4.785 .35 .8244 .8475 0.055 01
holes
4.785 .40.1-.032 .6
4
step
4.896 .15 .334 .45 0.107 0
1
top layer
5.135 .05 .25 .8375 0.251 0
1
Mirror
4.985 .1
0.3175 0.09 0
1
spring washer
4.972 .13 .19 .318 0.02 0
1
outter ring retainer
4.780 .192 .375 .425 0.04 0
1
inner ring
4.780 .1517 .159 .375 0.08 0
1
Table A.9: Model of proof mass: Tungsten balls, glass sphere, top hat cap. All
dimensions (position, height, rinner , routter , o set) are in centimeters. The masses
are in grams.

161

Description
Position Height rinner routter Mass O set #Base plate-4.1746 .1
0
1.4 1.592 0
1main sides-4.0746 1.95 1.125 1.474 14.375 0
1threads-3.8746 1.75 1.474 1.4975 0.497 0
1Part over sides -2.1246 .15 1.125 1.45 1.020 01
Stop-1.9746 .2
1.125 1.6 2.103 01
bit over stop
-1.7746 .1
1.125 1.47 0.728 0
1
2nd bit over stop -1.6746 .4
1.125 1.43 2.533 0
1
CC holder-1.2746 .5
1.28 1.43 1.652 01laser holes-4.1746 .1
0.4525 -.166 .8125 2
CC clearance slots -1.7996 .425 1.28 1.32 -.021 0
1
CC holder holesr -1.05 .1
0.1-.027 1.36 6
glue-1.7996 .4
1.26 1.32 0.352 01laser slice-4.0746 2.8 1.125 1.2 -.665 0
1main ring-3.303 .6
1.525 1.95 7.1251 0
1
threads-3.303 .6
1.4625 1.525 0.1034 0
1
top holes-2.963 .261 0.09 -.0177 1.74 2
bottom holes-3.303 .26 0.11 -.0265 1.74 2bolt long part-3.303 .60.09 0.080 1.74 2bolt head-3.503 .2.075 .19 0.180 1.74 2
Table A.10: Model of proof mass: Corner cube holder and counter weight. All
dimensions (position, height, rinner , routter , o set) are in centimeters. The masses
are in grams.

162

Description
Position Height rinner routter Mass O set #threads TM body -1.776 .5
1.675 1.6975 0.1596 01solid bottom-1.776 .5
1.47 1.675 2.692 0
1thread relief-1.276 .1
1.47 1.6 0.336 0
1body can-1.176 2.03 1.47 1.6975 12.304 01holes TM body
-1.126 .3
0.19 -.046 1.685 6
top of TM body can .854
.25 .3 1.697 5.867 01holes(V).854.25 0.15 -.0477 1.27 3
V portion of holes 1.04
.01 .25 .257 -.0837 1.27 3base of legs
1.105 .86 .125 .330 0.674 1.27 3
solid part of leg
1.965 .84 .15 .33 0.610 1.27 3
threads
1.965 .83 .125 .15 0.024 1.27 3
nibs
2.555 .25 0.04 -.008 .982 3
solid part-1.776 .5
1.725 1.9 2.671 0
1
thread-1.776 .5
1.662 1.677 0.035 0
1bottom ring
-2.118 .15 1.53 1.9 1.604 0
1thread relief-1.968 .2
1.68 1.9 1.170 01
holes-1.868 .001 0.001 -.015 1.5625 4
top of bolt
3.207 .30.275 0.599 1.27 3
solid of bolt
2.407 .80.10.211 1.27 3
thread
2.407 .8
.1 .15 0.132 1.27 3
allen key hole
3.307 .2
0.125 -.082 1.27 3
Table A.11: Model of proof mass: Test mass body, testmass nut and legs. All
dimensions (position, height, rinner , routter , o set) are in centimeters. The masses
are in grams.

163

Description Position Height rinner routter Mass O set #Bottom-1.2719 .127 0
1.265 1.72092 0
1-1.1449 0.09449 0
1.8898 1.52197 0
1-1.05041 0.09449 0
1.79531 1.37513 01-0.95592 0.09449 01.70082 1.16886 01-0.86143 0.09449 0
1.60633 1.00384 01-0.76694 0.09449 0
1.51184 .880089 01-0.67245 0.09449 0
1.41735 .770077 01-0.57796 0.09449 0
1.32286 .664191 01-0.48347 0.09449 01.22837 .569306 0
1Middle-0.38898 0.09449 0
1.13388 .481298 01-0.29449 0.09449 0
1.03939 .401540 01-0.2
0.09449 00.9449 .328657 01-0.10551 0.09449 00.85041 .262651 01-0.01102 0.09449 00.75592 .204895 010.08347 0.09449 00.66143 .154014 010.17796 0.09449 00.56694 .110010 010.27245 0.09449 00.47245 .072881 010.36694 0.09449 00.37796 .039790 010.46143 0.09449 00.28347 .019895 01Top0.55592 0.09449 00.18898 .009947 0
1
Table A.12: Model of proof mass: Corner Cube. All dimensions (position, height,
rinner , routter , o set) are in centimeters. The masses are in grams.

164

Figure A.2: A photograph of the proof mass. The proof mass is approximately 10
cm long. The \top hat" is upper portion of the mass, containing the glass sphereand the tungsten balls. The \test mass body" forms the center portion of themass, including the three legs. The \corner cube holder" is the bottom portion of
the mass, and has the counterweight treaded on its nether region.

165

Z, cm
5.50
5.00
4.50
4.00
3.50
3.00
2.50
2.00
1.50
1.00
0.50
0.00
-0.50
-1.00
-1.50
-2.00
-2.50
-3.00
-3.50
-4.00
-4.50

r, cm
-2.00

-1.00

0.00

1.00

2.00

Figure A.3: A drawing of the proof mass model. The vertical axis is verticalextent of the proof mass, in centimeters. The horizontal axis represents its radialextents, in centimeters.

APPENDIX B
ERROR CALCULATIONSThis appendix contains the details of the error analysis that was introducedin Section 5.1.

B.1

Positioning Errors

B.1.1

Source Mass Positioning Errors

The source mass is located vertically about the gravimeter with a steppermotor/shaft-encoder/screw-drive system. The lower mass position was measured
to  0.01 cm with relation to the top of the (3/8" thick) ange of the custom dropping chamber. The top position was referenced to the lower with an uncertaintyof  0.01 cm due to lead errors in the screws. Thermal expansion introduced
uncertainty of  0.006 cm. Wear in the support nuts introduced an additional
error of 0.01 cm. The root mean square uncertainty in the vertical position of thesource mass (with respect to the dropping chamber ange) was  0.025 cm forthe upper mass position, by direct measurement.
The distortion of the support plate in the 1997 experiment introduced avertical displacement of approximately 0.01 cm. This was accounted for in ourmodels, but still produced an uncertainty of 0.01 cm. The distortion of the plateraised some concern that the tungsten cylinders were being angled outwards. The
distortion changes the radial position of the cylinders by less than 0.003 cm from

Description
Magnitude Signal, Gal Error, ppm
2
Shaft Encoder positioning
0.04 cm
0.004
50 ppm1
Horizontal velocity+0.004 -0 cm
8e-6
8 ppb
Can setup1+0.15 -0 cm
0.003
33 ppmRotation errors
0.0004 radian
1e-6
1 ppb
2
Wear in tungsten vees, balls
 0.02cm
0.001
12 ppmreferencing to optical center
within proof mass2
 0.005 cm
0.0001|
2
referencing to gravimeter ange  0.01cm
0.00025 ppm

167

Table B.1: Uncertainties in proof mass position and orientation. 1 : A radialmeasurement. The net radial position uncertainty is 0.18 cm (0.007 Gal = 85ppm) in the 1997 experiment, and 0.15 cm (0.005 Gal = 60 ppm) in the 1998
experiment. 2 : This error is not included in the total, because it is absorbed into
the statistical uncertainty of the vertical position, from the direct extraction of
the start of drop position from the data.
top to bottom. In the 1998 experiment the support plates used were approximately
three times thicker than those of the 1997 run. Because the sti ness increases withthe cube of the thickness, these errors were negligible in the 1998 experiment.

B.1.2

Proof Mass Positioning errors

The signal magnitudes of test mass mis-positioning errors were calculatedwith a simpli ed test-mass model. The falling mass was modeled as a 8 cm long
cylinder of inner diameter 1 cm and wall thickness of 1 mm for the calculationsof errors. The net acceleration of this simpli ed model agrees with with the fullmodel acceleration to 5%. Table B.1 lists the errors associated with proof mass
positioning mechanisms.

B.1.3

Extracted Positions

It is possible to extract the average value for the start-of-drop position fromdata directly (Sec. 4.3), with statistical uncertainty of 0.05 cm. In the 1997
experiment this method gave a result in agreement with the \direct" measurement.

168
In the 1998 experiment, however, the two determinations di ered by 0.15 cm.
The directly extracted value was used for both data runs, with the statisticaluncertainty adopted as the error in this parameter. This method incorporates
vertical positioning errors of both the source and test masses.
It is possible to extract the average value for the start-of-drop position fromdata directly (Sec. 4.3), with statistical uncertainty of 0.05 cm. In the 1997
experiment this method gave a result in agreement with the \direct" measurement.
In the 1998 experiment, however, the two determinations di ered by 0.2 cm.
The directly extracted value was used for both data runs, with the statisticaluncertainty adopted as the error in this parameter. This method incorporates
vertical positioning errors of both the source and test masses.

B.2

Spurious Signals

B.2.1

Di erential Magnetic Forces.

The manner that magnetic forces a ect the results of the experiment are
discussed in Sections 5.2.7 and 5.2.8, but here we discuss the calculation of themagnitudes of these e ects. The information required to do this includes estimatesof the magnetic eld ( to calculate the direct attraction of the source mass to theproof mass) and eld gradient (to determine the eddy current damping forces
acting on the proof mass) carried by the changing position of our source mass.
Imagine the source mass as a thick-walled rod oriented vertically within
a volume of uniform external vertical magnetic eld. The eld will induce themagnetic surface pole density, or e ective charge, ofB  n^(B.1)

on the at faces of the cylinder. B is the magnetic eld,  is the susceptibilityof the material, and n^ is the unit vector normal to the surface. We integrate this169
density over the surfaces of the cylinder to nd the magnetic scalar potential dueto the presence of the source mass:
Z
0

B

n
^
da
1

'i (^r) = 4So jr^ , r^0 j
where So is the area of the two faces of the cylinder, r^ is the position of evaluation,and 'i is the induced magnetic scaler potential. For the magnetic eld on theaxis of symmetry, we evaluate the gradient of the scaler potential:

B(^r) = ,or' (^r)
where o is the permittivity of free space. The vertical component of the eld andits gradient are plotted in Figure B.1.
−7

x 10

2

Field gradient, gauss/cm

Vertical magnetic eld, gauss

−10

2

x 10

1.5

1.5

1

1

0.5

0.5

0

−0.5

0

−0.5

−1

−1.5

−1
−40

−2
−40

Position, cm
Position, cm
Figure B.1: The vertical magnetic eld and eld gradient induced by the susceptibility of a simple source mass model in a uniform vertical magnetic eld of 0.5
gauss.
−30

−20

−10

0

10

20

30

40

−30

−20

−10

0

10

20

30

40

A bound of 5  10,10 gauss and 5  10,7 gauss/cm for the eld and eld
gradient is reasonable. Modeling the proof mass as an aluminum cylinder [38] we
can calculate the expected worst case eddy damping e ect. From Eqn 5.5:vA2 ( dB )2
F = Rdzde nes the force, F , on a cylinder with surface area A perpendicular to a verticalmagnetic gradient dB
dz . R is the cylinder's resistance in a loop perpendicular to170
the gradient, and v is its velocity. Using the values in Table B.2 we nd that
eddy currents could only cause a di erential acceleration of 3X 10,9Gal. A quick
sanity check of this surprisingly small value can be made in the following way:
Assume a 1 gauss/ meter gradient. The source mass could produce a maximal
gradient on the order of 2 times this gradient. This gradient would result in an
acceleration of 10,8Gal | still far below the level that would compromise theresults of the experiment.
Variable Value
R
0.25
A
0.00018 m2
mass 0.13 kg
vmax
2 m/s
dB
5  10,7 gauss/cm
dz
Table B.2: A simple proof mass model for use with eddy current calculations.
The rst order direct attraction between the source mass and proof mass
may also be calculated. Using Equation B.1 we can calculate the attraction between the proof mass and just one pole of the \magnet" induced in the source
mass, thus bounding the actual e ect. Let us consider the magnetic \charge"
induced on the face of the source and proof mass by a 1 gauss vertical constantmagnetic eld:

Qs = AssB = (:176m2)(6:6  10,4)(1  10,4T ) = 1:16  10,7m2 T
Qp = AppB = ((:01)2m2 )(6:6  10,4)(1  10,4T ) = 2:1  10,11m2 T
The magneto-static force, F , is:p
Force = Qrs Q
2

Here the subscripts p and s refer to the proof and source mass. Q is the induced
e ective magnetic surface charge. For a separation of 5 cm the force is 1  10,18 N,

171
so the acceleration a is:
Acceleration = F = 1  10 N = 1  10,8Gal
m
:13kg
,18

(B.2)

Second order e ects arising from magnetic poles induced in the proof mass fromthe perturbation elds created by the presence of the source mass are on the orderof 2 smaller still. Only negligible torques were introduced by these elds.
These calculations were all based on a single hollow cylinder of tungsten,of dimensions comparable to the ring of tungsten in the source mass. In fact thesource mass also contains some stainless steel, bronze, and aluminum components.
In the 1997 experiment, 6 thin 304 stainless steel rods sandwiched the source
mass together. Stainless steel was chosen because of its great strength)1 . The
susceptibility of this alloy is 0.02 ([55]), thirty times greater than the susceptibilityof the tungsten alloy we used. The rods are the same height as the source mass, buthave a net surface area almost 2000 times smaller than that modeled in tungsten.
As the rods were at a radius from the drop comparable to that of the tungsten,
and as the magnetic eld was proportional to the product of the surface area withthe susceptibility, the magnetic elds due to these components were a factor of30
2000 smaller.
It is also necessary to check on the signal due to the stainless steel bolts used
to tie the smaller tungsten rods in place. These were cold worked, which had thee ect of increasing their magnetic susceptibility. When the height of the material
is decreased to a value consistent with the bolts used (but still preserving thesurface area of the tungsten), the magnetic eld induced in the mass is decreased
by a factor of two, as does (surprisingly) the gradient of the magnetic eld. Using
the same argument before, we can limit the e ect of these bolts to:
In the 1998 experiment 12 aluminum rods were used instead. Also the 1998 run the ttings
discussed later in this section were made of titanium, a non-magnetic material.
1172

Atungsten bolts
Bbolts = Btungsten
2A bolts tungsten

Bbolts  Btungsten=3

The calculations of the e ects of the AC magnetic elds produced by thedrive motor were based on measurements of the reading of the balance in the test
experiment, Sec. 5.2.8. Only a sample calculation is done here (all the real
numbers may be found in [56]), for a single proof mass position. Table B.3 showsthe eld generated by the drive motor at the upper position of the test mass.
Rotation frequency, Hz Vertical Bpp, Gauss
5.1
0.002569.2
0.00249
21.1
0.0021439.9
0.0016560
0.0014
Table B.3: Magnetic eld of the drive motor at the upper proof mass position.
Bpp is the vertical peak-to-peak magnetic eld.
Once I knew the magnitudes of the actual elds that would be sensed by theproof mass, I needed to generate much larger elds so that I could measure thecorresponding force with the balance system. By using a bit of algebra the two
could be related to give a limit on the actual force on the test mass. To increase
our signal the coil was placed midway between the test mass and the drive motor.
Table B.4 gives the calibration of one of the voice coils used to produce the larger
elds. The full elds of the coils could not be directly measured because theysaturated our sensor.
Once the coil was calibrated more current was driven through it and theforce on the test mass was measured. Table B.5 contains the current and force
measurements.

Rotation frequency, Hz Current, amps Vertical Bpp, Gauss
5.2
0.30
1.04
10.0
0.35
1.19
20.5
0.37
1.10
41.2
0.31
0.735
83.9
0.31
0.469
161
0.35
0.305
316
0.27
0.116
604
0.15
0.027

173

Table B.4: Calibration of the voice coil. This calibration was for the upper proof
mass position. Bpp is the vertical peak-to-peak magnetic eld.
Now the force measured with the coils can be compared to the force of thedrive motor. From Eqn. 5.8 the force at a given frequency is proportional to:2
Fr / Bh3

where h is the distance of the test mass from the dipole source. At a frequency ofapproximately 10 Hz, we can calculate the actual force error. From the calibration,a current of 0.35 amps corresponded to a eld of 1.19 gauss. Thus the actual
testing current of 6.6 amps must have made a eld of 1.19x6.6/0.35 = 22.4 gauss.
Recalling that the distance of the coil to the test mass was half the distance ofthe mass to the motor,

Fc = Bc2=(1=8)
Fm
Bm2
putting in the numbers gives:

Fc = 22:4  22:4  8
Fm
0:00249  0:00249= 6:5  108
So the force due to the motor was (5e-7 kg)(9.8 m/s2)/6.5e8 = 7e-15 N. This

Rotation frequency, Hz Current, amps Vertical Force, mg
5.6
6.0
0
10.1
6.6
0.5
20.9
6.15
1.0
37.7
4.95
1.3
82.2
3.56
0.5
165
2.8
0.1
309
1.18
0
610
0.61
0

174

Table B.5: Calibration of one of the voice coils used in the experiment. Thiscalibration was for the upper proof mass position. Bpp is the vertical peak-topeak magnetic eld.
corresponds to an acceleration of the 100 gram test mass of 7e-14 m/s2, or 7e-5
Gal. This is completely negligible.

B.3

Modeling Errors

B.3.1

Source Mass Modeling Errors

This section presents the errors associated with mass measurements andapproximations in modeling components of the source mass.
The large tungsten cylinders were weighed on a scale calibrated for stainlesssteel standard masses weighed in air (at 5200 feet altitude). This means thatthe scale, which measures a force, assumes both a value for g, and a buoyancy
correction. The buoyancy correction is only correct if the mass being weighed
really is stainless steel | in our case it wasn't. I corrected this problem, andbuoyancy e ects, in two steps. First I calculated the amount that the balance
overestimated the mass of the tungsten (the balance added in the weight of too
much air; tungsten is denser than stainless steel). This gave me the correct mass
value for the cylinders. Then I calculated the mass of air displaced by the tungsten.
Including this second correction in the model \automatically" compensate for theDescription
Signal, Gal
Eddy Current Damping < 0,7 Gal
Direct Magnetic attraction < 10,6 Gal
AC; Magnetic Forces< 10,4 Gal

175

Table B.6: Errors due to spurious forces. The e ects of the source mass are
discussed in Sec. 5.2.2.
gravitational attraction of the displaced air. The total correction to the reading on
the scale was -4.76 grams per cylinder. The uncertainty associated with the massof each of the 12 tungsten cylinders and the two aluminum plates (arising fromthe scale's accuracy) is 1 gram. The total mass uncertainty (for the cylinders)p
was taken as 1g  14 = 4 g.
In the same way a buoyancy correction was applied to the upper and lower
aluminum plates of the experiment. 1997: the Upper plate was modeled 0.62
grams lighter than its true mass, while the lower plate was made 0.76 grams
lighter. 1998: the lower plate was modeled 1.53 grams lighter and the upper was
made 1.55 grams lighter.
The error in the buoyancy corrections due to barometric pressure changes(approximately 1% magnitude) is far smaller than the error in weighing, and therefore is ignored. In some sense the time-varying barometric pressure correction,
which shifts with pressure, is a variable independent of our signal and is therefore
absorbed in the statistical errors of the experiment.
The mass uncertainty in all other components of the system (about 20 different objects, in which nuts and bolts were weighed as collections) is 0:1 gram.
This leads to a total error estimate of 0:5 g. Grease on the screws collected on thenuts and thus traveled with the source mass. The weight of a sample of grease ofabout the same volume as that on the nuts weighed 5 grams. I couldn't reliably
model this e ect because the screws were often cleaned. Rather than assume an176average perturbation, the whole grease e ect is accepted as an error.
The measurement of the position of the center-of-mass with respect to thegeometric centers of the tungsten cylinders allowed us to make estimates of errors
from angular and linear density variations within the cylinders. 11 of the 12
tungsten cylinders had an angular density variation producing a torque equivalent
to a 25 gram mass placed on their circumferences. The twelfth mass, however,(#4) displayed no angular density variation, and was therefore suspect (discussed
in Appendix A). We needed to include an estimate for the error introduced bythese characteristics.
Exactly how to calculate the errors due to these types of sources was not
immediately clear. The position of an unmodeled point mass distribution strongly
in uences the magnitude of the signal error it causes. We decided to assign avalue of 5g as the over all uncertainty in the mass of the source mass, distributed
uniformly about the source mass. We also modeled a 25 gram point mass in theposition that results in the largest signal bias. The possibility of a misplaced
point of mass of 25 grams is of greater probability than an error in the total massof this size because a misplaced point mass represents the uncertainty in density
inhomogeneities in the masses. It is interesting that a 25 gram mass placed on the
inner radius near the middle of the source mass results in a bias in the di erential
signal of 0.01 Gal, yet if the 25 grams were distributed uniformly around thesource mass, they would have a signal of:
25g total signal = 25g (80Gal) = 0:005Gal
perterbed signal = massof source mass 40kg  12
(B.3)
This result indicates that even if our source mass were in nitely dense, our signal
could only be improved by approximately a factor of two (assuming no major
redesign of the FG-5 and conserving the net mass of the source mass)!

Description
Signal, Gal Signal, ppm
Mass estimates errors
0.002
25
Unmodeled grease andunknown point mass
0.005
60
density variations:linear0.004
50
Radial0.002
25
Angular0.002
25
quadratic
0.005
60
TOTAL
0.011
110

177

Table B.7: Uncertainties in proof mass position and orientation
Other approximations in the source mass model were unimportant. The
error from modeling an array of screws as a ring of screws is approximately 2 
10,8Gal. the thread as part of the screw is approximately 4  10,9Gal. Table
B.7 displays the errors associated with modeling of the source mass.

B.3.2

Proof Mass Modeling errors

To scale errors of modeling the proof mass and to discover the degree ofprecision required for an adequate model to be constructed, I made three simpli ed
models of the proof mass. The rst was a vertical line charge of mass, the samelength as the proof mass. To test the sensitivity of the results to the vertical\granularity" of the model, the line charge was broken into n point charges, equidistant from each other. Figure B.2 displays the di erences in net accelerations
for the given models falling though a eld generated by a simpli ed source massmodel. The data shows that the convergence is better than linear, so with 32
layer model in the vertical direction we expect the error to be less than twice themaximum di erence between the acceleration of the 64 layer model and the 32.
This di erence is 0.0015 Gal. A vertical granularity corresponding to 64 layers(1.25 mm/step) was used in the nal proof mass model.

178

0.02

0.015

Acceleration difference, µGal

0.01

0.005

0

−0.005−0.01−0.015

−0.02
−40

−30

−20

−10

0
Position, cm

10

20

30

40

Figure B.2: Change in vertical acceleration as a function of the vertical granularity of model. (|) the di erence between calculated acceleration for a verticallyoriented line charge of the same length as the actual proof mass, modeled as 64
and 32 point masses. (--) modeled as 32 vs 16 point masses and (  ) modeled as
16 vs 8 point masses.
I performed a similar test of the horizontal granularity sensitivity by using a
1 cm line-charge radially oriented. Although the proof mass actually extends to a
1.5cm radius, modeling granularity in not an issue at these radii, because the proof
mass portions can be nearly perfectly described as cylinders. The corner cube,
the object most badly approximated by a cylinder, extends to a 0.5 cm radius.
Figure B.3 plots di erences in accelerations for various granularity models. Usinga radial model granularity of 0.125 cm is adequate for our purposes, introducing
only 0.0012 Gal error.
In one last test of the sensitivity, in both dimensions at the same time,

179

−3

5

x 10

4

Acceleration difference, µGal

3
2
1
0
−1
−2
−3
−4
−5
−40

−30

−20

−10

0
Position,cm

10

20

30

40

Figure B.3: Change in vertical acceleration as a function of the radial granularityof the model. (|) the di erence between calculated acceleration for a horizontallyoriented line charge extending from the axis of symmetry to 1 cm o axis, modeled
as 32 and 16 point masses. (  ) Modeled as 16 vs 8 point masses, (- -) 8 vs 4,
and ( , ) 4 vs 2 points.
I modeled an hourglass shaped mass with its axis of symmetry aligned in thevertical direction. The hourglass was the same height as the actual proof mass,and of 1 cm maximum radius. The results of this test show that a granularity of
32 slices across the height of 8 cm is satisfactory for the experiment, resulting inan error that (using the extrapolation to an continuous object) is good to better
than 0.01 Gal.
This granularity is so coarse that actual modeling errors of reasonable magnitude are not going to a ect the accuracy of the results strongly. For example,
the proof mass is made up of some objects that are very well modeled as cylinders| they only di er from perfect cylinders in that they are imperfectly machined.
When we model these pieces as perfect cylinders, the actual object may vary fromthe model by some small fraction of a millimeter: certainly by less than 0.003cm.
Since we only need the granularity to be better than about 0.1 cm, these errors

180
are lost in the glare of large modeling approximations. The fact that many piecesof the proof mass are very good approximation of cylinders means that adopting
the whole error due to granularity is unreasonable. Because well over half of themass is contained in objects which are perfectly described as cylinders to levelsof 0.01 cm, we might halve the error estimates for granularity errors. Since theerrors are so small, however, they are maintained.
The nal proof mass model was made with a vertical and horizontal granularity of better than 0.125 cm. This corresponds to an error of 0.0015 Gal dueto the vertical approximation, and 0.0006 Gal due to the radial approximation,
for a fully complex object, in a single source mass position.
The model of the proof mass was developed from machine drawings andmeasurements of the masses of individual proof mass components. There were
three elements of the model, the corner cube holder, corner cube, and glue that
we weren't able to weigh independently (because they're all glued together). We
weren't able to get their masses by weighing disassembled assemblies because they
aren't very consistent from piece to piece. For this reason we had to estimate2
the mass distribution between the three components. This error introduced by
the uncertainty in this distribution is greater than the combined uncertainty inall other components of the model.
Only the relative masses of each component of the test mass was importantin our models. Since the components were weighed in air a relative mass error
between materials of di erent density was introduced. The buoyancy correction
for the optical sphere, for example, is 0.002 grams. This is an order of magnitude
smaller uncertainty in the accuracy of the scale used in the measurements. We
We chose to use the greatest mass observed for both the corner cube and the corner cube
holder, and make up the mass di erence in the binding glue. Our original estimate of the glue
weight was based on a similar system, in which a corner cube was glued to a small aluminum
cylinder modeled on the corner cube holder. The glue mass used in this system was 0.29 grams,
only about 20% lighter than our nal glue estimate.
2

Description
Signal, Gal Signal, ppm
radial Model Granularity
0.001215 ppm
Vertical Granularity
0.003
35 ppmRelative Mass Errors(CC, CCholder,Glue)
0.003
35 ppmRelative Position Errors
0.007
80 ppmTotal
0.008
90 ppm181

Table B.8: Uncertainties in proof mass position and orientation
made the assumption that all the material of the proof mass was free of voids andappreciable density variations.
The positions of each element have uncertainties of as much as 0.01 cm. By
modifying the weight and position of individual elements of the the proof massmodel, and testing the output solution, I obtained error estimates of: 0.002 Galin the nal signal due to a 0.07 cm translation of any of the major proof masselements and and error of 0.0009 Gal due to a 0.1 gram mass error. The proof
mass is built of approximately 10 pieces, so the total error due to incorrect weights
and positions is about 0.007 Gal.
Table B.8 lists the proof mass modeling errors. The nal proof mass modelthat we used in Appendix A.

B.4
B.4.1

Numerical System
Absolute Accuracy of Quadrature: The Integration Overthe Source Mass

To insure the accuracy of our numerical results we used quadrature algorithms that calculated error estimates for the values they returned. The algorithms
were checked against analytic solutions whenever possible3 . We implemented theintegration algorithms twice, using a straight 4-D integration as a check to the3

We used analytic solutions for the the case of cylinders with a common axis of symmetry.

182
interpolation scheme discussed in Sec. 2.1. Unfortunately the 4-D integration
was incredibly slow | a full test of the integration over the complete source andproof mass models could not be performed. Once we were sure that the implementations were consistent and correct for small systems of masses the interpolation
system (that was used in the nal calculations) was tested for self-consistency.
The main source of concern was the required density of grid points in theinterpolation eld to achieve our required accuracy. A standard method of testingfor sucient density is to double the density until the variation in the nal values(integrated over the grid) becomes negligible. Figure B.4 is a plot of the net
acceleration of the proof mass (using a 44 piece model) as a function of the numberof grid points used in the vertical direction. The three plots were all calculatedusing the same radial grid density of 30 points/3 cm. The 500 to the 1000 point
accelerations di er by less than 0.001 /Ugal, as seen in Figure B.5. Results for aradial grid density of 15 points/3 cm are very similar.

42.725

Acceleration, uGal

42.72

42.71542.71

42.705

42.7

−2.75

−2.7

−2.65

−2.6

−2.55 −2.5 −2.45
Position, cm

−2.4

−2.35

−2.3

−2.25

Figure B.4: Calculated acceleration for a 44 piece proof mass model, using three
di erent vertical grid densities. The | line represents the acceleration resulting
from using 250 grid points in the vertical direction (over approximately 90 cm ).
The    line results from 500 points, and the -- line from 1000 points.

183

−3

x 10
2.5
2

Acceleration difference, uGal

1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
−40

−30

−20

−10

0
position, cm

10

20

30

40

Figure B.5: Di erences in calculated acceleration for a 44 piece proof mass model,using three di erent vertical grid densities. The solid line represents the di erencein accelerations calculated using 1000 grid points and 500 grid points. The dotted
plot is the di erence between the results of a 500 and a 250 point grid. All grids
used had 30 grid points o axis.
Testing the dependence of calculated acceleration on radial grid density inthe same way insured the appropriate choice of grid size. Figures B.6 and B.7 show
computed results for three di erent radial grid densities. Based on these results,
the nal solution was calculated using a vertical grid density of 1000 points/90
cm, and a radial density of 30 points/3 cm. These plots clearly indicate that theerror in the interpolation due to the radial grid density is bound by 0.0005 Gal .

B.4.2

Assumption of Cylindrical Symmetry

The two-dimensional interpolation grid algorithm assumes that the eld of
the source mass was cylindrically symmetric. Although the mass is 1-, 2-, 3-, and6-fold rotationally symmetric4 , it was not cylindrically symmetric. Figure B.8
displays the vertical eld as a function of angle for each of six cylinders making
up a simpli ed source mass model. The plots were made for a vertical position
4

To the degree that the masses were perfectly homogeneous. . .

184calculated acceleration, uGal

42.72842.72642.72442.72242.72

42.71842.716
−2.75

−2.7

−2.65

−2.6
−2.55
position, cm

−2.5

−2.45

−2.4

Figure B.6: Calculated acceleration for a 44 piece proof mass model, using three
di erent radial grid densities. The | line represents the acceleration resulting
from using 60 grid points over 3 cm. The    line results from 30 points, and the-- line from 15 points, over the same range.
−4

x 10

difference in calculated accelerations, uGal

0

−1

−2

−3

−4

−5

−6

−7
−40

−30

−20

−10

0
position, cm

10

20

30

40

Figure B.7: Di erences in calculated acceleration of a 44 piece proof mass model,using three di erent grid densities. The solid line represents the di erence inaccelerations calculated using 30 grid points and 15 grid points over 3 cm. The
dotted plot is the di erence between the results of a 60 and a 30 point grid. All
grids used had 500 grid points over 90 cm on the vertical axis.

185
at a maximum of the vertical eld. The dependence on vertical eld magnitude
on angle is more clearly seen in Figure B.9. The approximation of cylindricalsymmetry is clearly valid to better than 1 nGal, or a level of approximately 2parts in 105. At the zero in the eld, at the center of the source mass (but withthe same radial o set) the variations are two orders of magnitude smaller.

B.4.3

Di erential Precision of the System

Because there might be a non-linear error in the numerical integrations (whoknows?) I was concerned that the di erential accuracy of the method was suspect.
By testing the integration of the di erential equation describing the motion of thetest mass falling through various elds, I found that the numerical result wasgood to 2 parts in 106 for a 80 Gal signal. At this level the algorithms showedsome lack of stability. As the goal of the experiment was for a result good to 5
parts in 104, this is more than sucient. The di erential accuracy was tested ona constant eld and a linearly changing eld.
Description
Signal(Gal ) Error (ppm)Interpolation Density
0.001520
Assumption of Cylindrical Symmetry
0.001
10
Numerical Stability
0.0002
2
Table B.9: Errors due to the numerical system. Errors arising from the precisionof the quadrature algorithms are negligible.

186
9

8.5

Vertical acceleration, uGal

8

7.5

7

6.5

6

5.5

5
0

50

100

150
200
250
Offset angle, degree

300

350

400

Figure B.8: Vertical acceleration from a source mass model constructed of six
cylinders. Values are plotted as a function of angle from a line of symmetry, for apoint 2 cm o axis. Note that that contributions of the individual cylinders (  )
are not true cosines( as they would be if the vertical force on the proof mass werea function linear in the distance between the proof mass and the cylinder) because
they are modulated by the r12 factor in the gravitational interaction. The solidline represents the average acceleration due to each cylinder. As far as this curve
approximates a line, the eld is cylindrically symmetric.
−4

2

x 10

1.5

Vertical acceleration, uGal

1

0.5

0

−0.5

−1

−1.5

−2
0

50

100

150
200
250
Angular offset, degrees

300

350

400

Figure B.9: Variation of vertical acceleration as a function of angle when 2 cm
o axis. At a maximum in the eld, this plot shows that the eld is cylindricallysymmetric to 0.2 nGal, at a 2 cm radius.