A Measurement of Newton's Gravitational Constant
ABSTRACT A precision measurement of the gravitational constant $G$ has been made using a beam balance. Special attention has been given to determining the calibration, the effect of a possible nonlinearity of the balance and the zero-point variation of the balance. The equipment, the measurements and the analysis are described in detail. The value obtained for G is 6.674252(109)(54) 10^{-11} m3 kg-1 s-2. The relative statistical and systematic uncertainties of this result are 16.3 10^{-6} and 8.1 10^{-6}, respectively.
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ABSTRACT: We have recently completed a measurement of the Newtonian constant of gravitation G using atomic interferometry. Our result is G=6.67191(77)(62)×10(-11) m(3) kg(-1) s(-2) where the numbers in parenthesis are the type A and type B standard uncertainties, respectively. An evaluation of the measurement uncertainty is presented and the perspectives for improvement are discussed. Our result is approaching the precision of experiments based on macroscopic sensing masses showing that the next generation of atomic gradiometers could reach a total relative uncertainty in the 10 parts per million range.Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 10/2014; 372(2026). · 2.86 Impact Factor - [Show abstract] [Hide abstract]
ABSTRACT: The primary objective of the CODATA Task Group on Fundamental Constants is 'to periodically provide the scientific and technological communities with a self-consistent set of internationally recommended values of the basic constants and conversion factors of physics and chemistry based on all of the relevant data available at a given point in time'. I discuss why the availability of these recommended values is important and how it simplifies and improves science. I outline the process of determining the recommended values and introduce the principles that are used to deal with discrepant results. In particular, I discuss the specific challenges posed by the present situation of gravitational constant experimental results and how these principles were applied to the most recent 2010 recommended value. Finally, I speculate about what may be expected for the next recommended value of the gravitational constant scheduled for evaluation in 2014.Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 10/2014; 372(2026). · 2.86 Impact Factor -
Article: The attracting masses in measurements of G: an overview of physical characteristics and performance.
[Show abstract] [Hide abstract]
ABSTRACT: Simple spheres and cylinders have been the geometries employed most frequently, but not uniquely, for the attracting masses used historically in measurements of the Newtonian gravitational constant G. We present a brief overview of the range of sizes, materials and configurations of the attracting masses found in several representative experimental arrangements. As one particular case in point, we present details of the large tungsten spheres designed originally by Beams, which have been incorporated into several different apparatuses for measuring G over the past 50 years. We also consider the question of possible systematic dependence of the results and their precision on the size of the large masses/mass systems that have been employed to date. We close with some considerations for possible future work.Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 10/2014; 372(2026). · 2.86 Impact Factor
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arXiv:gr-qc/0609027v1 7 Sep 2006
A Measurement of Newton’s Gravitational Constant
St. Schlamminger,∗E. Holzschuh,†W. K¨ undig,‡F. Nolting,§R.E. Pixley,¶J. Schurr,∗∗and U. Straumann
Physik-Institut der Universit¨ at Z¨ urich, CH-8057 Z¨ urich, Switzerland
A precision measurement of the gravitational constant G has been made using a beam balance.
Special attention has been given to determining the calibration, the effect of a possible nonlinearity of
the balance and the zero-point variation of the balance. The equipment, the measurements and the
analysis are described in detail. The value obtained for G is 6.674252(109)(54) ×10−11m3kg−1s−2.
The relative statistical and systematic uncertainties of this result are 16.3×10−6and 8.1 ×10−6,
respectively.
PACS numbers: 04.80.-y, 06.20.Jr
I.INTRODUCTION
The gravitational constant G has proved to be a
very difficult quantity for experimenters to measure ac-
curately. In 1998, the Committee on Data Science
and Technology (CODATA) recommended a value of
6.673(10) × 10−11m3kg−1s−2. Surprisingly, the uncer-
tainty, 1,500 parts per million (ppm), had been increased
by a factor of 12 over the previously adjusted value of
1986. This was due to the fact that no explanation had
been found for the large differences obtained in the pre-
sumably more accurate measurements carried out since
1986. Obviously, the differences were due to very large
systematic errors. The most recent revision [1] of the
CODATA Task Group gives for the 2002 recommended
value G = 6.6742(10) × 10−11m3kg−1s−2. The uncer-
tainty (150 ppm) has been reduced by a factor of 10 from
the 1998 value, but the agreement among the measured
values considered in this compilation is still somewhat
worse than quoted uncertainties.
Initial interest in the gravitational constant at our in-
stitute had been motivated by reports [2] suggesting the
existence of a “fifth” force which was thought to be im-
portant at large distances. This prompted measurements
at a Swiss storage lake in which the water level varied by
44 m. The experiment involved weighing two test masses
(TM’s) suspended next to the lake at different heights.
No evidence [3, 4] was found for the proposed “fifth”
force, but, considering the large distances involved, a
reasonably accurate value (750 ppm) was obtained for
∗present address Univ. of Washington, Seattle, Washington, USA
†deceased
‡deceased; We dedicate this paper to our colleague Walter K¨ undig,
without whose untiring and persistent effort this ambitious exper-
iment would neither have been started nor brought to a successful
conclusion. Walter K¨ undig, died unexpectedly and prematurely of
a grave illness in May 2005. He conceived the set-up of this experi-
ment and worked on aspects of the analysis until a few days before
his death.
§Paul Scherrer Institut,Villigen, Switzerland
¶EMail address: ralph.pixley@freesurf.ch
∗∗present address Physikalisch Technische Bundesanstalt, Braun-
schweig, Germany
1 m
2.3 m
field masses
upper test mass
lower test mass
wires
mass exchanger
balance
Pos. TPos. A
FIG. 1: Principle of the measurement. The FM’s are shown
in the position together (Pos. T) and the position apart (Pos.
A).
G. It was realized that the same type of measurement
could be made in the laboratory with much better ac-
curacy with the lake being replaced by the well defined
geometry of a vessel containing a dense liquid such as
mercury. Equipment for this purpose was designed and
constructed in which two 1.1 kg TM’s were alternately
weighed in the presence of two moveable field masses
(FM’s) each with a mass of 7.5 t. A first series of mea-
surements [7, 8, 9, 10, 11] with this equipment resulted
in a value for G with an uncertainty of 220 ppm due pri-
marily to a possible nonlinearity of the balance response
function. A second series of measurements was under-
taken to eliminate this problem. A brief report of this
latter series of measurements has been given in ref. [12]
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2
and a more detailed description in a thesis [13]. Since
terminating the measurements, the following four years
have been spent in improving the analysis and checking
for possible systematic errors.
Following a brief overview of the experiment, the mea-
surement and the analysis of the data are presented in
Secs. III entitled Measurement of the Gravitational Sig-
nal and Sec. IV entitled Determination of the Mass-
Integration Constant. In Sec. V, the present result is
discussed and compared with other recent measurements
of the gravitational constant.
II.GENERAL CONSIDERATIONS
The design goal of this experiment was that the un-
certainty in the measured value of G should be less than
about 20 ppm. This is comparable to the quoted accu-
racy of recent G measurements made with a torsion bal-
ance. It is, however, several orders of magnitude better
than previous measurements of the gravitationalconstant
(made after 1898) employing a beam balance [4, 5, 6].
The experimental setup is illustrated in Fig. 1. Two
nearly identical 1.1 kg TM’s hanging on long wires are
alternately weighed on a beam balance in the presence of
the two movable FM’s weighing 7.5 t each. The position
of the FM’s relative to the TM’s influence the measured
weights. The geometry is such that when the FM are in
the position labelled ”together”, the weight of the upper
TM is increased and that of the lower TM is decreased.
The opposite change in the TM weights occurs when the
FM are in the position labelled ”apart”. One measures
the difference of TM weights first with one position of the
FM’s and then with the other. The difference between
the TM weight differences for the two FM positions is
the gravitational signal.
The use of two TM’s and two FM’s has several advan-
tages over a single TM and a single movable FM. Com-
paring two nearly equivalent TM’s tends to cancel slow
variations such as zero-point drift of the balance and the
effect of tidal variations. Using the difference of the two
TM weights doubles the signal. In addition, it causes the
influence of the FM motion on the counter weight of the
balance to be completely cancelled. Use of two FM’s with
equal and opposite motion reduces the power required to
that of overcoming friction. This also simplified some-
what the mechanical construction.
The geometry has been designed such that the TM
being weighed is positioned at (or near) an extremum of
the vertical force field in both the vertical and horizontal
directions for both positions of the FM’s. The extremum
is a maximum for the vertical position and a minimum for
the horizontal position. This double extremum greatly
reduces the positional accuracy required in the present
experiment.
The measurement took place at the Paul Scherrer In-
stitut (PSI) in Villigen. The apparatus was installed in a
pit with thick concrete walls which provided good ther-
14
15
16
13
12
11
10
9
8
2
3
4
5
6
7
1
1 m
FIG. 2: A side view of the experiment. Legend: 1=measur-
ing room enclosure, 2=thermally insulated chamber, 3=bal-
ance, 4=concrete walls of the pit, 5=granite plate, 6=steel
girder, 7=vacuum pumps, 8=gear drive, 9=motor, 10=work-
ing platform, 11=spindle, 12=steel girder of the main sup-
port, 13=upper TM, 14=FM’s, 15=lower TM, 16=vacuum
tube.
mal stability and isolation from vibrations. The arrange-
ment of the equipment is shown in Fig. 2. The system in-
volving the FM’s was supported by a rigid steel structure
mounted on the floor of the pit. Steel girders fastened to
the walls of the pit supported the balance, the massive
(200 kg) granite plate employed to reduce high frequency
vibrations and the vacuum system enclosing the balance
and the TM’s. A vacuum of better than 10−4Pa was pro-
duced by a turbomolecular pump located at a distance
of 2 m from the balance.
The pit was divided into an upper and a lower room
separated by a working platform 3.5 m above the floor
of the pit. All heat producing electrical equipment was
located in the upper measuring room. Both rooms had
their own separate temperature stabilizing systems. The
long term temperature stability in both rooms was better
than 0.1◦C. No one was allowed in either room during
the measurements in order to avoid perturbing effects.
Page 3
3
The equipment was fully automated. Measurements
lasting up to 7 weeks were essentially unattended. The
experiment was controlled from our Zurich office via the
internet with data transfer occurring once a day.
III. MEASUREMENT OF THE
GRAVITATIONAL SIGNAL
We begin this section with a description of the de-
vices employed in determining the gravitational signal.
Following the descriptions of these devices, the detailed
schedule of the various weighings and their analysis are
given. Balance weighings will be expressed in mass units
rather than force units. The value of local gravity was
determined for us by E. E. Klingel´ e of the geology de-
partment of the Swiss Federal Institute. The measure-
ment was made near the balance on Sept. 11, 1996 us-
ing a commercial gravimeter (model G #317 made by
the company LaCoste-Romberg). The value found was
9.8072335(6) ms−2. This value was used to convert the
balance readings into force units.
A. The Balance
The beam balance was a modified commercial mass
comparator of the type AT1006 produced by the Mettler-
Toledo company. The mass being measured is compen-
sated by a counter weight and a small magnetic force
between a permanent magnet and the current flowing in
a coil mounted on the balance arm. An opto-electrical
feedback system controlling the coil current maintains
the balance arm in essentially a fixed position indepen-
dent of the mass being weighed. The digitized coil cur-
rent is used as the output reading of the balance.
The balance arm is supported by two flexure strips
which act as the pivot. The pan of the balance is sup-
ported by a parallelogram guide attached to the balance
frame. This guide allows only vertical motion of the pan
to be coupled to the arm of the balance. Horizontal forces
produced by the load are transmitted to the frame and
have almost no influence on the arm.
As supplied by the manufacturer, the balance had a
measuring range of 24 g above the 1 kg offset determined
by the counter weight. The original readout resolution
was 1 µg and the specified reproducibility was 2 µg. The
balance was designed especially for weighing a 1 kg stan-
dard mass such as is maintained in many national metrol-
ogy institutes.
In the present experiment, the balance was modified
by removing some nonessential parts of the balance pan
which resulted in its weighing range being centered on
1.1 kg instead of the 1 kg of a standard mass. There-
fore, 1.1 kg TM’s were employed. In order to obtain
higher sensitivity required for measuring the approxi-
mately 0.8 mg difference between TM weighings, the
number of turns on the coil was reduced by a factor of
6, thus reducing the range to 4 g for the same maximum
coil current.The balance was operated at an output
value near 0.6 g which gave a good signal-to-noise ratio
with low internal heating. For the present measurements,
a mass range of only 0.2 g was required. The full read-
out resolution of the analog to digital converter (ADC)
measuring the coil current was employed which resulted
in a readout-mass resolution of 12.5 ng.
An 8th order low-pass, digital filter with various time
constants was available on the balance. Due to the many
weighings required by the procedure employed to can-
cel nonlinearity (see Sec. IIID), it was advantageous to
make the time taken for each weighing as short as possi-
ble. Therefore, the shortest filter time constant (approx-
imately 7.8 s) was employed and output readings were
taken at the maximum repetition rate allowed by the
balance (about 0.38 s between readings).
Pendulum oscillations were excited by the TM ex-
changes. Small oscillation amplitudes (less than 0.2 mm)
of the TM’s corresponding to one and two times the fre-
quency for pendulum oscillations (approximately 0.26 Hz
for the lower TM and 0.33 Hz for the upper TM’s) were
observed. They were essentially undamped with decay
times of several days. The unwanted output amplitudes
of these pendulum oscillation were not strongly attenu-
ated by the filter (half-power frequency of 0.13 Hz) and
therefore had to be taken into account in determining the
equilibrium value of a weighing.
The equilibrium value of a weighing was determined
in an on-line, 5-parameter, linear least-squares fit made
to 103 consecutive readings of the balance starting 40 s
after a load change. The parameters of the fit were 2 sine
amplitudes, 2 cosine amplitudes and the average weight.
The pendulum frequencies were known from other mea-
surements and were not parameters of the on-line fit. The
40 s delay before beginning data taking was required in
order to allow the balance to reach its equilibrium value
(except for oscillations) after a load change. This proce-
dure (including the 40 s wait) is what we call a ”weigh-
ing”. A weighing thus required about 80 s.
Data of a typical weighing and the fit function used to
describe their time distribution are shown in the upper
part of Fig. 3. The residuals δ divided by a normalization
constant σ are shown in the lower part of this figure.
The normalization constant has been chosen such that
the rms value of the residuals is 1. Since the balance
readings are correlated due to the action of the digital
filter, the value of σ does not represent the uncertainty
of the readings. It is seen that the residuals show only
rather wide peaks. These peaks are probably due to very
short random bursts of electronic noise which have been
widened by the digital filter. With the sensitivity of our
modified balance, they represent a sizable contribution
to the statistical variations of the weighings. They are of
no importance for the normal use of the AT1006 balance.
A direct calibration of the balance in the range of the
780 µg gravitation signal can not be made with the accu-
Page 4
4
0 10203040
time [sec
152448500
152449000
152449500
balance reading
-3
0
3
δ/σ
FIG. 3: Shown in the upper plot are the balance readings
for a typical weighing illustrating the oscillatory signal due
to pendulum oscillations. The output is the uncalibrated bal-
ance reading corresponding to approximately 1.1 kg with a
magnetic compensation of 0.6 g. The amplitude of the os-
cillatory signal corresponds to about 1.5 µg. The lower plot
shows the normalized residuals. The normalization has been
chosen such that their rms value is 1.
racy required in the present experiment (< 20 ppm) since
calibration masses of this size are not available with an
absolute accuracy of better than about 300 ppm. Instead,
we have employed a method in which an accurate, coarse
grain calibration was made using two 0.1 g calibration
masses (CM’s). The CM’s were each known with an ab-
solute accuracy of 4 ppm. A number of auxiliary masses
(AM’s) having approximate weights of either 783 µg or
16×783 = 12,528 µg were weighed along with each TM
in steps of 783 µg covering the 0.2 g range of the CM.
Although the AM’s were known with an absolute accu-
racy of only 800 ng (relative uncertainty 1,000 ppm), the
method allowed balance nonlinearity effects to be almost
entirely cancelled. Thus, the effective calibration accu-
racy for the average of the TM difference measurements
was essentially that of the CM’s. A detailed description
of this method is given in Sec. IIIJ.
In our measurements, the balance was operated in vac-
uum. The balance proved to be extremely temperature
sensitive which was exacerbated by the lack of convec-
tion cooling in vacuum. The measured zero-point drift
was 5.5 mg/◦C. The sensitivity of the balance changed
by 220 ppm/◦C. To reduce these effects, the air tem-
perature of the room was stabilized to about 0.1◦C. A
second stabilized region near the balance was maintained
at a constant temperature to 0.01◦C. Inside the vacuum,
the balance was surrounded by a massive (45 kg) copper
box which resulted in a temperature stability of about
1 mK. Although zero-point drift under constant load for
a 1 mK temperature change was only 5.5 µg , the effects
of self heating of the balance due to load changes during
the measurement of the gravitational signal were much
FIG. 4: Drawing of TM inside the vacuum tube. Dimensions
are given in mm.
larger. Details of this effect and how they were corrected
are described in Sec.IIIG.
B.The Test Masses
One series of measurements was made using copper
TM’s and two with tantalum TM’s. Various problems
with the mass handler occurred during the measurements
with the tantalum TM’s which resulted in large system-
atic errors. Although the tantalum results were consis-
tent with the measurements with the copper TM’s, the
large systematic errors resulted in large total errors. The
tantalum measurements were included in our first pub-
lication, but we now believe that better accuracy is ob-
tained overall with the copper measurements alone. We
therefore describe only the measurements made with the
copper TM’s in the present work.
A drawing of a copper TM is shown in Fig. 4. The
45 mm diameter, 77 mm high copper cylinders were
plated with a 10 µm gold layer to avoid oxidation. The
gold plating was made without the use of nickel in or-
der to avoid magnetic effects. Near the top of each TM
on opposite sides of the cylinder were two short horizon-
tal posts. The posts were made of Cu-Be (Berylco 25).
The tungsten wires used to attach the TM’s to the bal-
ance were looped around these posts in grooves provided
for this purpose. The wires had a diameter of 0.1 mm
and lengths of 2.3 m for the upper TM and 3.7 m for
the lower. The loop was made by crimping the tungsten
wire together in a thin copper tube. A thin, accurately
machined, copper washer was placed in a cylindrical in-
dentation on the top surface of the lower TM in order
to trim its weight (including suspension) to within about
400 µg of that of the upper TM and suspension.
Measurement of the TM’s dimensions was made with
an accuracy of 5 µm using the coordinate measuring ma-
chine (CMM) at PSI. The weight of the gold plating
Page 5
5
was determined from the specified thickness of the layer.
The weight of the tungsten wires was determined from
the dimensions and the density of tungsten. The thin
tubing used to crimp the tungsten wires was weighed
directly. The weight of the complete TM’s was deter-
mined at the Mettler-Toledo laboratory with an accuracy
of 25 µg (0.022 ppm) before and after the gravitational
measurement. It was found that the mass of both TM’s
had increased by a negligible amount (0.5 ppm) during
the measurement.
An estimate of possible density gradients in the TM’s
was determined by measuring the density of copper sam-
ples bordering the material used for making the TM’s.
It was found that the variation of the relative density
gradients over the dimensions of either TM was less than
2 × 10−4in both the longitudinal and the radial direc-
tions.
C.TM Exchanger
In weighing the TM’s, it was necessary to remove the
suspension supporting one TM from the balance and
replace it by the other supporting the other TM. The
exchange was accomplished by a step-motor driven hy-
draulic systems to raise the suspension of one TM while
lowering the other. A piezo-electric transducer mounted
above the pan of the balance was used to keep the load on
the balance during the exchange as constant as possible.
This was done in order to avoid excessive heating due
to the coil current and to reduce anelastic effects in the
flexure strips supporting the balance arm. The output
excursions were typically less than 0.1 g. The exchange
of the TM’s required about 4 min.
The TM suspension rested on a thin metal arm de-
signed to bend through 0.6 mm when loaded with 1.1 kg.
Therefore, the transfer of TM’s was accomplished with
a vertical movement of typically 2 mm (0.6 mm bending
of the spring plus an additional 1.4 mm to avoid electro-
static forces). The metal arm was attached to a paral-
lelogram guide (similar to that of the balance) to assure
only vertical motion.
Although the parallelogram reduced the error resulting
from the positioning of the load, it was nevertheless im-
portant to have the TM load always suspended from the
same point on the balance pan. This was accomplished
by means of a kinematic coupling [14, 15]. The coupling
consisted of three pointed titanium pins attached to each
TM suspension which would come to rest in three tita-
nium V grooves mounted on the balance pan. The repro-
ducibility of this positioning was 10 µm. The pieces of
the coupling were coated with tungsten carbide to avoid
electrical charging and reduce friction.
D.Auxiliary Masses
In order to correct for any nonlinearity of the balance
in the range of the signal, use was made of many auxiliary
masses (AM’s) spanning the 200 mg range of the CM’s in
steps of approximately 783 µg. Although the AM’s could
not be measured with sufficient accuracy to calibrate the
balance absolutely, they were accurate enough to correct
the measured gravitational signal for a possible nonlin-
earity of the balance. Each TM was weighed along with
various combinations of AM’s. One essentially averaged
the nonlinearity over the 200 mg range of the CM’s in
256 load steps of 200 mg/256=783 µg. A weighing of
both 100 mg CM’s was then used to determine the ab-
solute calibration of the balance which is valid for the
TM weighings averaged over this range. The effect of
any nonlinearity essentially cancels due to the averaging
process. The accuracy of the nonlinearity correction is
described in Sec. IIIJ.
The 256 load steps were accomplished using 15 AM’s
with a mass of approximately 783 µg called AM1’s and 15
AM’s with 16 times this mass (12,528 µg) called AM16’s.
They were made from short pieces of stainless steel wire
with diameters of 0.1 mm and 0.3 mm. The wires were
bent through about 70◦on both ends leaving a straight
middle section of about 6 mm. The mass of the AM’s
were electrochemically etched to obtain as closely as pos-
sible the desired masses. The RMS deviation was 1.5 µg
for the AM1’s and 2.3 µg for the AM16’s.
By weighing a TM together with various AM combina-
tions, one obtains the value of the TM weight simultane-
ously with the linearity information. The only additional
time required for this procedure over that of weighing
only the TM’s is the time necessary to change an AM
combination (10 to 30 s).
E.Mass Handler
The mass handler is the device which placed the AM’s
and the CM’s on the balance or removed them from the
balance. The mass handler was designed by the firm
Metrotec AG. The operation of this device is illustrated
in the somewhat simplified drawing of Fig. 5 showing
how the AM1’s and the CM1 are placed on the metal
strip attached to the balance pan. Only 6 of the 14 steps
are shown in this illustration for clarity. The portion
of the handler used for the AM’16 and the CM2 (not
shown) is similar except that the AM’16 are placed on
a metal strip located below the one used for the AM1’s.
All of the AM1’s pictured in Fig. 5 are lying on the steps
of a pair of parallel double staircases.
are separated by 6 mm which is the width of the AM’s
between the bent regions on both ends.
between the staircases is such that they could pass on
either side of the horizontal metal strip fastened to the
balance pan as the staircases were moved up or down.
The motion of each staircase pair was constrained to the
The staircases
The spacing
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6
1
3
2
6
7
4
5
8
6
9
FIG. 5: Simplified drawing of the mass handler illustrating
the principle of operation.Legend: 1=pivoted lever pair
holding a CM, 2=narrow strip to receive the CM, 3=dou-
ble stair case pair holding AM’s, 4=narrow strip to receive
AM’s, 5=balance pan, 6 flat spring, 7=frame, 8=stepmotor
driven cogwheel, and 9=coil spring. The pivoted-lever pair
and the double-staircase pair are spaced such that they can
pass on either side of the narrow strips 2 and 4 fastened to
the balance pan. The two flat springs 6 form two sides of
a parallelogram which assures vertical motion of the double
stair case pair.
vertical direction by a parallelogram (similar to those of
the balance) fastened to the frame of the mass handler.
The staircases for AM1’s and AM16’s were moved by two
separate step motors located outside the vacuum system.
The step motors were surrounded by mu metal shielding
to reduce the magnetic field in the neighborhood of the
balance. Moving the staircases down deposited one AM
after another onto the metal strip. Moving the staircases
up removed the AM’s lying on the strip. The steps of
the staircase had hand filed, saddle shaped indentations
to facilitate the positioning of the AM’s. The heights of
the steps were 2 mm and the steps on the left side of the
double staircase were displaced in height by 1 mm from
those on the right. Thus, the AM1’s were alternately
placed on the balance to the left and to the right of the
center of the main pan in order to minimize the torque
which they produced on the balance.
Raising the staircase structure above the position
shown in the figure caused a rod to push against a piv-
oted lever holding CM1. With this operation, CM1 was
placed on the upper strip attached to the balance. Re-
versing the operation allowed the spring to move the lever
in the opposite direction and remove CM1 from the bal-
ance.
Due to the very small mass of the AM1’s, difficulty was
occasionally experienced with the AM1’s sticking to one
side of the staircase or the other. The staircases were
made of aluminum and were coated with a conductive
layer of tungsten carbide to reduce the sticking probabil-
ity. Sticking nevertheless did occur. The sticking would
cause an AM1 to rest partly on the staircase and partly
on the pan, thus giving a false balance reading. In ex-
treme cases, the AM1 would fall from the holder and
therefore be lost for the rest of the measurement. No
problem was experienced with the heavier AM16’s and
the CM’s.
F.Weighing Schedule
The experiment was planned so that the zero-point
(ZP) drift and the linearity of the balance could be deter-
mined while weighing a TM. In principle one needs just
4 weighings (upper and lower TM with FM’s together
and apart) to determine the signal for each AM placed
on the balance. Repeating these 4 weighings allows one
to determine how much the zero point has changed and
thereby correct for the drift. Since there are 256 AM val-
ues required to correct the nonlinearity of the balance,
a minimum of 2048 weighings is needed for a complete
determination of the signal corrected for ZP drift and lin-
earity. One also wishes to make a number of calibration
measurements during the series of measurements.
The order in which the measurements are performed
influences greatly the ZP drift correction of measure-
ments. Changing AM’s requires only 6 to 30 s, while
exchanging TM on the balance takes about 230 s and
moving the FM from one position to the other requires
about 600 s. These times are to be compared with the 80
s required for a weighing and about 1 hr for a complete
calibration measurement (see Sec. IIII). One therefore
wishes to measure a number of AM values before ex-
changing TM, and repeat these measurements for the
other TM before changing the FM positions or making a
calibration.
The schedule of weighing adopted is based on several
basic series for the weighing of the different TM’s with
different FM positions. The series are defined as follows:
1. An S4 series is defined as the weighing of four suc-
cessive AM values with a particular TM and with
all weighing made for the same FM positions.
2. An S12 series involves three S4 series all with the
same four AM values and the same FM positions.
The S4 series are measured first for one TM, then
the other TM and finally with the original TM.
Page 7
7
3. An S96 is eight S12 series, all made with the same
FM positions and with the AM values incremented
by four units between each S12 series. A TM ex-
change is also made between each S12 series. An
S96 series represents the weighings with 32 succes-
sive AM values for both TM all with the same FM
positions.
4. An S288 series is three S96 series, first with one
FM position, then the other and finally with the
original FM position. A calibration measurement
is made at the beginning of each S288 series. Thus,
the S288 series represents the weighings with 32
successive AM values for both TM’s and both FM
positions and includes its own calibration.
5. An S2304 series is made up of eight S288 series
with the AM values incremented by 32 between
each S288 series. An S2304 series completes the
full 256 AM values with weighings of both TM’s
and both FM positions.
A total of eight valid S2304 series was made over a pe-
riod of 43 days. Alternate S2304 series were intended to
be made with increasing and decreasing AM values. Un-
fortunately, the restart after a malfunction of the tem-
perature stabilization in the measuring room was made
with the wrong incrementing sign. This resulted in five
S2304 series being made with increasing AM values and
three with decreasing.
G.Analysis of the Weighings
In ref. [12, 13], the so called ABA method was used to
analyze the data obtained from the balance and thereby
obtain the difference between the mass of the A and B
TM. This method assumes a linear time dependence of
the weight that would be obtained for the A TM at the
time when the B TM was measured based on the weights
measured for A at an earlier and a later time. However,
a careful examination of the data showed that the curva-
ture of the ZP drift was quite large and was influenced
by the previous load history of the balance. This indi-
cated that the linear approximation was not a particu-
larly good approximation. We have therefore reanalyzed
the data using a fitting procedure to determine a contin-
uous ZP function of time for each S96 series. The data
and fit function for a typical S96 series starting with a
calibration measurement is shown in Fig. 6. The proce-
dure used to determine the ZP data and fit are described
in the following. The criterion for a valid weighing is
described in Sec. IIIH.
The data of Fig. 6 show a slow rise during the first
hour after the calibration measurement followed by a con-
tinuous decrease with a time constant of several hours.
These slow variations are attributed to thermal varia-
tions resulting principally from the different loading of
the balance during the calibration measurement. Super-
posed on the slow variations are rapid variations which
218.60218.70218.80
day of 2001
-2
-1
0
1
2
3
zero point [µg]
-3
0
3
δ/σ
FIG. 6: The zero-point variation as a function of time for a
typical S96 series including calibration is shown in the upper
part of this figure. The solid curve is the fit function starting
after the last dummy weighing. The fit function for this S96
series has 76 degrees of freedom. The normalized residuals
δ/σ are shown in the lower plot. The normalization of the
residuals has been chosen such that their rms value is 1.
are synchronous with the exchange of the TM’s. The
rapid variations peak immediately after the TM exchange
and decrease thereafter with a typical slope of 0.3 µg /hr.
The cause of the rapid variations is unknown.
The data employed in the ZP determination were the
weighings of the upper and lower TM’s for the S96 se-
ries. The known AM load for each weighing was first
subtracted to obtain a net weight for either TM plus the
unknown zero-point function at the time of each weigh-
ing. A series of Legendre polynomials was used to de-
scribe the slow variations of the zero-point function. A
separate P0 coefficient was employed for each TM. The
rapid variations were described by a sawtooth function
starting at the time of each TM exchange. The fit pa-
rameters were the coefficient of the Legendre polynomials
and the amplitude of the sawtooth function. The saw-
tooth amplitude was assumed to be the same for all rapid
peaks of an S96 series. The sawtooth function was used
principally to reduce the χ2of the fit and had almost no
effect on the results obtained when using the ZP func-
tion. All parameters are linear parameters so that no
iteration is required. The actual ZP function is the saw-
tooth function and the polynomial series exclusive of the
time independent terms (i.e. the sum of the coefficients
times Pn(0) for even n).
Such calculations were made for various numbers of
Legendre coefficients in the ZP function. It was found
that the gravitational signal was essentially constant for
a maximum order of Legendre polynomials between 8
and 36. In this range of polynomials, the minimum cal-
culated signal was 784.8976(91) µg for a maximum order
equal to 22 and a maximum signal of 784.9025(93) µg
for a maximum order equal to 36 (i.e. a very small dif-
ference). In all following results, we shall use the signal
Page 8
8
211.8211.9212.0212.1
-500
-400
-300
-200
-100
0
100
200
300
400
500
day of 2001
apart together
1 µg
10 min
FIG. 7: The measured weight difference in µg between TM’s
obtained from an S288 series. The magnified insert shows the
individual TM differences which are not resolved in the main
part of the figure.
784.8994(91) µg obtained with a maximum polynomial
order of 15.
It has been implicitly assumed in the above ZP deter-
mination that the AM load values were known with much
better accuracy than the reproducibility of the balance
producing the data used in the ZP fit. Although the AM
values were sufficiently accurate for determining the gen-
eral shape of the ZP function, their relative uncertainties
were comparable to the uncertainties of the balance data
used in the fit. The P0 mass parameters of the TM’s
obtained from the fit were therefore not used for the TM
differences at the two FM positions which are needed in
order to determine the gravitational signal. Instead, the
value of the ZP function was subtracted from each weigh-
ing, and an ABA mass difference was determined for each
triplet of weighings having the same AM load. Since the
mass of the AM’s do not occur in this TM difference, they
do not influence the calculation. The ABA calculation is
valid for this purpose since the ZP corrected weighings
have essentially no curvature. Such TM differences de-
termined for the apart-together positions of the FM’s are
then used to calculate the gravitational signal.
The TM differences for the apart-together positions
of the FM’s as a function of time are shown for a ZP
corrected S288 series in Fig. 7. Individual data points
are resolved in the magnified insert of this figure. Each
data point is the B member of a TM difference obtained
from an ABA triplet in which all weighings have the same
load value.
All of the TM differences (ZP corrected) for the entire
experiment are shown in Fig. 8. The data labelled apart
have been shifted by 782 µg in order to allow both data
sets to be presented in the same figure. A slow variation
of 2.5 µg in both TM differences occurred during the 43
day measurement. Also seen in this figure is a 0.7 µg
210220230240250
355
357
359
361
day of 2001
apart together
FIG. 8: The measured weight difference in µg between TM’s
for the FM’s positions apart and together.
values for the FM apart have been displaced 782 µg in order
to show both types of data in the same figure.
The measured
jump which occurred in the data for both the apart and
together positions of the FM’s on day 222. The slow vari-
ation is probably due to sorption-effect differences of the
upper and lower TM’s. The jump was caused by the loss
or gain of a small particle such as a dust particle by one
of the TM’s. In order to determine the gravitational sig-
nal, an ABA difference was calculated for apart-together
values having the same AM load. The slow variation seen
in Fig. 8 is sufficiently linear so that essentially no error
results from the use of the ABA method. The jump in
the apart-together differences caused no variation of the
gravitational signal.
In Fig. 9 is shown a plot of the binned difference be-
tween the FM apart-together positions for all valid data
(see Sec. IIIH. The differences were determined using the
ABA method applied to weighings made with the same
AM loads. Also shown in the figure is a Gaussian func-
tion fit to the data. The data are seen to agree well with
the Gaussian shape which is a good test for the qual-
ity of experimental data. The root-mean-square (RMS)
width of the data is 1.03 time the width of the Gaussian
function. The true resolution for these weighings may be
somewhat different than shown in Fig. 9 due to the fact
that the data have not been corrected for nonlinearity of
the balance (see Sec. IIIJ) and for correlations due to the
common ZP function . Nevertheless, these effects would
not be expected to influence the general Gaussian form
of the distribution.
A plot of the signal obtained for the S2304 series with
increasing and decreasing load is shown in Fig. 10. The
average signal for increasing load is 784.9121(125) µg and
the average for decreasing load is 784.8850(133) µg. The
common average for both is 784.8994(91) µg. The aver-
ages for increasing load and for decreasing load lie within
the uncertainty of the combined average. This shows that
Page 9
9
-1.0-0.5 0.00.51.0
0
100
200
difference [µg]
number/bin
FIG. 9: Binned data for the FM apart-together weight differ-
ences (points) and a fitted Gaussian function (curve) shown
as a deviation from the mean difference. Poisson statistics
were used to determine the uncertainties.
0123
series number
45678
784.85
784.90
784.95
785.00
signal [µg]
FIG. 10: The average signal for each of the eight S2304 series.
Series with increasing load are shown as circles. Series with
decreasing load are shown as squares. The dashed line is the
average of all eight series.
the direction of load incrementing did not appreciably in-
fluence the result.
Although the weighings making up an S96 series are
correlated due to the common ZP function determined
for each S96 series, the results of each S96 series, in par-
ticular the TM parameter, are independent. The 32 sig-
nal values obtained from the three S96 series making each
S288 series are also independent. However, since the non-
linearity correction (see Sec. IIIJ) being employed is ap-
plicable only to an entire S2304 series (not to individual
S288 series), it is only the eight S2304 series which should
be compared with one another. This restricts the way in
which the average signal is to be calculated for the entire
measurement, namely the way in which the data are to
be weighted.
We have investigated two weighting procedures. In the
first, each S2304 series average was weighted by the num-
ber of valid triplets in that series. This assumes that the
weighings measured in all S2304 series have the same a
priori accuracy. In the second method, it was assumed
that the accuracy for each weighing in a series was the
same but might be different for different series. We be-
lieve the second method is the better method since it
takes into account changes that occur during the long,
43 day measurement (e.g. the not completely compen-
sated effects of vibration, tidal forces and temperature).
The averages obtained with the two methods differ by
approximately 6 ng with the second method giving the
smaller average signal. This is a rather large effect. It is
only slightly smaller than the statistical uncertainty of 9
to 10 ng obtained for either method. In the rest of this
work, we shall discus only the results obtained with the
second method.
H.Criterion for Valid Data
Two tests were used to determine whether a measured
weighing was valid. An on-line test checked whether the
χ2value of the fit to the pendulum oscillations was rea-
sonable. A large value caused a repeat of the weighing.
After two repeats with large χ2, the measurement for this
AM value was aborted. An aborted weighing usually in-
dicated that the AM was resting on the mass handler and
on the balance pan in an unstable way.
A more frequent occurrence was that of an AM which
rested on both the mass handler and the balance was al-
most stable thereby giving a reasonable χ2. In order to
reject such weighings, an off-line calculation was made
to check whether the measured weight was within 10 µg
of the expected weight. The statistical noise of a valid
weighing was typically about 0.15 µg (see Fig. 9 show-
ing ABA difference involving 3 weighings). Excursions of
more than 10 µg were thus a clear indication of a mal-
function.
This off-line test is somewhat more restrictive than the
off-line test employed in our original analysis.
original analysis, a check was made only to see that the
weight difference between the TM’s for equal AM load-
ings was reasonable. The more restrictive test used in the
present analysis resulted in the rejection of the S2304 se-
ries at the time when the room temperature stabilizer
was just beginning to fail. It was also the reason for not
including the tantalum TM-measurements in the present
analysis. In the eight S2304 series accepted for the de-
termination of the gravitational constant, approximately
8% of the expected zero-point values could not be deter-
mined due to at least one of the three weighings at each
load value being rejected by the test for valid weighings.
In the
Page 10
10
I.Calibration Measurements
A coarse calibration of the balance was made periodi-
cally during the gravitational measurement (before each
S288 series) using two calibration masses each with a
weight of approximately 100 mg. A correction to the
coarse calibration constant due to the nonlinearity of the
balance will be discussed in Sec. IIIJ. The two CM’s used
for the coarse calibration were short sections of stainless
steel wire. The diameter of CM1 was 0.50(1) mm and
that of CM2 was 0.96(1) mm. The surface area of CM1
was approximately 1 cm2and that of CM2 was 0.5 cm2.
The CM’s were electrochemically etched to the desired
mass and then cleaned in an ethanol ultrasonic bath. The
mass of each CM was determined at METAS (Metrology
and Accreditation Switzerland) in air with an absolute
accuracy of 0.4 µg or a relative uncertainty of 4 ppm. The
absolute determinations of the CM masses were made be-
fore and after the gravitational measurement with copper
TM’s and after the second measurement with tantalum
TM’s. Only the first measurement was used to evaluate
the coarse calibration constant employed in the measure-
ment with copper TM’s. As will be discussed below, the
second and third measurement were used for the mea-
surement with copper TM’s only to check the stability of
the CM’s.
A calibration measurement involved either TM and one
of following seven additional loads: (1) CM1 alone, (2)
CM2 alone, (3) again CM1 alone, (4) empty balance,
(5) CM1+CM2, (6) empty balance and (7) CM1+CM2
and nine so called dummy weighings. These measure-
ments were made with no AM’s on the balance. After
the seventh weighing, a series of nine dummy weighings
alternating between upper and lower TM’s were made
with the AM load set to the value for the next TM
weighing. The dummy weighings were made in order
to allow the balance to recover from the large load vari-
ations experienced during the calibration measurement
and thereby come to an approximate equilibrium value
before the next TM weighing. Calibrations were made
alternately with the upper and lower TM’s as load. Cali-
bration measurements were made about twice a day. In-
cluding the dummy weighings, each calibration required
about 50 min.
A three-parameter least squares fit was made to the
calibration weighings labelled 4,5,6 and 7 above. The
fit thereby determined best values for the balance ZP,
the slope of the ZP and a parameter representing the
effective ZP corrected reading of the balance for the
load CM1+CM2. This third parameter is of particu-
lar interest since the coarse calibration constant is deter-
mined from the known mass of CM1+CM2 (measured by
METAS) divided by this parameter. Therefore, the re-
sults of the least-squares fit to each set of calibration data
gave a value for the coarse calibration constant which
then was used to convert the balance output of the S288
series to approximate mass values. An ABA analysis of
the first three weighings of each set of calibration data
was also made in order to determine the difference in
mass between CM1 and CM2.
The absolute masses obtained for CM1 and CM2 as
determined by METAS are given in columns 2 and 3 of
Table I. Also shown in Table I (column 4) are the mass
difference between CM1 and CM2 as obtained from the
METAS measurement in air and the average of our CM
measurement in vacuum. The mass differences between
CM1 and CM2 measured in vacuum are particularly use-
ful in checking for any mass variation of the CM’s.
TABLE I: The mass of the CM’s as measured by METAS and
the CM1-CM2 mass differences measured in air at METAS
and in vacuum during the gravitational measurements at PSI.
All values are given in µg.
Date CM1 CM2Difference
6.40(60)
5.853(19)
7.30(50)
7.269(29)
7.496(25)
7.04(50)
Feb 6, 01
Jul. - Sep., 01
Nov. 29, 01
Jan. - Mar., 02
Apr. - May, 02
May 27, 02
100,270.30(40)100,263.90(40)
in vacuum
100,262.90(35)
in vacuum
in vacuum
100,262.97(35)
100,270.20(35)
100,270.01(35)
It is seen that CM2 mass decreased by 1.00(53) µg be-
tween the first and second METAS measurements while
the mass of CM1 was essentially the same in all three
measurements. From the mass difference values in air
and vacuum it is clear that the change occurred after
the measurements with copper TM’s ended in Sept. 2001
and before the weighing at METAS in Nov. 2001 which
preceded the start of the tantalum measurements. We
ascribe this change of CM2 to either the loss of a dust
particle or perhaps a piece of the wire itself. The loss
of a piece of the wire was possible since the wire used
for the CM’s had been cut with a wire cutter and there
could have been a small broken piece that was not bound
tightly to the wire. For this reason only the values given
for the first weighing of the CM’s were used to determine
the coarse calibration constant used for the measurement
with copper TM’s.
A plot of the relative change of the effective ZP
corrected balance reading corresponding to the load
CM1+CM2 is shown in Fig. 11. It is seen that it changed
by only a few ppm over the 43 days of the measurement.
A linear fit made to these data results in a slope equal
to -0.044(6) ppm/day which is equivalent to a mass rate
variation of -0.0088(12) µg cm−2d−1. The uncertainty
was obtained by normalizing χ2of the fit to the degrees
of freedom (DOF).
The slow variation of the effective balance reading for
the load CM1+CM2 seen in Fig. 11 could be due either
to a change of the balance sensitivity, to a decrease in the
mass of CM1+CM2 due to the removal of a contaminant
layer from the CM’s in vacuum or to a combination of
both causes. A variation of the balance sensitivity would
have essentially no effect on the analysis of the weighing
for the gravitational measurement as the coarse calibra-
Page 11
11
tion constant used for the analysis was determined from
the balance parameter for each S288 series. However, a
variation of the mass of CM1+CM2 would result in an
error in the analysis since the mass would not be the
value measured by METAS shown in Table I.
In order to investigate this problem, we have examined
the difference between the balance readings for CM1 and
CM2. This difference is proportional to the surface ar-
eas of CM1 and CM2 which differ by approximately a
factor of 2 (CM1 area=1 cm2and CM2 area=0.5 cm2).
The balance reading difference is only slightly dependent
upon the coarse calibration constant so that it repre-
sents essentially the mass difference itself. In Fig 12 is
shown the measured mass difference as a function of time
during the gravitational measurement. Also shown is a
linear function fit to these data. The slope parameter
of the fit results in a rate of increase per area equal to
0.0021(18) µg cm−2d−1. The uncertainty has been de-
termined by normalizing χ2to the DOF. The sign of the
slope is such that the CM with the larger area has the
larger rate of increase. A mass difference variation (CM1-
CM2) would require a slope of -0.0088(12) µg cm−2d−1.
The measured slope of the effective balance reading for
the load CM1+CM2 clearly excludes such a large nega-
tive slope as assumed for a mass variation. We therefore
conclude that the variation of this parameter is due pri-
marily to the sensitivity variation of the balance.
We note that Schwartz [17] has also found a mass in-
crease for stainless steel samples in a vacuum system in-
volving a rotary pump, a turbomolecular pump and a
liquid nitrogen cold trap. His samples were 1 kg masses
with surface areas differing by a factor of 1.8. He mea-
sured the thickness of a contaminant layer using ellip-
sometry as well as the increase in weight of the sample
during pumping periods of 1.2 d and 0.36 d. The rate of
210220230
day of 2001
240 250
-3
-2
-1
0
difference [ppm]
FIG. 11: The change of the effective balance reading for the
load CM1+CM2 as a function of time relative to its value on
the first day. No valid measurements were made between day
229 and 235.
210220230
day of 2001
240250
5.25
5.50
5.75
6.00
6.25
CM1-CM2 [µg]
FIG. 12: The mass difference of the CM’s as a function of
time and the linear fit function.
mass increase per area which he reports is approximately
a factor of 5 larger than the value we find. No explana-
tion for this difference can be made without a detailed
knowledge of the partial pressures of the various contam-
inant gases in the two systems and the surface properties
of the samples employed.
There still remains the possibility that a rapid re-
moval of an adsorbed layer such as water might have
occurred between the absolute determination of the CM
masses in air at METAS and the gravitational measure-
ment in vacuum (i.e. during the pump down of the sys-
tem). Schwartz [16] has measured the mass variation per
unit area of 1 kg stainless steel objects in air with rel-
ative humidity between 3% and 77%. He [17] also has
measured the additional mass variation per area due to
pumping the system from atmospheric pressure at 3%
relative humidity down to 5×10−3Pa. His samples were
first cleaned by wiping them with a linen cloth soaked
in ethanol and diethylether and then ultrasonic clean-
ing in ethanol.After cleaning, they were dried in a
vacuum oven at 50◦C. For these cleaned samples, the
weight change found for 3% to 50% humidity variation
was 11.5 ng cm−2with an additional change of 29 ng
cm−2in going from 3% relative humidity in air to vac-
uum (total change of 40.5 ng cm−2). Similar measure-
ments with ”uncleaned” samples gave a total change of
80 ng cm−2. The variation due to the cleanliness of the
samples was much larger than the difference found for
the two types of stainless steel investigated and the ef-
fect of improving the surface smoothness (average peak-
to-valley height equal to 0.1 µm and 0.024 µm). Since the
cleaning procedure used for our CM’s and their smooth-
ness were different than the samples used by Schwartz,
we have employed the average of Schwartz’s ”cleaned”
and ”uncleaned” objects for estimating the mass change
of our CM’s. Based on these data, the relative mass dif-
ference found for both CM’s together as measured in air
Page 12
12
0 10203040506070
-3
10
-2
10
-1
10
0
10
number of parameters
χ2 probability
FIG. 13: The χ2probability as a function of the number of
parameters.
having 50% humidity and in vacuum was 0.5 ppm. We
assign a relative systematic uncertainty of this correction
equal to the correction itself.
J.Nonlinearity Correction
By nonlinearity of the balance, one is referring to the
variation of the balance response function with load, that
is, the degree to which the balance output is not a linear
function of the load. The nonlinearity of a mass compara-
tor similar to the one employed in the present work has
been investigated [18] by the firm Mettler-Toledo. It was
found that besides nonlinearity effects in 10 g load inter-
vals, there was also a fine structure of the nonlinearity
in the 0.1 mg load interval which would be important for
the accuracy of the present measurement. It is the non-
linearity of our mass comparator in the particular load
interval less than 0.2 g involved in the present experiment
that we wish to determine.
One expects the nonlinearity of the balance used in
this experiment to be small; however, it should be real-
ized that a 200 mg test mass (two 100 mg CM’s) required
for having an accurately known test mass for calibration
purposes is over 250 times the size of the gravitational
signal that one wishes to determine. In addition, the sta-
tistical accuracy of the measured gravitational signal is
some 30 times better than the specified accuracy (2 µg)
of the unmodified commercial balance. One therefore has
no reason to expect the nonlinearity of the balance to be
negligible with this precision. In Sec. IIIA we have pre-
sented the general idea that the measurements with 256
AM values tends to average out the effect of any nonlin-
earity. We wish now to give a more detailed analysis of
this problem.
The correction for nonlinearity makes use of an arbi-
trary response as a function of the load. Since the two
TM’s are essentially equal (< 400 µg difference), the vari-
ation of the response function can be thought of as being
a function of the additional load due to the AM’s. Al-
though a power series or any polynomial series would
suffice for this function, we have for convenience used a
series of Legendre polynomials
f(u) =
Lmax
?
ℓ=0
aℓPℓ(2u/umax− 1).
The coefficients of Pℓare chosen subject to the two con-
ditions that (1) f(u) = 0 for no load and (2) f(u) = C
for u = C where C is the weight of the two CM’s to-
gether. These two conditions represent the sensitivity
of the balance over the 0.2 g range of the calibration
(i.e. the coarse calibration). The value of the maximum
load umaxin the present measurements was very nearly
C. Substituting the above conditions into the response
function, one obtains for the lowest two coefficients the
expressions
a0= C/2 −
Lmax
?
even ℓ=2
aℓ
and
a1= C/2 −
Lmax
?
odd ℓ=3
aℓ.
One can then minimize
χ2=
N
?
n=1
[f(un+ s) − f(un) − yn]2σ−2
n
and thereby determine best values for the parameters s
and aℓ for ℓ = 1 to Lmax. The yn are the measured
balance signal for the load values un, s is the load in-
dependent signal and N is the number of different loads
with valid measurements. The error σnfor the load value
unis the load-independent intrinsic noise of the balance
σ0for a single weighing divided by the square root of the
number of weighings for the load un. The value of Lmax
must be chosen large enough to describe the response
function accurately. All of the parameters in the fit are
linear parameters with the exception of s. Thus, there
is no difficulty in extending the fit to a large number of
parameters since only the nonlinear parameter must be
determined by a search method.
In order to determine Lmax, we calculate the χ2prob-
ability [19] (often referred to as confidence level) as a
function of Lmax. This requires an approximate value
for the intrinsic noise of the balance σ0. The value of σ0
sets the scale of the χ2probability but does not change
the general shape of the function. One can obtain a rea-
sonable approximation for σ0by setting χ2equal to the
DOF obtained for a large number of parameter. We have
arbitrarily set χ2equal to the DOF for 61 parameters.
Page 13
13
0 50000100000
load [µg]
150000200000
784.50
784.60
784.70
784.80
784.90
785.00
785.10
785.20
signal [µg]
-3
0
3
δ/σ
FIG. 14: Signal and fit function employing 60 parameters as
a function of load. The data are shown as a stepped line. The
fit is the smooth curve. The lower plot shows the normalized
residuals. Residuals were divided by the relative uncertainty
of each point. The normalization has been chosen such that
the rms value of the residuals is 1.
The χ2probability as a function of the maximum num-
ber of parameters is shown in Fig. 13. It is seen that
the χ2probability reaches a plateau near this maximum
number of parameters.
Starting from a low value of 10−4for one parameter,
the χ2probability rises rapidly to a value of 0.05 for three
parameters. It remains approximately constant at this
value up to 57 parameters where it rises sharply to reach a
plateau of approximately 0.5 at 60 parameters and above.
The fit parameter representing the signal corrected for
nonlinearity of the balance was essentially constant over
the entire range of parameters with a variation of less
than ±1.3 ng. The signal for one parameter representing
complete linearity was 784.8994 µg. The signal of the
plateau region from 60 to 67 parameters was 784.9005 µg
with a statistical uncertainty of 5.5 ng. In this region
the signal varied by less than 0.2 ng. We therefore take
the nonlinearity correction of the measured signal to be
1.1(5.5) ng (i.e. the difference between the signal using
one parameter as would be obtained with no correction
and the average value obtained for 60 to 67 parameters).
The nonlinear signal and fit as a function of load de-
termined for 60 parameter is shown in Fig. 14. The func-
tion shows many narrow peaks with widths of 3 to 10 load
steps and with amplitudes of roughly 0.1 µg. In principle
one could use this response function to correct the indi-
vidual weighings with various loads; however, we prefer
to use the signal as corrected for nonlinearity over the
entire range of measurements as described above. The
variation of the response function indicates that a mea-
surement made at an arbitrary load value could be in
error by as much as ±130 ng assuming the response to
be linear. This is to be compared with the assumed un-
certainty in ref. [8] due to nonlinearity of ±200 ng.
212.0 212.5213.0
21.300
21.350
21.400
day of 2001
Temperature [deg C]
FIG. 15: Temperatures of the vacuum tube measured at the
position of the TM’s. The upper curve is the temperature
at the position of the upper TM. The square wave in the
middle section of the plot indicates the FM motion. The data
(crosses) for the lower TM and fit function (solid line) are
shown in the lowest section of the figure.
K. Correction of the TM-Sorption Effect
Moving the FM’s changed slightly the temperature of
the vacuum tube surrounding the TM’s. These tempera-
ture variations were due to changes in the air circulation
in the region of the vacuum tube as obstructed by the
FM’s. An increase of the wall temperature of the tube
caused adsorbed gases to be released which were then
condensed onto the TM. Since the temperature variation
was different in the regions near the upper and lower
TM’s, this resulted in a variation of the weight differ-
ence between the upper and lower TM’s (i.e. a ”false”
gravitational signal).
The temperature variation at the positions of the up-
per and lower TM’s during one day of the gravitational
measurement is shown in Fig. 15 along with a curve rep-
resenting the FM motion. The peak-to-peak tempera-
ture variation was approximately 0.04◦C at the upper
position and 0.01◦C at the lower position. The shape
of the temperature variation at the upper position was
used as a fit function (employing an offset and an ampli-
tude parameter) to obtain a better determination of the
temperature variation at the lower position. There were
32 one-day measurements of the temperature variations
during the gravitational measurement. The average am-
plitude at the lower position determined from these 32
measurements was 0.0138(2)◦C.
The signal produced by these temperature variations
was small and therefore not directly measurable with the
balance in a reasonable length of time. The procedure
that was employed to determine this temperature depen-
dent signal was to use four electrical heater bands to pro-
duce a variation of the temperature distribution along the
Page 14
14
312.70 312.90313.10
day of 2001
313.30313.50 313.70
11.0
11.5
12.0
12.5
signal[µ g]
FIG. 16: Weight difference between TM’s as a function of
time for a temperature variation roughly 10 times that of the
gravitational measurement. The solid curve is the best fit of
the temperature variation difference at upper and lower TM
positions. For the purpose of this plot, an arbitrary offset of
the weight difference between upper and lower TM has been
employed.
vacuum tube that was a factor of approximately seven
larger than the variation resulting from the motion of
the FM’s. The bands were positioned 30 cm above and
below the positions of the upper and lower TM’s. The
heater windings were bifilar to avoid magnetic effects.
The heater power (less than 3 W total) was turned off and
on with the same 8-hour period as the FM motion and
produced essentially no change in the average tempera-
ture of the vacuum tube in the day-long measurement.
The FM’s were not moved during the measurements with
heaters. The signal (TM weight difference as determined
with the balance) obtained during a one day measure-
ment with heaters is shown in Fig. 16. The shape of
the fit function (employing an offset and an amplitude
parameter) shown in this figure was obtained from the
variation of the temperature difference at the upper and
lower positions of the TM’s. The signal obtained from
the fitting procedure was 0.114(40) µg.
In order to scale the heater produced signal to that
resulting from the FM motion during the gravitational
measurement, we make the simplifying assumption that
the signal variation is proportional to the temperature
variation at the upper TM position minus the tempera-
ture variation at the lower TM position. The term vari-
ation in this statement refers to the variation about its
mean value. One uses the temperature difference since
the signal is defined as the difference between TM weigh-
ings.
With just four heater bands it was not possible to ob-
tain a variation of the temperature distribution along the
vacuum tube that was exactly a constant factor times
that of the FM motion. For the best adjustment that we
were able to obtain, the ratio of the heater produced tem-
perature variation to the FM produced variation was 7.1
at the upper position and 9.2 at the lower. The ratio for
the variation of the temperature difference at the upper
and lower positions relative to the gravitational values
was 6.8. These ratios are based on the peak-to-peak am-
plitudes obtained for the fitted functions. The scaling
factor for the temperature difference ratio is the recip-
rocal of the temperature difference ratio or 0.147. This
results in a scaled signal of 0.0168(58) µg where the un-
certainty is the statistical uncertainties of the measured
signal and the scaling factor. The scaled signal (”false”
signal) is to be subtracted from the total signal measured
in the gravitational experiment.
In order to check our assumption regarding the scal-
ing factor, we have made four additional one-day mea-
surements in which the temperature variations were very
different from that produced by the FM motion. The
object of these measurements was to determine whether
the scaled signals obtained with the heaters were consis-
tent with one another when calculated with the assumed
scaling factors. The most extreme distribution involved
a temperature variation of the lower TM which was even
larger (factor of 4) than that of the upper TM. The sig-
nals obtained in all of the test measurements were con-
sistent with each other within their statistical uncertain-
ties (relative uncertainties of approximately 30 %). We
therefore conclude that the assumption used for scaling
the signals was sufficiently accurate for the present pur-
pose. Nevertheless, we assign a systematic uncertainty
to the scaled signal equal to its statistical uncertainty
of 5.8 ng (relative systematic uncertainty of the ”false”
signal is 35 %).
L.Magnetic Forces on the Test Masses
In the absence of a permanent magnetization, the z
component of force on the TM due to a magnetic field
can be calculated from
Fz= −µ0χmV H∂H
∂z
where V denotes the volume of the TM, χmis its mag-
netic susceptibility and H is the magnetic field inten-
sity. The magnetic properties of the TM’s were measured
by METAS. No permanent magnetization was found
(< 0.08 A/m). The magnetic susceptibility was 4×10−6
for the copper TM’s. The magnetic field intensity for
both positions of the FM’s was measured at cm intervals
along the axis of the vacuum tube at the positions occu-
pied by the TM’s using a flux gate magnetometer. The
difference of Fzfor the FM positions obtained from these
data was 0.01 ng which is a negligible correction to the
measured gravitational signal.
Page 15
15
M.Tilt Angle of Balance
Since the weight of the TM’s and the weight of the
CM’s both produce forces on the balance arm in the ver-
tical direction, a small angle between the balance weigh-
ing direction and the vertical produces no error in the
weighing of the TM’s. However, if the balance weigh-
ing direction is correlated with the motion of the FM’s,
a systematic error in the measured gravitational signal
will result. Sensitive angle monitors were mounted on
the base of the balance. No angle variation correlated
with the motion of the FM’s was found with a sensitivity
of 100 nrad. Since the sensitivity of the balance varies
with the cosine of the angle (near 0 rad), this limit is com-
pletely negligible. For a balance misalignment of 0.01 rad
relative to vertical and a correlated variation of 100 nrad
with respect to this angle due to the FM motion, the
relative signal variation is approximately 0.001 ppm.
IV.
MASS-INTEGRATION CONSTANT
DETERMINATION OF THE
One must relate the gravitational constant to the mea-
sured gravitation signal. This involves integrating an in-
verse square force over the mass distribution of the TM’s
and FM’s. The gravitational force Fzin the z (vertical)
direction on a single TM produced by both FM’s is given
by
Fz= G
? ?
ez· (r2− r1) dm1dm2
|r2− r1|3
(1)
where ez is a unit vector in the z direction, r1 and r2
are vectors from the origin to the mass elements dm1of
the TM and dm2of the FM’s and G is the gravitational
constant to be determined. The mass-integration con-
stant is the double integral in Eq. (1) multiplying G.
Actually, the mass-integration constant for the present
experiment is composed of four different mass-integration
constants, namely those for the upper TM and lower TM
with the FM’s together and apart. We shall use as mass-
integration constant the actual constant multiplied by
the 1986 CODATA value of G (6.67259 m3kg−1s−2and
give the result in dimensions of grams ”force” (i.e. the
same dimensions as used for the weighings).
The objects contributing most to Fz(TM’s, FM tanks
and the mercury) have very nearly axial symmetry which
greatly simplifies the integration. Parts which do not
have axial symmetry were represented by single point
masses for small parts and multiple point masses for
larger parts. For axial symmetric objects, we employ the
standard method of electrostatics for determining the off-
axis potential in terms of the potential and its derivatives
on axis (see e.g.[20]). The force on a cylindrical TM in
the z direction produced by an axially symmetric FM can
be conveniently expressed as (see Eq. 10, Sec. VII)
Fz= 2MTM×
∞
?
n=0
V(2n+1)
0
n
?
i=0
1
(−4)i
1
i!(i + 1)!
1
(2n − 2i + 1)!b2n−2ir2i
(2)
where MTM is the mass of a cylindrical TM with radius
r and height b, and V(2n+1)
0
the gravitational potential with respect to z evaluated at
the center of mass of the TM (r = 0, z = z0).
The potential V (r = 0,z) of the various FM compo-
nents having axial symmetry was determined analytically
for three types of axially symmetric bodies, namely a hol-
low ring with rectangular cross section, one with trian-
gular cross section and one with circular cross section.
This allows one to calculate the gravitation potential of
the tank walls and the mercury content of the tank as
a sum of such bodies. For example, the region between
measured heights on the top plate and z = 0 at two values
of the radius was represented by a cylindrical shell com-
posed of a right triangular torus and a rectangular torus
(i.e. a linear interpolation between the points describing
the cross section of the rings). O-rings were calculated
employing the equation for rings with circular cross sec-
tions. A total of nearly 1200 objects (point masses and
rings of various shapes) were required to describe the two
FM’s.
The derivatives of the potential were evaluated using a
numerical method called “automatic differentiation” (see
e.g. [22]). For the geometry of the present experiment,
the terms in the summation over n decrease rapidly so
that 8 terms were sufficient for an accuracy of 0.02 ppm
in the mass-integration constant.
is the 2n + 1st derivative of
A.Positions of TM’s and FM’s
In order to carry out the mass integration, one needs
accurate weight and dimension measurements of the
TM’s and FM’s as well as distances defining their relative
positions. The dimension and weight measurements for
TM’s were described in Sec. IIIB. The measurement of
the TM positions shown in Fig. 17 will now be addressed.
A special tool was made to adjust the length of the
tungsten wires under tension. Each wire made a single
loop around the post on either side of the TM and a thin
tube was crimped onto the wires to hold them together
thereby forming the loop (see Fig. 4). The position of
the TM could only be measured with the vacuum tube
vented. The vacuum tube was removed below a flange
located at a point just above the upper TM. The TM
hanging from the balance was then viewed through the
telescope of an optical measuring device to determine its
position.
The vertical position of the TM’s and FM’s was mea-
sured relative to a surveyor’s rod which was adjusted to
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