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arXiv:grqc/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
PhysikInstitut der Universit¨ at Z¨ urich, CH8057 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 zeropoint 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 setup 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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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 zeropoint 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.
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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 LaCosteRomberg). 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 optoelectrical
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 signaltonoise 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 readoutmass resolution of 12.5 ng.
An 8th order lowpass, 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 (halfpower 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 online, 5parameter, linear leastsquares 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 online 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
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0 10 20 3040
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 zeropoint 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 zeropoint 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 CuBe (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
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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 MettlerToledo 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 stepmotor driven hy
draulic systems to raise the suspension of one TM while
lowering the other. A piezoelectric 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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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 pivotedlever pair
and the doublestaircase 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 zeropoint
(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.
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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
218.60 218.70218.80
day of 2001
2
1
0
1
2
3
zero point [µg]
3
0
3
δ/σ
FIG. 6: The zeropoint 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 zeropoint function at the time of each weigh
ing. A series of Legendre polynomials was used to de
scribe the slow variations of the zeropoint 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 aparttogether positions of the FM’s are
then used to calculate the gravitational signal.
The TM differences for the aparttogether 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
210220 230240 250
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 sorptioneffect 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 aparttogether
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 aparttogether differences caused no variation of the
gravitational signal.
In Fig. 9 is shown a plot of the binned difference be
tween the FM aparttogether 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 rootmeansquare (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
1.00.50.00.51.0
0
100
200
difference [µg]
number/bin
FIG. 9: Binned data for the FM aparttogether 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 online 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 offline 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 offline test is somewhat more restrictive than the
offline 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 TMmeasurements in the present
analysis. In the eight S2304 series accepted for the de
termination of the gravitational constant, approximately
8% of the expected zeropoint 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 threeparameter 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 leastsquares 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 CM1CM2 mass differences measured in air at METAS
and in vacuum during the gravitational measurements at PSI.
All values are given in µg.
DateCM1CM2 Difference
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
210220 230
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
240 250
5.25
5.50
5.75
6.00
6.25
CM1CM2 [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
tovalley 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 102030 40 506070
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 MettlerToledo. 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 loadindependent 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
050000 100000
load [µg]
150000 200000
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.
212.0212.5213.0
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 TMSorption 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 peaktopeak 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 oneday 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
312.70312.90313.10
day of 2001
313.30313.50313.70
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 8hour period as the FM motion and
produced essentially no change in the average tempera
ture of the vacuum tube in the daylong 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 peaktopeak 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 oneday 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.
MASSINTEGRATION 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− r13
(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 massintegration con
stant is the double integral in Eq. (1) multiplying G.
Actually, the massintegration constant for the present
experiment is composed of four different massintegration
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). Orings 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 massintegration 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
Page 16
16
FIG. 17: Drawing showing the measured vertical distances to
TM and FM surfaces for the two FM positions (T=Together
and A=Apart).
be vertical. The bottom of the surveyor’s rod was po
sitioned to just touch a special marker mounted on the
floor of the pit. The surveyor’s rod had accurate mark
ings every cm along its length. A precision levelling de
vice in which the optical axis of the telescope could be
displaced by somewhat more than a cm was then used to
compare the position of the upper surface of a TM with a
mark on the surveyor’s rod. Similar measurements were
made for the FM’s. These measurements were made be
fore and after each of the three gravitational measure
ment(Cu, Ta 1 and Ta 2 TM’s). Although the measuring
device including the surveyor’s rod was removed from the
pit after each of these measurements, the reproducibil
ity of each position measurement was found to be better
than 35 µm. The accuracy of the average of the two sets
of position measurement for each type of TM including
systematic uncertainty was 35 µm.
A small vertical displacement of the TM’s occurred
when the system was evacuated. This was measured dur
ing the evacuation of the system by observing the TM’s
through the windows on the side of the vacuum tube
using the levelling device that was also used for measur
ing the TM position in air. The vertical displacement
was measured several times and found to be 0.10(3) mm.
This displacement is shown as ε in Fig. 17).
The angle of the TM axis relative to the vertical direc
tion was also determined with the same precision levelling
instruments by measuring the height on the top surface
of the TM at two opposite points near the outer radius of
the TM. This was done for each TM from a viewing direc
tion almost perpendicular to the plane of the supporting
wires. The angle of the axis relative to the vertical was
found to be less than 1◦for both TM’s.
The horizontal positions of the TM’s were determined
using a theodolite. The left and right sides of the TM
were viewed through the telescope of the theodolite rel
ative to an arbitrary zero angle. The horizontal position
of the central tube was determined relative to the same
zero angle. These measurements were made for each TM
and FM from two nearly perpendicular viewing angles.
The measurements were made before and after each grav
itational measurement. The radial positions of the upper
and lower TM’s relative to the common axis of the FM’s
were found to be 0.45 mm and 0.50 mm, respectively.
The overall accuracy of the radial positions of the TM’s
from these measurements was 0.1 mm. This uncertainty
results in only a small uncertainty in the value of G de
termined in this experiment due to the extremum of the
force field in the radial direction. No problem was expe
rience with pendulum oscillation during these measure
ments as the amplitudes were strongly damped in air.
B.Dimensions and Weight of the FM’s
The individual parts of the FM’s were weighed at
METAS with an accuracy of 1 g. The weights were cor
rected for buoyancy to obtain the masses.
The narrow confines of the pit made measurement of
the FM’s deformed by the mercury load difficult. Al
though measurements of the individual pieces before as
sembly were in principle more accurate, the loading and
temperature difference between dimension measurements
and gravitational measurement reduced the accuracy of
these measurements. In addition, it is known that long
term loading can release tensions in the material which
result in inelastic deformation of the material. Therefore,
the measurements made with mercury load were always
used in the analysis when available.
The uncertainty in the height of the central piece
proved to be very important in determining the uncer
tainty of the massintegration constant. Due to the var
ious types of measurements for this dimension with dif
ferent systematic effects, we decided that the best value
would be an equally weighted average of the four avail
able measurements with the uncertainty being deter
mined from the rms (rootmeansquare) deviation from
the mean. The measurements employed were the follow
ing: (1) a Coordinate Measuring Machine (CMM) mea
surement before the tanks were assembled, (2) a Laser
Tracker (LT) measurement with mercury loading dur
ing the experiment, (3) a LT measurement in the ma
chine shop with no loading after completion of the ex
periment and (4) a CMM measurement after the tanks
had been disassembled at the end of the experiment. The
two CMM measurements were independent in that they
were made with different CMM devices and with differ
ent temperature sensors. The uncertainty in the height
as determined from the rms deviation was 19 µm for the
upper tank and 9 µm for the lower tank.
A cutaway drawing of a FM tank is shown in Fig.18.
All pieces were made from stainless steel type No. 1.4301
which is resistant to mercury and has a low magnetic sus
Page 17
17
? ? ? ?
? ? ?
? ?
? ?
? ?
? ? ?
? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ?
? ? ?
? ? ?
figure
FIG. 18: Drawing of the field mass. All dimensions are given
in mm.
ceptibility. The pieces were sealed to one another with
mercury resistant Perbunan Orings. The top and bot
tom plates were fastened to the central piece with 12
screws. The top and bottom plates were screwed to the
outer casing with 24 screws.
Due to the nearness of the central piece to the TM’s, es
pecially close tolerances were specified for this piece. The
central piece was annealed before machining to remove
tensions which could deform the piece during machin
ing. A surface roughness value of < 1 µm was obtained
by grinding the surface after machining. The roughness
value is defined as the average height of the peaks times
the area of peaks relative to the total area of the surface.
The top and bottom plates were less critical than the
central piece. A surface roughness value of 6 to 10 µm
was specified for the machining of the inner surface of
these pieces. The inner surface of the casing was even
less critical and a surface roughness value of 15 µm was
specified for it.
Since the mercury was filled into the tanks under vac
uum conditions, it filled more or less the exact surface
profile (i.e. the region between the grooves caused by
machining). The mercury in the filled tank was under
positive pressure on all surfaces including the top plate.
The over pressure ranged from about 0.1 bar on the top
plates (due to the height of mercury in the compensa
tion vessel shown in Fig. 18) and 1 bar on the bottom
plates. The bulging of the central cylinder and the outer
casing due to the mercury pressure was calculated using
the equations for thin cylindrical shells [21]. The inward
bulging of the central tube was found to have a maximum
value of 0.17 µm which is completely negligible for the
present experiment. The maximum outward bulging of
the outer casing was approximately 4 µm and resulted in
a small (8 ppm) correction of the volume. The loading of
the tanks produced a 2 µm elongation of the outer casing
corresponding to a relative volume increase of 1 ppm.
Before the tanks were assembled, measurements of the
individual pieces were made with a coordinate measuring
100 200 300400500
0.20
0.00
0.20
0.40
radius[mm]
zvalues[mm]
FIG. 19: Data and fit for the upper surface of the lower tank
as a function of radius. The solid curve is the fit function
based on the theory of circular plates. The offset has been
chosen such that the zvalue is 0 mm at a radius of 60 mm.
machine (CMM). The inner diameters of the central tube
were checked with a special dial gauge and found to agree
with the CMM values. The wall thickness of the outer
casing, top and bottom plates were measured with a wide
jaw micrometer having a dial gauge readout.
After the tanks were filled with mercury, measurements
of the outer surface of the top and bottom plates were
made with a lasertracker (LT). It was not possible to
measure the outer surface of the casing due to the close
confines of the pit. The LT measurements were made in
the dynamic mode in which the retro reflector is moved
on the surface and readings are taken as fast as possible
(1000 per s). An example of these data for the top surface
of the lower tank is shown in Fig. 19. The data for the
other plates are similar. The force due to the mercury
loading which tended to stretch the central tube, and to a
lesser extent the outer casing, could be determined from
these data using equations based on thin axial symmet
ric plates and shells [21]. The force on the central tube
was calculated to be (17 ± 8 kN) which resulted in an
elongation of 15 µm.
Finally, after the mercury had been removed from the
tanks, additional LT measurements were made of the out
side height of the central tube in order to clear up a dis
crepancy of this dimension as measured with the CMM
and LT.
The form of the central tube and outer casing as
determined by the analysis of the measurements was
very nearly circular; however, the deviation from per
fect roundness was larger than the expected accuracy of
the measurements. For the purpose of determining the
actual accuracy of the measurements, a leastsquares fit
was made to the data employing a number of Fourier
terms. The uncertainty of the measured data was then
determined by setting the χ2of this fit equal to the DOF.
Page 18
18
TABLE II: Estimated uncertainty in µm of the effective values
for the radius r of the central tube, the radius R of the outer
casing and the height h of the central tube due to various
effects. The uncertainties apply to the inside dimensions of
the tank.
Description
measurement
thickness
temperature
elongation
bulge
roughness
Total
σr
1.0
σR
σh
205

4

2
6
9
5
6
7

7
0.3

0
0
1.124
For the purpose of determining the uncertainty of the
volume and the massintegration constant, we have em
ployed effective dimensions describing a hollow, right cir
cular cylinder. The effective value of the small radius r
and height h of the central piece were assigned the ap
proximate values of 60 mm and 650 mm, respectively.
The effective inner radius R of the casing was then de
termined such that the inside volume of the tank was the
value determined from measurements.
Besides the accuracy of the directly measured dimen
sions, one has also to consider the effect of the expansion
coefficient of the stainless steel times the temperature dif
ference between measurement of the piece and the tem
perature during the gravitational measurement, accuracy
of the surface roughness and the deformation due to the
load. The accuracy of the temperature, roughness and
deformations effects are assumed to be one half of the
change caused by these effects. An estimate of these ac
curacies for the inside effective dimensions of the tanks
is given in Table II.
C.Weighing the Mercury
After a preliminary measurement in which the tanks
were filled with water, the water was drained from the
tanks and they were ventilated with warm dry air. The
tanks were then evacuated to a pressure of 10−2mbar
using a rotary pump with an oil filter to prevent back
streaming in preparation for being filled with mercury.
Since the tanks were to be filled with mercury only
once, every effort was made to weigh the mercury as
accurately as possible during the filling. The mercury
(specified purity 99.99%) was purchased in 395 flasks
each weighing 36.5 kg and containing 34.5 kg of mercury.
The general procedure for filling each tank was as follows.
The outer surface of the flasks were first cleaned with
ethanol. Half of the flasks were brought into a measuring
room near the experiment and allowed to come into equi
librium with the room atmosphere for a few days. These
flasks were then weighed over a period of one week. Then,
one after another, the flasks were attached to the transfer
device. Most of the mercury in a flask was transferred
via compressed nitrogen, first into an intermediate ves
sel used as a vacuum lock and then into the evacuated
tank. A small amount of mercury was intentionally left
in each flask in order not to transfer any of the thin ox
ide layer floating on the surface. The filling of a tank
required one week. After completion of the filling, the
flasks with their small remaining mercury content were
weighed again. The entire process was then repeated for
the second tank.
Various precautions and test measurements were un
dertaken to insure the accuracy of the weighings. The
balance employed for these measurements was type SR
30002 made by MettlerToledo.
erated in the differential mode with accurately known
(5 mg) standards for weights less than 2 kg (for empty
flasks) and a stainless steel mass of approximately 36.6 kg
made in our machine shop (for full flasks). The mass
of the 36.6 kg weight was calibrated several times at
METAS and remained constant within the 18 mg (the
certified accuracy of the weighings) during the weighing
of the mercury. The reproducibility of the weighing of an
almost empty flask or full flask was 20 mg. The averageof
three weighings was made for empty and full flasks with
the balance output transferred directly to a computer
via an RS232 interface.The balance was checked for
nonlinearity and none was found within the accuracy of
the standard weights. A centering table was used which
allowed the flasks to be off center by as much as 2 cm
without influencing the measurement. Atmospheric data
used for buoyancy corrections were taken several times a
day. A 10 kg calibration test was made once a day and
the balance zero was checked every hour. It was found
that the flasks were magnetized along their symmetry
axis. Rotating the flask on the balance did not change
the measured weight but inverting the flask resulted in
a 100 mg difference. Since weighings were always made
with the flasks in an upright position, no magnetization
error occurred in the difference between full and empty
flasks. The variation of the weight for 12 flasks was mon
itored over a period of 8 weeks. The variation was similar
for all 12 flasks and amounted to about 20 mg per flask
during the 3 weeks required to weigh the flasks and fill
the tank. Mercury droplets which had not reached the
tanks and small flakes of paint which had accidentally
been removed from the outer surface of the flasks during
the transfer process were collected and weighed.
The total uncertainty in the approximately 6760 kg of
mercury in each tank was 12 g for the upper tank and
15 g for the lower tank which gives a relative accuracy
of 2.2 ppm for the mass of mercury in the tanks.
listing of the various weighing uncertainties is given in
Table 3. An estimate of the mercury and flask residue
that had not been collected (and the uncertainty of the
amount collected) was assumed to be 20% of the amount
collected. The pressurizing of the flasks with nitrogen
during transfer resulted in a buoyancy correction of the
almost empty flasks due to the difference in density be
tween air and nitrogen. The flasks were sealed after use,
The balance was op
A
Page 19
19
TABLE III: A listing of the weighing uncertainties for the
upper and lower tanks. All uncertainties except that of the
standard masses are independent for each tank. The total
mass of mercury in each tank was approximately 6760 kg.
Type of UncertaintyUpper
Tank[g]
8
7
4
3.7
1.2
0.4
12
Lower
Tank[g]
11
8
4
3.7
1.2
0.4
15
loss of mercury and residue from flasks
approximate equation
mass variation of flasks
uncertainty of standard masses
buoyancy correction
weighing statistics
total uncertainty
but air gradually replaced the nitrogen. The assumption
of a constant density (approximate equation) for the gas
left in each empty flask during the time of the measure
ment amounted to an uncertainty of 7 and 8 g for the
upper and lower tanks, respectively. The change in mass
of the flasks during the three week measurements gave
an uncertainty of 4 g for each tank. The accuracy of the
standard masses caused an uncertainty of 3.7 g for each
tank. Air buoyancy uncertainties resulted in an uncer
tainty of 1.2 g for each tank. The statistical uncertainty
due to reproducibility of the balance was 0.4 g for each
tank.
D.Mass Distribution of the Central Piece
Although the principle mass making up the FM’s was
mercury and therefore had only a negligible density vari
ation due to its pressure gradient, the density variation
of the walls of the tanks was important in determining
the massintegration constant.
central pieces which were very near to the TM’s and
which were composed of three different pieces of stainless
steel welded together were critical for this determination.
Therefore after the gravitational measurements had been
completed, the central pieces were cut into a number of
rings in order to determine the density of these rings and
thus the mass distribution of the central pieces.
As shown in Fig. 18, each central piece was composed
of upper and lower flanges and a central tube. The ma
terial of the flanges extended about 40 mm beyond the
surface of the flanges in the form of the central tube. In
order to determine the vertical mass distribution in the
flanges, three rings of 10 mm height were cut from each
of these 40 mm sections with the last ring straddling the
weld between flange and central tube. The weight and
dimensions of the various pieces (12 rings, 4 flanges and
two central sections of the tube) were used to determine
the densities of these pieces. The densities of the flanges
were found to be between 7.9138 to 7.9147 g cm−3and
the densities of the central section of tubes were 7.9062
and 7.9101 g cm−3. The accuracy of the absolutedensity
In particular, the two
determinations was somewhat better than 0.001 g cm−3.
The density of the weld regions did not differ signifi
cantly from that of the flanges.
flange densities over the 65 mm of the flange and adjoin
ing section of the tube was found to be less than 0.005 g
cm−3. From these measurements, it was not possible to
determine a radial density gradient of the flanges. For
calculating the effect of a radial gradient on the mass
integration constant, we shall make the assumption that
the radial density variation was less than 0.005 g cm−3
over the radial dimension of the flange (160 mm). The
vertical density gradients of the central tubes were not
important since their effects on the gravitational signal
are almost completely cancelled due to the symmetry of
central tube relative to the aparttogether measurements
with the FM’s.
The variation of the
E. Using the Measured Dimensions
The first step in calculating the massintegration con
stant was to enter the measured dimensions and masses
of the various pieces in a computer program. For pieces
that had essentially axial symmetry such as the central
tube and the outer casing, an average radius was deter
mined from the data measured at each height and used in
further calculations. For horizontal surfaces which were
nearly planar such as the top and bottom plates, an av
erage height was determined from the data at each mea
sured radius and used in further calculations. Since the
original data were normally available only in Cartesian
coordinates, it was necessary to determine the symmetry
axis and make the conversion to cylindrical coordinates.
For data without axial symmetry such as screws, screw
holes and linear objects, single or multiple point masses
were used. With this reduced set of dimensions, approxi
mately 580 data elements were necessary to describe each
tank.
As a preliminary calculation related to the mass
integration constant, the volume of the individual pieces
and the inside volume of the tanks were calculated from
the reduced dimensions. Using the known weight of the
piece, the calculated volume allowed the density of the
material to be determined. This was a valuable test to
check whether the input data for the piece was reason
able.
The volume for pieces with axial symmetry was de
termined by making a linear interpolation between the
points of the reduced data. The volume of a piece was
thus composed of a sum of cylindrical rings with rectan
gular and right triangular form. A cylindrical ring with
circular cross section was used for the volume of Orings.
The accuracy of the linear approximation in the volume
determination was checked relative to a quadratic ap
proximation of the surface. The linear approximation
was found to be sufficient for all calculations.
The original CMM measurements had been made for
12 ϕ angles at 14 heights on the central tube, at 4 radii on
Page 20
20
the horizontal surfaces of the central flange, at 11 radii on
the horizontal surfaces of the top and bottom plates and
at 7 heights on the outer casing. Although many more
points of the horizontal surfaces were measured with the
LT, they were reduced to the original CMM points for
the purpose of volume integration and mass integration
by fitting functions to the LT data. Only outside surfaces
were measured with the LT. Inside dimensions were ob
tained from the LT data by subtracting the micrometer
thickness values. The only measurements of the casing
radius were the CMM measurements of the inner radius.
The outer surface of the casing was determined from the
CMM values combined with the micrometer data.
F. Density Constraint
Since the mercury represented roughly 90% of the to
tal tank mass, special attention was given to its contribu
tion to the signal. The density of mercury samples from
each tank was measured at the PhysikalischTechnische
Bundesanstalt, Braunschweig with an accuracy of 3 ppm.
One can use this density and mass measurements of the
mercury (see Sec. IVC) to obtain better values for the ef
fective tank dimensions and thus for the massintegration
constant. This results in a correlation among the effective
parameters, r,R,h,m. The method employed to deter
mine the best parameters representing the effective val
ues (determined from measurements as described in the
previous section) is based on minimizing a χ2function of
the form
χ2=
?r − r0
σr
?2
+
?R − R0
σR
?2
?2
+
?h − h0
σh
?2
?2
+
?m − m0
σm
+
?̺ − ̺0
σ̺
(3)
subject to the density constraint
̺ =
m
π(R2− r2)h.
(4)
After substituting ̺ from Eq. (4) in Eq. (3), χ2becomes
a function of the four parameters r, R, h, m, the five
measured quantities r0, R0, h0, m0, ̺0and their uncer
tainties (see Table IV for the uncertainties of the effective
values). The simplex method was used to minimize χ2
and thereby obtain best values for the fit parameters.
Although ̺ is not explicitly one of the fit parameters, a
best value for ̺ can be obtained by substituting best fit
parameters in Eq. (4).
The difference between best fit parameters and the
measured values are shown in Table IV along with the
resulting minimum χ2. It is seen that the difference be
tween parameters and measured values is less than the
error in all cases and that χ2is consistent with the ex
pected χ2for a leastsquares fit with one DOF.
In order to obtain the uncertainty of the mass
integration signal one needs the parametererror matrix
TABLE IV: The correlated measured values, their uncertain
ties and the difference between the best fit parameters and the
measured values for upper and lower tanks labelled 1 and 2.
Only approximate values are shown for the measured quanti
ties.
Measured
60 mm
498 mm
650 mm
Uncertainty
1.1 µm
9.0 µm
24.0 µm
14.8 g
Difference 1
0.007 µm
3.5 µm
9.5 µm
0.3 g
1.3 µg/cm3
0.27
Difference 2
0.013 µm
7.1 µm
19.0 µm
0.7 g
2.5 µg/cm3
1.18
r0
R0
h0
m0 6760 kg
̺0 13.54 g/cm340.6 µg/cm3
χ2
involving r, R, h, m and ̺ multiplied by partial deriva
tives of the signal with respect to the these quantities.
The partial derivative with respect to ̺ is zero since it
does not occur explicitly in the expression for the signal.
The error of the signal S is given by
σS=
?
i,j
∂S
∂xi
∂S
∂xjerr(xi,xj)
1/2
where the xiand xjare any pair of the measured quanti
ties and err(xi,xj) is the 5 by 5 parametererror matrix.
Assuming that, for small variations about the measured
values, the fit parameters represented by the 5dimension
vector y can be expressed as a linear function of the mea
sured values x of the form y = Tx + a. The parameter
error matrix can be written as
err(yi,yj) = TV Tt
where T is the Jacobi derivative of y with respect to x,
Ttis its transpose, V is the 5 by 5 matrix covariance ma
trix (i.e. Vi,j= err(yi,yj)) with all zero elements except
along its diagonal and a is a constant vector. The ele
ments of the matrix T and the vector a were determined
numerically by solving a system of linear equations in
which the fit parameters were determined for measured
values incremented by small amounts (σx).
The partial derivatives were also determined numeri
cally by calculating the signal for measured values with
small increments (σx). The signal was calculated using
actual dimensions of the deformed tanks corrected by
factors relating the r,R,h parameters to the effective di
mensions r0,R0,h0. The resulting covariance matrix rep
resenting the square of the uncertainty in the calculated
massintegration signal is shown in Table V. It is seen
that the elements involving R and h are much larger than
those for r and m. For the upper tank (tank 1), the rel
ative uncertainty of the calculated massintegration con
stant due to the correlated dimensions is 2.14 ppm. For
tank 2, it is 2.41 ppm. The large cancellation occurring
in the sum of the elements results in the uncertainty for
these constrained parameters being approximately a fac
tor of seven smaller than the uncertainty that would be
obtained without the density information.
Page 21
21
TABLE V: The covariance matrix involving r,R,h,m for the
uncertainty of the calculated signal. Values are given in units
of ng2. Only the upper part of the symmetric matrix is shown.
The sum of all element in the full matrix is 2.84 ng2. The sum
of a similar diagonal matrix for uncorrelated r,R,h,m values
(not shown) is 175 ng2.
rRhm
r
R
h
m
0.520.04
35.77
0.050.0003
−0.20
−0.25
0.66
−42.52
51.62
G. The Effect of Air Density
Since air is not present in the region of the FM’s, the
motions of the FM’s results in a force on the TM’s due
to the mass of the air elsewhere. It is as if there were a
negative contribution to the mass of the FM’s due to the
lack of air in this region. This effect depends upon the
density of the air in the region surrounding the FM’s.
The air pressure, the relative humidity and the air tem
perature were recorded every 12 min during the gravita
tional measurement thereby providing the information
necessary to determine the air density.
air density on the calculated massintegration constant
was approximately 100 ppm. The variation of the mass
integration constant for this effect was only about 1 ppm.
Thus, it was sufficient to employ only the average value
of the air density during the entire gravitational measure
ment. The average density employed was 1.156 kg/m3.
The effect of
H. The Effect of Mercury Expansion
Due to the small temperature variations of the FM’s,
the volume of the mercury relative to the volume of the
tanks changed slightly during the gravitational measure
ment. This resulted in a variation of the mercury height
in the compensation vessels. The height of the mercury
in each compensation vessel was recorded every 12 min
during the experiment. The calculated massintegration
constant varied by only 0.3 ppm due to this effect. There
fore, only an average value of the mercury height in each
compensation vessel was employed in determining the
massintegration constant for the entire measurement.
I.Uncertainties Affecting the MassIntegration
Constant
The relative uncertainties of the massintegration con
stant due to the various measured and estimated quan
tities relating to either the upper or lower TM or to ei
ther the upper or lower FM are given in Table VI. The
signs of the estimated quantities have been chosen to give
the largest uncertainty of the massintegration constant.
TABLE VI: Massintegration constant relative uncertainties
(ppm) associated with the measured quantities. ’Upper’ and
’Lower’ refer to the upper and lower FM or TM quantities.
The values in parentheses are the uncertainty of the measured
quantities. Where two measured values are listed, the first ap
plies to the upper object and the second to the lower object.
Quantities marked with a∗are obtained from estimated lim
iting values. All uncertainties are independent except for the
constrained quantities r,R,h,m. However, these constrained
quantities are independent for the upper and lower tanks.
Measured Quantity UpperLower
FM Quantities
r,R,h,m constrained
position=2460 or 1042 mm (35 µm)
inner radius=50 mm (1.1 µm)
travel=709 mm (10 µm)
upper plate mass=153 kg (0.9 g)
lower plate mass=154 kg (0.45 g)
central piece mass=46 kg (0.18 g)
outer tube mass=412 kg (0.83 g)
central piece no density gradient
central piece z density gradient∗
central piece r density gradient∗
1.20
2.05
0.91
0.95
0.01
0.02
0.03
0.01
0.03
1.09
2.99
0.91
1.22
0.01
0.02
0.03
0.01
0.03
< 0.03
< 1.1
< 0.03
< 1.1
TM Quantities
radius=23 mm (5 µm)
height=77 mm (5 µm)
position=2495, 1077 mm (35 µm)
mass=1.1 kg (300 µg)
off center=0.44 or 0.51 mm (0.1 mm)
angle relative to vertical∗
relative z density gradient∗
relative r density gradient∗
0.57
0.49
0.45
0.14
1.03
0.57
0.87
0.32
0.14
1.03
< 1.85
< 0.9
< 0.02
< 1.85
< 0.7
< 0.02
With the exception of the constrained quantities r, R,
h, and m, all measured quantities of this table are in
dependent (i.e. uncorrelated). All estimated quantities
are also independent. The total uncertainty of the mass
integration constant due to the measured quantities listed
in Table VI results in a relative statistical uncertainty of
4.89 ppm. The total uncertainty of the massintegration
constant due to estimated quantities results in a relative
systematic uncertainty of 3.25 ppm.
In addition to the uncertainties related to either TM
alone, there is the common vertical displacement (shown
as ε in Fig. 17) of both TM’s due to evacuating the sys
tem. The uncertainty of this displacement results in a rel
ative uncertainty of the massintegration constant equal
to 0.78 ppm which is added to the other uncertainties as
an independent relative uncertainty. Including the un
certainty of ε results in a relative statistical uncertainty
of the massintegration constant equal to 4.95 ppm.
Page 22
22
TABLE VII: Relative statistical and systematic uncertainties
of G as determined in this experiment.
Description
Measured Signal
Weighings
TMsorption
Linearity
Calibration
Mass Integration
Total
Statistical(ppm) Systematic(ppm)
11.6
7.4
6.1
4.0
5.0
16.3
7.4
0.5
3.3
8.1
V. DISCUSSION OF MEASUREMENTS
The measured gravitationalsignal discussedin
Secs. IIIG to IIIK is 784,883.3(12.2)(5.8) ng.
calculated massintegration constant determined in
Secs. IVA to IVE is 784,687.8(3.9)(2.6) ng. Using these
values, we obtain the value for the gravitational constant
The
G=6.674252(109)(54)×10−11m3kg−1s−2.
A summary of the relative uncertainties contribut
ing to this result is given in Table VII.
The relative statistical and systematic uncertainties
of this result are 16.3 ppm and 8.1 ppm, respectively.
The two largest contributions to the total relative un
certainty are the statistical uncertainty of the weighings
(11.6 ppm) and the combined statistical and systematic
uncertainty due to the TMsorption effect (10.3 ppm).
All uncertainties have been given as one sigma uncertain
ties. Statistical and systematic uncertainties have been
combined to give a total uncertainty by taking the square
root of the sum of their squares.
A. Comparison with Our Previous Analysis
Our previously published value [12] for G was
6.674070(220) × 10−11m3kg−1s−2which was based
on the measurements of both the copper and tanta
lum TM’s. The value for the copper TM’s alone was
6.674040(210)× 10−11m3kg−1s−2. The value obtained
for G in the present analysis for only the copper TM’s
(6.674252((124) × 10−11m3kg−1s−2) is in reasonable
agreement with the previous value. The difference be
tween the present and previous result is due primarily to
the correction for the ZP curvature which was not taken
into account in the previous analysis. A minor difference
is also due to a slightly different selection of the analyzed
data.
The uncertainty given for the present result is appre
ciably smaller than that of our previous result.
is due to a better method used in computing the non
linearity correction (Sec. IIIJ) and a calculation of the
massintegration constant (Sec. IVE) using the mercury
density as a constraint. The uncertainty of the linearity
This
correction was reduced from 18 ppm to 6.1 ppm and the
uncertainty of the massintegration constant was reduced
from 20.6 ppm to 6.7 ppm. The statistical uncertainty of
the weighings in the present analysis is somewhat larger
than the previous value (9.1 ng vs 5.4 ng). This is due
to the correlation of the ZPcorrected data of the present
analysis. The previous ABA analysis involved only un
correlated data.
B. Comparison with Other Measurements
Recent measurements [23, 24, 25] of the gravitational
constant with relative errors less than 50 ppm are listed in
Table VIII and shown in Fig. 20. We list only the latest
publication of each group. It is seen that the present
result is in good agreement with those of Gundlach and
Merkowitz [23] and Fitzgerald [25]. It is in disagreement
(3.6 times the sum of the uncertainties) with the result
of Quinn et al. [24].
ref_23(2000)
ref_24(2001)
ref_25(2003)
Present_work
6.6740 6.6750 6.6760
G [1011m3 kg1 s2]
FIG. 20: Plot of recent measurements with relative errors less
than 50 ppm
All of the measurements listed in Table VIII with the
exception of our own were performed using torsion bal
ances. It is therefore instructive to compare the problems
encountered in the different types of measurements.
In the measurements being discussed, the statistical
accuracy in determining the gravitational signal was ob
tained in measurements lasting one to six weeks. How
ever, as in the case of all precision measurements, the
time required for obtaining a good statistical accuracy of
the measurement is less than the time required to obtain
calibration accuracy of the equipment and the time nec
essary to investigate and eliminate systematic errors. All
of the measurements listed in Table VIII have been long,
ongoing investigations which have lasted for periods up
to ten years.
Although the beambalance measurement was made
Page 23
23
with more massive FM’s (15 t) than were employed in
the torsionbalance measurements (< 60 kg), the larger
gravitational signal had to be measured in the presence
of the TM weight. In our experiment the gravitational
signal was roughly 0.7 ppm of the total weight on the
balance. This small ratio of signal to total weight on the
balance resulted in larger effects of zeropoint drift as
well as larger statistical noise in the beambalance data
than in the torsionbalance data. In the torsionbalance
measurements, the deflection of the balance arm is due
entirely to the gravitational force to be determined with
only small perturbations from distant moving objects.
A similar problem has to do with the change of TM
weight that is produced by an adsorbed water layer. In
our measurement, this varied with the temperature of the
vacuum tube produced by the FM motion. This resulted
in one of the largest contribution to the uncertainty of
the gravitational signal (see Sec. IIIJ). In the beam
balance measurement the weight change adds directly to
the signal whereas in the torsionbalance it adds only to
the mass of the torsionarm and is therefore a negligible
effect.
The calibration of the beam balance, while simple in
principle, is difficult in practice due to the lack of calibra
tion masses having the required mass and accuracy. We
have used an averaging method to correct for the non
linearity of the balance (see Sec. IIIK) involving a large
number of auxiliary masses. This allowed the compari
son of the gravitational signal with a heavier, accurately
known calibration mass. The accurate calibration of the
torsion balance also presents a problem. Various methods
involving either electric forces or an angular acceleration
of the measuring table to compensate the gravitational
force have been used.
In our measurement, the TM was positioned at a dou
ble extremum of the force field produced by the FM’s.
This greatly reduces the relative accuracy of the dis
tance measurements required to determine the mass
integration constant. It also reduces the problem result
ing from a density gradient in the TM. It is difficult to
compare the problems involved in determining the mass
integration constant for the two types of experiments. It
appears that the distance between the field masses at
tracting the small mass of the torsion balance must be
measured with very high absolute accuracy (1 µm is the
accuracy given for this distance in the torsion balance
experiments).
The use of liquid mercury as the principle component
of the FM’s reduces the problem associated with the den
sity gradients of the FM’s. There is still the density gra
dient of the vessel walls which has to be considered. The
field masses employed in the torsionbalance experiments
were either spheres or cylinders. The FM’s were rotated
between measurements in order to compensate gradient
effects.
The large mercury mass resulted in deformations of the
vessel which had to be accurately determined. The FM’s
although, nearly cylindrical in form, required more than
TABLE VIII: Recent measurements of the gravitational con
stant.Statistical and systematic uncertainties have been
added as if the were independent quantities.
Reference
Gundlach and Merkowitz [23]
Quinn et al.[24]
Armstrong and Fitzgerald [25]
Present analysis
G[10−11m3kg−1s−2]
6.674215( 92)
6.675590(270)
6.673870(270)
6.674252(124)
1000 ring and pointmass elements in order to determine
the massintegration constant. A similar problem (but
on a smaller scale) occurs in the torsion experiments in
accounting for small imperfections of the spheres, cylin
ders or plates and in determining their relative positions.
The determination of G using a beam balance is be
set with a number of problems which we have tried to
describe in detail. We have been able to reduce the
uncertainty in G resulting from these problems to val
ues comparable to the statistical reproducibility of the
weighings determining the gravitational signal. The to
tal uncertainty for G which we obtain with a beam bal
ance is comparable to the uncertainty quoted in recent
torsionbalance determinations of G. We believe that the
beambalance measurement involving a number of quite
different problems than encountered in torsion balance
measurements can therefore provide a useful contribu
tion to the accuracy of the gravitational constant.
VI. ACKNOWLEDGEMENTS
The present research has been generously supported by
the Swiss National Fund, the Kanton of Z¨ urich, and as
sistantship grants provided by Dr. Tomalla Foundation
for which we are very grateful. The experiment could
not have been carried out without the close support of
the MettlerToledo company which donated the balance
for this measurement and made their laboratory available
for our use. We are also extremely grateful to the Paul
Scherrer Institut for providing a suitable measuring room
and the help of their staff in determining the geometry of
our experiment. We also wish to thank the Swiss Metro
logical Institute and the PhysikalischTechnische Bundes
anstalt, Braunschweig, Germany for making a number
of certified precision measurements for us. Our heartfelt
thanks are also extended to the staff of our machine shop
for their advice and precision work in producing many
of the pieces for this experiment. We wish to thank E.
Klingel´ e for determining the value of local gravity at the
measuring site. We are also indebted to the firms Al
maden Co., Metrotec A.G. and Reishauer A.G. for the
services they provided.
Page 24
24
VII. APPENDIX
The general idea in determining the z component of
force on a small volume element of a TM is first to cal
culate the potential along the z axis due to the FM. The
offaxis potential can then be obtained by making a Tay
lor series expansion for small r and substituting this in
the Laplace equation. This is the procedure which is of
ten used in electrostatic calculations [20]. The force in
the z direction is just the negative derivative of this po
tential with respect to z. We present first a derivation of
the force for the axially symmetric case and then describe
the modification required for a nonaxially symmetric po
tential.
The gravitational potential on the z axis of a homo
geneous, torus of rectangular cross section with density
̺FM, inner radius R1, outer radius R2and half height Z
is given by the equation
Φ(r = 0,z) = 2π̺FMG
?Z/2
−Z/2
?R2
R1
r′dr′dz′
?
r′2+ (z′− z)2
(5)
where r and z are the radial and axial coordinates of a
point within the TM expressed in cylindrical polar co
ordinates. For convenience, one chooses the zero of the
potential at the center of mass of a TM. For simplicity,
the cross section of the FM employed in Eq. 5 has been
chosen to be a rectangle. Besides the torus with rect
angular cross section, analytic expressions for the two
dimensional integrals with triangular and circular cross
sections were also employed.
The potential at points close to the axis can be calcu
lated as a power series in r
Φ(r,z) = a0+ a1r + a2r2+ a3r3+ ... =
∞
?
i=0
airi
(6)
with unknown coefficients a1,a2,... which are functions of
z alone. For r=0, one has a0= Φ(0,z). The gravitational
field satisfies the Laplace equation ∇2Φ = 0. Applying
the Laplace operator in cylindrical coordinates to Eq. 6
leads to
∇2Φ = a1r−1+
∞
?
i=0
ri
?d2ai
dz2+ (i + 2)2ai+2
?
= 0 .
This equation is valid for all r. Since ∇2Φ = 0 for r = 0,
a1must be identically zero. Thus, the values for aican
be calculated recursively from
ai+2= −
1
(i + 2)2
d2ai
dz2
starting with either a0 or a1. Since a1 = 0, all terms
with odd i are zero and ai for even i can be obtained
starting with a0. By induction, it is easily shown that
the coefficient a2nis
?n
a2n=
?
−1
4
1
(n!)2V(2n)(z)
with
V(2n)(z) =d2nΦ(0,z)
dz2n
.
Here, V(0)is just Φ(0,z) which can be easily calculated
using Eq. 5.
Using this expression for the coefficients in the expan
sion 6, the gravitational potential can be calculated from
the sum
?i
Φ(r,z) =
∞
?
i=0
?
−1
4
1
(i!)2V(2i)(z) r2i. (7)
The gravitational field gz in zdirection is given by
−∂Φ/∂z or
gz(r,z) = −
∞
?
i=0
?
−1
4
?i
1
(i!)2V(2i+1)(z) r2i.
Integrating gz over the volume of the TM, one obtains
the force on the TM in the z direction
Fz= −2π̺TM×
∞
?
i=0
?
−1
4
?i
1
(i!)2
?+b
−b
dz′
?r
0
V(2i+1)(z) r′2ir′dr′
(8)
where the origin has been taken to be the center of the
TM. The integration over r is trivial. The integration
over z is
?+b
−b
V(2i+1)dz′= V(2i)(b) − V(2i)(−b) .
Making a Taylor expansion for small b on the right side
of equation, one obtains
?+b
−b
V(2i+1)dz′=
V(2i)(0) + bV(2i+1)(0) +1
−V(2i)(0) + bV(2i+1)(0) −1
2!b2V(2i+2)(0) + ...
2!b2V(2i+2)(0) + ... .
Adding similar terms results in
?+b
−b
V(2i+1)dz′=2bV(2i+1)(0) +2
3!b3V(2i+3)(0)
+2
5!b5V(2i+5)(0) + ...
or
?+b
−b
V(2i+1)dz′=
∞
?
j=0
2
(2j + 1)!b2j+1V(2i+2j+1)(0) . (9)
The final equation for the gravitationalforce on a cylin
der can then be calculated by combining Eq. 8 and 9 to
obtain
Fz= −2πbr2̺TM
∞
?
n=0
V(2n+1)(0)×
n
?
i=0
1
(−4)i
1
i!(i + 1)!
1
(2n − 2i + 1)!b2n−2ir2i.(10)
Page 25
25
Only odd derivatives of the potential are required for
this case involving complete axial symmetry. The con
vergence of the series can be improved by dividing the
TM into two or more shorter cylinders. The algebraic
expressions for the V2n(0), as determined using ”auto
matic differentiation” [22], are very long and will not be
given here.
The above derivation has assumed that the TM and
FM have a common axis of cylindrical symmetry. We
will now show how essentially the same equations can be
employed for an arbitrary FM potential. This allows one
to calculate the potential of a FM with cylindrical sym
metry but with its axis displaced and/or tilted relative
to that of the TM.
Again, in order to facilitate integration over the vol
ume of the cylindrical TM, one employs cylindrical po
lar coordinates with the z axis along the symmetry axis
of the TM. The potential is now a potential of the
form ψ(x,y,z) = ψ (rcos(ϕ),r sin(ϕ),z) which satisfies
∇2ψ = 0. The center of mass of the TM is chosen as the
zero of potential.
One defines a function Ψ(r,z) such that
Ψ(r,z) = (2π)−1
?2π
ϕ=0
ψ(r,ϕ,z)dϕ.(11)
One can then show that this function satisfies the same
assumption that were made for the function Φ(r,z),
namely that Ψ has axial symmetry, that it is zero on
the z axis and that its Laplacian is zero. It can therefore
be used in the Eqs. 7 through 10 instead of Φ to obtain
the gravitational force integrated over the TM.
The axial symmetry of Ψ is obvious since ϕ has been
removed by the integration over ϕ. The zero potential
was chosen to be at the TM center of mass. The Lapla
cian of Ψ can be shown to be zero by allowing the Lapla
cian to operate on Ψ as defined in Eq. 11. Reversing the
integration and differentiation operations one obtains
∇2Ψ(r,z) = (2π)−1
?2π
ϕ=0
(∇2ψ − r−2∂2ψ
∂ϕ2)dϕ.
The Laplacian of ψ(r,ϕ,z) is zero for an inverse r po
tential. The integral of the second term is r−2∂ψ/∂ϕ
evaluated at ϕ = 0 and 2π which is also zero. Thus, one
obtains the desired result that ∇2Ψ(r,z) = 0.
In order to use this property of a potential which does
not have axial symmetry about the TM axis (the z axis),
one must determine the potential and its derivative with
respect to z along the z axis. This is not difficult for the
case of a FM which has axial symmetry about an axis not
coincident with the TM axis. One merely uses Eq. 7 to
determine the potential at radial distances from the FM
axis corresponding to points on the TM axis. The force
in the z direction (TM axis) is then determined as before
using Eqs. 8 and 10. This procedure is particularly useful
for the case of TM and FM axes which are parallel but
which are displaced relative to one another.
In principle, one can determine the derivatives with
respect to r which are required for the force on a TM
tilted relative to the vertical; however, in this case it
is simpler to approximate the TM by a number of thin
slabs displaced from the vertical axis. This completes the
discussion of nonaxialsymmetric potentials.
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