# A Measurement of Newton's Gravitational Constant

**ABSTRACT** A precision measurement of the gravitational constant $G$ has been made using a beam balance. Special attention has been given to determining the calibration, the effect of a possible nonlinearity of the balance and the zero-point variation of the balance. The equipment, the measurements and the analysis are described in detail. The value obtained for G is 6.674252(109)(54) 10^{-11} m3 kg-1 s-2. The relative statistical and systematic uncertainties of this result are 16.3 10^{-6} and 8.1 10^{-6}, respectively.

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**ABSTRACT:**In the Randall-Sundrum model with one brane, we found the approximate and exact solutions for gravitational potentials and accelerations of test bodies in these potentials for different geometrical configurations. We applied these formulas for calculation of the gravitational interaction between two spheres and found the approximate and exact expressions for the relative force corrections to the Newton's gravitational force. We demonstrated that the difference between relative force corrections for the approximate and exact cases increases with the parameter $l$ (for the fixed distance $r$ between centers of the spheres). On the other hand, this difference increases with decreasing of the distance between the centers of the spheres (for the fixed curvature scale parameter $l$). We got the upper limit for the curvature scale parameter $l\lesssim 10\, \mu$m. For these values of $l$, the difference between the approximate and exact solutions is negligible.General Relativity and Gravitation 11/2011; 44(9). · 1.90 Impact Factor - SourceAvailable from: Takeshi Chiba[Show abstract] [Hide abstract]

**ABSTRACT:**The current observational and experimental bounds on the time variation of the constants of nature (the fine structure constant $\alpha$, the gravitational constant $G$ and the proton-electron mass ratio $\mu=m_p/m_e$) are reviewed.Progress of Theoretical Physics 10/2011; 126. · 2.48 Impact Factor - SourceAvailable from: ArXiv[Show abstract] [Hide abstract]

**ABSTRACT:**We determined the Newtonian constant of gravitation G by interferometrically measuring the change in spacing between two free-hanging pendulum masses caused by the gravitational field from large tungsten source masses. We find a value for G of (6.672 34±0.000 14)×10(-11) m3 kg(-1) s(-2). This value is in good agreement with the 1986 Committee on Data for Science and Technology (CODATA) value of (6.672 59±0.000 85)×10(-11) m3 kg(-1) s(-2) [Rev. Mod. Phys. 59, 1121 (1987)] but differs from some more recent determinations as well as the latest CODATA recommendation of (6.674 28±0.000 67)×10(-11) m3 kg(-1) s(-2) [Rev. Mod. Phys. 80, 633 (2008)].Physical Review Letters 09/2010; 105(11):110801. · 7.73 Impact Factor

Page 1

arXiv:gr-qc/0609027v1 7 Sep 2006

A Measurement of Newton’s Gravitational Constant

St. Schlamminger,∗E. Holzschuh,†W. K¨ undig,‡F. Nolting,§R.E. Pixley,¶J. Schurr,∗∗and U. Straumann

Physik-Institut der Universit¨ at Z¨ urich, CH-8057 Z¨ urich, Switzerland

A precision measurement of the gravitational constant G has been made using a beam balance.

Special attention has been given to determining the calibration, the effect of a possible nonlinearity of

the balance and the zero-point variation of the balance. The equipment, the measurements and the

analysis are described in detail. The value obtained for G is 6.674252(109)(54) ×10−11m3kg−1s−2.

The relative statistical and systematic uncertainties of this result are 16.3×10−6and 8.1 ×10−6,

respectively.

PACS numbers: 04.80.-y, 06.20.Jr

I.INTRODUCTION

The gravitational constant G has proved to be a

very difficult quantity for experimenters to measure ac-

curately. In 1998, the Committee on Data Science

and Technology (CODATA) recommended a value of

6.673(10) × 10−11m3kg−1s−2. Surprisingly, the uncer-

tainty, 1,500 parts per million (ppm), had been increased

by a factor of 12 over the previously adjusted value of

1986. This was due to the fact that no explanation had

been found for the large differences obtained in the pre-

sumably more accurate measurements carried out since

1986. Obviously, the differences were due to very large

systematic errors. The most recent revision [1] of the

CODATA Task Group gives for the 2002 recommended

value G = 6.6742(10) × 10−11m3kg−1s−2. The uncer-

tainty (150 ppm) has been reduced by a factor of 10 from

the 1998 value, but the agreement among the measured

values considered in this compilation is still somewhat

worse than quoted uncertainties.

Initial interest in the gravitational constant at our in-

stitute had been motivated by reports [2] suggesting the

existence of a “fifth” force which was thought to be im-

portant at large distances. This prompted measurements

at a Swiss storage lake in which the water level varied by

44 m. The experiment involved weighing two test masses

(TM’s) suspended next to the lake at different heights.

No evidence [3, 4] was found for the proposed “fifth”

force, but, considering the large distances involved, a

reasonably accurate value (750 ppm) was obtained for

∗present address Univ. of Washington, Seattle, Washington, USA

†deceased

‡deceased; We dedicate this paper to our colleague Walter K¨ undig,

without whose untiring and persistent effort this ambitious exper-

iment would neither have been started nor brought to a successful

conclusion. Walter K¨ undig, died unexpectedly and prematurely of

a grave illness in May 2005. He conceived the set-up of this experi-

ment and worked on aspects of the analysis until a few days before

his death.

§Paul Scherrer Institut,Villigen, Switzerland

¶EMail address: ralph.pixley@freesurf.ch

∗∗present address Physikalisch Technische Bundesanstalt, Braun-

schweig, Germany

1 m

2.3 m

field masses

upper test mass

lower test mass

wires

mass exchanger

balance

Pos. TPos. A

FIG. 1: Principle of the measurement. The FM’s are shown

in the position together (Pos. T) and the position apart (Pos.

A).

G. It was realized that the same type of measurement

could be made in the laboratory with much better ac-

curacy with the lake being replaced by the well defined

geometry of a vessel containing a dense liquid such as

mercury. Equipment for this purpose was designed and

constructed in which two 1.1 kg TM’s were alternately

weighed in the presence of two moveable field masses

(FM’s) each with a mass of 7.5 t. A first series of mea-

surements [7, 8, 9, 10, 11] with this equipment resulted

in a value for G with an uncertainty of 220 ppm due pri-

marily to a possible nonlinearity of the balance response

function. A second series of measurements was under-

taken to eliminate this problem. A brief report of this

latter series of measurements has been given in ref. [12]

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and a more detailed description in a thesis [13]. Since

terminating the measurements, the following four years

have been spent in improving the analysis and checking

for possible systematic errors.

Following a brief overview of the experiment, the mea-

surement and the analysis of the data are presented in

Secs. III entitled Measurement of the Gravitational Sig-

nal and Sec. IV entitled Determination of the Mass-

Integration Constant. In Sec. V, the present result is

discussed and compared with other recent measurements

of the gravitational constant.

II.GENERAL CONSIDERATIONS

The design goal of this experiment was that the un-

certainty in the measured value of G should be less than

about 20 ppm. This is comparable to the quoted accu-

racy of recent G measurements made with a torsion bal-

ance. It is, however, several orders of magnitude better

than previous measurements of the gravitationalconstant

(made after 1898) employing a beam balance [4, 5, 6].

The experimental setup is illustrated in Fig. 1. Two

nearly identical 1.1 kg TM’s hanging on long wires are

alternately weighed on a beam balance in the presence of

the two movable FM’s weighing 7.5 t each. The position

of the FM’s relative to the TM’s influence the measured

weights. The geometry is such that when the FM are in

the position labelled ”together”, the weight of the upper

TM is increased and that of the lower TM is decreased.

The opposite change in the TM weights occurs when the

FM are in the position labelled ”apart”. One measures

the difference of TM weights first with one position of the

FM’s and then with the other. The difference between

the TM weight differences for the two FM positions is

the gravitational signal.

The use of two TM’s and two FM’s has several advan-

tages over a single TM and a single movable FM. Com-

paring two nearly equivalent TM’s tends to cancel slow

variations such as zero-point drift of the balance and the

effect of tidal variations. Using the difference of the two

TM weights doubles the signal. In addition, it causes the

influence of the FM motion on the counter weight of the

balance to be completely cancelled. Use of two FM’s with

equal and opposite motion reduces the power required to

that of overcoming friction. This also simplified some-

what the mechanical construction.

The geometry has been designed such that the TM

being weighed is positioned at (or near) an extremum of

the vertical force field in both the vertical and horizontal

directions for both positions of the FM’s. The extremum

is a maximum for the vertical position and a minimum for

the horizontal position. This double extremum greatly

reduces the positional accuracy required in the present

experiment.

The measurement took place at the Paul Scherrer In-

stitut (PSI) in Villigen. The apparatus was installed in a

pit with thick concrete walls which provided good ther-

14

15

16

13

12

11

10

9

8

2

3

4

5

6

7

1

1 m

FIG. 2: A side view of the experiment. Legend: 1=measur-

ing room enclosure, 2=thermally insulated chamber, 3=bal-

ance, 4=concrete walls of the pit, 5=granite plate, 6=steel

girder, 7=vacuum pumps, 8=gear drive, 9=motor, 10=work-

ing platform, 11=spindle, 12=steel girder of the main sup-

port, 13=upper TM, 14=FM’s, 15=lower TM, 16=vacuum

tube.

mal stability and isolation from vibrations. The arrange-

ment of the equipment is shown in Fig. 2. The system in-

volving the FM’s was supported by a rigid steel structure

mounted on the floor of the pit. Steel girders fastened to

the walls of the pit supported the balance, the massive

(200 kg) granite plate employed to reduce high frequency

vibrations and the vacuum system enclosing the balance

and the TM’s. A vacuum of better than 10−4Pa was pro-

duced by a turbomolecular pump located at a distance

of 2 m from the balance.

The pit was divided into an upper and a lower room

separated by a working platform 3.5 m above the floor

of the pit. All heat producing electrical equipment was

located in the upper measuring room. Both rooms had

their own separate temperature stabilizing systems. The

long term temperature stability in both rooms was better

than 0.1◦C. No one was allowed in either room during

the measurements in order to avoid perturbing effects.

Page 3

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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 LaCoste-Romberg). The value found was

9.8072335(6) ms−2. This value was used to convert the

balance readings into force units.

A. The Balance

The beam balance was a modified commercial mass

comparator of the type AT1006 produced by the Mettler-

Toledo company. The mass being measured is compen-

sated by a counter weight and a small magnetic force

between a permanent magnet and the current flowing in

a coil mounted on the balance arm. An opto-electrical

feedback system controlling the coil current maintains

the balance arm in essentially a fixed position indepen-

dent of the mass being weighed. The digitized coil cur-

rent is used as the output reading of the balance.

The balance arm is supported by two flexure strips

which act as the pivot. The pan of the balance is sup-

ported by a parallelogram guide attached to the balance

frame. This guide allows only vertical motion of the pan

to be coupled to the arm of the balance. Horizontal forces

produced by the load are transmitted to the frame and

have almost no influence on the arm.

As supplied by the manufacturer, the balance had a

measuring range of 24 g above the 1 kg offset determined

by the counter weight. The original readout resolution

was 1 µg and the specified reproducibility was 2 µg. The

balance was designed especially for weighing a 1 kg stan-

dard mass such as is maintained in many national metrol-

ogy institutes.

In the present experiment, the balance was modified

by removing some nonessential parts of the balance pan

which resulted in its weighing range being centered on

1.1 kg instead of the 1 kg of a standard mass. There-

fore, 1.1 kg TM’s were employed. In order to obtain

higher sensitivity required for measuring the approxi-

mately 0.8 mg difference between TM weighings, the

number of turns on the coil was reduced by a factor of

6, thus reducing the range to 4 g for the same maximum

coil current.The balance was operated at an output

value near 0.6 g which gave a good signal-to-noise ratio

with low internal heating. For the present measurements,

a mass range of only 0.2 g was required. The full read-

out resolution of the analog to digital converter (ADC)

measuring the coil current was employed which resulted

in a readout-mass resolution of 12.5 ng.

An 8th order low-pass, digital filter with various time

constants was available on the balance. Due to the many

weighings required by the procedure employed to can-

cel nonlinearity (see Sec. IIID), it was advantageous to

make the time taken for each weighing as short as possi-

ble. Therefore, the shortest filter time constant (approx-

imately 7.8 s) was employed and output readings were

taken at the maximum repetition rate allowed by the

balance (about 0.38 s between readings).

Pendulum oscillations were excited by the TM ex-

changes. Small oscillation amplitudes (less than 0.2 mm)

of the TM’s corresponding to one and two times the fre-

quency for pendulum oscillations (approximately 0.26 Hz

for the lower TM and 0.33 Hz for the upper TM’s) were

observed. They were essentially undamped with decay

times of several days. The unwanted output amplitudes

of these pendulum oscillation were not strongly attenu-

ated by the filter (half-power frequency of 0.13 Hz) and

therefore had to be taken into account in determining the

equilibrium value of a weighing.

The equilibrium value of a weighing was determined

in an on-line, 5-parameter, linear least-squares fit made

to 103 consecutive readings of the balance starting 40 s

after a load change. The parameters of the fit were 2 sine

amplitudes, 2 cosine amplitudes and the average weight.

The pendulum frequencies were known from other mea-

surements and were not parameters of the on-line fit. The

40 s delay before beginning data taking was required in

order to allow the balance to reach its equilibrium value

(except for oscillations) after a load change. This proce-

dure (including the 40 s wait) is what we call a ”weigh-

ing”. A weighing thus required about 80 s.

Data of a typical weighing and the fit function used to

describe their time distribution are shown in the upper

part of Fig. 3. The residuals δ divided by a normalization

constant σ are shown in the lower part of this figure.

The normalization constant has been chosen such that

the rms value of the residuals is 1. Since the balance

readings are correlated due to the action of the digital

filter, the value of σ does not represent the uncertainty

of the readings. It is seen that the residuals show only

rather wide peaks. These peaks are probably due to very

short random bursts of electronic noise which have been

widened by the digital filter. With the sensitivity of our

modified balance, they represent a sizable contribution

to the statistical variations of the weighings. They are of

no importance for the normal use of the AT1006 balance.

A direct calibration of the balance in the range of the

780 µg gravitation signal can not be made with the accu-

Page 4

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0 10203040

time [sec

152448500

152449000

152449500

balance reading

-3

0

3

δ/σ

FIG. 3: Shown in the upper plot are the balance readings

for a typical weighing illustrating the oscillatory signal due

to pendulum oscillations. The output is the uncalibrated bal-

ance reading corresponding to approximately 1.1 kg with a

magnetic compensation of 0.6 g. The amplitude of the os-

cillatory signal corresponds to about 1.5 µg. The lower plot

shows the normalized residuals. The normalization has been

chosen such that their rms value is 1.

racy required in the present experiment (< 20 ppm) since

calibration masses of this size are not available with an

absolute accuracy of better than about 300 ppm. Instead,

we have employed a method in which an accurate, coarse

grain calibration was made using two 0.1 g calibration

masses (CM’s). The CM’s were each known with an ab-

solute accuracy of 4 ppm. A number of auxiliary masses

(AM’s) having approximate weights of either 783 µg or

16×783 = 12,528 µg were weighed along with each TM

in steps of 783 µg covering the 0.2 g range of the CM.

Although the AM’s were known with an absolute accu-

racy of only 800 ng (relative uncertainty 1,000 ppm), the

method allowed balance nonlinearity effects to be almost

entirely cancelled. Thus, the effective calibration accu-

racy for the average of the TM difference measurements

was essentially that of the CM’s. A detailed description

of this method is given in Sec. IIIJ.

In our measurements, the balance was operated in vac-

uum. The balance proved to be extremely temperature

sensitive which was exacerbated by the lack of convec-

tion cooling in vacuum. The measured zero-point drift

was 5.5 mg/◦C. The sensitivity of the balance changed

by 220 ppm/◦C. To reduce these effects, the air tem-

perature of the room was stabilized to about 0.1◦C. A

second stabilized region near the balance was maintained

at a constant temperature to 0.01◦C. Inside the vacuum,

the balance was surrounded by a massive (45 kg) copper

box which resulted in a temperature stability of about

1 mK. Although zero-point drift under constant load for

a 1 mK temperature change was only 5.5 µg , the effects

of self heating of the balance due to load changes during

the measurement of the gravitational signal were much

FIG. 4: Drawing of TM inside the vacuum tube. Dimensions

are given in mm.

larger. Details of this effect and how they were corrected

are described in Sec.IIIG.

B.The Test Masses

One series of measurements was made using copper

TM’s and two with tantalum TM’s. Various problems

with the mass handler occurred during the measurements

with the tantalum TM’s which resulted in large system-

atic errors. Although the tantalum results were consis-

tent with the measurements with the copper TM’s, the

large systematic errors resulted in large total errors. The

tantalum measurements were included in our first pub-

lication, but we now believe that better accuracy is ob-

tained overall with the copper measurements alone. We

therefore describe only the measurements made with the

copper TM’s in the present work.

A drawing of a copper TM is shown in Fig. 4. The

45 mm diameter, 77 mm high copper cylinders were

plated with a 10 µm gold layer to avoid oxidation. The

gold plating was made without the use of nickel in or-

der to avoid magnetic effects. Near the top of each TM

on opposite sides of the cylinder were two short horizon-

tal posts. The posts were made of Cu-Be (Berylco 25).

The tungsten wires used to attach the TM’s to the bal-

ance were looped around these posts in grooves provided

for this purpose. The wires had a diameter of 0.1 mm

and lengths of 2.3 m for the upper TM and 3.7 m for

the lower. The loop was made by crimping the tungsten

wire together in a thin copper tube. A thin, accurately

machined, copper washer was placed in a cylindrical in-

dentation on the top surface of the lower TM in order

to trim its weight (including suspension) to within about

400 µg of that of the upper TM and suspension.

Measurement of the TM’s dimensions was made with

an accuracy of 5 µm using the coordinate measuring ma-

chine (CMM) at PSI. The weight of the gold plating

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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 Mettler-Toledo laboratory with an accuracy

of 25 µg (0.022 ppm) before and after the gravitational

measurement. It was found that the mass of both TM’s

had increased by a negligible amount (0.5 ppm) during

the measurement.

An estimate of possible density gradients in the TM’s

was determined by measuring the density of copper sam-

ples bordering the material used for making the TM’s.

It was found that the variation of the relative density

gradients over the dimensions of either TM was less than

2 × 10−4in both the longitudinal and the radial direc-

tions.

C.TM Exchanger

In weighing the TM’s, it was necessary to remove the

suspension supporting one TM from the balance and

replace it by the other supporting the other TM. The

exchange was accomplished by a step-motor driven hy-

draulic systems to raise the suspension of one TM while

lowering the other. A piezo-electric transducer mounted

above the pan of the balance was used to keep the load on

the balance during the exchange as constant as possible.

This was done in order to avoid excessive heating due

to the coil current and to reduce anelastic effects in the

flexure strips supporting the balance arm. The output

excursions were typically less than 0.1 g. The exchange

of the TM’s required about 4 min.

The TM suspension rested on a thin metal arm de-

signed to bend through 0.6 mm when loaded with 1.1 kg.

Therefore, the transfer of TM’s was accomplished with

a vertical movement of typically 2 mm (0.6 mm bending

of the spring plus an additional 1.4 mm to avoid electro-

static forces). The metal arm was attached to a paral-

lelogram guide (similar to that of the balance) to assure

only vertical motion.

Although the parallelogram reduced the error resulting

from the positioning of the load, it was nevertheless im-

portant to have the TM load always suspended from the

same point on the balance pan. This was accomplished

by means of a kinematic coupling [14, 15]. The coupling

consisted of three pointed titanium pins attached to each

TM suspension which would come to rest in three tita-

nium V grooves mounted on the balance pan. The repro-

ducibility of this positioning was 10 µm. The pieces of

the coupling were coated with tungsten carbide to avoid

electrical charging and reduce friction.

D.Auxiliary Masses

In order to correct for any nonlinearity of the balance

in the range of the signal, use was made of many auxiliary

masses (AM’s) spanning the 200 mg range of the CM’s in

steps of approximately 783 µg. Although the AM’s could

not be measured with sufficient accuracy to calibrate the

balance absolutely, they were accurate enough to correct

the measured gravitational signal for a possible nonlin-

earity of the balance. Each TM was weighed along with

various combinations of AM’s. One essentially averaged

the nonlinearity over the 200 mg range of the CM’s in

256 load steps of 200 mg/256=783 µg. A weighing of

both 100 mg CM’s was then used to determine the ab-

solute calibration of the balance which is valid for the

TM weighings averaged over this range. The effect of

any nonlinearity essentially cancels due to the averaging

process. The accuracy of the nonlinearity correction is

described in Sec. IIIJ.

The 256 load steps were accomplished using 15 AM’s

with a mass of approximately 783 µg called AM1’s and 15

AM’s with 16 times this mass (12,528 µg) called AM16’s.

They were made from short pieces of stainless steel wire

with diameters of 0.1 mm and 0.3 mm. The wires were

bent through about 70◦on both ends leaving a straight

middle section of about 6 mm. The mass of the AM’s

were electrochemically etched to obtain as closely as pos-

sible the desired masses. The RMS deviation was 1.5 µg

for the AM1’s and 2.3 µg for the AM16’s.

By weighing a TM together with various AM combina-

tions, one obtains the value of the TM weight simultane-

ously with the linearity information. The only additional

time required for this procedure over that of weighing

only the TM’s is the time necessary to change an AM

combination (10 to 30 s).

E.Mass Handler

The mass handler is the device which placed the AM’s

and the CM’s on the balance or removed them from the

balance. The mass handler was designed by the firm

Metrotec AG. The operation of this device is illustrated

in the somewhat simplified drawing of Fig. 5 showing

how the AM1’s and the CM1 are placed on the metal

strip attached to the balance pan. Only 6 of the 14 steps

are shown in this illustration for clarity. The portion

of the handler used for the AM’16 and the CM2 (not

shown) is similar except that the AM’16 are placed on

a metal strip located below the one used for the AM1’s.

All of the AM1’s pictured in Fig. 5 are lying on the steps

of a pair of parallel double staircases.

are separated by 6 mm which is the width of the AM’s

between the bent regions on both ends.

between the staircases is such that they could pass on

either side of the horizontal metal strip fastened to the

balance pan as the staircases were moved up or down.

The motion of each staircase pair was constrained to the

The staircases

The spacing

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1

3

2

6

7

4

5

8

6

9

FIG. 5: Simplified drawing of the mass handler illustrating

the principle of operation.Legend: 1=pivoted lever pair

holding a CM, 2=narrow strip to receive the CM, 3=dou-

ble stair case pair holding AM’s, 4=narrow strip to receive

AM’s, 5=balance pan, 6 flat spring, 7=frame, 8=stepmotor

driven cogwheel, and 9=coil spring. The pivoted-lever pair

and the double-staircase pair are spaced such that they can

pass on either side of the narrow strips 2 and 4 fastened to

the balance pan. The two flat springs 6 form two sides of

a parallelogram which assures vertical motion of the double

stair case pair.

vertical direction by a parallelogram (similar to those of

the balance) fastened to the frame of the mass handler.

The staircases for AM1’s and AM16’s were moved by two

separate step motors located outside the vacuum system.

The step motors were surrounded by mu metal shielding

to reduce the magnetic field in the neighborhood of the

balance. Moving the staircases down deposited one AM

after another onto the metal strip. Moving the staircases

up removed the AM’s lying on the strip. The steps of

the staircase had hand filed, saddle shaped indentations

to facilitate the positioning of the AM’s. The heights of

the steps were 2 mm and the steps on the left side of the

double staircase were displaced in height by 1 mm from

those on the right. Thus, the AM1’s were alternately

placed on the balance to the left and to the right of the

center of the main pan in order to minimize the torque

which they produced on the balance.

Raising the staircase structure above the position

shown in the figure caused a rod to push against a piv-

oted lever holding CM1. With this operation, CM1 was

placed on the upper strip attached to the balance. Re-

versing the operation allowed the spring to move the lever

in the opposite direction and remove CM1 from the bal-

ance.

Due to the very small mass of the AM1’s, difficulty was

occasionally experienced with the AM1’s sticking to one

side of the staircase or the other. The staircases were

made of aluminum and were coated with a conductive

layer of tungsten carbide to reduce the sticking probabil-

ity. Sticking nevertheless did occur. The sticking would

cause an AM1 to rest partly on the staircase and partly

on the pan, thus giving a false balance reading. In ex-

treme cases, the AM1 would fall from the holder and

therefore be lost for the rest of the measurement. No

problem was experienced with the heavier AM16’s and

the CM’s.

F.Weighing Schedule

The experiment was planned so that the zero-point

(ZP) drift and the linearity of the balance could be deter-

mined while weighing a TM. In principle one needs just

4 weighings (upper and lower TM with FM’s together

and apart) to determine the signal for each AM placed

on the balance. Repeating these 4 weighings allows one

to determine how much the zero point has changed and

thereby correct for the drift. Since there are 256 AM val-

ues required to correct the nonlinearity of the balance,

a minimum of 2048 weighings is needed for a complete

determination of the signal corrected for ZP drift and lin-

earity. One also wishes to make a number of calibration

measurements during the series of measurements.

The order in which the measurements are performed

influences greatly the ZP drift correction of measure-

ments. Changing AM’s requires only 6 to 30 s, while

exchanging TM on the balance takes about 230 s and

moving the FM from one position to the other requires

about 600 s. These times are to be compared with the 80

s required for a weighing and about 1 hr for a complete

calibration measurement (see Sec. IIII). One therefore

wishes to measure a number of AM values before ex-

changing TM, and repeat these measurements for the

other TM before changing the FM positions or making a

calibration.

The schedule of weighing adopted is based on several

basic series for the weighing of the different TM’s with

different FM positions. The series are defined as follows:

1. An S4 series is defined as the weighing of four suc-

cessive AM values with a particular TM and with

all weighing made for the same FM positions.

2. An S12 series involves three S4 series all with the

same four AM values and the same FM positions.

The S4 series are measured first for one TM, then

the other TM and finally with the original TM.

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7

3. An S96 is eight S12 series, all made with the same

FM positions and with the AM values incremented

by four units between each S12 series. A TM ex-

change is also made between each S12 series. An

S96 series represents the weighings with 32 succes-

sive AM values for both TM all with the same FM

positions.

4. An S288 series is three S96 series, first with one

FM position, then the other and finally with the

original FM position. A calibration measurement

is made at the beginning of each S288 series. Thus,

the S288 series represents the weighings with 32

successive AM values for both TM’s and both FM

positions and includes its own calibration.

5. An S2304 series is made up of eight S288 series

with the AM values incremented by 32 between

each S288 series. An S2304 series completes the

full 256 AM values with weighings of both TM’s

and both FM positions.

A total of eight valid S2304 series was made over a pe-

riod of 43 days. Alternate S2304 series were intended to

be made with increasing and decreasing AM values. Un-

fortunately, the restart after a malfunction of the tem-

perature stabilization in the measuring room was made

with the wrong incrementing sign. This resulted in five

S2304 series being made with increasing AM values and

three with decreasing.

G.Analysis of the Weighings

In ref. [12, 13], the so called ABA method was used to

analyze the data obtained from the balance and thereby

obtain the difference between the mass of the A and B

TM. This method assumes a linear time dependence of

the weight that would be obtained for the A TM at the

time when the B TM was measured based on the weights

measured for A at an earlier and a later time. However,

a careful examination of the data showed that the curva-

ture of the ZP drift was quite large and was influenced

by the previous load history of the balance. This indi-

cated that the linear approximation was not a particu-

larly good approximation. We have therefore reanalyzed

the data using a fitting procedure to determine a contin-

uous ZP function of time for each S96 series. The data

and fit function for a typical S96 series starting with a

calibration measurement is shown in Fig. 6. The proce-

dure used to determine the ZP data and fit are described

in the following. The criterion for a valid weighing is

described in Sec. IIIH.

The data of Fig. 6 show a slow rise during the first

hour after the calibration measurement followed by a con-

tinuous decrease with a time constant of several hours.

These slow variations are attributed to thermal varia-

tions resulting principally from the different loading of

the balance during the calibration measurement. Super-

posed on the slow variations are rapid variations which

218.60218.70218.80

day of 2001

-2

-1

0

1

2

3

zero point [µg]

-3

0

3

δ/σ

FIG. 6: The zero-point variation as a function of time for a

typical S96 series including calibration is shown in the upper

part of this figure. The solid curve is the fit function starting

after the last dummy weighing. The fit function for this S96

series has 76 degrees of freedom. The normalized residuals

δ/σ are shown in the lower plot. The normalization of the

residuals has been chosen such that their rms value is 1.

are synchronous with the exchange of the TM’s. The

rapid variations peak immediately after the TM exchange

and decrease thereafter with a typical slope of 0.3 µg /hr.

The cause of the rapid variations is unknown.

The data employed in the ZP determination were the

weighings of the upper and lower TM’s for the S96 se-

ries. The known AM load for each weighing was first

subtracted to obtain a net weight for either TM plus the

unknown zero-point function at the time of each weigh-

ing. A series of Legendre polynomials was used to de-

scribe the slow variations of the zero-point function. A

separate P0 coefficient was employed for each TM. The

rapid variations were described by a sawtooth function

starting at the time of each TM exchange. The fit pa-

rameters were the coefficient of the Legendre polynomials

and the amplitude of the sawtooth function. The saw-

tooth amplitude was assumed to be the same for all rapid

peaks of an S96 series. The sawtooth function was used

principally to reduce the χ2of the fit and had almost no

effect on the results obtained when using the ZP func-

tion. All parameters are linear parameters so that no

iteration is required. The actual ZP function is the saw-

tooth function and the polynomial series exclusive of the

time independent terms (i.e. the sum of the coefficients

times Pn(0) for even n).

Such calculations were made for various numbers of

Legendre coefficients in the ZP function. It was found

that the gravitational signal was essentially constant for

a maximum order of Legendre polynomials between 8

and 36. In this range of polynomials, the minimum cal-

culated signal was 784.8976(91) µg for a maximum order

equal to 22 and a maximum signal of 784.9025(93) µg

for a maximum order equal to 36 (i.e. a very small dif-

ference). In all following results, we shall use the signal

Page 8

8

211.8211.9212.0212.1

-500

-400

-300

-200

-100

0

100

200

300

400

500

day of 2001

apart together

1 µg

10 min

FIG. 7: The measured weight difference in µg between TM’s

obtained from an S288 series. The magnified insert shows the

individual TM differences which are not resolved in the main

part of the figure.

784.8994(91) µg obtained with a maximum polynomial

order of 15.

It has been implicitly assumed in the above ZP deter-

mination that the AM load values were known with much

better accuracy than the reproducibility of the balance

producing the data used in the ZP fit. Although the AM

values were sufficiently accurate for determining the gen-

eral shape of the ZP function, their relative uncertainties

were comparable to the uncertainties of the balance data

used in the fit. The P0 mass parameters of the TM’s

obtained from the fit were therefore not used for the TM

differences at the two FM positions which are needed in

order to determine the gravitational signal. Instead, the

value of the ZP function was subtracted from each weigh-

ing, and an ABA mass difference was determined for each

triplet of weighings having the same AM load. Since the

mass of the AM’s do not occur in this TM difference, they

do not influence the calculation. The ABA calculation is

valid for this purpose since the ZP corrected weighings

have essentially no curvature. Such TM differences de-

termined for the apart-together positions of the FM’s are

then used to calculate the gravitational signal.

The TM differences for the apart-together positions

of the FM’s as a function of time are shown for a ZP

corrected S288 series in Fig. 7. Individual data points

are resolved in the magnified insert of this figure. Each

data point is the B member of a TM difference obtained

from an ABA triplet in which all weighings have the same

load value.

All of the TM differences (ZP corrected) for the entire

experiment are shown in Fig. 8. The data labelled apart

have been shifted by 782 µg in order to allow both data

sets to be presented in the same figure. A slow variation

of 2.5 µg in both TM differences occurred during the 43

day measurement. Also seen in this figure is a 0.7 µg

210220230240250

355

357

359

361

day of 2001

apart together

FIG. 8: The measured weight difference in µg between TM’s

for the FM’s positions apart and together.

values for the FM apart have been displaced 782 µg in order

to show both types of data in the same figure.

The measured

jump which occurred in the data for both the apart and

together positions of the FM’s on day 222. The slow vari-

ation is probably due to sorption-effect differences of the

upper and lower TM’s. The jump was caused by the loss

or gain of a small particle such as a dust particle by one

of the TM’s. In order to determine the gravitational sig-

nal, an ABA difference was calculated for apart-together

values having the same AM load. The slow variation seen

in Fig. 8 is sufficiently linear so that essentially no error

results from the use of the ABA method. The jump in

the apart-together differences caused no variation of the

gravitational signal.

In Fig. 9 is shown a plot of the binned difference be-

tween the FM apart-together positions for all valid data

(see Sec. IIIH. The differences were determined using the

ABA method applied to weighings made with the same

AM loads. Also shown in the figure is a Gaussian func-

tion fit to the data. The data are seen to agree well with

the Gaussian shape which is a good test for the qual-

ity of experimental data. The root-mean-square (RMS)

width of the data is 1.03 time the width of the Gaussian

function. The true resolution for these weighings may be

somewhat different than shown in Fig. 9 due to the fact

that the data have not been corrected for nonlinearity of

the balance (see Sec. IIIJ) and for correlations due to the

common ZP function . Nevertheless, these effects would

not be expected to influence the general Gaussian form

of the distribution.

A plot of the signal obtained for the S2304 series with

increasing and decreasing load is shown in Fig. 10. The

average signal for increasing load is 784.9121(125) µg and

the average for decreasing load is 784.8850(133) µg. The

common average for both is 784.8994(91) µg. The aver-

ages for increasing load and for decreasing load lie within

the uncertainty of the combined average. This shows that

Page 9

9

-1.0-0.5 0.00.51.0

0

100

200

difference [µg]

number/bin

FIG. 9: Binned data for the FM apart-together weight differ-

ences (points) and a fitted Gaussian function (curve) shown

as a deviation from the mean difference. Poisson statistics

were used to determine the uncertainties.

0123

series number

45678

784.85

784.90

784.95

785.00

signal [µg]

FIG. 10: The average signal for each of the eight S2304 series.

Series with increasing load are shown as circles. Series with

decreasing load are shown as squares. The dashed line is the

average of all eight series.

the direction of load incrementing did not appreciably in-

fluence the result.

Although the weighings making up an S96 series are

correlated due to the common ZP function determined

for each S96 series, the results of each S96 series, in par-

ticular the TM parameter, are independent. The 32 sig-

nal values obtained from the three S96 series making each

S288 series are also independent. However, since the non-

linearity correction (see Sec. IIIJ) being employed is ap-

plicable only to an entire S2304 series (not to individual

S288 series), it is only the eight S2304 series which should

be compared with one another. This restricts the way in

which the average signal is to be calculated for the entire

measurement, namely the way in which the data are to

be weighted.

We have investigated two weighting procedures. In the

first, each S2304 series average was weighted by the num-

ber of valid triplets in that series. This assumes that the

weighings measured in all S2304 series have the same a

priori accuracy. In the second method, it was assumed

that the accuracy for each weighing in a series was the

same but might be different for different series. We be-

lieve the second method is the better method since it

takes into account changes that occur during the long,

43 day measurement (e.g. the not completely compen-

sated effects of vibration, tidal forces and temperature).

The averages obtained with the two methods differ by

approximately 6 ng with the second method giving the

smaller average signal. This is a rather large effect. It is

only slightly smaller than the statistical uncertainty of 9

to 10 ng obtained for either method. In the rest of this

work, we shall discus only the results obtained with the

second method.

H.Criterion for Valid Data

Two tests were used to determine whether a measured

weighing was valid. An on-line test checked whether the

χ2value of the fit to the pendulum oscillations was rea-

sonable. A large value caused a repeat of the weighing.

After two repeats with large χ2, the measurement for this

AM value was aborted. An aborted weighing usually in-

dicated that the AM was resting on the mass handler and

on the balance pan in an unstable way.

A more frequent occurrence was that of an AM which

rested on both the mass handler and the balance was al-

most stable thereby giving a reasonable χ2. In order to

reject such weighings, an off-line calculation was made

to check whether the measured weight was within 10 µg

of the expected weight. The statistical noise of a valid

weighing was typically about 0.15 µg (see Fig. 9 show-

ing ABA difference involving 3 weighings). Excursions of

more than 10 µg were thus a clear indication of a mal-

function.

This off-line test is somewhat more restrictive than the

off-line test employed in our original analysis.

original analysis, a check was made only to see that the

weight difference between the TM’s for equal AM load-

ings was reasonable. The more restrictive test used in the

present analysis resulted in the rejection of the S2304 se-

ries at the time when the room temperature stabilizer

was just beginning to fail. It was also the reason for not

including the tantalum TM-measurements in the present

analysis. In the eight S2304 series accepted for the de-

termination of the gravitational constant, approximately

8% of the expected zero-point values could not be deter-

mined due to at least one of the three weighings at each

load value being rejected by the test for valid weighings.

In the

Page 10

10

I.Calibration Measurements

A coarse calibration of the balance was made periodi-

cally during the gravitational measurement (before each

S288 series) using two calibration masses each with a

weight of approximately 100 mg. A correction to the

coarse calibration constant due to the nonlinearity of the

balance will be discussed in Sec. IIIJ. The two CM’s used

for the coarse calibration were short sections of stainless

steel wire. The diameter of CM1 was 0.50(1) mm and

that of CM2 was 0.96(1) mm. The surface area of CM1

was approximately 1 cm2and that of CM2 was 0.5 cm2.

The CM’s were electrochemically etched to the desired

mass and then cleaned in an ethanol ultrasonic bath. The

mass of each CM was determined at METAS (Metrology

and Accreditation Switzerland) in air with an absolute

accuracy of 0.4 µg or a relative uncertainty of 4 ppm. The

absolute determinations of the CM masses were made be-

fore and after the gravitational measurement with copper

TM’s and after the second measurement with tantalum

TM’s. Only the first measurement was used to evaluate

the coarse calibration constant employed in the measure-

ment with copper TM’s. As will be discussed below, the

second and third measurement were used for the mea-

surement with copper TM’s only to check the stability of

the CM’s.

A calibration measurement involved either TM and one

of following seven additional loads: (1) CM1 alone, (2)

CM2 alone, (3) again CM1 alone, (4) empty balance,

(5) CM1+CM2, (6) empty balance and (7) CM1+CM2

and nine so called dummy weighings. These measure-

ments were made with no AM’s on the balance. After

the seventh weighing, a series of nine dummy weighings

alternating between upper and lower TM’s were made

with the AM load set to the value for the next TM

weighing. The dummy weighings were made in order

to allow the balance to recover from the large load vari-

ations experienced during the calibration measurement

and thereby come to an approximate equilibrium value

before the next TM weighing. Calibrations were made

alternately with the upper and lower TM’s as load. Cali-

bration measurements were made about twice a day. In-

cluding the dummy weighings, each calibration required

about 50 min.

A three-parameter least squares fit was made to the

calibration weighings labelled 4,5,6 and 7 above. The

fit thereby determined best values for the balance ZP,

the slope of the ZP and a parameter representing the

effective ZP corrected reading of the balance for the

load CM1+CM2. This third parameter is of particu-

lar interest since the coarse calibration constant is deter-

mined from the known mass of CM1+CM2 (measured by

METAS) divided by this parameter. Therefore, the re-

sults of the least-squares fit to each set of calibration data

gave a value for the coarse calibration constant which

then was used to convert the balance output of the S288

series to approximate mass values. An ABA analysis of

the first three weighings of each set of calibration data

was also made in order to determine the difference in

mass between CM1 and CM2.

The absolute masses obtained for CM1 and CM2 as

determined by METAS are given in columns 2 and 3 of

Table I. Also shown in Table I (column 4) are the mass

difference between CM1 and CM2 as obtained from the

METAS measurement in air and the average of our CM

measurement in vacuum. The mass differences between

CM1 and CM2 measured in vacuum are particularly use-

ful in checking for any mass variation of the CM’s.

TABLE I: The mass of the CM’s as measured by METAS and

the CM1-CM2 mass differences measured in air at METAS

and in vacuum during the gravitational measurements at PSI.

All values are given in µg.

Date CM1 CM2Difference

6.40(60)

5.853(19)

7.30(50)

7.269(29)

7.496(25)

7.04(50)

Feb 6, 01

Jul. - Sep., 01

Nov. 29, 01

Jan. - Mar., 02

Apr. - May, 02

May 27, 02

100,270.30(40)100,263.90(40)

in vacuum

100,262.90(35)

in vacuum

in vacuum

100,262.97(35)

100,270.20(35)

100,270.01(35)

It is seen that CM2 mass decreased by 1.00(53) µg be-

tween the first and second METAS measurements while

the mass of CM1 was essentially the same in all three

measurements. From the mass difference values in air

and vacuum it is clear that the change occurred after

the measurements with copper TM’s ended in Sept. 2001

and before the weighing at METAS in Nov. 2001 which

preceded the start of the tantalum measurements. We

ascribe this change of CM2 to either the loss of a dust

particle or perhaps a piece of the wire itself. The loss

of a piece of the wire was possible since the wire used

for the CM’s had been cut with a wire cutter and there

could have been a small broken piece that was not bound

tightly to the wire. For this reason only the values given

for the first weighing of the CM’s were used to determine

the coarse calibration constant used for the measurement

with copper TM’s.

A plot of the relative change of the effective ZP

corrected balance reading corresponding to the load

CM1+CM2 is shown in Fig. 11. It is seen that it changed

by only a few ppm over the 43 days of the measurement.

A linear fit made to these data results in a slope equal

to -0.044(6) ppm/day which is equivalent to a mass rate

variation of -0.0088(12) µg cm−2d−1. The uncertainty

was obtained by normalizing χ2of the fit to the degrees

of freedom (DOF).

The slow variation of the effective balance reading for

the load CM1+CM2 seen in Fig. 11 could be due either

to a change of the balance sensitivity, to a decrease in the

mass of CM1+CM2 due to the removal of a contaminant

layer from the CM’s in vacuum or to a combination of

both causes. A variation of the balance sensitivity would

have essentially no effect on the analysis of the weighing

for the gravitational measurement as the coarse calibra-

Page 11

11

tion constant used for the analysis was determined from

the balance parameter for each S288 series. However, a

variation of the mass of CM1+CM2 would result in an

error in the analysis since the mass would not be the

value measured by METAS shown in Table I.

In order to investigate this problem, we have examined

the difference between the balance readings for CM1 and

CM2. This difference is proportional to the surface ar-

eas of CM1 and CM2 which differ by approximately a

factor of 2 (CM1 area=1 cm2and CM2 area=0.5 cm2).

The balance reading difference is only slightly dependent

upon the coarse calibration constant so that it repre-

sents essentially the mass difference itself. In Fig 12 is

shown the measured mass difference as a function of time

during the gravitational measurement. Also shown is a

linear function fit to these data. The slope parameter

of the fit results in a rate of increase per area equal to

0.0021(18) µg cm−2d−1. The uncertainty has been de-

termined by normalizing χ2to the DOF. The sign of the

slope is such that the CM with the larger area has the

larger rate of increase. A mass difference variation (CM1-

CM2) would require a slope of -0.0088(12) µg cm−2d−1.

The measured slope of the effective balance reading for

the load CM1+CM2 clearly excludes such a large nega-

tive slope as assumed for a mass variation. We therefore

conclude that the variation of this parameter is due pri-

marily to the sensitivity variation of the balance.

We note that Schwartz [17] has also found a mass in-

crease for stainless steel samples in a vacuum system in-

volving a rotary pump, a turbomolecular pump and a

liquid nitrogen cold trap. His samples were 1 kg masses

with surface areas differing by a factor of 1.8. He mea-

sured the thickness of a contaminant layer using ellip-

sometry as well as the increase in weight of the sample

during pumping periods of 1.2 d and 0.36 d. The rate of

210220230

day of 2001

240 250

-3

-2

-1

0

difference [ppm]

FIG. 11: The change of the effective balance reading for the

load CM1+CM2 as a function of time relative to its value on

the first day. No valid measurements were made between day

229 and 235.

210220230

day of 2001

240250

5.25

5.50

5.75

6.00

6.25

CM1-CM2 [µg]

FIG. 12: The mass difference of the CM’s as a function of

time and the linear fit function.

mass increase per area which he reports is approximately

a factor of 5 larger than the value we find. No explana-

tion for this difference can be made without a detailed

knowledge of the partial pressures of the various contam-

inant gases in the two systems and the surface properties

of the samples employed.

There still remains the possibility that a rapid re-

moval of an adsorbed layer such as water might have

occurred between the absolute determination of the CM

masses in air at METAS and the gravitational measure-

ment in vacuum (i.e. during the pump down of the sys-

tem). Schwartz [16] has measured the mass variation per

unit area of 1 kg stainless steel objects in air with rel-

ative humidity between 3% and 77%. He [17] also has

measured the additional mass variation per area due to

pumping the system from atmospheric pressure at 3%

relative humidity down to 5×10−3Pa. His samples were

first cleaned by wiping them with a linen cloth soaked

in ethanol and diethylether and then ultrasonic clean-

ing in ethanol.After cleaning, they were dried in a

vacuum oven at 50◦C. For these cleaned samples, the

weight change found for 3% to 50% humidity variation

was 11.5 ng cm−2with an additional change of 29 ng

cm−2in going from 3% relative humidity in air to vac-

uum (total change of 40.5 ng cm−2). Similar measure-

ments with ”uncleaned” samples gave a total change of

80 ng cm−2. The variation due to the cleanliness of the

samples was much larger than the difference found for

the two types of stainless steel investigated and the ef-

fect of improving the surface smoothness (average peak-

to-valley height equal to 0.1 µm and 0.024 µm). Since the

cleaning procedure used for our CM’s and their smooth-

ness were different than the samples used by Schwartz,

we have employed the average of Schwartz’s ”cleaned”

and ”uncleaned” objects for estimating the mass change

of our CM’s. Based on these data, the relative mass dif-

ference found for both CM’s together as measured in air

Page 12

12

0 10203040506070

-3

10

-2

10

-1

10

0

10

number of parameters

χ2 probability

FIG. 13: The χ2probability as a function of the number of

parameters.

having 50% humidity and in vacuum was 0.5 ppm. We

assign a relative systematic uncertainty of this correction

equal to the correction itself.

J.Nonlinearity Correction

By nonlinearity of the balance, one is referring to the

variation of the balance response function with load, that

is, the degree to which the balance output is not a linear

function of the load. The nonlinearity of a mass compara-

tor similar to the one employed in the present work has

been investigated [18] by the firm Mettler-Toledo. It was

found that besides nonlinearity effects in 10 g load inter-

vals, there was also a fine structure of the nonlinearity

in the 0.1 mg load interval which would be important for

the accuracy of the present measurement. It is the non-

linearity of our mass comparator in the particular load

interval less than 0.2 g involved in the present experiment

that we wish to determine.

One expects the nonlinearity of the balance used in

this experiment to be small; however, it should be real-

ized that a 200 mg test mass (two 100 mg CM’s) required

for having an accurately known test mass for calibration

purposes is over 250 times the size of the gravitational

signal that one wishes to determine. In addition, the sta-

tistical accuracy of the measured gravitational signal is

some 30 times better than the specified accuracy (2 µg)

of the unmodified commercial balance. One therefore has

no reason to expect the nonlinearity of the balance to be

negligible with this precision. In Sec. IIIA we have pre-

sented the general idea that the measurements with 256

AM values tends to average out the effect of any nonlin-

earity. We wish now to give a more detailed analysis of

this problem.

The correction for nonlinearity makes use of an arbi-

trary response as a function of the load. Since the two

TM’s are essentially equal (< 400 µg difference), the vari-

ation of the response function can be thought of as being

a function of the additional load due to the AM’s. Al-

though a power series or any polynomial series would

suffice for this function, we have for convenience used a

series of Legendre polynomials

f(u) =

Lmax

?

ℓ=0

aℓPℓ(2u/umax− 1).

The coefficients of Pℓare chosen subject to the two con-

ditions that (1) f(u) = 0 for no load and (2) f(u) = C

for u = C where C is the weight of the two CM’s to-

gether. These two conditions represent the sensitivity

of the balance over the 0.2 g range of the calibration

(i.e. the coarse calibration). The value of the maximum

load umaxin the present measurements was very nearly

C. Substituting the above conditions into the response

function, one obtains for the lowest two coefficients the

expressions

a0= C/2 −

Lmax

?

even ℓ=2

aℓ

and

a1= C/2 −

Lmax

?

odd ℓ=3

aℓ.

One can then minimize

χ2=

N

?

n=1

[f(un+ s) − f(un) − yn]2σ−2

n

and thereby determine best values for the parameters s

and aℓ for ℓ = 1 to Lmax. The yn are the measured

balance signal for the load values un, s is the load in-

dependent signal and N is the number of different loads

with valid measurements. The error σnfor the load value

unis the load-independent intrinsic noise of the balance

σ0for a single weighing divided by the square root of the

number of weighings for the load un. The value of Lmax

must be chosen large enough to describe the response

function accurately. All of the parameters in the fit are

linear parameters with the exception of s. Thus, there

is no difficulty in extending the fit to a large number of

parameters since only the nonlinear parameter must be

determined by a search method.

In order to determine Lmax, we calculate the χ2prob-

ability [19] (often referred to as confidence level) as a

function of Lmax. This requires an approximate value

for the intrinsic noise of the balance σ0. The value of σ0

sets the scale of the χ2probability but does not change

the general shape of the function. One can obtain a rea-

sonable approximation for σ0by setting χ2equal to the

DOF obtained for a large number of parameter. We have

arbitrarily set χ2equal to the DOF for 61 parameters.

Page 13

13

0 50000100000

load [µg]

150000200000

784.50

784.60

784.70

784.80

784.90

785.00

785.10

785.20

signal [µg]

-3

0

3

δ/σ

FIG. 14: Signal and fit function employing 60 parameters as

a function of load. The data are shown as a stepped line. The

fit is the smooth curve. The lower plot shows the normalized

residuals. Residuals were divided by the relative uncertainty

of each point. The normalization has been chosen such that

the rms value of the residuals is 1.

The χ2probability as a function of the maximum num-

ber of parameters is shown in Fig. 13. It is seen that

the χ2probability reaches a plateau near this maximum

number of parameters.

Starting from a low value of 10−4for one parameter,

the χ2probability rises rapidly to a value of 0.05 for three

parameters. It remains approximately constant at this

value up to 57 parameters where it rises sharply to reach a

plateau of approximately 0.5 at 60 parameters and above.

The fit parameter representing the signal corrected for

nonlinearity of the balance was essentially constant over

the entire range of parameters with a variation of less

than ±1.3 ng. The signal for one parameter representing

complete linearity was 784.8994 µg. The signal of the

plateau region from 60 to 67 parameters was 784.9005 µg

with a statistical uncertainty of 5.5 ng. In this region

the signal varied by less than 0.2 ng. We therefore take

the nonlinearity correction of the measured signal to be

1.1(5.5) ng (i.e. the difference between the signal using

one parameter as would be obtained with no correction

and the average value obtained for 60 to 67 parameters).

The nonlinear signal and fit as a function of load de-

termined for 60 parameter is shown in Fig. 14. The func-

tion shows many narrow peaks with widths of 3 to 10 load

steps and with amplitudes of roughly 0.1 µg. In principle

one could use this response function to correct the indi-

vidual weighings with various loads; however, we prefer

to use the signal as corrected for nonlinearity over the

entire range of measurements as described above. The

variation of the response function indicates that a mea-

surement made at an arbitrary load value could be in

error by as much as ±130 ng assuming the response to

be linear. This is to be compared with the assumed un-

certainty in ref. [8] due to nonlinearity of ±200 ng.

212.0 212.5213.0

21.300

21.350

21.400

day of 2001

Temperature [deg C]

FIG. 15: Temperatures of the vacuum tube measured at the

position of the TM’s. The upper curve is the temperature

at the position of the upper TM. The square wave in the

middle section of the plot indicates the FM motion. The data

(crosses) for the lower TM and fit function (solid line) are

shown in the lowest section of the figure.

K. Correction of the TM-Sorption Effect

Moving the FM’s changed slightly the temperature of

the vacuum tube surrounding the TM’s. These tempera-

ture variations were due to changes in the air circulation

in the region of the vacuum tube as obstructed by the

FM’s. An increase of the wall temperature of the tube

caused adsorbed gases to be released which were then

condensed onto the TM. Since the temperature variation

was different in the regions near the upper and lower

TM’s, this resulted in a variation of the weight differ-

ence between the upper and lower TM’s (i.e. a ”false”

gravitational signal).

The temperature variation at the positions of the up-

per and lower TM’s during one day of the gravitational

measurement is shown in Fig. 15 along with a curve rep-

resenting the FM motion. The peak-to-peak tempera-

ture variation was approximately 0.04◦C at the upper

position and 0.01◦C at the lower position. The shape

of the temperature variation at the upper position was

used as a fit function (employing an offset and an ampli-

tude parameter) to obtain a better determination of the

temperature variation at the lower position. There were

32 one-day measurements of the temperature variations

during the gravitational measurement. The average am-

plitude at the lower position determined from these 32

measurements was 0.0138(2)◦C.

The signal produced by these temperature variations

was small and therefore not directly measurable with the

balance in a reasonable length of time. The procedure

that was employed to determine this temperature depen-

dent signal was to use four electrical heater bands to pro-

duce a variation of the temperature distribution along the

Page 14

14

312.70 312.90313.10

day of 2001

313.30313.50 313.70

11.0

11.5

12.0

12.5

signal[µ g]

FIG. 16: Weight difference between TM’s as a function of

time for a temperature variation roughly 10 times that of the

gravitational measurement. The solid curve is the best fit of

the temperature variation difference at upper and lower TM

positions. For the purpose of this plot, an arbitrary offset of

the weight difference between upper and lower TM has been

employed.

vacuum tube that was a factor of approximately seven

larger than the variation resulting from the motion of

the FM’s. The bands were positioned 30 cm above and

below the positions of the upper and lower TM’s. The

heater windings were bifilar to avoid magnetic effects.

The heater power (less than 3 W total) was turned off and

on with the same 8-hour period as the FM motion and

produced essentially no change in the average tempera-

ture of the vacuum tube in the day-long measurement.

The FM’s were not moved during the measurements with

heaters. The signal (TM weight difference as determined

with the balance) obtained during a one day measure-

ment with heaters is shown in Fig. 16. The shape of

the fit function (employing an offset and an amplitude

parameter) shown in this figure was obtained from the

variation of the temperature difference at the upper and

lower positions of the TM’s. The signal obtained from

the fitting procedure was 0.114(40) µg.

In order to scale the heater produced signal to that

resulting from the FM motion during the gravitational

measurement, we make the simplifying assumption that

the signal variation is proportional to the temperature

variation at the upper TM position minus the tempera-

ture variation at the lower TM position. The term vari-

ation in this statement refers to the variation about its

mean value. One uses the temperature difference since

the signal is defined as the difference between TM weigh-

ings.

With just four heater bands it was not possible to ob-

tain a variation of the temperature distribution along the

vacuum tube that was exactly a constant factor times

that of the FM motion. For the best adjustment that we

were able to obtain, the ratio of the heater produced tem-

perature variation to the FM produced variation was 7.1

at the upper position and 9.2 at the lower. The ratio for

the variation of the temperature difference at the upper

and lower positions relative to the gravitational values

was 6.8. These ratios are based on the peak-to-peak am-

plitudes obtained for the fitted functions. The scaling

factor for the temperature difference ratio is the recip-

rocal of the temperature difference ratio or 0.147. This

results in a scaled signal of 0.0168(58) µg where the un-

certainty is the statistical uncertainties of the measured

signal and the scaling factor. The scaled signal (”false”

signal) is to be subtracted from the total signal measured

in the gravitational experiment.

In order to check our assumption regarding the scal-

ing factor, we have made four additional one-day mea-

surements in which the temperature variations were very

different from that produced by the FM motion. The

object of these measurements was to determine whether

the scaled signals obtained with the heaters were consis-

tent with one another when calculated with the assumed

scaling factors. The most extreme distribution involved

a temperature variation of the lower TM which was even

larger (factor of 4) than that of the upper TM. The sig-

nals obtained in all of the test measurements were con-

sistent with each other within their statistical uncertain-

ties (relative uncertainties of approximately 30 %). We

therefore conclude that the assumption used for scaling

the signals was sufficiently accurate for the present pur-

pose. Nevertheless, we assign a systematic uncertainty

to the scaled signal equal to its statistical uncertainty

of 5.8 ng (relative systematic uncertainty of the ”false”

signal is 35 %).

L.Magnetic Forces on the Test Masses

In the absence of a permanent magnetization, the z

component of force on the TM due to a magnetic field

can be calculated from

Fz= −µ0χmV H∂H

∂z

where V denotes the volume of the TM, χmis its mag-

netic susceptibility and H is the magnetic field inten-

sity. The magnetic properties of the TM’s were measured

by METAS. No permanent magnetization was found

(< 0.08 A/m). The magnetic susceptibility was 4×10−6

for the copper TM’s. The magnetic field intensity for

both positions of the FM’s was measured at cm intervals

along the axis of the vacuum tube at the positions occu-

pied by the TM’s using a flux gate magnetometer. The

difference of Fzfor the FM positions obtained from these

data was 0.01 ng which is a negligible correction to the

measured gravitational signal.

Page 15

15

M.Tilt Angle of Balance

Since the weight of the TM’s and the weight of the

CM’s both produce forces on the balance arm in the ver-

tical direction, a small angle between the balance weigh-

ing direction and the vertical produces no error in the

weighing of the TM’s. However, if the balance weigh-

ing direction is correlated with the motion of the FM’s,

a systematic error in the measured gravitational signal

will result. Sensitive angle monitors were mounted on

the base of the balance. No angle variation correlated

with the motion of the FM’s was found with a sensitivity

of 100 nrad. Since the sensitivity of the balance varies

with the cosine of the angle (near 0 rad), this limit is com-

pletely negligible. For a balance misalignment of 0.01 rad

relative to vertical and a correlated variation of 100 nrad

with respect to this angle due to the FM motion, the

relative signal variation is approximately 0.001 ppm.

IV.

MASS-INTEGRATION CONSTANT

DETERMINATION OF THE

One must relate the gravitational constant to the mea-

sured gravitation signal. This involves integrating an in-

verse square force over the mass distribution of the TM’s

and FM’s. The gravitational force Fzin the z (vertical)

direction on a single TM produced by both FM’s is given

by

Fz= G

? ?

ez· (r2− r1) dm1dm2

|r2− r1|3

(1)

where ez is a unit vector in the z direction, r1 and r2

are vectors from the origin to the mass elements dm1of

the TM and dm2of the FM’s and G is the gravitational

constant to be determined. The mass-integration con-

stant is the double integral in Eq. (1) multiplying G.

Actually, the mass-integration constant for the present

experiment is composed of four different mass-integration

constants, namely those for the upper TM and lower TM

with the FM’s together and apart. We shall use as mass-

integration constant the actual constant multiplied by

the 1986 CODATA value of G (6.67259 m3kg−1s−2and

give the result in dimensions of grams ”force” (i.e. the

same dimensions as used for the weighings).

The objects contributing most to Fz(TM’s, FM tanks

and the mercury) have very nearly axial symmetry which

greatly simplifies the integration. Parts which do not

have axial symmetry were represented by single point

masses for small parts and multiple point masses for

larger parts. For axial symmetric objects, we employ the

standard method of electrostatics for determining the off-

axis potential in terms of the potential and its derivatives

on axis (see e.g.[20]). The force on a cylindrical TM in

the z direction produced by an axially symmetric FM can

be conveniently expressed as (see Eq. 10, Sec. VII)

Fz= 2MTM×

∞

?

n=0

V(2n+1)

0

n

?

i=0

1

(−4)i

1

i!(i + 1)!

1

(2n − 2i + 1)!b2n−2ir2i

(2)

where MTM is the mass of a cylindrical TM with radius

r and height b, and V(2n+1)

0

the gravitational potential with respect to z evaluated at

the center of mass of the TM (r = 0, z = z0).

The potential V (r = 0,z) of the various FM compo-

nents having axial symmetry was determined analytically

for three types of axially symmetric bodies, namely a hol-

low ring with rectangular cross section, one with trian-

gular cross section and one with circular cross section.

This allows one to calculate the gravitation potential of

the tank walls and the mercury content of the tank as

a sum of such bodies. For example, the region between

measured heights on the top plate and z = 0 at two values

of the radius was represented by a cylindrical shell com-

posed of a right triangular torus and a rectangular torus

(i.e. a linear interpolation between the points describing

the cross section of the rings). O-rings were calculated

employing the equation for rings with circular cross sec-

tions. A total of nearly 1200 objects (point masses and

rings of various shapes) were required to describe the two

FM’s.

The derivatives of the potential were evaluated using a

numerical method called “automatic differentiation” (see

e.g. [22]). For the geometry of the present experiment,

the terms in the summation over n decrease rapidly so

that 8 terms were sufficient for an accuracy of 0.02 ppm

in the mass-integration constant.

is the 2n + 1st derivative of

A.Positions of TM’s and FM’s

In order to carry out the mass integration, one needs

accurate weight and dimension measurements of the

TM’s and FM’s as well as distances defining their relative

positions. The dimension and weight measurements for

TM’s were described in Sec. IIIB. The measurement of

the TM positions shown in Fig. 17 will now be addressed.

A special tool was made to adjust the length of the

tungsten wires under tension. Each wire made a single

loop around the post on either side of the TM and a thin

tube was crimped onto the wires to hold them together

thereby forming the loop (see Fig. 4). The position of

the TM could only be measured with the vacuum tube

vented. The vacuum tube was removed below a flange

located at a point just above the upper TM. The TM

hanging from the balance was then viewed through the

telescope of an optical measuring device to determine its

position.

The vertical position of the TM’s and FM’s was mea-

sured relative to a surveyor’s rod which was adjusted to

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