Measurement of Newton’s gravitational constant

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 $6.674252(109)(54)\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$. The relative statistical and systematic uncertainties of this result are $16.3\times {10}^{-6}$ and $8.1\times {10}^{-6}$, respectively.

Figure 1

Principle of the measurement. The FM’s are shown in the position together (Pos. T) and the position apart (Pos. A).

Figure 2

A side view of the experiment. Legend: 1 measuring room enclosure, 2 thermally insulated chamber, 3 balance, 4 concrete walls of the pit, 5 granite plate, 6 steel girder, 7 vacuum pumps, 8 gear drive, 9 motor, 10 working platform, 11 spindle, 12 steel girder of the main support, 13 upper TM, 14 FM’s, 15 lower TM, 16 vacuum tube.

Figure 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 balance reading corresponding to approximately 1.1 kg with a magnetic compensation of 0.6 g. The amplitude of the oscillatory signal corresponds to about $1.5\mu \mathrm{g}$. The lower plot shows the normalized residuals. The normalization has been chosen such that their rms value is 1.

Figure 4

Drawing of TM inside the vacuum tube. Dimensions are given in mm.

Figure 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 double-staircase pair holding AM’s, 4 narrow strip to receive AM’s, 5 balance pan, 6 flat spring, 7 frame, 8 step-motor 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-staircase pair.

Figure 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 $\delta /\sigma$ are shown in the lower plot. The normalization of the residuals has been chosen such that their rms value is 1.

Figure 7

The measured weight difference in $\mu \mathrm{g}$ 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.

Figure 8

The measured weight difference in $\mu \mathrm{g}$ between TM’s for the FM’s positions apart and together. The measured values for the FM apart have been displaced $782\mu \mathrm{g}$ in order to show both types of data in the same figure.

Figure 9

Binned data for the FM apart-together weight differences (points) and a fitted Gaussian function (curve) shown as a deviation from the mean difference. Poisson statistics were used to determine the uncertainties.

Figure 10

The average signal for each of the eight S2304 series. Series with increasing loads are shown as circles. Series with decreasing loads are shown as squares. The dashed line is the average of all eight series.

Figure 11

The change of the effective balance reading for the load $\mathrm{CM}1+\mathrm{CM}2$ as a function of time relative to its value on the first day. No valid measurements were made between day 229 and 235.

Figure 12

The mass difference of the CM’s as a function of time and the linear fit function.

Figure 13

The ${\chi }^2$ probability as a function of the number of parameters.

Figure 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.

Figure 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.

Figure 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.

Figure 17

Drawing showing the measured vertical distances to TM and FM surfaces for the two FM positions (T—Together and A—Apart).

Figure 18

Drawing of the field mass. All dimensions are given in mm.

Figure 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 $z$ value is 0 mm at a radius of 60 mm.

Figure 20

Plot of recent measurements with relative errors less than 50 ppm.