**6. Accuracy budget**

**5. Attitude control**

68 Metrology

angular noise contributions.

excited by ocean waves.

One final problem must be addressed here to give a complete picture of the complexity of this apparently very simple experiment: the attitude of the whole apparatus with respect to the vertical direction, as defined by the Earth's local gravity vector, can affect the operation of pendulums and must therefore be guaranteed to be adequately accurate and stable in time, if necessary by active attitude control. Two different problems must be addressed in this respect

The absolute verticality is important because the two cylinders must be guaranteed to be horizontal for the symmetry of the swing, which in turn guarantees the positioning of the minimum period in amplitude space (the curve of **Figure 10** was calculated for the case of two cylinders at the same level). Because the pendulums are two, this issue is complicated by the

The attitude stability is particularly critical in the case of low sampling-rate detection, like simple flyby time stamping at half periods, because of the heavy aliasing of seismic angular noise [1] that it produces. In **Figure 12**, a series of background seismic power spectrums is reported, as collected in different locations of the global seismographic network [21]. A peak at about 0.2 Hz appears in all of them, which is produced by low damping surface Rayleigh waves excited by ocean waves hammering the shores, extended with reduced intensity at higher frequencies. Because of their low frequency, it is very difficult to filter out such seismic

Work done to attack this problem includes passive and active attitude control [22, 23]. However, passive filtering was quickly understood to be inadequate for the purpose, not only because of its awkwardness at such low frequencies but also because of the need for stiffness of the structure holding pendulums to prevent detrimental effects of recoil from pendulums on attitude stability and damping itself. Active stabilization was then decided to be necessary, and work

**Figure 12.** Background seismic power spectrums, as collected in different locations of the global seismographic network. High noise at periods above 1s (i.e. Fourier frequencies below 1 Hz) is caused by far travelling Rayleigh surface-waves

need to have both pairs of suspension cylinders aligned on the same horizontal plane.

as both the absolute tilt and its stability are relevant, in different ways.

A tentative accuracy budget for the experiment described here is given in [1]. Because of the highly efficient time and frequency metrology approach, only geometrical uncertainties are expected to be relevant at the level of 10−5, provided the necessary differential stability of 10−12 can be achieved. This is clearly a big "if," as discussed above, because it assumes that seismic and mode leakage problems are adequately solved. However, it can be in principle obtained if the limitation is electronic noise. It must be noted here for completeness that the


proportional to τ−3/2. This fact is unique among experiments for the determination of *G* and offsets the poor signal size problem allowing to focus the design on accuracy rather than S/N ratio. It remains to be shown that differential stability in the 10−12 region can be obtained with consistency for two similar pendulums of the design which has been sketched here. This seems to be a long shot when considering the absolute stability achieved by the best Shortt clock [13], because it requires an improvement of more than three orders of magnitude with respect to it, at the target few hours (*T*R) averaging time. However, it is not unreasonable to think that two adequately similar pendulums can be realized, and if they are within 100 mm of each other, it can be expected that *g* uniformity may be adequately stable in time to support the assumption. A description of the apparatus and a discussion of pendulum design optimization for this experiment were offered in detail, pointing out problems and possible solutions. Work is in progress on the preparation of the experiment, considering both a free decay solution and pendulum operation with active support of oscillations and amplitude control. It is expected that an accuracy of 10−5 may be obtained for *G* with the proposed approach, limited only by the accuracy of field masses' size and positioning, and that it may be possible in a metrology laboratory to reduce limiting geometrical uncertainties enough to push it into the 10−6 range.

Measuring 'Big G', the Newtonian Constant, with a Frequency Metrology Approach

http://dx.doi.org/10.5772/intechopen.75635

71

The author wishes to thank for encouragement and discussions Robert Drullinger, Stephan Schlamminger, and Bill Phillips of NIST and Valter Giaretto, Mario Lavella, and Lamberto Rondoni of the Politecnico di Torino. Special thanks go to Luca Maffioli for his master's thesis on the pendulum analysis and to Meccanica Mori of Parma for the TIG welding of the thin steel tubes to the experimental chamber. The author also wishes to acknowledge the support of the US Department of Commerce and NIST through the Precision Measurements Grant

Department of Electronics and Telecommunications, Politecnico di Torino, Torino, Italy

[1] De Marchi A. A frequency metrology approach to Newtonian constant *G* determination using a pair of extremely high *Q* simple pendulums in free decay. Journal of Physics:

**Acknowledgements**

**Author details**

Andrea De Marchi

**References**

Program, Award ID number 70NANB15H348.

Address all correspondence to: andrea.demarchi@polito.it

Conference Series. 2016. DOI: 10.1088/1742-6596/723/012046

**Table 1.** Accuracy budget projection based on 1-m-long 4 μm tungsten fibres, 6-mm-diameter suspension cylindrical profiles, a swing amplitude of 0.01 rad, and a 5 mm tungsten bob. The position of field masses' gravity center is assumed known with <300 nm uncertainty.

best pendulum clocks ever realized [13] were probably not differentially stable better than 10−9 at the target 104 s averaging time, which means that 60 dB improvement is necessary. Although this is granted on paper by projected S/N and *Q*, actually achieving it is still a big challenge. On the positive side, it is worth pointing out here that energy-induced amplitude changes [11] do not affect frequency if operation is kept at the minimum period isochronous point and that an approach to oscillation support aimed at overcoming pulse stability problems by moving to a sine-wave excitation system similar to that employed in high-stability quartz oscillators will remove one of the worst contributions to instability.

This said, it can be seen in **Table 1** that most geometrical contributions to uncertainty impose quite loose requirements at the target accuracy level of 10−5, with the sole exception of size and positioning of field masses, which must be guaranteed at high accuracy. While other contributions enjoy relaxed specifications granted by parabolic minima which are specific of this configuration, the latter do not and must comply with specs which are similar to any other big G experiment. However, the expectation that accuracy on G may be limited by control on this single geometry contribution ushers the possibility of doing even better than 10−5 if resources were to become available to improve the accuracy of field masses. A summary of such uncertainties is reported in **Table 1**, as listed in Ref. [1], where the reader can find more details and a deeper discussion on accuracy.
