## Abstract

The Newtonian gravitational constant, *G*, is one of the most fundamental constants of nature, but we still do not have an accurate value for it. Despite two centuries of experimental effort, the value of *G* remains the least precisely known of the fundamental constants. A discrepancy of up to 0.05 per cent in recent determinations of *G* suggests that there may be undiscovered systematic errors in the various existing methods. One way to resolve this issue is to measure *G* using a number of methods that are unlikely to involve the same systematic effects. Here we report two independent determinations of *G* using torsion pendulum experiments with the time-of-swing method and the angular-acceleration-feedback method. We obtain *G* values of 6.674184 × 10^{−11} and 6.674484 × 10^{−11} cubic metres per kilogram per second squared, with relative standard uncertainties of 11.64 and 11.61 parts per million, respectively. These values have the smallest uncertainties reported until now, and both agree with the latest recommended value within two standard deviations.

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## Acknowledgements

We are grateful to R. Newman, T. Quinn, C. Speake, J. E. Faller, J. H. Gundlach, H. J. Paik, Z. H. Lu, J. Luo and S. H. Fan for discussions and suggestions. We thank Q. T. Fan, Y. T. Zhang, B. P. Wang, X. D. Fan, M. Ke, L. Zhao, Y. Tu, J. Q. Guo, D. C. Chen, W. M. Wang, X. L. Wang, X. J. Luo, X. H. Fu, J. Tang and Y. B. Cheng for their early works on *G* measurement. We thank the National Institute of Metrology (NIM) of China for the calibration of some measuring instruments, source masses and the length gauges. This work is partly supported by the National Natural Science Foundation of China under grants number 91536223, 11722542, 11325523 and 11605295, the National Basic Research Program of China under grant number 2010CB832801 and the National Precise Gravity Measurement Facility.

### Reviewer information

*Nature* thanks S. Schlamminger and the other anonymous reviewer(s) for their contribution to the peer review of this work.

## Author information

### Author notes

These authors contributed equally: Qing Li, Chao Xue, Jian-Ping Liu, Jun-Fei Wu

### Affiliations

#### MOE Key Laboratory of Fundamental Physical Quantities Measurements, Hubei Key Laboratory of Gravitation and Quantum Physics, School of Physics, Huazhong University of Science and Technology, Wuhan, China

- Qing Li
- , Jian-Ping Liu
- , Jun-Fei Wu
- , Shan-Qing Yang
- , Cheng-Gang Shao
- , Wen-Hai Tan
- , Liang-Cheng Tu
- , Hao Xu
- , Zhong-Kun Hu
- , Ze-Bing Zhou
- , Peng-Shun Luo
- , Shu-Chao Wu
- & Jun Luo

#### TianQin Research Center for Gravitational Physics, Sun Yat-sen University, Zhuhai, China

- Chao Xue
- , Liang-Cheng Tu
- , Qi Liu
- & Jun Luo

#### School of Physics and Astronomy, Sun Yat-sen University, Zhuhai, China

- Chao Xue
- , Qi Liu
- & Jun Luo

#### College of Engineering, Huzhou University, Huzhou, China

- Li-Di Quan

#### Teaching Research and Assessment Center, Henan Institute of Technology, Xinxiang, China

- Lin-Xia Liu

#### School of Science, Hubei University of Automotive Technology, Shiyan, China

- Qing-Lan Wang

#### Sternberg Astronomical Institute, Moscow State University, Moscow, Russia

- Vadim Milyukov

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### Contributions

J.L. had the idea for the experiment. J.L. and S.-Q.Y. supervised all the experiments. Q.Li and J.-P.L. performed the experiment with the TOS method and analysed the data. C.X. and J.-F.W. performed the experiment with the AAF method and analysed the data. L.-D.Q. designed and built the feedback control system of the two turntables in the AAF method. C.-G.S. analysed all the errors and data independently. W.-H.T., H.X., L.-C.T., Q.Liu, L.-X.L., Q.-L.W., Z.-K.H., Z.-B.Z., P.-S.L., S.-C.W. and V.M. contributed to the analysis and discussion. S.-Q.Y., Q.Li and C.X. wrote the manuscript.

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to Shan-Qing Yang or Cheng-Gang Shao or Jun Luo.

## Extended data figures and tables

### Extended Data Fig. 1 Photographs of the experimental apparatus.

**a**, Apparatus 1, used in TOS-I.**b**, Apparatus 2, used in TOS-II.**c**, The suspended pendulum and source masses in the vacuum chamber used in the TOS method. The copper tube around the fibre is used to reduce the temperature gradient. The electrostatic shield (here elevated to show the pendulum), the three-point mounts, the ULE-glass disk and the turntable are also shown.**d**, The preliminary apparatus used to perform the proof-of-principle measurements^{24,25}of*G*using the AAF method.**e**, The improved apparatus used in the present work. The apparatus was completely rebuilt to reduce several sources of uncertainty encountered in the proof-of-principle experiments (see text for details).**f**, The suspended pendulum and the optical path system used in the AAF method.### Extended Data Fig. 2 Fabrication of the silica fibre and measurement of its

*Q*factor.**a**, Photograph of a silica fibre pulled from a rod over an oxygen–natural gas flame.**b**, Magnetron sputtering equipment used for the coating of the silica fibres.**c**, The Bi target, with a height of ∼1 m. The Ge target (not shown here) is similar. The two targets are installed on opposite sides of the coating equipment, with the fibre located between the two targets and rotated continuously. The surfaces of the fibres were coated with a 5-nm-thick Ge layer and then a 10-nm-thick Bi layer.**d**, Typical decay curves of the torsional amplitude of a pendulum suspended by a ∼45-μm-diameter fibre. Curve A represents the uncoated silica fibre, with a*Q*factor of 2.6 × 10^{5}. Curve B corresponds to the coated silica fibre, with a*Q*factor of 5.5 × 10^{4}. The dot-dashed lines denote fitting curves of the exponential function*A*=*A*_{0}exp(−π*f*_{0}*t*/*Q*), where*A*_{0}is the initial amplitude,*f*_{0}is the free oscillation frequency and*t*is the time. Source data### Extended Data Fig. 3 Schematic diagram of the source masses.

**a**,**b**, In the TOS method, spheres 1 and 3 are used in apparatus 1 (TOS-I;**a**) and spheres 2 and 4 are used in apparatus 2 (TOS-II;**b**).*S*_{1,3}and*S*_{2,4}are the horizontal distances of the geometric centres of the spheres in apparatus 1 and 2, respectively.**c**, In the AAF method, spheres 7, 9, 10 and 12 are used.*S*_{7,9}and*S*_{10,12}are the horizontal distances and*S*_{7,10}and*S*_{9,12}are the vertical distances between the geometric centres of the spheres.### Extended Data Fig. 4 Effect of temperature on the measurement of the angular acceleration in the AAF method.

A modulation experiment was carried out by increasing the temperature variation in the room to about 1 °C. Solid circles represent the average angular acceleration of the pendulum turntable over 12-h data taking periods. The dashed line with a slope of (2.2 ± 3.6) × 10

^{−12}rad s^{−2}°C^{−1}represents the least-squares fitting curve. The result indicates that the apparatus is insensitive to the temperature variation and that a temperature variation of less than 0.1 °C during each experimental run contributes an uncertainty of less than 0.91 p.p.m. to the*G*measurement. Source data### Extended Data Fig. 5 Electrostatic effect on the measurement of the pendulum period in the TOS method.

Different voltages are applied on the shield in the sequence: ground, 0.1 V, −0.1 V, ground. For each voltage, 4–5 sets of measurements of the pendulum period are performed at the ‘near’ and ‘far’ configurations. The corresponding change of the frequency squared (Δ

*ω*^{2}) for the steps of the sequence is determined to be 1.662192(8) × 10^{−6}s^{−2}, 1.662184(16) × 10^{−6}s^{−2}, 1.662181(15) × 10^{−6}s^{−2}and 1.662200(13) × 10^{−6}s^{−2}, respectively. The results show that the period changes with the applied voltage, but the Δ*ω*^{2}values for the ‘near’ and ‘far’ configurations are consistent with each other within the statistical uncertainty. The dot-dashed lines are polynomial fitting curves that represent the period drift due to the ‘aging’ effect of the fibre. Source data

## Supplementary information

### Supplementary Information

This file contains the Supplementary Discussion sections about the experimental method, the alignment and positioning, the nonlinear effects in the ToS method, the data acquisition and correction factors, compensation of the background gravitational gradient in the AAF method, the combination of the values of G, and discussion of the results. The file also contains Supplementary Figure 1 showing the PSD of the pendulum twist angle in AAF-I and the photo of the ULE shelf, Supplementary Figure 2 showing the photo of the lead blocks used to compensate the lab-fixed background gradient field, Supplementary Table 1 showing the parameters of the two-stage pendulum system, Supplementary Table 2 showing the determined Δ

*C*_{g}/*I*and the average value of <Δ*ω*^{2}> in the ToS method, Supplementary Table 3 showing the determined*P*_{g,l,m}and the average value of <*α*_{t}> in the AAF method, and Supplementary Table 4, showing the nonlinearity effects of the gravitational field and the fiber in the ToS method.### Supplementary Data

This file contains the source data for Supplementary Figure 1.

### Source Data Fig. 2

### Source Data Fig. 3

### Source Data Extended Data Fig. 2

### Source Data Extended Data Fig. 4

### Source Data Extended Data Fig. 5

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