# Gravitational Constant

#### The story of the gravitational constant, Big G:

In 1686 Isaac Newton realized that the motion of the planets and the moon as well as that of a falling apple could be explained by his Law of Universal Gravitation, which states that any two objects attract each other with a force equal to the product of their masses divided by the square of their separation times a constant of proportionality. Newton estimated this constant of proportionality, often called Big G, perhaps from the gravitational acceleration of the falling apple and an inspired guess for the average density of the Earth. However, more than 100 years elapsed before G was first measured in the laboratory; in 1798 Cavendish and co-workers obtained a value accurate to about 1%. When asked why he was measuring G, Cavendish replied that he was "weighing the Earth"; once G is known the mass of the Earth can be obtained from the 9.8 m/s2 gravitational acceleration on the Earth surface and the Sun's mass can be obtained from the size and period of the Earth orbit around the sun. Early in this century Albert Einstein developed his theory of gravity called General Relativity in which the gravitational attraction is explained as a result of the curvature of space-time. This curvature is proportional to Big G.

Measurements of Big G in the past

Naturally, the value of the fundamental constant G has interested physicists for over 300 years and, except for the speed of light, it has the longest history of measurements. Most measurements of G have used variations of the torsion balance technique first used by Cavendish. Torsion balance experiments to measure G consisted of a 'dumbbell' (two masses connected by a horizontal rod) suspended by a very thin fiber. When two heavy attracting bodies were placed on opposite sides near the masses on the dumbbell pendulum, it would rotate by a very small amount, twisting the fiber. The attracting bodies were then moved to the other side of the dumbbell twisting it in the in the opposite direction. The magnitude of these twists is used to find G. In a variation of this technique, the dumbbell was set into an oscillatory motion and the frequency of the oscillation was measured. The gravitational interaction between the dumbbell and the attracting bodies caused the oscillation frequency to change slightly when the attractors are moved to a different position and this frequency change was used to determine G. This frequency shift method was by Gabe Luther and William Towler from the National Bureau of Standards and the University of Virginia. It was published in 1982. Based on their measurement, the Committee on Data for Science and Technology (CODATA), which gathers and critically analyzes data on the fundamental constants, assigned an uncertainty of 0.0128% to G. Although this seems quite precise, the fractional uncertainty in G is thousands of times larger than those of other important physical constants, such as Planck's constant or the charge on the electron. Following 1982 measurement the value of G was called into question by measurements from respected research teams in Germany, New Zealand, and Russia. The new values disagreed wildly. For example, a team from the German Institute of Standards obtained a value for G that was 0.6% larger than the accepted value; a group in Wuppertal, Germany found a value that is 0.06% lower, and a group in New Zealand measured a value that was 0.1% lower. A Russian group found a curious space and time variation of G of up to 0.7%. The collection of these new results suggests that the uncertainty in G could be much larger than originally thought.

One of the greatest difficulties in any G measurement was determining with sufficient accuracy the dimensions and density distribution of the torsion pendulum body, e.g. the dumbbell. A second limitation was associated with the suspension fiber. The Japanese physicist Kazuaki Kuroda had pointed out that internal friction in the torsion fiber, which had previously been neglected, may have caused a significant bias in the existing torsion balance measurements.

The University of Washington Big G Measurement

At the University of Washington’s Eöt-Wash research group, Jens Gundlach invented a torsion balance method that elegantly sidesteps the limitations of all previous measurements. The first thing he noted was that if the usual dumbbell pendulum were replaced by a thin, flat plate hung by its edge, neither the pendulum's dimensions nor its density distribution would have to be known with very high precision. In principle, one can obtain G by measuring the angular acceleration of a flat pendulum without even knowing its mass or dimensions. This simple fact had not been recognized in 200 years of gravitational experiments! Then he eliminated the problems with the torsion fiber by placing the torsion balance on a turntable that continuously rotates between a set of attracting spheres. The turntable is controlled by a feedback loop that speeds  up or slows the turntable rotation rate exactly so that the suspension fiber never has to twist; G can then be accurately inferred from the rotation rate of the turntable. Since the torsion fiber does not twist, the bias that Kuroda had warned about simply does not occur.  By placing the attractor spheres on another (concentric) turntable that rotates at a much faster frequency than the frequency at which previous mass exchanges had been done, Gundlach’s method reduced the so-called 1/f-noise. This is noise associated with slow changes over time and afflicts many delicate measurements. By measuring G at a higher frequency the fluctuating noise of the measurement was significantly reduced.

For these ideas Gundlach received a prestigious Precision Measurement Grant from the National Institute of Standards and Technology, NIST, and he began together with his colleague Stephen Merkowitz to build the instrument and then carry out the measurement. Gundlach’s tricks paid off and the result was a much more accurate measurement of Big G with an uncertainty of only 14 ppm (parts per million); published in 2000. The measurement moved the value of G higher than the CODATA value so that the accepted value was redefined, mostly based on the University of Washington measurement.

The University of Washington value was followed by measurements from other groups that were in good agreement, confirming the higher value of G. Most notably, a group led by Jun Luo in China built a copy of Gundlach’s experiment, and measured a very similar value for Big G.

Photograph of the big G apparatus. The spheres are 12.5 cm in diameter. (JPG image)

Photograph of the pendulum with several mirrors directing the light beam. A penny was placed in the foreground for scale. (JPG image)

Schematic cut-open drawing of the big G apparatus. The inner turntable rotates at about 1rev/20 min, the outer turntable rotates at about 1rev/5 min.

### Relevant literature

1. G.G. Luther and W.R. Towler, Phys. Rev. Lett., 48, 121 (1982). [The previously best measurement]
2. W. Michaelis, H. Haars, and R. Augustin, Metrologia 32, 267 (1995). [The result from the German Bureau of Standards]
3. M. Fitzgerald and T. R. Armstrong, IEEE Trans. on Inst. and Meas. 44, 494 (1995). [The result from New Zealand]
4. H. Walesch, H. Meyer, H. Piehl, and J. Schurr, IEEE Trans. on Inst. and Meas. 44, 491 (1995). [The result from Wuppertal]
5. V.P. Izmailov, O.V. Karagioz, V.A. Kuznetsov, V.N. Mel'nikov, and A.E. Roslyakov, Measurement Techniques 36, 1065 (1993). [The Russian result]
6. Kazuaki Kuroda, Phys. Rev. Lett. 75, 2796 (1995). [The torsion fiber bias]
7. J.H. Gundlach, E.G. Adelberger, B.R. Heckel, and H.E. Swanson, Phys. Rev. D 54, 1256R (1996).
8. J.H. Gundlach, Measurement Sci. and Tech. 10 454 (1999).
9. J.H. Gundlach and S.M. Merkowitz, Phys. Rev. Lett. 85 2869 (2000). [The measurement]
10. R. Newman, M. Bantel, E. Berg, and W. Cross, Philos. Trans. R. Soc., A372 20140025 (2014). [UC Irvine, uses plate trick]
11. S. Schlamminger, et al. Phys Rev D74:, 082001 (2006). [Zurich]
12. Qing Li, et al. Nature 560 582-588. (2018). [HUST, China]