# Gravitational Constant

#### The Controversy over Newton's Gravitational Constant

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, called 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 9.8m/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 G.

Naturally, the value of the fundamental constant G has interested physicists for over 300 years and, except for the the speed of light, it has the longest history of measurements. Almost all measurements of G have used variations of the torsion balance technique pioneered by Cavendish. The usual torsion balance consists of a 'dumbbell' (two masses connected by a horizontal rod) suspended by a very thin fiber. When two heavy attracting bodies are placed on opposite sides of the dumbbell, the dumbbell twists by a very small amount. The attracting bodies are then moved to the other side of the dumbbell and the dumbbell twists in the opposite direction. The magnitude of these twists is used to find G. In a variation of the technique, the dumbbell is set into an oscillatory motion and the frequency of the oscillation is measured. The gravitational interaction between the dumbbell and the attracting bodies causes the oscillation frequency to change slightly when the attractors are moved to a different position and this frequency change determines G. This frequency shift method was used in the most precise measurement of G to date (reported in 1982) 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, 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 fundamental constants, such as Planck's constant or the charge on the electron. As a result, the mass of the Earth is known far less precisely than, for instance, its diameter. In fact, if the Earth's diameter were known as poorly as its mass, it would be uncertain by one mile. This should be compared to the 3 cm uncertainty in the distance between the Earth and Moon, which is determined using laser ranging and the well-known speed of light!

Recently the value of G has been called into question by new measurements from respected research teams in Germany, New Zealand, and Russia. The new values disagree wildly. For example, a team from the German Institute of Standards led by W. Michaelis obtained a value for G that is 0.6% larger than the accepted value; a group from the University of Wuppertal in Germany led by Hinrich Meyer found a value that is 0.06% lower, and Mark Fitzgerald and collaborators at Measurement Standards Laboratory of New Zealand measured a value that is 0.1% lower. The 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. This controversy has spurred several efforts to make a more reliable measurement of G.

One of the greatest difficulties in any G measurement is determining with sufficient accuracy the dimensions and density distribution of the torsion pendulum body (the dumbbell). A second limitation is in knowing the properties of the suspension fiber with sufficient accuracy. The Japanese physicist Kazuaki Kuroda recently pointed out that internal friction in the torsion fiber, which had previously been neglected, may have caused some of the problems in the existing measurements.

Jens Gundlach, Eric Adelberger, and Blayne Heckel from the University of Washington Eöt-Wash research group have pioneered a method that elegantly sidesteps these uncertainties. They noted that if the usual dumbbell is replaced by a thin, flat plate hung by its edge, neither the pendulum's dimensions nor its density distribution 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! The Seattle researchers eliminate the problems with the torsion fiber by placing the torsion balance on a turntable that continuously rotates between a set of attracting bodies. The turntable is controlled by a feedback loop that speeds it up or slows it down so that the suspension fiber never has to twist; G can then be accurately inferred from the rotation rate of the turntable. This new method uses eight, rather than two, attracting bodies and these are strategically placed on a second turntable that rotates in the opposite sense from the first turntable. This novel technique is discussed in the July 15 issue of Physical Review D.

At the University of California at Irvine, Riley Newman and graduate student Michael Bantel are refining the frequency shift method. They plan to operate their balance at a temperature only 4 degrees above absolute zero to reduce the internal friction in the suspension fiber and to make its properties more constant. Their apparatus will also use a flat pendulum.

The fact that this famous fundamental constant is still so uncertain testifies to the difficulty of gravitational measurements. The recent flurry of new ideas for measuring G would surely have pleased Isaac Newton (quite a clever experimenter himself) who started this whole enterprise over 300 years ago.

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/20min, the outer turntable rotates at about 1rev/5min. (PDF image)

### Relevant literature

1. C.C Speake and G.T. Gillies Z. Naturforsch., 42a 663 (1987). "Why is G the least precisely known physical constant?"
2. E.R. Cohen and B.N. Taylor, Rev. Mod. Phys., 59 1121 (1987). [The Committee on Data for Science and Technology report]
3. G.G. Luther and W.R. Towler, Phys. Rev. Lett., 48, 121 (1982).[The most reliable measurement]
4. W. Michaelis, H. Haars, and R. Augustin, Metrologia 32, 267 (1995). [The result from the German Bureau of Standards]
5. M. Fitzgerald and T. R. Armstrong, IEEE Trans. on Inst. and Meas. 44, 494 (1995). [The result from New Zealand]
6. H. Walesch, H. Meyer, H. Piehl, and J. Schurr, IEEE Trans. on Inst. and Meas. 44, 491 (1995). [The result from Wuppertal]
7. 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]
8. Kazuaki Kuroda, Phys. Rev. Lett. 75, 2796 (1995).
9. J.H. Gundlach, E.G. Adelberger, B.R. Heckel, and H.E. Swanson, Phys. Rev. D 54, 1256R (1996).
10. J.H. Gundlach, Measurement Sci. and Tech. 10 454 (1999).
11. J.H. Gundlach and S.M. Merkowitz, Phys. Rev. Lett. 85 2869 (2000).