Gravitational constant

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According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them.

The constant of proportionality is called <math> {G} \ </math>, the gravitational constant, the universal gravitational constant, Newton's constant, and colloquially big G. The gravitational constant is a physical constant which appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. In some other theories the constant is replaced with a scalar value. See Rosen bi-metric theory of gravity.

The gravitational constant is perhaps the most difficult physical constant to measure. In SI units, the 2002 CODATA recommended value of the gravitational constant is

<math> G = \left(6.6742 \plusmn 0.001 \right) \times 10^{-11} \ \mbox{N} \ \mbox{m}^2 \ \mbox{kg}^{-2} \,</math>

<math> = \left(6.6742 \plusmn 0.001 \right) \times 10^{-11} \ \mbox{m}^3 \ \mbox{s}^{-2} \ \mbox{kg}^{-1} \,</math>

Another authoritative estimate is given by the International Astronomical Union (see Standish, 1995).

In Natural units, of which Planck units are perhaps the best example, G and other physical constants such as c (the speed of light) may be set equal to 1.

When considering forces of fundamental particles, the gravitational force can appear extremely weak compared with other fundamental forces. For example, the gravitational force between an electron and proton 1 metre apart is approximately 10^-67 newton, while the electromagnetic force between the same two particles still 1 metre apart is approximately 10^-28 newton. Both these forces are weak when compared with the forces we are able to experience directly, but the electromagnetic force in this example is some 39 orders of magnitude (i.e. 39 = 67-28) greater than the force of gravity - that's even greater than the ratio between the mass of a human and the mass of the Solar System!

When <math> {G} \ </math> is represented in Natural Units, on the other hand, it is not the force of gravity that appears extremely small but rather the masses of the proton and electron.

1 Measurement of the gravitational constant
2 The GM product
3 The dimensions of G
4 See also
5 References
6 External links

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Measurement of the gravitational constant

<math> {G} \ </math> was first implicitly measured by Henry Cavendish (Philosophical Transactions 1798). He used a horizontal torsion beam with lead balls whose inertia (in relation to the torsion constant) he could tell by timing the beam's oscillation. Their faint attraction to other balls placed alongside the beam was detectable by the deflection it caused. However, it is worth mentioning that the aim of Cavendish was not to measure the gravitational constant but rather to measure the mass of the Earth through the precise knowledge of the gravitational interaction.

The accuracy of the measured value of <math> {G} \ </math> has increased only modestly since the original experiment of Cavendish. <math> {G} \ </math> is quite difficult to measure, as gravity is much weaker than other fundamental forces, and an experimental apparatus cannot be separated from the gravitational influence of other bodies. Furthermore, gravity has no established relation to other fundamental forces, so it does not appear possible to measure it indirectly. A recent review (Gillies, 1997) shows that published values of <math> {G} \ </math> have varied rather broadly, and some recent measurements of high precision are, in fact, mutually exclusive.

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The GM product

The <math> {GM} \ </math> product is the standard gravitational parameter <math> {\mu} \ </math>, according to the case also called the geocentric or heliocentric gravitational constant, among others. This gives a convenient simplification of various gravity-related formulas. Also, for the Earth and the Sun, the value of the product is known more accurately than each factor. (As a result, the accuracy to which the masses of the Earth and the Sun are known correspond to the accuracy to which <math> {G} \ </math> is known.)

In calculations of gravitational force in the solar system, it is the products which appear, so computations are more accurate using the standard gravitational parameters directly (or, correspondingly, using values for the masses and the gravitational constant which correspond, i.e., result in an accurate product, though not very accurate individually). In other words, because <math> GM \ </math> appear together, there really is no need to substitute values for each; rather use the more accurate measurement of their product, <math> \mu \ </math>, in place of <math> GM \ </math>.

<math> \mu = GM = 398 600.4418 \plusmn 0.0008 \ \mbox{km}^{3} \ \mbox{s}^{-2} </math> (for earth)

Also, calculations in celestial mechanics can be carried out using the unit of solar mass rather than the standard SI unit kilogram. In this case we use the Gaussian gravitational constant which is <math> {k^2} \ </math>, where

and

<math> {A} \ </math> is the astronomical unit

<math> {D} \ </math> is the mean solar day

<math> {S} \ </math> is the solar mass.

If instead of mean solar day we use the sidereal year as our time unit, the value is very close to <math>2 \pi \ </math>.

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The dimensions of G

The dimensions assigned to the gravitational constant (length cubed, divided by mass and by time squared) are those needed to make gravitational equations 'come out right'. However, these dimensions have fundamental significance in terms of Planck units: when expressed in SI units, the gravitational constant is dimensionally and numerically equal to the cube of the Planck length divided by the Planck mass and by the square of Planck time.

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References

George T. Gillies. "The Newtonian gravitational constant: recent measurements and related studies". Reports on Progress in Physics, 60:151-225, 1997. (A lengthy, detailed review. See Figure 1 and Table 2 in particular. Available online: PDF)

E. Myles Standish. "Report of the IAU WGAS Sub-group on Numerical Standards". In Highlights of Astronomy, I. Appenzeller, ed. Dordrecht: Kluwer Academic Publishers, 1995. (Complete report available online: PostScript. Tables from the report also available: Astrodynamic Constants and Parameters)

Jens H. Gundlach and Stephen M. Merkowitz. "Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback". Physical Review Letters, 85(14):2869-2872, 2000. (Also available online: PDF)

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