Discover

(Redirected from Universal gravitational constant)
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 $\{G\}$, 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.
The gravitational constant is perhaps the most difficult physical constant to measure. In SI units, the 2006 CODATA recommended value of the gravitational constant is
:$G\; =\; left(6.67428\; plusmn\; 0.0007$
ight) imes 10^{-11} mbox{m}^3 mbox{kg}^{-1} mbox{s}^{-2} , ^[1]
::$=\; left(6.67428\; plusmn\; 0.0007$
ight) imes 10^{-11} mbox{N} mbox{m}^2 mbox{kg}^{-2} ,
::$=\; left(6.67428\; plusmn\; 0.0007$
ight) imes 10^{-8} mbox{cm}^3 mbox{g}^{-1} mbox{s}^{-2} ,
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. In galactic scales, where distances are measured in (kilo-)parsecs, velocities in kilometers per second and masses in solar units, it is useful to express ''G'' as :
:$G\; =\; 4.5\; imes\; 10^\{-3\}\; mbox\{pc\}\; mbox\{M\_odot\}^\{-1\}\; mbox\{(km/s)\}^2\; ,$
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 meter apart is approximately 10^-67 newton, while the electromagnetic force between the same two particles still 1 meter 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. 10³⁹) greater than the force of gravity — which is even greater than the ratio between the mass of a human and the mass of the Solar System.

Contents

Measurement of the gravitational constant

The ''GM'' product

The dimensions of ''G''

Recent measurement

See also

References

External links

Measurement of the gravitational constant

The gravitational constant appears in Newton's law of universal gravitation, but it was not measured until 1798 — 71 years after Newton's death — by Henry Cavendish (''Philosophical Transactions'' 1798). Cavendish measured $\{G\}$ implicitly, using a torsion balance invented by Rev. geologist John Michell. 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 and density relative to water of the Earth through the precise knowledge of the gravitational interaction.
The accuracy of the measured value of $\{G\}$ has increased only modestly since the original experiment of Cavendish.
$\{G\}$ 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 $\{G\}$ have varied rather broadly, and some recent measurements of high precision are, in fact, mutually exclusive.^[2]

The ''GM'' product

The $\{GM\}$ product is the standard gravitational parameter $\{mu\}$, 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 $\{G\}$ 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 $GM$ appear together, there really is no need to substitute values for each; rather use the more accurate measurement of their product, $mu$, in place of $GM$.
: $mu\; =\; GM\; =\; 398\; 600.4418\; plusmn\; 0.0008\; mbox\{km\}^\{3\}\; mbox\{s\}^\{-2\}$ (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 $\{k^2\}$, where
:
: and
::$\{A\}$ is the astronomical unit
::$\{D\}$ is the mean solar day
::$\{S\}$ 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 $2\; pi$.

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.

Recent measurement

In the January 5, 2007 issue of ''Science'' (page 74), the report "Atom Interferometer Measurement of the Newtonian Constant of Gravity" (J. B. Fixler, G. T. Foster, J. M. McGuirk, and M. A. Kasevich) describes a new measurement of the gravitational constant. According to the abstract: "Here, we report a value of G = 6.693 x 10^–11 cubic meters per kilogram second squared, with a standard error of the mean of ±0.027 x 10^–11 and a systematic error of ±0.021 x 10^–11 cubic meters per kilogram second squared.".^[3]

See also

★ Cavendish experiment

★ Standard gravity

★ Planck units

★ Dirac large numbers hypothesis

★ Accelerating Universe

★

References

1. http://www.physics.nist.gov/cgi-bin/cuu/Value?bg
2. http://www.iop.org/EJ/abstract/0034-4885/60/2/001
3. http://www.sciencemag.org/cgi/content/abstract/315/5808/74?rss=1%5D%5B%5BREPORTS%5D


★ 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 link broken, as of 2007-09-08)''

★ CODATA recommended values of the fundamental physical constants: 2002, Peter J. Mohr and Barry N. Taylor, , , Reviews of Modern Physics, January 2005 Section Q (pp. 42–47) describes the mutually inconsistent measurement experiments from which the CODATA value for ''G'' was derived.

External links

★ CODATA Internationally recommended values of the Fundamental Physical Constants ''(at The NIST References on Constants, Units, and Uncertainty)''

★ The Controversy over Newton's Gravitational Constant — additional commentary on measurement problems

★ The Antikythera Calculator (Italian and English versions)

★ The Gravitational Constant