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.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = G \frac{m_1 m_2}{r^2} }

The constant of proportionality is called Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {G} \ } , the gravitational constant, the universal gravitational constant, Newton's constant, and colloquially big G. The gravitational constant is a fundamental 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.

In SI units, the 2002 CODATA recommended value of the gravitational constant is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = \left(6.6742 \plusmn 0.001 \right) \times 10^{-11} \ \mbox{N} \ \mbox{m}^2 \ \mbox{kg}^{-2} \,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left(6.6742 \plusmn 0.001 \right) \times 10^{-11} \ \mbox{m}^3 \ \mbox{s}^{-2} \ \mbox{kg}^{-1} \,}

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

The gravitational force is relatively weak. As an example, two SUVs, each with a mass of 3000 kilograms and placed with their centers of gravity 3 metres apart, will attract each other with a force of about 67 micronewtons. This force is approximately equal to the weight of a large grain of sand.

Measurement of the gravitational constant

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {G} \ } 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {G} \ } has increased only modestly since the original experiment of Cavendish. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {G} \ } have varied rather broadly, and some recent measurements of high precision are, in fact, mutually exclusive.

The GM product

The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {GM} \ } product is the standard gravitational parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle GM \ } appear together, there really is no need to substitute values for each; rather use the more accurate measurement of their product, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu \ } , in place of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle GM \ } .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {k^2} \ } , where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {k = 0.01720209895 \ A^{\frac{3}{2}} \ D^{-1} \ S^{-\frac{1}{2}} } \ }
and
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {A} \ } is the astronomical unit
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {D} \ } is the mean solar day
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \pi \ } .

Planck units

By combining the gravitational constant with Planck's constant and the speed of light in vacuum, it is possible to create a system of units known as Planck units. The gravitational constant, the reduced Planck's constant (or Dirac's constant), and the speed of light all take the numerical value 1 in this system.

See also

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)

External links


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