# 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.

**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.

## Contents

## 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.

## 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

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

- 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)*

ca:Constant de la gravitació
da:Den universelle gravitationskonstant
de:Gravitationskonstante
es:Constante de gravitación universal
eo:Gravita konstanto
ko:중력상수
it:Costante di gravitazione universale
he:קבוע הכבידה
nl:Gravitatieconstante
ja:万有引力定数
pl:Stała grawitacji
pt:Constante gravitacional universal
ru:Гравитационная постоянная
sl:Gravitacijska konstanta
sv:Gravitationskonstant
zh:万有引力常数