# 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.

$\displaystyle F = G \frac{m_1 m_2}{r^2}$

The constant of proportionality is called $\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

$\displaystyle G = \left(6.6742 \plusmn 0.001 \right) \times 10^{-11} \ \mbox{N} \ \mbox{m}^2 \ \mbox{kg}^{-2} \,$
$\displaystyle = \left(6.6742 \plusmn 0.001 \right) \times 10^{-11} \ \mbox{m}^3 \ \mbox{s}^{-2} \ \mbox{kg}^{-1} \,$

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

$\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 $\displaystyle {G} \$ has increased only modestly since the original experiment of Cavendish. $\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 $\displaystyle {G} \$ have varied rather broadly, and some recent measurements of high precision are, in fact, mutually exclusive.

## The GM product

The $\displaystyle {GM} \$ product is the standard gravitational parameter $\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 $\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 $\displaystyle GM \$ appear together, there really is no need to substitute values for each; rather use the more accurate measurement of their product, $\displaystyle \mu \$ , in place of $\displaystyle GM \$ .

$\displaystyle \mu = GM = 398 600.4418 \plusmn 0.0008 \ \mbox{km}^{3} \ \mbox{s}^{-2}$ (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 $\displaystyle {k^2} \$ , where

$\displaystyle {k = 0.01720209895 \ A^{\frac{3}{2}} \ D^{-1} \ S^{-\frac{1}{2}} } \$
and
$\displaystyle {A} \$ is the astronomical unit
$\displaystyle {D} \$ is the mean solar day
$\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 $\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.