Gravitational constant

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The gravitational constant G is a key quantity in Newton's law of universal gravitation.

The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal gravitational constant, Newton's constant, and colloquially Big G. It should not be confused with "little g" (g), which is the local gravitational field (equivalent to the local acceleration due to gravity), especially that at the Earth's surface; see Earth's gravity and Standard gravity.

According to the law of universal gravitation, the attractive force (F) between two bodies is proportional to the product of their masses (m₁ and m₂), and inversely proportional to the square of the distance (r) between them:

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

The constant of proportionality, G, is the gravitational constant.

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:

$G = \left(6.67428 \plusmn 0.00067 \right) \times 10^{-11} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}.$

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

Dimensions, units and magnitude

The dimensions assigned to the gravitational constant in the equation above — length cubed, divided by mass and by time squared (in SI units, metres cubed per kilogram per second squared) — are those needed to balance the units of measurements in gravitational equations. 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.

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 many secondary school texts, the dimensions of G are derived from force in order to assist student comprehension:

$G \approx 6.674 \times 10^{-11} {\rm \ N}\, {\rm (m/kg)^2}.$

In cgs, G can be written as:

$G\approx 6.674 \times 10^{-8} {\rm \ cm}^3 {\rm g}^{-1} {\rm s}^{-2}.$

In astrophysical use, when distances are measured in parsecs (pc), velocities in kilometers per second (km/s) and masses in solar units ( $M_\odot$ ), it is useful to express G as:

$G \approx 4.3 \times 10^{-3} {\rm \ pc}\, M_\odot^{-1} \, {\rm (km/s)}^2. \,$

The gravitational force is extremely weak compared with other fundamental forces. For example, the gravitational force between an electron and proton 1 meter apart is approximately 10 newton, while the electromagnetic force between the same two particles is approximately 10 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. 10) greater than the force of gravity — roughly the same ratio as the mass of the Sun compared to a microgram mass.

History of measurement

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 the geologist Rev. 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 value that he calculated, in SI units, was 6.754 × 10 m/kg/s
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. Published values of G have varied rather broadly, and some recent measurements of high precision are, in fact, mutually exclusive.

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 × 10 cubic meters per kilogram second squared, with a standard error of the mean of ±0.027 × 10 and a systematic error of ±0.021 × 10 cubic meters per kilogram second squared."

The GM product

The quantity GM — the product of the gravitational constant and the mass of a given astronomical body such as the Sun or the Earth — is known as the standard gravitational parameter and is denoted μ. Depending on the body concerned, it may also be called the geocentric or heliocentric gravitational constant, among other names.

This quantity gives a convenient simplification of various gravity-related formulas. Also, for many celestial bodies such as the Earth and the Sun, the value of the product GM is known more accurately than each factor independently. Indeed, the limited accuracy available for G often limits the accuracy of scientific determination of such masses in the first place.

For Earth, using M_⊕ as the symbol for the mass of the Earth, we have

$\mu = GM_\oplus = ( 398 600.4418 \plusmn 0.0008 ) \ \mbox{km}^{3} \ \mbox{s}^{-2}.$

Calculations in celestial mechanics can also 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, where

${k = 0.01720209895 \ A^{\frac{3}{2}} \ D^{-1} \ S^{-\frac{1}{2}} } \$

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 of k is very close to 2π (k = 6.28315).

The standard gravitational parameter GM appears as above in Newton's law of universal gravitation, as well as in formulas for the deflection of light caused by gravitational lensing, in Kepler's laws of planetary motion, and in the formula for escape velocity.

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Published in July 2009.

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