Gravitational constant

Value of

The gravitational constant is an empirical physical constant that gives the strength of the gravitational field induced by a mass. It is involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's theory of general relativity. It is also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant, denoted by the capital letter . It is contrastable with the Einstein gravitational constant, denoted by lowercase kappa.
In Newton's law, it is the proportionality constant connecting the gravitational force between two bodies with the product of their masses and the inverse square of their distance. In the Einstein field equations, it quantifies the relation between the geometry of spacetime and the stress–energy tensor.
The measured value of the constant is known with some certainty to four significant digits. In SI units, its value is approximately
The modern notation of Newton's law involving was introduced in the 1890s by C. V. Boys. The first implicit measurement with an accuracy within about 1% is attributed to Henry Cavendish in a 1798 experiment.

Definition

According to Newton's law of universal gravitation, the magnitude of the attractive force between two bodies each with a spherically symmetric density distribution is directly proportional to the product of their masses, and, and inversely proportional to the square of the distance,, directed along the line connecting their centres of mass:
The constant of proportionality,, in this non-relativistic formulation is the gravitational constant. Colloquially, the gravitational constant is also called "Big G", distinct from "small g", which is the local gravitational field of Earth. Where is the mass of Earth and is the radius of Earth, the two quantities are related by:
The gravitational constant is a constant term in the Einstein field equations of general relativity,
where is the Einstein tensor, is the cosmological constant, is the metric tensor, is the stress–energy tensor, and is the Einstein gravitational constant, a constant originally introduced by Einstein that is directly related to the Newtonian constant of gravitation:

Meaning

The units of, m³ kg⁻¹ s⁻², can be arranged for meaning in more than one way, to indicate the meaning of as the strength of gravity in the universe. One arrangement is N/ to indicate the force between two masses at a distance according to the inverse-square law. Another, cancelling a common mass unit, is / to indicate the acceleration due to gravity due to a mass at a distance, or the acceleration resulting at a point in space due to the amount of influence there from a distant mass.

Value and uncertainty

The gravitational constant is a physical constant that is difficult to measure with high accuracy. This is because the gravitational force is an extremely weak force as compared to other fundamental forces at the laboratory scale.
In SI units, the CODATA-recommended value of the gravitational constant is:
The relative standard uncertainty is.

Natural units

Due to its use as a defining constant in some systems of natural units, particularly geometrized unit systems such as Planck units and Stoney units, the value of the gravitational constant will generally have a numeric value of 1 or a value close to it when expressed in terms of those units. Due to the significant uncertainty in the measured value of G in terms of other known fundamental constants, a similar level of uncertainty will show up in the value of many quantities when expressed in such a unit system.

Orbital mechanics

In astrophysics, it is convenient to measure distances in parsecs, velocities in kilometres per second and masses in solar units. In these units, the gravitational constant is:
For situations where tides are important, the relevant length scales are solar radii rather than parsecs. In these units, the gravitational constant is:
In orbital mechanics, the period of an object in circular orbit around a spherical object obeys
where is the volume inside the radius of the orbit, and is the total mass of the two objects. It follows that
This way of expressing shows the relationship between the average density of a planet and the period of a satellite orbiting just above its surface.
For elliptical orbits, applying Kepler's third law, expressed in units characteristic of Earth's orbit:
where distance is measured in terms of the semi-major axis of Earth's orbit, time in years, and mass in the total mass of the orbiting system.
The above equation is exact only within the approximation of the Earth's orbit around the Sun as a two-body problem in Newtonian mechanics, the measured quantities contain corrections from the perturbations from other bodies in the Solar System and from general relativity.
From 1964 until 2012, however, it was used as the definition of the astronomical unit and thus held by definition:
Since 2012, the au is defined as exactly, and the equation can no longer be taken as holding precisely.
The quantity —the product of the gravitational constant and the mass of a given astronomical body such as the Sun or Earth—is known as the standard gravitational parameter. The standard gravitational parameter 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.
This quantity gives a convenient simplification of various gravity-related formulas. The product is known much more accurately than either factor is.

Body	Value	Relative uncertainty
Sun
Earth

Calculations in celestial mechanics can also be carried out using the units of solar masses, mean solar days and astronomical units rather than standard SI units. For this purpose, the Gaussian gravitational constant was historically in widespread use,, expressing the mean angular velocity of the Sun–Earth system. The use of this constant, and the implied definition of the astronomical unit discussed above, has been deprecated by the IAU since 2012.

History of measurement

Early history

The existence of the constant is implied in Newton's law of universal gravitation as published in the 1680s, but is not calculated in his Philosophiæ Naturalis Principia Mathematica where it postulates the inverse-square law of gravitation. In the Principia, Newton considered the possibility of measuring gravity's strength by measuring the deflection of a pendulum in the vicinity of a large hill, but thought that the effect would be too small to be measurable. Nevertheless, he had the opportunity to estimate the order of magnitude of the constant when he surmised that "the mean density of the earth might be five or six times as great as the density of water", which is equivalent to a gravitational constant of the order:
A measurement was attempted in 1738 by Pierre Bouguer and Charles Marie de La Condamine in their "Peruvian expedition". Bouguer downplayed the significance of their results in 1740, suggesting that the experiment had at least proved that the Earth could not be a hollow shell, as some thinkers of the day, including Edmond Halley, had suggested.
The Schiehallion experiment, proposed in 1772 and completed in 1776, was the first successful measurement of the mean density of the Earth, and thus indirectly of the gravitational constant. The result reported by Charles Hutton suggested a density of , about 20% below the modern value. This immediately led to estimates on the densities and masses of the Sun, Moon and planets, sent by Hutton to Jérôme Lalande for inclusion in his planetary tables. As discussed above, establishing the average density of Earth is equivalent to measuring the gravitational constant, given Earth's mean radius and the mean gravitational acceleration at Earth's surface, by setting
Based on this, Hutton's 1778 result is equivalent to.
File:Cavendish Torsion Balance Diagram.svg|thumb|Diagram of torsion balance used in the Cavendish experiment performed by Henry Cavendish in 1798, to measure G, with the help of a pulley, large balls hung from a frame were rotated into position next to the small balls.
The first direct measurement of gravitational attraction between two bodies in the laboratory was performed in 1798, seventy-one years after Newton's death, by Henry Cavendish. He determined a value for implicitly, using a torsion balance invented by the geologist Rev. John Michell. He used a horizontal torsion beam with lead balls whose inertia 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. In spite of the experimental design being due to Michell, the experiment is now known as the Cavendish experiment for its first successful execution by Cavendish.
Cavendish's stated aim was the "weighing of Earth", that is, determining the average density of Earth and the Earth's mass. His result,, corresponds to value of. It is remarkably accurate, being about 1% above the modern CODATA recommended value, consistent with the claimed relative standard uncertainty of 0.6%.

19th century

The accuracy of the measured value of has increased only modestly since the original Cavendish experiment. is quite difficult to measure because gravity is much weaker than other fundamental forces, and an experimental apparatus cannot be separated from the gravitational influence of other bodies.
Measurements with pendulums were made by Francesco Carlini, Edward Sabine, Carlo Ignazio Giulio and George Biddell Airy.
Cavendish's experiment was first repeated by Ferdinand Reich, who found a value of, less accurate than Cavendish's result, differing from the modern value by 1.5%. Cornu and Baille, found.
Cavendish's experiment proved to result in more reliable measurements than pendulum experiments of the "Schiehallion" type or "Peruvian" type. Pendulum experiments still continued to be performed, by Robert von Sterneck and Thomas Corwin Mendenhall.
Cavendish's result was first improved upon by John Henry Poynting, who published a value of, differing from the modern value by 0.2%, but compatible with the modern value within the cited relative standard uncertainty of 0.55%. In addition to Poynting, measurements were made by C. V. Boys and Carl Braun, with compatible results suggesting =. The modern notation involving the constant was introduced by Boys in 1894 and becomes standard by the end of the 1890s, with values usually cited in the cgs system. Richarz and Krigar-Menzel attempted a repetition of the Cavendish experiment using 100,000 kg of lead for the attracting mass. The precision of their result of was, however, of the same order of magnitude as the other results at the time.
Arthur Stanley Mackenzie in The Laws of Gravitation reviews the work done in the 19th century. Poynting is the author of the article "Gravitation" in the Encyclopædia Britannica Eleventh Edition. Here, he cites a value of = with a relative uncertainty of 0.2%.