TheInfoList

OR: The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an
empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences ...
physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, ...
involved in the calculation of gravitational effects in
Sir Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
's
law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance ...
and in
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's theory of general relativity. In Newton's law, it is the proportionality constant connecting the gravitational force between two bodies with the product of their
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
es and the inverse square of their
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over") ...
. In the Einstein field equations, it quantifies the relation between the geometry of spacetime and the energy–momentum tensor (also referred to as the stress–energy tensor). The measured value of the constant is known with some certainty to four significant digits. In
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
, 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 Henry Cavendish ( ; 10 October 1731 – 24 February 1810) was an English natural philosopher and scientist who was an important experimental and theoretical chemist and physicist. He is noted for his discovery of hydrogen, which he termed "infl ...
in a 1798 experiment.

# Definition

According to Newton's law of universal gravitation, the attractive
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
() between two point-like bodies is directly proportional to the product of their
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
es ( and ) and inversely proportional to the square of the distance, , between their centers of mass: $F=G\frac.$ The constant of proportionality, , 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 (equivalent to the free-fall acceleration). Where $M_\oplus$ is the mass of the Earth and $r_\oplus$ is the
radius of the Earth Earth radius (denoted as ''R''🜨 or R_E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly (equatorial radius, deno ...
, the two quantities are related by: $g = \frac.$ The gravitational constant appears in the Einstein field equations of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physic ...
, $G_ + \Lambda g_ = \kappa T_ \,,$ where is the Einstein tensor, is the
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field e ...
, is the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
, 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: $\kappa = \frac \approx 2.076647442844 \times 10^ \mathrm.$

# Value and uncertainty

The gravitational constant is a physical constant that is difficult to measure with high accuracy.. A lengthy, detailed review. See Figure 1 and Table 2 in particular. This is because the gravitational force is an extremely weak force as compared to other
fundamental forces In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: the gravitational and electro ...
at the laboratory scale. In SI units, the 2018 Committee on Data for Science and Technology (CODATA)-recommended value of the gravitational constant (with standard uncertainty in parentheses) is: $G = 6.67430(15) \times 10^$ This corresponds to a relative standard
uncertainty Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable o ...
of (22 ppm).

## Natural units

The gravitational constant is a defining constant in some systems of
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ...
, particularly geometrized unit systems, such as Planck units and Stoney units. When expressed in terms of such units, the value of the gravitational constant will generally have a numeric value of 1 or a value close to it. 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 Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the ...
, it is convenient to measure distances in
parsec The parsec (symbol: pc) is a unit of length used to measure the large distances to astronomical objects outside the Solar System, approximately equal to or (au), i.e. . The parsec unit is obtained by the use of parallax and trigonometry, a ...
s (pc), velocities in kilometres per second (km/s) and masses in solar units . In these units, the gravitational constant is: $G \approx 4.3009 \times 10^ \ ^ .$ For situations where tides are important, the relevant length scales are solar radii rather than parsecs. In these units, the gravitational constant is: $G \approx 1.90809\times 10^ \mathrm \, R_\odot M_\odot^ .$ In orbital mechanics, the period of an object in circular orbit around a spherical object obeys $GM=\frac ,$ where is the volume inside the radius of the orbit. It follows that :$P^2=\frac\frac\approx 10.896 \, \mathrm\frac.$ 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 3rd law, expressed in units characteristic of
Earth's orbit Earth orbits the Sun at an average distance of 149.60 million km (92.96 million mi) in a counterclockwise direction as viewed from above the Northern Hemisphere. One complete orbit takes  days (1 sidereal year), during which time Ear ...
: :$G = 4 \pi^2 \mathrm \ M^ \approx 39.478 \mathrm \ M_\odot^ ,$ where distance is measured in terms of the semi-major axis of Earth's orbit (the
astronomical unit The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbi ...
, AU), time in
year A year or annus is the orbital period of a planetary body, for example, the Earth, moving in its orbit around the Sun. Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by change in weather, the ...
s, 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 classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting eac ...
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: $1\ \mathrm = \left( \frac \mathrm^2 \right)^ \approx 1.495979 \times 10^\ \mathrm.$ 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 (also denoted ). 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 In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular or ...
, and in the formula for
escape velocity In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically ...
. This quantity gives a convenient simplification of various gravity-related formulas. The product is known much more accurately than either factor is. Calculations in
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
can also be carried out using the units of
solar mass The solar mass () is a standard unit of mass in astronomy, equal to approximately . It is often used to indicate the masses of other stars, as well as stellar clusters, nebulae, galaxies and black holes. It is approximately equal to the mass ...
es,
mean solar day Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of solar time is the day, based on the synodic rotation period. Two types of solar time are apparent solar time (sundial t ...
s and
astronomical unit The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbi ...
s rather than standard SI units. For this purpose, the Gaussian gravitational constant was historically in widespread use, , expressing the mean
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
of the Sun–Earth system measured in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s per
day A day is the time period of a full rotation of the Earth with respect to the Sun. On average, this is 24 hours, 1440 minutes, or 86,400 seconds. In everyday life, the word "day" often refers to a solar day, which is the length between two so ...
. The use of this constant, and the implied definition of the
astronomical unit The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbi ...
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 (although its notation as dates to the 1890s), but is not calculated in his '' Philosophiæ Naturalis Principia Mathematica'' where it postulates the
inverse-square law In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be underst ...
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 Edmond (or Edmund) Halley (; – ) was an English astronomer, mathematician and physicist. He was the second Astronomer Royal in Britain, succeeding John Flamsteed in 1720. From an observatory he constructed on Saint Helena in 1676–77, H ...
Schiehallion experiment The Schiehallion experiment was an 18th-century experiment to determine the mean density of the Earth. Funded by a grant from the Royal Society, it was conducted in the summer of 1774 around the Scottish mountain of Schiehallion, Perthshir ...
, 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 Charles Hutton FRS FRSE LLD (14 August 1737 – 27 January 1823) was a British mathematician and surveyor. He was professor of mathematics at the Royal Military Academy, Woolwich from 1773 to 1807. He is remembered for his calculation of the ...
(1778) suggested a density of ( times the density of water), about 20% below the modern value. This immediately led to estimates on the densities and masses of the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrare ...
,
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width ...
and planets, sent by Hutton to
Jérôme Lalande Joseph Jérôme Lefrançois de Lalande (; 11 July 1732 – 4 April 1807) was a French astronomer, freemason and writer. Biography Lalande was born at Bourg-en-Bresse (now in the département of Ain) to Pierre Lefrançois and Marie‐Anne� ...
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 settingBoys 1894
p.330 In this lecture before the Royal Society, Boys introduces ''G'' and argues for its acceptance. See
Poynting 1894
p. 4
MacKenzie 1900
p.vi
$G = g\frac = \frac.$ Based on this, Hutton's 1778 result is equivalent to ≈ . 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 Henry Cavendish ( ; 10 October 1731 – 24 February 1810) was an English natural philosopher and scientist who was an important experimental and theoretical chemist and physicist. He is noted for his discovery of hydrogen, which he termed "infl ...
. He determined a value for implicitly, using a torsion balance invented by the geologist Rev. John Michell (1753). He used a horizontal
torsion beam The twist-beam rear suspension (also torsion-beam axle, deformable torsion beam or compound crank) is a type of automobile suspension based on a large H or C-shaped member. The front of the H attaches to the body via rubber bushings, and the r ...
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. In spite of the experimental design being due to Michell, the experiment is now known as the
Cavendish experiment The Cavendish experiment, performed in 1797–1798 by English scientist Henry Cavendish, was the first experiment to measure the force of gravity between masses in the laboratory and the first to yield accurate values for the gravitational cons ...
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, ''ρ''🜨 = , corresponds to value of = . It is surprisingly accurate, about 1% above the modern value (comparable to the claimed 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 (1821, ),
Edward Sabine Sir Edward Sabine ( ; 14 October 1788 – 26 June 1883) was an Irish astronomer, geophysicist, ornithologist, explorer, soldier and the 30th president of the Royal Society. He led the effort to establish a system of magnetic observatories in ...
(1827, ), Carlo Ignazio Giulio (1841, ) and
George Biddell Airy Sir George Biddell Airy (; 27 July 18012 January 1892) was an English mathematician and astronomer, and the seventh Astronomer Royal from 1835 to 1881. His many achievements include work on planetary orbits, measuring the mean density of the ...
(1854, ). Cavendish's experiment was first repeated by Ferdinand Reich (1838, 1842, 1853), who found a value of , which is actually worse than Cavendish's result, differing from the modern value by 1.5%. Cornu and Baille (1873), found . Cavendish's experiment proved to result in more reliable measurements than pendulum experiments of the "Schiehallion" (deflection) type or "Peruvian" (period as a function of altitude) type. Pendulum experiments still continued to be performed, by Robert von Sterneck (1883, results between 5.0 and ) and
Thomas Corwin Mendenhall Thomas Corwin Mendenhall (October 4, 1841 – March 23, 1924) was an American autodidact physicist and meteorologist. He was the first professor hired at Ohio State University in 1873 and the superintendent of the U.S. Coast and Geodetic Surve ...
(1880, ). Cavendish's result was first improved upon by
John Henry Poynting John Henry Poynting FRS (9 September 185230 March 1914) was an English physicist. He was the first professor of physics at Mason Science College from 1880 to 1900, and then the successor institution, the University of Birmingham until his de ...
(1891), who published a value of , differing from the modern value by 0.2%, but compatible with the modern value within the cited standard uncertainty of 0.55%. In addition to Poynting, measurements were made by C. V. Boys (1895) and Carl Braun (1897), 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 (1898) 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.Sagitov, M. U., "Current Status of Determinations of the Gravitational Constant and the Mass of the Earth", Soviet Astronomy, Vol. 13 (1970), 712–718, translated from ''Astronomicheskii Zhurnal'' Vol. 46, No. 4 (July–August 1969), 907–915 (table of historical experiments p. 715). Arthur Stanley Mackenzie in ''The Laws of Gravitation'' (1899) reviews the work done in the 19th century. Poynting is the author of the article "Gravitation" in the ''Encyclopædia Britannica'' Eleventh Edition (1911). Here, he cites a value of = with an uncertainty of 0.2%.

## Modern value

Paul R. Heyl (1930) published the value of (relative uncertainty 0.1%), improved to (relative uncertainty 0.045% = 450 ppm) in 1942. Published values of derived from high-precision measurements since the 1950s have remained compatible with Heyl (1930), but within the relative uncertainty of about 0.1% (or 1,000 ppm) have varied rather broadly, and it is not entirely clear if the uncertainty has been reduced at all since the 1942 measurement. Some measurements published in the 1980s to 2000s were, in fact, mutually exclusive. Section Q (pp. 42–47) describes the mutually inconsistent measurement experiments from which the CODATA value for was derived. Establishing a standard value for with a standard uncertainty better than 0.1% has therefore remained rather speculative. By 1969, the value recommended by the
National Institute of Standards and Technology The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sc ...
(NIST) was cited with a standard uncertainty of 0.046% (460 ppm), lowered to 0.012% (120 ppm) by 1986. But the continued publication of conflicting measurements led NIST to considerably increase the standard uncertainty in the 1998 recommended value, by a factor of 12, to a standard uncertainty of 0.15%, larger than the one given by Heyl (1930). The uncertainty was again lowered in 2002 and 2006, but once again raised, by a more conservative 20%, in 2010, matching the standard uncertainty of 120 ppm published in 1986. For the 2014 update, CODATA reduced the uncertainty to 46 ppm, less than half the 2010 value, and one order of magnitude below the 1969 recommendation. The following table shows the NIST recommended values published since 1969: In the January 2007 issue of ''
Science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence ...
'', Fixler et al. described a measurement of the gravitational constant by a new technique,
atom interferometry An atom interferometer is an interferometer which uses the wave character of atoms. Similar to optical interferometers, atom interferometers measure the difference in phase between atomic matter waves along different paths. Atom interferometers ...
, reporting a value of , 0.28% (2800 ppm) higher than the 2006 CODATA value. An improved cold atom measurement by Rosi et al. was published in 2014 of . Although much closer to the accepted value (suggesting that the Fixler ''et al.'' measurement was erroneous), this result was 325 ppm below the recommended 2014 CODATA value, with non-overlapping standard uncertainty intervals. As of 2018, efforts to re-evaluate the conflicting results of measurements are underway, coordinated by NIST, notably a repetition of the experiments reported by Quinn et al. (2013). In August 2018, a Chinese research group announced new measurements based on torsion balances, and based on two different methods. These are claimed as the most accurate measurements ever made, with a standard uncertainties cited as low as 12 ppm. The difference of 2.7 σ between the two results suggests there could be sources of error unaccounted for.

# Suggested time-variation

A controversial 2015 study of some previous measurements of , by Anderson et al., suggested that most of the mutually exclusive values in high-precision measurements of ''G'' can be explained by a periodic variation. The variation was measured as having a period of 5.9 years, similar to that observed in length-of-day (LOD) measurements, hinting at a common physical cause that is not necessarily a variation in . A response was produced by some of the original authors of the measurements used in Anderson et al. This response notes that Anderson et al. not only omitted measurements, but that they also used the time of publication rather than the time the experiments were performed. A plot with estimated time of measurement from contacting original authors seriously degrades the length of day correlation. Also, consideration of the data collected over a decade by Karagioz and Izmailov shows no correlation with length of day measurements. As such, the variations in most likely arise from systematic measurement errors which have not properly been accounted for. Under the assumption that the physics of type Ia supernovae are universal, analysis of observations of 580 of them has shown that the gravitational constant has varied by less than one part in ten billion per year over the last nine billion years according to Mould et al. (2014).

* Gravity of Earth * Standard gravity * Gaussian gravitational constant * Orbital mechanics *
Escape velocity In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically ...
*
Gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work ( energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
* Gravitational wave * Strong gravitational constant * Dirac large numbers hypothesis * Accelerating universe * Lunar Laser Ranging experiment *
Cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field e ...

# References

Footnotes Citations

## Sources

* ''(Complete report available online
PostScriptPDF
Tables from the report also available
Astrodynamic Constants and Parameters
'' *