View source for Gravitational constant

{{short description|Physical constant relating the gravitational force between objects to their mass and distance}}
{{Use dmy dates|date=August 2019}}
{{Distinguish|text = g, the [[gravity of Earth]]}}
{{2019 SI redefinition|df=y}}
{|  class="wikitable" style="float: right;"
! Values of {{mvar|G}}
! Units
|-
| {{physconst|G|unit=no}} || [[metre|m]]<sup>3</sup>⋅[[kilogram|kg]]<sup>−1</sup>⋅[[second|s]]<sup>−2</sup>
|-
| {{val|4.30091|(25)|e=-3}} || [[parsec|pc]]⋅[[Solar mass|M<sub>⊙</sub>]]<sup>−1</sup>⋅([[kilometre|km]]/[[second|s]])<sup>2</sup>
|}

[[File:NewtonsLawOfUniversalGravitation.svg|thumb|right|300px|The gravitational constant {{math|''G''}} is a key quantity in [[Newton's law of universal gravitation]].]]

The '''gravitational constant''' (also known as the '''universal gravitational constant''', the '''Newtonian constant of gravitation''', or the '''Cavendish gravitational constant'''),{{efn|"Newtonian constant of gravitation" is the name introduced for ''G'' by Boys (1894). Use of the term by  T.E. Stern (1928) was misquoted as "Newton's constant of gravitation" in ''Pure Science Reviewed for Profound and Unsophisticated Students'' (1930), in what is apparently the first use of that term. Use of "Newton's constant" (without specifying "gravitation" or "gravity") is more recent, as "Newton's constant" was also
used for the [[heat transfer coefficient]] in [[Newton's law of cooling]], but has by now become quite common, e.g.
Calmet et al, ''Quantum Black Holes'' (2013), p. 93;  P. de Aquino, ''Beyond Standard Model Phenomenology at the LHC'' (2013), p. 3.

The name "Cavendish gravitational constant", sometimes "Newton–Cavendish gravitational constant", appears to have been common in the 1970s to 1980s, especially in (translations from) Soviet-era Russian literature, 
e.g. Sagitov (1970 [1969]), ''Soviet Physics: Uspekhi'' 30 (1987), Issues 1–6, p. 342 [etc.].
"Cavendish constant" and "Cavendish gravitational constant" is also used in 
Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, "Gravitation",  (1973), 1126f.

Colloquial use of "Big G", as opposed to "[[little g]]" for gravitational acceleration dates to the 1960s (R.W.  Fairbridge, ''The encyclopedia of atmospheric sciences and astrogeology'', 1967, p. 436; 
note use of "Big G's" vs. "little g's" as early as the 1940s of the [[Einstein tensor]] ''G''<sub>''μν''</sub> vs. the [[metric tensor]] ''g''<sub>''μν''</sub>,
''Scientific, medical, and technical books published in the United States of America: a selected list of titles in print with annotations: supplement of books published 1945–1948'', Committee on American Scientific and Technical Bibliography National Research Council, 1950, p. 26).}} denoted by the letter {{math|''G''}}, is an [[empirical]] [[physical constant]] involved in the calculation of [[gravitational]] effects  in [[Sir Isaac Newton]]'s [[Newton's law of universal gravitation|law of universal gravitation]] and in [[Albert Einstein]]'s [[general relativity|general theory of relativity]].

In Newton's law, it is the proportionality constant connecting the [[gravitational force]] between two bodies with the product of their [[mass]]es and the [[inverse-square law|inverse square]] of their [[distance]]. In the [[Einstein field equations]], it quantifies the relation between the geometry of spacetime and the energy&ndash;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]] its value is approximately <!--{{math|''G''}} = -->{{physconst|G|after=.|round=3}}

The modern notation of Newton's law involving {{math|''G''}} 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 [[Cavendish experiment|1798 experiment]].<ref>Cavendish determined the value of ''G'' indirectly, by reporting a value for the [[Earth's mass]], or the average density of Earth, as {{val|5.448|u=g.cm-3}}</ref>

==Definition==
According to  [[Newton's law of universal gravitation]], the attractive [[force]] ({{math|''F''}}) between two point-like bodies is directly proportional to the product of their [[mass]]es ({{math|''m''<sub>1</sub>}} and {{math|''m''<sub>2</sub>}}), and inversely proportional to the square of the distance, {{math|''r''}}, ([[inverse-square law]]) between them:
:<math>F =  G\frac{m_1m_2}{r^2} \,.</math>
The [[Proportionality (mathematics)|constant of proportionality]], {{math|''G''}}, is the gravitational constant.
Colloquially, the gravitational constant is also called "Big G", for disambiguation with "small g" ({{math|''g''}}), which is the [[Gravity of Earth|local gravitational field of Earth]] (equivalent to the free-fall acceleration).<ref>{{cite web |first1=Jens H. |last1=Gundlach |first2=Stephen M. |last2=Merkowitz |title=University of Washington Big G Measurement |work=Astrophysics Science Division |publisher=Goddard Space Flight Center |date=23 December 2002 |url=http://asd.gsfc.nasa.gov/Stephen.Merkowitz/G/Big_G.html |quote=Since Cavendish first measured Newton's Gravitational constant 200 years ago, "Big G" remains one of the most elusive constants in physics }}</ref><ref>{{cite book|title=Fundamentals of Physics|edition=8th |last1=Halliday |first1=David |last2=Resnick |first2=Robert |last3=Walker |first3=Jearl |isbn=978-0-470-04618-0 |page=336|title-link=Fundamentals of Physics |date=September 2007 }}</ref>
Where {{math|''M''<sub>⊕</sub>}} is the [[mass of the Earth]] and {{math|''r''<sub>⊕</sub>}} is the [[Earth radius|radius of the Earth]], the two quantities are related by:
:{{math|''g'' {{=}} {{sfrac|''GM''<sub>⊕</sub>|''r''{{sub|⊕}}{{sup|2}}}}}}.

In the [[Einstein field equations]] of [[general relativity]],<ref>{{cite book |title=Einstein's General Theory of Relativity: With Modern Applications in Cosmology |edition=illustrated |first1=Øyvind |last1=Grøn |first2=Sigbjorn |last2=Hervik |publisher=Springer Science & Business Media |year=2007 |isbn=978-0-387-69200-5 |page=180 |url=https://books.google.com/books?id=IyJhCHAryuUC&pg=PA180}}</ref><ref name="ein">{{cite journal |last=Einstein |first=Albert |title=The Foundation of the General Theory of Relativity |journal=[[Annalen der Physik]] |volume=354 |issue=7 |pages=769–822 |year=1916 |url=http://www.alberteinstein.info/gallery/science.html |doi=10.1002/andp.19163540702 |format=[[PDF]] |id= |accessdate= |bibcode=1916AnP...354..769E |archive-url=https://web.archive.org/web/20120206225139/http://www.alberteinstein.info/gallery/gtext3.html |archive-date=6 February 2012}}</ref>
:<math>R_{\mu \nu} - \tfrac{1}{2}R \, g_{\mu \nu}  = \frac{8 \pi G }{c^4} T_{\mu \nu} \,,</math> <!--Lambda set to 0 for simplicity  because it isn't relevant here-->
Newton's constant appears in the proportionality between the [[Ricci curvature tensor|spacetime curvature]] and the energy density component of the [[stress–energy tensor]]. The scaled gravitational constant, or [[Einstein's constant]], is:<ref>{{cite book |title= Introduction to General Relativity |first1=Ronald |last1=Adler |first2=Maurice |last2=Bazin |first3=Menahem |last3=Schiffer |publisher= McGraw-Hill |location= New York |year= 1975 |edition= 2nd |isbn= 978-0-07-000423-8 |page= 345}}</ref>{{efn|Depending on the choice of definition of the stress–energy tensor it can also be normalized as {{nowrap|{{math|''κ'' {{=}} {{sfrac|8π|''c''<sup>2</sup>}}''G''}} ≈ {{val|1.866|e=-26|u=m⋅kg<sup>−1</sup>}}}}.}}
:{{nowrap|{{math|''κ'' {{=}} {{sfrac|8π|''c''<sup>4</sup>}}''G''}} ≈ {{val|2.071|e=-43|u=s<sup>2</sup>⋅m<sup>−1</sup>⋅kg<sup>−1</sup>}}}}.

==Value and dimensions==
The gravitational constant is a physical constant that is difficult to measure with high accuracy.<ref name=gillies>{{cite journal|first=George T. |last=Gillies |title=The Newtonian gravitational constant: recent measurements and related studies |journal=Reports on Progress in Physics |date=1997 |volume=60 |issue=2 |pages= 151–225 |url=http://www.iop.org/EJ/abstract/0034-4885/60/2/001 |doi=10.1088/0034-4885/60/2/001|bibcode = 1997RPPh...60..151G }}. A lengthy, detailed review. See Figure 1 and Table 2 in particular.</ref> 
This is because the gravitational force is extremely weak as compared to other [[fundamental forces]].{{efn|For example, the gravitational force between an [[electron]] and [[proton]] one meter apart is approximately {{val|e=−67|ul=N}}, whereas the [[electromagnetic force]] between the same two particles is approximately {{val|e=−28|u=N}}. The electromagnetic force in this example is some 39 [[orders of magnitude]] (i.e. 10<sup>39</sup>) greater than the force of gravity—roughly the same ratio as the [[Solar mass|mass of the Sun]] to a microgram.}}

In [[International System of Units|SI]] units, the 2018 [[Committee on Data for Science and Technology|CODATA]]-recommended value of the gravitational constant (with [[standard uncertainty]] in parentheses) is:<ref name="2014 CODATA">{{cite journal |last1=Mohr |first1=Peter J. |last2=Newell |first2=David B. |last3=Taylor |first3=Barry N. |arxiv=1507.07956 |title=CODATA Recommended Values of the Fundamental Physical Constants: 2014 |date=21 July 2015 |doi=10.1103/RevModPhys.88.035009 |volume=88 |issue=3 |pages=035009 |journal=Reviews of Modern Physics |bibcode=2016RvMP...88c5009M}}</ref><ref>{{cite web|title=Newtonian constant of gravitation G|url=http://physics.nist.gov/cgi-bin/cuu/Value?bg |publisher=CODATA, NIST}}</ref>
:<math> G = 6.67430(15) \times 10^{-11} {\rm \ m^3 {\cdot} kg^{-1} {\cdot} s^{-2} }</math>
This corresponds to a relative standard [[Measurement uncertainty|uncertainty]] of {{val|2.2|e=-5}} (22 [[Parts per million|ppm]]).

The dimensions assigned to the gravitational constant  are [[force]] times [[length]] squared divided by [[mass]] squared; this is equivalent to  length cubed, divided by mass and by [[time]] squared:
:<math>[G] = \frac{[F][L]^2}{[M]^{2}} = \frac{[L]^3 }{[M][T]^{2}}</math>

In [[SI base units]], this amounts to  [[meter]]s cubed per [[kilogram]] per [[second]] squared:
:<math>{\rm N {\cdot} m^{2} {\cdot} kg^{-2} } =  {\rm m^3 {\cdot} kg^{-1} {\cdot} s^{-2} } .</math>
In [[centimetre-gram-second system of units|cgs]], {{math|''G''}} can be written as {{math|''G''}} ≈ {{val|6.674|e=-8|u=cm<sup>3</sup>⋅g<sup>−1</sup>⋅s<sup>−2</sup>}}.
<!-- "cm-g-hr" seems pretty random, is this used at all in any context?
{{math|''G''}} can also be given as: <math> G\approx 0.8650 {\rm \ cm^3 {\cdot} g^{-1} {\cdot} hr^{-2} }.</math>
-->

===Natural units===
{{further|Natural units|Gravitational coupling constant}}
The gravitational constant is taken as the basis of the [[Planck units]]: it is equal to the cube of the [[Planck length]] divided by the product of the [[Planck mass]] and the square of [[Planck time]]:
:<math>G= \frac{l_{\rm P}^3}{m_{\rm P} t_{\rm P}^2}.</math>
In other words, in Planck units, {{math|''G''}} has the numerical value of {{val|1}}.

Thus, in Planck units, and other natural units taking  {{math|''G''}} as their basis, the value of the gravitational constant cannot be measured as this is set by definition. Depending on the choice of units, uncertainty in the value of a physical constant as expressed in one system of units shows up as uncertainty of the value of another constant in another system of units.  Where there is [[time-variation of physical constants|variation]] in [[dimensionless physical constant]]s, no matter which choice of physical "constants" is used to define the units, this variation is preserved independently of the choice of units; in the case of the gravitational constant, such a dimensionless value is the [[gravitational coupling constant]] of the electron,
:<math>\alpha_\text{G} = \frac{G m_\text{e}^2}{\hbar c} = \left( \frac{m_\text{e}}{m_\text{P}} \right)^2 \approx 1.7518 \times 10^{-45} </math>,
a measure for the gravitational attraction between a pair of [[electron]]s, proportional to the square of the [[electron rest mass]].

===Orbital mechanics===
{{see|Standard gravitational parameter|orbital mechanics|celestial mechanics|Gaussian gravitational constant|Earth mass|Solar mass}}
In [[astrophysics]], it is convenient to measure  distances  in [[parsec]]s (pc), velocities in kilometers per second (km/s) and masses in [[Solar mass|solar units]] {{math|''M''{{sub|⊙}}}}. In these units, the gravitational constant is:
:<math> G \approx 4.302 \times 10^{-3} {\rm \ pc}\ M_\odot^{-1} {\rm \ (km/s)^2 }. \, </math>
For situations where tides are important, the relevant length scales are [[solar radius|solar radii]] rather than parsecs. In these units, the gravitational constant is:
:<math> G \approx 1.90809\times 10^{5} R_\odot M_\odot^{-1} {\rm \ (km/s)^2 }. \, </math>
In [[orbital mechanics]], the period {{math|''P''}} of an object in circular orbit around a spherical object obeys
:<math> GM=\frac{3\pi V}{P^2}</math>
where {{math|''V''}} is the volume inside the radius of the orbit.
It follows that
:<math> P^2=\frac{3\pi}{G}\frac{V}{M}\approx 10.896 \ {\rm hr^2 {\cdot} g {\cdot} cm^{-3} }\frac{V}{M}.</math>
This way of expressing {{math|''G''}} 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 laws of planetary motion#Third law|Kepler's 3rd law]], expressed in units characteristic of  [[Earth's orbit]]:
:<math> G = 4 \pi^2  {\rm \ AU^3 {\cdot} yr^{-2}} \ M^{-1} \approx 39.478 {\rm \ AU^3 {\cdot} yr^{-2}} \ M_\odot^{-1} ,</math> 
where distance is measured in terms of the [[semi-major axis]] of Earth's orbit (the [[astronomical unit]], AU), time in [[solar year|year]]s, and mass in the total mass of the orbiting system ({{math|1=''M'' = {{solar mass}} + {{earth mass}} + {{lunar mass|sym=yes}}}}<ref>
{{mvar|M}}  ≈ 1.000003040433 {{math|{{solar mass}}}}, so that {{mvar|M}} {{=}} {{math|{{solar mass}}}} can be used for accuracies of five or fewer significant digits.</ref>).

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: 
:<math> 1\ {\rm AU} = \left( \frac{GM}{4 \pi^2} {\rm yr}^2 \right)^{\frac{1}{3}} \approx 1.495979 \times 10^{11} {\rm m}.</math> <!--(1.3271244002e+20 * 1.000003040433 * 86400^2 * 365.25636^2)^(1/3) = 1.4959788e+11-->
Since 2012, the AU is defined as {{val|1.495978707|e=11|u=m}} exactly, and the equation can no longer be taken as holding precisely.

The quantity {{math|''GM''}}—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 and (also denoted {{math|''μ''}}). 
The standard gravitational parameter {{math|''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]].

This quantity gives a convenient simplification of various gravity-related formulas. The product {{math|''GM''}} is known much more accurately than either factor is.
:{| class=wikitable
|+ Values for '''GM'''
! Body || {{math|1=''μ'' = ''GM''}} || Value || Precision
|-
| [[Sun]] || {{math|''G''{{solar mass}}}} || {{val|1.32712440018|(9)|e=20|u=m<sup>3</sup>⋅s<sup>−2</sup>}}<ref name="Astrodynamic Constants"> {{cite web
| title = Astrodynamic Constants
| date = 27 February 2009
| publisher = [[NASA]]/[[Jet Propulsion Laboratory|JPL]]
| url = http://ssd.jpl.nasa.gov/?constants
| accessdate = 27 July 2009
}}
</ref>||10 digits
|-
| [[Earth]] || {{math|''G''{{earth mass}}}} || {{val|3.986004418|(8)|e=14|u=m<sup>3</sup>⋅s<sup>−2</sup>}}<ref name="IAU best estimates">{{cite web|title=Geocentric gravitational constant |work=Numerical Standards for Fundamental Astronomy |url=http://maia.usno.navy.mil/NSFA/NSFA_cbe.html#GME2009 |via=maia.usno.navy.mil |publisher=IAU Working Group|accessdate=31 October 2017}} Citing
*{{cite journal|vauthors=Ries JC, Eanes RJ, Shum CK, Watkins MM, |title=Progress in the determination of the gravitational coefficient of the Earth |journal=Geophysical Research Letters|date=20 March 1992 |volume=19 |issue=6 |doi=10.1029/92GL00259 |bibcode=1992GeoRL..19..529R |pages=529–531}}</ref> || 9 digits
|}

Calculations in [[celestial mechanics]] can also be carried out using the units of [[solar mass]]es, [[mean solar day]]s and [[astronomical unit]]s rather than standard SI units. For this purpose, the 
[[Gaussian gravitational constant]] was historically in widespread use, {{nowrap|{{mvar|k}} {{=}} {{val|0.01720209895}}}}, expressing the mean [[angular velocity]] of the Sun–Earth system measured in [[radian]]s per [[day]]. The use of this constant, and the implied definition of the [[astronomical unit]] discussed above, has been deprecated by the [[IAU]] in 2012.

==History of measurement==
{{see|Earth mass|Schiehallion experiment|Cavendish experiment}}

===Early history===
Between 1640 and 1650, [[Francesco Maria Grimaldi|Grimaldi]] and [[Giovanni Battista Riccioli|Riccioli]] had discovered that the distance covered by objects in [[free fall]] was proportional to the square of the time taken, which led them to attempt a calculation of the gravitational constant by recording the oscillations of a [[pendulum]].<ref>J.L. Heilbron, ''Electricity in the 17th and 18th Centuries: A Study of Early Modern Physics'' (Berkeley: University of California Press, 1979), 180.</ref>

The existence of the constant is implied in  [[Newton's law of universal gravitation]] as published in the 1680s (although its notation as {{math|''G''}} dates to the 1890s),<ref name=BoysG/> but is not [[Algebra#Algebra as a branch of mathematics|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.<ref name="Davies">{{Cite journal|last=Davies|first=R.D.|title=A Commemoration of Maskelyne at Schiehallion|journal=Quarterly Journal of the Royal Astronomical Society|volume=26|issue=3|pages=289–294|bibcode=1985QJRAS..26..289D|date=1985}}</ref>  
Nevertheless, he estimated 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:<ref>"Sir Isaac Newton thought it probable, that the mean density of the earth might be five or six times as great as the density of water; and we have now found, by experiment, that it is very little less than what he had  thought it to be: so much justness was even in the surmises of this wonderful man!" Hutton (1778), p. 783</ref>

:{{math|''G''}} ≈ {{val|7|1|e=-11|u=m<sup>3</sup>⋅kg<sup>–1</sup>⋅s<sup>−2</sup>}}

A measurement was attempted in 1738 by  [[Pierre Bouguer]] and [[Charles Marie de La Condamine]], in their "[[French Geodesic Mission|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 Earth|hollow shell]], as some thinkers of the day, including [[Edmond Halley]], had suggested.<ref name="Poynting_p50-56">{{Cite book|last=Poynting|first=J.H.|title=The Earth: its shape, size, weight and spin|publisher=Cambridge|date=1913 |pages=50–56 |url=https://books.google.com/books?id=whA9AAAAIAAJ&pg=PA50}}</ref>

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]] (1778) suggested a density of  {{val|4.5|u=g/cm<sup>3</sup>}} (4{{frac|1|2}} times the density of water), about 20% below the modern value.<ref name="Hutton">{{Cite journal|last=Hutton|first=C. |date=1778 |title=An Account of the Calculations Made from the Survey and Measures Taken at Schehallien |journal=Philosophical Transactions of the Royal Society |volume=68 |pages=689–788 |doi=10.1098/rstl.1778.0034|url=http://rstl.royalsocietypublishing.org/content/68/689.citation}}</ref>
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 radius#Mean radius|Earth's mean radius]] and the [[little g|mean gravitational acceleration]] at Earth's surface, by setting
<!--modern values: g=9.80665 ms^-2, Re=  6.3781e+6 m
3*g/(4*pi*Re)=3.6706e-7
3.6706e-7/5.448e3=6.7375e-11
the "correct" value (for G=6.674e-11) would be 5.500 gcm^-3.
-->
:<math>G = g\frac{R_\oplus^2}{M_\oplus} = \frac{3g}{4\pi R_\oplus\rho_\oplus}.</math><ref name=BoysG>[https://books.google.com/books?id=ZrloHemOmUEC&pg=PA353 Boys 1894], p.330 In this lecture before the Royal Society, Boys introduces ''G'' and argues for its acceptance. See:
[https://books.google.com/books?id=dg0RAAAAIAAJ&pg=PA4 Poynting 1894], p.&nbsp;4, [https://books.google.com/books?id=O58mAAAAMAAJ&pg=PA1 MacKenzie 1900], p.vi</ref>
Based on this, Hutton's 1778 result is equivalent to  {{math|''G''}} ≈ {{val|8|e=-11|u=m<sup>3</sup>⋅kg<sup>–1</sup>⋅s<sup>−2</sup>}}.

[[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]].<ref>Published in ''[[Philosophical Transactions of the Royal Society]]'' (1798); reprint: Cavendish, Henry (1798). "Experiments to Determine the Density of the Earth". In MacKenzie, A. S., ''Scientific Memoirs'' Vol. 9: ''The Laws of Gravitation''. American Book Co. (1900), pp. 59–105.</ref>
He  determined a value for {{math|''G''}} implicitly, using a [[Torsion spring#Torsion balance|torsion balance]] invented by the geologist Rev. [[John Michell]] (1753). 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. 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, ''&rho;''<sub>⊕</sub> = {{val|5.448|(33)|u=g·cm<sup>&minus;3</sup>}}, corresponds to value of {{math|''G''}} = {{val|6.74|(4)|e=-11|u=m<sup>3</sup>⋅kg<sup>–1</sup>⋅s<sup>−2</sup>}}. It is surprisingly accurate, about 1% above the modern value (comparable to the claimed standard uncertainty of 0.6%).<ref>2014 [[CODATA]] value {{nowrap|{{val|6.674|e=−11|u=m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup>}}}}.</ref>

===19th century===
The accuracy of the measured value of {{math|''G''}} has increased only modestly since the original Cavendish experiment.<ref>
{{cite book|last1=Brush |first1=Stephen G. |last2=Holton |first2=Gerald James |title=Physics, the human adventure: from Copernicus to Einstein and beyond |publisher=Rutgers University Press |location=New Brunswick, NJ |date=2001 |pages= 137 |isbn=978-0-8135-2908-0 |oclc= |doi= }}
{{cite web |first=Jennifer Lauren |last=Lee |title=Big G Redux: Solving the Mystery of a Perplexing Result |date=16 November 2016 |publisher=NIST |url=https://www.nist.gov/news-events/news/2016/11/big-g-redux-solving-mystery-perplexing-result}}</ref>
{{math|''G''}} 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. Furthermore, gravity has no established relation to other fundamental forces, so it does not appear possible to calculate it indirectly from other constants that can be measured more accurately, as is done in some other areas of physics.

Measurements with pendulums were made by  [[Francesco Carlini]] (1821,  {{val|4.39|u=g/cm<sup>3</sup>}}), [[Edward Sabine]] (1827, {{val|4.77|u=g/cm<sup>3</sup>}})  Carlo Ignazio Giulio (1841, {{val|4.95|u=g/cm<sup>3</sup>}}) and  [[George Biddell Airy]] (1854, {{val|6.6|u=g/cm<sup>3</sup>}}).<ref>{{cite book
  | last = Poynting
  | first = John Henry
    | title = The Mean Density of the Earth
  | publisher = Charles Griffin
  | date = 1894
  | location = London
  | pages = 22–24
  | url = https://books.google.com/books?id=dg0RAAAAIAAJ&pg=PA23
  | doi = 
  | id = 
  | isbn = }}</ref>

Cavendish's experiment was first repeated by [[Ferdinand Reich]] (1838, 1842, 1853), who found a value of {{val|5.5832|(149)|u=g·cm<sup>&minus;3</sup>}},<ref>F. Reich,  ''On the Repetition of the Cavendish Experiments for Determining the  mean density of the Earth" ''Philosophical Magazine''  12: 283–284.''</ref> which is actually worse than Cavendish's result, differing from the modern value by 1.5%.
Cornu and Baille (1873), found {{val|5.56|u=g·cm<sup>&minus;3</sup>}}.<ref>Mackenzie (1899), p. 125.</ref>

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 {{val|6.3|u=g/cm<sup>3</sup>}}) and   [[Thomas Corwin Mendenhall]] (1880,  {{val|5.77|u=g/cm<sup>3</sup>}}).<ref>A.S. Mackenzie , ''The Laws of Gravitation'' (1899), [https://archive.org/stream/lawsgravitation01newtgoog#page/n140/mode/2up 127f.]</ref>

Cavendish's result was first improved upon by [[John Henry Poynting]] (1891)<ref>{{Cite book|url=http://archive.org/details/meandensityofear00poynuoft|title=The mean density of the earth|last=Poynting|first=John Henry|date=1894|publisher=London|others=Gerstein - University of Toronto}}</ref>, who published a value of {{val|5.49|(3)|u=g·cm<sup>&minus;3</sup>}}, 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)<ref>C.V. Boys, ''Phil. Trans. Roy. Soc.'' A. Pt. 1.  (1895).</ref> and [[Carl Braun (astronomer)|Carl Braun]]<!--[[:de:Carl Braun (Astronom)]]--> (1897),<ref>Carl Braun, ''Denkschriften der k. Akad. d. Wiss. (Wien), math. u. naturwiss. Classe'', 64 (1897).
Braun (1897) quoted an optimistic standard uncertainty of 0.03%, {{val|6.649|(2)|e=−11|u=m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup>}} but his result was significantly worse than the 0.2% feasible at the time.</ref> with compatible results suggesting {{math|''G''}} = {{val|6.66|(1)|e=−11|u=m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup>}}.
The modern notation involving the constant {{math|''G''}} was introduced by Boys in 1894<ref name=BoysG/> 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&nbsp;kg of lead for the attracting mass. The precision of their result of {{val|6.683|(11)|e=-11|u=m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup>}}
was, however,  of the same order of magnitude as the other results at the time.<ref name=Sagitov>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).</ref>

[[Arthur Stanley Mackenzie]] in ''The Laws of Gravitation'' (1899) reviews the work done in the 19th century.<ref>Mackenzie, A. Stanley, ''[https://archive.org/stream/lawsgravitation01newtgoog#page/n6/mode/2up The laws of gravitation; memoirs by Newton, Bouguer and Cavendish, together with abstracts of other important memoirs]'', American Book Company (1900 [1899]).</ref>
Poynting is the author of the article  "Gravitation" in the [[Encyclopædia Britannica Eleventh Edition|''Encyclopædia Britannica'' Eleventh Edition]] (1911). Here, he cites a value of  {{math|''G''}} = {{val|6.66|e=−11|u=m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup>}} with an uncertainty of 0.2%.

===Modern value===
[[Paul R. Heyl]] (1930) published the value of {{val|6.670|(5)|e=−11|u=m<sup>3</sup>⋅kg<sup>–1</sup>⋅s<sup>−2</sup>}} (relative uncertainty 0.1%),<ref>{{cite journal |first=P. R. |last=Heyl |authorlink=Paul R. Heyl |url=https://nvlpubs.nist.gov/nistpubs/jres/5/jresv5n6p1243_A2b.pdf |title=A redetermination of the constant of gravitation |journal=National Bureau of Standards Journal of Research |volume=5 |issue=6 |year=1930 |pages=1243–1290|doi=10.6028/jres.005.074 }}<!--Also  https://archive.org/details/redeterminationo56124heyl, and a shorter version at https://europepmc.org/articles/PMC1085130--></ref>
improved to  {{val|6.673|(3)|e=−11|u=m<sup>3</sup>⋅kg<sup>–1</sup>⋅s<sup>−2</sup>}} (relative uncertainty 0.045% = 450 ppm) in 1942.<ref>P. R. Heyl and P. Chrzanowski (1942), cited after Sagitov (1969:715).</ref>

Published values of {{mvar|G}} 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.<ref name=gillies/><ref name=codata2002>{{cite journal|first1=Peter J. |last1=Mohr |first2=Barry N. |last2=Taylor |title=CODATA recommended values of the fundamental physical constants: 2002 |journal=Reviews of Modern Physics |date=January 2005 |volume=77 |issue=1 | pages=1–107 |url=http://www.atomwave.org/rmparticle/ao%20refs/aifm%20refs%20sorted%20by%20topic/other%20rmp%20articles/CODATA2005.pdf |accessdate=1 July 2006 |doi=10.1103/RevModPhys.77.1 |bibcode=2005RvMP...77....1M |citeseerx=10.1.1.245.4554 |url-status=dead |archive-url=https://web.archive.org/web/20070306174141/http://www.atomwave.org/rmparticle/ao%20refs/aifm%20refs%20sorted%20by%20topic/other%20rmp%20articles/CODATA2005.pdf |archive-date=6 March 2007}} Section Q (pp. 42–47) describes the mutually inconsistent measurement experiments from which the CODATA value for {{mvar|G}} was derived.</ref>  
Establishing a standard value for {{mvar|G}} 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]] (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 radically 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&nbsp;ppm published in 1986.<ref>{{Cite journal|url = http://physics.nist.gov/cuu/pdf/RevModPhysCODATA2010.pdf|title = CODATA recommended values of the fundamental physical constants: 2010|date = 13 November 2012|journal = Reviews of Modern Physics |doi = 10.1103/RevModPhys.84.1527|pmid = |access-date = |bibcode=2012RvMP...84.1527M|arxiv = 1203.5425 |volume=84 |issue = 4|pages=1527–1605|last1 = Mohr|first1 = Peter J.|last2 = Taylor|first2 = Barry N.|last3 = Newell|first3 = David B.|citeseerx = 10.1.1.150.3858}}</ref> 
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:
[[File:Gravitational constant historical.png|thumb|350px|Timeline of measurements and recommended values for ''G'' since 1900: values recommended based on a literature review are shown in red, individual torsion balance experiments in blue, other types of experiments in green.]]
{|class=wikitable
! Year !! ''G'' <br />(10<sup>−11</sup>·m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup>) ||  Standard<br />uncertainty<br />(ppm) || Ref.
|-
|1969|| 6.6732(31)  || 460 || <ref>B. N. Taylor, W. H. Parker, and D. N. Langenberg (1969) Rev. Mod. Phys. '''41'''(3), 375–496</ref>
|-
|1973|| 6.6720(49)  || 730 || <ref>E. R. Cohen and B. N. Taylor (1973) J. Phys. Chem. Ref. Data '''2'''(4), 663–734 [699]</ref>
|-
|1986|| 6.67449(81) || 120 || <ref>E. R. Cohen and B. N. Taylor (1987) Rev. Mod. Phys. '''59'''(4), 1121–1148</ref>
|-
|1998|| 6.673(10) || 1,500 || <ref>P. J. Mohr and B. N. Taylor (2000) Rev. Mod. Phys. '''72'''(2), 351–495</ref>
|-
|2002|| 6.6742(10)  || 150 || <ref>P. J. Mohr and B. N. Taylor (2005) Rev. Mod. Phys. '''77'''(1), 1–107</ref>
|-
|2006|| 6.67428(67) || 100 || <ref>P. J. Mohr, B. N. Taylor, and D. B. Newell (2008) J. Phys. Chem. Ref. Data '''37'''(3), 1187–1284</ref>
|-
|2010|| 6.67384(80) || 120 || <ref>P. J. Mohr, B. N. Taylor, and D. B. Newell (2012) J. Phys. Chem. Ref. Data '''41'''</ref>
|-
|2014|| 6.67408(31) ||  46 || <ref>P. J. Mohr, D. B. Newell, and B. N. Taylor (2016) J. Phys. Chem. Ref. Data '''45'''</ref>
|-
|2018|| 6.67430(15) ||  22 || <ref>Eite Tiesinga, Peter J. Mohr, David B. Newell, and Barry N. Taylor (2019), "[http://physics.nist.gov/constants The 2018 CODATA Recommended Values of the Fundamental Physical Constants]" (Web Version 8.0). Database developed by J. Baker, M. Douma, and S. Kotochigova. National Institute of Standards and Technology, Gaithersburg, MD 20899.</ref>
|-
|}
In the January 2007 issue of ''[[Science (journal)|Science]]'', Fixler et al. described a measurement of the gravitational constant by a new technique, [[atom interferometry]], reporting a value of {{nowrap|1={{mvar|G}} = {{val|6.693|(34)|e=−11|u=m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup>}}}}, 0.28% (2800 ppm) higher than the 2006 CODATA value.<ref>{{Cite journal |first1=J. B. |last1=Fixler |first2=G. T. |last2=Foster |first3=J. M. |last3=McGuirk |first4=M. A. |last4=Kasevich |title=Atom Interferometer Measurement of the Newtonian Constant of Gravity |date=5 January 2007 |volume=315 |issue=5808 |pages=74–77 |doi=10.1126/science.1135459 |journal=Science |pmid=17204644 |bibcode=2007Sci...315...74F }}</ref>  An improved cold atom measurement by Rosi et al. was published in 2014 of {{nowrap|1={{mvar|G}} = {{val|6.67191|(99)|e=−11|u=m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup>}}}}.<ref>{{cite journal
 |last1=Rosi |first1=G. |last2=Sorrentino |first2=F. |last3=Cacciapuoti |first3=L.
 |last4=Prevedelli |first4=M. |last5=Tino |first5=G. M.
 |title=Precision measurement of the Newtonian gravitational constant using cold atoms
 |journal=Nature |volume=510 |issue=7506  |date=26 June 2014 |pages=518–521
 |doi=10.1038/nature13433 |pmid=24965653  |arxiv=1412.7954
 |url=http://www2.fisica.unlp.edu.ar/materias/FisGral2semestre2/Rosi.pdf}}<br />
{{cite journal
 |last1=Schlamminger |first1=Stephan
 |title=Fundamental constants: A cool way to measure big G
 |journal=Nature |volume=510 |issue=7506 |pages=478–480 |date=18 June 2014
 |doi=10.1038/nature13507 |doi-access=free |bibcode=2014Natur.510..478S |pmid=24965646
 |url=https://www.nature.com/articles/nature13507.pdf
}}</ref>  Although much closer to the accepted value (suggesting that the Fixler ''et. al.'' measurement was erroneous), this result was 325&nbsp;ppm below the recommended 2014 CODATA value, with non-overlapping [[standard uncertainty]] intervals.
<!-- 6.67191(99) vs. 6.67408(31) [2014], a difference of 0.00217(104). Also *barely* not overlapping with the
2010 interval, 6.67384(80) [2010] (differences 0.00193(127) and 0.00024(86)). 
This doesn't mean anything beyond "2-sigma effect" until the experiment is repeated.
-->

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).<ref>{{cite journal |author1=C. Rothleitner |author2=S. Schlamminger |title=Invited Review Article: Measurements of the Newtonian constant of gravitation, G |journal=Review of Scientific Instruments |volume=88 |id=111101 |year=2017 |doi=10.1063/1.4994619
|quote=However, re-evaluating or repeating experiments that have already been performed may provide insights into hidden biases or dark uncertainty. NIST has the unique opportunity to repeat the experiment of Quinn et al. [2013] with an almost identical setup. By mid-2018, NIST researchers will publish their results and assign a number as well as an uncertainty to their value.}} Referencing:
* {{cite journal |author1=T. Quinn |author2=H. Parks |author3=C. Speake |author4=R. Davis |title=Improved determination of G using two methods |journal=Phys. Rev. Lett. |volume=111 |id=101102 |year=2013 |doi=10.1103/PhysRevLett.111.101102 |bibcode=2013PhRvL.111j1102Q |url=https://www.bipm.org/utils/en/pdf/PhysRevLett.111.101102.pdf}}
The 2018 experiment was described by {{cite conference |author=C. Rothleitner |url=https://www.bipm.org/cc/CODATA-TGFC/Allowed/2015-02/Rothleitner.pdf |title=Newton’s Gravitational Constant 'Big' G – A proposed Free-fall Measurement |conference=CODATA Fundamental Constants Meeting, Eltville – 5 February 2015}}</ref>

In August 2018, a Chinese research group announced new measurements based on torsion balances, 
{{val|6.674184|(78)|e=−11|u=m<sup>3</sup>⋅kg<sup>–1</sup>⋅s<sup>−2</sup>}} and {{val|6.674484|(78)|e=−11|u=m<sup>3</sup>⋅kg<sup>–1</sup>⋅s<sup>−2</sup>}} based on two different methods.<ref>{{cite journal|first=Qing |last=Li |display-authors=etal |title=Measurements of the gravitational constant using two independent methods |journal=Nature |volume=560 |issue=7720 |pages=582–588 |year=2018 |doi=10.1038/s41586-018-0431-5|pmid=30158607 }}.
See also: {{cite news|url=https://www.techexplorist.com/physicists-precise-measurement-ever-gravitys-strength/16643/|title=Physicists just made the most precise measurement ever of Gravity’s strength|date=31 August 2018|accessdate=13 October 2018}}</ref> 
These are claimed as the most accurate measurements ever made, with a standard uncertainties cited as low as 12&nbsp;ppm. The difference of 2.7[[standard deviation|σ]] between the two results suggests there could be sources of error unaccounted for.

==Suggested time-variation==
{{see|Time-variation of fundamental constants}}
A controversial 2015 study of some previous measurements of {{math|''G''}}, by Anderson et al., suggested that most of the mutually exclusive values in high-precision measurements of ''G'' can be explained by a [[time-variation of physical constants|periodic variation]].<ref name=anderson2015>{{cite journal|first1=J. D. |last1=Anderson |first2=G. |last2=Schubert|first3=3=V. |last3=Trimble|first4=M. R. |last4=Feldman |title=Measurements of Newton's gravitational constant and the length of day |journal=EPL |date=April 2015 |volume=110 |issue=1 |url=http://iopscience.iop.org/0295-5075/110/1/10002/pdf/0295-5075_110_1_10002.pdf |doi=10.1209/0295-5075/110/10002|arxiv = 1504.06604 |bibcode = 2015EL....11010002A |pages=10002}}</ref> 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 which is not necessarily a variation in {{math|''G''}}. A response was produced by some of the original authors of the {{math|''G''}} measurements used in Anderson et al.<ref name="SchlammingerGundlach2015">{{cite journal|last1=Schlamminger|first1=S.|last2=Gundlach|first2=J. H.|last3=Newman|first3=R. D.|title=Recent measurements of the gravitational constant as a function of time|journal=Physical Review D|volume=91|issue=12|pages=121101|year=2015|issn=1550-7998|doi=10.1103/PhysRevD.91.121101|arxiv = 1505.01774 |bibcode = 2015PhRvD..91l1101S }}</ref> This response notes that Anderson et al. not only omitted measurements, 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 taking the data collected over a decade by Karagioz and Izmailov shows no correlation with length of day measurements.<ref name=SchlammingerGundlach2015/><ref name="KaragiozIzmailov1996">{{cite journal|last1=Karagioz|first1=O. V.|last2=Izmailov|first2=V. P.|title=Measurement of the gravitational constant with a torsion balance|journal=Measurement Techniques|volume=39|issue=10|year=1996|pages=979–987|issn=0543-1972|doi=10.1007/BF02377461}}</ref> As such the variations in {{math|''G''}} 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 type Ia supernovae 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).<ref>{{cite journal|first1=J. |last1=Mould |first2=S. A. |last2=Uddin |title=Constraining a Possible Variation of G with Type Ia Supernovae |url=http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9198037&fulltextType=RA&fileId=S1323358014000095 |date=10 April 2014 |volume=31 |pages=e015 |doi=10.1017/pasa.2014.9 |journal=Publications of the Astronomical Society of Australia|arxiv = 1402.1534 |bibcode = 2014PASA...31...15M }}</ref>

==See also==
{{Portal|Physics}}
{{div col|colwidth=18em}}
*[[Gravity of Earth]]
*[[Standard gravity]]
*[[Gaussian gravitational constant]]
*[[Orbital mechanics]]
*[[Escape velocity]]
*[[Gravitational coupling constant]]
*[[Gravitational potential]]
*[[Gravitational wave]]
*[[Strong gravitational constant]]
*[[Dirac large numbers hypothesis]]
*[[Accelerating universe]]
*[[Lunar Laser Ranging experiment]]
*[[Cosmological constant]]
{{div col end}}

==References==
'''Footnotes'''
{{Notelist}}

'''Citations'''
{{Reflist|30em}}

===Sources===
{{Refbegin}}
*{{cite book|first=E. Myles |last=Standish. |contribution=Report of the IAU WGAS Sub-group on Numerical Standards |title=Highlights of Astronomy |editor-first=I. |editor-last=Appenzeller |location=Dordrecht |publisher=Kluwer Academic Publishers |date=1995}} ''(Complete report available online: [http://ssd.jpl.nasa.gov/iau-comm4/iausgnsrpt.ps PostScript]; [http://iau-comm4.jpl.nasa.gov/iausgnsrpt.pdf PDF]. Tables from the report also available: [http://ssd.jpl.nasa.gov/?constants  Astrodynamic Constants and Parameters])''
* {{cite journal|first1=Jens H. |last1=Gundlach |first2=Stephen M. |last2=Merkowitz |title=Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback |journal=Physical Review Letters |volume=85 |issue=14 |pages= 2869–2872 |date=2000 |doi=10.1103/PhysRevLett.85.2869|pmid=11005956 |bibcode=2000PhRvL..85.2869G|arxiv = gr-qc/0006043 |title-link=arXiv:gr-qc/0006043v1 }}
{{Refend}}

==External links==
* [http://physics.nist.gov/cgi-bin/cuu/Value?bg Newtonian constant of gravitation {{math|''G''}}] at the [[National Institute of Standards and Technology]] [http://physics.nist.gov/cuu  References on Constants, Units, and Uncertainty]
* [https://www.npl.washington.edu/eotwash/gravitational-constant The Controversy over Newton's Gravitational Constant] — additional commentary on measurement problems

{{Isaac Newton}}
{{Scientists whose names are used in physical constants}}
{{Authority control}}

[[Category:Gravity]]
[[Category:Fundamental constants]]