Gravitational constant

Value of $G$	Unit
6.67430(15)×10⁻¹¹‍^[1]	N⋅m²⋅kg⁻²
6.67430(15)×10⁻⁸	dyn⋅cm²⋅g⁻²
4.3009172706(3)×10⁻³	pc⋅M_⊙⁻¹⋅(km/s)²

Thegravitational constantis anempirical physical constantinvolved in the calculation ofgravitationaleffects inSir Isaac Newton'slaw of universal gravitationand inAlbert Einstein'stheory of general relativity.It is also known as theuniversal gravitational constant,theNewtonian constant of gravitation,or theCavendish gravitational constant,^[a]denoted by the capital letter $G$ .

In Newton's law, it is the proportionality constant connecting thegravitational forcebetween two bodies with the product of theirmassesand theinverse squareof theirdistance.In theEinstein field equations,it quantifies the relation between the geometry of spacetime and the energy–momentum tensor (also referred to as thestress–energy tensor).

The measured value of the constant is known with some certainty to four significant digits. InSI units,its value is approximately6.6743×10⁻¹¹N⋅m²/kg².^[1]

The modern notation of Newton's law involving $G$ was introduced in the 1890s byC. V. Boys.The first implicit measurement with an accuracy within about 1% is attributed toHenry Cavendishin a1798 experiment.^[b]

Definition

According toNewton's law of universal gravitation,themagnitudeof the attractiveforce( $F$ ) between two bodies each with a spherically symmetricdensitydistribution is directly proportional to the product of theirmasses, $m 1$ and $m 2$ ,andinversely proportional to the square of the distance, $r$ ,directed along the line connecting theircentres of mass: $F=G{\frac {m_{1}m_{2}}{r^{2}}}.$ Theconstant of proportionality, $G$ ,in this non-relativistic formulation is the gravitational constant. Colloquially, the gravitational constant is also called "Big G", distinct from "small g" ( $g$ ), which is thelocal gravitational field of Earth(also referred to as free-fall acceleration).^[2]^[3]Where $M_{\oplus }$ is themass of the Earthand $r_{\oplus }$ is theradius of the Earth,the two quantities are related by: $g=G{\frac {M_{\oplus }}{r_{\oplus }^{2}}}.$

The gravitational constant appears in theEinstein field equationsofgeneral relativity,^[4]^[5] $G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }\,,$ where $G μν$ is theEinstein tensor(not the gravitational constant despite the use of $G$ ), $Λ$ is thecosmological constant, $g μν$ is themetric tensor, $T μν$ is thestress–energy tensor,and $κ$ is theEinstein gravitational constant,a constant originally introduced byEinsteinthat is directly related to the Newtonian constant of gravitation:^[5]^[6]^[c] $\kappa ={\frac {8\pi G}{c^{4}}}\approx 2.076647(46)\times 10^{-43}\mathrm {\,N^{-1}}.$

Value and uncertainty

The gravitational constant is a physical constant that is difficult to measure with high accuracy.^[7]This is because the gravitational force is an extremely weak force as compared to otherfundamental forcesat the laboratory scale.^[d]

InSIunits, theCODATA-recommended value of the gravitational constant is:^[1]

G

=6.67430(15)×10⁻¹¹m³⋅kg⁻¹⋅s⁻²

The relative standarduncertaintyis2.2×10⁻⁵.

Natural units

Due to its use as a defining constant in some systems ofnatural units,particularlygeometrized unit systemssuch asPlanck unitsandStoney 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 ofGin 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

Inastrophysics,it is convenient to measure distances inparsecs(pc), velocities in kilometres per second (km/s) and masses insolar units $M ⊙$ .In these units, the gravitational constant is: $G\approx 4.3009\times 10^{-3}\ {\mathrm {pc{\cdot }(km/s)^{2}} \,M_{\odot }}^{-1}.$ For situations where tides are important, the relevant length scales aresolar radiirather than parsecs. In these units, the gravitational constant is: $G\approx 1.90809\times 10^{5}\mathrm {\ (km/s)^{2}} \,R_{\odot }M_{\odot }^{-1}.$ Inorbital mechanics,the period $P$ of an object in circular orbit around a spherical object obeys $GM={\frac {3\pi V}{P^{2}}},$ where $V$ is the volume inside the radius of the orbit, and $M$ is the total mass of the two objects. It follows that

P^{2}={\frac {3\pi }{G}}{\frac {V}{M}}\approx 10.896\,\mathrm {h^{2}{\cdot }g{\cdot }cm^{-3}\,} {\frac {V}{M}}.

This way of expressing $G$ shows the relationship between the average density of a planet and the period of a satellite orbiting just above its surface.

For elliptical orbits, applyingKepler's 3rd law,expressed in units characteristic ofEarth's orbit:

G=4\pi ^{2}\mathrm {\ AU^{3}{\cdot }yr^{-2}} \ M^{-1}\approx 39.478\mathrm {\ AU^{3}{\cdot }yr^{-2}} \ M_{\odot }^{-1},

where distance is measured in terms of thesemi-major axisof Earth's orbit (theastronomical unit,AU), time inyears,and mass in the total mass of the orbiting system ( $M = M ☉ + M E + M ☾$ ^[e]).

The above equation is exact only within the approximation of the Earth's orbit around the Sun as atwo-body problemin 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: $1\ \mathrm {AU} =\left({\frac {GM}{4\pi ^{2}}}\mathrm {yr} ^{2}\right)^{\frac {1}{3}}\approx 1.495979\times 10^{11}\ \mathrm {m}.$ Since 2012, the AU is defined as1.495978707×10¹¹mexactly, and the equation can no longer be taken as holding precisely.

The quantity $GM$ —the product of the gravitational constant and the mass of a given astronomical body such as the Sun or Earth—is known as thestandard gravitational parameter(also denoted $μ$ ). 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 bygravitational lensing,inKepler's laws of planetary motion,and in the formula forescape velocity.

This quantity gives a convenient simplification of various gravity-related formulas. The product $GM$ is known much more accurately than either factor is.

Values forGM
Body	$μ = GM$	Value	Relative uncertainty
Sun	$G M ☉$	1.32712440018(8)×10²⁰m³⋅s⁻²^[8]	6×10⁻¹¹
Earth	$G M E$	3.986004418(8)×10¹⁴m³⋅s⁻²^[9]	2×10⁻⁹

Calculations incelestial mechanicscan also be carried out using the units ofsolar masses,mean solar daysandastronomical unitsrather than standard SI units. For this purpose, theGaussian gravitational constantwas historically in widespread use, $k = 0.017 20209895 radians per day$ ,expressing the meanangular velocityof the Sun–Earth system.^{[citation needed]}The use of this constant, and the implied definition of theastronomical unitdiscussed above, has been deprecated by theIAUsince 2012.^{[citation needed]}

History of measurement

Early history

The existence of the constant is implied inNewton's law of universal gravitationas published in the 1680s (although its notation as $G$ dates to the 1890s),^[10]but is notcalculatedin hisPhilosophiæ Naturalis Principia Mathematicawhere it postulates theinverse-square lawof gravitation. In thePrincipia,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.^[11]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:^[12]

G

≈(6.7±0.6)×10⁻¹¹m³⋅kg⁻¹⋅s⁻²

A measurement was attempted in 1738 byPierre BouguerandCharles Marie de La Condaminein 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 ahollow shell,as some thinkers of the day, includingEdmond Halley,had suggested.^[13]

TheSchiehallion 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 byCharles Hutton(1778) suggested a density of4.5 g/cm³(⁠4+1/2⁠times the density of water), about 20% below the modern value.^[14]This immediately led to estimates on the densities and masses of theSun,Moonandplanets,sent by Hutton toJérôme Lalandefor inclusion in his planetary tables. As discussed above, establishing the average density of Earth is equivalent to measuring the gravitational constant, givenEarth's mean radiusand themean gravitational accelerationat Earth's surface, by setting^[10] $G=g{\frac {R_{\oplus }^{2}}{M_{\oplus }}}={\frac {3g}{4\pi R_{\oplus }\rho _{\oplus }}}.$ Based on this, Hutton's 1778 result is equivalent to $G \approx 8 \times 10 -11 m 3 \cdotkg -1 \cdots -2$ .

The first direct measurement of gravitational attraction between two bodies in the laboratory was performed in 1798, seventy-one years after Newton's death, byHenry Cavendish.^[15]He determined a value for $G$ implicitly, using atorsion balanceinvented by the geologist Rev.John Michell(1753). He used a horizontaltorsion beamwith 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 theCavendish experimentfor its first successful execution by Cavendish.

Cavendish's stated aim was the "weighing of Earth", that is, determining the average density of Earth and theEarth's mass.His result, $ρ 🜨 = 5.448(33) g\cdotcm -3$ ,corresponds to value of $G = 6.74(4) \times 10 -11 m 3 \cdotkg -1 \cdots -2$ .It is surprisingly accurate, about 1% above the modern value (comparable to the claimed relative standard uncertainty of 0.6%).^[16]

19th century

The accuracy of the measured value of $G$ has increased only modestly since the original Cavendish experiment.^[17] $G$ 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.

Measurements with pendulums were made byFrancesco Carlini(1821,4.39 g/cm³),Edward Sabine(1827,4.77 g/cm³), Carlo Ignazio Giulio (1841,4.95 g/cm³) andGeorge Biddell Airy(1854,6.6 g/cm³).^[18]

Cavendish's experiment was first repeated byFerdinand Reich(1838, 1842, 1853), who found a value of5.5832(149) g⋅cm⁻³,^[19]which is actually worse than Cavendish's result, differing from the modern value by 1.5%. Cornu and Baille (1873), found5.56 g⋅cm⁻³.^[20]

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, byRobert von Sterneck(1883, results between 5.0 and6.3 g/cm³) andThomas Corwin Mendenhall(1880,5.77 g/cm³).^[21]

Cavendish's result was first improved upon byJohn Henry Poynting(1891),^[22]who published a value of5.49(3) g⋅cm⁻³,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 byC. V. Boys(1895)^[23]andCarl Braun(1897),^[24]with compatible results suggesting $G$ =6.66(1)×10⁻¹¹m³⋅kg⁻¹⋅s⁻².The modern notation involving the constant $G$ was introduced by Boys in 1894^[10]and becomes standard by the end of the 1890s, with values usually cited in thecgssystem. 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 of6.683(11)×10⁻¹¹m³⋅kg⁻¹⋅s⁻²was, however, of the same order of magnitude as the other results at the time.^[25]

Arthur Stanley MackenzieinThe Laws of Gravitation(1899) reviews the work done in the 19th century.^[26]Poynting is the author of the article "Gravitation" in theEncyclopædia BritannicaEleventh Edition(1911). Here, he cites a value of $G$ =6.66×10⁻¹¹m³⋅kg⁻¹⋅s⁻²with a relative uncertainty of 0.2%.

Modern value

Paul R. Heyl(1930) published the value of6.670(5)×10⁻¹¹m³⋅kg⁻¹⋅s⁻²(relative uncertainty 0.1%),^[27]improved to6.673(3)×10⁻¹¹m³⋅kg⁻¹⋅s⁻²(relative uncertainty 0.045% = 450 ppm) in 1942.^[28]

However, Heyl used the statistical spread as his standard deviation, and he admitted himself that measurements using the same material yielded very similar results while measurements using different materials yielded vastly different results. He spent the next 12 years after his 1930 paper to do more precise measurements, hoping that the composition-dependent effect would go away, but it did not, as he noted in his final paper from the year 1942.

Published values of $G$ derived from high-precision measurements since the 1950s have remained compatible with Heyl (1930), but within the relative uncertainty of about 0.1% (or 1000 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.^[7]^[29]Establishing a standard value for $G$ with a relative standard uncertainty better than 0.1% has therefore remained rather speculative.

By 1969, the value recommended by theNational Institute of Standards and Technology(NIST) was cited with a relative 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 relative standard uncertainty of 120 ppm published in 1986.^[30]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:

Recommended values forG
Year	G (10⁻¹¹m³⋅kg⁻¹⋅s⁻²)	Relative standard uncertainty	Ref.
1969	6.6732(31)	460 ppm	^[31]
1973	6.6720(49)	730 ppm	^[32]
1986	6.67449(81)	120 ppm	^[33]
1998	6.673(10)	1500 ppm	^[34]
2002	6.6742(10)	150 ppm	^[35]
2006	6.67428(67)	100 ppm	^[36]
2010	6.67384(80)	120 ppm	^[37]
2014	6.67408(31)	46 ppm	^[38]
2018	6.67430(15)	22 ppm	^[39]
2022	6.67430(15)	22 ppm	^[40]

In the January 2007 issue ofScience,Fixler et al. described a measurement of the gravitational constant by a new technique,atom interferometry,reporting a value of $G$ =6.693(34)×10⁻¹¹m³⋅kg⁻¹⋅s⁻²,0.28% (2800 ppm) higher than the 2006 CODATA value.^[41]An improved cold atom measurement by Rosi et al. was published in 2014 of $G$ =6.67191(99)×10⁻¹¹m³⋅kg⁻¹⋅s⁻².^[42]^[43]Although much closer to the accepted value (suggesting that the Fixleret al.measurement was erroneous), this result was 325 ppm below the recommended 2014 CODATA value, with non-overlappingstandard uncertaintyintervals.

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).^[44]

In August 2018, a Chinese research group announced new measurements based on torsion balances,6.674184(78)×10⁻¹¹m³⋅kg⁻¹⋅s⁻²and6.674484(78)×10⁻¹¹m³⋅kg⁻¹⋅s⁻²based on two different methods.^[45]These are claimed as the most accurate measurements ever made, with standard uncertainties cited as low as 12 ppm. The difference of 2.7σbetween the two results suggests there could be sources of error unaccounted for.

Constancy

Analysis of observations of 580type Ia supernovaeshows that the gravitational constant has varied by less than one part in ten billion per year over the last nine billion years.^[46]

References

Footnotes

^"Newtonian constant of gravitation" is the name introduced forGby Boys (2000). Use of the term by T.E. Stern (1928) was misquoted as "Newton's constant of gravitation" inPure 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 theheat transfer coefficientinNewton'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: Uspekhi30 (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 theEinstein tensorG_μνvs. themetric tensorg_μν,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).
^Cavendish determined the value ofGindirectly, by reporting a value for theEarth's mass,or the average density of Earth, as5.448 g⋅cm⁻³.
^Depending on the choice of definition of the Einstein tensor and of the stress–energy tensor it can alternatively be defined as $κ = ⁠ 8π G / c 2 ⁠ \approx 1.866 \times 10 -26 m\cdotkg -1$
^For example, the gravitational force between anelectronand aproton1 m apart is approximately10⁻⁶⁷N,whereas theelectromagnetic forcebetween the same two particles is approximately10⁻²⁸N.The electromagnetic force in this example is in the order of 10³⁹times greater than the force of gravity—roughly the same ratio as themass of the Sunto a microgram.
^ $M$ ≈ 1.000003040433 $M ☉$ ,so that $M$ = $M ☉$ can be used for accuracies of five or fewer significant digits.

Citations

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^Gundlach, Jens H.; Merkowitz, Stephen M. (23 December 2002)."University of Washington Big G Measurement".Astrophysics Science Division.Goddard Space Flight Center.Since Cavendish first measured Newton's Gravitational constant 200 years ago, 'Big G' remains one of the most elusive constants in physics
^Halliday, David; Resnick, Robert; Walker, Jearl (September 2007).Fundamentals of Physics(8th ed.). John Wiley & Sons, Limited. p. 336.ISBN 978-0-470-04618-0.
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^^a ^bEinstein, Albert (1916)."The Foundation of the General Theory of Relativity".Annalen der Physik.354(7): 769–822.Bibcode:1916AnP...354..769E.doi:10.1002/andp.19163540702.Archived fromthe original(PDF)on 6 February 2012.
^Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975).Introduction to General Relativity(2nd ed.). New York: McGraw-Hill. p.345.ISBN 978-0-07-000423-8.
^^a ^bGillies, George T. (1997). "The Newtonian gravitational constant: recent measurements and related studies".Reports on Progress in Physics.60(2): 151–225.Bibcode:1997RPPh...60..151G.doi:10.1088/0034-4885/60/2/001.S2CID 250810284..A lengthy, detailed review. See Figure 1 and Table 2 in particular.
^"Astrodynamic Constants".NASA/JPL.27 February 2009.Retrieved27 July2009.
^
"Geocentric gravitational constant".Numerical Standards for Fundamental Astronomy.IAU Division I Working Group on Numerical Standards for Fundamental Astronomy.Retrieved24 June2021– via iau-a3.gitlab.io.Citing
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^^a ^b ^cBoys 1894,p.330 In this lecture before the Royal Society, Boys introducesGand argues for its acceptance. See: Poynting 1894,p. 4,MacKenzie 1900,p.vi
^Davies, R.D. (1985). "A Commemoration of Maskelyne at Schiehallion".Quarterly Journal of the Royal Astronomical Society.26(3): 289–294.Bibcode:1985QJRAS..26..289D.
^"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
^Poynting, J.H. (1913).The Earth: its shape, size, weight and spin.Cambridge. pp. 50–56.
^Hutton, C. (1778)."An Account of the Calculations Made from the Survey and Measures Taken at Schehallien".Philosophical Transactions of the Royal Society.68:689–788.doi:10.1098/rstl.1778.0034.
^Published inPhilosophical Transactions of the Royal Society(1798); reprint: Cavendish, Henry (1798). "Experiments to Determine the Density of the Earth". In MacKenzie, A. S.,Scientific MemoirsVol. 9:The Laws of Gravitation.American Book Co. (1900), pp. 59–105.
^2014CODATAvalue $6.674 \times 10 -11 m 3 \cdotkg -1 \cdots -2$ .
^ Brush, Stephen G.; Holton, Gerald James (2001).Physics, the human adventure: from Copernicus to Einstein and beyond.New Brunswick, NJ: Rutgers University Press. pp.137.ISBN 978-0-8135-2908-0. Lee, Jennifer Lauren (16 November 2016)."Big G Redux: Solving the Mystery of a Perplexing Result".NIST.
^Poynting, John Henry (1894).The Mean Density of the Earth.London: Charles Griffin. pp.22–24.
^F. Reich,On the Repetition of the Cavendish Experiments for Determining the mean density of the Earth "Philosophical Magazine12: 283–284.
^Mackenzie (1899), p. 125.
^A.S. Mackenzie,The Laws of Gravitation(1899),127f.
^Poynting, John Henry (1894).The mean density of the earth.Gerstein - University of Toronto. London.
^Boys, C. V. (1 January 1895)."On the Newtonian Constant of Gravitation".Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.186.The Royal Society: 1–72.Bibcode:1895RSPTA.186....1B.doi:10.1098/rsta.1895.0001.ISSN 1364-503X.
^Carl Braun,Denkschriften der k. Akad. d. Wiss. (Wien), math. u. naturwiss. Classe,64 (1897). Braun (1897) quoted an optimistic relative standard uncertainty of 0.03%,6.649(2)×10⁻¹¹m³⋅kg⁻¹⋅s⁻²but his result was significantly worse than the 0.2% feasible 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 fromAstronomicheskii ZhurnalVol. 46, No. 4 (July–August 1969), 907–915 (table of historical experiments p. 715).
^Mackenzie, A. Stanley,The laws of gravitation; memoirs by Newton, Bouguer and Cavendish, together with abstracts of other important memoirs,American Book Company (1900 [1899]).
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^P. R. Heyl and P. Chrzanowski (1942), cited after Sagitov (1969:715).
^Mohr, Peter J.; Taylor, Barry N. (2012)."CODATA recommended values of the fundamental physical constants: 2002"(PDF).Reviews of Modern Physics.77(1): 1–107.arXiv:1203.5425.Bibcode:2005RvMP...77....1M.CiteSeerX10.1.1.245.4554.doi:10.1103/RevModPhys.77.1.Archived fromthe original(PDF)on 6 March 2007.Retrieved1 July2006.Section Q (pp. 42–47) describes the mutually inconsistent measurement experiments from which the CODATA value for $G$ was derived.
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^Eite Tiesinga, Peter J. Mohr, David B. Newell, and Barry N. Taylor (2019), "The 2018 CODATA Recommended Values of the Fundamental Physical Constants"(Web Version 8.0). Database developed by J. Baker, M. Douma, andS. Kotochigova.National Institute of Standards and Technology, Gaithersburg, MD 20899.
^Mohr, P.; Tiesinga, E.; Newell, D.; Taylor, B. (8 May 2024),Codata Internationally Recommended 2022 Values of the Fundamental Physical Constants,retrieved15 May2024
^Fixler, J. B.; Foster, G. T.; McGuirk, J. M.; Kasevich, M. A. (5 January 2007). "Atom Interferometer Measurement of the Newtonian Constant of Gravity".Science.315(5808): 74–77.Bibcode:2007Sci...315...74F.doi:10.1126/science.1135459.PMID 17204644.S2CID 6271411.
^ Rosi, G.; Sorrentino, F.; Cacciapuoti, L.; Prevedelli, M.; Tino, G. M. (26 June 2014)."Precision measurement of the Newtonian gravitational constant using cold atoms"(PDF).Nature.510(7506): 518–521.arXiv:1412.7954.Bibcode:2014Natur.510..518R.doi:10.1038/nature13433.PMID 24965653.S2CID 4469248.Archived(PDF)from the original on 9 October 2022.
^ Schlamminger, Stephan (18 June 2014)."Fundamental constants: A cool way to measure big G"(PDF).Nature.510(7506): 478–480.Bibcode:2014Natur.510..478S.doi:10.1038/nature13507.PMID 24965646.Archived(PDF)from the original on 9 October 2022.
^
C. Rothleitner; S. Schlamminger (2017)."Invited Review Article: Measurements of the Newtonian constant of gravitation, G".Review of Scientific Instruments.88(11): 111101.Bibcode:2017RScI...88k1101R.doi:10.1063/1.4994619.PMC8195032.PMID 29195410.111101.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:
- T. Quinn; H. Parks; C. Speake; R. Davis (2013)."Improved determination of G using two methods"(PDF).Phys. Rev. Lett.111(10): 101102.Bibcode:2013PhRvL.111j1102Q.doi:10.1103/PhysRevLett.111.101102.PMID 25166649.101102. Archived fromthe original(PDF)on 4 December 2020.Retrieved4 August2019.
The 2018 experiment was described byC. Rothleitner.Newton's Gravitational Constant 'Big' G – A proposed Free-fall Measurement(PDF).CODATA Fundamental Constants Meeting, Eltville – 5 February 2015.Archived(PDF)from the original on 9 October 2022.
^Li, Qing; et al. (2018). "Measurements of the gravitational constant using two independent methods".Nature.560(7720): 582–588.Bibcode:2018Natur.560..582L.doi:10.1038/s41586-018-0431-5.PMID 30158607.S2CID 52121922.. See also:"Physicists just made the most precise measurement ever of Gravity's strength".31 August 2018.Retrieved13 October2018.
^Mould, J.; Uddin, S. A. (10 April 2014). "Constraining a Possible Variation of G with Type Ia Supernovae".Publications of the Astronomical Society of Australia.31:e015.arXiv:1402.1534.Bibcode:2014PASA...31...15M.doi:10.1017/pasa.2014.9.S2CID 119292899.

Sources

Standish., E. Myles (1995). "Report of the IAU WGAS Sub-group on Numerical Standards". In Appenzeller, I. (ed.).Highlights of Astronomy.Dordrecht: Kluwer Academic Publishers.(Complete report available online:PostScript;PDF.Tables from the report also available:Astrodynamic Constants and Parameters)
Gundlach, Jens H.; Merkowitz, Stephen M. (2000)."Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback".Physical Review Letters.85(14): 2869–2872.arXiv:gr-qc/0006043.Bibcode:2000PhRvL..85.2869G.doi:10.1103/PhysRevLett.85.2869.PMID 11005956.S2CID 15206636.

External links

Newtonian constant of gravitation $G$ at theNational Institute of Standards and Technology References on Constants, Units, and Uncertainty
The Controversy over Newton's Gravitational Constant— additional commentary on measurement problems

[2] "Newtonian constant of gravitation" is the name introduced forGby Boys (2000). Use of the term by T.E. Stern (1928) was misquoted as "Newton's constant of gravitation" inPure 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 theheat transfer coefficientinNewton'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: Uspekhi30 (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 theEinstein tensorG_μνvs. themetric tensorg_μν,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).

[3] Cavendish determined the value ofGindirectly, by reporting a value for theEarth's mass,or the average density of Earth, as5.448 g⋅cm⁻³.

[9] Depending on the choice of definition of the Einstein tensor and of the stress–energy tensor it can alternatively be defined as $κ = ⁠ 8π G / c 2 ⁠ \approx 1.866 \times 10 -26 m\cdotkg -1$

[11] For example, the gravitational force between anelectronand aproton1 m apart is approximately10⁻⁶⁷N,whereas theelectromagnetic forcebetween the same two particles is approximately10⁻²⁸N.The electromagnetic force in this example is in the order of 10³⁹times greater than the force of gravity—roughly the same ratio as themass of the Sunto a microgram.

[12] $M$ ≈ 1.000003040433 $M ☉$ ,so that $M$ = $M ☉$ can be used for accuracies of five or fewer significant digits.

[physconst-G-1] "2022 CODATA Value: Newtonian constant of gravitation".The NIST Reference on Constants, Units, and Uncertainty.NIST.May 2024.Retrieved18 May2024.

[4] Gundlach, Jens H.; Merkowitz, Stephen M. (23 December 2002)."University of Washington Big G Measurement".Astrophysics Science Division.Goddard Space Flight Center.Since Cavendish first measured Newton's Gravitational constant 200 years ago, 'Big G' remains one of the most elusive constants in physics

[5] Halliday, David; Resnick, Robert; Walker, Jearl (September 2007).Fundamentals of Physics(8th ed.). John Wiley & Sons, Limited. p. 336.ISBN 978-0-470-04618-0.

[6] Grøn, Øyvind; Hervik, Sigbjorn (2007).Einstein's General Theory of Relativity: With Modern Applications in Cosmology(illustrated ed.). Springer Science & Business Media. p. 180.ISBN 978-0-387-69200-5.

[ein-7] Einstein, Albert (1916)."The Foundation of the General Theory of Relativity".Annalen der Physik.354(7): 769–822.Bibcode:1916AnP...354..769E.doi:10.1002/andp.19163540702.Archived fromthe original(PDF)on 6 February 2012.

[8] Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975).Introduction to General Relativity(2nd ed.). New York: McGraw-Hill. p.345.ISBN 978-0-07-000423-8.

[gillies-10] Gillies, George T. (1997). "The Newtonian gravitational constant: recent measurements and related studies".Reports on Progress in Physics.60(2): 151–225.Bibcode:1997RPPh...60..151G.doi:10.1088/0034-4885/60/2/001.S2CID 250810284..A lengthy, detailed review. See Figure 1 and Table 2 in particular.

[Astrodynamic_Constants-13] "Astrodynamic Constants".NASA/JPL.27 February 2009.Retrieved27 July2009.

[IAU_best_estimates-14] "Geocentric gravitational constant".Numerical Standards for Fundamental Astronomy.IAU Division I Working Group on Numerical Standards for Fundamental Astronomy.Retrieved24 June2021– via iau-a3.gitlab.io.Citing
Ries JC, Eanes RJ, Shum CK, Watkins MM (20 March 1992). "Progress in the determination of the gravitational coefficient of the Earth".Geophysical Research Letters.19(6): 529–531.Bibcode:1992GeoRL..19..529R.doi:10.1029/92GL00259.S2CID 123322272.

[15] Ries JC, Eanes RJ, Shum CK, Watkins MM (20 March 1992). "Progress in the determination of the gravitational coefficient of the Earth".Geophysical Research Letters.19(6): 529–531.Bibcode:1992GeoRL..19..529R.doi:10.1029/92GL00259.S2CID 123322272.

[BoysG-15] Boys 1894,p.330 In this lecture before the Royal Society, Boys introducesGand argues for its acceptance. See: Poynting 1894,p. 4,MacKenzie 1900,p.vi

[Davies-16] Davies, R.D. (1985). "A Commemoration of Maskelyne at Schiehallion".Quarterly Journal of the Royal Astronomical Society.26(3): 289–294.Bibcode:1985QJRAS..26..289D.

[17] "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

[Poynting_p50-56-18] Poynting, J.H. (1913).The Earth: its shape, size, weight and spin.Cambridge. pp. 50–56.

[Hutton-19] Hutton, C. (1778)."An Account of the Calculations Made from the Survey and Measures Taken at Schehallien".Philosophical Transactions of the Royal Society.68:689–788.doi:10.1098/rstl.1778.0034.

[20] Published inPhilosophical Transactions of the Royal Society(1798); reprint: Cavendish, Henry (1798). "Experiments to Determine the Density of the Earth". In MacKenzie, A. S.,Scientific MemoirsVol. 9:The Laws of Gravitation.American Book Co. (1900), pp. 59–105.

[21] 2014CODATAvalue $6.674 \times 10 -11 m 3 \cdotkg -1 \cdots -2$ .

[22] Brush, Stephen G.; Holton, Gerald James (2001).Physics, the human adventure: from Copernicus to Einstein and beyond.New Brunswick, NJ: Rutgers University Press. pp.137.ISBN 978-0-8135-2908-0. Lee, Jennifer Lauren (16 November 2016)."Big G Redux: Solving the Mystery of a Perplexing Result".NIST.

[23] Poynting, John Henry (1894).The Mean Density of the Earth.London: Charles Griffin. pp.22–24.

[24] F. Reich,On the Repetition of the Cavendish Experiments for Determining the mean density of the Earth "Philosophical Magazine12: 283–284.

[25] Mackenzie (1899), p. 125.

[26] A.S. Mackenzie,The Laws of Gravitation(1899),127f.

[27] Poynting, John Henry (1894).The mean density of the earth.Gerstein - University of Toronto. London.

[28] Boys, C. V. (1 January 1895)."On the Newtonian Constant of Gravitation".Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.186.The Royal Society: 1–72.Bibcode:1895RSPTA.186....1B.doi:10.1098/rsta.1895.0001.ISSN 1364-503X.

[29] Carl Braun,Denkschriften der k. Akad. d. Wiss. (Wien), math. u. naturwiss. Classe,64 (1897). Braun (1897) quoted an optimistic relative standard uncertainty of 0.03%,6.649(2)×10⁻¹¹m³⋅kg⁻¹⋅s⁻²but his result was significantly worse than the 0.2% feasible at the time.

[Sagitov-30] 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 fromAstronomicheskii ZhurnalVol. 46, No. 4 (July–August 1969), 907–915 (table of historical experiments p. 715).

[31] Mackenzie, A. Stanley,The laws of gravitation; memoirs by Newton, Bouguer and Cavendish, together with abstracts of other important memoirs,American Book Company (1900 [1899]).

[32] Heyl, P. R.(1930)."A redetermination of the constant of gravitation".Bureau of Standards Journal of Research.5(6): 1243–1290.doi:10.6028/jres.005.074.

[33] P. R. Heyl and P. Chrzanowski (1942), cited after Sagitov (1969:715).

[codata2002-34] Mohr, Peter J.; Taylor, Barry N. (2012)."CODATA recommended values of the fundamental physical constants: 2002"(PDF).Reviews of Modern Physics.77(1): 1–107.arXiv:1203.5425.Bibcode:2005RvMP...77....1M.CiteSeerX10.1.1.245.4554.doi:10.1103/RevModPhys.77.1.Archived fromthe original(PDF)on 6 March 2007.Retrieved1 July2006.Section Q (pp. 42–47) describes the mutually inconsistent measurement experiments from which the CODATA value for $G$ was derived.

[35] Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (13 November 2012)."CODATA recommended values of the fundamental physical constants: 2010"(PDF).Reviews of Modern Physics.84(4): 1527–1605.arXiv:1203.5425.Bibcode:2012RvMP...84.1527M.CiteSeerX10.1.1.150.3858.doi:10.1103/RevModPhys.84.1527.S2CID 103378639.

[36] Taylor, B. N.; Parker, W. H.; Langenberg, D. N. (1 July 1969). "Determination ofe/h,Using Macroscopic Quantum Phase Coherence in Superconductors: Implications for Quantum Electrodynamics and the Fundamental Physical Constants ".Reviews of Modern Physics.41(3). American Physical Society (APS): 375–496.Bibcode:1969RvMP...41..375T.doi:10.1103/revmodphys.41.375.ISSN 0034-6861.

[37] Cohen, E. Richard; Taylor, B. N. (1973). "The 1973 Least-Squares Adjustment of the Fundamental Constants".Journal of Physical and Chemical Reference Data.2(4). AIP Publishing: 663–734.Bibcode:1973JPCRD...2..663C.doi:10.1063/1.3253130.hdl:2027/pst.000029951949.ISSN 0047-2689.

[38] Cohen, E. Richard; Taylor, Barry N. (1 October 1987). "The 1986 adjustment of the fundamental physical constants".Reviews of Modern Physics.59(4). American Physical Society (APS): 1121–1148.Bibcode:1987RvMP...59.1121C.doi:10.1103/revmodphys.59.1121.ISSN 0034-6861.

[39] Mohr, Peter J.; Taylor, Barry N. (2012). "CODATA recommended values of the fundamental physical constants: 1998".Reviews of Modern Physics.72(2): 351–495.arXiv:1203.5425.Bibcode:2000RvMP...72..351M.doi:10.1103/revmodphys.72.351.ISSN 0034-6861.

[40] Mohr, Peter J.; Taylor, Barry N. (2012). "CODATA recommended values of the fundamental physical constants: 2002".Reviews of Modern Physics.77(1): 1–107.arXiv:1203.5425.Bibcode:2005RvMP...77....1M.doi:10.1103/revmodphys.77.1.ISSN 0034-6861.

[41] Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2012). "CODATA recommended values of the fundamental physical constants: 2006".Journal of Physical and Chemical Reference Data.37(3): 1187–1284.arXiv:1203.5425.Bibcode:2008JPCRD..37.1187M.doi:10.1063/1.2844785.ISSN 0047-2689.

[42] Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2012). "CODATA Recommended Values of the Fundamental Physical Constants: 2010".Journal of Physical and Chemical Reference Data.41(4): 1527–1605.arXiv:1203.5425.Bibcode:2012JPCRD..41d3109M.doi:10.1063/1.4724320.ISSN 0047-2689.

[43] Mohr, Peter J.; Newell, David B.; Taylor, Barry N. (2016). "CODATA Recommended Values of the Fundamental Physical Constants: 2014".Journal of Physical and Chemical Reference Data.45(4): 1527–1605.arXiv:1203.5425.Bibcode:2016JPCRD..45d3102M.doi:10.1063/1.4954402.ISSN 0047-2689.

[44] Eite Tiesinga, Peter J. Mohr, David B. Newell, and Barry N. Taylor (2019), "The 2018 CODATA Recommended Values of the Fundamental Physical Constants"(Web Version 8.0). Database developed by J. Baker, M. Douma, andS. Kotochigova.National Institute of Standards and Technology, Gaithersburg, MD 20899.

[45] Mohr, P.; Tiesinga, E.; Newell, D.; Taylor, B. (8 May 2024),Codata Internationally Recommended 2022 Values of the Fundamental Physical Constants,retrieved15 May2024

[46] Fixler, J. B.; Foster, G. T.; McGuirk, J. M.; Kasevich, M. A. (5 January 2007). "Atom Interferometer Measurement of the Newtonian Constant of Gravity".Science.315(5808): 74–77.Bibcode:2007Sci...315...74F.doi:10.1126/science.1135459.PMID 17204644.S2CID 6271411.

[47] Rosi, G.; Sorrentino, F.; Cacciapuoti, L.; Prevedelli, M.; Tino, G. M. (26 June 2014)."Precision measurement of the Newtonian gravitational constant using cold atoms"(PDF).Nature.510(7506): 518–521.arXiv:1412.7954.Bibcode:2014Natur.510..518R.doi:10.1038/nature13433.PMID 24965653.S2CID 4469248.Archived(PDF)from the original on 9 October 2022.

[48] Schlamminger, Stephan (18 June 2014)."Fundamental constants: A cool way to measure big G"(PDF).Nature.510(7506): 478–480.Bibcode:2014Natur.510..478S.doi:10.1038/nature13507.PMID 24965646.Archived(PDF)from the original on 9 October 2022.

[49] C. Rothleitner; S. Schlamminger (2017)."Invited Review Article: Measurements of the Newtonian constant of gravitation, G".Review of Scientific Instruments.88(11): 111101.Bibcode:2017RScI...88k1101R.doi:10.1063/1.4994619.PMC8195032.PMID 29195410.111101.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:
T. Quinn; H. Parks; C. Speake; R. Davis (2013)."Improved determination of G using two methods"(PDF).Phys. Rev. Lett.111(10): 101102.Bibcode:2013PhRvL.111j1102Q.doi:10.1103/PhysRevLett.111.101102.PMID 25166649.101102. Archived fromthe original(PDF)on 4 December 2020.Retrieved4 August2019.
The 2018 experiment was described byC. Rothleitner.Newton's Gravitational Constant 'Big' G – A proposed Free-fall Measurement(PDF).CODATA Fundamental Constants Meeting, Eltville – 5 February 2015.Archived(PDF)from the original on 9 October 2022.

[51] T. Quinn; H. Parks; C. Speake; R. Davis (2013)."Improved determination of G using two methods"(PDF).Phys. Rev. Lett.111(10): 101102.Bibcode:2013PhRvL.111j1102Q.doi:10.1103/PhysRevLett.111.101102.PMID 25166649.101102. Archived fromthe original(PDF)on 4 December 2020.Retrieved4 August2019.

[50] Li, Qing; et al. (2018). "Measurements of the gravitational constant using two independent methods".Nature.560(7720): 582–588.Bibcode:2018Natur.560..582L.doi:10.1038/s41586-018-0431-5.PMID 30158607.S2CID 52121922.. See also:"Physicists just made the most precise measurement ever of Gravity's strength".31 August 2018.Retrieved13 October2018.

[51] Mould, J.; Uddin, S. A. (10 April 2014). "Constraining a Possible Variation of G with Type Ia Supernovae".Publications of the Astronomical Society of Australia.31:e015.arXiv:1402.1534.Bibcode:2014PASA...31...15M.doi:10.1017/pasa.2014.9.S2CID 119292899.

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v t e Sir Isaac Newton
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