Gravitational constant: Difference between revisions

The '''gravitational constant''' (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|theory of general relativity]]. It is 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 (2000). 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.

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''_''μν'' vs. the [[metric tensor]] ''g''_''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).|name=|group=}} denoted by the capital 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|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]]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–momentumenergy–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|round=4|unit=no|after=&nbsp;N⋅m²/kg².|round=3}}

According to [[Newton's law of universal gravitation]], the [[Norm (mathematics)#Euclidean norm|magnitude]] of the attractive [[force]] ({{math|''F''}}) between two [[Pointbodies particle|point-likeeach bodieswith a spherically symmetric [[density]] distribution is directly proportional to the product of their [[mass]]es, {{math|''m''₁}} and {{math|''m''₂}}, and [[Inverse-square law|inversely proportional to the square of the distance]], {{math|''r''}}, directed along the line connecting their centers[[centre of mass|centres of mass]]:

The [[Proportionality (mathematics)|constant of proportionality]], {{math|''G''}}, in this non-relativistic formulation is the gravitational constant. Colloquially, the gravitational constant is also called "Big G", distinct from "small g" ({{math|''g''}}), which is the [[Gravity of Earth|local gravitational field of Earth]] (equivalentalso toreferred theas 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 |publisher=John Wiley & Sons, Limited }}</ref> Where <math>M_\oplus</math> is the [[mass of the Earth]] and <math>r_\oplus</math> is the [[Earth radius|radius of the Earth]], the two quantities are related by:

<math display="block">g = G\frac{GM_M_\oplus}{r_\oplus^2}.</math>

where {{math|''G''{{sub|''μν''}}}} is the [[Einstein tensor]] (not the gravitational constant despite the use of {{mvar|G}}), {{math|Λ}} is the [[cosmological constant]], {{mvar|g{{sub|μν}}}} is the [[metric tensor (general relativity)|metric tensor]], {{mvar|T{{sub|μν}}}} is the [[stress–energy tensor]], and {{math|''κ''}} is the [[Einstein gravitational constant]], a constant originally introduced by [[Albert Einstein|Einstein]] that is directly related to the Newtonian constant of gravitation:<ref name="ein" /><ref>{{cite book |title= Introduction to General Relativity |url= https://archive.org/details/introductiontoge00adle |url-access= limited |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= [https://archive.org/details/introductiontoge00adle/page/n360 345]}}</ref>{{efn|Depending on the choice of definition of the Einstein tensor and of the stress–energy tensor it can alternatively be defined as {{math|1=''κ'' = {{sfrac|8π''G''|''c''²}} ≈ {{val|1.866|e=-26|u=m⋅kg⁻¹}}}}}}

<math display="block">\kappa = \frac{8\pi G}{c^4} \approx 2.0766076647(46) \times 10^{-43} \mathrm{\,N^{-1}}.</math>

This corresponds to aThe relative standard [[Measurement uncertainty|uncertainty]] ofis {{valphysconst|2.2G|erunc=-5yes|ref=no}}. (22 [[Parts per million|ppm]])

{{seefurther|Standard gravitational parameter|orbital mechanics|celestial mechanics|Gaussian gravitational constant|Earth mass|Solar mass}}

where {{math|''V''}} is the volume inside the radius of the orbit, and {{math|''M''}} is the total mass of the two objects. It follows that

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

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

{{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.}}).

From 1964 until 2012, however, it was used as the definition of the astronomical unit and thus held by definition:

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]] (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]].

!scope="col"| Body

* {{cite journal|vauthors=Ries JC, Eanes RJ, Shum CK, Watkins MM |s2cid=123322272 |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>

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|{{mvarmath|''k}}'' {{=}} {{val|0.01720209895}} [[radian]]s per [[day]]}}, expressing the mean [[angular velocity]] of the Sun–Earth system measured in [[radian]]s per [[day]].{{Citation needed|date=September 2020}} The use of this constant, and the implied definition of the [[astronomical unit]] discussed above, has been deprecated by the [[IAU]] since 2012.{{Citation needed|date=September 2020}}

{{seefurther|Earth mass|Schiehallion experiment|Cavendish experiment}}

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/cm³cm3}} ({{sfrac|4|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|doi-access=free }}</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<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:

Based on this, Hutton's 1778 result is equivalent to {{nowrap|{{math|''G''}} ≈ {{val|8|e=-11|u=m³⋅kg⁻¹⋅s⁻²}}}}.

Cavendish's stated aim was the "weighing of Earth", that is, determining the average density of Earth and the [[Earth's mass]]. His result, {{nowrapmath|1=''&rho;ρ''_🜨 = {{val|5.448|(33)|u=g·.cm⁻³-3}}}}, corresponds to value of {{nowrapmath|1={{math|''G''}} = {{val|6.74|(4)|e=-11|u=m³⋅kg⁻¹⋅s⁻²}}}}. It is surprisingly accurate, about 1% above the modern value (comparable to the claimed relative standard uncertainty of 0.6%).<ref>2014 [[CODATA]] value {{nowrapmath|{{val|6.674|e=−11|u=m³⋅kg⁻¹⋅s⁻²}}}}.</ref>

Measurements with pendulums were made by [[Francesco Carlini]] (1821, {{val|4.39|u=g/cm³cm3}}), [[Edward Sabine]] (1827, {{val|4.77|u=g/cm³cm3}}), Carlo Ignazio Giulio (1841, {{val|4.95|u=g/cm³cm3}}) and [[George Biddell Airy]] (1854, {{val|6.6|u=g/cm³cm3}}).<ref>{{cite book | last = Poynting | first = John Henry | title = The Mean Density of the Earth | publisher = Charles Griffin | date = 1894 | location = London | pages = [https://archive.org/details/meandensityeart00poyngoog/page/n44 22]–24 | url = https://archive.org/details/meandensityeart00poyngoog }}</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^&minus;-3}},<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^&minus;-3}}.<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/cm³cm3}}) and [[Thomas Corwin Mendenhall]] (1880, {{val|5.77|u=g/cm³cm3}}).<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=https://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^&minus;-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 by [[C. V. Boys]] (1895)<ref>{{cite journal | last=Boys | first=C. V. | title=On the Newtonian Constant of Gravitation | journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences | publisher=The Royal Society | volume=186 | date=1895-01-01 | issn=1364-503X | doi=10.1098/rsta.1895.0001 | bibcode=1895RSPTA.186....1B | pages=1–72| doi-access=free }}</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 relative standard uncertainty of 0.03%, {{val|6.649|(2)|e=−11|u=m³⋅kg⁻¹⋅s⁻²}} 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=m³⋅kg⁻¹⋅s⁻²}}. 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=m³⋅kg⁻¹⋅s⁻²}} 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=m³⋅kg⁻¹⋅s⁻²}} with an relative uncertainty of 0.2%.

[[Paul R. Heyl]] (1930) published the value of {{val|6.670|(5)|e=−11|u=m³⋅kg⁻¹⋅s⁻²}} (relative uncertainty 0.1%),<ref>{{cite journal |first=P. R. |last=Heyl |author-link=Paul R. Heyl |title=A redetermination of the constant of gravitation |journal= Bureau of Standards Journal of Research|volume=5 |issue=6 |year=1930 |pages=1243–1290|doi=10.6028/jres.005.074 |doi-access=free }}<!--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=m³⋅kg⁻¹⋅s⁻²}} (relative uncertainty 0.045% = 450 &nbsp;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 1000&nbsp;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 |year=2012 |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 |access-date=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|arxiv=1203.5425 }} 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 relative 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 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&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|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.|s2cid = 103378639|citeseerx = 10.1.1.150.3858}}</ref> For the 2014 update, CODATA reduced the uncertainty to 46 &nbsp;ppm, less than half the 2010 value, and one order of magnitude below the 1969 recommendation.

!scope="col"| ''G'' <br />(10{{sup|−11}}·&nbsp;m{{sup|3}}⋅kg{{sup|−1}}⋅s{{sup|−2}})

! scope="col"|StandardRelative standard uncertainty

| 6.6732(31) || 460 ppm || <ref>{{cite journal | last1=Taylor | first1=B. N. | last2=Parker | first2=W. H. | last3=Langenberg | first3=D. N. | title=Determination of ''e''/''h'', Using Macroscopic Quantum Phase Coherence in Superconductors: Implications for Quantum Electrodynamics and the Fundamental Physical Constants | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=41 | issue=3 | date=1969-07-01 | issn=0034-6861 | doi=10.1103/revmodphys.41.375 | bibcode=1969RvMP...41..375T | pages=375–496}}</ref>

| 6.6720(49) || 730 ppm || <ref>{{cite journal | last1=Cohen | first1=E. Richard | last2=Taylor | first2=B. N. | title=The 1973 Least‐SquaresLeast-Squares Adjustment of the Fundamental Constants | journal=Journal of Physical and Chemical Reference Data | publisher=AIP Publishing | volume=2 | issue=4 | year=1973 | issn=0047-2689 | doi=10.1063/1.3253130 | bibcode=1973JPCRD...2..663C | pages=663–734| hdl=2027/pst.000029951949 | hdl-access=free }}</ref>

| 6.673(10) || 1,5001500 ppm || <ref>{{cite journal | last1=Mohr | first1=Peter J. | last2=Taylor | first2=Barry N. | title=CODATA recommended values of the fundamental physical constants: 1998 | journal=Reviews of Modern Physics | volume=72 | issue=2 | year=2012 | issn=0034-6861 | doi=10.1103/revmodphys.72.351 | bibcode=2000RvMP...72..351M | pages=351–495| arxiv=1203.5425 }}</ref>

| 6.67408(31) || 46 ppm || <ref>{{cite journal | last1=Mohr | first1=Peter J. | last2=Newell | first2=David B. | last3=Taylor | first3=Barry N. | title=CODATA Recommended Values of the Fundamental Physical Constants: 2014 | journal=Journal of Physical and Chemical Reference Data | volume=45 | issue=4 | year=2016 | pages=1527–1605 | issn=0047-2689 | doi=10.1063/1.4954402 | bibcode=2016JPCRD..45d3102M | arxiv=1203.5425 }}</ref>

| 6.67430(15) || 22 ppm || <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 [[Svetlana Kotochigova|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={{mvarmath|''G''}} = {{val|6.693|(34)|e=−11|u=m³⋅kg⁻¹⋅s⁻²}}}}, 0.28% (2800 &nbsp;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 |s2cid=6271411 |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={{mvarmath|''G''}} = {{val|6.67191|(99)|e=−11|u=m³⋅kg⁻¹⋅s⁻²}}}}.<ref>
{{cite journal

|last1=Rosi |first1=G.
|last2=Sorrentino |first2=F.
|last3=Cacciapuoti |first3=L.

|last4=Prevedelli |first4=M.
|last5=Tino |first5=G. M.

|s2cid=4469248 |title=Precision measurement of the Newtonian gravitational constant using cold atoms

|journal=Nature |volume=510 |issue=7506 |pages=478–480
|date=18 June 2014

|url=https://www.nature.com/articles/nature13507.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.nature.com/articles/nature13507.pdf |archive-date=2022-10-09
|url-status=live

2010 interval, 6.67384(80) [2010] (differences 0.00193(127) and 0.00024(86)).

{{seefurther|Time-variation of fundamental constants}}

'''; Footnotes''' :

{{Notelistnotelist|45em}}

'''; Citations''' :

{{Reflistreflist|30em}}

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