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| {{further|Gravity of Earth|Standard gravity}}
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| [[Image:NewtonsLawOfUniversalGravitation.svg|thumb|right|300px|The gravitational constant ''G'' is a key quantity in [[Newton's law of universal gravitation]].]]
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| The '''gravitational constant''', approximately 6.67×10<sup>−11</sup> N·(m/kg)<sup>2</sup> and denoted by letter ''G'', is an [[empirical]] [[physical constant]] involved in the calculation(s) of [[gravitational]] force between two bodies. It usually appears in [[Sir Isaac Newton]]'s [[Newton's law of universal gravitation|law of universal gravitation]], and in [[Albert Einstein| Albert Einstein's]] [[theory of general relativity]]. It is also known as the '''universal gravitational constant''', '''Newton's constant''', and colloquially as '''Big G'''.<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=2002-12-23 |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> It should not be confused with "little g" (''[[Gravity of Earth|g]]''), which is the local gravitational field (equivalent to the free-fall acceleration<ref>Fundamentals of Physics 8ed, Halliday/Resnick/Walker, ISBN 978-0-470-04618-0 p336</ref>), especially that at the Earth's surface.
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| ==Laws and constants==
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| According to the law of universal gravitation, the attractive [[force]] (''F'') between two bodies is proportional to the product of their [[mass]]es (''m''<sub>1</sub> and ''m''<sub>2</sub>), and inversely proportional to the square of the distance, ''r'', ([[inverse-square law]]) between them:
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| :<math>F = \frac{Gm_1 m_2}{r^2}\ </math>
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| The [[constant of proportionality]], ''G'', is the gravitational constant.
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| The gravitational constant is a physical constant that is difficult to measure with high accuracy.<ref name=gillies>{{Citation |author=George T. Gillies |title=The Newtonian gravitational constant: recent measurements and related studies |journal=Reports on Progress in Physics |year=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> In [[International System of Units|SI]] units, the 2010 [[Committee on Data for Science and Technology|CODATA]]-recommended value of the gravitational constant (with [[standard uncertainty]] in parentheses) is:<ref name="2010 CODATA">P.J. Mohr, B.N. Taylor, and D.B. Newell (2011), "The 2010 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 6.0). This database was developed by J. Baker, M. Douma, and S. Kotochigova. Available: http://physics.nist.gov/constants [Thursday, 02-Jun-2011 21:00:12 EDT]. National Institute of Standards and Technology, Gaithersburg, MD 20899.</ref>
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| :<math> G = 6.67384(80) \times 10^{-11} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2} = 6.67384(80) \times 10^{-11} \ {\rm N}\, {\rm (m/kg)^2}</math>
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| with relative standard uncertainty {{val|1.2|e=-4}}.<ref name="2010 CODATA" />
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| ==Dimensions, units, and magnitude==
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| The dimensions assigned to the gravitational constant in the equation above—[[length]] cubed, divided by [[mass]], and by [[time]] squared (in SI units, meters cubed per [[gram|kilogram]] per second squared)—are those needed to balance the units of measurements in gravitational equations. However, these dimensions have fundamental significance in terms of [[Planck units]]; when expressed in SI units, the gravitational constant is dimensionally and numerically equal to the cube of the [[Planck length]] divided by the product of the [[Planck mass]] and the square of [[Planck time]].
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| In [[natural units]], of which [[Planck units]] are a common example, ''G'' and other physical constants such as ''c'' (the [[speed of light]]) may be set equal to 1.
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| In many secondary school texts, the dimensions of G are derived from force in order to assist student comprehension:
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| :<math> G \approx 6.674 \times 10^{-11} {\rm \ N}\, {\rm (m/kg)^2}.</math>
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| In [[centimetre-gram-second system of units|cgs]], G can be written as:
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| :<math> G\approx 6.674 \times 10^{-8} {\rm \ cm}^3 {\rm g}^{-1} {\rm s}^{-2}.</math>
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| G can also be given as:
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| :<math> G\approx 0.8650 {\rm \ cm}^3 {\rm g}^{-1} {\rm hr}^{-2}.</math>
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| Given the fact that the period ''P'' of an object in circular orbit around a spherical object obeys
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| :<math> GM=3\pi V/P^2</math>
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| where V is the volume inside the radius of the orbit, we see that
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| :<math> P^2=\frac{3\pi}{G}\frac{V}{M}\approx 10.896 {\rm\ hr}^2 {\rm g\ }{\rm cm}^{-3}\frac{V}{M}.</math>
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| 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.
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| In some fields of [[astrophysics]], where distances are measured in [[parsec]]s (pc), velocities in kilometers per second (km/s) and masses in [[Solar mass|solar units]] {{nowrap|(<math>M_\odot</math>)}}, it is useful to express ''G'' as:
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| :<math> G \approx 4.302 \times 10^{-3} {\rm \ pc}\, M_\odot^{-1} \, {\rm (km/s)}^2. \, </math>
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| The gravitational force is extremely weak compared with other [[fundamental forces]]. For example, the gravitational force between an [[electron]] and [[proton]] one meter apart is approximately 10<sup>−67</sup> [[newton (unit)|N]], whereas the [[electromagnetic force]] between the same two particles is approximately 10<sup>−28</sup> N. Both these forces are weak when compared with the forces we are able to experience directly, but 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 [[mass of the Sun]] compared to a microgram.
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| ==History of measurement==
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| The gravitational constant appears in [[Newton's law of universal gravitation]], but it was not measured until seventy-one years after Newton's death by [[Henry Cavendish]] with his [[Cavendish experiment]], performed in 1798 (''Philosophical Transactions'' 1798). Cavendish measured ''G'' implicitly, using a [[Torsion spring#Torsion balance|torsion balance]] invented by the geologist Rev. [[John Michell]]. 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. Cavendish's aim was not actually to measure the gravitational constant, but rather to measure the Earth's density relative to water, through the precise knowledge of the gravitational interaction. In retrospect, the density that Cavendish calculated implies a value for ''G'' of {{nowrap|6.754 × 10<sup>−11</sup> m<sup>3</sup> kg<sup>−1</sup> s<sup>−2</sup>}}.<ref>{{Citation |author=Brush, Stephen G.; Holton, Gerald James |title=Physics, the human adventure: from Copernicus to Einstein and beyond |publisher=Rutgers University Press |location=New Brunswick, N.J |year=2001 |pages= 137 |isbn=0-8135-2908-5 |oclc= |doi= |accessdate=}}</ref>
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| The accuracy of the measured value of ''G'' has increased only modestly since the original Cavendish experiment. ''G'' is quite difficult to measure, as 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. Published values of ''G'' have varied rather broadly, and some recent measurements of high precision are, in fact, mutually exclusive.<ref name=gillies/><ref name=codata2002>{{Citation |author1=Peter J. Mohr |author2=Barry N. Taylor |title=CODATA recommended values of the fundamental physical constants: 2002 |journal=Reviews of Modern Physics |year=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 |format=PDF |accessdate=2006-07-01 |doi=10.1103/RevModPhys.77.1 |bibcode=2005RvMP...77....1M}}. Section Q (pp. 42–47) describes the mutually inconsistent measurement experiments from which the CODATA value for ''G'' was derived.</ref>
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| In the January 5, 2007 issue of ''[[Science (journal)|Science]]'' (page 74), the report "Atom Interferometer Measurement of the Newtonian Constant of Gravity" (J. B. Fixler, G. T. Foster, J. M. McGuirk, and M. A. Kasevich) describes a new measurement of the gravitational constant. According to the abstract: "Here, we report a value of G = 6.693 × 10<sup>−11</sup> cubic meters per kilogram second squared, with a standard error of the mean of ±0.027 × 10<sup>−11</sup> and a systematic error of ±0.021 × 10<sup>−11</sup> cubic meters per kilogram second squared."<ref>{{Citation |author1=J. B. Fixler |author2=G. T. Foster |author3=J. M. McGuirk |author4=M. A. Kasevich |title=Atom Interferometer Measurement of the Newtonian Constant of Gravity |url=http://www.sciencemag.org/cgi/content/abstract/315/5808/74 |date=2007-01-05 |volume=315 |issue=5808 |pages= 74–77 |doi=10.1126/science.1135459|journal=Science |pmid=17204644|bibcode = 2007Sci...315...74F }}</ref>
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| <!--
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| Debunked because of the wrong use of the Planck Constant, resulting in wrong dimensional analysis, in the arXiv document.
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| See:eprint arXiv:1111.6941
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| In this study the Newton's gravity constant G has been measured at scale of atoms by the quantum resonance method other than Cavendish's torsion-balance way. The analyzed and experiment model is based on the "hyperfine splitting" of the ground energy state of hydrogen atom. The result is that only in empty space the G is constant, in hydrogen atom (near the distance of [[Bohr radius]]), the gravitational constant G=6.7192878(13)\times10-11 N\cdotm2/kg2; in vacuum, G=6.6722779(13)\times10-11 N\cdotm2/kg2. This method of experiment has eliminated all sources of possible experimental errors associated with the classical and existed setup, and with a precision better than existing experiments.
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| -->
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| ==The ''GM'' product==
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| {{Main|Standard gravitational parameter}}
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| The quantity ''GM''—the product of the gravitational constant and the mass of a given astronomical body such as the Sun or the Earth—is known as the standard gravitational parameter and is denoted <math>\scriptstyle \mu\!</math>. Depending on the body concerned, it may also be called the geocentric or heliocentric gravitational constant, among other names.
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| This quantity gives a convenient simplification of various gravity-related formulas. Also, for celestial bodies such as the Earth and the Sun, the value of the product ''GM'' is known much more accurately than each factor independently. Indeed, the limited accuracy available for ''G'' often limits the accuracy of scientific determination of such masses in the first place.
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| For [[Earth]], using <math>M_\oplus</math> as the symbol for the mass of the Earth, we have
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| : <math> \mu = GM_\oplus = ( 398 600.4418 \pm 0.0008 ) \ \mbox{km}^{3} \ \mbox{s}^{-2}.</math>
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| Calculations in [[celestial mechanics]] can also be carried out using the unit of [[solar mass]] rather than the standard SI unit kilogram. In this case we use the [[Gaussian gravitational constant]] ''k'', where
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| :<math> {k = 0.01720209895 \ A^{\frac{3}{2}} \ D^{-1} \ S^{-\frac{1}{2}} } \ </math>
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| and
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| :<math>A\!</math> is the [[astronomical unit]];
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| :<math>D\!</math> is the [[mean solar day]];
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| :<math>S\!</math> is the [[solar mass]]. | |
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| If instead of mean solar day we use the [[sidereal year]] as our time unit, the value of ''ks'' is very close to 2[[pi|π]] (''k'' = 6.28315).
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| 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 by [[gravitational lensing]], in [[Kepler's laws of planetary motion]], and in the formula for [[escape velocity]].
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| ==See also==
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| {{Portal|Physics}}
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| <div style="column-count:2;-moz-column-count:2;-webkit-column-count:2">
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| *[[Dirac large numbers hypothesis]]
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| *[[Accelerating universe]]
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| *[[Orbital period#Small body orbiting a central body|Gravity expressed in terms of orbital period]]
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| *[[Lunar Laser Ranging experiment]]
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| *[[Cosmological constant]]
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| *[[Gravitational coupling constant]]
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| *[[Strong gravitational constant]]
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| </div>
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| {{Refbegin}}
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| * E. Myles Standish. "Report of the IAU WGAS Sub-group on Numerical Standards". In ''Highlights of Astronomy'', I. Appenzeller, ed. Dordrecht: Kluwer Academic Publishers, 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])''
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| * {{Citation |author1=Jens H. Gundlach |author2=Stephen M. Merkowitz |title=[[arXiv:gr-qc/0006043v1|Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback]] |journal=Physical Review Letters |volume=85 |issue=14 |pages= 2869–2872 |year=2000 |doi=10.1103/PhysRevLett.85.2869|pmid=11005956 |bibcode=2000PhRvL..85.2869G|arxiv = gr-qc/0006043 }}
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| {{Refend}}
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| ==External links==
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| * [http://physics.nist.gov/cgi-bin/cuu/Value?bg Newtonian constant of gravitation ''G''] at the [[National Institute of Standards and Technology]] [http://physics.nist.gov/cuu References on Constants, Units, and Uncertainty]
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| * [http://www.npl.washington.edu/eotwash/bigG The Controversy over Newton's Gravitational Constant] — additional commentary on measurement problems
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| *[http://www.physik.uni-wuerzburg.de/~rkritzer/grav.pdf The Gravitational Constant]
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| [[Category:Gravitation]]
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| [[Category:Fundamental constants]]
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