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Physical constant relating the gravitational force betwixt objects to their mass and distance

Notations for the gravitational constant

Values of 1000	Units
6.67430(15)×10−11 [1]	N m2⋅kg–2
6.674thirty(fifteen)×ten−8	dyne cm2⋅g–2
four.30091(25)×10−3	pc⋅M ⊙ –1⋅(km/due south)2

The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant),^[a] denoted by the upper-case letter G , is an empirical physical constant involved in the calculation of gravitational furnishings in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general theory of relativity.

In Newton's law, information technology is the proportionality constant connecting the gravitational forcefulness between two bodies with the product of their masses and the changed square of their distance. In the Einstein field equations, information technology quantifies the relation between the geometry of spacetime and the energy–momentum tensor (as well referred to every bit the stress–free energy tensor).

The measured value of the constant is known with some certainty to four significant digits. In SI units, its value is approximately vi.674×x⁻¹¹ 1000³⋅kg⁻¹⋅s⁻² .^[ane]

The modernistic notation of Newton'south police involving K was introduced in the 1890s by C. 5. Boys. The outset implicit measurement with an accuracy within near 1% is attributed to Henry Cavendish in a 1798 experiment.^[b]

Definition [edit]

Co-ordinate to Newton's law of universal gravitation, the attractive force ( F ) between two point-like bodies is direct proportional to the product of their masses ( m _ane and 1000 ₂ ) and inversely proportional to the square of the distance, r , betwixt their centers of mass.:

$${\displaystyle F=Grand{\frac {m_{one}m_{2}}{r^{2}}}.}$$

The constant of proportionality, G , is the gravitational constant. Colloquially, the gravitational abiding is also called "Big 1000", singled-out from "pocket-sized g" ( chiliad ), which is the local gravitational field of Earth (equivalent to the free-fall acceleration).^{[two] [3]} Where ${\displaystyle M_{\oplus }}$ is the mass of the World and ${\displaystyle r_{\oplus }}$ is the radius of the Earth, the ii quantities are related by:

$${\displaystyle yard={\frac {GM_{\oplus }}{r_{\oplus }^{2}}}.}$$

The gravitational abiding appears in the Einstein field equations of general relativity,^{[4] [five]}

$${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }\,,}$$

where Thousand _μν is the Einstein tensor, Λ is the cosmological constant, k_μν is the metric tensor, T_μν is the stress–energy tensor, and κ is a abiding originally introduced past Einstein that is directly related to the Newtonian constant of gravitation:^{[5] [6] [c]}.

$${\displaystyle \kappa ={\frac {8\pi Yard}{c^{two}}}\approx one.866\times x^{-26}\mathrm {\,thou{\cdot }kg^{-1}} .}$$

Value and uncertainty [edit]

The gravitational abiding is a concrete constant that is difficult to measure with loftier accuracy.^[vii] This is because the gravitational strength is an extremely weak force as compared to other fundamental forces.^[d]

In SI units, the 2018 CODATA-recommended value of the gravitational abiding (with standard doubtfulness in parentheses) is:^{[1] [8]}

$${\displaystyle Chiliad=half-dozen.67430(xv)\times 10^{-11}{\rm {\ one thousand^{3}{\cdot }kg^{-one}{\cdot }s^{-2}}}}$$

This corresponds to a relative standard incertitude of 2.ii×10⁻⁵ (22 ppm).

Natural units [edit]

The gravitational constant is a defining constant in some systems of natural units, particularly geometrized unit systems, such every bit Planck units and Stoney units. When expressed in terms of such units, the value of the gravitational abiding volition generally take a numeric value of i or a value close to it. Due to the meaning dubiety in the measured value of 1000 in terms of other known fundamental constants, a similar level of dubiousness will show upward in the value of many quantities when expressed in such a unit arrangement.

Orbital mechanics [edit]

In astrophysics, it is user-friendly to measure distances in parsecs (pc), velocities in kilometres per second (km/southward) and masses in solar units M _⊙ . In these units, the gravitational constant is:

$${\displaystyle G\approx 4.3009\times ten^{-3}{\rm {}}{\frac {pc}{M_{\odot }}}{\rm {\ (km/s)^{2}}}.\,}$$

For situations where tides are important, the relevant length scales are solar radii rather than parsecs. In these units, the gravitational abiding is:

$${\displaystyle Thousand\approx one.90809\times x^{5}R_{\odot }M_{\odot }^{-ane}{\rm {\ (km/due south)^{2}}}.\,}$$

In orbital mechanics, the period P of an object in circular orbit effectually a spherical object obeys

$${\displaystyle GM={\frac {three\pi Five}{P^{2}}}}$$

where V is the volume inside the radius of the orbit. It follows that

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

This mode of expressing G shows the human relationship betwixt the boilerplate density of a planet and the period of a satellite orbiting just above its surface.

For elliptical orbits, applying Kepler'due south 3rd law, expressed in units characteristic of Earth's orbit:

${\displaystyle K=4\pi ^{2}{\rm {\ AU^{three}{\cdot }yr^{-2}}}\ M^{-i}\approx 39.478{\rm {\ AU^{three}{\cdot }yr^{-2}}}\ M_{\odot }^{-ane},}$

where distance is measured in terms of the semi-major axis of Earth's orbit (the astronomical unit, AU), time in years, and mass in the total mass of the orbiting organization ( M = Grand _☉ + M _Globe + M _☾ ^[eastward]).

The above equation is exact simply within the approximation of the Earth's orbit around the Dominicus as a two-torso problem in Newtonian mechanics, the measured quantities incorporate corrections from the perturbations from other bodies in the solar organisation and from general relativity.

From 1964 until 2012, all the same, information technology was used as the definition of the astronomical unit and thus held by definition:

$${\displaystyle one\ \mathrm {AU} =\left({\frac {GM}{4\pi ^{two}}}{\rm {yr}}^{2}\right)^{\frac {1}{3}}\approx 1.495979\times 10^{11}{\rm {g}}.}$$

Since 2012, the AU is defined as 1.495978 707 ×ten^eleven m exactly, and the equation can no longer be taken as holding precisely.

The quantity GM —the production of the gravitational constant and the mass of a given astronomical torso such every bit the Dominicus or Globe—is known as the standard gravitational parameter (besides denoted μ ). The standard gravitational parameter GM appears as above in Newton's constabulary of universal gravitation, too as in formulas for the deflection of light caused by gravitational lensing, in Kepler'south laws of planetary motion, and in the formula for escape 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 for GM

Body	μ = GM	Value	Relative uncertainty
Sun	Thou M ☉	1.327124 400 18(8)×1020 giii⋅southward−2 [ix]	six×10−11
Earth	G M Earth	three.986004 418(eight)×x14 m3⋅s−two [10]	two×10−nine

Calculations in celestial mechanics can as well exist carried out using the units of solar masses, hateful solar days and astronomical units rather than standard SI units. For this purpose, the Gaussian gravitational constant was historically in widespread utilise, chiliad = 0.017202 098 95 , expressing the mean angular velocity of the Lord's day–Earth system measured in radians per day.^{[ commendation needed ]} The use of this abiding, and the implied definition of the astronomical unit discussed above, has been deprecated by the IAU since 2012.^{[ citation needed ]}

History of measurement [edit]

Early history [edit]

The existence of the constant is implied in Newton'southward law of universal gravitation as published in the 1680s (although its notation as G dates to the 1890s),^[11] but is not calculated in his Philosophiæ Naturalis Principia Mathematica where it postulates the changed-square law of gravitation. In the Principia, Newton considered the possibility of measuring gravity's forcefulness past measuring the deflection of a pendulum in the vicinity of a large loma, simply thought that the effect would be too small to be measurable.^[12] Nevertheless, he estimated the order of magnitude of the constant when he surmised that "the hateful density of the globe might be v or six times as groovy every bit the density of water", which is equivalent to a gravitational constant of the order:^[13]

Chiliad ≈ (half dozen.7±0.6)×10^−xi m³⋅kg^–1⋅s⁻ⁱⁱ

A measurement was attempted in 1738 by Pierre Bouguer and Charles Marie de La Condamine in their "Peruvian expedition". Bouguer downplayed the significance of their results in 1740, suggesting that the experiment had at least proved that the Earth could not be a hollow trounce, as some thinkers of the day, including Edmond Halley, had suggested.^[fourteen]

The Schiehallion experiment, proposed in 1772 and completed in 1776, was the showtime successful measurement of the mean density of the World, and thus indirectly of the gravitational abiding. The consequence reported past Charles Hutton (1778) suggested a density of iv.5 g/cm^three (4+ ane / 2 times the density of water), about 20% below the modern value.^[xv] 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. Every bit discussed in a higher place, establishing the average density of Earth is equivalent to measuring the gravitational constant, given Earth's mean radius and the mean gravitational acceleration at Earth'south surface, by setting^[xi]

$${\displaystyle M=grand{\frac {R_{\oplus }^{2}}{M_{\oplus }}}={\frac {3g}{four\pi R_{\oplus }\rho _{\oplus }}}.}$$

Based on this, Hutton's 1778 result is equivalent to G ≈ viii×10⁻¹¹ m³⋅kg^–one⋅s⁻² .

Diagram of torsion balance used in the Cavendish experiment performed by Henry Cavendish in 1798, to measure 1000, with the assistance of a pulley, large balls hung from a frame were rotated into position next to the small balls.

The start direct measurement of gravitational attraction between two bodies in the laboratory was performed in 1798, seventy-1 years later Newton's decease, past Henry Cavendish.^[sixteen] He determined a value for G implicitly, using a torsion balance invented by the geologist Rev. John Michell (1753). He used a horizontal torsion beam with lead assurance whose inertia (in relation to the torsion abiding) he could tell by timing the axle's oscillation. Their faint attraction to other assurance placed alongside the beam was detectable by the deflection it caused. In spite of the experimental design being due to Michell, the experiment is at present known every bit the Cavendish experiment for its first successful execution past Cavendish.

Cavendish'southward stated aim was the "weighing of Earth", that is, determining the average density of World and the Earth'southward mass. His result, ρ _🜨 = 5.448(33) thou·cm^−three , corresponds to value of 1000 = half-dozen.74(iv)×x^−xi g³⋅kg^–1⋅southward⁻² . It is surprisingly authentic, about i% above the modern value (comparable to the claimed standard doubt of 0.6%).^[17]

19th century [edit]

The accurateness of the measured value of G has increased but modestly since the original Cavendish experiment.^[18] 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 by Francesco Carlini (1821, iv.39 g/cm³ ), Edward Sabine (1827, 4.77 g/cm³ ), Carlo Ignazio Giulio (1841, four.95 g/cm³ ) and George Biddell Airy (1854, 6.half-dozen k/cmⁱⁱⁱ ).^[19]

Cavendish's experiment was first repeated past Ferdinand Reich (1838, 1842, 1853), who found a value of 5.5832(149) g·cm⁻³ ,^[20] which is really worse than Cavendish's result, differing from the modern value by 1.v%. Cornu and Baille (1873), found v.56 g·cm⁻³ .^[21]

Cavendish's experiment proved to result in more reliable measurements than pendulum experiments of the "Schiehallion" (deflection) type or "Peruvian" (menses every bit a function of altitude) type. Pendulum experiments still connected to be performed, by Robert von Sterneck (1883, results between 5.0 and six.3 grand/cm³ ) and Thomas Corwin Mendenhall (1880, 5.77 g/cm³ ).^[22]

Cavendish's event was commencement improved upon by John Henry Poynting (1891),^[23] who published a value of 5.49(3) thousand·cm⁻³ , differing from the modern value by 0.2%, but compatible with the modern value within the cited standard uncertainty of 0.55%. In improver to Poynting, measurements were made by C. V. Boys (1895)^[24] and Carl Braun (1897),^[25] with compatible results suggesting Yard = half-dozen.66(1)×ten⁻¹¹ one thousand³⋅kg^−ane⋅s⁻ⁱⁱ . The modern notation involving the abiding G was introduced by Boys in 1894^[11] and becomes standard by the end of the 1890s, with values usually cited in the cgs system. Richarz and Krigar-Menzel (1898) attempted a repetition of the Cavendish experiment using 100,000 kg of lead for the attracting mass. The precision of their upshot of 6.683(11)×ten⁻¹¹ g³⋅kg^−one⋅s⁻ⁱⁱ was, however, of the aforementioned social club of magnitude every bit the other results at the fourth dimension.^[26]

Arthur Stanley Mackenzie in The Laws of Gravitation (1899) reviews the work washed in the 19th century.^[27] Poynting is the writer of the article "Gravitation" in the Encyclopædia Britannica Eleventh Edition (1911). Here, he cites a value of G = half-dozen.66×10⁻¹¹ mⁱⁱⁱ⋅kg⁻¹⋅s⁻² with an dubiousness of 0.ii%.

Modern value [edit]

Paul R. Heyl (1930) published the value of vi.670(v)×10⁻¹¹ g³⋅kg^–1⋅southward⁻² (relative doubt 0.1%),^[28] improved to vi.673(3)×x^−xi m³⋅kg^–1⋅s⁻² (relative doubtfulness 0.045% = 450 ppm) in 1942.^[29]

Published values of Thou derived from high-precision measurements since the 1950s take remained uniform with Heyl (1930), but within the relative dubiousness of about 0.1% (or ane,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.^{[seven] [xxx]} Establishing a standard value for Chiliad with a standard uncertainty better than 0.i% 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. Just the continued publication of conflicting measurements led NIST to considerably increment the standard uncertainty in the 1998 recommended value, past a factor of 12, to a standard doubt of 0.15%, larger than the i given by Heyl (1930).

The uncertainty was again lowered in 2002 and 2006, but again raised, by a more conservative 20%, in 2010, matching the standard uncertainty of 120 ppm published in 1986.^[31] For the 2014 update, CODATA reduced the dubiousness to 46 ppm, less than half the 2010 value, and ane order of magnitude below the 1969 recommendation.

The following table shows the NIST recommended values published since 1969:

Timeline of measurements and recommended values for Thousand since 1900: values recommended based on a literature review are shown in red, individual torsion balance experiments in bluish, other types of experiments in green.

Recommended values for G

Year	One thousand (10−eleven·miii⋅kg−i⋅due south−2)	Standard uncertainty	Ref.
1969	6.6732(31)	460 ppm	[32]
1973	6.6720(49)	730 ppm	[33]
1986	6.67449(81)	120 ppm	[34]
1998	6.673(10)	1,500 ppm	[35]
2002	6.6742(10)	150 ppm	[36]
2006	6.67428(67)	100 ppm	[37]
2010	half dozen.67384(lxxx)	120 ppm	[38]
2014	vi.67408(31)	46 ppm	[39]
2018	6.67430(15)	22 ppm	[40]

In the Jan 2007 issue of Science, Fixler et al. described a measurement of the gravitational abiding past a new technique, atom interferometry, reporting a value of G = half dozen.693(34)×10⁻¹¹ thou³⋅kg⁻ⁱ⋅southward⁻² , 0.28% (2800 ppm) college than the 2006 CODATA value.^[41] An improved cold cantlet measurement by Rosi et al. was published in 2014 of 1000 = 6.67191(99)×10⁻¹¹ grand³⋅kg⁻¹⋅s⁻ⁱⁱ .^{[42] [43]} Although much closer to the accustomed value (suggesting that the Fixler et. al. measurement was erroneous), this result was 325 ppm beneath the recommended 2014 CODATA value, with not-overlapping standard uncertainty intervals.

As of 2018, efforts to re-evaluate the alien results of measurements are underway, coordinated past NIST, notably a repetition of the experiments reported by Quinn et al. (2013).^[44]

In August 2018, a Chinese research group appear new measurements based on torsion balances, 6.674184(78)×ten⁻¹¹ m³⋅kg^–1⋅s⁻² and 6.674484(78)×10⁻¹¹ mⁱⁱⁱ⋅kg^–one⋅southward⁻² based on ii dissimilar methods.^[45] These are claimed as the most authentic measurements ever made, with a standard uncertainties cited as depression as 12 ppm. The difference of 2.7σ between the two results suggests there could be sources of error unaccounted for.

Suggested time-variation [edit]

A controversial 2015 study of some previous measurements of G , by Anderson et al., suggested that most of the mutually exclusive values in loftier-precision measurements of G can be explained by a periodic variation.^[46] The variation was measured every bit having a period of 5.nine years, similar to that observed in length-of-day (LOD) measurements, hinting at a common concrete crusade that is non necessarily a variation in 1000 . A response was produced past some of the original authors of the K measurements used in Anderson et al.^[47] This response notes that Anderson et al. not only omitted measurements, simply that they also used the time of publication rather than the fourth dimension the experiments were performed. A plot with estimated time of measurement from contacting original authors seriously degrades the length of day correlation. Likewise, consideration of the data collected over a decade past Karagioz and Izmailov shows no correlation with length of day measurements.^{[47] [48]} As such, the variations in Yard well-nigh probable ascend from systematic measurement errors which take not properly been deemed for. Under the assumption that the physics of type Ia supernovae are universal, assay of observations of 580 of them has shown that the gravitational constant has varied past less than one part in x billion per year over the last nine billion years according to Mould et al. (2014).^[49]

Come across also [edit]

• Gravity of Earth
• Standard gravity
• Gaussian gravitational constant
• Orbital mechanics
• Escape velocity
• Gravitational potential
• Gravitational wave
• Stiff gravitational constant
• Dirac large numbers hypothesis
• Accelerating universe
• Lunar Light amplification by stimulated emission of radiation Ranging experiment
• Cosmological constant

References [edit]

Footnotes

1. ^ "Newtonian constant of gravitation" is the proper noun introduced for G by Boys (1894). Apply of the term by T.Eastward. Stern (1928) was misquoted equally "Newton'due south abiding of gravitation" in Pure Science Reviewed for Profound and Unsophisticated Students (1930), in what is apparently the first use of that term. Utilize of "Newton's constant" (without specifying "gravitation" or "gravity") is more than recent, as "Newton'southward abiding" was besides used for the rut transfer coefficient in Newton'southward law of cooling, but has by now become quite mutual, eastward.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 abiding", appears to accept been common in the 1970s to 1980s, especially in (translations from) Soviet-era Russian literature, e.g. Sagitov (1970 [1969]), Soviet Physics: Uspekhi xxx (1987), Bug ane–6, p. 342 [etc.]. "Cavendish abiding" and "Cavendish gravitational constant" is also used in Charles W. Misner, Kip South. Thorne, John Archibald Wheeler, "Gravitation", (1973), 1126f. Colloquial use of "Big G", every bit opposed to "piddling g" for gravitational acceleration dates to the 1960s (R.West. Fairbridge, The encyclopedia of atmospheric sciences and astrogeology, 1967, p. 436; note use of "Big G'south" vs. "footling g's" as early as the 1940s of the Einstein tensor G _μν vs. the metric tensor yard _μν , Scientific, medical, and technical books published in the U.s.a.: 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).
2. ^ Cavendish determined the value of Grand indirectly, by reporting a value for the Earth'due south mass, or the average density of Earth, as 5.448 1000⋅cm⁻³ .
3. ^ Depending on the pick of definition of the Einstein tensor and of the stress–energy tensor it tin alternatively be defined as κ = 8πK / c ⁴ ≈ 2.077×10⁻⁴³ s²⋅m⁻¹⋅kg^−ane
4. ^ For instance, the gravitational strength between an electron and a proton 1 m apart is approximately 10⁻⁶⁷ N, whereas the electromagnetic forcefulness between the same two particles is approximately ten⁻²⁸ N. The electromagnetic force in this instance is in the order of ten³⁹ times greater than the strength of gravity—roughly the same ratio as the mass of the Sun to a microgram.
5. ^ M ≈ 1.000003040433 M _☉ , so that Thousand = M _☉ can exist used for accuracies of five or fewer pregnant digits.

Citations

1. ^ ^{a b c} "2018 CODATA Value: Newtonian abiding of gravitation". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
2. ^ Gundlach, Jens H.; Merkowitz, Stephen M. (23 December 2002). "University of Washington Big G Measurement". Astrophysics Science Division. Goddard Infinite Flight Center. Since Cavendish get-go measured Newton'south Gravitational constant 200 years ago, "Large One thousand" remains one of the most elusive constants in physics
3. ^ Halliday, David; Resnick, Robert; Walker, Jearl (September 2007). Fundamentals of Physics (eighth ed.). p. 336. ISBN978-0-470-04618-0.
4. ^ Grøn, Øyvind; Hervik, Sigbjorn (2007). Einstein's General Theory of Relativity: With Modern Applications in Cosmology (illustrated ed.). Springer Science & Business Media. p. 180. ISBN978-0-387-69200-five.
5. ^ ^{a b} Einstein, Albert (1916). "The Foundation of the Full general Theory of Relativity". Annalen der Physik. 354 (7): 769–822. Bibcode:1916AnP...354..769E. doi:ten.1002/andp.19163540702. Archived from the original (PDF) on vi February 2012.
6. ^ Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975). Introduction to Full general Relativity (2d ed.). New York: McGraw-Loma. p. 345. ISBN978-0-07-000423-8.
7. ^ ^{a b} Gillies, George T. (1997). "The Newtonian gravitational constant: contempo measurements and related studies". Reports on Progress in Physics. 60 (2): 151–225. Bibcode:1997RPPh...sixty..151G. doi:x.1088/0034-4885/threescore/ii/001. . A lengthy, detailed review. Encounter Figure 1 and Table ii in item.
8. ^ Mohr, Peter J.; Newell, David B.; Taylor, Barry N. (21 July 2015). "CODATA Recommended Values of the Fundamental Concrete Constants: 2014". Reviews of Modern Physics. 88 (3): 035009. arXiv:1507.07956. Bibcode:2016RvMP...88c5009M. doi:10.1103/RevModPhys.88.035009. S2CID 1115862.
9. ^ "Astrodynamic Constants". NASA/JPL. 27 February 2009. Retrieved 27 July 2009.
10. ^ "Geocentric gravitational abiding". Numerical Standards for Fundamental Astronomy. IAU Division I Working Group on Numerical Standards for Primal Astronomy. Retrieved 24 June 2021 – 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 Messages. 19 (six): 529–531. Bibcode:1992GeoRL..19..529R. doi:10.1029/92GL00259. S2CID 123322272.
11. ^ ^{a b c} Boys 1894, p.330 In this lecture before the Regal Society, Boys introduces G and argues for its acceptance. Encounter: Poynting 1894, p. 4, MacKenzie 1900, p.vi
12. ^ Davies, R.D. (1985). "A Commemoration of Maskelyne at Schiehallion". Quarterly Journal of the Royal Astronomical Society. 26 (iii): 289–294. Bibcode:1985QJRAS..26..289D.
13. ^ "Sir Isaac Newton thought it probable, that the mean density of the earth might be five or 6 times as great as the density of water; and we accept at present found, past experiment, that it is very little less than what he had thought it to be: so much justness was fifty-fifty in the surmises of this wonderful man!" Hutton (1778), p. 783
14. ^ Poynting, J.H. (1913). The Earth: its shape, size, weight and spin. Cambridge. pp. 50–56.
15. ^ Hutton, C. (1778). "An Account of the Calculations Made from the Survey and Measures Taken at Schehallien". Philosophical Transactions of the Regal Society. 68: 689–788. doi:10.1098/rstl.1778.0034.
16. ^ Published in Philosophical Transactions of the Royal Club (1798); reprint: Cavendish, Henry (1798). "Experiments to Determine the Density of the Globe". In MacKenzie, A. S., Scientific Memoirs Vol. ix: The Laws of Gravitation. American Book Co. (1900), pp. 59–105.
17. ^ 2014 CODATA value vi.674×10^−eleven thousand^three⋅kg⁻¹⋅s⁻² .
18. ^ Brush, Stephen G.; Holton, Gerald James (2001). Physics, the human being adventure: from Copernicus to Einstein and across . New Brunswick, NJ: Rutgers University Press. pp. 137. ISBN978-0-8135-2908-0. Lee, Jennifer Lauren (sixteen November 2016). "Large Grand Redux: Solving the Mystery of a Perplexing Result". NIST.
19. ^ Poynting, John Henry (1894). The Mean Density of the Earth. London: Charles Griffin. pp. 22–24.
20. ^ F. Reich, On the Repetition of the Cavendish Experiments for Determining the mean density of the Earth" Philosophical Magazine 12: 283–284.
21. ^ Mackenzie (1899), p. 125.
22. ^ A.S. Mackenzie , The Laws of Gravitation (1899), 127f.
23. ^ Poynting, John Henry (1894). The hateful density of the globe. Gerstein - University of Toronto. London.
24. ^ Boys, C. V. (1 January 1895). "On the Newtonian Abiding of Gravitation". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. The Royal Society. 186: 1–72. Bibcode:1895RSPTA.186....1B. doi:10.1098/rsta.1895.0001. ISSN 1364-503X.
25. ^ Carl Braun, Denkschriften der k. Akad. d. Wiss. (Wien), math. u. naturwiss. Classe, 64 (1897). Braun (1897) quoted an optimistic standard doubtfulness of 0.03%, 6.649(ii)×10⁻¹¹ 1000³⋅kg⁻¹⋅s⁻² but his result was significantly worse than the 0.two% feasible at the time.
26. ^ Sagitov, Thou. U., "Electric current Condition 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. iv (July–August 1969), 907–915 (tabular array of historical experiments p. 715).
27. ^ Mackenzie, A. Stanley, The laws of gravitation; memoirs past Newton, Bouguer and Cavendish, together with abstracts of other important memoirs, American Book Company (1900 [1899]).
28. ^ Heyl, P. R. (1930). "A redetermination of the abiding of gravitation". Bureau of Standards Periodical of Enquiry. five (vi): 1243–1290. doi:ten.6028/jres.005.074.
29. ^ P. R. Heyl and P. Chrzanowski (1942), cited after Sagitov (1969:715).
30. ^ Mohr, Peter J.; Taylor, Barry N. (2012). "CODATA recommended values of the cardinal physical constants: 2002" (PDF). Reviews of Mod Physics. 77 (i): 1–107. arXiv:1203.5425. Bibcode:2005RvMP...77....1M. CiteSeerX10.ane.1.245.4554. doi:10.1103/RevModPhys.77.1. Archived from the original (PDF) on vi March 2007. Retrieved 1 July 2006. Section Q (pp. 42–47) describes the mutually inconsistent measurement experiments from which the CODATA value for G was derived.
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Sources [edit]

• Standish., Eastward. Myles (1995). "Report of the IAU WGAS Sub-group on Numerical Standards". In Appenzeller, I. (ed.). Highlights of Astronomy. Dordrecht: Kluwer Bookish Publishers. (Complete written report available online: PostScript; PDF. Tables from the study also available: Astrodynamic Constants and Parameters)
• Gundlach, Jens H.; Merkowitz, Stephen M. (2000). "Measurement of Newton's Constant Using a Torsion Balance with Angular Dispatch Feedback". Physical Review Letters. 85 (14): 2869–2872. arXiv:gr-qc/0006043. Bibcode:2000PhRvL..85.2869G. doi:ten.1103/PhysRevLett.85.2869. PMID 11005956. S2CID 15206636.

External links [edit]

• Newtonian constant of gravitation G at the National Found of Standards and Engineering References on Constants, Units, and Uncertainty
• The Controversy over Newton's Gravitational Constant — boosted commentary on measurement problems

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