Gravitational constant

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

In Newton's law, it is the proportionality constant connecting the gravitational force between two bodies with the product of their masses and the inverse square of their distance. In the Einstein field equations, it quantifies the relation between the geometry of spacetime and the energy–momentum tensor.

The measured value of the constant is known with some certainty to four significant digits. In SI units its value is approximately 6.674×10−11 m3⋅kg−1⋅s−2.[1]

The modern notation of Newton's law involving G was introduced in the 1890s by C. V. Boys. The first implicit measurement with an accuracy within about 1% is attributed to Henry Cavendish in a 1798 experiment.[2]

Values of G Units
6.67408(31)×10−11[1] m3kg−1s−2
4.30091(25)×10−3 pcM−1⋅(km/s)2
The gravitational constant G is a key quantity in Newton's law of universal gravitation.


According to Newton's law of universal gravitation, the attractive force (F) between two point-like bodies is directly proportional to the product of their masses (m1 and m2), and inversely proportional to the square of the distance, r, (inverse-square law) between them:

The constant of proportionality, G, is the gravitational constant. Colloquially, the gravitational constant is also called "Big G", for disambiguation with "small g" (g), which is the local gravitational field of Earth (equivalent to the free-fall acceleration).[3][4] Where M is the mass of the Earth and r is the radius of the Earth, the two quantities are related by:

g = GM/r2.

In the Einstein field equations of general relativity,[5][6]

Newton's constant appears in the proportionality between the spacetime curvature and the energy density component of the stress–energy tensor. The scaled gravitational constant, or Einstein's constant, is:[7][b]

κ = /c4G2.071×10−43 s2⋅m−1⋅kg−1.

Value and dimensions

The gravitational constant is a physical constant that is difficult to measure with high accuracy.[8] This is because the gravitational force is extremely weak as compared to other fundamental forces.[c]

In SI units, the 2014 CODATA-recommended value of the gravitational constant (with standard uncertainty in parentheses) is:[9][10]

This corresponds to a relative standard uncertainty of 4.6×10−5 (46 ppm).

The dimensions assigned to the gravitational constant are force times length squared divided by mass squared; this is equivalent to length cubed, divided by mass and by time squared:

In SI base units, this amounts to meters cubed per kilogram per second squared:

In cgs, G can be written as G6.674×10−8 cm3⋅g−1⋅s−2.

Natural units

The gravitational constant is taken as the basis of the Planck units: it is equal to the cube of the Planck length divided by the product of the Planck mass and the square of Planck time:

In other words, in Planck units, G has the numerical value of 1.

Thus, in Planck units, and other natural units taking G as their basis, the value of the gravitational constant cannot be measured as this is set by definition. Depending on the choice of units, uncertainty in the value of a physical constant as expressed in one system of units shows up as uncertainty of the value of another constant in another system of units. Where there is variation in dimensionless physical constants, no matter which choice of physical "constants" is used to define the units, this variation is preserved independently of the choice of units; in the case of the gravitational constant, such a dimensionless value is the gravitational coupling constant of the electron,


a measure for the gravitational attraction between a pair of electrons, proportional to the square of the electron rest mass.

Orbital mechanics

In astrophysics, it is convenient to measure distances in parsecs (pc), velocities in kilometers per second (km/s) and masses in solar units M. In these units, the gravitational constant is:

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

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

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

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, applying Kepler's 3rd law, expressed in units characteristic of Earth's orbit:

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 system (M = M + M + M[11]).

The above equation is exact only within the approximation of the Earth's orbit around the Sun as a two-body problem in Newtonian mechanics, the measured quantities contain corrections from the perturbations from other bodies in the solar system and from general relativity.

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

Since 2012, the AU is defined as 1.495978707×1011 m exactly, 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 the standard gravitational parameter and (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 by gravitational lensing, in Kepler's laws of planetary motion, and in the formula for escape velocity.

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

Values for GM
Body μ = GM Value Precision
Sun GM 1.32712440018(9)×1020 m3⋅s−2[12] 10 digits
Earth GM 3.986004418(8)×1014 m3⋅s−2[13] 9 digits

Calculations in celestial mechanics can also be carried out using the units of solar masses, mean solar days and astronomical units rather than standard SI units. For this purpose, the Gaussian gravitational constant was historically in widespread use, k = 0.01720209895, expressing the mean angular velocity of the Sun–Earth system measured in radians per day. The use of this constant, and the implied definition of the astronomical unit discussed above, has been deprecated by the IAU in 2012.

History of measurement

Early history

Between 1640 and 1650, Grimaldi and Riccioli had discovered that the distance covered by objects in free fall was proportional to the square of the time taken, which led them to attempt a calculation of the gravitational constant by recording the oscillations of a pendulum.[14]

The existence of the constant is implied in Newton's law of universal gravitation as published in the 1680s (although its notation as G dates to the 1890s),[15] but is not calculated in his Philosophiæ Naturalis Principia Mathematica where it postulates the inverse-square law of gravitation. In the Principia, Newton considered the possibility of measuring gravity's strength by measuring the deflection of a pendulum in the vicinity of a large hill, but thought that the effect would be too small to be measurable.[16] Nevertheless, he estimated the order of magnitude of the constant when he surmised that "the mean density of the earth might be five or six times as great as the density of water", which is equivalent to a gravitational constant of the order:[17]

G(7±1)×10−11 m3⋅kg–1⋅s−2

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 shell, as some thinkers of the day, including Edmond Halley, had suggested.[18]

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 4.5 g/cm3 (4​12 times the density of water), about 20% below the modern value.[19] 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's mean radius and the mean gravitational acceleration at Earth's surface, by setting


Based on this, Hutton's 1778 result is equivalent to G8×10−11 m3⋅kg–1⋅s−2.

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

The first direct measurement of gravitational attraction between two bodies in the laboratory was performed in 1798, seventy-one years after Newton's death, by Henry Cavendish.[20] 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 balls whose inertia (in relation to the torsion constant) he could tell by timing the beam's oscillation. Their faint attraction to other balls placed alongside the beam was detectable by the deflection it caused. In spite of the experimental design being due to Michell, the experiment is now known as the Cavendish experiment for its first successful execution by Cavendish.

Cavendish's stated aim was the "weighing of Earth", that is, determining the average density of Earth and the Earth's mass. His result, ρ = 5.448(33) g·cm−3, corresponds to value of G = 6.74(4)×10−11 m3⋅kg–1⋅s−2. It is surprisingly accurate, about 1% above the modern value (comparable to the claimed standard uncertainty of 0.6%).[21]

19th century

The accuracy of the measured value of G has increased only modestly since the original Cavendish experiment.[22] G is quite difficult to measure because gravity is much weaker than other fundamental forces, and an experimental apparatus cannot be separated from the gravitational influence of other bodies. Furthermore, gravity has no established relation to other fundamental forces, so it does not appear possible to calculate it indirectly from other constants that can be measured more accurately, as is done in some other areas of physics.

Measurements with pendulums were made by Francesco Carlini (1821, 4.39 g/cm3), Edward Sabine (1827, 4.77 g/cm3) Carlo Ignazio Giulio (1841, 4.95 g/cm3) and George Biddell Airy (1854, 6.6 g/cm3).[23]

Cavendish's experiment was first repeated by Ferdinand Reich (1838, 1842, 1853), who found a value of 5.5832(149) g·cm−3,[24] which is actually worse than Cavendish's result, differing from the modern value by 1.5%. Cornu and Baille (1873), found 5.56 g·cm−3.[25]

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 6.3 g/cm3) and Thomas Corwin Mendenhall (1880, 5.77 g/cm3).[26]

Cavendish's result was first improved upon by John Henry Poynting (1891)[27], who published a value of 6.69(84) g·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 addition to Poynting, measurements were made by C. V. Boys (1895)[28] and Carl Braun (1897),[29] with compatible results suggesting G = 6.66(1)×10−11 m3⋅kg−1⋅s−2. The modern notation involving the constant G was introduced by Boys in 1894[15] 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 result of 6.683(11)×10−11 m3⋅kg−1⋅s−2 was, however, of the same order of magnitude as the other results at the time.[30]

Arthur Stanley Mackenzie in The Laws of Gravitation (1899) reviews the work done in the 19th century.[31] Poynting is the author of the article "Gravitation" in the Encyclopædia Britannica Eleventh Edition (1911). Here, he cites a value of G = 6.66×10−11 m3⋅kg−1⋅s−2 with an uncertainty of 0.2%.

Modern value

Paul R. Heyl (1930) published the value of 6.670(5)×10−11 m3⋅kg–1⋅s−2 (relative uncertainty 0.1%),[32] improved to 6.673(3)×10−11 m3⋅kg–1⋅s−2 (relative uncertainty 0.045% = 450 ppm) in 1942.[33]

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 1,000 ppm) have varied rather broadly, and it is not entirely clear if the uncertainty has been reduced at all since the 1942 measurement. Some measurements published in the 1980s to 2000s were, in fact, mutually exclusive.[8][34] Establishing a standard value for G with a standard uncertainty better than 0.1% has therefore remained rather speculative.

By 1969, the value recommended by the National Institute of Standards and Technology (NIST) was cited with a standard uncertainty of 0.046% (460 ppm), lowered to 0.012% (120 ppm) by 1986. But the continued publication of conflicting measurements led NIST to radically increase the standard uncertainty in the 1998 recommended value, by a factor of 12, to a standard uncertainty of 0.15%, larger than the one given by Heyl (1930).

The uncertainty was again lowered in 2002 and 2006, but once again raised, by a more conservative 20%, in 2010, matching the standard uncertainty of 120 ppm published in 1986.[35] 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:

Gravitational constant historical
Timeline of measurements and recommended values for G since 1900: values recommended based on a literature review are shown in red, individual torsion balance experiments in blue, other types of experiments in green.
year G
1969 6.6732(31) 460 [36]
1973 6.6720(49) 730 [37]
1986 6.67449(81) 120 [38]
1998 6.673(10) 1,500 [39]
2002 6.6742(10) 150 [40]
2006 6.67428(67) 100 [41]
2010 6.67384(80) 120 [42]
2014 6.67408(31) 46 [43]

In the January 2007 issue of Science, Fixler et al. described a measurement of the gravitational constant by a new technique, atom interferometry, reporting a value of G = 6.693(34)×10−11 m3⋅kg−1⋅s−2, 0.28% (2800 ppm) higher than the 2006 CODATA value.[44] An improved cold atom measurement by Rosi et al. was published in 2014 of G = 6.67191(99)×10−11 m3⋅kg−1⋅s−2.[45] Although much closer to the accepted value (suggesting that the Fixler et. al. measurement was erroneous), this result was 325 ppm below the recommended 2014 CODATA value, with non-overlapping standard uncertainty intervals.

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).[46]

In August 2018, a Chinese research group announced new measurements based on torsion balances, 6.674184(78)×10−11 m3⋅kg–1⋅s−2 and 6.674484(78)×10−11 m3⋅kg–1⋅s−2 based on two different methods.[47] These are claimed as the most accurate measurements ever made, with a standard uncertainties cited as low as 12 ppm. The difference of 2.7σ between the two results suggests there could be sources of error unaccounted for.

Suggested time-variation

A controversial 2015 study of some previous measurements of G, by Anderson et al., suggested that most of the mutually exclusive values in high-precision measurements of G can be explained by a periodic variation.[48] The variation was measured as having a period of 5.9 years, similar to that observed in length-of-day (LOD) measurements, hinting at a common physical cause which is not necessarily a variation in G. A response was produced by some of the original authors of the G measurements used in Anderson et al.[49] This response notes that Anderson et al. not only omitted measurements, they also used the time of publication not the time the experiments were performed. A plot with estimated time of measurement from contacting original authors seriously degrades the length of day correlation. Also taking the data collected over a decade by Karagioz and Izmailov shows no correlation with length of day measurements.[49][50] As such the variations in G most likely arise from systematic measurement errors which have not properly been accounted for. Under the assumption that the physics of type Ia supernovae are universal, analysis of observations of 580 type Ia supernovae has shown that the gravitational constant has varied by less than one part in ten billion per year over the last nine billion years according to Mould et al. (2014).[51]

See also



  1. ^ "Newtonian constant of gravitation" is the name introduced for G by Boys (1894). Use of the term by T.E. Stern (1928) was misquoted as "Newton's constant of gravitation" in Pure Science Reviewed for Profound and Unsophisticated Students (1930), in what is apparently the first use of that term. Use of "Newton's constant" (without specifying "gravitation" or "gravity") is more recent, as "Newton's constant" was also used for the heat transfer coefficient in Newton's law of cooling, but has by now become quite common, e.g. Calmet et al, Quantum Black Holes (2013), p. 93; P. de Aquino, Beyond Standard Model Phenomenology at the LHC (2013), p. 3. The name "Cavendish gravitational constant", sometimes "Newton–Cavendish gravitational constant", appears to have been common in the 1970s to 1980s, especially in (translations from) Soviet-era Russian literature, e.g. Sagitov (1970 [1969]), Soviet Physics: Uspekhi 30 (1987), Issues 1–6, p. 342 [etc.]. "Cavendish constant" and "Cavendish gravitational constant" is also used in Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, "Gravitation", (1973), 1126f. Colloquial use of "Big G", as opposed to "little g" for gravitational acceleration dates to the 1960s (R.W. Fairbridge, The encyclopedia of atmospheric sciences and astrogeology, 1967, p. 436; note use of "Big G's" vs. "little g's" as early as the 1940s of the Einstein tensor Gμν vs. the metric tensor gμν, 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).
  2. ^ Depending on the choice of definition of the stress–energy tensor it can also be normalized as κ = /c2G1.866×10−26 m⋅kg−1.
  3. ^ For example, the gravitational force between an electron and proton one meter apart is approximately 10−67 N, whereas the electromagnetic force between the same two particles is approximately 10−28 N. The electromagnetic force in this example is some 39 orders of magnitude (i.e. 1039) greater than the force of gravity—roughly the same ratio as the mass of the Sun to a microgram.


  1. ^ a b "CODATA Value: Newtonian constant of gravitation". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
  2. ^ Cavendish determined the value of G indirectly, by reporting a value for the Earth's mass, or the average density of Earth, as 5.448
  3. ^ Gundlach, Jens H.; Merkowitz, Stephen M. (2002-12-23). "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
  4. ^ Halliday, David; Resnick, Robert; Walker, Jearl (September 2007). Fundamentals of Physics (8th ed.). p. 336. ISBN 978-0-470-04618-0.
  5. ^ 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.
  6. ^ 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 from the original (PDF) on 2012-02-06.
  7. ^ 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.
  8. ^ a b 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.. A lengthy, detailed review. See Figure 1 and Table 2 in particular.
  9. ^ Mohr, Peter J.; Newell, David B.; Taylor, Barry N. (2015-07-21). "CODATA Recommended Values of the Fundamental Physical Constants: 2014". Reviews of Modern Physics. 88 (3): 035009. arXiv:1507.07956. Bibcode:2016RvMP...88c5009M. doi:10.1103/RevModPhys.88.035009.
  10. ^ "Newtonian constant of gravitation G". CODATA, NIST.
  11. ^ M ≈ 1.000003040433 M, so that M = M can be used for accuracies of five or fewer significant digits.
  12. ^ "Astrodynamic Constants". NASA/JPL. 27 February 2009. Retrieved 27 July 2009.
  13. ^ "Numerical Standards for Fundamental Astronomy". IAU Working Group. Retrieved 31 October 2017., citing Ries, J. C., Eanes, R. J., Shum, C. K., and Watkins, M. M., 1992, "Progress in the Determination of the Gravitational Coefficient of the Earth," Geophys. Res. Lett., 19(6), pp. 529–531. Ries, J. C.; Eanes, R. J.; Shum, C. K.; Watkins, M. M. (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.
  14. ^ J.L. Heilbron, Electricity in the 17th and 18th Centuries: A Study of Early Modern Physics (Berkeley: University of California Press, 1979), 180.
  15. ^ a b c Boys 1894, p.330 In this lecture before the Royal Society, Boys introduces G and argues for its acceptance. See: Poynting 1894, p. 4, MacKenzie 1900,
  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
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  20. ^ Published in Philosophical Transactions of the Royal Society (1798); reprint: Cavendish, Henry (1798). "Experiments to Determine the Density of the Earth". In MacKenzie, A. S., Scientific Memoirs Vol. 9: The Laws of Gravitation. American Book Co. (1900), pp. 59–105.
  21. ^ 2014 CODATA value 6.674×10−11 m3⋅kg−1⋅s−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. p. 137. ISBN 978-0-8135-2908-0. Lee, Jennifer Lauren (November 16, 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 Magazine 12: 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. ^ C.V. Boys, Phil. Trans. Roy. Soc. A. Pt. 1. (1895).
  29. ^ Carl Braun, Denkschriften der k. Akad. d. Wiss. (Wien), math. u. naturwiss. Classe, 64 (1897). Braun (1897) quoted an optimistic standard uncertainty of 0.03%, 6.649(2)×10−11 m3⋅kg−1⋅s−2 but his result was significantly worse than the 0.2% feasible at the time.
  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 from Astronomicheskii Zhurnal Vol. 46, No. 4 (July–August 1969), 907-915 (table of historical experiments p. 715).
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  33. ^ P. R. Heyl and P. Chrzanowski (1942), cited after Sagitov (1969:715).
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  43. ^ P. J. Mohr, D. B. Newell, and B. N. Taylor, J. Phys. Chem. Ref. Data 45 (2016)
  44. ^ Fixler, J. B.; Foster, G. T.; McGuirk, J. M.; Kasevich, M. A. (2007-01-05). "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.
  45. ^ 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. doi:10.1038/nature13433. PMID 24965653. Schlamminger, Stephan (18 June 2014). "Fundamental constants: A cool way to measure big G". Nature. 510 (7506): 478–480. Bibcode:2014Natur.510..478S. doi:10.1038/nature13507. PMID 24965646.
  46. ^ C. Rothleitner, S. Schlamminger, "Invited Review Article: Measurements of the Newtonian constant of gravitation, G", Review of Scientific Instruments 88, 111101 (2017) doi:10.1063/1.4994619. "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, and R. Davis, "Improved determination of G using two methods," Phys. Rev. Lett. 111, 101102 (2013).) The 2018 experiment was described by C. Rothleitner, "Newton’s Gravitational Constant ‚Big‘ G – A proposed Free-fall Measurement", CODATA Fundamental Constants Meeting, Eltville, 5 February 2015].
  47. ^ Li, Qing; et al. (2018). "Measurements of the gravitational constant using two independent methods". Nature. 560 (7720): 582–588. doi:10.1038/s41586-018-0431-5. PMID 30158607.. See also: "Physicists just made the most precise measurement ever of Gravity's strength". August 31, 2018. Retrieved October 13, 2018.
  48. ^ Anderson, J. D.; Schubert, G.; Trimble, 3=V.; Feldman, M. R. (April 2015). "Measurements of Newton's gravitational constant and the length of day" (PDF). EPL. 110 (1): 10002. arXiv:1504.06604. Bibcode:2015EL....11010002A. doi:10.1209/0295-5075/110/10002.
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Astronomical unit

The astronomical unit (symbol: au, ua, or AU) is a unit of length, roughly the distance from Earth to the Sun. However, that distance varies as Earth orbits the Sun, from a maximum (aphelion) to a minimum (perihelion) and back again once a year. Originally conceived as the average of Earth's aphelion and perihelion, since 2012 it has been defined as exactly 149597870700 metres or about 150 million kilometres (93 million miles). The astronomical unit is used primarily for measuring distances within the Solar System or around other stars. It is also a fundamental component in the definition of another unit of astronomical length, the parsec.

Cavendish experiment

The Cavendish experiment, performed in 1797–1798 by British scientist Henry Cavendish, was the first experiment to measure the force of gravity between masses in the laboratory and the first to yield accurate values for the gravitational constant. Because of the unit conventions then in use, the gravitational constant does not appear explicitly in Cavendish's work. Instead, the result was originally expressed as the specific gravity of the Earth, or equivalently the mass of the Earth. His experiment gave the first accurate values for these geophysical constants.

The experiment was devised sometime before 1783 by geologist John Michell, who constructed a torsion balance apparatus for it. However, Michell died in 1793 without completing the work. After his death the apparatus passed to Francis John Hyde Wollaston and then to Henry Cavendish, who rebuilt the apparatus but kept close to Michell's original plan. Cavendish then carried out a series of measurements with the equipment and reported his results in the Philosophical Transactions of the Royal Society in 1798.

Earth mass

Earth mass (ME or M⊕, where ⊕ is the standard astronomical symbol for planet Earth) is the unit of mass equal to that of Earth.

The current best estimate for Earth mass is M⊕ = 5.9722×1024 kg, with a standard uncertainty of

6×1020 kg (relative uncertainty 10−4).

It is equivalent to an average density of 5515 kg⋅m−3.

The Earth mass is a standard unit of mass in astronomy that is used to indicate the masses of other planets, including rocky terrestrial planets and exoplanets. One Solar mass is close to 333,000 Earth masses.

The Earth mass excludes the mass of the Moon. The mass of the Moon is about 1.2% of that of the Earth, so that the mass of the Earth+Moon system is close to 6.0456×1024 kg.

Most of the mass is accounted for by iron and oxygen (c. 32% each), magnesium and silicon (c. 15% each), calcium, aluminium and nickel (c. 1.5% each).

Precise measurement of the Earth mass is difficult, as it is equivalent to measuring the gravitational constant, which is the fundamental physical constant known with least accuracy, due to the relative weakness of the gravitational force.

The mass of the Earth was first measured with any accuracy (within about 20% of the correct value) in the Schiehallion experiment in the 1770s, and within 1% of the modern value in the Cavendish experiment of 1798.

Expanding Earth

The expanding Earth or growing Earth hypothesis asserts that the position and relative movement of continents is at least partially due to the volume of Earth increasing. Conversely, geophysical global cooling was the hypothesis that various features could be explained by Earth contracting.

Although it was suggested historically, since the recognition of plate tectonics in the 1970s, scientific consensus has rejected any significant expansion or contraction of Earth.

Gauss's constant

In mathematics, Gauss's constant, denoted by G, is defined as the reciprocal of the arithmetic–geometric mean of 1 and the square root of 2:

The constant is named after Carl Friedrich Gauss, who on May 30, 1799 discovered that

so that

where Β denotes the beta function.

Gauss's constant should not be confused with the Gaussian gravitational constant.

Gaussian gravitational constant

The Gaussian gravitational constant (symbol k) is a parameter used in the orbital mechanics of the solar system.

It relates the orbital period to the orbit's semi-major axis and the mass of the orbiting body in Solar masses.

The value of k historically expresses the mean angular velocity of the system of Earth+Moon and the Sun considered as a two body problem,

with a value of about 0.986 degrees per day, or about 0.0172 radians per day.

As a consequence of Newton's law of gravitation and Kepler's third law,

k is directly proportional to the square root of the standard gravitational parameter of the Sun,

and its value in radians per day follows by setting Earth's semi-major axis (the astronomical unit, a.u.) to unity, k:(rad/day) = (GM☉)0.5·(a.u.)-1.5A value of k = 0.01720209895 rad/day was determined by Carl Friedrich Gauss in his 1809 work Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientum ("Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections").

Gauss' value was introduced as a fixed, defined value by the IAU (adpoted in 1938, formally defined in 1964),

which detached it from its immediate representation of the (observable) mean angular velocity of the Sun-Earth system. Instead, the astronomical unit now became a measurable quantity slightly different from unity.

This was useful in 20th-century celestial mechanics to prevent the constant adaptation of orbital parameters to updated measured values, but it came at the expense of intuitiveness, as the astronomical unit, ostensibly a unit of length, was now dependent on the measurement of the strength of the gravitational force.

The IAU abandoned the defined value of k in 2012 in favour of a defined value of the astronomical unit of 1.495978707×1011 m exactly, while the strength of the gravitational force is now to be expressed in the separate standard gravitational parameter GM☉, measured in SI units of m3 s−2.

Gravitational energy

Gravitational energy is the potential energy a body with mass has in relation to another massive object due to gravity. It is potential energy associated with the gravitational field. Gravitational energy is dependent on the masses of two bodies, their distance apart and the gravitational constant (G).

In everyday cases only one body is accelerating measurably, and its acceleration is constant (for example, dropping a ball on Earth). For such scenarios the Newtonian formula can – for the potential energy of the accelerating body with respect to the stationary – be reduced to:

where is the gravitational potential energy, is the mass of the object accelerating, is the acceleration of the object, and is the distance between the bodies. Note that this formula treats the potential energy as a positive quantity.

Jeans instability

In stellar physics, the Jeans instability causes the collapse of interstellar gas clouds and subsequent star formation, named after James Jeans. It occurs when the internal gas pressure is not strong enough to prevent gravitational collapse of a region filled with matter. For stability, the cloud must be in hydrostatic equilibrium, which in case of a spherical cloud translates to:


where is the enclosed mass, is the pressure, is the density of the gas (at radius ), is the gravitational constant, and is the radius. The equilibrium is stable if small perturbations are damped and unstable if they are amplified. In general, the cloud is unstable if it is either very massive at a given temperature or very cool at a given mass; under these circumstances, the gas pressure cannot overcome gravity, and the cloud will collapse.

Jupiter mass

Jupiter mass, also called Jovian mass, is the unit of mass equal to the total mass of the planet Jupiter. This value may refer to the mass of the planet alone, or the mass of the entire Jovian system to include the moons of Jupiter. Jupiter is by far the most massive planet in the Solar System. It is approximately 2.5 times more massive than all of the other planets in the Solar System combined.Jupiter mass is a common unit of mass in astronomy that is used to indicate the masses of other similarly-sized objects, including the outer planets and extrasolar planets. It may also be used to describe the masses of brown dwarfs, as this unit provides a convenient scale for comparison.

Mean motion

In orbital mechanics, mean motion (represented by n) is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the actual body. The concept applies equally well to a small body revolving about a large, massive primary body or to two relatively same-sized bodies revolving about a common center of mass. While nominally a mean, and theoretically so in the case of two-body motion, in practice the mean motion is not typically an average over time for the orbits of real bodies, which only approximate the two-body assumption. It is rather the instantaneous value which satisfies the above conditions as calculated from the current gravitational and geometric circumstances of the body's constantly-changing, perturbed orbit.

Mean motion is used as an approximation of the actual orbital speed in making an initial calculation of the body's position in its orbit, for instance, from a set of orbital elements. This mean position is refined by Kepler's equation to produce the true position.

N-body units

N-body units are a completely self-contained system of units used for N-body simulations of self-gravitating systems in astrophysics. In this system, the base physical units are chosen so that the total mass, M, the gravitational constant, G, and the virial radius, R, are normalized. The underlying assumption is that the system of N objects (stars) satisfies the virial theorem. The consequence of standard N-body units is that the velocity dispersion of the system, v, is and that the dynamical or crossing time, t, is . The use of standard N-body units was advocated by Michel Hénon in 1971. Early adopters of this system of units included H. Cohn in 1979 and D. Heggie and R. Mathieu in 1986. At the conference MODEST14 in 2014, D. Heggie proposed that the community abandon the name "N-body units" and replace it with the name "Hénon units" to commemorate the originator.

Newton's law of universal gravitation

Newton's law of universal gravitation states that every particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning. It is a part of classical mechanics and was formulated in Newton's work Philosophiæ Naturalis Principia Mathematica ("the Principia"), first published on 5 July 1687. When Newton presented Book 1 of the unpublished text in April 1686 to the Royal Society, Robert Hooke made a claim that Newton had obtained the inverse square law from him.

In today's language, the law states that every point mass attracts every other point mass by a force acting along the line intersecting the two points. The force is proportional to the product of the two masses, and inversely proportional to the square of the distance between them.

The equation for universal gravitation thus takes the form:

where F is the gravitational force acting between two objects, m1 and m2 are the masses of the objects, r is the distance between the centers of their masses, and G is the gravitational constant.

The first test of Newton's theory of gravitation between masses in the laboratory was the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798. It took place 111 years after the publication of Newton's Principia and approximately 71 years after his death.

Newton's law of gravitation resembles Coulomb's law of electrical forces, which is used to calculate the magnitude of the electrical force arising between two charged bodies. Both are inverse-square laws, where force is inversely proportional to the square of the distance between the bodies. Coulomb's law has the product of two charges in place of the product of the masses, and the electrostatic constant in place of the gravitational constant.

Newton's law has since been superseded by Albert Einstein's theory of general relativity, but it continues to be used as an excellent approximation of the effects of gravity in most applications. Relativity is required only when there is a need for extreme accuracy, or when dealing with very strong gravitational fields, such as those found near extremely massive and dense objects, or at very close distances (such as Mercury's orbit around the Sun).

Planck length

In physics, the Planck length, denoted ℓP, is a unit of length that is the distance light travels in one unit of Planck time. It is equal to 1.616229(38)×10−35 m. It is a base unit in the system of Planck units, developed by physicist Max Planck. The Planck length can be defined from three fundamental physical constants: the speed of light in a vacuum, the Planck constant, and the gravitational constant.

Planck time

In quantum mechanics, the Planck time (tP) is the unit of time in the system of natural units known as Planck units. A Planck time unit is the time required for light to travel a distance of 1 Planck length in a vacuum, which is a time interval of approximately 5.39 × 10 −44 s. The unit is named after Max Planck, who was the first to propose it.

The Planck time is defined as:


ħ = ​h2π is the reduced Planck constant (sometimes h is used instead of ħ in the definition)
G = gravitational constant
c = speed of light in vacuum

Using the known values of the constants, the approximate equivalent value in terms of the SI unit, the second, is

where the two digits between parentheses denote the standard error of the approximated value.

Solar mass

The solar mass (M) is a standard unit of mass in astronomy, equal to approximately 2×1030 kg. It is used to indicate the masses of other stars, as well as clusters, nebulae, and galaxies. It is equal to the mass of the Sun (denoted by the solar symbol ⊙︎). This equates to about two nonillion (two quintillion in the long scale) kilograms:

M = (1.98847±0.00007)×1030 kg

The above mass is about 332946 times the mass of Earth (M), or 1047 times the mass of Jupiter (MJ).

Because Earth follows an elliptical orbit around the Sun, the solar mass can be computed from the equation for the orbital period of a small body orbiting a central mass. Based upon the length of the year, the distance from Earth to the Sun (an astronomical unit or AU), and the gravitational constant (G), the mass of the Sun is given by:

The value of G is difficult to measure and is only known with limited accuracy in SI units (see Cavendish experiment). The value of G times the mass of an object, called the standard gravitational parameter, is known for the Sun and several planets to much higher accuracy than G alone. As a result, the solar mass is used as the standard mass in the astronomical system of units.

Standard gravitational parameter

In celestial mechanics, the standard gravitational parameter μ of a celestial body is the product of the gravitational constant G and the mass M of the body.

For several objects in the Solar System, the value of μ is known to greater accuracy than either G or M. The SI units of the standard gravitational parameter are m3 s−2. However, units of km3 s−2 are frequently used in the scientific literature and in spacecraft navigation.

Standard gravity

The standard acceleration due to gravity (or standard acceleration of free fall), sometimes abbreviated as standard gravity, usually denoted by ɡ0 or ɡn, is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is defined by standard as 9.80665 m/s2 (about 32.17405 ft/s2). This value was established by the 3rd CGPM (1901, CR 70) and used to define the standard weight of an object as the product of its mass and this nominal acceleration. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from the rotation of the Earth (but which is small enough to be neglected for most purposes); the total (the apparent gravity) is about 0.5% greater at the poles than at the Equator.Although the symbol ɡ is sometimes used for standard gravity, ɡ (without a suffix) can also mean the local acceleration due to local gravity and centrifugal acceleration, which varies depending on one's position on Earth (see Earth's gravity). The symbol ɡ should not be confused with G, the gravitational constant, or g, the symbol for gram. The ɡ is also used as a unit for any form of acceleration, with the value defined as above; see g-force.

The value of ɡ0 defined above is a nominal midrange value on Earth, originally based on the acceleration of a body in free fall at sea level at a geodetic latitude of 45°. Although the actual acceleration of free fall on Earth varies according to location, the above standard figure is always used for metrological purposes. In particular, it gives the conversion factor between newton and kilogram-force, two units of force.

Stoney units

In physics the Stoney units form a system of units named after the Irish physicist George Johnstone Stoney, who first proposed them in 1881. They are the first historical example of natural units, i.e. units of measurement designed so that certain fundamental physical constants serve as base units. The set of constants that Stoney used as base units is the following:

This means that, in terms of Stoney units, the numerical values of all these constants equal one:

Stoney's set of base units is similar to the one used in Planck units, proposed independently by Planck thirty years later, but Planck normalized the reduced Planck constant in place of the elementary charge. In Stoney units, the numerical value of the reduced Planck constant is not 1, but is

where α is the fine-structure constant. Planck units are more commonly used than Stoney units in modern physics, especially quantum gravity (including string theory). Rarely, Planck units are referred to as Planck–Stoney units.

Strong gravity

Strong gravity is a non-mainstream theoretical approach to particle confinement having both a cosmological scale and a particle scale gravity. In the 1960s, it was taken up as an alternative to the then young QCD theory by several theorists, including Abdus Salam, who showed that the particle level gravity approach can produce confinement and asymptotic freedom while not requiring a force behavior differing from an inverse-square law, as does QCD. Sivaram published a review of this bimetric theory approach.Although this approach has not so far led to a recognizably successful unification of strong and other forces, the modern approach of string theory is characterized by a close association between gauge forces and spacetime geometry. In some cases, string theory recognizes important duality between gravity-like and QCD-like theories, most notably the AdS/QCD correspondence.

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