Gravitational constant

The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant),[lower-alpha 1] denoted by the capital 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 theory of general relativity.

Not to be confused with g, the gravity of Earth.

Value of G Unit
6.67430(15)×10−11[1] N⋅m2⋅kg−2
6.67430(15)×10−8 dyn⋅cm2⋅g−2
4.3009172706(3)×10−3 pc⋅M⊙−1⋅(km/s)2
The gravitational constant G is a key quantity in Newton's law of universal gravitation.

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 (also referred to as the stress–energy tensor).

The measured value of the constant is known with some certainty to four significant digits. In SI units, its value is approximately 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.[lower-alpha 2]

Definition

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, between their centers of mass:

The constant of proportionality, G, is the gravitational constant. Colloquially, the gravitational constant is also called "Big G", distinct from "small g" (g), which is the local gravitational field of Earth (equivalent to the free-fall acceleration).[2][3] Where is the mass of the Earth and is the radius of the Earth, the two quantities are related by:

The gravitational constant appears in the Einstein field equations of general relativity,[4][5]

where Gμν is the Einstein tensor, Λ is the cosmological constant, gμν is the metric tensor, Tμν is the stress–energy tensor, and κ is the Einstein gravitational constant, a constant originally introduced by Einstein that is directly related to the Newtonian constant of gravitation:[5][6][lower-alpha 3]

Value and uncertainty

The gravitational constant is a physical constant that is difficult to measure with high accuracy.[7] This is because the gravitational force is an extremely weak force as compared to other fundamental forces at the laboratory scale.[lower-alpha 4]

In SI units, the 2018 Committee on Data for Science and Technology (CODATA)-recommended value of the gravitational constant (with standard uncertainty in parentheses) is:[1][8]

This corresponds to a relative standard uncertainty of 2.2×10−5 (22 ppm).

Natural units

The gravitational constant is a defining constant in some systems of natural units, particularly geometrized unit systems, such as Planck units and Stoney units. When expressed in terms of such units, the value of the gravitational constant will generally have a numeric value of 1 or a value close to it. Due to the significant uncertainty in the measured value of G in terms of other known fundamental constants, a similar level of uncertainty will show up in the value of many quantities when expressed in such a unit system.

Orbital mechanics
Further information: Standard gravitational parameter, orbital mechanics, celestial mechanics, Gaussian gravitational constant, Earth mass, and Solar mass

In astrophysics, it is convenient to measure distances in parsecs (pc), velocities in kilometres 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☉ + MEarth + M☾[lower-alpha 5]).

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 (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 Relative uncertainty
Sun GM☉ 1.32712440018(8)×1020 m3⋅s−2[9] 6×10−11
Earth GMEarth 3.986004418(8)×1014 m3⋅s−2[10] 2×10−9

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 since 2012.

History of measurement
Further information: Earth mass, Schiehallion experiment, and Cavendish experiment

Early history

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),[11] 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.[12] Nevertheless, he had the opportunity to estimate the order of magnitude of the constant when he surmised that "the mean density of the earth might be five or six times as great as the density of water", which is equivalent to a gravitational constant of the order:[13]

G ≈ (6.7±0.6)×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.[14]

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+1/2 times the density of water), about 20% below the modern value.[15] 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[11]

Based on this, Hutton's 1778 result is equivalent to G ≈ 8×10−11 m3⋅kg−1⋅s−2.

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.[16] 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%).[17]

19th century

The accuracy of the measured value of G has increased only 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, 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).[19]

Cavendish's experiment was first repeated by Ferdinand Reich (1838, 1842, 1853), who found a value of 5.5832(149) g·cm−3,[20] 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.[21]

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).[22]

Cavendish's result was first improved upon by John Henry Poynting (1891),[23] who published a value of 5.49(3) 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)[24] and Carl Braun (1897),[25] 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[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 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.[26]

Arthur Stanley Mackenzie in The Laws of Gravitation (1899) reviews the work done in the 19th century.[27] 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%),[28] improved to 6.673(3)×10−11 m3⋅kg−1⋅s−2 (relative uncertainty 0.045% = 450 ppm) in 1942.[29]

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.[7][30] 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 considerably increase the standard uncertainty in the 1998 recommended value, by a factor of 12, to a standard uncertainty of 0.15%, larger than the one given by Heyl (1930).

The uncertainty was again lowered in 2002 and 2006, but once again raised, by a more conservative 20%, in 2010, matching the standard uncertainty of 120 ppm published in 1986.[31] 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:

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.
Recommended values for G
Year G
(10−11·m3⋅kg−1⋅s−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 6.67384(80)120 ppm[38]
2014 6.67408(31)46 ppm[39]
2018 6.67430(15)22 ppm[40]

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.[41] 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.[42][43] 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).[44]

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.[45] 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.[46] 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 that 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.[47] This response notes that Anderson et al. not only omitted measurements, but that they also used the time of publication rather than 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, consideration of the data collected over a decade by Karagioz and Izmailov shows no correlation with length of day measurements.[47][48] 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 of them 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).[49]

See also

References

Footnotes

  1. "Newtonian constant of gravitation" is the name introduced for G by Boys (2000). Use of the term by T.E. Stern (1928) was misquoted as "Newton's constant of gravitation" in Pure Science Reviewed for Profound and Unsophisticated Students (1930), in what is apparently the first use of that term. Use of "Newton's constant" (without specifying "gravitation" or "gravity") is more recent, as "Newton's constant" was also used for the heat transfer coefficient in Newton's law of cooling, but has by now become quite common, e.g. 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. 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 g⋅cm−3.
  3. Depending on the choice of definition of the Einstein tensor and of the stress–energy tensor it can alternatively be defined as κ = 8πG/c2 ≈ 1.866×10−26 m⋅kg−1
  4. For example, the gravitational force between an electron and a proton 1 m 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 in the order of 1039 times greater than the force of gravity—roughly the same ratio as the mass of the Sun to a microgram.
  5. M ≈ 1.000003040433 M☉, so that M = M☉ can be used for accuracies of five or fewer significant digits.

Citations

  1. "2018 CODATA Value: Newtonian constant 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 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
  3. Halliday, David; Resnick, Robert; Walker, Jearl (September 2007). Fundamentals of Physics (8th ed.). p. 336. ISBN 978-0-470-04618-0.
  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. ISBN 978-0-387-69200-5.
  5. 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 6 February 2012.
  6. 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.
  7. Gillies, George T. (1997). "The Newtonian gravitational constant: recent measurements and related studies". Reports on Progress in Physics. 60 (2): 151–225. Bibcode:1997RPPh...60..151G. doi:10.1088/0034-4885/60/2/001. S2CID 250810284.. A lengthy, detailed review. See Figure 1 and Table 2 in particular.
  8. Mohr, Peter J.; Newell, David B.; Taylor, Barry N. (21 July 2015). "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. S2CID 1115862.
  9. "Astrodynamic Constants". NASA/JPL. 27 February 2009. Retrieved 27 July 2009.
  10. "Geocentric gravitational constant". Numerical Standards for Fundamental Astronomy. IAU Division I Working Group on Numerical Standards for Fundamental Astronomy. Retrieved 24 June 2021 – via iau-a3.gitlab.io. Citing
  11. 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, p.vi
  12. 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.
  13. "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
  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 Royal Society. 68: 689–788. doi:10.1098/rstl.1778.0034.
  16. 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.
  17. 2014 CODATA value 6.674×10−11 m3⋅kg−1⋅s−2.
  18. Brush, Stephen G.; Holton, Gerald James (2001). Physics, the human adventure: from Copernicus to Einstein and beyond. New Brunswick, NJ: Rutgers University Press. pp. 137. ISBN 978-0-8135-2908-0. Lee, Jennifer Lauren (16 November 2016). "Big G Redux: Solving the Mystery of a Perplexing Result". NIST.
  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 mean density of the earth. Gerstein - University of Toronto. London.
  24. Boys, C. V. (1 January 1895). "On the Newtonian Constant 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 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.
  26. 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).
  27. Mackenzie, A. Stanley, The laws of gravitation; memoirs by Newton, Bouguer and Cavendish, together with abstracts of other important memoirs, American Book Company (1900 [1899]).
  28. Heyl, P. R. (1930). "A redetermination of the constant of gravitation". Bureau of Standards Journal of Research. 5 (6): 1243–1290. doi:10.6028/jres.005.074.
  29. P. R. Heyl and P. Chrzanowski (1942), cited after Sagitov (1969:715).
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