The measured value of the constant is known with some certainty to four significant digits. InSI units, its value is approximately6.6743×10−11 m3⋅kg−1⋅s−2.[1]
The modern notation of Newton's law involvingG was introduced in the 1890s byC. V. Boys. The first implicit measurement with an accuracy within about 1% is attributed toHenry Cavendish in a1798 experiment.[b]
According to Newton's law of universal gravitation, themagnitude of the attractiveforce (F) between two bodies each with a spherically symmetricdensity distribution is directly proportional to the product of theirmasses,m1 andm2, and inversely proportional to the square of the distance,r, directed along the line connecting theircentres of mass:Theconstant of proportionality,G, in this non-relativistic formulation 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 (also referred to as free-fall acceleration).[2][3] Where is themass of Earth and is theradius of Earth, the two quantities are related by:
The units ofG, m3 kg−1 s−2 (in SI), can be arranged for meaning in more than one way, to indicate the meaning ofG as the strength of gravity in the universe. One arrangement is N/(kg2/m2) to indicate the force between two masses at a distance according to the inverse-square law. Another, cancelling a common mass unit, is (m/s2)/(kg/m2) to indicate the acceleration due to gravity due to a mass at a distance, or the acceleration resulting at a point in space due to the amount of influence there from a distant mass.
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 otherfundamental forces at the laboratory scale.[d]
InSI units, theCODATA-recommended value of the gravitational constant is:[1]
Due to its use as a defining constant in some systems ofnatural units,[8][9] particularlygeometrized unit systems such asPlanck units andStoney units, the value of the gravitational constant will generally have a numeric value of 1 or a value close to it when expressed in terms of those units. Due to the significant uncertainty in the measured value ofG 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.
Inastrophysics, it is convenient to measure distances inparsecs (pc), velocities in kilometres per second (km/s) and masses in solar unitsM⊙. In these units, the gravitational constant is:For situations where tides are important, the relevant length scales aresolar radii rather than parsecs. In these units, the gravitational constant is:Inorbital mechanics, the periodP of an object in circular orbit around a spherical object obeyswhereV is the volume inside the radius of the orbit, andM is the total mass of the two objects. It follows that
This way of expressingG shows the relationship between the average density of a planet and the period of a satellite orbiting just above its surface.
where distance is measured in terms of thesemi-major axis of Earth's orbit (theastronomical unit, au), time inyears, and mass in the total mass of the orbiting system (M =M☉ +M🜨 +M☾[e]).
The above equation is exact only within the approximation of the Earth's orbit around the Sun as atwo-body problem in Newtonian mechanics, the measured quantities contain corrections from the perturbations from other bodies in theSolar 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 as1.495978707×1011 m exactly, and the equation can no longer be taken as holding precisely.
The quantityGM—the product of the gravitational constant and the mass of a given astronomical body such as the Sun or Earth—is known as thestandard gravitational parameter (also denotedμ). The standard gravitational parameterGM appears as above in Newton's law of universal gravitation, as well as in formulas for the deflection of light caused bygravitational lensing, inKepler's laws of planetary motion, and in the formula forescape velocity.
This quantity gives a convenient simplification of various gravity-related formulas. The productGM is known much more accurately than either factor is.
The existence of the constant is implied inNewton's law of universal gravitation as published in the 1680s (although its notation asG dates to the 1890s),[12] but is notcalculated in hisPhilosophiæ Naturalis Principia Mathematica where it postulates theinverse-square law of gravitation. In thePrincipia, 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.[13] 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:[14]
TheSchiehallion 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 byCharles Hutton (1778) suggested a density of4.5 g/cm3 (4+1/2 times the density of water), about 20% below the modern value.[16] This immediately led to estimates on the densities and masses of theSun,Moon andplanets, sent by Hutton toJérôme Lalande for inclusion in his planetary tables. As discussed above, establishing the average density of Earth is equivalent to measuring the gravitational constant, givenEarth's mean radius and themean gravitational acceleration at Earth's surface, by setting[12]Based on this, Hutton's 1778 result is equivalent toG ≈8×10−11 m3⋅kg−1⋅s−2.
Diagram of torsion balance used in theCavendish experiment performed byHenry 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.[17] He determined a value forG implicitly, using atorsion balance invented by the geologist Rev.John Michell (1753). He used a horizontaltorsion 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 theEarth's mass. His result,ρ🜨 =5.448(33) g⋅cm−3, corresponds to value ofG =6.74(4)×10−11 m3⋅kg−1⋅s−2. It is remarkably accurate, being about 1% above the modernCODATA recommended value6.674×10−11 m3⋅kg−1⋅s−2, consistent with the claimed relative standard uncertainty of 0.6%.
The accuracy of the measured value ofG 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.
Cavendish's experiment was first repeated byFerdinand Reich (1838, 1842, 1853), who found a value of5.5832(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), found5.56 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, byRobert von Sterneck (1883, results between5.0 and 6.3 g/cm3) andThomas Corwin Mendenhall (1880,5.77 g/cm3).[22]
Cavendish's result was first improved upon byJohn Henry Poynting (1891),[23] who published a value of5.49(3) g⋅cm−3, differing from the modern value by 0.2%, but compatible with the modern value within the cited relative standard uncertainty of 0.55%. In addition to Poynting, measurements were made byC. V. Boys (1895)[24] andCarl Braun (1897),[25] with compatible results suggestingG =6.66(1)×10−11 m3⋅kg−1⋅s−2. The modern notation involving the constantG was introduced by Boys in 1894[12] and becomes standard by the end of the 1890s, with values usually cited in thecgs 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 of6.683(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 inThe Laws of Gravitation (1899) reviews the work done in the 19th century.[27] Poynting is the author of the article "Gravitation" in theEncyclopædia Britannica Eleventh Edition (1911). Here, he cites a value ofG =6.66×10−11 m3⋅kg−1⋅s−2 with a relative uncertainty of 0.2%.
Paul R. Heyl (1930) published the value of6.670(5)×10−11 m3⋅kg−1⋅s−2 (relative uncertainty 0.1%),[28] improved to6.673(3)×10−11 m3⋅kg−1⋅s−2 (relative uncertainty 0.045% = 450 ppm) in 1942.[29]
However, Heyl used the statistical spread as his standard deviation, and he admitted himself that measurements using the same material yielded very similar results while measurements using different materials yielded vastly different results. He spent the next 12 years after his 1930 paper to do more precise measurements, hoping that the composition-dependent effect would go away, but it did not, as he noted in his final paper from the year 1942.
Published values ofG derived from high-precision measurements since the 1950s have remained compatible with Heyl (1930), but within the relative uncertainty of about 0.1% (or 1000 ppm) have varied rather broadly, and it is not entirely clear whether 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 forG with a relative standard uncertainty better than 0.1% has therefore remained rather speculative.
By 1969, the value recommended by theNational Institute of Standards and Technology (NIST) was cited with a relative standard uncertainty of 0.046% (460 ppm), lowered to 0.012% (120 ppm) by 1986. But the continued publication of conflicting measurements led NIST to considerably increase the standard uncertainty in the 1998 recommended value, by a factor of 12, to a standard uncertainty of 0.15%, larger than the one given by Heyl (1930).
The uncertainty was again lowered in 2002 and 2006, but once again raised, by a more conservative 20%, in 2010, matching the relative standard uncertainty of 120 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 forG 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.
In the January 2007 issue ofScience, Fixler et al. described a measurement of the gravitational constant by a new technique,atom interferometry, reporting a value ofG =6.693(34)×10−11 m3⋅kg−1⋅s−2, 0.28% (2800 ppm) higher than the 2006 CODATA value.[42] An improved cold atom measurement by Rosi et al. was published in 2014 ofG =6.67191(99)×10−11 m3⋅kg−1⋅s−2.[43][44] Although much closer to the accepted value (suggesting that the Fixleret al. measurement was erroneous), this result was 325 ppm below the recommended 2014 CODATA value, with non-overlappingstandard 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).[45]
In August 2018, a Chinese research group announced new measurements based on torsion balances,6.674184(78)×10−11 m3⋅kg−1⋅s−2 and6.674484(78)×10−11 m3⋅kg−1⋅s−2 based on two different methods.[46] These are claimed as the most accurate measurements ever made, with standard uncertainties cited as low as 12 ppm. The difference of 2.7 σ between the two results suggests there could be sources of error unaccounted for.
Analysis of observations of 580type Ia supernovae shows that the gravitational constant has varied by less than one part in ten billion per year over the last nine billion years.[47]
^"Newtonian constant of gravitation" is the name introduced forG by Boys (1984). Use of the term by T.E. Stern (1928) was misquoted as "Newton's constant of gravitation" inPure 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 theheat transfer coefficient inNewton'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 theEinstein tensorGμν vs. themetric tensorgμν,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).
^Cavendish determined the value ofG indirectly, by reporting a value for theEarth's mass, or the average density of Earth, as5.448 g⋅cm−3.
^Depending on the choice of definition of the Einstein tensor and of the stress–energy tensor it can alternatively be defined asκ =8πG/c2 ≈1.866×10−26 m⋅kg−1
^For example, the gravitational force between anelectron and aproton 1 m apart is approximately10−67N, whereas theelectromagnetic force between the same two particles is approximately10−28 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 themass of the Sun to a microgram.
^M ≈1.000003040433M☉, so thatM =M☉ can be used for accuracies of five or fewer significant digits.
^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
^abGillies, 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.S2CID250810284.. A lengthy, detailed review. See Figure 1 and Table 2 in particular.
^"Geocentric gravitational constant".Numerical Standards for Fundamental Astronomy. IAU Division I Working Group on Numerical Standards for Fundamental Astronomy. Retrieved24 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 Letters.19 (6):529–531.Bibcode:1992GeoRL..19..529R.doi:10.1029/92GL00259.S2CID123322272.
^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.
^"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
^Published inPhilosophical 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.
^Carl Braun,Denkschriften der k. Akad. d. Wiss. (Wien), math. u. naturwiss. Classe, 64 (1897).Braun (1897) quoted an optimistic relative standard uncertainty of 0.03%,6.649(2)×10−11 m3⋅kg−1⋅s−2 but his result was significantly worse than the 0.2% feasible at the time.
^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 fromAstronomicheskii Zhurnal Vol. 46, No. 4 (July–August 1969), 907–915 (table of historical experiments p. 715).
^Taylor, B. N.; Parker, W. H.; Langenberg, D. N. (1 July 1969). "Determination ofe/h, Using Macroscopic Quantum Phase Coherence in Superconductors: Implications for Quantum Electrodynamics and the Fundamental Physical Constants".Reviews of Modern Physics.41 (3). American Physical Society (APS):375–496.Bibcode:1969RvMP...41..375T.doi:10.1103/revmodphys.41.375.ISSN0034-6861.
^C. Rothleitner; S. Schlamminger (2017)."Invited Review Article: Measurements of the Newtonian constant of gravitation, G".Review of Scientific Instruments.88 (11): 111101.Bibcode:2017RScI...88k1101R.doi:10.1063/1.4994619.PMC8195032.PMID29195410. 111101.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:
Standish., E. Myles (1995). "Report of the IAU WGAS Sub-group on Numerical Standards". In Appenzeller, I. (ed.).Highlights of Astronomy. Dordrecht: Kluwer Academic Publishers.(Complete report available online:PostScript;PDF. Tables from the report also available:Astrodynamic Constants and Parameters)
