gravitational constant
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The gravitational constant is an
empirical Empirical evidence is evidence obtained through sense experience or experimental procedure. It is of central importance to the sciences and plays a role in various other fields, like epistemology and law. There is no general agreement on how t ...
physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that cannot be explained by a theory and therefore must be measured experimentally. It is distinct from a mathematical constant, which has a ...
involved in the calculation of gravitational effects in
Sir Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Enlightenment that followed. His book (''Mathe ...
's law of universal gravitation and in
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
's theory of general relativity. It is also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant, denoted by the capital letter . In Newton's law, it is the proportionality constant connecting the gravitational force between two bodies with the product of their
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
es and the
inverse square In science, an inverse-square law is any scientific law stating that the observed "intensity" of a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental ca ...
of their
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
. In the
Einstein field equations In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...
, 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 The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official st ...
, its value is approximately The modern notation of Newton's law involving 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.


Definition

According to Newton's law of universal gravitation, the magnitude of the attractive
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
() between two bodies each with a spherically symmetric
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
distribution is directly proportional to the product of their
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
es, and , and inversely proportional to the square of the distance, , directed along the line connecting their centres of mass: F=G\frac. The constant of proportionality, , in this non-relativistic formulation is the gravitational constant. Colloquially, the gravitational constant is also called "Big G", distinct from "small g" (), which is the local gravitational field of Earth (also referred to as free-fall acceleration). Where M_\oplus is the mass of Earth and r_\oplus is the radius of Earth, the two quantities are related by: g = G\frac. The gravitational constant is a constant term in the
Einstein field equations In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...
of
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, G_ + \Lambda g_ = \kappa T_ \,, where is the Einstein tensor (not the gravitational constant despite the use of ), is the cosmological constant, is the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
, is the stress–energy tensor, and is the Einstein gravitational constant, a constant originally introduced by Einstein that is directly related to the Newtonian constant of gravitation: \kappa = \frac \approx 2.076\,647(46) \times 10^ \mathrm.


Value and uncertainty

The gravitational constant is a physical constant that is difficult to measure with high accuracy.. A lengthy, detailed review. See Figure 1 and Table 2 in particular. This is because the gravitational force is an extremely weak force as compared to other
fundamental forces In physics, the fundamental interactions or fundamental forces are interactions in nature that appear not to be reducible to more basic interactions. There are four fundamental interactions known to exist: * gravity * electromagnetism * weak int ...
at the laboratory scale. In SI units, the CODATA-recommended value of the gravitational constant is: : G = The relative standard
uncertainty Uncertainty or incertitude refers to situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown, and is particularly relevant for decision ...
is .


Natural units

Due to its use as a defining constant in some systems of
natural units In physics, natural unit systems are measurement systems for which selected physical constants have been set to 1 through nondimensionalization of physical units. For example, the speed of light may be set to 1, and it may then be omitted, equa ...
, particularly geometrized unit systems such as
Planck units In particle physics and physical cosmology, Planck units are a system of units of measurement defined exclusively in terms of four universal physical constants: ''Speed of light, c'', ''Gravitational constant, G'', ''Reduced Planck constant, ħ ...
and Stoney 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 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

In
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline, James Keeler, said, astrophysics "seeks to ascertain the ...
, it is convenient to measure distances in
parsec The parsec (symbol: pc) is a unit of length used to measure the large distances to astronomical objects outside the Solar System, approximately equal to or (AU), i.e. . The parsec unit is obtained by the use of parallax and trigonometry, and ...
s (pc), velocities in kilometres per second (km/s) and masses in solar units . In these units, the gravitational constant is: G \approx 4.3009 \times 10^ ^ . For situations where tides are important, the relevant length scales are
solar radii Solar may refer to: Astronomy * Of or relating to the Sun ** Solar telescope, a special purpose telescope used to observe the Sun ** A device that utilizes solar energy (e.g. "solar panels") ** Solar calendar, a calendar whose dates indicat ...
rather than parsecs. In these units, the gravitational constant is: G \approx 1.908\ 09 \times 10^ \mathrm \, R_\odot M_\odot^ . In
orbital mechanics Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the law of universal ...
, the period of an object in circular orbit around a spherical object obeys GM=\frac , where is the volume inside the radius of the orbit, and is the total mass of the two objects. It follows that : P^2=\frac\frac\approx 10.896 \mathrm\frac. This way of expressing 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 Earth orbits the Sun at an astronomical unit, average distance of , or 8.317 light-second, light-minutes, in a retrograde and prograde motion, counterclockwise direction as viewed from above the Northern Hemisphere. One complete orbit takes & ...
: : G = 4 \pi^2 \mathrm \ M^ \approx 39.478 \mathrm \ M_\odot^ , where distance is measured in terms of the
semi-major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longe ...
of Earth's orbit (the
astronomical unit The astronomical unit (symbol: au or AU) is a unit of length defined to be exactly equal to . Historically, the astronomical unit was conceived as the average Earth-Sun distance (the average of Earth's aphelion and perihelion), before its m ...
, AU), time in
year A year is a unit of time based on how long it takes the Earth to orbit the Sun. In scientific use, the tropical year (approximately 365 Synodic day, solar days, 5 hours, 48 minutes, 45 seconds) and the sidereal year (about 20 minutes longer) ...
s, and mass in the total mass of the orbiting system (). 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 The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...
and from general relativity. From 1964 until 2012, however, it was used as the definition of the astronomical unit and thus held by definition: 1\ \mathrm = \left( \frac \mathrm^2 \right)^ \approx 1.495\,979 \times 10^\mathrm. Since 2012, the AU is defined as exactly, and the equation can no longer be taken as holding precisely. The quantity —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 The standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of that body. For two bodies, the parameter may be expressed as , or as when one body is much larger than the ...
(also denoted ). The standard gravitational parameter 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 In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler in 1609 (except the third law, which was fully published in 1619), describe the orbits of planets around the Sun. These laws replaced circular orbits and epicycles in ...
, and in the formula for
escape velocity In celestial mechanics, escape velocity or escape speed is the minimum speed needed for an object to escape from contact with or orbit of a primary body, assuming: * Ballistic trajectory – no other forces are acting on the object, such as ...
. This quantity gives a convenient simplification of various gravity-related formulas. The product is known much more accurately than either factor is. Calculations in
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
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 The Gaussian gravitational constant (symbol ) 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 histor ...
was historically in widespread use, , expressing the mean
angular velocity In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...
of the Sun–Earth system. The use of this constant, and the implied definition of the
astronomical unit The astronomical unit (symbol: au or AU) is a unit of length defined to be exactly equal to . Historically, the astronomical unit was conceived as the average Earth-Sun distance (the average of Earth's aphelion and perihelion), before its m ...
discussed above, has been deprecated by the IAU since 2012.


History of measurement


Early history

The existence of the constant is implied in
Newton's law of universal gravitation Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is Proportionality (mathematics)#Direct proportionality, proportional to the product ...
as published in the 1680s (although its notation as dates to the 1890s), but is not calculated in his ''
Philosophiæ Naturalis Principia Mathematica (English: ''The Mathematical Principles of Natural Philosophy''), often referred to as simply the (), is a book by Isaac Newton that expounds Newton's laws of motion and his law of universal gravitation. The ''Principia'' is written in Lati ...
'' 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. 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: : ≈ 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 Edmond (or Edmund) Halley (; – ) was an English astronomer, mathematician and physicist. He was the second Astronomer Royal in Britain, succeeding John Flamsteed in 1720. From an observatory he constructed on Saint Helena in 1676–77, Hal ...
, had suggested. 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 ( times the density of water), about 20% below the modern value. This immediately led to estimates on the densities and masses of the Sun,
Moon The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...
and
planets A planet is a large, rounded astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets by the most restrictive definition of the te ...
, sent by Hutton to
Jérôme Lalande Joseph Jérôme Lefrançois de Lalande (; 11 July 1732 – 4April 1807) was a French astronomer, freemason and writer. He is known for having estimated a precise value of the astronomical unit (the distance from the Earth to the Sun) using measu ...
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 settingBoys 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
G = g\frac = \frac. Based on this, Hutton's 1778 result is equivalent to . 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. He determined a value for implicitly, using a torsion balance invented by the geologist Rev.
John Michell John Michell (; 25 December 1724 – 21 April 1793) was an English natural philosopher and clergyman who provided pioneering insights into a wide range of scientific fields including astronomy, geology, optics, and gravitation. Considered "on ...
(1753). He used a horizontal
torsion beam The twist-beam rear suspension (also torsion-beam axle, deformable torsion beam, or compound crank) is a type of automobile suspension based on a large H- or C-shaped member. The front of the H attaches to the body via rubber bushings, and th ...
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, , corresponds to value of . It is remarkably accurate, being about 1% above the modern CODATA recommended value , consistent with the claimed relative standard uncertainty of 0.6%.


19th century

The accuracy of the measured value of has increased only modestly since the original Cavendish experiment. 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, ), Edward Sabine (1827, ), Carlo Ignazio Giulio (1841, ) and George Biddell Airy (1854, ). Cavendish's experiment was first repeated by
Ferdinand Reich Ferdinand Reich (19 February 1799 – 27 April 1882) was a German chemist who co-discovered indium in 1863 with Hieronymous Theodor Richter. Reich was born in Bernburg, Anhalt-Bernburg, Holy Roman Empire and died in Freiberg, Saxony, Freibe ...
(1838, 1842, 1853), who found a value of , which is actually worse than Cavendish's result, differing from the modern value by 1.5%. Cornu and Baille (1873), found . 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 ) and
Thomas Corwin Mendenhall Thomas Corwin Mendenhall (October 4, 1841 – March 23, 1924) was an American autodidact physicist and meteorologist. He was the first professor hired at Ohio State University in 1873 and the superintendent of the United States Coast and Geodeti ...
(1880, ). Cavendish's result was first improved upon by
John Henry Poynting John Henry Poynting Fellow of the Royal Society, FRS (9 September 185230 March 1914) was an English physicist. He was the first professor of physics at Mason Science College from 1880 to 1900, and then the successor institution, the University ...
(1891), who published a value of , 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 by C. V. Boys (1895) and Carl Braun (1897), with compatible results suggesting = . The modern notation involving the constant was introduced by Boys in 1894 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 was, however, of the same order of magnitude as the other results 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 from ''Astronomicheskii Zhurnal'' Vol. 46, No. 4 (July–August 1969), 907–915 (table of historical experiments p. 715). Arthur Stanley Mackenzie in ''The Laws of Gravitation'' (1899) reviews the work done in the 19th century. Poynting is the author of the article "Gravitation" in the ''Encyclopædia Britannica'' Eleventh Edition (1911). Here, he cites a value of = with a relative uncertainty of 0.2%.


Modern value

Paul R. Heyl (1930) published the value of (relative uncertainty 0.1%), improved to (relative uncertainty 0.045% = 450 ppm) in 1942. 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 of 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. Section Q (pp. 42–47) describes the mutually inconsistent measurement experiments from which the CODATA value for was derived. Establishing a standard value for with a relative standard uncertainty better than 0.1% has therefore remained rather speculative. By 1969, the value recommended by the
National Institute of Standards and Technology The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into Outline of p ...
(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. 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: In the January 2007 issue of ''
Science Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...
'', Fixler et al. described a measurement of the gravitational constant by a new technique,
atom interferometry An atom interferometer uses the wave-like nature of atoms in order to produce interference. In atom interferometers, the roles of matter and light are reversed compared to the laser based interferometers, i.e. the beam splitter and mirrors are lase ...
, reporting a value of , 0.28% (2800 ppm) higher than the 2006 CODATA value. An improved cold atom measurement by Rosi et al. was published in 2014 of . 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 Uncertainty or incertitude refers to situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown, and is particularly relevant for decision ...
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). In August 2018, a Chinese research group announced new measurements based on torsion balances, and based on two different methods. 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.


Constancy

Analysis of observations of 580 type 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.


See also

*
Gravity of Earth The gravity of Earth, denoted by , is the net force, net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation). It is a Eucl ...
*
Standard gravity The standard acceleration of gravity or standard acceleration of free fall, often called simply standard gravity and denoted by or , is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is a constant ...
*
Gaussian gravitational constant The Gaussian gravitational constant (symbol ) 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 histor ...
*
Orbital mechanics Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the law of universal ...
*
Escape velocity In celestial mechanics, escape velocity or escape speed is the minimum speed needed for an object to escape from contact with or orbit of a primary body, assuming: * Ballistic trajectory – no other forces are acting on the object, such as ...
*
Gravitational potential In classical mechanics, the gravitational potential is a scalar potential associating with each point in space the work (energy transferred) per unit mass that would be needed to move an object to that point from a fixed reference point in the ...
*
Gravitational wave Gravitational waves are oscillations of the gravitational field that Wave propagation, travel through space at the speed of light; they are generated by the relative motion of gravity, gravitating masses. They were proposed by Oliver Heaviside i ...
* Strong gravity * Dirac large numbers hypothesis * Accelerating expansion of the universe *
Lunar Laser Ranging experiment Lunar Laser Ranging (LLR) is the practice of measuring Lunar distance (astronomy), the distance between the surfaces of the Earth and the Moon using Lidar, laser ranging. The distance can be calculated from the Round-trip delay, round-trip time ...
* Cosmological constant


References

; Footnotes : ; Citations :


Sources

* ''(Complete report available online
PostScriptPDF
Tables from the report also available
Astrodynamic Constants and Parameters
'' *


External links


Newtonian constant of gravitation
at the
National Institute of Standards and Technology The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into Outline of p ...
br>References on Constants, Units, and Uncertainty

The Controversy over Newton's Gravitational Constant
— additional commentary on measurement problems {{Authority control Gravity Fundamental constants