Gravitational constant

The gravitational constant G is a key quantity in Newton's law of universal gravitation.

The gravitational constant is approximately 6.6741×10⁻¹¹ N⋅m²/kg^2[1] and denoted by the letter G, is an empirical physical constant involved in the calculation(s) of gravitational force between two bodies. It usually appears in Sir Isaac Newton's law of universal gravitation, and in Albert Einstein's general theory of relativity. It is also known as the universal gravitational constant, Newton's constant, and colloquially as Big G.^[2] It should not be confused with "small g" (g), which is the local gravitational field of Earth (equivalent to the free-fall acceleration^[3]).

Laws and constants

According to the law of universal gravitation, the attractive force (F) between two 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:

$F = G\frac{m_1 m_2}{r^2}\$

The constant of proportionality, G, is the gravitational constant.

The gravitational constant is a physical constant that is difficult to measure with high accuracy.^[4] In SI units, the 2014 CODATA-recommended value of the gravitational constant (with standard uncertainty in parentheses) is:,^[1][5]

$G = 6.674\ 08(31) \times 10^{-11} {\rm \ m^3 \ kg^{-1} \ s^{-2} } = 6.674\ 08(31) \times 10^{-11} {\rm \ N \ m^{2} \ kg^{-2} },$

with relative standard uncertainty 4.7×10⁻⁵.

Dimensions, units, and magnitude

The dimensions assigned to the gravitational constant in the equation above—length cubed, divided by mass, and by time squared—

$[G] = \frac{[L]^3 }{[M][T]^{2}}$

(in SI units, meters cubed per kilogram per second squared)—are those needed to balance the units of measurements in gravitational equations. However, these dimensions have fundamental significance in terms of Planck units; when expressed in SI units, the gravitational constant is dimensionally and numerically equal to the cube of the Planck length divided by the product of the Planck mass and the square of Planck time.

In natural units, of which Planck units are a common example, G and other physical constants such as c (the speed of light) may be set equal to 1.

In many secondary school texts, the dimensions of G are derived from force in order to assist student comprehension:

$G \approx 6.674 \times 10^{-11} {\rm \ N \ (m/kg)^2}.$

In cgs, G can be written as:

$G\approx 6.674 \times 10^{-8} {\rm \ cm^3 \ g^{-1} \ s^{-2} }.$

Applying Kepler's 3rd law, in Newtonian form, to Earth's orbit:

$G = 4 \pi^2 {\rm \ AU^3} {\rm \ yr^{-2}} \ M_\odot^{-1}\,$,

where distance is measured in astronomical units (AU), time in years, and mass in solar masses (M_⊙).

In other fields of astrophysics, where distances are measured in parsecs (pc), velocities in kilometers per second (km/s) and masses again in solar units M_⊙, it is useful to express G as:

$G \approx 4.302 \times 10^{-3} {\rm \ pc}\ M_\odot^{-1} {\rm \ (km/s)^2 }. \,$

G can also be given as:

$G\approx 0.8650 {\rm \ cm^3 \ g^{-1} \ hr^{-2} }.$

Given the fact that the period P of an object in circular orbit around a spherical object obeys

$GM=\frac{3\pi V}{P^2}$

where V is the volume inside the radius of the orbit, we see that

$P^2=\frac{3\pi}{G}\frac{V}{M}\approx 10.896 \ {\rm hr^2 \ g \ cm^{-3} }\frac{V}{M}.$

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.

The gravitational force is extremely weak compared with other fundamental forces. 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. Both these forces are weak when compared with the forces we are able to experience directly, but 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.

History of measurement

The gravitational constant appears in Newton's law of universal gravitation, but it was not measured until seventy-one years after Newton's death by Henry Cavendish with his Cavendish experiment, performed in 1798 (Philosophical Transactions 1798). Cavendish measured G implicitly, using a torsion balance invented by the geologist Rev. John Michell. 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. Cavendish's aim was not actually to measure the gravitational constant, but rather to measure Earth's density relative to water, through the precise knowledge of the gravitational interaction. In retrospect, the density that Cavendish calculated implies a value for G of 6.754×10⁻¹¹ m³ kg⁻¹ s⁻².^[6]

The accuracy of the measured value of G has increased only modestly since the original Cavendish experiment. 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. Published values of G have varied rather broadly, and some recent measurements of high precision are, in fact, mutually exclusive.^[4][7] This led to the 2010 CODATA value by NIST having 20% increased uncertainty than in 2006.^[8] For the 2014 update, CODATA reduced the uncertainty to less than half the 2010 value.

In the January 2007 issue of Science, Fixler et al. described a new measurement of the gravitational constant by atom interferometry, reporting a value of G = 6.693(34)×10⁻¹¹ m³ kg⁻¹ s⁻².^[9] 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.^[10]

A controversial 2015 study of some previous measurements of G, by Anderson et al., suggested that most of the mutually exclusive values can be explained by a periodic variation.^[11] 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.^[12] 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.^[12][13] 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.^[14]

The GM product

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 is denoted μ. Depending on the body concerned, it may also be called the geocentric or heliocentric gravitational constant, among other names.

This quantity gives a convenient simplification of various gravity-related formulas. Also, for celestial bodies such as Earth and the Sun, the value of the product GM is known much more accurately than each factor independently. Indeed, the limited accuracy available for G limits the accuracy of scientific determination of such masses in the first place.

For Earth, using M_⊕ as the symbol for the mass of Earth, we have

$\mu = GM_\oplus = ( 398\ 600.4415 \pm 0.0008 ) {\rm \ km^{3} \ s^{-2} } .$^[15]

For Sun, we have

$\mu = GM_\odot = (1.327\ 124\ 400 \times 10^{11}) {\rm \ km^{3} \ s^{-2} } .$

Calculations in celestial mechanics can also be carried out using the unit of solar mass rather than the standard SI unit kilogram. In this case we use the Gaussian gravitational constant k, where

$k = 0.017\ 202\ 098\ 95 \ A^{\frac{3}{2}} \ D^{-1} \ S^{-\frac{1}{2}}$

and

A is the astronomical unit;

D is the mean solar day;

S is the solar mass.

If instead of mean solar day we use the sidereal year as our time unit, the value of ks is very close to 2π (k = 6.28315).

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.

See also

• Dirac large numbers hypothesis
• Accelerating universe
• Gravity expressed in terms of orbital period
• Lunar Laser Ranging experiment
• Cosmological constant
• Gravitational coupling constant
• Strong gravitational constant

Notes

1. Mohr, Peter J.; Newell, David B.; Taylor, Barry N. (2015-07-21). "CODATA Recommended Values of the Fundamental Physical Constants: 2014". arXiv:1507.07956 [physics.atom-ph].
2. 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
3. Fundamentals of Physics 8ed, Halliday/Resnick/Walker, ISBN 978-0-470-04618-0 p336
4. George T. Gillies (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.
5. "Newtonian constant of gravitation G". CODATA, NIST.
6. Brush, Stephen G.; Holton, Gerald James (2001), Physics, the human adventure: from Copernicus to Einstein and beyond, New Brunswick, N.J: Rutgers University Press, p. 137, ISBN 0-8135-2908-5
7. Peter J. Mohr; Barry N. Taylor (January 2005), "CODATA recommended values of the fundamental physical constants: 2002" (PDF), Reviews of Modern Physics 77 (1): 1–107, Bibcode:2005RvMP...77....1M, doi:10.1103/RevModPhys.77.1, retrieved 2006-07-01. Section Q (pp. 42–47) describes the mutually inconsistent measurement experiments from which the CODATA value for G was derived.
8. "CODATA recommended values of the fundamental physical constants: 2010" (PDF). Rev Mod Phys 84: 1527–1605. 13 November 2012. arXiv:1203.5425. Bibcode:2012RvMP...84.1527M. doi:10.1103/RevModPhys.84.1527.
9. J. B. Fixler; G. T. Foster; J. M. McGuirk; M. A. Kasevich (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
10. Schlamminger, Stephan (18 June 2014). "Fundamental constants: A cool way to measure big G". Nature 510: 478–480. Bibcode:2014Natur.510..478S. doi:10.1038/nature13507.
11. J.D. Anderson; G. Schubert; V. Trimble; M.R. Feldman (April 2015), "Measurements of Newton’s gravitational constant and the length of day" (PDF), EPL 110: 10002, arXiv:1504.06604, Bibcode:2015EL....11010002A, doi:10.1209/0295-5075/110/10002
12. Schlamminger, S.; Gundlach, J. H.; Newman, R. D. (2015). "Recent measurements of the gravitational constant as a function of time". Physical Review D 91 (12). doi:10.1103/PhysRevD.91.121101. ISSN 1550-7998.
13. Karagioz, O. V.; Izmailov, V. P. (1996). "Measurement of the gravitational constant with a torsion balance". Measurement Techniques 39 (10): 979–987. doi:10.1007/BF02377461. ISSN 0543-1972.
14. J. Mould; S. A. Uddin (2014-04-10), "Constraining a Possible Variation of G with Type Ia Supernovae", Publications of the Astronomical Society of Australia 31: e015, arXiv:1402.1534, Bibcode:2014PASA...31...15M, doi:10.1017/pasa.2014.9
15. 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. Retrieved 5 February 2016.

References

• E. Myles Standish. "Report of the IAU WGAS Sub-group on Numerical Standards". In Highlights of Astronomy, I. Appenzeller, ed. Dordrecht: Kluwer Academic Publishers, 1995. (Complete report available online: PostScript; PDF. Tables from the report also available: Astrodynamic Constants and Parameters)
• Jens H. Gundlach; Stephen M. Merkowitz (2000), "Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback", Physical Review Letters 85 (14): 2869–2872, arXiv:gr-qc/0006043, Bibcode:2000PhRvL..85.2869G, doi:10.1103/PhysRevLett.85.2869, PMID 11005956

External links

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