Editing Gravitational constant (section)

== Value and uncertainty ==
The gravitational constant is a physical constant that is difficult to measure with high accuracy.<ref name=gillies>{{cite journal|first=George T. |last=Gillies |title=The Newtonian gravitational constant: recent measurements and related studies |journal=Reports on Progress in Physics |date=1997 |volume=60 |issue=2 |pages=151–225 |doi=10.1088/0034-4885/60/2/001|bibcode = 1997RPPh...60..151G |s2cid=250810284 }}. A lengthy, detailed review. See Figure 1 and Table 2 in particular.</ref> This is because the gravitational force is an extremely weak force as compared to other [[fundamental forces]] at the laboratory scale.{{efn|For example, the gravitational force between an [[electron]] and a [[proton]] 1&nbsp;m apart is approximately {{val|e=−67|ul=N}}, whereas the [[electromagnetic force]] between the same two particles is approximately {{val|e=−28|u=N}}. The electromagnetic force in this example is in the order of 10<sup>39</sup> times greater than the force of gravity—roughly the same ratio as the [[Solar mass|mass of the Sun]] to a microgram.|name=|group=}}

In [[International System of Units|SI]] units, the [[CODATA]]-recommended value of the gravitational constant is:{{physconst|G|ref=only}}
: <math>G</math> = {{physconst|G|ref=no}}

The relative standard [[Measurement uncertainty|uncertainty]] is {{physconst|G|runc=yes|ref=no}}.

=== Natural units ===
Due to its use as a defining<!--sic in cited sources--> constant in some systems of [[natural units]],<ref>{{cite book |title=The Foundation of Reality: Fundamentality, Space, and Time |author1=David Glick |author2=George Darby |author3=Anna Marmodoro |edition= |publisher=Oxford University Press |year=2020 |isbn=978-0-19-883150-1 |page=99 |url=https://books.google.com/books?id=sqXaDwAAQBAJ}} [https://books.google.com/books?id=sqXaDwAAQBAJ&pg=PA99 Extract of page 99]</ref><ref>{{cite book |title=Relativistic Celestial Mechanics of the Solar System |author1=Sergei Kopeikin |author2=Michael Efroimsky |author3=George Kaplan |edition= |publisher=John Wiley & Sons |year=2011 |isbn=978-3-527-63457-6 |page=820 |url=https://books.google.com/books?id=uN5_DQWSR14C}} [https://books.google.com/books?id=uN5_DQWSR14C&pg=PA820 Extract of page 820]</ref> particularly [[geometrized unit system]]s such as [[Planck units]] 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 ===
{{further|Standard gravitational parameter|orbital mechanics|celestial mechanics|Gaussian gravitational constant|Earth mass|Solar mass}}
In [[astrophysics]], it is convenient to measure distances in [[parsec]]s (pc), velocities in kilometres per second (km/s) and masses in solar units {{math|''M''{{sub|⊙}}}}. In these units, the gravitational constant is:
<math display="block"> G \approx 4.3009 \times 10^{-3} {\mathrm{~pc{\cdot}(km/s)^2} \, M_\odot}^{-1} .</math>
For situations where tides are important, the relevant length scales are [[solar radius|solar radii]] rather than parsecs. In these units, the gravitational constant is:
<math display="block"> G \approx 1.908\ 09 \times 10^{5} \mathrm{~(km/s)^2 } \, R_\odot M_\odot^{-1} .</math>
In [[orbital mechanics]], the period {{math|''P''}} of an object in circular orbit around a spherical object obeys
<math display="block"> GM=\frac{3\pi V}{P^2} ,</math>
where {{math|''V''}} is the volume inside the radius of the orbit, and {{math|''M''}} is the total mass of the two objects. It follows that
: <math> P^2=\frac{3\pi}{G}\frac{V}{M}\approx 10.896 \mathrm{~ h^2 {\cdot} g {\cdot} cm^{-3} \,}\frac{V}{M}.</math>
This way of expressing {{math|''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 laws of planetary motion#Third law|Kepler's 3rd law]], expressed in units characteristic of [[Earth's orbit]]:
: <math> G = 4 \pi^2 \mathrm{\ AU^3 {\cdot} yr^{-2}} \ M^{-1} \approx 39.478 \mathrm{\ AU^3 {\cdot} yr^{-2}} \ M_\odot^{-1} ,</math>
where distance is measured in terms of the [[semi-major axis]] of Earth's orbit (the [[astronomical unit]], AU), time in [[solar year|year]]s, and mass in the total mass of the orbiting system ({{math|1=''M'' = {{solar mass}} + {{earth mass|sym=y}} + {{lunar mass|sym=yes}}}}{{efn|
{{mvar|M}} ≈ {{val|1.000003040433}} {{math|{{solar mass}}}}, so that {{mvar|M}} {{=}} {{math|{{solar mass}}}} can be used for accuracies of five or fewer significant digits.}}).

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:
<math display="block"> 1\ \mathrm{AU} = \left( \frac{GM}{4 \pi^2} \mathrm{yr}^2 \right)^{\frac{1}{3}} \approx 1.495\,979 \times 10^{11}\mathrm{~m}.</math> <!--(1.3271244002e+20 * 1.000003040433 * 86400^2 * 365.25636^2)^(1/3) = 1.4959788e+11-->
Since 2012, the AU is defined as {{val|1.495978707|e=11|u=m}} exactly, and the equation can no longer be taken as holding precisely.

The quantity {{math|''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 {{math|''μ''}}). The standard gravitational parameter {{math|''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 {{math|''GM''}} is known much more accurately than either factor is.
{| class=wikitable
|+ Values for ''GM''
!scope="col"| Body
!scope="col"| {{math|1=''μ'' = ''GM''}}
!scope="col"| Value
!scope="col"| Relative uncertainty
|-
!scope="row"| [[Sun]]
| {{math|''G''{{solar mass}}}}
| {{val|1.32712440018|(8)|e=20|u=m{{sup|3}}⋅s{{sup|−2}}}}<ref name="Astrodynamic Constants">{{cite web
| title = Astrodynamic Constants
| date = 27 February 2009
| publisher = [[NASA]]/[[Jet Propulsion Laboratory|JPL]]
| url = http://ssd.jpl.nasa.gov/?constants
| access-date = 27 July 2009
}}
</ref>
| {{val|6|e=-11}}
|-
!scope="row"|[[Earth]]
| {{math|''G''{{earth mass|sym=y}}}}
| {{val|3.986004418|(8)|e=14|u=m{{sup|3}}⋅s{{sup|−2}}}}<ref name="IAU best estimates">{{cite web |title=Geocentric gravitational constant |work=Numerical Standards for Fundamental Astronomy |url=https://iau-a3.gitlab.io/NSFA/NSFA_cbe.html#GME2009 |via=iau-a3.gitlab.io |publisher=IAU Division I Working Group on Numerical Standards for Fundamental Astronomy |access-date=24 June 2021}} Citing
* {{cite journal|vauthors=Ries JC, Eanes RJ, Shum CK, Watkins MM |s2cid=123322272 |title=Progress in the determination of the gravitational coefficient of the Earth |journal=Geophysical Research Letters | date=20 March 1992 |volume=19 |issue=6 |doi=10.1029/92GL00259 |bibcode=1992GeoRL..19..529R |pages=529–531}}</ref>
| {{val|2|e=-9}}
|}

Calculations in [[celestial mechanics]] can also be carried out using the units of solar masses, [[mean solar day]]s and astronomical units rather than standard SI units. For this purpose, the [[Gaussian gravitational constant]] was historically in widespread use, {{math|''k'' {{=}} {{val|0.01720209895}} [[radian]]s per [[day]]}}, expressing the mean [[angular velocity]] of the Sun–Earth system.{{citation needed|date=September 2020}} The use of this constant, and the implied definition of the [[astronomical unit]] discussed above, has been deprecated by the [[IAU]] since 2012.{{citation needed|date=September 2020}}