Editing Standard gravitational parameter

{{Short description|Concept in celestial mechanics}}
{| class="wikitable floatright"
|-
! Body
! colspan="2"|'''''μ''''' [m<sup>3</sup> s<sup>−2</sup>]
|-
| [[Sun]]
| style="border-right:none;"|{{val|1.32712440018|(9)}}
| style="border-left :none;"|× 10<sup>20</sup> <ref name="Astrodynamic Constants" />
|-
| [[Mercury (planet)|Mercury]]
| style="border-right:none;"|{{val|2.20320|(9)}}
| style="border-left :none;"|× 10<sup>13</sup> <ref name="Anderson" />
|-
| [[Venus]]
| style="border-right:none;"|{{val|3.24858592|(6)}}
| style="border-left :none;"|× 10<sup>14</sup> <ref name="Konopliv99" />
|-
| [[Earth]]
| style="border-right:none;"|{{val|3.986004418|(8)}}
| style="border-left :none;"|× 10<sup>14</sup> <ref name="IAU best estimates"/> 
|-
| [[Moon]]
| style="border-right:none;"|{{val|4.9048695|(9)}}
| style="border-left :none;"|× 10<sup>12</sup>
|-
| [[Mars]]
| style="border-right:none;"|{{val|4.282837|(2)}}
| style="border-left :none;"|× 10<sup>13</sup> <ref>{{cite web|title=Mars Gravity Model 2011 (MGM2011)|url=https://ddfe.curtin.edu.au/gravitymodels/MGM2011/MGM2011_website_copy.pdf|date=2015-03-26<!--from PDF source-->|publisher=Western Australian Geodesy Group|url-status=live|archive-url=https://web.archive.org/web/20130410022448/http://geodesy.curtin.edu.au/research/models/mgm2011/|archive-date=2013-04-10}}</ref>
|-
| [[1 Ceres|Ceres]]
| style="border-right:none;"|{{val|6.26325}}
| style="border-left :none;"|× 10<sup>10</sup> <ref name="SPICE" /><ref name="Pitjeva2005" /><ref name="Britt2002" />
|-
| [[Jupiter]]
| style="border-right:none;"|{{val|1.26686534|(9)}}
| style="border-left :none;"|× 10<sup>17</sup>
|-
| [[Saturn]]
| style="border-right:none;"|{{val|3.7931187|(9)}}
| style="border-left :none;"|× 10<sup>16</sup>
|-
| [[Uranus]]
| style="border-right:none;"|{{val|5.793939|(9)}}
| style="border-left :none;"|× 10<sup>15</sup> <ref name="Jacobson1992" />
|-
| [[Neptune]]
| style="border-right:none;"|{{val|6.836529|(9)}}
| style="border-left :none;"|× 10<sup>15</sup>
|-
| [[Pluto]]
| style="border-right:none;"|{{val|8.71|(9)}}
| style="border-left :none;"|× 10<sup>11</sup> <ref name="Buie06" />
|-
| [[Eris (dwarf planet)|Eris]]
| style="border-right:none;"|{{val|1.108|(9)}}
| style="border-left :none;"|× 10<sup>12</sup> <ref name="Brown Schaller 2007" />
|}

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 {{math|''G''(''m''<sub>1</sub> + ''m''<sub>2</sub>)}}, or as {{math|''GM''}} when one body is much larger than the other:
<math display="block">\mu=G(M+m)\approx GM .</math>

For several objects in the [[Solar System]], the value of ''μ'' is known to greater accuracy than either ''G'' or ''M''. The [[SI]] unit of the standard gravitational parameter is {{nowrap|[[metre|m]]<sup>3</sup>⋅[[second|s]]<sup>−2</sup>}}. However, the unit {{nowrap|[[kilometre|km]]<sup>3</sup>⋅[[second|s]]<sup>−2</sup>}} is frequently used in the scientific literature and in spacecraft navigation.

== Definition ==
=== Small body orbiting a central body ===
{{solar_system_orbital_period_vs_semimajor_axis.svg}}
The [[central body]] in an orbital system can be defined as the one whose mass (''M'') is much larger than the mass of the [[orbiting body]] (''m''), or {{nowrap|''M'' ≫ ''m''}}. This approximation is standard for planets orbiting the [[Sun]] or most moons and greatly simplifies equations. Under [[Newton's law of universal gravitation]], if the distance between the bodies is ''r'', the force exerted on the smaller body is:
<math display="block">F = \frac{G M m}{r^2} = \frac{\mu m}{r^2}</math>

Thus only the product of ''G'' and ''M'' is needed to predict the motion of the smaller body. Conversely, measurements of the smaller body's orbit only provide information on the product, ''μ'', not ''G'' and ''M'' separately. The gravitational constant, ''G'', is difficult to measure with high accuracy,<ref name=gillies>{{Citation |author=George T. 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 |url=https://iopscience.iop.org/article/10.1088/0034-4885/60/2/001/pdf |doi=10.1088/0034-4885/60/2/001 | bibcode = 1997RPPh...60..151G |s2cid=250810284 |url-access=subscription }}. A lengthy, detailed review.</ref> while orbits, at least in the solar system, can be measured with great precision and used to determine ''μ'' with similar precision.

For a [[circular orbit]] around a central body, where the [[Centripetal Force#Formula|centripetal force]] provided by gravity is {{nowrap|1=''F'' = ''mv''{{sup|2}}''r''{{sup|−1}}}}:
<math display="block">\mu = rv^2 = r^3\omega^2 = \frac{4\pi^2r^3}{T^2} ,</math>
where ''r'' is the orbit [[radius]], ''v'' is the [[orbital speed]], ''ω'' is the [[angular speed]], and ''T'' is the [[orbital period]].

This can be generalized for [[elliptic orbit]]s:
<math display="block">\mu = \frac{4\pi^2a^3}{T^2} ,</math>
where ''a'' is the [[semi-major axis]], which is [[Kepler's laws of planetary motion#Kepler's third law|Kepler's third law]].

For [[parabolic trajectory|parabolic trajectories]] ''rv''<sup>2</sup> is constant and equal to 2''μ''. For elliptic and hyperbolic orbits magnitude of ''μ'' = 2 times the magnitude of ''a''  times the magnitude of ''ε'', where ''a'' is the semi-major axis and ''ε'' is the [[specific orbital energy]].

=== General case ===
In the more general case where the bodies need not be a large one and a small one, e.g. a [[binary star]] system, we define:
* the vector '''r''' is the position of one body relative to the other
* ''r'', ''v'', and in the case of an [[elliptic orbit]], the [[semi-major axis]] ''a'', are defined accordingly (hence ''r'' is the distance)
* ''μ'' = ''Gm''<sub>1</sub> + ''Gm''<sub>2</sub> = ''μ''<sub>1</sub> + ''μ''<sub>2</sub>, where ''m''<sub>1</sub> and ''m''<sub>2</sub> are the masses of the two bodies.

Then:
* for [[circular orbit]]s, ''rv''<sup>2</sup> = ''r''<sup>3</sup>''ω''<sup>2</sup> = 4π<sup>2</sup>''r''<sup>3</sup>/''T''<sup>2</sup> = ''μ''
* for [[elliptic orbit]]s, {{nowrap|1=4π<sup>2</sup>''a''<sup>3</sup>/''T''<sup>2</sup> = ''μ''}} (with ''a'' expressed in AU; ''T'' in years and ''M'' the total mass relative to that of the Sun, we get {{nowrap|1=''a''<sup>3</sup>/''T''<sup>2</sup> = ''M''}})
* for [[parabolic trajectory|parabolic trajectories]], ''rv''<sup>2</sup> is constant and equal to 2''μ''
* for elliptic and hyperbolic orbits, ''μ'' is twice the semi-major axis times the negative of the [[specific orbital energy]], where the latter is defined as the total energy of the system divided by the [[reduced mass]].

=== In a pendulum ===
The standard gravitational parameter can be determined using a [[pendulum]] oscillating above the surface of a body as:<ref>
{{citation
 | last1 = Lewalle
 | first1 = Philippe
 | last2 = Dimino
 | first2 = Tony
 | year = 2014
 | title = Measuring Earth's Gravitational Constant with a Pendulum
 | page = 1
 | url = http://teacher.pas.rochester.edu/phy141/Laboratory/SampleReports/Sample141lab_pendulumg.pdf
}}</ref>

<math display="block">\mu \approx \frac{4 \pi^2 r^2 L}{T^2} </math>
where ''r'' is the radius of the gravitating body, ''L'' is the length of the pendulum, and ''T'' is the [[Frequency|period]] of the pendulum (for the reason of the approximation see [[Pendulum (mechanics)|Pendulum in mechanics]]).

== Solar system ==
{{Further|Gaussian gravitational constant}}

=== Geocentric gravitational constant ===
{{Further|Earth mass}}
''G''{{Earth mass}}, the gravitational parameter for the [[Earth]] as the central body, is called the '''geocentric gravitational constant'''. It equals {{val|3.986004418|0.000000008|e=14|u=m<sup>3</sup>⋅s<sup>−2</sup>}}.<ref name="IAU best estimates">{{cite web|title=IAU Astronomical Constants: Current Best Estimates|url=https://iau-a3.gitlab.io/NSFA/NSFA_cbe.html#GME2009|website=iau-a2.gitlab.io|publisher=IAU Division I Working Group on Numerical Standards for Fundamental Astronomy|access-date=25 June 2021}}, citing Ries, J. C., Eanes, R. J., Shum, C. K., and Watkins, M. M., 1992, "Progress in the Determination of the Gravitational Coefficient of the Earth," Geophys. Res. Lett., 19(6), pp. 529-531.</ref>

The value of this constant became important with the beginning of [[spaceflight]] in the 1950s, and great effort was expended to determine it as accurately as possible during the 1960s. Sagitov (1969) cites a range of values reported from 1960s high-precision measurements, with a relative uncertainty of the order of 10<sup>−6</sup>.<ref name=Sagitov>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.</ref>

During the 1970s to 1980s, the increasing number of [[artificial satellite]]s in Earth orbit further facilitated high-precision measurements, 
and the relative uncertainty was decreased by another three orders of magnitude, to about {{val|2|e=-9}} (1 in 500 million) as of 1992. 
Measurement involves observations of the distances from the satellite to Earth stations at different times, which can be obtained to high accuracy using radar or laser ranging.<ref>{{cite journal|last1=Lerch|first1=Francis J.|last2=Laubscher|first2=Roy E.|last3=Klosko|first3=Steven M.|last4=Smith|first4=David E.|last5=Kolenkiewicz|first5=Ronald|last6=Putney|first6=Barbara H.|last7=Marsh|first7=James G.|last8=Brownd|first8=Joseph E.|title=Determination of the geocentric gravitational constant from laser ranging on near-Earth satellites|journal=Geophysical Research Letters|date=December 1978|volume=5|issue=12|pages=1031–1034|doi=10.1029/GL005i012p01031|bibcode=1978GeoRL...5.1031L}}</ref>

=== Heliocentric gravitational constant ===
{{Further|Solar mass}}
''G''{{Solar mass}}, the gravitational parameter for the [[Sun]] as the central body, 
is called the '''heliocentric gravitational constant''' or ''geopotential of the Sun'' and equals {{nowrap|{{val|1.32712440042|0.0000000001|e=20|u=m<sup>3</sup>⋅s<sup>−2</sup>}}.<ref name="AIP Helio">{{cite journal|last1=Pitjeva|first1=E. V.|title=Determination of the Value of the Heliocentric Gravitational Constant from Modern Observations of Planets and Spacecraft|journal=Journal of Physical and Chemical Reference Data|date=September 2015|volume=44|issue=3|pages=031210|doi=10.1063/1.4921980|bibcode=2015JPCRD..44c1210P}}</ref>}}

The relative uncertainty in ''G''{{Solar mass}}, cited at below 10<sup>−10</sup> as of 2015,<!--7.5e-11 as of 2015--> is smaller than the uncertainty in ''G''{{Earth mass}}  <!--2e-9 as of 1992-->
because ''G''{{Solar mass}} is derived from the ranging of interplanetary probes, and the absolute error of the distance measures to them is about the same as the earth satellite ranging measures, while the absolute distances involved are much bigger.{{citation needed|date=August 2016}}<!-- OR maybe it is because one source is dated 1992 and the other is dated 2015? -->

== See also ==
* [[Astronomical system of units]]
* [[Planetary mass]]

== References ==
{{reflist|30em
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<ref name="Astrodynamic Constants">
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<ref name="Pitjeva2005">
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<ref name="Britt2002">
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<ref name="SPICE">
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}}</ref>

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<ref name="Brown Schaller 2007">
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</ref>

<!--
<ref name="Lunar Constants and Models Document">
{{cite web
 | title = Lunar Constants and Models Document
 | date = 23 September 2005
 | publisher = [[NASA]]/[[Jet Propulsion Laboratory|JPL]]
 | url = http://www.hq.nasa.gov/alsj/lunar_cmd_2005_jpl_d32296.pdf
 | access-date = 1 April 2013
}}
</ref>-->

}}

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