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|-
! Body
! ''μ'' (km<sup>3</sup>s<sup>−2</sup>)
|-
| [[Sun]]
| {{val|132712440018|(9)}}<ref name="Astrodynamic Constants" />
|-
| [[Mercury (planet)|Mercury]]
| {{val|22032}}
|-
| [[Venus]]
| {{val|324859}}
|-
| [[Earth]]
| {{val|398600.4418|(9)}}
|-
| [[Moon]]
| {{val|4902.8000}}
|-
| [[Mars]]
| {{val|42828}}
|-
| [[1 Ceres|Ceres]]
| {{val|63.1|(3)}}<ref name="Pitjeva2005" /><ref name="Britt2002" />
|-
| [[Jupiter]]
| {{val|126686534}}
|-
| [[Saturn]]
| {{val|37931187}}
|-
| [[Uranus]]
| {{val|5793939|(13)}}<ref name="Jacobson1992" />
|-
| [[Neptune]]
| {{val|6836529}}
|-
| [[Pluto]]
| {{val|871|(5)}}<ref name="Buie06" />
|-
| [[Eris (dwarf planet)|Eris]]
| {{val|1108|(13)}}<ref name="Brown Schaller 2007" />
|}
 
In [[celestial mechanics]], the '''standard gravitational parameter''' ''μ'' of a [[celestial body]] is the product of the [[gravitational constant]] ''G'' and the mass ''M'' of the body.
 
:<math>\mu=GM \ </math>
 
For several objects in the solar system, the value of ''μ'' is known to greater accuracy than ''G'' or ''M.'' The [[SI]] units of the standard gravitational parameter are [[Kilometre|km]]<sup>3</sup>[[second|s]]<sup>−2</sup>.
 
== Small body orbiting a central body ==
{{Properties_of_mass}}
Under standard assumptions in astrodynamics we have:
: <math>m \ll M \ </math>
where ''m'' is the mass of the [[orbiting body]], ''M'' is the mass of the [[central body]].
 
For all [[circular orbit]]s around a given central body:
: <math>\mu = rv^2 = r^3\omega^2 = 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]].
 
The last equation has a very simple generalization to [[elliptic orbit]]s:
: <math>\mu=4\pi^2a^3/T^2 \ </math>
where ''a'' is the [[semi-major axis]]. (See [[Kepler's laws of planetary motion#Kepler's third law|Kepler's third law]]).
 
For all [[parabolic trajectory|parabolic trajectories]] ''rv''<sup>2</sup> is constant and equal to 2''μ''. For elliptic and hyperbolic orbits ''μ''&nbsp;=&nbsp;2''a''|''ε''|, where ''ε'' is the [[specific orbital energy]].
 
== Two bodies orbiting each other == <!-- This section is linked from [[Alpha Centauri]] -->
 
In the more general case where the bodies need not be a large one and a small one (the [[two-body problem]]), 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, 4π<sup>2</sup>''a''<sup>3</sup>/''T''<sup>2</sup> = ''μ'' (with ''a'' expressed in AU; ''T'' in seconds and ''M'' the total mass relative to that of the Sun, we get ''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 absolute value of the [[specific orbital energy]], where the latter is defined as the total energy of the system divided by the [[reduced mass]].
 
== Terminology and accuracy ==
 
Note that the [[reduced mass]] is also denoted by <math>\mu.\!\,</math>.
 
The value for the [[Earth]] is called the '''geocentric gravitational constant''' and equals {{val|398600.4418|0.0008|u=km<sup>3</sup>s<sup>−2</sup>}}. Thus the uncertainty is 1 to {{val|500000000}}, much smaller than the uncertainties in ''G'' and ''M'' separately (1 to {{val|7000}} each).
 
The value for the [[Sun]] is called the '''heliocentric gravitational constant''' or
''geopotential of the sun'' and equals {{val|1.32712440018|e=20|u=m<sup>3</sup>s<sup>−2</sup>}}.
 
== References ==
 
{{reflist
| refs =
 
<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
| accessdate = 27 July 2009
}}
</ref>
 
<ref name="Pitjeva2005">
{{cite journal
| doi = 10.1007/s11208-005-0033-2
| author = E.V. Pitjeva
| year = 2005
| title = High-Precision Ephemerides of Planets&nbsp;— EPM and Determination of Some Astronomical Constants
| url = http://iau-comm4.jpl.nasa.gov/EPM2004.pdf
| journal = [[Solar System Research]]
| volume = 39
| issue=3
| pages=176
| bibcode = 2005SoSyR..39..176P
}}
</ref>
 
<ref name="Britt2002">
{{cite book
| author = D. T. Britt, D. Yeomans, K. Housen, G. Consolmagno
| year = 2002
| chapter = Asteroid density, porosity, and structure
| title = [http://www.lpi.usra.edu/books/AsteroidsIII/download.html Asteroids III]
| editor = W. Bottke, A. Cellino, P. Paolicchi, R.P. Binzel
| page = 488
| publisher = [[University of Arizona Press]]
| url = http://www.lpi.usra.edu/books/AsteroidsIII/pdf/3022.pdf
}}
</ref>
 
<ref name="Jacobson1992">
{{cite journal
| doi = 10.1086/116211
| author = R.A. Jacobson, J.K. Campbell, A.H. Taylor, S.P. Synnott
| year = 1992
| title = The masses of Uranus and its major satellites from Voyager tracking data and Earth-based Uranian satellite data
| journal = [[Astronomical Journal]]
| volume = 103
| issue = 6
| pages = 2068–2078
| bibcode = 1992AJ....103.2068J
}}
</ref>
 
<ref name="Buie06">
{{cite journal
| doi = 10.1086/504422
| author = M.W. Buie, W.M. Grundy, E.F. Young, L.A. Young, S.A. Stern
| year = 2006
| title = Orbits and photometry of Pluto's satellites: Charon, S/2005 P1, and S/2005 P2
| journal = [[Astronomical Journal]]
| volume = 132
| pages = 290
| bibcode = 2006AJ....132..290B
| arxiv = astro-ph/0512491
}}
</ref>
 
<ref name="Brown Schaller 2007">
{{cite journal
| doi = 10.1126/science.1139415
| author = M.E. Brown, E.L. Schaller
| year = 2007
| title = The Mass of Dwarf Planet Eris
| journal = [[Science (journal)|Science]]
| volume = 316
| issue = 5831
| pages = 1586
| bibcode = :2007Sci...316.1585B
}}
</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
| accessdate = 1 April 2013
}}
</ref>-->
 
}}
 
== See also ==
 
* [[Astronomical system of units]]
* [[Planetary mass]]
 
{{orbits}}
 
[[Category:Orbits]]

Latest revision as of 20:35, 26 December 2014

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