# Standard gravitational parameter

Body | (mμ^{3} s^{−2}) |
---|---|

Sun | 12440018(9)×10 1.327^{20}^{[1]} |

Mercury | ×10 2.2032(9)^{13} |

Venus | 59(9)×10 3.248^{14} |

Earth | 004418(9)×10 3.986^{14} |

Moon | 8695(9)×10 4.904^{12} |

Mars | ×10 4.2828(9)^{13} |

Ceres | 25×10 6.263^{10}^{[2]}^{[3]}^{[4]} |

Jupiter | 86534(9)×10 1.266^{17} |

Saturn | 1187(9)×10 3.793^{16} |

Uranus | 939(9)×10 5.793^{15}^{[5]} |

Neptune | 529(9)×10 6.836^{15} |

Pluto | ×10 8.71(9)^{11}^{[6]} |

Eris | ×10 1.108(9)^{12}^{[7]} |

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.

For all objects in the Solar System, the value of *μ* is known to greater accuracy than either *G* or *M*. This is because *μ* is calculated directly from the speed of a small test mass in orbit about the body, which can be done very accurately. The mass *M* is then inferred by using *G*.^{[8]} The SI units of the standard gravitational parameter are m^{3} s^{−2}.

## Contents

## Small body orbiting a central body

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 *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:

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,^{[9]} 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 *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 orbits:

where *a* is the semi-major axis, which is Kepler's third law.

For parabolic trajectories *rv*^{2} is constant and equal to 2*μ*. For elliptic and hyperbolic orbits *μ* = 2*a*| *ε* |, where *ε* is the specific orbital energy.

## Two bodies orbiting each other

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*_{1}+*Gm*_{2}=*μ*_{1}+*μ*_{2}, where*m*_{1}and*m*_{2}are the masses of the two bodies.

Then:

- for circular orbits,
*rv*^{2}=*r*^{3}*ω*^{2}= 4π^{2}*r*^{3}/*T*^{2}=*μ* - for elliptic orbits, 4π
^{2}*a*^{3}/*T*^{2}=*μ*(with*a*expressed in AU;*T*in seconds and*M*the total mass relative to that of the Sun, we get*a*^{3}/*T*^{2}=*M*) - for parabolic trajectories,
*rv*^{2}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 *μ*.

The value for the Earth is called the **geocentric gravitational constant** and equals 600.4418±0.0008 km^{3} s^{−2}. Thus the uncertainty is 1 to 398000000, much smaller than the uncertainties in 500*G* and *M* separately (1 to each). 7000

The value for the Sun is called the **heliocentric gravitational constant** or *geopotential of the Sun* and equals 12440018×10^{20} m^{3} s^{−2}. 1.327

## See also

## References

- ↑ "Astrodynamic Constants". NASA/JPL. 27 February 2009. Retrieved 27 July 2009.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Asteroid Ceres P_constants (PcK) SPICE kernel file". Retrieved 5 November 2015.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
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*Solar System Research*.**39**(3): 176. Bibcode:2005SoSyR..39..176P. doi:10.1007/s11208-005-0033-2.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ D. T. Britt; D. Yeomans; K. Housen; G. Consolmagno (2002). "Asteroid density, porosity, and structure" (PDF). In W. Bottke; A. Cellino; P. Paolicchi; R.P. Binzel (eds.).
*Asteroids III*. University of Arizona Press. p. 488.CS1 maint: display-editors (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ R.A. Jacobson; J.K. Campbell; A.H. Taylor; S.P. Synnott (1992). "The masses of Uranus and its major satellites from Voyager tracking data and Earth-based Uranian satellite data".
*Astronomical Journal*.**103**(6): 2068–2078. Bibcode:1992AJ....103.2068J. doi:10.1086/116211.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ M.W. Buie; W.M. Grundy; E.F. Young; L.A. Young; et al. (2006). "Orbits and photometry of Pluto's satellites: Charon, S/2005 P1, and S/2005 P2".
*Astronomical Journal*.**132**: 290. arXiv:astro-ph/0512491. Bibcode:2006AJ....132..290B. doi:10.1086/504422.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ M.E. Brown; E.L. Schaller (2007). "The Mass of Dwarf Planet Eris".
*Science*.**316**(5831): 1586. Bibcode::2007Sci...316.1585B Check`|bibcode=`

length (help). doi:10.1126/science.1139415. PMID 17569855.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ This is mostly because
*μ*can be measured by observational astronomy alone, as it has been for centuries. Decoupling it into*G*and*M*must be done by measuring the force of gravity in sensitive laboratory conditions, as first done in the Cavendish experiment. - ↑ 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<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>. A lengthy, detailed review.