Standard_gravitational_parameter

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Body	$\backslash mu$ (km3s-2)
Sun	132,712,440,018
Mercury	22,032
Venus	324,859
Earth	398,600	.4418	±0.0008
Moon	4902	.7779
Mars	42,828
Ceres	63	.1	±0.3Pitjeva, E. V. (2005). "High-Precision Ephemerides of Planets — EPM and Determination of Some Astronomical Constants" (PDF). Solar System Research 39 (3): 176. doi:10.1007/s11208-005-0033-2. D. T. Britt et al Asteroid density, porosity, and structure, pp. 488 in Asteroids III, University of Arizona Press (2002).
Jupiter	126,686,534
Saturn	37,931,187
Uranus	5,793,939		± 13Jacobson, R.A.; Campbell, J.K.; Taylor, A.H.; Synnott, S.P. (1992). "The masses of Uranus and its major satellites from Voyager tracking data and Earth-based Uranian satellite data". The Astronomical Journal 103 (6): 2068–2078. doi:10.1086/116211.
Neptune	6,836,529
Pluto	871		±5M. W. Buie, W. M. Grundy, E. F. Young, L. A. Young, S. A. Stern (2006). "Orbits and photometry of Pluto\'s satellites: Charon, S/2005 P1, and S/2005 P2". Astronomical Journal 132: 290. arXiv:astro-ph/0512491.
Eris	1,108		±13M.E. Brown and E.L. Schaller (2007). "The Mass of Dwarf Planet Eris". Science 316 (5831). doi:10.1126/science.1139415.

In astrodynamics, the standard gravitational parameter $\backslash mu\; \backslash$ of a celestial body is the product of the gravitational constant $G$ and the mass $M$:

$\backslash mu=GM\; \backslash$

The units of the standard gravitational parameter are km³s^-2

Contents 1 Small body orbiting a central body 2 Two bodies orbiting each other 3 Terminology and accuracy 4 References

Small body orbiting a central body

Under standard assumptions in astrodynamics we have:

where:

• $m\; \backslash$ is the mass of the orbiting body,
• $M\; \backslash$ is the mass of the central body,

and the relevant standard gravitational parameter is that of the larger body.

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For all circular orbits around a given central body:

$\backslash mu\; =\; rv^2\; =\; r^3\backslash omega^2\; =\; 4\backslash pi^2r^3/T^2\; \backslash$

where:

• $r\; \backslash$ is the orbit radius,
• $v\; \backslash$ is the orbital speed,
• $\backslash omega\; \backslash$ is the angular speed,
• $T\; \backslash$ is the orbital period.

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The last equality has a very simple generalization to elliptic orbits:

$\backslash mu=4\backslash pi^2a^3/T^2\; \backslash$

where:

• $a\; \backslash$ is the semi-major axis.

See Kepler\'s third law.

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For all parabolic trajectories $r\; v^2\; \backslash$ is constant and equal to $2\; \backslash mu\; \backslash$;.

For elliptic and hyperbolic orbits $\backslash mu\; \backslash$ is twice the semi-major axis times the absolute value of 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, we define:

• the vector $\backslash mathbf\{r\}\; \backslash$ is the position of one body relative to the other
• $r\; \backslash$, $v\; \backslash$, and in the case of an elliptic orbit, the semi-major axis $a\; \backslash$, are defined accordingly (hence $r\; \backslash$ is the distance)
• $\backslash mu=\{G\}(m\_1\; +\; m\_2)\; \backslash$ (the sum of the two $\backslash mu\; \backslash$ values)

where:

• $m\_1\; \backslash$ and $m\_2\; \backslash$ are the masses of the two bodies.

Then:

• for circular orbits $rv^2\; =\; r^3\; \backslash omega^2\; =\; 4\; \backslash pi^2\; r^3/T^2\; =\; \backslash mu\backslash !\backslash ,$
• for elliptic orbits: $4\; \backslash pi^2\; a^3/T^2\; =\; \backslash mu\; \backslash$ (with a expressed in AU and T in years, and with M the total mass relative to that of the Sun, we get $a^3/T^2\; =\; M$)
• for parabolic trajectories $r\; v^2\; \backslash$ is constant and equal to $2\; \backslash mu\; \backslash$
• for elliptic and hyperbolic orbits $\backslash mu\; \backslash$ 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

The value for the Earth is called geocentric gravitational constant and equal to 398 600.441 8 ± 0.000 8 km³s^-2. Thus the uncertainty is 1 to 500 000 000, much smaller than the uncertainties in $G$ and $M$ separately (1 to 7000 each).

The value for the Sun is called heliocentric gravitational constant and equals 1.32712440018×10²⁰ m³s^-2.

References

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