Standard gravitational parameter

Body	μ (m^3 s^−2)
Sun	1.32712440018(9)×10^20[1]
Mercury	2.2032(9)×10^13
Venus	3.24859(9)×10^14
Earth	3.986004418(9)×10^14
Moon	4.9048695(9)×10^12
Mars	4.282837(2)×10^13[2]
Ceres	6.26325×10^10[3][4][5]
Jupiter	1.26686534(9)×10^17
Saturn	3.7931187(9)×10^16
Uranus	5.793939(9)×10^15[6]
Neptune	6.836529(9)×10^15
Pluto	8.71(9)×10^11[7]
Eris	1.108(9)×10^12[8]

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.

$\mu=GM \$

For several objects in the Solar System, the value of μ is known to greater accuracy than either G or M.^[9] The SI units of the standard gravitational parameter are m³ s⁻².

Small body orbiting a central body

The relation between properties of mass and their associated physical constants. Every massive object is believed to exhibit all five properties. However, due to extremely large or extremely small constants, it is generally impossible to verify more than two or three properties for any object.

The Schwarzschild radius (r_s) represents the ability of mass to cause curvature in space and time.

The standard gravitational parameter (μ) represents the ability of a massive body to exert Newtonian gravitational forces on other bodies.

Inertial mass (m) represents the Newtonian response of mass to forces.

Rest energy (E₀) represents the ability of mass to be converted into other forms of energy.

The Compton wavelength (λ) represents the quantum response of mass to local geometry.

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:

$F = \frac{G M m}{r^2} = \frac{\mu m}{r^2}$

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,^[10] 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:

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

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:

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

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

For parabolic trajectories rv² is constant and equal to 2μ. For elliptic and hyperbolic orbits μ = 2a| 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 = Gm₁ + Gm₂ = μ₁ + μ₂, where m₁ and m₂ are the masses of the two bodies.

Then:

• for circular orbits, rv² = r³ω² = 4π²r³/T² = μ
• for elliptic orbits, 4π²a³/T² = μ (with a expressed in AU; T in seconds and M the total mass relative to that of the Sun, we get a³/T² = M)
• for parabolic trajectories, rv² is constant and equal to 2μ
• 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 398600.4418±0.0008 km³ s⁻². Thus the uncertainty is 1 to 500000000, much smaller than the uncertainties in G and M separately (1 to 7000 each).

The value for the Sun is called the heliocentric gravitational constant or geopotential of the Sun and equals 1.32712440018×10²⁰ m³ s⁻².

See also

• Astronomical system of units
• Planetary mass

References

1. "Astrodynamic Constants". NASA/JPL. 27 February 2009. Retrieved 27 July 2009.
2. "Mars Gravity Model 2011 (MGM2011)". Western Australian Geodesy Group.
3. "Asteroid Ceres P_constants (PcK) SPICE kernel file". Retrieved 5 November 2015.
4. E.V. Pitjeva (2005). "High-Precision Ephemerides of Planets — EPM and Determination of Some Astronomical Constants" (PDF). Solar System Research 39 (3): 176. Bibcode:2005SoSyR..39..176P. doi:10.1007/s11208-005-0033-2.
5. 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. Asteroids III. University of Arizona Press. p. 488.
6. 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.
7. 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.
8. M.E. Brown; E.L. Schaller (2007). "The Mass of Dwarf Planet Eris". Science 316 (5831): 1586. Bibcode::2007Sci...316.1585B. doi:10.1126/science.1139415. PMID 17569855.
9. 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.
10. 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. A lengthy, detailed review.