standard gravitational parameter : definition of standard gravitational parameter and synonyms of standard gravitational parameter (English)

Body	μ (km³s⁻²)
Sun	132,712,440,018(8)^[1]
Mercury	22,032
Venus	324,859
Earth	398,600.4418(9)
Moon	4,902.7779
Mars	42,828
Ceres	63.1(3)^[2]^[3]
Jupiter	126,686,534
Saturn	37,931,187
Uranus	5,793,939(13)^[4]
Neptune	6,836,529
Pluto	871(5)^[5]
Eris	1,108(13)^[6]

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 G or M. The SI units of the standard gravitational parameter are m³s⁻².

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

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.

Under standard assumptions in astrodynamics we have:

$m \ll M \$

where m is the mass of the orbiting body, M is the mass of the central body, and G is the standard gravitational parameter of the larger body.

For all circular orbits around a given 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.

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

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

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

For all 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 and T in years, and with 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 $\mu.\!\,$ .

The value for the Earth is called the geocentric gravitational constant and equals 398,600.4418±0.0008 km³s⁻². Thus the uncertainty is 1 to 500,000,000, much smaller than the uncertainties in G and M separately (1 to 7,000 each).

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

References

^ "Astrodynamic Constants". NASA/JPL. 27 February 2009. http://ssd.jpl.nasa.gov/?constants. Retrieved 27 July 2009.
^ E.V. Pitjeva (2005). "High-Precision Ephemerides of Planets — EPM and Determination of Some Astronomical Constants". Solar System Research 39 (3): 176. Bibcode 2005SoSyR..39..176P. DOI:10.1007/s11208-005-0033-2. http://iau-comm4.jpl.nasa.gov/EPM2004.pdf.
^ D. T. Britt, D. Yeomans, K. Housen, G. Consolmagno (2002). "Asteroid density, porosity, and structure". In W. Bottke, A. Cellino, P. Paolicchi, R.P. Binzel. Asteroids III. University of Arizona Press. p. 488. http://www.lpi.usra.edu/books/AsteroidsIII/pdf/3022.pdf.
^ 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.
^ M.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. Bibcode 2006AJ....132..290B. DOI:10.1086/504422.
^ Brown, M. E.; Schaller, E. L. (15 June 2007). "The Mass of Dwarf Planet Eris". Science 316 (5831): 1585. Bibcode 2007Sci...316.1585B. DOI:10.1126/science.1139415. PMID 17569855. edit

General	Box Capture Circular Elliptical / Highly elliptical Escape Graveyard Hyperbolic trajectory Inclined / Non-inclined Osculating Parabolic trajectory Parking Synchronous semi sub

Geocentric	Geosynchronous Geostationary Sun-synchronous Low Earth Medium Earth High Earth Molniya Near-equatorial Orbit of the Moon Polar Tundra Two-line elements

About other points	Areosynchronous Areostationary Halo Lissajous Lunar Heliocentric Heliosynchronous

Parameters

Shape/Size	$e\,\!$ Eccentricity $a\,\!$ Semi-major axis $b\,\!$ Semi-minor axis $Q,q\,\!$ Apsides

Orientation	$i\,\!$ Inclination $\Omega\,\!$ Longitude of the ascending node $\omega\,\!$ Argument of periapsis $\varpi\,\!$ Longitude of the periapsis

Position	$M\,\!$ Mean anomaly $\nu\,\!$ True anomaly $E\,\!$ Eccentric anomaly $L\,\!$ Mean longitude $l\,\!$ True longitude

Variation	$T\,\!$ Orbital period $n\,\!$ Mean motion $v\,\!$ Orbital speed $t_0\,\!$ Epoch

Maneuvers

Collision avoidance (spacecraft) Delta-v Delta-v budget Bi-elliptic transfer Geostationary transfer Gravity assist Gravity turn Hohmann transfer Low energy transfer Oberth effect Inclination change Phasing Rocket equation Rendezvous Transposition, docking, and extraction

Other orbital mechanics topics

Celestial coordinate system Characteristic energy Ephemeris Equatorial coordinate system Ground track Hill sphere Interplanetary Transport Network Kepler's laws of planetary motion Lagrangian point n-body problem Orbit equation Orbital state vectors Perturbation Retrograde motion Specific orbital energy Specific relative angular momentum

List of orbits

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Arabic (ar)	Bulgarian (bg)	Chinese (zh)	Czech (cs)
Danish (da)	Dutch (nl)	English (en)	Estonian (et)
Finnish (fi)	French (fr)	German (de)	Greek (el)
Hungarian (hu)	Italian (it)	Japanese (ja)	Korean (ko)
Norwegian (no)	Polish (pl)	Portuguese (pt)	Romanian (ro)
Russian (ru)	Serbian (sr)	Slovenian (sl)	Spanish (es)
Swedish (sv)	Turkish (tr)	Latvian (lv)	Persian (fa)
Lithuanian (lt)	Icelandic (is)	Slovak (sk)	Croatian (hr)
Hindi (hi)	Vietnamese (vi)	Hebrew (he)	Thai (th)
Indonesian (id)

Wikipedia

Standard gravitational parameter

Contents

Small body orbiting a central body

Two bodies orbiting each other

Terminology and accuracy

References

See also