Standard gravitational parameter
Body | μ [m3 s−2] | |
---|---|---|
Sun | 1.32712440018(9) | × 1020 [1] |
Mercury | 2.2032(9) | × 1013 [2] |
Venus | 3.24859(9) | × 1014 |
Earth | 3.986004418(8) | × 1014 [3] |
Moon | 4.9048695(9) | × 1012 |
Mars | 4.282837(2) | × 1013 [4] |
Ceres
|
6.26325 | × 1010 [5][6][7] |
Jupiter | 1.26686534(9) | × 1017 |
Saturn | 3.7931187(9) | × 1016 |
Uranus | 5.793939(9) | × 1015 [8] |
Neptune | 6.836529(9) | × 1015 |
Pluto | 8.71(9) | × 1011 [9] |
Eris | 1.108(9) | × 1012 [10] |
In
For several objects in the
−2 is frequently used in the scientific literature and in spacecraft navigation.Definition
Small body orbiting a central body
The
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,[11] while orbits, at least in the solar system, can be measured with great precision and used to determine μ with similar precision.
For a
This can be generalized for elliptic orbits:
For parabolic trajectories rv2 is constant and equal to 2μ. For elliptic and hyperbolic orbits magnitude of μ = 2 times the magnitude of a times the magnitude of ε, where a is the semi-major axis and ε is the specific orbital energy.
General case
In the more general case where the bodies need not be a large one and a small one, e.g. a binary star system, we define:
- the vector r is the position of one body relative to the other
- r, v, and in the case of an semi-major axisa, 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 years 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 negative of the specific orbital energy, where the latter is defined as the total energy of the system divided by the reduced mass.
In a pendulum
The standard gravitational parameter can be determined using a pendulum oscillating above the surface of a body as:[12]
Solar system
Geocentric gravitational constant
GME, the gravitational parameter for the Earth as the central body, is called the geocentric gravitational constant. It equals (3.986004418±0.000000008)×1014 m3⋅s−2.[3]
The value of this constant became important with the beginning of spaceflight in the 1950s, and great effort was expended to determine it as accurately as possible during the 1960s. Sagitov (1969) cites a range of values reported from 1960s high-precision measurements, with a relative uncertainty of the order of 10−6.[13]
During the 1970s to 1980s, the increasing number of
Heliocentric gravitational constant
GM☉, the gravitational parameter for the Sun as the central body, is called the heliocentric gravitational constant or geopotential of the Sun and equals (1.32712440042±0.0000000001)×1020 m3⋅s−2.[15]
The relative uncertainty in GM☉, cited at below 10−10 as of 2015, is smaller than the uncertainty in GME because GM☉ is derived from the ranging of interplanetary probes, and the absolute error of the distance measures to them is about the same as the earth satellite ranging measures, while the absolute distances involved are much bigger.[citation needed]
See also
References
- ^ "Astrodynamic Constants". NASA/JPL. 27 February 2009. Retrieved 27 July 2009.
- ^ Anderson, John D.; Colombo, Giuseppe; Esposito, Pasquale B.; Lau, Eunice L.; Trager, Gayle B. (September 1987). "The mass, gravity field, and ephemeris of Mercury". Icarus. 71 (3): 337–349. .
- ^ a b "IAU Astronomical Constants: Current Best Estimates". iau-a2.gitlab.io. IAU Division I Working Group on Numerical Standards for Fundamental Astronomy. Retrieved 25 June 2021., citing Ries, J. C., Eanes, R. J., Shum, C. K., and Watkins, M. M., 1992, "Progress in the Determination of the Gravitational Coefficient of the Earth," Geophys. Res. Lett., 19(6), pp. 529-531.
- ^ "Mars Gravity Model 2011 (MGM2011)" (PDF). Western Australian Geodesy Group. 2015-03-26. Archived from the original on 2013-04-10.
- ^ Raymond, Carol; Semenov, Boris (October 16, 2015). Asteroid Ceres P_constants (PcK) SPICE kernel file (Report). Version 0.5.
- ^
E.V. Pitjeva (2005). "High-Precision Ephemerides of Planets — EPM and Determination of Some Astronomical Constants" (PDF). S2CID 120467483. Archived from the original(PDF) on 2006-08-22.
- ^ 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.
- ^
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". doi:10.1086/116211.
- ^
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". S2CID 119386667.
- ^
M.E. Brown; E.L. Schaller (2007). "The Mass of Dwarf Planet Eris". S2CID 21468196.
- S2CID 250810284. A lengthy, detailed review.
- ^ Lewalle, Philippe; Dimino, Tony (2014), Measuring Earth's Gravitational Constant with a Pendulum (PDF), p. 1[dead link]
- ^ Sagitov, M. U., "Current Status of Determinations of the Gravitational Constant and the Mass of the Earth", Soviet Astronomy, Vol. 13 (1970), 712–718, translated from Astronomicheskii Zhurnal Vol. 46, No. 4 (July–August 1969), 907–915.
- .
- .