Standard gravitational parameter
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.
For a circular orbit around a central body:
This can be generalized for elliptic orbits:
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.
- 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 .
The value for the Earth is called the geocentric gravitational constant and equals 398600.4418±0.0008 km3s−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 m3s−2.
- "Astrodynamic Constants". NASA/JPL. 27 February 2009. 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.
- 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.
- 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.
- 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.
- 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.