Standard gravitational parameter

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Body μ (km3s−2)
Sun 132712440018(9)[1]
Mercury 22032
Venus 324859
Earth 398600.4418(9)
Moon 4902.8000
Mars 42828
Ceres 63.1(3)[2][3]
Jupiter 126686534
Saturn 37931187
Uranus 5793939(13)[4]
Neptune 6836529
Pluto 871(5)[5]
Eris 1108(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.

Small body orbiting a central body[edit]

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 (rs) 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.

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 equation 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[edit]

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[edit]

Note that the reduced mass is also denoted by \mu.\!\,.

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.

References[edit]

  1. ^ "Astrodynamic Constants". NASA/JPL. 27 February 2009. Retrieved 27 July 2009. 
  2. ^ 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. 
  3. ^ 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. 
  4. ^ 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. 
  5. ^ 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. 
  6. ^ 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. 

See also[edit]