Standard gravitational parameter
Body | μ (km^{3}s^{−2}) |
---|---|
Sun | 712440018(8)^{[1]} 132 |
Mercury | 032 22 |
Venus | 859 324 |
Earth | 600.4418(9) 398 |
Moon | 902.7779 4 |
Mars | 828 42 |
Ceres | ^{[2]}^{[3]} 63.1(3) |
Jupiter | 686534 126 |
Saturn | 931187 37 |
Uranus | 793939(13)^{[4]} 5 |
Neptune | 836529 6 |
Pluto | ^{[5]} 871(5) |
Eris | ^{[6]} 1108(13) |
In astrodynamics, the standard gravitational parameter μ of a celestial body is the product of the gravitational constant G and the mass M of the body.
- '"`UNIQ--postMath-00000001-QINU`"'
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 m^{3}s^{−2}.
Contents
Small body orbiting a central body
Under standard assumptions in astrodynamics we have:
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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:
- '"`UNIQ--postMath-00000003-QINU`"'
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:
- '"`UNIQ--postMath-00000004-QINU`"'
where a is the semi-major axis. See Kepler's third law.
For all parabolic trajectories rv^{2} 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 = Gm_{1} + Gm_{2} = μ_{1} + μ_{2}, where m_{1} and m_{2} are the masses of the two bodies.
Then:
- for circular orbits, rv^{2} = r^{3}ω^{2} = 4π^{2}r^{3}/T^{2} = μ
- for elliptic orbits, 4π^{2}a^{3}/T^{2} = μ (with a expressed in AU and T in years, and with M the total mass relative to that of the Sun, we get a^{3}/T^{2} = M)
- for parabolic trajectories, rv^{2} 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 '"`UNIQ--postMath-00000005-QINU`"'.
The value for the Earth is called the geocentric gravitational constant and equals 600.4418±0.0008 km^{3}s^{−2}. Thus the uncertainty is 1 to 398000000, much smaller than the uncertainties in G and M separately (1 to 500 each). 7000
The value for the Sun is called the heliocentric gravitational constant and equals 12440018×10^{20} m^{3}s^{−2}. 1.327
References
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