Standard gravitational parameter
Body | μ (km^{3}s^{−2}) |
---|---|
Sun | Template:Val^{[1]} |
Mercury | Template:Val |
Venus | Template:Val |
Earth | Template:Val |
Moon | Template:Val |
Mars | Template:Val |
Ceres | Template:Val^{[2]}^{[3]} |
Jupiter | Template:Val |
Saturn | Template:Val |
Uranus | Template:Val^{[4]} |
Neptune | Template:Val |
Pluto | Template:Val^{[5]} |
Eris | Template:Val^{[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.
For several objects in the Solar System, the value of μ is known to greater accuracy than either G or M.^{[7]} The SI units of the standard gravitational parameter are m^{3}s^{−2}.
Small body orbiting a central body
Template:Properties of mass 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:
where r is the orbit radius, v is the orbital speed, ω is the angular speed, and T is the orbital period.
This can be generalized for elliptic orbits:
where a is the semi-major axis, which is Kepler's third law.
For 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; T in seconds and 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 .
The value for the Earth is called the geocentric gravitational constant and equals Template:Val. Thus the uncertainty is 1 to Template:Val, much smaller than the uncertainties in G and M separately (1 to Template:Val each).
The value for the Sun is called the heliocentric gravitational constant or geopotential of the sun and equals Template:Val.
References
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- ↑ 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.