Molecular geometry

Body	μ (km3s−2)
Sun	Template:Val
Mercury	Template:Val
Venus	Template:Val
Earth	Template:Val
Moon	Template:Val
Mars	Template:Val
Ceres	Template:Val
Jupiter	Template:Val
Saturn	Template:Val
Uranus	Template:Val
Neptune	Template:Val
Pluto	Template:Val
Eris	Template:Val

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 km³s⁻².

Small body orbiting a central body

Template:Properties of mass 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:

$\mu =r{v}^2=r^3{\omega }^2=4{\pi }^2{r}^3/T^2$

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:

$\mu =4{\pi }^2{a}^3/T^2$

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

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

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

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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See also

• Astronomical system of units
• Planetary mass

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