Standard gravitational parameter

Body	μ (m3 s−2)
Sun	1.32712440018(9)×1020
Mercury	2.2032(9)×1013
Venus	3.24859(9)×1014
Earth	3.986004418(9)×1014
Moon	4.9048695(9)×1012
Mars	4.2828(9)×1013
Ceres	6.26325×1010
Jupiter	1.26686534(9)×1017
Saturn	3.7931187(9)×1016
Uranus	5.793939(9)×1015
Neptune	6.836529(9)×1015
Pluto	8.71(9)×1011
Eris	1.108(9)×1012

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 all objects in the Solar System, the value of μ is known to greater accuracy than either G or M. This is because μ is calculated directly from the speed of a small test mass in orbit about the body, which can be done very accurately. The mass M is then inferred by using G. The SI units of the standard gravitational parameter are m³ s⁻².

Small body orbiting a central body

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 (r_s) 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.

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. Under Newton's law of universal gravitation, if the distance between the bodies is r, the force exerted on the smaller body is:

$F = \frac{G M m}{r^2} = \frac{\mu m}{r^2}$

Thus only the product of G and M is needed to predict the motion of the smaller body. Conversely, measurements of the smaller body's orbit only provide information on the product, μ, not G and M separately. The gravitational constant, G, is difficult to measure with high accuracy, while orbits, at least in the solar system, can be measured with great precision and used to determine μ with similar precision.

For a circular orbit around a 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.

This can be generalized for elliptic orbits:

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

where a is the semi-major axis, which is Kepler's third law.

For 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 μ.

The value for the Earth is called the geocentric gravitational constant and equals 398600.4418±0.0008 km³ s⁻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 m³ s⁻².

See also

• Astronomical system of units
• Planetary mass