Standard gravitational parameter

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Body	$μ$ (km³s^-2)
Sun	132,712,440,018
Mercury	22,032
Venus	324,859
Earth	398,600	.4418	±0.0008
Moon	4902	.7779
Mars	42,828
Ceres	63	.1	±0.3^[1]^[2]
Jupiter	126,686,534
Saturn	37,931,187
Uranus	5,793,939		± 13^[3]
Neptune	6,836,529
Pluto	871		±5^[4]
Eris	1,108		±13^[5]

In astrodynamics, the standard gravitational parameter $\mu \$ of a celestial body is the product of the gravitational constant $G$ and the mass $M$ :

$\mu=GM \$

The units of the standard gravitational parameter are km³s^-2

[edit] Small body orbiting a central body

The above diagram illustrates five interrelated properties of mass together with the proportionality constants that relate these properties. Every sample of mass is believed to exhibit all five properties, however, due to extremely large proportionality constants, it is generally impossible to verify more than two or three properties for a specific sample of mass.

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 ( $E 0$ ) 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 << M \$

where:

$m \$ is the mass of the orbiting body,
$M \$ is the mass of the central body,

and the relevant standard gravitational parameter is that of the larger 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,
$\omega \$ is the angular speed,
$T \$ is the orbital period.

The last equality 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 $r v^2 \$ is constant and equal to $2 \mu \$ ;.

For elliptic and hyperbolic orbits $\mu \$ is twice the semi-major axis times the absolute value of the specific orbital energy.

[edit] Two bodies orbiting each other

In the more general case where the bodies need not be a large one and a small one, we define:

the vector $\mathbf{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)
$\mu={G}(m_1 + m_2) \$ (the sum of the two $\mu \$ values)

where:

$m_1 \$ and $m_2 \$ are the masses of the two bodies.

Then:

for circular orbits $rv^2 = r^3 \omega^2 = 4 \pi^2 r^3/T^2 = \mu\!\,$
for elliptic orbits: $4 \pi^2 a^3/T^2 = \mu \$ (with a expressed in AU and T in seconds, and with M the total mass relative to that of the Sun, we get $a 3 / T 2 = M$ )
for parabolic trajectories $r v^2 \$ is constant and equal to $2 \mu \$
for elliptic and hyperbolic orbits $\mu \$ 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.

[edit] Terminology and accuracy

The value for the Earth is called geocentric gravitational constant and equal to 398 600.441 8 ± 0.000 8 km³s^-2. Thus the uncertainty is 1 to 500 000 000, much smaller than the uncertainties in $G$ and $M$ separately (1 to 7000 each).

The value for the Sun is called heliocentric gravitational constant and equals 1.32712440018 × 10²⁰ m³s^-2.