Standard gravitational parameter

From Wikipedia, the free encyclopedia

Body	$μ$ (km³s^-2)
Sun	132,712,440,018
Mercury	22,032
Venus	324,859
Earth	398,600
Mars	42,828
Ceres	63
Jupiter	126,686,534
Saturn	37,931,187
Uranus	5,793,947
Neptune	6,836,529
Pluto	1,001

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

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.

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 years, 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.

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