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Standard gravitational parameter

From Biocrawler, the free encyclopedia.

Body	$μ$
-	[km³s^-2]
Sun	132,712,440,000
Mercury	22,032
Venus	324,859
Earth	398,600
Mars	42,828
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 $M\!\,$ :

$\mu=G*M\!\,$

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

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Small body orbiting a central body

Under standard assumptions in astrodynamics we have:

$m_1 << m_2\!\,$

where:

$m_1\!\,$ is the mass of the orbiting body,
$m_2\!\,$ 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 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.

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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 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 μ-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: $4 \pi^2 a^3/T^2 = \mu\!\,$
for parabolic trajectories $r v^2\!\,$ is constant and equal to $2 \mu\!\,$
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.

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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.

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Categories: Astrodynamics | Celestial mechanics