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

1 Small body orbiting a central body
2 Two bodies orbiting each other
3 Terminology and accuracy
4 References

[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.

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 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.

v • d • e

Orbits

Types

General	Box · Capture · Circular · Elliptical / Highly elliptical · Escape · Graveyard · Hyperbolic trajectory · Inclined / Non-inclined · Osculating · Parabolic trajectory · Parking · Synchronous (semi · sub)

Geocentric	Geosynchronous · Geostationary · Sun-synchronous · Low Earth · Medium Earth · Molniya · Near-equatorial · Orbit of the Moon · Polar · Tundra

Other	Areosynchronous · Areostationary · Halo · Lissajous · Lunar · Heliocentric · Heliosynchronous

Parameters

Classical				Other
$i\,\!$	Inclination	$\omega\,\!$	Argument of periapsis	$\nu\,$	True anomaly	$L\,$	Mean longitude
$\Omega\,\!$	Longitude of the ascending node	$a\,\!$	Semi-major axis	$b\,$	Semi-minor axis	$l\,$	True longitude
$e\,\!$	Eccentricity	$M_o\,\!$	Mean anomaly at epoch	$\epsilon\,$	Linear eccentricity	$T\,$	Orbital period
				$E\,$	Eccentric anomaly

Maneuvers

Bi-elliptic transfer · Geostationary transfer · Gravity assist · Hohmann transfer · Inclination change · Phasing · Rendezvous · Transposition, docking, and extraction

[edit] References

^ Pitjeva, E. V. (2005). "High-Precision Ephemerides of Planets — EPM and Determination of Some Astronomical Constants" (PDF). Solar System Research 39 (3): 176. doi:10.1007/s11208-005-0033-2.
^ D. T. Britt et al Asteroid density, porosity, and structure, pp. 488 in Asteroids III, University of Arizona Press (2002).
^ Jacobson, R.A.; Campbell, J.K.; Taylor, A.H.; Synnott, S.P. (1992). "The masses of Uranus and its major satellites from Voyager tracking data and Earth-based Uranian satellite data". The Astronomical Journal 103 (6): 2068–2078. doi:10.1086/116211.
^ M. W. Buie, W. M. Grundy, E. F. Young, L. A. Young, S. A. Stern (2006). "Orbits and photometry of Pluto's satellites: Charon, S/2005 P1, and S/2005 P2". Astronomical Journal 132: 290. doi:10.1086/504422. arXiv:astro-ph/0512491.
^ M.E. Brown and E.L. Schaller (2007). "The Mass of Dwarf Planet Eris". Science 316 (5831): 1585. doi:10.1126/science.1139415.

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Standard gravitational parameter

From Wikipedia, the free encyclopedia

Contents

[edit] Small body orbiting a central body

[edit] Two bodies orbiting each other

[edit] Terminology and accuracy

[edit] References

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