Standard gravitational parameter: Information from Answers.com

Body	$μ$ (km³s^-2)
Sun	132,712,440,018		±8^[1]
Mercury	22,032
Venus	324,859
Earth	398,600	.4418	±0.0008
Moon	4902	.7779
Mars	42,828
Ceres	63	.1	±0.3^[2]^[3]
Jupiter	126,686,534
Saturn	37,931,187
Uranus	5,793,939		± 13^[4]
Neptune	6,836,529
Pluto	871		±5^[5]
Eris	1,108		±13^[6]

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 History
5 References
6 See also

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.

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.

Terminology and accuracy

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

History

Johannes Kepler was the first to give an accurate description of planetary motion, and by doing so, he was the first to calculate the standard gravitational parameter of the Sun. Kepler determined that the planets follow elliptical orbits under the Sun’s influence, and in 1609, he published the rules known as Kepler's laws of planetary motion. Some of the underlying values involved in Kepler’s calculations; however, had existed prior to the publication of Kepler’s three laws. The orbital periods of the planets, in particular, appear to have been discovered at a very early stage of human history.

Orbital periods of the planets

$\mu = \frac{4\pi^2 r^3}{\color{Red}T^2} \$

The Divine Planets
English Name	Babylonian deity	Hindu Navagraha	Chinese Wu Xing	Sidereal orbital period
Mercury	Nabu	Budha	Black Tortoise	0.240 842 sidereal year
Venus	Ishtar	Shukra	White Tiger	0.615 187 sidereal year
Mars	Nergal	Mangala	Vermilion Bird	1.880 816 sidereal year
Jupiter	Marduk	Brihaspati	Azure Dragon	11.861 776 sidereal year
Saturn	Ninurta	Shani	Yellow Dragon	29.456 626 sidereal year

The word planet comes from the Greek verb πλανώμαι planōmai^[7] which means to wander around. The planets appear to move through the night sky and are thus distinguished from the stars, which appear to maintain a fixed position with respect to each other ^[8]. This supposed ability to move freely may have given the planets the appearance of self-determination. Consequently, many cultures have either directly worshipped the planets as deities or at least associated them with divinity^[9]. The modern English names for the planets were in fact derived from their names as Roman gods.

Zodiac in a 6th century synagogue at Beit Alpha, Israel.

As the Earth orbits the Sun, to an observer on Earth the Sun appears to move with respect to the background stars. A sidereal year is the time required for the Earth to complete one orbit around the Sun, or equivalently, the time required for the Sun to appear to complete one orbit and to return to the same relative position with respect to the stars. In the western zodiac, as depicted in the image to the right, the ecliptic is divided into twelve equal zones of celestial longitude. Ancient Europeans used the zodiac to track the Earth’s orbit, and were thus aware of the duration of the sidereal year. Recording the periods of other planets in sidereal years allows a direct comparison to the Earth’s orbital period.

Stone carving of Chinese zodiac

A stone carving of the Chinese zodiac is depicted in the image to the left. Chinese astronomers built this system (know as the earthly branches) from observations of the orbit of Jupiter (歳星 Suìxīng, the Year Star), which has an 11.86 yr period. Chinese astronomers divided the celestial circle into 12 sections to follow the orbit of Jupiter, and assigned an animal to each year. These earthly branches were cyclically paired with celestial stems, a base ten numeral system, to produce a 60 year sexagenary cycle, and each year was assigned a Tai Sui deity to be worshipped, or at least respected during that year ^[10].

The sun stone also called the Aztec calendar on display at the National Museum of Anthropology in Mexico City.

A stone carving of the Aztec calendar is depicted in the image to the right. This astronomical system, used by some early Americans, has surprising similarities with the Asian system. The Asians obtained their 60 year sexagenary cycle by cyclically pairing the base ten celestial stems with the base twelve earthly branches, the least common multiple of 10 and 12 being 60. The Americans obtained a 260 day tonalpohualli (Mayan Tzolkin) cycle by pairing their base twenty numeral system with a base thirteen trecena cycle, the least common multiple of 20 and 13 being 260 ^[11]. The exact origin of the Mayan calendar is uncertain, but some scholars speculate that it may have been derived form the orbit of Venus, which held special significance within Mayan culture. ^[12]

Early astronomers were limited by their inability to measure large distances. They could accurately measure the duration of time required for each planet to complete its cycle, but they couldn’t accurately measure the distances and path traveled by the planet during an orbit. Hence, early astronomers failed to accurately describe planetary motion, and many envisioned the planets as following circular paths around the Earth. Although drawn from disparate and geographically isolated cultures, these images all use a similar circular motif to represent the passage of time.

Orbital distances

$\mu = \frac{4\pi^2 {\color{Red}r^3}}{T^2} \$

Orbital paths

$\mu = \frac{\color{Red}4\pi^2 a^3}{T^2} \$

References

^ "Astrodynamic Constants". NASA JPL. 27 February 2009. http://ssd.jpl.nasa.gov/?constants. Retrieved 27 July 2009.
^ 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. http://iau-comm4.jpl.nasa.gov/EPM2004.pdf.
^ 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. http://adsabs.harvard.edu/abs/1992AJ....103.2068J.
^ 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. arΧiv:astro-ph/0512491. http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2006AJ....132..290B&db_key=AST&data_type=HTML&format=&high=444b66a47d27727.
^ M.E. Brown and E.L. Schaller (2007). "The Mass of Dwarf Planet Eris". Science 316 (5831): 1585. doi:10.1126/science.1139415. PMID 17569855. http://www.sciencemag.org/cgi/content/full/316/5831/1585.
^ H. G. Liddell and R. Scott, A Greek–English Lexicon, ninth edition, (Oxford: Clarendon Press, 1940).
^ "Definition of planet". Merriam-Webster OnLine. http://www.merriam-webster.com/dictionary/planet. Retrieved 2007-07-23.
^ Evans, James (1998). "The History and Practice of Ancient Astronomy". Oxford University Press. pp. 296–7. http://books.google.com/books?id=nS51_7qbEWsC&pg=PA17&lpg=PA17&dq=babylon+greek+astronomy&source=web&ots=c1afKhoAt6&sig=A4cDSCcvWmd6B9e9YPZ9T1I91GM#PPA15,M1. Retrieved 2008-02-04.
^ Heavenly Stems and Earthly Branches - Hong Kong Observatory
^ Discussion of origin of the 260-day cycle
^ Aveni, Anthony F. (2000). Empires of Time: Calendars, Clocks, and Cultures (reprint of 1990 original ed.). London: Tauris Parke. ISBN 1-86064-602-6. OCLC 45144264.

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Standard gravitational parameter

Contents

Small body orbiting a central body

Two bodies orbiting each other

Terminology and accuracy

History

Orbital periods of the planets

Orbital distances

Orbital paths

References

See also