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

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

Small body orbiting a central body

Under standard assumptions in astrodynamics we have:

where:

• $m\; \backslash$ is the mass of the orbiting body,

• $M\; \backslash$ is the mass of the central body,

and the relevant standard gravitational parameter is that of the larger body.

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For all circular orbits around a given central body:

where:

• $r\; \backslash$ is the orbit radius,

• $v\; \backslash$ is the orbital speed,

• $\backslash omega\; \backslash$ is the angular speed,

• $T\; \backslash$ is the orbital period.

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The last equality has a very simple generalization to elliptic orbits:

where:

• $a\; \backslash$ is the semi-major axis.

See Kepler's third law.

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For all parabolic trajectories $r\; v^2\; \backslash$ is constant and equal to $2\; \backslash mu\; \backslash$.

For elliptic and hyperbolic orbits $\backslash mu\; \backslash$ 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 $\backslash mathbf\{r\}\; \backslash$ is the position of one body relative to the other

• $r\; \backslash$, $v\; \backslash$, and in the case of an elliptic orbit, the semi-major axis $a\; \backslash$, are defined accordingly (hence $r\; \backslash$ is the distance)

• $\backslash mu=\{G\}(m\_1\; +\; m\_2)\; \backslash$ (the sum of the two $\backslash mu\; \backslash$ values)

where:

• $m\_1\; \backslash$ and $m\_2\; \backslash$ are the masses of the two bodies.

Then:

• for circular orbits $rv^2\; =\; r^3\; \backslash omega^2\; =\; 4\; \backslash pi^2\; r^3/T^2\; =\; \backslash mu\backslash !\backslash ,$

• for elliptic orbits: $4\; \backslash pi^2\; a^3/T^2\; =\; \backslash mu\; \backslash$ (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 $r\; v^2\; \backslash$ is constant and equal to $2\; \backslash mu\; \backslash$

• for elliptic and hyperbolic orbits $\backslash mu\; \backslash$ 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 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 thee 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

The word planet comes from the Greek verb planōmai 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 . 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. The modern English names for the planets were in fact derived from their names as Roman gods.
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.
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 .
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 . 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.

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

Orbital paths