In
celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
, the standard gravitational parameter ''μ'' of a
celestial body is the product of the
gravitational constant ''G'' and the mass ''M'' of the bodies. For two bodies the parameter may be expressed as G(m
1+m
2), or as GM when one body is much larger than the other.
For several objects in the
Solar System
The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
, the value of ''μ'' is known to greater accuracy than either ''G'' or ''M''. The
SI units of the standard gravitational parameter are . However, units of are frequently used in the scientific literature and in spacecraft navigation.
Definition
Small body orbiting a central body
The
central body
A primary (also called a gravitational primary, primary body, or central body) is the main physical body of a gravitationally bound, multi-object system. This object constitutes most of that system's mass and will generally be located near the syst ...
in an orbital system can be defined as the one whose mass (''M'') is much larger than the mass of the
orbiting body
In astrodynamics, an orbiting body is any physical body that orbits a more massive one, called the primary body. The orbiting body is properly referred to as the secondary body (m_2), which is less massive than the primary body (m_1).
Thus, m_2 ...
(''m''), or . This approximation is standard for planets orbiting the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
or most moons and greatly simplifies equations. Under
Newton's law of universal gravitation
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
, if the distance between the bodies is ''r'', the force exerted on the smaller body is:
Thus only the product of G and M is needed to predict the motion of the smaller body. Conversely, measurements of the smaller body's orbit only provide information on the product, μ, not G and M separately. The gravitational constant, G, is difficult to measure with high accuracy,
[. A lengthy, detailed review.] while orbits, at least in the solar system, can be measured with great precision and used to determine μ with similar precision.
For a
circular orbit around a central body:
where ''r'' is the orbit
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
, ''v'' is the
orbital speed
In gravitationally bound systems, the orbital speed of an astronomical body or object (e.g. planet, moon, artificial satellite, spacecraft, or star) is the speed at which it orbits around either the barycenter or, if one body is much more mas ...
, ''ω'' is the
angular speed
Angular may refer to:
Anatomy
* Angular artery, the terminal part of the facial artery
* Angular bone, a large bone in the lower jaw of amphibians and reptiles
* Angular incisure, a small anatomical notch on the stomach
* Angular gyrus, a regio ...
, and ''T'' is the
orbital period
The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets ...
.
This can be generalized for
elliptic orbit
In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, i ...
s:
where ''a'' is the
semi-major axis, which is
Kepler's third law
In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbi ...
.
For
parabolic trajectories ''rv''
2 is constant and equal to 2''μ''. For elliptic and hyperbolic orbits , where ''ε'' is the
specific orbital energy
In the gravitational two-body problem, the specific orbital energy \varepsilon (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy (\varepsilon_p) and their total kinetic energy (\varepsilon_k), divide ...
.
General case
In the more general case where the bodies need not be a large one and a small one, e.g. a
binary star system, 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
In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, i ...
, the
semi-major axis ''a'', are defined accordingly (hence ''r'' is the distance)
* ''μ'' = ''Gm''
1 + ''Gm''
2 = ''μ''
1 + ''μ''
2, where ''m''
1 and ''m''
2 are the masses of the two bodies.
Then:
* for
circular orbits, ''rv''
2 = ''r''
3''ω''
2 = 4π
2''r''
3/''T''
2 = ''μ''
* for
elliptic orbit
In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, i ...
s, (with ''a'' expressed in AU; ''T'' in years and ''M'' the total mass relative to that of the Sun, we get )
* for
parabolic trajectories, ''rv''
2 is constant and equal to 2''μ''
* for elliptic and hyperbolic orbits, ''μ'' is twice the semi-major axis times the negative of the
specific orbital energy
In the gravitational two-body problem, the specific orbital energy \varepsilon (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy (\varepsilon_p) and their total kinetic energy (\varepsilon_k), divide ...
, where the latter is defined as the total energy of the system divided by the
reduced mass
In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass ...
.
In a pendulum
The standard gravitational parameter can be determined using a
pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
oscillating above the surface of a body as:
where ''r'' is the radius of the gravitating body, ''L'' is the length of the pendulum, and ''T'' is the
period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
of the pendulum (for the reason of the approximation see
Pendulum in mechanics).
Solar system
Geocentric gravitational constant
, the gravitational parameter for the
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
as the central body, is called the geocentric gravitational constant. It equals .
[, citing Ries, J. C., Eanes, R. J., Shum, C. K., and Watkins, M. M., 1992, "Progress in the Determination of the Gravitational Coefficient of the Earth," Geophys. Res. Lett., 19(6), pp. 529-531.]
The value of this constant became important with the beginning of
spaceflight
Spaceflight (or space flight) is an application of astronautics to fly spacecraft into or through outer space, either with or without humans on board. Most spaceflight is uncrewed and conducted mainly with spacecraft such as satellites in o ...
in the 1950s, and great effort was expended to determine it as accurately as possible during the 1960s. Sagitov (1969) cites a range of values reported from 1960s high-precision measurements, with a relative uncertainty of the order of 10
−6.
[Sagitov, M. U., "Current Status of Determinations of the Gravitational Constant and the Mass of the Earth", ''Soviet Astronomy'', Vol. 13 (1970), 712-718, translated from ''Astronomicheskii Zhurnal'' Vol. 46, No. 4 (July–August 1969), 907-915.]
During the 1970s to 1980s, the increasing number of
artificial satellite
A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioisoto ...
s in Earth orbit further facilitated high-precision measurements,
and the relative uncertainty was decreased by another three orders of magnitude, to about (1 in 500 million) as of 1992.
Measurement involves observations of the distances from the satellite to Earth stations at different times, which can be obtained to high accuracy using radar or laser ranging.
Heliocentric gravitational constant
, the gravitational parameter for the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
as the central body,
is called the heliocentric gravitational constant or ''geopotential of the Sun'' and equals
The relative uncertainty in , cited at below 10
−10 as of 2015, is smaller than the uncertainty in
because is derived from the ranging of interplanetary probes, and the absolute error of the distance measures to them is about the same as the earth satellite ranging measures, while the absolute distances involved are much bigger.
See also
*
Astronomical system of units
The astronomical system of units, formerly called the IAU (1976) System of Astronomical Constants, is a system of measurement developed for use in astronomy. It was adopted by the International Astronomical Union (IAU) in 1976 via Resolution No. ...
*
Planetary mass
In astronomy, planetary mass is a measure of the mass of a planet-like astronomical object. Within the Solar System, planets are usually measured in the astronomical system of units, where the unit of mass is the solar mass (), the mass of the Su ...
References
{{Portal bar, Physics, Astronomy, Stars, Spaceflight, Outer space, Solar System
Orbits