In
celestial mechanics, the standard gravitational parameter ''μ'' of a
celestial body
An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical object, physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''bod ...
is the product of the
gravitational constant
The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
''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 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 in an orbital system can be defined as the one whose mass (''M'') is much larger than the mass of the
orbiting body (''m''), or . This approximation is standard for planets orbiting the
Sun or most moons and greatly simplifies equations. Under
Newton's law of universal gravitation, 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
A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.
Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is ...
around a central body:
where ''r'' is the orbit
radius, ''v'' is the
orbital speed, ''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.
This can be generalized for
elliptic orbits:
where ''a'' is the
semi-major axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the long ...
, which is
Kepler's third law.
For
parabolic trajectories ''rv''
2 is constant and equal to 2''μ''. For elliptic and hyperbolic orbits , where ''ε'' is the
specific orbital energy.
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
A binary star is a system of two stars that are gravitationally bound to and in orbit around each other. Binary stars in the night sky that are seen as a single object to the naked eye are often resolved using a telescope as separate stars, in wh ...
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, the
semi-major axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the long ...
''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 orbit
A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.
Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is ...
s, ''rv''
2 = ''r''
3''ω''
2 = 4π
2''r''
3/''T''
2 = ''μ''
* for
elliptic orbits, (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, where the latter is defined as the total energy of the system divided by the
reduced mass.
In a pendulum
The standard gravitational parameter can be determined using a
pendulum 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 of the pendulum (for the reason of the approximation see
Pendulum in mechanics).
Solar system
Geocentric gravitational constant
, the gravitational parameter for the
Earth 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 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 satellites 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 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
*
Planetary mass
References
{{Portal bar, Physics, Astronomy, Stars, Spaceflight, Outer space, Solar System
Orbits