Barycenter
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The barycenter (or barycentre; from the Greek βαρύς heavy + κέντρον centre^{[1]}) is the center of mass of two or more bodies that are orbiting each other, or the point around which they both orbit. It is an important concept in fields such as astronomy and astrophysics. The distance from a body's center of mass to the barycenter can be calculated as a simple twobody problem.
In cases where one of the two objects is considerably more massive than the other (and relatively close), the barycenter will typically be located within the more massive object. Rather than appearing to orbit a common center of mass with the smaller body, the larger will simply be seen to "wobble" slightly. This is the case for the Earth–Moon system, where the barycenter is located on average 4,671 km from the Earth's center, well within the planet's radius of 6,378 km. When the two bodies are of similar masses, the barycenter will generally be located between them and both bodies will follow an orbit around it. This is the case for Pluto and Charon, as well as for many binary asteroids and binary stars. It is also the case for Jupiter and the Sun, despite the 1,000fold difference in mass, due to the relatively large distance between them.
In astronomy, barycentric coordinates are nonrotating coordinates with the origin at the center of mass of two or more bodies. The International Celestial Reference System is a barycentric one, based on the barycenter of the Solar System.
In geometry, the term "barycenter" is synonymous with centroid, the geometric center of a twodimensional shape.
Twobody problem
The barycenter is one of the foci of the elliptical orbit of each body. This is an important concept in the fields of astronomy and astrophysics. If a is the distance between the centers of the two bodies (the semimajor axis of the system), r_{1} is the semimajor axis of the primary's orbit around the barycenter, and r_{2} = a − r_{1} is the semimajor axis of the secondary's orbit. When the barycenter is located within the more massive body, that body will appear to "wobble" rather than to follow a discernible orbit. In a simple twobody case, r_{1}, the distance from the center of the primary to the barycenter is given by:
where :
 r_{1} is the distance from body 1 to the barycenter
 a is the distance between the centers of the two bodies
 m_{1} and m_{2} are the masses of the two bodies.
Primary–secondary examples
The following table sets out some examples from the Solar System. Figures are given rounded to three significant figures. The last two columns show R_{1}, the radius of the first (more massive) body, and r_{1} / R_{1}, the ratio of the distance to the barycenter and that radius: a value less than one shows that the barycenter lies inside the first body. The term primary–secondary is used to distinguish between the different degrees of relationship of the involved participants.
Primary–secondary examples Larger
bodym_{1}
(M_{⊕})Smaller
bodym_{2}
(M_{⊕})a
(km)r_{1}
(km)R_{1}
(km)r_{1} / R_{1} Earth 1 Moon 0.0123 384,000 4,670 6,380 0.732 The Earth has a perceptible "wobble". Also see tides. Pluto 0.0021 Charon 19,600 2,110 1,150 1.83 Pluto and Charon have distinct orbits around their barycenter, and as such they were considered as a double planet by many before the redefinition of a planet in 2006. Sun 333,000 Earth 1 150,000,000
(1 AU)449 696,000 0.000646 The Sun's wobble is barely perceptible. Sun 333,000 Jupiter 778,000,000
(5.20 AU)742,000 696,000 1.07 The Sun orbits a barycenter just above its surface.^{[2]}
Inside or outside the Sun?
If m_{1} ≫ m_{2} — which is true for the Sun and any planet — then the ratio r_{1}/R_{1} approximates to:
Hence, the barycenter of the Sun–planet system will lie outside the Sun only if:
That is, where the planet is massive and far from the Sun.
If Jupiter had Mercury's orbit (57,900,000 km, 0.387 AU), the Sun–Jupiter barycenter would be approximately 55,000 km from the center of the Sun (r_{1}/R_{1} ~ 0.08). But even if the Earth had Eris' orbit (68 AU), the Sun–Earth barycenter would still be within the Sun (just over 30,000 km from the center).
To calculate the actual motion of the Sun, you would need to sum all the influences from all the planets, comets, asteroids, etc. of the Solar System (see nbody problem). If all the planets were aligned on the same side of the Sun, the combined center of mass would lie about 500,000 km above the Sun's surface.
The calculations above are based on the mean distance between the bodies and yield the mean value r_{1}. But all celestial orbits are elliptical, and the distance between the bodies varies between the apses, depending on the eccentricity, e. Hence, the position of the barycenter varies too, and it is possible in some systems for the barycenter to be sometimes inside and sometimes outside the more massive body. This occurs where:
Note that the Sun–Jupiter system, with e_{Jupiter} = 0.0484, just fails to qualify: 1.05 ≯ 1.07 > 0.954.
Gallery
Images are representative (made by hand), not simulated.

Two bodies with the same mass orbiting a common barycenter (similar to the 90 Antiope system)

Two bodies with the same mass orbiting a common barycenter, external to both bodies, with eccentric elliptic orbits (a common situation for binary stars)

Scale model of the Pluto system: Pluto and its five moons, including the location of the system's barycenter. Sizes, distances and apparent magnitude of the bodies are to scale.

Sideview of a star orbiting the barycenter of a planetary system. The radialvelocity method makes use of the star's wobble to detect extrasolar planets
Relativistic corrections
In classical mechanics, this definition simplifies calculations and introduces no known problems. In general relativity, problems arise because, while it is possible, within reasonable approximations, to define the barycenter, the associated coordinate system does not fully reflect the inequality of clock rates at different locations. Brumberg explains how to set up barycentric coordinates in general relativity.^{[3]}
The coordinate systems involve a worldtime, i.e. a global time coordinate that could be set up by telemetry. Individual clocks of similar construction will not agree with this standard, because they are subject to differing gravitational potentials or move at various velocities, so the worldtime must be slaved to some ideal clock that is assumed to be very far from the whole selfgravitating system. This time standard is called Barycentric Coordinate Time, "TCB".
Selected barycentric orbital elements
Barycentric osculating orbital elements for some objects in the Solar System:^{[4]}
Object 
Semimajor axis (in AU) 
Apoapsis (in AU) 
Orbital period (in years) 

C/2006 P1 (McNaught)  2,050  4,100  92,600 
Comet Hyakutake  1,700  3,410  70,000 
C/2006 M4 (SWAN)  1,300  2,600  47,000 
(308933) 2006 SQ372  799  1,570  22,600 
(87269) 2000 OO67  549  1,078  12,800 
90377 Sedna  506  937  11,400 
2007 TG422  501  967  11,200 
For objects at such high eccentricity, the Sun's barycentric coordinates are more stable than heliocentric coordinates.^{[5]}
See also
Wikimedia Commons has media related to barycenter animations. 
References
 ↑ Oxford English Dictionary, Second Edition.
 ↑ "What's a Barycenter?". Space Place @ NASA. 20050908. Archived from the original on 23 December 2010. Retrieved 20110120.
 ↑ Essential Relativistic Celestial Mechanics by Victor A. Brumberg (Adam Hilger, London, 1991) ISBN 0750300620.
 ↑ Horizons output (20110130). "Barycentric Osculating Orbital Elements for 2007 TG422". Retrieved 20110131. (Select Ephemeris Type:Elements and Center:@0)
 ↑ Kaib, Nathan A.; Becker, Andrew C.; Jones, R. Lynne; Puckett, Andrew W.; Bizyaev, Dmitry; Dilday, Benjamin; Frieman, Joshua A.; Oravetz, Daniel J.; Pan, Kaike; Quinn, Thomas; Schneider, Donald P.; Watters, Shannon (2009). "2006 SQ372: A Likely LongPeriod Comet from the Inner Oort Cloud". The Astrophysical Journal 695 (1): 268–275. arXiv:0901.1690. Bibcode:2009ApJ...695..268K. doi:10.1088/0004637X/695/1/268.