# Barycenter (Two-body problem)

## Description

barycentre; from the Greek βαρύ-ς heavy + κέντρ-ον centre) 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 two-body 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 1,000-fold difference in mass, due to the relatively large distance between them.

In astronomy, barycentric coordinates are non-rotating 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 two-dimensional shape.

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 semi-major axis of the system), r1 is the semi-major axis of the primary’s orbit around the barycenter, and r2 = a − r1 is the semi-major 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 two-body case, r1, the distance from the center of the primary to the barycenter is given by the formula shown here.

Related formulas## Variables

r_{1} | distance from body 1 to the barycenter (m) |

a | distance between the centers of the two bodies (m) |

m_{1} | mass of body 1 (kg) |

m_{2} | mass of body 2 (kg) |