Barycenter

A barycenter is the center of mass of two or more bodies. In physics and astronomy, it is the point where the system’s mass can be considered to be concentrated for the purpose of analyzing motion. The barycenter is found by the mass-weighted average of the position vectors of all bodies: r_B = (m1 r1 + m2 r2 + …) / (m1 + m2 + …). When expressed relative to the barycenter, the bodies satisfy m1 r1' + m2 r2' + … = 0.

For a two-body system, the barycenter lies on the line joining the two bodies. The distances a1 and a2 from the barycenter satisfy m1 a1 = m2 a2, and the total separation a = a1 + a2. If one mass greatly exceeds the other, the barycenter is close to the larger body and may lie inside it. A well-known example is the Sun–Jupiter system, where the barycenter lies just outside the Sun’s surface at certain configurations, reflecting Jupiter’s substantial mass relative to the Sun.

In astronomical practice, the barycenter serves as the reference point for describing orbital motion. Planets and stars orbit about the system’s barycenter, not about a single body. In the solar system, the Sun–planet barycenter can be located within or outside the Sun’s radius, depending on planetary configurations. For systems with more than two bodies, the barycenter remains the mass-weighted center of the entire system, though its instantaneous location can be complex to compute.

The concept extends to celestial mechanics and spacecraft navigation, where barycentric coordinates provide a convenient framework for tracking motion in a multi-body gravitational field.