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Barycenter

The Barycenter is the center of mass of orbiting bodies; all the bodies in an orbiting system will orbit around the barycenter. The barycenter of two bodies can be in the middle of two equal masses, close to a large mass, or even inside a very large mass as is shown in Figure 1.

Five two body systems with orbits presented as blue spheres or ellipses, and barycenters shown as red crosses. First system shows two spherical bodies with equal sizes and shapes orbiting on the same spherical orbit with a barycenter in the middle of the orbit. Second system has two spherical bodies with one being bigger than the other. The bigger sphere orbits on the inner spherical orbit, and the smaller one orbits on the outer, bigger spherical orbit, with barycenter being in the centre of the two spherical orbits. The third system has one very big spherical body with a tiny spherical body orbiting around it. The barycenter is placed inside the bigger body. The fourth system is similar two the third one, however the big spherical body is a bit smaller and has a small circular orbit. The barycenter is placed in the middle of the smaller and bigger orbit, on the edge of the big spherical body. The last system has two identical spherical bodies moving on two elliptical orbits that overlap each other at one third of the ellipses. The barycenter is placed in the middle of that cross over.

Figure 1: Different two-body systems with indicated orbits (blue) and barycenters (red).

The barycenter of our solar system lies inside our sun, but not at its center, meaning the sun ‘wobbles’ as it orbits around the solar system barycenter. We observe this wobble around other stars in the universe with a technique called Doppler Spectroscopy which shows that other stars also have orbiting planets.

Referred from:

Article
Free-body diagrams
Article
Planetary Motion (Kepler's Laws)