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Understanding the Concept of the Barycenter
Every object has a center of mass. This center of mass is defined as the barycenter of the object. It is commonly also known as the center of gravity of the object. However, when we have a system of massive bodies i.e., two or more bodies, they also have a fixed center of mass or barycenter. This system orbits around the common center. In a system of bodies with identical masses, the barycenter lies directly in the center of the two bodies. However, in a system of bodies with unequal masses the center of mass lies close to the body with the greater mass. If one of the bodies is far greater in mass, the center may even lie within this massive body. This is important, as we will see in the next few sections when we look at the center of mass of the Solar System and of the Earth-Moon system.
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Center of Mass of the Solar System
We generally tend to agree with the notion that the planets revolve around the Sun. But what is actually happening is that the Sun and the planets are involved in a "cosmic dance" around a common center of mass. The Sun is the most massive object in the Solar System and contains over 99% of the mass of the Solar System.
The Sun is far more massive than all of the planets put together, because of this the barycenter of the Solar System is close to the surface of the Sun, and at times the center of mass is located within the Sun itself. The position of the planets is what determines whether the barycenter will be near the surface of the Sun or within the Sun. In particular, the two of the largest planets, Jupiter and Saturn, influence the outcome to a great extent.
Another thing to note is that if one of the objects in the system of bodies is far more massive, it will have a very small orbit around the barycenter, sometimes to the point of being barely visible. This is the reason why the Sun does not appear to move around a common center, but in reality it actually is. The Solar System barycenter forces depend upon the size of the bodies involved and the distance between them. Knowing these parameters we can calculate where the common center of mass will be.
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Just like the planets orbiting the Sun, moons orbiting planets also comprise a system with a center of mass. For the Earth-Moon system, the barycenter is located 1,710 km below the surface of the Earth. This is because the Earth is far more massive than the Moon and it is this common center of mass around which the Earth and the Moon seem to go around. The International Astronomical Union (IAU) does not consider the Earth-Moon system as a double-planet system, since the center of mass does not lie between them but rather within the Earth.
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A Method to Locate Planets
Astronomers are using the concept of the barycenter to locate new planets around stars outside our Solar System. The way this is done is that since a planet and its star orbit a common center and due to the star being more massive, the center of mass is located within the star itself. This actually causes the star to "wobble" in its path and it is this tell-tale signature of wobbling of stars that tells astronomers that a planet is causing the effect. Many extrasolar planets have been located around various stars in our own galaxy making the barycentric approach a useful and practical means of planet hunting.
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Image Credits: WikimediaCommons/Alcandre(http://commons.wikimedia.org/wiki/File:Triangle_barycentre.png)/Rubik-wuerfel(http://en.wikipedia.org/wiki/File:Solar_system_barycenter.svg)/EnEdC(http://commons.wikimedia.org/wiki/File:L4_diagram.svg)