, where total mass .
In , . Each of the center of mass's components can be computed by
When talking about celestial bodies, the center of mass has a special relevance: when a moon orbits around planet, or a planet orbits around a star, both of them are actually orbiting around their center of mass, called the barycenter. There are some interesting consequences:
- Earth-Moon system: the Moon's mass is 1/81 of Earth. Put Earth in position 0, mass 1 (here we use an arbitrary mass unit. It does not matter, provided that we use the same unit for the Moon). Moon position 400,000 km, mass 1/81. Center of mass is at:
- Sun-Earth system: put Sun in position 0, mass=333,000 times the Earth. Earth in position 150,000,000 km, mass=1. Center of mass is 450 km from the Sun center. Here, the large mass difference between the two bodies makes the center of mass lie almost where we were expecting it.
- Sun-Jupiter system: put Sun in position 0, mass = 333,000 Earths. Jupiter in position 778,000,000 km, mass=318 Earths. Center of mass is 742,000 km from the Sun center. It's actually outside its surface! As Jupiter does its 11 year orbit, the Sun majestically does a full 1.5 million km orbit around the center of mass.
- 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. But, only Jupiter manages to pull the center of mass so far, thanks to its large mass. If all the planets would align on the same side of the Sun, the combined center of mass would lie about 500,000 km outside the Sun surface.