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xkcd #2898: Orbital Argument
(imgs.xkcd.com)
Alt text:
"Some people say light is waves, and some say it's particles, so I bet light is some in-between thing that's both wave and particle depending on how you look at it. Am I right?" "YES, BUT YOU SHOULDN'T BE!"
No, they orbit around a common barycenter. Even if there were no other bodies in the solar system, that wouldn't be exactly at the center of mass of the sun.