Can two stars orbit each other without forming a binary system? If so, how would this look like?

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This is what most people think of as a binary system. This is just an exaggerated planetary orbit and is even capable of having outer planets.

This is a binary twin system and is not stable enough to have anything except oort cloud style planetoids in the distance.

A binary system describes the number of stars, not the motion of the stars. Yes, in the vast majority of cases you will not find a twin binary. The lesser gravity object will orbit the the greater gravity object. Size is not the issue. Stars will orbit black holes.

Then you chance across a binary twin system.

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