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no the planets are not dragged in the wake of the star. its all fairly coplanar (as in not a cone). everything orbits the solar center of mass, and that center of mass orbits the galaxy. it's not like the sun is the only thing orbiting the galaxy, with the planets "just coming along".
it's like how you can freely move around a plane while in flight. you're not separate to the plane, and the plane isn't separate to the rotation of the earth, and so on. it's all stable.
