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The first main difference is that the force governing the motion of the planets is the gravitational force. The gravitational force is always attractive (it depends on the mass of the objects and the distance).
The movement of electrons in the atom is governed by the electrostatic force. While this force is an inverse square force as gravity, it can ...
268 words point out the important things that planet movement and atomic movement do not share, since the former is often used as an analogy of the latter.