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This question already has an answer here:

Let’s consider two planets of mass $m$ and $M$. They both will attract one another with a constant force $F$ according to the law of attraction $F=G m M/r^2$. The force and masses are constant. $G$ is also constant. So therefore the distance between them would also be constant. If this distance will be assumed as a radius then the orbit formed will be a circle. So why is it elliptical?

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marked as duplicate by Qmechanic Mar 24 at 5:34

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Related web article pdf. $\endgroup$ – ja72 Mar 24 at 3:52
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    $\begingroup$ Duplicate(?): physics.stackexchange.com/q/69997 and links therein. $\endgroup$ – dmckee Mar 24 at 3:55
  • $\begingroup$ As my2cts explains, your mistake is your false assumption that the force should be constant. If the initial relative velocity of the two particles has any radial component, it can’t be, because the next instant they will be closer together or farther apart. $\endgroup$ – G. Smith Mar 24 at 5:28
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"with a constant force F" The force is only constant if the object moves in a perfect circle. If it moves in an ellipse than the force, the distance and the speed vary.

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The equation you're using to describe this system is describing the magnitude of the force between the two objects for some given distance - it's not really telling you anything about the orbits necessarily. What you really want to describe the motion of two bodies is some vector equation that also describes an aspect of their orbit. To start to get at this, you'll be considering something called the "two-body problem" (at least what I've heard it called). If you want a full briefing on the entire problem, just give "two body problem" a google and (by my cursory glance) you should find some worked examples of the physics behind it.

If you want a quick answer as to why they follow ellipses you can think of it like this: a circle is really just a special case of an ellipse, so for a general system, the orbital motion is likely to be the most general case (an ellipse).

Note that this doesn't even scratch the surface of orbital mechanics or really explains any of the physics behind it, but for the level of the question you asked I thought it might be satisfying. If you actually want to learn the gritty details and full derivations, an introductory textbook is "Foundations of Astrophysics (Ryden and Peterson)" which should get you started.

Edit: I should mention that in general orbits follow conic sections which could be an ellipse, circle, parabola and hyperbola. These depend on the parameter 'e' called eccentricity, that is a geometric property of conic sections.

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