And how exactly did we find out how they move?
In: Planetary Science
Because the solution for the possible paths in the 1/r potential motion are the conic sections of which the closed orbits are ellipses and the circle is just a special case of an ellipse.
A good way of thinking about it is to imagine a spaceship on a circular orbit that begins to accelerate, once it reaches a high enough velocity it’ll leave the closed orbit and enter a parabolic or hyperbolic orbit. This transition (as long as you continuously increase your velocity) is smooth so you have to go from a circle to something like a parabola. So the circular orbit gets more and more elongated (its eccentricity increases) until the kinetic energy of the spaceship is great enough at which point the ellipse “snaps” and the spaceship enters an outbound orbit never to return.
How have we figured this out? First a guy called Brahe recorded lot’s of data on where the planets were at any given time. Then Kepler worked some 30 years to organise that data and essentially fitted their trajectories and that’s where the laws of planetary motion come from. Then Newton came up with gravity as a force from which Kepler’s laws can be derived.
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