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If we look at the well known means of drawing an ellipse with two tacks and a string, you can keep moving your pencil around and around and it will draw over and over the same line.

The same cannot be said of an orbit. The center of all orbits is constantly traveling. The orbiting body will never retrace a previous path.

My question is: are there some orbiting bodies that would end up describing circles if the centers of their orbits would just stay still, and others that would actually describe ellipses, or do all orbits always describe ellipses because of the fact all centers of orbit are continuously moving?