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Alright guys, so I attracted you with the title. Now, to preface the question, I am fully aware of the work of Copernicus, as well as the concepts of Heliocentrism and Barycentric Coordinates. I have a master's degree in Engineering, and have taken many a course in dynamics and kinematics. It is with this experience that this question arises.
When studying bodies in motion, it is customary to pick a stationary reference frame for the basis of your calculations. Since the Sun is extremely massive compared to the rest of the planetary bodies, the barycenter of the Solar System (the stationary origin of choice) is extremely close to (and often inside) the sun. Hence why we commonly say that the planets all revolve around the Sun.
Although this certainly simplifies drawings of the planetary orbits, allowing for mostly non-intersecting ellipsoids in modeling, the base reference frame is, for all intents and purposes, arbitrary. All motion is relative to it's observer, so who's to say we cannot define the origin to be at the center of Earth? Sure, children would no longer be able to make working models out of hangers and Styrofoam balls, but wouldn't the equations of motion remain the same?
I have been searching online for a video, or gif, that illustrated this principle, but was unable to find anyone who took the time to do so. I'd be very interested in seeing what the orbits actually looked like if we re-defined the stationary reference frame from the barycenter to the center of the earth. I'm sure the orbits would look pretty rad! Much like 1:40 into this Solar System Orbit Video.
Is there something I'm missing that would theoretically preclude us from doing so?
Ah hah! Finally found a video that shows this simulation.