What precisely do you mean by “the same problem with epicycles”? You can fix Jupiter at the origin. You use exactly the same equations, except that the sun is u

What precisely do you mean by “the same problem with epicycles”? You can fix Jupiter at the origin. You use exactly the same equations, except that the sun is using the equation that used to describe Jupiter (with an extra negative sign) and the other planets add in the term describing the Sun’s motion. Earth is no different than any other planet (or other celestial body in a stable orbit relative to the solar system) in this respect. Hell, you could make the Moon the center of the solar system - the math works just fine.