Are epicycles and mean Earth/Sun actually useful astrophysical concepts? Copernicus put the Sun in the center of the Solar system. This turns the Moon's orbit into an epicycle around Earth orbit. And the Jovian moons (later discovered) have orbits which are epicycles around Jupiter. Since Copernicus created a physical epicycle in the Moon, it maybe wasn't so strange to think about "epicycles of epicycles" as he did to improve the model's fit to calendar data.

Actually, Ptolemy and Brahe did not put Earth in the center of their Solar system models. They used a fixed point in empty space near the Earth which they called "mean Earth". And Copernicus used a corresponding "mean Sun". And as a matter of fact, planets do turn around a barycenter which is not in the center of the Sun, but nearby.

Do epicycles and "mean Earth/Sun" actually have physical correspondents in the orbits of moons and the barycenter respectively? That'd be a bit funny since they are purely geometric constructions with the purpose to make data fit better, created even without a thought about gravity or anything physical at all.
I'll say yes here, but only if the epicycles are used as approximations. Here's how they would be useful:


The eccentricity of the celestial body's orbit is low, i.e. $e&lt;&lt;0$. This means that it would be nearly circular, and epicycles could therefore be reasonably accurate. For example, it would not be a good idea to use epicycles to model Mercury's motion, because it has a high eccentricity. Comets would also be poor choices. Triton, on the other hand, has low eccentricity.
Only a vague approximation is needed. If you aren't looking for calculations with a high degree of accuracy, you could use epicycles. It would be a bad idea to use epicycles to put a spacecraft into a precision orbit, but a better idea to use them in a toy simulator. NASA, however, has higher standards.
You don't need to use general relativity. One of general relativity's most important predictions was the precession of Mercury's orbit. Over long periods of time, general relativity triumphs over classical theories when it comes to this accuracy. It would be hard to use epicycles to model this - although potentially possible.