Orbits in After Effects
I would probably start from the smallest orbit and work my way out.  If you adjust the anchor point for each object that is rotating, a simple rotation keyframe should make it rotate around the center.

Once you have the smallest thing rotating, you include that composition in another composition and adjust the anchor point similarly to make it rotate around the center of the new composition.

If you need oblong orbits, you can also use a path instead and animate it along the path.

You can keep nesting them indefinitely for as many nestings as you need. It isn't a 100% accurate simulation since the actual orbital mechanics are a bit more complicated, but it gives a simple simulation of them and can easily support various periods for each part.