It comes as a surprise to many that the orbit of the moon around the sun has no loops in it. Indeed, it is a convex curve not very different from the orbit of Earth around the sun.
How far away from Earth would our moon have to be for the moon's orbit around the sun to have a loop? How far away for it to be nonconvex?
Assume all orbits are circular and all lie in the same plane (so that "loop" and "convex" have clear planar meanings), the Earthsun distance is 93 million miles, the Earth's orbit requires 365 days, and the moon's orbit around Earth takes 27 days (and that is constant in this problem). Using such approximations has negligible impact on the problem.
Note that the moon's orbit is "prograde": in the same direction as Earth moves around the sun. Both motions are counterclockwise, viewed from our north pole.
I accept the three previous solutions, as well as the "official" solution. However I submit that the constant 27 day lunar orbit period is not at all negligible (as implied by Steve Lord). In fact, in Steve's solution, the parameter "p" (ratio of Earth period to Moon period) actually varies (using appropriate physics) proportionally to the lunar orbit radius to the 3/2 power. This means the rate of change of "p" can be faster than that of "d". I *think* this flips the solution space on it's head, meaning, (for Earth/Moon masses):
Loops only occur for very small orbital radii. As the lunar orbit radius increases, you then transition to wiggles, then convex. I think the transition points are: Loops below approx. 453 km lunar radius, Loop to Wiggles transition  about 453km, Wiggles to Convex  about 260600km. Convex above 260600km lunar radius.

Posted by Kenny M
on 20210524 17:05:38 