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.
(In reply to
re: Problem is oversimplified? by Steven Lord)
I'm thinking I did a poor job of explaining myself in my previous post.
What I was trying to say is that the assumption of a constant 27 day period for the lunar orbital period is an assumption that drastically changes the solution vs. the real physics. It's still a really interesting problem as is. And the solutions that are given are correct for the problem as stated. And it's also cool that we are having this extended discussion. Good one, Danish!
But, just for fun, I played around with the equation for orbital period as a function of orbital radius using Earth/Moon masses, and then applied the solution methodology that SL gives, using p's and d's. The answers for that problem are *really* different.
Example. For the stated problem, the orbit shape is convex if the moon's orbital radius is LESS THAN 0.52 M Miles (per SL). However the solution using varying orbital period as a function(orbital radius) is convex only if the moon's orbital radius is GREATER THAN approx. 161900 miles.
This surprised me until I noted that orbital period varies proportionally to the 3/2 power of orbital radius. This means that SL's "p", which is constant in the posted problem, varies, for real physics, proportionally with the 3/2 power of the moon's orbital radius. Thus the solution to my "playing around" revised problem must be VERY different.

Posted by Kenny M
on 20210525 14:41:22 