12 December 2018 markie: Delusions about the reasoning of HappySkeptic99
HappySkeptic99 presented the textbook explanation of the net force between a sphere and an internal body being zero for any inverse square force.
The same basic physics and math applies to a ring and an internal body.
Not quite, Reality Check. A ring does not have a net zero force. It must be a shell. The ring, in its plane, gets increasingly more attractive to the side the is closest to the central mass. A ring, out of the plane, actually does have a restoring force, and would be a stable oscillator.
The geometrical reason why a shell has zero net force, is that a small solid angle subtended from the central mass, in opposite directions, hits the shell on opposite sides at radii R1 and R2. The areas (and therefore masses) that the solid angle interacts with with are proportional to R1^2 and R2^2. The gravitational force from each equal-area patch on the shell in the opposing regions is proportional to 1/R1^2 and 1/R2^2. The net force will be the multiple, which is just 1, or equal forces on each side.
A solid angle will not do this with a ring, because it will go outside the bounds of the ring on one side, and not the other. The math doesn't work for a ring. It is inherently unstable within the plane, and only has zero force when the mass is centered.
As Markie pointed out from a paper, the combination of the in-plane attractive force when off-center and restoring force off-plane can indeed give a stable rotating ring, if the ring is inclined and precessing, at a very narrow range of starting conditions. This wasn't obvious to me, but I can see that it could certainly be possible.
However, Mills' orbitsphere is definitely a sphere, regardless of whether it is made up of a long string, or a bunch of locked hoops.