NOTES
Now for the first bad news. Although moving the Ring so the sun is out of plane leads to a nice gravitational restoring force, moving the Ring so the sun is off center doesn’t. This can be demonstrated without calculating the whole force on the Ring, just by calculating the forces due to gravity from small arcs of the Ring. On the near side, the arc contains a mass proportional to (R-x) and the effect of this mass is divided by a distance of (R-x) squared. On the far side, the opposite arc has mass proportional to R+x and the effect is divided by (R+x) squared. These two effects don’t balance - any slight offcentering will get worse and worse. Note that a Dyson sphere is much better off. The mass contained in an arc at distance (R-x) is proportional to (R-x) squared, so when this is divided by (R-x) squared to get a force that exactly balances the force due to the opposite arc. The sun can be placed anywhere inside the sphere and will feel no force toward any particular part of the sphere. In Ringworld Engineers Larry Niven revealed that the Ringworld has attitude jets along its edge which convert solar wind into thrust, making the Ringworld stable by providing a restoring force that increases properly as the off-centering gets bigger. The only problem is that someone has stolen most of the jets.
By the way, Pournelle and Sheffield's "Higher Education," which I enjoyed quite a bit and noticed a cute touch (spoiler space below) - one of the questions that the space-faring students have to answer is this:
A solid ring made of strong material is attached by strings (like spokes) to a rotating massive object (and therefore is rotating at the same rate as the massive object). When all the strings are simultaneously cut, what happens to the ring. The answer is, of course, that the ring will shortly collide with the object in the middle – for the same reason that Ringworld is unstable. I wonder if this is a deliberate reference to Ringworld.
This is, by the way, the way that James Clerk Maxwell determined that Saturn’s Rings could not be solid – since a solid ring would be unstable.
Let’s talk a little more about stability, since I mentioned it.