After Niven published Ringworld in 1970 he was flooded with letters from physics students carefully explaining that the Ringworld isn't gravitationally stable. So he wrote some more books (and oh how that pained him ;) with rockets to fix the problem. But Grayson and I were discussing solar sails today and the thought struck me that perhaps the ring world could be stable if it were arranged like solar sails.
You see, the pressure required to recenter the ring might be quite small compared to the weight of the star, and the pressure will decay as 1/r^2 (the particles get spread over an expanding sphere), so the ring will experience a force decaying like gravity, but in the opposite direction. I futzed around with the maths for this, but couldn't get a tractable equation. Here's my diagram anyway:
Say that the sun drifts to position x, then the force produced by the solar wind is integral around the circle of <x-t, sqrt(1-t^2)>*(1-2xt+x^2)^(-3/2) which is beyond me at this hour (left as an exercise etc). Let's do a hand-wavy differential analysis instead: star drifts dx, force on ring is int (cos t - dx)/((c-dx)^2 + s^2)^(3/2) from 0 to pi, denominator = 1, so about pi dx, in other words, pushing it a bit harder that way (hand-wave hand-wave) and re-centering.
Feel free to tell me the correct analysis.