A Dyson sphere is a hypothetical megastructure made up of a finite number of thin sun-shades in free-fall orbit around a central star. The goal is to harvest as much of the sun's radiation power as possible. In this regard, let's agree that a Dyson sphere is "fully-fledged" iff all rays emanating from the star's center must intersect one of the sphere's shades (at all times). Suppose in addition that all shades must keep a distance of at least $R$ from the star's center (in order not to evaporate, say). What is the minimal surface area of shade-material that must be allocated in creating such a fully fledged Dyson-sphere?
To have my question well-posed, I should briefly discuss the rules of motion in orbital mechanics. For the occasion, let me state them as follows: a single sun-shade $S(t)\subset \mathbb{R}^3$ is a relatively open subset of a sphere $\partial B(0,d) \subset \mathbb{R}^3$ which is contained within a ribbon $r(\varepsilon,C)\subset B(0,d)$ of a (small) thickness $\varepsilon$ around a great circle $C$ of the sphere $\partial B(0,d).$ $S(t)$ changes with time $t$ through rigid rotation along the great circle $C$ with rotation speed $v=1/\sqrt{d}$. As such, the question of the minimal required surface area $A_{\varepsilon}$ retains a dependence on the thickness-parameter $\varepsilon$. Obviously, $\varepsilon \mapsto A_{\varepsilon}$ is a decreasing function so that I'm ready to ask my question: what might $A:=\lim_{\varepsilon \to 0+} A_{\varepsilon}$ might be?
Allow me also to briefly discuss a few obvious bounds: clearly we always have that $A\geq 4\pi R^2$. On the other hand, by arranging a number of circular ribbons in the way of the picture below(*), it is clear that $A \leq 2\pi^2 R^2$: in this setup, at all time around 36% of the shade-area is in the shadow cast by other shades.
Picture courtesy by wikipedia commons
(*) The picture does not do a perfect job of depicting the configuration that I'm referring to. Specifically, the ribbons in the picture approach each-other closest at distinct locations in space, while I rather am referring to a configurations where these ribbons are great circles going from a shared north pole to a shared south pole (with slightly different radii to avoid collision at these poles) along the meridians suggested by this pair of poles.