Freeman Dyson proposed that the power needs of an advanced civilisation would eventually require the entire energy output of a star to be collected, so that the star would end up surrounded by a dense network of satellites extracting power from the radiation. In science fiction this idea has often mutated into a solid shell that completely surrounds the star. My question is, if it were possible to build such a shall, what effect would it have on the dynamics of the star inside?
I suspect the effect would be catastrophic, but I'd like more details. Here's my reasoning: Wikipedia implies that the temperature gradient between the core and outer layers of a star plays an important role in its stability. The shell around the star would reflect or re-emit a lot of the star's radiation. (I'm assuming the shell isn't composed of perfect solar collectors that would simply absorb all the radiation.) This reflected radiation would reduce the loss of heat from the outer layers of the star. This would reduce or even eliminate the temperature gradient, which I guess would cause the star to expand. My question is whether this is correct, and if so whether it would be enough to disrupt the process of fusion in the star's core. Or would there be some other, less obvious effect on the star's dynamics?
A closely related question is, would stars be stable in a static or contracting universe? In this case all of space would be filled with the radiation emitted from other stars, and I'd be interested to know what effect this would have on stellar dynamics.