1) The size of a late class M star is closer to that of Jupiter than that of the Sun.

2) The whole point of a Dyson sphere is to capture and use the energy of a star. Reflecting it back to the star defeats the whole purpose.

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I think Kaptain K is correct on both points, but I find it hard to imagine how a closed sphere could avoid reflecting 1% of the energy back into the photosphere. How long would it take for the photosphere to heat from 2000 k to 7000k with 1% of the energy being returned to the photosphere? Forever perhaps?
The 99% needs to go somewhere. Laser beams powering space craft? million gigawatt radio transmitters? 1000 times the number of far infrared photons our Sun radiates, radiated from the outer surface of the dyson sphere?
Is a "late" class M star cooler than an average class M star? Neil

Yes, but the Dyson swarm, as originally envisaged by Dyson, is relatively simple technology which has no requirement for speculative technology. In theory any civilisation could build one eventually.

On the other hand no method of converting mass to energy efficiently is known to exist. In order to create enough antimatter to annihilate the Earth you would need a Dyson swarm just to collect enough energy to manufacture the antimatter. Even the use of microscopic black holes to generate power has a whole range of associated engineering difficulties which may be impossible to overcome. So it may be the case that no civilization, however advanced, can ever develop mass -> energy technology which creates a surplus of energy, and the Dyson swarm is the only practical option for large scale energy generation.

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A lightweight Dyson bubble would be supported by light pressure; it could have the mass of a large asteroid and still cover the star. A Dyson bubble supported by light is one of the more plausible options for a shell design, but to me the swarm is much more likely. A partial swarm would look very like a shell of dust, or maybe a disk of dust similar to those already observed around many stars.

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Unless you want to increase the temperature of the star, and thereby increase its output. Use half of the energy emitted by the star, and return the rest to heat the star up; eventually you will increase the luminosity of the star severalfold.

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I'm having problems with the theory. It seems reasonable that feeding energy back to the photosphere would increase the luminocity several fold from class M to class G?
But the core temperature will not increase significantly for thousands, perhaps millions of years due to the feed back, so the core output will not increase. Over centuries, the photosphere output can only increase 50% if you are returning 1/2 of the energy. How much does the luminosity increase with a 50% energy increase? 50%? Neil

The photosphere could get quite a bit hotter than you might think if you reflect 50% of the energy; after all, almost exactly 100% of the energy emitted by the star is currently lost. Heating the outer layers increases the temperature, there fore the luminosity- you then reflect 50% of the increased luminosity back to the star.

By reflecting that energy back to the star you are heating the star up from the surface inwards. The photosphere will get hotter, therefore more luminous, well before the increase in temperature reaches the core.

Since the star expands when it is heated in this way, that would affect the density- I wonder how that would affect the rate of reaction at the core?

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