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How efficient can a Dyson sphere be?

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The shell variant of a Dyson sphere consists of an artificially-made shell of material about 1 AU in radius encircling a star. The sphere captures most of the star's energy and stores it for future use. Unfortunately, the megastructure will lose energy. It has a non-zero temperature, and therefore it will radiate energy in the form of black body radiation. As with any heat engine, we can assign an efficiency, $\eta$, to it: $$\eta\equiv1-\frac{T_{\text{DS}}}{T_*}$$ with $T_{\text{DS}}$ the temperature of the shell and $T_*$ the temperature of the star. An old paper I found thinks that a temperature of $T_{\text{DS}}=300\text{ K}$ might be realistic (giving an efficiency $\eta=0.95$), and that the ultimate lower-temperature limit is set by the cosmic microwave background, at $T_{\text{DS}}=2.7\text{ K}$ and $\eta=0.99955$, all assuming a Sun-like star.

I'd bet anything that the true limit is higher and depends on the composition of the shell, but I have no idea what that limit is. Assuming that the structure is built by a Type II civilization but that they don't have access to handwavium or any other magical material, what's the maximum efficiency of a Dyson sphere of this nature?

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I think a reasonable assumption is that you don't want to accumulate entropy from the Dyson sphere. That is, the entropy you get from the star must not be lower than the entropy you send to outer space (indeed, you'll have to send out more entropy because on one hand you have other sources of entropy like the cosmological microwave background, energy released by radioactive substances, inefficiencies of the machinery used inside the sphere, and last not least the entropy produced by the humans living inside the sphere).

I'm going to assume that the star's entropy is predominantly emitted in the form of (thermal) radiation. I'm also going to assume that thermal radiation is the only significant way for the Dyson sphere to get rid of entropy.

According to the Stefan-Boltzmann law, the total power emitted by a hot body is given by $$P = \sigma A T^4$$ where $\sigma = 5.67\cdot 10^{-8}\ \rm W m^{-2} K^{-4}$ is the Stefan Boltzmann constant. Since Power is Energy over time, and the stars radiation is thermal radiation (so it's all thermal energy), the corresponding entropy rate is $$\dot S = \frac{P}{T} = \sigma A T^3$$ Since we are talking about spherical objects (the star and the Dyson sphere), we have in both cases $$A = \frac{4\pi}{3} r^3$$ where $r$ is the corresponding object (star or Dyson sphere).

Now according to the condition above, we have $$\dot S_{\rm DS} \ge \dot S_* $$ Inserting the formulas above and cancelling all the constants then gives us $$ r_{\rm DS}^2 T_{\rm DS}^3 \ge r_*^2 T_*^3 $$ which implies for the ideal efficiency $$\eta_{\rm Carnot} = 1-\frac{T_{\rm DS}}{T_*} \le 1-\left(\frac{r_*}{r_{\rm DS}}\right)^{\frac{2}{3}}$$ Assuming the sun as star, $1\ \rm AU$ is about 215 times the radius of the sun and therefore the corresponding efficiency is $97\%$.

However the civilization would likely not go for the Carnot efficiency (which only can be achieved with negligible power throughput), but for the point of maximal power output, which is given by the Curzon-Ahlborn formula: $$\eta_{\rm CA} = 1 - \sqrt{\frac{T_{\rm DS}}{T_*}} \le 1-\left(\frac{r_*}{r_{\rm DS}}\right)^{\frac{1}{3}} $$ For the solar Dyson sphere, this gives an efficiency of about $83\ \%$.

I think that the outside material of the Dyson sphere does not matter much here, except that it should be a good heat conductor and its outside should be as black as possible (but we are already quite good in that regard and by the time we can make Dyson spheres, we'll surely have improved on that).

Now of course there will also be inefficiencies in whatever machinery is used inside the Dyson sphere, and that will lower the overall efficiency (and the amount of entropy to get rid of). Unfortunately I have no idea how efficient/inefficient the technology of a Type II civilisation will be.

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