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So, as a bit of background, my world takes the assumption that Planet Nine is a rotating primordial black hole, and humanity, a Type I civilisation, is able to build a sphere of mirrors around it (with a little window so it doesn't explode). In this case, how much energy would the system put out, and in what place would it put humanity in the Kardashev scale?

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  • $\begingroup$ Can you link to a page about the Kardashev scale for those of us that don't know what that is? $\endgroup$
    – overlord
    Nov 13, 2019 at 22:49

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On the assumption that by "Planet Nine" you mean an object in our own solar system, I'm going to assume that it doesn't have a massive accretion disk... there's probably not enough stuff available for one to have remained intact until the present day. Probably. Anyway, Vogon Poet has a perfectly good answer for you if it does have an accretion disk, or you construct one by throwing things at it.

Black holes radiate energy via Hawking radiation. The intensity of this radiation is inversely proportional to the mass of the black hole. Most black holes, being much larger than our sun, don't put out much in the way of Hawking radiation. A primordial black hole can of course have substellar mass, and in this case it is probably required to have quite a low mass otherwise it would have seriously disrupted the early solar system, and your alternate-universe humans wouldn't be around to harvest it.

Very roughly speaking, the power output via Hawking radiation can be approximated by $P \approx 3.56345\times10^{32}\left(\frac{1}{M_0}^2\right)$ (where $M_0$ is the initial mass of the black hole) so in order to get a power output of more than a couple of watts, your black hole must mass less than 1016kg, eg. lighter than Phobos.

To reach Kardashev II power levels, you need to get at least 1026W out of your black hole, which means your black hole would have to weigh 1832kg.

Problem! Because of the old mass-energy equivalence, as your black hole radiates energy, its mass has to decrease! The evaporation time is given by $T_{ev} \approx 8.41092\times 10^{-17}M_0^3$, which for the sort of miniscule black hole you're after is about half a microsecond.

Your civilisation will therefore be at K2 level for less than a microsecond, on the assumption that their power collectors can harvest that much juice. During that time the remainder of the black hole's mass will be converted to gamma rays and particles somewhat resembling a nuclear explosion of about 40GT TNT equivalent yield.

I hope they enjoyed their brief moment of power.

Alternatively, they should be happy remaining mere K1s, and having something as fascinating and useful as a substellar black hole in their nighbourhood to play with.

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By definition, if harvested, it would make the civilization a Type I civilization and produce that much energy.

You would be harvesting 40% of the rest mass accumulated on the accretion disk around the black hole as energy.

You are assuming the mirrors consume zero energy to construct or maintain, and they are 100% efficient in directing all black body radiation to the collectors, which consume zero energy. And transportation around the mirror array consumes zero energy as well.

In this perfect case, the energy would need to start with the mass of the matter in the accretion disk, $m_a$.

Your energy will be: $$E=m_ac^2\times 0.4 \times \eta_m \times \eta_c$$

Where $\eta_m$ is your mirror efficiency and $\eta_c$ is your collector efficiency.

Find your starting mass, this gives you the answer. When E = 1016 Watts you have a type I.

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The only energy that can be emitted from a black hole itself (and not the surrounding accretion disc) is Hawking radiation. Hawking radiation is a tiny amount of energy, so harvesting it probably isn't worthwhile.

In addition, the sphere of mirrors reflects the radiation back into the black hole, making it so you can't harvest any.

However, if you add other mass or light, you could maybe construct a Penrose mechanism to collect infinite energy.

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