Black Hole Power!

[Mr Skeptic]

Given the assumptions that:

1) Black Holes, as described by general relativity, exist, even at small sizes.

2) Hawking Radiation exists and follows the formula even at small sizes.

 

Source for equations: http://en.wikipedia.org/wiki/Hawking_radiation

The schwarzschild radius of a black hole depends on its mass: [math]r_s = \frac{2GM}{c^2}[/math]

The Hawking Radiation can be written as a relation between power and mass: [math]P = \frac{h c^6}{7680 G^2 M^2}[/math]

The time it would take for a black hole to evaporate can also be written in terms of mass: [math]t_{ev} = \frac{2560 G^2 M^3}{h c^4}[/math]

Therefore, if P is the power output of the Hawking Radiation:

[math]M = \sqrt{\frac{h c^6}{7680 G^2} \frac{1}{P}} = 1.186 \times 10^{17} \times P^{-1/2}[/math]

[math]r_s = 1.761 \times 10^{-10} \times P^{-1/2}[/math]

[math]t_{ev} = 3.553 \times 10^{33} \times P^{-3/2}[/math]

 

Then, for a black hole to produce a certain amount of power, it would have the attributes:

Black Holes by Hawking radiative power output

Power(W)   Mass(kg)    Size(m)    Evaporation time(s)  Power Comparable to
10[sup]3[/sup]        3.8X10[sup]15[/sup]    5.6X10[sup]-12[/sup]   1.1X10[sup]29[/sup]             peak output of healthy human
10[sup]6[/sup]        1.2X10[sup]14[/sup]    1.8X10[sup]-13[/sup]   3.6X10[sup]24[/sup]             P-51 Mustang fighter aircraft
10[sup]9[/sup]        3.8X10[sup]12[/sup]    5.6X10[sup]-15[/sup]   1.1X10[sup]20[/sup]             half of Hoover Dam
10[sup]12[/sup]       1.2X10[sup]11[/sup]    1.8X10[sup]-16[/sup]   3.6X10[sup]15[/sup]             1/4 of US energy consumption
10[sup]13[/sup]       3.8X10[sup]10[/sup]    5.6X10[sup]-17[/sup]   1.1X10[sup]14[/sup]             world energy consumption
10[sup]14[/sup]       1.2X10[sup]10[/sup]    1.8X10[sup]-17[/sup]   3.6X10[sup]12[/sup]             medium sized hurricane
10[sup]15[/sup]       3.8X10[sup]9[/sup]     5.6X10[sup]-18[/sup]   1.1X10[sup]11[/sup]             heat flux of Gulf Stream
10[sup]16[/sup]       1.2X10[sup]9[/sup]     1.8X10[sup]-18[/sup]   3.6X10[sup]9[/sup]              Type-I civilization on the Kardashev scale
10[sup]17[/sup]       3.8X10[sup]8[/sup]     5.6X10[sup]-19[/sup]   1.1X10[sup]8[/sup]              Type-I civilization on the Kardashev scale
10[sup]18[/sup]       1.2X10[sup]8[/sup]     1.8X10[sup]-19[/sup]   3.6X10[sup]6[/sup]             
10[sup]19[/sup]       3.8X10[sup]7[/sup]     5.6X10[sup]-20[/sup]   1.1X10[sup]5[/sup]             
10[sup]20[/sup]       1.2X10[sup]7[/sup]     1.8X10[sup]-20[/sup]   3.6X10[sup]4[/sup]             
10[sup]21[/sup]       3.8X10[sup]6[/sup]     5.6X10[sup]-21[/sup]   1.1X10[sup]2[/sup]             
10[sup]22[/sup]       1.2X10[sup]6[/sup]     1.8X10[sup]-21[/sup]   3.6X10[sup]0[/sup]             
10[sup]23[/sup]       3.8X10[sup]5[/sup]     5.6X10[sup]-22[/sup]   1.1X10[sup]-2[/sup]             luminosity of Wolf 359
10[sup]24[/sup]       1.2X10[sup]5[/sup]     1.8X10[sup]-22[/sup]   3.6X10[sup]-3[/sup]             
10[sup]25[/sup]       3.8X10[sup]4[/sup]     5.6X10[sup]-23[/sup]   1.1X10[sup]-5[/sup]             
10[sup]26[/sup]       1.2X10[sup]4[/sup]     1.8X10[sup]-23[/sup]   3.6X10[sup]-6[/sup]             luminosity of the sun, Type-II civilization
10[sup]27[/sup]       3.8X10[sup]3[/sup]     5.6X10[sup]-24[/sup]   1.1X10[sup]-8[/sup]             
10[sup]28[/sup]       1.2X10[sup]3[/sup]     1.8X10[sup]-24[/sup]   3.6X10[sup]-9[/sup]             
10[sup]29[/sup]       3.8X10[sup]2[/sup]     5.6X10[sup]-25[/sup]   1.1X10[sup]-11[/sup]             
10[sup]30[/sup]       1.2X10[sup]2[/sup]     1.8X10[sup]-25[/sup]   3.6X10[sup]-12[/sup]             
10[sup]31[/sup]       3.8X10[sup]1[/sup]     5.6X10[sup]-26[/sup]   1.1X10[sup]-13[/sup]            luminosity of Beta Centauri

 

The reason evaporation time is an important consideration is that black holes get hotter the smaller they are. If your black hole evaporates it gets really hot and will release all the energy it has. Which might vaporize a planet. The size is important because you need to feed as much mass into the black hole as it converts into energy.

 

If the assumptions hold, then a single black hole could provide as much power as you like, so long as you can feed an equal amount of energy. I'd assume that smaller black holes would spit out some particles as well, so the efficiency would not be 100%. Still, you could throw any sort of matter in and it should convert it into energy, right?

Edited March 15, 2009 by Mr Skeptic

[cameron marical]

where does the energy from black holes in space go to?

 

and dont black holes suck in everything, not only mass?

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Consecutive posts merged

either way man, if your right, thats pretty crazy. alot of energy.

[MysteriBoi]

I got a question.

 

Is it possible for a black hole to suck it's own self? i know stupid question but just want to know more.

 

Any suggestions?

[swansont]

If your black hole evaporates it gets really hot and will release all the energy it has. Which might vaporize a planet.

 

Power isn't the right thing to look at here, because the time for which that power is emitted is so short.

[Klaynos]

where does the energy from black holes in space go to?

 

and dont black holes suck in everything, not only mass?

 

The energy released as blackbody radiation (so photons with a spectra that looks like a black body).

 

They only don't let things go that are inside the event horizon, outside that they act like normal gravitational objects.

 

Hawking radiation has methods for getting around the event horizon.

[Mr Skeptic]

Power isn't the right thing to look at here, because the time for which that power is emitted is so short.

 

No, for the what happens if it is allowed to evaporate scenario, mass and evaporation time are a better guide. So long as you can feed matter into it at the same rate as it emits energy, you could keep it running at that power indefinitely. However, I could definitely see some problems feeding that much matter into something smaller than a proton.

[swansont]

No, for the what happens if it is allowed to evaporate scenario, mass and evaporation time are a better guide. So long as you can feed matter into it at the same rate as it emits energy, you could keep it running at that power indefinitely. However, I could definitely see some problems feeding that much matter into something smaller than a proton.

 

I was referring to the "vaporize a planet" remark. Something that releases energy at, say 10^20 Watts but only runs for an attosecond releases all of 100 Joules of energy. The power is indeed prodigious, but by itself is meaningless in terms of how much destruction it will generate. Such a device could be contained by a styrofoam cup of sufficient volume.

[Sisyphus]

It seems like the main problem is that any black hole long-lived enough to be useful would be too long-lived to be safe and too massive to be able to be manipulated. And any black hole small enough to be safe and manageable would be so short-lived that it would be probably impossible to keep it in equilibrium by feeding it mass, and even if you could, it would give off way too much radiation. A black hole of appropriate size to supply all of the Earth’s energy needs would surely kill us all.

[Mr Skeptic]

I was referring to the "vaporize a planet" remark. Something that releases energy at, say 10^20 Watts but only runs for an attosecond releases all of 100 Joules of energy. The power is indeed prodigious, but by itself is meaningless in terms of how much destruction it will generate. Such a device could be contained by a styrofoam cup of sufficient volume.

 

Hm, you're right. The only black holes with enough mass-energy to vaporize a planet would be the heavier ones on the list, and they will take their sweet time doing so. Mostly I just wanted to point out that there would be safety concerns -- if the black hole were to not be fed enough matter, its intensity would increase. Eventually this would destroy the facility containing it, possibly with the exception of black holes of weighing less than 100 tons, in which case the power spike would release about as much energy as the facility can handle, but might fry the surface components..

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It seems like the main problem is that any black hole long-lived enough to be useful would be too long-lived to be safe and too massive to be able to be manipulated. And any black hole small enough to be safe and manageable would be so short-lived that it would be probably impossible to keep it in equilibrium by feeding it mass, and even if you could, it would give off way too much radiation.

 

I think that the black holes would have to be in space. As you said, the heavier ones would be hard to hold against gravity. However, I'd say that there would most likely be electromagnetic containment as well as controlling it with a matter stream.

 

And yes, I realize that for charged black holes the equations would be slightly different. So long as the charge/mass ratio is small they should be close approximations though.

 

A black hole of appropriate size to supply all of the Earth’s energy needs would surely kill us all.

 

That would be one of the most dangerous sizes to have on earth, incidentally -- massive enough to release a lot of energy, light enough to release it quickly. (Though the ones far heavier than on my list would be more dangerous in the sense that rather than evaporating they would eat up the earth.)

 

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I think that it would be more the laws of engineering and economics than the laws of physics that would make my idea fail. To take a specific example, making a black hole that is 10^-15 m -- about the size of a proton (calculating that size, not one on the table), would require stuffing 673 426 629 metric tons into about the size of a proton. And if you want it to generate more energy than a couple dozen gigawatts, you will have to wait a very, very long time. Like 20 billion years long. And if you want the black hole to be bigger (so you could shove the requisite amount of mass into the requisite volume), then your power output will be reduced, and your wait time for the power to increase drastically increased. On the bright side, the power source will outlast the sun -- and this without the bother of feeding matter into it.

Edited March 17, 2009 by Mr Skeptic
Consecutive posts merged.

[Mr Skeptic]

On the other hand, the engineering difficulties in making one are no problem for hard science fiction. I would think this would be an excellent power source for a ship, and some of the drawbacks in real life would make for excellent plot. As such, I thought I might revive this thread to consider the uses of tiny black holes for science fiction.

 

Also, I have considered using all this as a basis for a paper to publish on arxive.

 

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Uses of black holes in science fiction:

Power source: as discussed above, making it smaller makes it brighter. I propose calling it a Hawking reactor, after Stephen Hawking's theoretical discovery of the radiation it should emit.

Anti-matter generation: If fed on protons, I'm pretty sure it should spit out positrons. Even at 0.05% efficiency, this would be far better than what we have now, and would still function as a power source.

Rocket propulsion: As pure energy, the radiation of a black hole would, if directed in one direction only, produce thrust = Power/c as above, which also would put it at the same specific impulse as a matter/antimatter rocket. An analogue to a Bussard ramjet would mean it can get as close as it likes to the speed of light, given enough time.

Weaponry: No, it won't suck up everything, it would just be really bright and able to penetrate any armor. Putting one in orbit near a planet's surface (or inside the planet to melt it) would be particularly nasty.

 

Plot devices:

They would probably be really hard to make, requiring an enormous accelerator, so it is quite possible that the means of production becomes unavailable and no more can be made.

They cannot be turned off nor broken. If the mechanisms to convert the radiation to useful electricity or shunt it away break down, it will cheerfully keep emitting just as much energy, potentially melting the whole ship.

The longer they go without matter being thrown in, the brighter they get. The most powerful ones would be extremely delicate, but could replace a star in terms of energy output. However, though it would get really bright for a while, that wouldn't last very long and the bigger problem would not be an explosion, but a major blackout. The heavier ones could go for years with little increased output.

Containment: The black hole could conveniently be held in place by the particle stream that keeps them from evaporating. Which would have to be very, very carefully aimed considering the size of the black holes. An excellent target for sabotage, and could also force a ship to maintain its current heading because otherwise they leave their power source behind (not to mention having it pass through their ship).

A free black hole would be very hard to catch, as they are so small, so heavy, and so bright.

 

Cheating:

The numbers in post 1 assume a black hole that follows both general relativity and Hawking radiation. Since these black holes are so small, it is likely that quantum gravity will be needed to explain them, so that a different formula might apply. As for Hawking radiation, there may be a maximum limit to it due to the density of virtual particles being limited, which would make smaller black holes much more manageable (ie, not necessarily as much power output as a sun, and take longer to evaporate, which would make them much easier to make and also the only way they could be used in hand-held devices.) One or both of these could be used as an excuse to largely disregard the numbers in post 1, and for scifi writers to invent their own.

Edited October 18, 2009 by Mr Skeptic

[Edtharan]

Uses of black holes in science fiction:

Black Holes (in a sci-fi sense) might be a good source of power. If one could harvest the Hawking Radiation from a Microscopic black hole, one could feed the hole enough matter to counteract the loss from the Hawking Radiation and the black hole would "convert" it into the hawking radiation.

 

So long as one has a convenient source of matter (any matter) one could turn it into energy for use in a power station (on a space ship for instance).

 

Also, if you fed it enough charged matter (protons, or electrons) to give it an electrical charge, then you could move it around quite easily and (relatively) safely but placing it in a electromagnetic trap.

 

It would also be a way for you to dispose of wast (dangerous chemicals in the environment, just dump them into the Hawking Generator. CO2 choking your atmosphere? Try the new Hawking Generator. Sequester your carbon and generate clean renewable energy today! ).

 

then of course, there is all the plot that can happen (terrorists plan to dump a large amount mass into it to cause the planet to be swallowed up by the hole, etc )

[Moontanman]

Arthur C Clark used to idea of a black hole as a propulsion in his later space Odyssey books. I don't think he took into account hawking radiation.