Every supervillain worth their salt needs to credibly threaten to extinguish the sun at least once: you don't really deserve the prefix super unless you do can things of that magnitude. What are some vaguely scientifically realistic ways to extinguish the sun, thereby extinguishing most (but surprisingly not all, because chemosynthesis) life on Earth?
Easiest, perhaps too easy, is to put a huge shade at the L1 Lagrange point between the Earth and the sun, blocking out the sun by casting a shadow. Make it mirrored on the front side so less heat leaks out the back. Similar: blot out the sun with objects in the atmosphere or earth orbit, as seen in The Matrix.
Disassemble the sun, taking all its matter and spreading it out. Maybe use the sun's own power to do this: induce huge mass ejections in the style of coronal mass ejections or intense solar wind in the style of Wolf-Rayet stars. What is the gravitational binding energy of the sun? How long does it take for the sun to produce that much energy? Two approaches: collect and store all the needed energy before starting disassembly. Or use the sun's power online as you disassemble. The latter had to deal with the sun's declining power as its mass decreases. There will be a remnant brown dwarf.
The lazy supervillain might only need to decrease the sun's mass, thereby decreasing its power output (and, because of its weaker gravity, shifting the earth to an orbit more distant from the sun), enough to make earth freeze. But clever humans will probably use technology to concentrate the decreased solar power enough for some to survive: Mars colony on Earth. (Not all of us will survive.) Actually, by harnessing nuclear power, some humans could probably survive a very long time even with the sun completely extinguished.
Launching earth out of the solar system at escape velocity would also freeze the earth. Let's dismiss this. Maybe we are a K2 civilization with also zillions of space stations in orbit around the sun, and it's too tedious to fling all of them away.
Steer a white dwarf through the core of the sun. The white dwarf will punch right through but will disrupt the sun so much that the sun will explode or something. Hopefully the explosion won't destroy earth, because the supervillain's point is to have the residents of earth suffer for a long time as they slowly freeze to death. (Merely destroying the earth in vaguely scientifically plausible ways is a topic others have discussed.)
Steer a black hole to the core of the sun and then make it stop it there. It will gobble up the sun from the inside. How long will it take? One way to make a black hole stop at the center of the sun is to hit it with two black holes from opposite directions which merge at the center. Will the gravitational waves from the merger violently destroy the sun?
One could also gobble up the sun with a neutron star with enough mass that it transitions into a black hole after it collects enough of the sun's mass. Is there a supernova-like explosion at the transition that will blow away the rest of the sun or destroy earth (which we don't want to do)?
Transmute the material of the sun into elements that won't undergo fusion (at the sun's mass). The sun is of course already naturally doing this; the supervillain just makes it go faster and suppresses the resulting energy or shunts it elsewhere. Maybe elsewhere could be the supervillain's space station at the L3 point on the far side of the sun.
Alter the laws of nuclear physics so that hydrogen fusion does not release energy, i.e., make hydrogen do the thing that happens with iron that causes a supernova.
Alter the laws of nuclear physics so that the gravitational pressure of the mass of the sun is insufficient to cause hydrogen-hydrogen nuclear fusion. Or make the reaction rate even slower (than the sun's already surprisingly low fusion rate).
Some of these could be prequels to the movie "Sunshine".
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