atomic_fungus (atomic_fungus) wrote,

#5657: Well, less three orders of magnitude.

So, had it pointed out to me that when you're talking E=mc2 the "m" is in kilograms, so the power delivered by a 1 ug black hole vaporizing is in fact the equivalent of 21.5 kg of TNT, not 215 tons (215,000 kg) as previously stated.

So: the explosion would wreck the collision chamber of the LHC, taking it off-line for more than a few months. But it would most assuredly not mean the doom of Earth.

That's the energy released by a microgram of matter turning into energy.

Watch me use this information to calculate how much of the Hiroshima bomb's mass was converted into energy to produce its 60 terajoule yield! It turns out that it's 0.67 grams.

6.7x10^-4 kg times 9x10^16 (speed of light in meters/sec squared) is 60.3x10^12 joules...or about sixty terajoules, more or less.

That's two-thirds of a gram, converted into energy, out of a plutonium core weighing approximately 10 kilograms.

The problem with a nuclear bomb is containing the core. The longer you can contain it, the longer the chain reaction can continue, and the more energy will be released. Fat Man--the Hiroshima bomb--had an enormous amount of U238 surrounding the core simply to contain the explosion, and it was incredibly inefficient. Out of its 4700 kilogram weight, 10 kg was fissile material, and of that 10 kg not even a gram turned into energy.

At issue here is what happens during detonation. You have the plutonium core, compacted into a smaller space by the implosion. The chain reaction starts, and it gets so hot so quickly that the core is vaporized and blown apart by the energy it's emitting. The total reaction time is about thirty nanoseconds; that's from the start of the chain reaction to its end due to molar disruption. Not all of the plutonium fissions, not even most of it, because before that can happen the core is being blown apart by its own radiated energy.

On this time scale, the shock wave from the implosion moves at approximately glacial speed. It's moving at the speed of sound; once the implosion forces the core supercritical, the shock wave of the implosion simply stops mattering as the nuclear reaction proceeds at a vastly higher speed than mere chemistry can.

The entire reaction is done in 30 ns; the rest of it is just distributing energy.

You know--I find it interesting that, years and years ago, I couldn't wrap my head around things that happened that fast. I don't know how, but somewhere along the line I learned how to do it, how to change gears and think about how things work in such short time scales. It's probably just a matter of patience.

...but this kind of thing is what I love about physics: you can take a few bits of information and, by subjecting them to the right analysis, understand how something works better than you did before. You may miss a few decimal places (as I did) but your understanding remains. Okay, when the whole "CERN built a black hole machine that will destroy the Earth!" cropped up, I knew going in that was fatuous, because the energies involved were far too low; but by learning a few things about black holes and Hawking radiation, I was able to build a test case and describe why the foretold doom of Earth can't happen.

And although my error magnified the size of the hypothetical explosion, it did not change the fact that the resulting explosion is not "an Earth-shattering 'kaboom'." 21.5 kg of high explosive is nothing to fool around with; that's enough TNT to move a lot of anything. A satchel charge, enough to destroy a medium tank, is 4 kg of high explosive. This explosion is five times bigger. Yeah. And still no danger to Earth.

So, I was wrong--but not badly, and my error did not invalidate my point. And I learned a few things, too, so in the spirit of the scientific method the way it's supposed to be applied, I'll call this one a "win".

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