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There is a common statement running around that we as a species has enough nuclear weapons to blow up the earth several times over. What I want to know is: by how many orders of magnitude is that a wrong statement? (Just getting a lower bound would be fine.)

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Giving a rhetorical statement by George Carlin : "It is the greatest arrogance of mankind to think we can harm the planet. The planet's been here for a billion years and has gone through worse. Are we in our right minds to think that our tiny little weapons can mess it up?" Just a soft-point answer –  Cheeku Mar 12 '13 at 12:57
    
More than ten trillion tonnes of TNT –  user21900 Mar 12 '13 at 18:42

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up vote 9 down vote accepted

The most obvious interpretation of the phrase blow up the Earth is to dismantle it into tiny particles headed off to infinity. If you're prepared to accept this definition then the calculation is easy because it's (approximately) the gravitational binding energy for matter with the mass of the Earth falling into a sphere the size of the Earth. I say approximately because I'm ignoring the ellipticity of the Earth and the fact it's rotating, and I'm assuming it's a uniform density throughout.

The gravitational binding energy of a uniform sphere is:

$$ U = \frac{3GM^2}{5r} $$

and for the Earth this works out as 2.24 $\times$ 10$^{32}$J.

According to the Wikipedia article on nuclear weapons, the Federation of American Scientists estimates there are more than 17,000 nuclear warheads in the world as of 2012, with around 4,300 of them considered "operational", ready for use. Let's take the average yield to be one megaton (almost certainly an overestimate), which is 4.184 $\times$ 10$^{15}$J. In that case the total energy of all 17,000 bombs is about 7 $\times$ 10$^{19}$J or about a factor of 3 $\times$ 10$^{-13}$J smaller than the gravitational binding energy.

I suppose another interpretation of blow up the Earth would be to render it uninhabitable. An obvious reference point for this is the meteor collision that caused the extinction of the dinosaurs. Wikipedia estimates this as 4.2 $\times$ 10$^{23}$J, or about a factor of ten thousand greater than all the current nuclear bombs.

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You exhibit the gravitational binding energy of a sphere of uniform density. I believe the Earth will be slightly more bound than that. –  dmckee Mar 12 '13 at 15:30
    
Noted, though I doubt it makes a big dent in that factor of 10$^{-13}$ :-) –  John Rennie Mar 12 '13 at 15:43
    
I always thought that saying was referring to evenly distributing the bombs across the land area of the Earth, so that every point was within 50-200 kilometers of a detonation. Surely that would lead to habitability issues. –  Chris White Mar 12 '13 at 20:05
    
@ChrisWhite I've no doubt the world stockpiles are enough to catastrophically crash civilization, and perhaps to threaten the survival of the species, but you do occasionally hear people use "blow up the Earth". It's hard to say exactly what they mean, but the OP is clearly asking about the gravitational binding energy. –  dmckee Mar 12 '13 at 20:11
    
qntm.org/data gives the same figure. –  b_jonas Mar 12 '13 at 21:17

protected by Qmechanic Mar 12 '13 at 18:51

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