Megatons and milligrams

A reader writes:

In one of Alastair Reynolds' books, someone sets off a "pinhead-sized" antimatter bomb, and it explodes with a yield of about two kilotons of TNT. Is that accurate? Would you really only need that much?


I think you're talking about Revelation Space, the first book in that series, written slightly before Futurama debuted and so forgivable for its inclusion of a captain named Brannigan.

First, note that in a matter-antimatter explosion, you're not just converting the mass of the antimatter into energy. You're also converting an equal mass of matter, because if that matter was not around there'd be no annihilation and no explosion.

The energy yield of matter annihilation is a simple case of mass-energy equivalence, and thus subject to the famous e equals mc squared. Which is to say, energy in joules equals the mass being annihilated in kilograms times the square of the speed of light in metres per second.

The dominant number there is obviously c-squared; the speed of light in vacuum is 299,792,458m/s, and squaring that gives you 89,875,517,873,681,800. Or, in less-cumbersome scientific notation about 8.99E+16 - 8.99 times ten to the power of 16.

"TNT equivalent" bomb-yield numbers are tightly defined, too; one ton of TNT is defined as 4.184 gigajoules.

Now, what's a pinhead weigh?

I just grabbed some ordinary one-inch dressmakers' pins and found there were about fifteen whole pins to the gram. I'm not about to snip off enough pinheads to get them to add up to the minimum resolution of my triple-beam balance, but I'd guess the mass of these pins' heads to be ten milligrams, at most.

Fortunately, the mass of the Revelation Space bomb is mentioned in the book; it's described as containing "only a twentieth of a gramme of antilithium". That's fifty milligrams, but that doesn't sound like a crazy weight for the head of a stouter pin than the ones I weighed.

Plugging fifty milligrams, 0.00005 kilograms, into e=mc^2 gives

e = 0.00005 * c^2

= 4.49378E+12 joules

= 4494 gigajoules

...which at 4.184 gigajoules per ton of TNT, adds up to 1.074 kilotons. Double that to take into account the matter that's annihilating with the antimatter, and you get 2.148 kilotons. Which is indeed close enough to two kilotons for horseshoes, hand grenades and tactical nuclear weapons.

The biggest thermonuclear explosion ever created by humans, the immense and impractical Soviet "Tsar Bomba", had a possible yield of about 100 megatons, but was dialled down to 50. 50 megatons at 4.184 gigajoules per ton is 2.092E+17 joules. Turning e=mc^2 around to solve for mass, m = e/c^2, gives:

m = 2.092E+17 / c^2

= 2.33 kilograms of matter converted into energy, for the biggest bomb we've ever made, and possibly the biggest bomb we ever will make.

Around the weight of a healthy adult chihuahua.

(See also solid blocks of electrons, which knock antimatter energy density into a cocked hat and which may be a technology within the reach of some entities in the Revelation Space universe. Oh, and see also, also, the fun you could have whacking lumps of plutonium together by hand.)

Psycho Science is a regular feature here. Ask me your science questions, and I'll answer them. Probably.

And then commenters will, I hope, correct at least the most obvious flaws in my answer.

2 Responses to “Megatons and milligrams”

  1. max00 Says:

    I imagine the astrophysics PhD was careful to check his numbers.

    He was probably least careful about an incredibly literal pinhead, given that even a couple dozen grams is pretty much a pinhead compared to a few million kilograms of whatever.

  2. jam Says:

    Talking of fission-by-hand, have you ever watched Edge of Darkness ( ?

    In the final episode a guy does this, and it plays out pretty much as you described in that article.

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