What If? Wyrmworld Style!

Not only is XKCD a wonderfully enjoyable webcomic, but every Thursday its author, Randal Munroe, answers crazy physics problems submitted by readers in his What If? section.

For a while now, I’ve been trying to think of something to send in, and just recently came up with one. But then I realised it was a question that I was perfectly capable of answering myself, if I got off my arse and did some research and some maths. So I did.

If all the excess carbon released into the atmosphere since the start of the industrial revolution was compressed into a sphere of pure diamond, how big would it be, and if it were placed into orbit would it focus a death ray of concentrated sunlight down onto the planet?

First step, how much carbon has been added to the atmosphere? According to Wikipedia about 12 Gigatons was released from 1751 to 1900, then a further 334 Gigatons from 1900 to 2008. Adding these together comes to 346 Gigatons, which is as good a figure as any. (It’s important to note that this is just the carbon – not the carbon dioxide containing the carbon. If it were the carbon dioxide we’d have to divide the weight by 3.67 to get just the weight of the carbon.)

The next step is to determine the weight to volume ratio of diamond, so we can figure out how much space 346 Gigatons of carbon would take up when arranged into its crystaline form. Some more poking around online provides a density figure for diamond of 3.52 grams per cubic centimetre. There are 1,000,000 cubic centimetres to a cubic metre and 1,000,000 grams in a ton, so the maths is nice and simple (gotta love the metric system) telling us that 1 cubic metre of diamond weighs 3.52 tons.

To get a volume for our 356 Gigatons of diamond we simply need to divide 346,000,000,000 tons by 3.52 tons – which leaves us with a volume of 98,295,454,545.45455 cubic metres, or 98.29545454545455 cubic kilometres.

So we now know just how much space our chunk of diamond takes up, but so far it’s just sitting around in a roughly shaped blob. We need to reshape it into a sphere.

The formula for the volume of a sphere is v = (4/3)πr^3, where r is the radius of said sphere. Turning this inside out we can derive r = (3v/4π)^(1/3). Plugging the volume figure in gives us a radius of 2.86296 kilometres. Doubling this for the diameter gives us sphere of pure diamond 5.72592 kilometres across – roughly the distance from New York City’s Battery Park to 33rd Street or from London’s Tower Bridge to the cafe in Hyde Park.

That’s one big diamond.

On to the second part of the question – would this diamond project a death ray? To figure this out we need to discover the focal length of the sphere – that is the distance from its centre to its focal point – the point where the light passing through the sphere is focused. The formula for this is pretty simple – EFL = nD/4(n-1) where EFL is Effective Focal Length, n is the refractive index of the material the sphere is made from, and D is the diameter of the sphere. We already know the diameter and Wikipedia assures us the refractive index of  diamond is 2.419. Solving the equation gives us a focal length of… 2.440274926004228 kilometres. Wut?

Yes folks! It turns out that the focal length of a sphere made of diamond is always less that its radius, meaning that the focal point is always inside the sphere! No death ray for you!

So in conclusion, if you could pull all the excess carbon out of the atmosphere, turn it into a diamond and launch it into orbit you would save the planet’s climate, but you couldn’t use it blackmail major population centres. Hardly seems worth it does it? 🙂

(yes, yes, you could shape the diamond into a lens instead and blackmail all the cities you want, but the maths required is just horrible ;))

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