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I read this article on xkcd.xom, from its spin-off blog "What If?" and that answer made me wonder.

It was mentioned in that article that a very hot but indestructible box would eventually be able to instantly superheat the air around it, which would then cause massive drafts that lift it high up into the atmosphere.

My question is, although he did mention 18.7 GW of energy inside the 1x1x1 box, how hot would a sphere have to be to do the same, and what is the relation of its size and shape to be able to superheat the air in one big burst (at 1 atm at the surface, and then I imagine it would have to fall back down into the ground)?

And how hot would a disc have to be to do the same, but remain stable? What factors in real life (aside from the massive amounts of energy required) would cause this to be unsustainable/wobbly?

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Wait, why should a disc ever be stable with this configuration? You could quite simply compare the drag coefficients, but of course the disc will rotate eventually. I don't get the expectation that that will ever change. – Alan Rominger Mar 6 '13 at 13:55
Well, if we're looking at it in theory, then wouldn't an equal and uniform application of pressure across the entire underside surface of the disc (if it's already flat) cause it to not wobble? Since the heat is coming from the disc itself, it could be conceivable. So my follow up question asks what are the things in real life that could upset this balance? A small breeze? Or a soft push? – markovchain Mar 6 '13 at 14:08
@markovchain If it were balanced perfectly, like a pencil balanced perfectly on its tip. But the slightest disturbance would cause it to start tumbling, surely. – Michael Brown Mar 6 '13 at 14:19

Whether it's a box or a sphere or a disk wouldn't make very much difference at all, although its overall size might make a bit of a difference. As the article kind-of explains, the only thing that really matters is the amount of energy coming out of the object.

The xkcd article doesn't describe the box heating the air around it so much as heating the ground, which then heats the air. The radiation coming from the object would heat the air as well, since air isn't completely transparent in the infrared, but the region of heated air surrounding it would be very large. Each of these effects would lead to very large-scale movements of air, so we're talking very strong, turbulent updrafts air that would be strong enough to lift the object, rather than a nice smooth flow. These winds would be very gusty and chaotic, so we shouldn't expect any object to be stable in them - however it's shaped, its going to be tubled around by such conditions.

The ease with which an object can catch these updrafts would depend mostly on its size. A large, light object will be better at catching them than a small, heavy one. I suppose if you had a very large disk that was rapidly rotating, it might be sort-of stable due to gyroscopic effects.

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