# What's the largest cube of some material that would not collapse under its own gravity?

Another pointless and naive physics question from me. I know that planets and larger things are spherical because of gravity, that is to say; the combined mass/size/strength of the material the object is made from means it’s impossible to diverge significantly from a sphere.

So, something about the size of Sedna or larger would necessarily end up spherical if it was made of some relatively massive material like iron. But what about polystyrene, for example? Its mass/volume is a lot less. There are probably even better materials to choose from, but is it possible to arrange polystyrene into a cube with the same volume as Sedna and not have it collapse into a sphere? How large could such a structure get? I’d like the structure to be stable for at least a couple of hundred thousand years, longer if possible (I imagine that in the very long run, random collisions would deform any structure you can imagine).

Please note that I’m talking about volume, not mass, with the assumption that “hollow” structures can be built on that scale. Now that I’m thinking about it, that would probably mean that any size “empty” cube would count as well.

ETA: Obviously, I’m positing this structure to be out in “open space”. Not on or near any planet/star/other massive object.

WAG here. I don’t think an empty cube with sides made of anything but heavy matter would collapse under its own gravity for couple of hundred thousand years out in open space at any size unless the sides were made unnecessarily thick. There are probably lots of materials that won’t hold together for 5 minutes, but not collapsing from their own gravity. It just doesn’t seem that you could get enough mass into something just a little thicker than a 2 dimensional area to have it’s own gravity cause a collapse. It might deform if it’s too flimsy. But you could make something light years across out of thin steel, and I don’t think the gravity from opposing sides would pull the sides together in just a couple of hundred thousand years.

If that’s the case, when you specify stable, and if you mean maintaining a cubic shape within some reasonable tolerance, I don’t know. Something like an aerogel I guess.

I think a solid cube would be a much more difficult problem.

What did you need this for by the way? Some kind of art thing? A tomb? Intergalactic billboard?

A need? I’m not Skald - I’m just in it for the curiosity value

ETA: And yes, I was thinking of something like aerogel - something relatively “empty” but still sturdy enough to hold its shape when banged around a bit.

Oh good! I thought you were looking for a quote.

A back of the envelope (and it is in the most literal sense perfomed back of the envelope) calculation gives me the approximate equation of:

d = side length of cube, ρ = density, S = compressive strength G = universal gravitational constant
d = sqrt[6S/(Gρ[sup]2[/sup](1 - π/6))]

plugging in typical figures for concrete I get a cube of sides of 1,280 km. Which actually seems on the low side to me, but not suprising given the simplifications I made.

So it’s a cube made out of envelopes?

Um, that’s for a solid cube right? I think you’d need tensile strength to figure out an empty cube.

ETA: Ok, based on intervening post, sounds like solid cube, not necessarily made out of envelopes.

Yes it’s a solid cube and by literally back of the enevlope I mean I did the calculation on the back on an envelope.

Infact I think I mispoke, the figure for a concrete cube doesn’t sound too far off or if anything a little too large. According to wikipedia the largest object in the solar system not in hyrdostatic equilibrium has a diamter of 532 km.

Looking at the solar system, the smallest object known to be gravitationally rounded is the moon Mimas, while the largest irregular object is the slightly larger moon Proteus. You might want to check your calculations against these real-world examples. Oddly, Proteus has a higher density, but it’s thought that Mimas formed at a higher temperature, giving it greater plasticity.

That does sound pretty small - even for a solid heavy object - then again, a cube with a 500 km diameter would be pretty amazing to watch. Still waiting for someone to give some estimates for a polystyrene/aerogel cube though.

Comparing rela-life examples it seems I’m certainly in the right ballpark.

I make a cube of polystyrene to have a maximum length of about 121,000 km under my approximate formula.

Pure aerogel?
Missed Superfluous Parentheses’s reference.

Is that the diameter or the width of a side? In any case, it’s pretty damn big.

I think a big factor is heat history. Hot things deform easier than cold. Hot polystyrene deforms easily. Cold is more rigid. How about a foam made out of titanium? I don’t have to pay for it do I? Or even figure out how to make it?

No. He cleared that up before. Something to do with Skald I think.

4 Vesta at 529 km (mean diameter) is the largest out-of-round object in the Solar System. Most of its out-of-roundness is due to a large crater at the south pole, and the rest is thought to be round due to hydrostatic equilibrium. If that’s the case, then it must have been in equilibrium when it formed and was probablly hotter than now. If it were still in equilibrium, that crater would have slumped and filled in.

Impressive, pants. When I do a back-of-the-envelope approximation, I usually leave out the pis and other dimensionless constants.

You don’t just say pi=3? Aren’t you off by orders of magnitude commonly?

Yep I don’t think the 1-pi/6 term really adds much given the approximate nature!

Though in this case I couldn’t be 100% sure there wasn’t a pi^3 or simlair term lurking in there until I divided everything out.