That's no moon! It's a space station! How big can a space ship be before it collapses on itself?

Question

Now a solid body of rock will collapse itself to a round shape when it hits about 600 km in diameter (400 km for ice). Now, the Second Death Star is estimated to be between 160 and 900 km. How big can a space ship be made of metal but still with "large open" living spaces. I assume it would still be filled with gases which would have its own gravity. Can a a spaceship be much larger than a 600 km sphere? To do so, would the infrastructure need to be primarily aluminum?

Adding from the comments.

a Dyson Sphere doesn't count, it should have internal structure.

Metal was suggested, but any material strong enough to build the self same craft is allowed.

Answer 1

The defining equation of hydrostatic equilibrium - the state a celestial body must be in to maintain some semblance of a spherical shape - is $$\frac{dP}{dr}=-\frac{GM(r)\rho(r)}{r^2}$$ where $P$ is pressure, $r$ is radius, $M$ is mass, $\rho$ is density, and $G$ is the universal gravitational constant. Assuming constant density here - which is actually a problem because there are gaps - we say that $$\frac{d\rho}{dr}=0$$ and, after a quick derivation (see here for an example), we find $$P(r)=\frac{2\pi G\rho^2}{3}\left(R^2-r^2\right)$$ where $R$ is the radius of the body. At $r=0$, we have $$P(0)=\frac{2\pi GR^2\rho^2}{3}$$ Given that $$\frac{dP}{dr}<0$$ it is clear that $P$ is at a maximum at $r=0$. Re-arranging, we have $$R=\frac{1}{\rho}\sqrt{\frac{3P(0)}{2\pi G}}$$ $$$$ For $R$ to be maximized, we want the ratio $\frac{\sqrt{P(0)}}{\rho}$ to be maximized. We can say that $P(0)$ is the ultimate compression strength of a material.

Let's take a look at the strengths of various materials. The metal with the highest ratio is pre-stressed steel, at $$R=\frac{\sqrt{3,757,000,000}}{{1440}}\cdot\sqrt{\frac{3}{2\pi G}}=3,600\text{ kilometers}$$ That sounds pretty good to me.

Answer 2

Going a slightly different direction from HDE226868, I'm going to design my ship to be as big of a sphere as possible. To do this, I'm going to put all of the living space on the outer surface of a big hollow steel sphere full of vacuum.

I'm going to have a lot more steel sphere, mass wise, per square meter than I will living accomodations on the outside of it, so my question essentially becomes this: how big can I make a hollow steel sphere before it is crushed by its own gravity? Now it's time for equations.

Gravity

$g = GM_{sphere}/r^2$

Where $g$ is acceleration due to gravity, $G$ is the gravitational constant, $M_{sphere}$ is the mass of the sphere, and $r$ is the radius of the sphere.

Mass of sphere

$M_{sphere}=4\rho\pi r^2t$

Where $t$ is the thickness of the sphere and $\rho$ is the density of steel.

Pressure on the sphere

$p = g\rho t$

This is a conservative estimate, since only the outermost portion of the sphere actually feels the full weight of its gravity. The actual pressure involves solving a simple integral that I don't feel like doing right now .

Stress

$\sigma = pr/2t$

This is the equation for stress in a thin walled pressure vessel.

Final equation

Putting this all together, we get:

$\sigma = 4\pi G{\rho}^2r^3t^2/r^2t $

Or, simplified and solved for $r$,

$r = \frac{\sigma}{4\pi t G{\rho}^2}$

Plugging in values for the density of steel (8000$kg/m^3$), the ultimate stress of steel (3,757,000,000), and G ($6.67408 \times 10^{-11}$), we get a total maximum size of around 70,030,000km, given a thickness of 1m. Our ship's radius is inversely proportional to its thickness, so we can make it bigger if we make it thinner.

Of course, our giant sphere-ship will only be able to lurk about in deep space. Tidal Forces (Differences in the force of gravity between one side of the ship and the other) would destroy it if it came close to a large body like a planet or a star.

Answer 3

Even though they are not solid, does the concept of a Dyson Sphere fit within your question?

http://www.technologyreview.com/view/536171/physicists-describe-new-class-of-dyson-sphere/

Ahh, seeing the response...

It seems that most superstructure mega ships in theory have to deal not only with their own gravity of the structure, but with creating it for the inhabitants. I could see upwards of 900km depending on the solutions of internal and structural strain. The sphere of course comes to mind as the collapsed and near rest structure, one way of dealing with the stresses of gravity is to create pockets of open space that would essentially reduce the over all gravity stress as it's reduced by the open space.

I believe there has been discussion around these concepts on http://hieroglyph.asu.edu/ but I cannot search it right now.