Size limits of a hollow planet.

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Sockmonkey
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Size limits of a hollow planet.

Postby Sockmonkey » Mon Dec 30, 2013 8:54 pm UTC

In theory, if one were to construct a hollow iron sphere with the goal of making a planetoid with a surface gravity as close to one Gee as possible, how large and massive could you make it before it collapsed?

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yurell
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Re: Size limits of a hollow planet.

Postby yurell » Tue Dec 31, 2013 8:14 am UTC

How thick can it be before it's no longer hollow?
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Re: Size limits of a hollow planet.

Postby Tass » Tue Dec 31, 2013 9:59 am UTC

Sockmonkey wrote:In theory, if one were to construct a hollow iron sphere with the goal of making a planetoid with a surface gravity as close to one Gee as possible, how large and massive could you make it before it collapsed?


With a vacuum inside, you couldn't. Anything big enough to produce one gee would collapse under its own weight.

With hydrogen inside to provide pressure to lift the shell, you could. However, when you go too big it starts to not really be "hollow" in any sense of the word, since the gas would compress under its own weight.

Saturn has a surface gravity of about one gee, but enclosing it in an iron "balloon" would hardly make it a hollow planet. It would still be awesome, though.

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Re: Size limits of a hollow planet.

Postby lgw » Tue Dec 31, 2013 7:09 pm UTC

yurell wrote:How thick can it be before it's no longer hollow?


That's really the question. Long before you get to one g, the iron will make its way to the center, likely as a liquid. Under enough pressure it will be solid again, but you need a planet more or less the size of the Earth to manage that. Iron is about 50% denser than the Earth, so a solid iron ball with 1g surface gravity would be 2/3s the Earth's radius (surface gravity is directly proportional to density and radius). I'm not sure where the triple point of iron is, but I suspect you'd still have a liquid mantel and a solid core, even if it were constructed with no heat of settling to warm the core. (And so would probably be a bit denser on average.)

Could you have a small hollow space, or several bubbles, in the core - likely, given the technology to make the thing in the first place, but they'd be quite small in comparison. If the Earth's core had a bubble, would that make Earth a hollow planet?
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Sockmonkey
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Re: Size limits of a hollow planet.

Postby Sockmonkey » Tue Dec 31, 2013 9:49 pm UTC

Yep, I'm thinking in terms of a construct. How high could you have the gravity before it started to sag? Lunar gravity? I know that if the diameter is big enough, being hollow means that a lot of the mass will be far enough away from the rest of the mass that opposite sides of the planet won't pull on each other very much.
The main thing I'm concerned about is being able to hold on to an atmosphere of some sort.

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Re: Size limits of a hollow planet.

Postby eternauta3k » Wed Jan 01, 2014 12:04 am UTC

We need to know how a sheet's crumple resistance scales depending on curvature and thickness.
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Re: Size limits of a hollow planet.

Postby Tass » Thu Jan 02, 2014 9:58 am UTC

Okay, lets say that we can prevent sheet buckling by a complicated fractal trussed structure. Then we only need to worry about the compressive strength of the material.

If you simply double all linear dimensions then mass increased by a factor of eight. Since radius is increased by a factor two and weight by a factor eight, gravity is doubled (m/r^2), and in turn the weight is increased by a factor of sixteen. But the strength to resist this weight is only increased by a factor of four. We need to reduce the weight of this up scaled shell. By pulling out 3/4 of the material we reduce gravity by a factor four, down to half what it was before we scaled, weight and strength is now the same.

So you need to keep it small in order to get gravity high. However you specifically asked for the gravity to retain an atmosphere. What determines that is escape velocity it scales with gravity but also with radius. When we doubled radius we halved gravity so escape velocity stays the same. The escape velocity you can achieve with a hollow shell is simply a function of the material you are using, similar to breaking length. Unfortunately iron does not have the required specific compressive strength to hold on to an atmosphere for any considerable time, nor does any other known material.

If you want a planet big enough to retain an atmosphere then you need something providing hydrostatic pressure inside. On earth that is done by rock and iron at pressures above what they can hold except when it is hydrostatic (no bubbles). If you want your planet bigger then you need a lighter pressurant. Hydrogen would be best. There is nothing theoretical to prevent a Saturn encased in a balloon-like structure upon which you could walk build and breathe. Then you'd have a planet with a surface 83 times as large as Earths, with the same gravity. But it would not be hollow, dig down and you would not fall into a vacuum, but rather release a hydrogen geyser.

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Re: Size limits of a hollow planet.

Postby lgw » Thu Jan 02, 2014 7:35 pm UTC

Again, what counts as "hollow"?

Let's say we're aiming for surface gravity about like Mars - that would retain an atmosphere for a little while.
  • An iron ball roughly 1600 kilometers in radius would have a similar gravity to Mars.
  • A radius of 2000 km and a ~40% thickness of the shell, or 800 km thick, gives about the same gravity.
  • A radius of 2400 km and a ~30% thickness of the shell, or 720 km thick, works.
  • A radius of 3200 km and a ~20% thickness, or 640 km thick, also works.
  • As the sphere grows from there, the thickness needed remains basically the same, (I believe an infinite plane would be about that thickness for Mars gravity, and would retain atmosphere really well!).

I doubt this is what you had in mind. It would take a non-trivial percentage of all the iron in the Earth to make this work. However, I suspect that at these pressures the iron would remain solid, even at the likely extreme temperatures. Not sure what you'd do about the temperature increase that would naturally occur from the heat of settling: it would be a challenge to cool the interior before you lost the atmosphere, but given magical technology, why not?

ETA: So how many orders of magnitude is that beyond what the structural strength of iron will support?
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Re: Size limits of a hollow planet.

Postby Tass » Thu Jan 02, 2014 8:40 pm UTC

lgw wrote:Let's say we're aiming for surface gravity about like Mars - that would retain an atmosphere for a little while.


Again, atmospheric escape depends on escape velocity, not surface gravity.

Edit:

lgw wrote:As the sphere grows from there, the thickness needed remains basically the same, (I believe an infinite plane would be about that thickness for Mars gravity, and would retain atmosphere really well!).


That is true. (It would retain it infinitely well with an infinite escape velocity. It would actually be a black hole.)

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Re: Size limits of a hollow planet.

Postby Sockmonkey » Fri Jan 03, 2014 12:37 am UTC

So, with the ability to hold an atmosphere we can make the interior livable without worrying about leaks and the inside would be an awesome null-G place to do cool stuff.

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Re: Size limits of a hollow planet.

Postby Schrollini » Fri Jan 03, 2014 6:03 pm UTC

Surprisingly, a hollow iron sphere with a surface gravity of 1g isn't completely ruled out in a simple thin-shell analysis, although it does require a rather liberal interpretation of "thin".

First, as lgw points out, the surface gravity of a thin shell depends only on the shell's thickness h, not its radius R. The surface gravity is g = GM/R2, but the mass of the shell is M = 4πR2h, so g = 4πGhρ. Using the Earth's gravity, I get h = 1481 km.

I don't know anything about the stability of self-gravitating spheres, but the stability of spheres under pressure is well-understood. This paper by Carlson, Sendelbeck, and Hoff quotes a theoretical critical overpressure of pc = 2E/√[3 (1-ν2)] (h/R)2, and shows that experiments can reach over 80% of that value. (E is the Young's modulus, and ν is the Poisson ratio.) In our case, the "pressure" is actually the force per area of gravity, ρgh/2. (The 1/2 comes from the force growing from 0 at the inner edge to g on the outer edge.) Thus, for stability, we must have

ρgh/2 ≤ pc = 2E/√[3 (1-ν2)] (h/R)2

Using the above expression for g, this reduces to

πGρ2E/√[3 (1-ν2)] (1/R)2

This leads to the somewhat surprising conclusion that the stability of this shell does not depend on the thickness! Instead, stability will occur when

R2E/(πGρ2 √[3 (1-ν2)])

I imagine that the modulus of iron may depend on the type of iron used. Wikipedia cites a value of 211 GPa. I'm just going to use that. It also gives a Poisson ratio of 0.29. Together, these tell me that for stability, our planet must have R ≤ 3129 km. This suggests that we could have a hollow planet, albeit one with the void half the radius, and thus one eighth the volume, of the planet.

Edit: Actually, the radius is to the midplane of the shell, so the inner radius of the shell would be about 2400 km and the outer radius about 3900 km. The void would be roughly 1/4 of the total volume.

Of course, working at h = R/2 means that all of those thin-shell approximations I've been making aren't particularly valid. I'm also ignoring the fact that the large stresses within the shell will undoubtedly push the material out of its linear elastic regime, if not liquefy it completely. If you try building your own hollow planet, I'd allow for a safety factor of 10, and build it with a radius of 300 km and a thickness of 1481 km, for a void radius of -440 km. This is just a minor architectural problem.
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Re: Size limits of a hollow planet.

Postby PolakoVoador » Mon Jan 06, 2014 12:05 pm UTC

Surely the guys at Magrathea can work this out.


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