This post expands on concepts introduced in a previous post about living in the voids in asteroids, and it is the writeup of the fractal image I presented in the post right before this one. I've come to call this scenario "under-pressure" because it entails using a pressure below what's necessary for the rocks to fully float against the atmosphere pressure. In the present context, the gravitational balance is really the upper limit to the pressure and radial extent for using the voids in the way I'm talking about here.

In that previous point I laid out a very vague idea of how a collection of smaller rocks (within a lattice of larger rocks) might form the backstop for the airtight lining. I'll look into this with a little more detail in this post. Rubble piles are likely to be a collection of (former) asteroids which are resting in contact with each other, held by gravity. We don't know much about these structures yet, but the general idea is something like this, from a 1999 Nature article, "Survival of the weakest"

As the different scenarios in the above image suggest, this illustration is essentially a guess at the interior of asteroids. The constituent parts could be small, large, or a combination of them all. Furthermore, there are a large number of candidate asteroids. There could be a great diversity in their different interiors.

In order to build a space habitat in a cavity within one of these rubble piles, I would imagine that the ideal interior structure would be composed of rocks on the 1 km scale. This is approximately the minimum scale needed to produce artificial gravity at acceptably low rotation rates. It is reasonable to believe that an ideal candidate asteroid exists out there somewhere, although we can't say which asteroid that is. It's certain that many of them have unusable interiors for this type of habitat, so the problem is just narrowing down the sample space.

Another issue is how one might

**seal**off the pressure boundary around the empty space to be transformed into a habitat. If you plan to ship in material to cover a kilometer scale gap, then you've obviously defeated the impetus of the concept altogether. We are obligated to think of approaches more clever than that.

Lattice Structure of Unmovable Rocks

As a best guess, we might as well imagine the interior to be some lattice structure. We might also imagine that the rubble is roughly spherical rock. The 3D packing of spheres is a well-studied problem, particularly in material science. There are several configurations that an infinite lattice of this sort might have, but the most efficient structures are only 2, which are "face centered cubic" and "hexagonal closed packed". For the purposes of this study, there is no reason to distinguish between them. They are different geometries, but the important numbers such as coordination number and packing density are all the same. The FCC structure looks like:

packing density = 0.740

Compare this packing density to the measured asteroid macro-porosity values. You can see that asteroid porosity is all over the place, but seems to be roughly constrained by the FCC density as a maximum.

This makes a lot of sense. Since FCC is has the most efficient packing density, and we expect that not all asteroids have differing degrees of porosity. It also makes it clear that if you want to find the kind of ideal body I'm referencing, you really will have to cherry pick. Of course, the above data set isn't comprehensive either, so there's plenty of space to find a body which approximates an FCC structure of the desired size.

Movable Rock for Boundary Formation

So we have our perfect candidate, what now? The habitat (or at least the early version of it) will go in the interstitial space. That space is an odd one, because it doesn't start out enclosed. That leaves the builders with the burden of enclosing it. This is a self-obvious reality of the gravity balloon idea. If no excavation is necessary to get to the center, they the center obviously shouldn't be expected to have a well-defined cavity. From there, we have to make it into a well-defined empty space. That means sealing off a boundary within this contiguous interstitial space. That would be very difficult if you used manufactured materials. In fact, it would defeat the entire point.

The obvious solution is to seal off the surrounding pathways between the rocks with other rocks. Refer to the FCC diagram. The interstitial space has 4 close-by rocks, for which the centers are arranged like a tetrahedron. Looking toward one "exit" of this space, we see 3 rocks arranged in a triangle. This strange triangle space is what needs to be blocked. It's not hard to set a minimum amount of rock needed to accomplish this, because we could just fill it in in 2D. I took a fractal approach:

Specifically, I used a program to place circles in the empty spaces between the previous level of circles. This has an obvious branching of 1->3. With each level you'll increase the number of circles you're adding by a factor of 3, but not all of these will be the same size circle. Because of this, I found it easier to add circles in order of their size. Then, using the handy svg markup, place them on a palate. The presentation of this graphic does falter a little bit, because the circles can only be specified by integer values for their location and radius. This causes some of the smaller ones to be displaced a bit from where they should be.

This is useful because it provides a long list (as long as you want it) of circles that allows us to map the connections between volume and area. Importantly, this theoretically lifts the requirement for the liner which hermetically seals the air inside the habitat to have any material strength as the number of rocks diverge to infinity. In practice, the areas very quickly fall off to imperceptibly small values, which is expected in these fractal scenarios.

But now how do we resolve the problem of carrying this out in the real world? In the previous illustration you would have to commit more resources to secure these rocks in place. In practice, you would prefer to use a rock which is larger than than the hole, and sufficiently large so that its compressive strength holds up against the atmospheric pressure. This is a function of the contact angle and more complicated structural engineering, which I will not get into. For a vague idea of what we would do, I just multiplied the size of all the circles by a constant factor, which would seem to be mathematically sound. So collapsed to a 2D view, it would be something like this:

Now returning to the volume vs. area correlation, we can produce some graphs from the realistic over-sized rock scenario. It will then tell us essentially what volume of rock we need to move into place, based on the assumption that we can find any size of perfectly spherical rock just laying around inside the asteroid voids.

Obviously that last assumption was stretching, so let me talk about where I intended to go with this.

Blasting or Cutting or Shattering?

Originally, I had wanted to get a figure in terms of tons TNT that would be needed to blast rock in order to create the size distribution of rock that I worked with in the above figures and graphs. However, once it was finished, it became apparent that this was a fool's errand. Firstly, how the heck do we quantify a metric for mass of TNT per fracture area of rock? Actually I found some useful references for this (pdf link).

All kinds of variables are involved, but you can still get a metric for mass of TNT per square foot of the fracture you're creating. At first, the reference gives mass of TNT per foot of borehole, but this is a function of the spacing of the boreholes and the burden above it. Obviously, combining the linear mass density of TNT of the borehole with the distance between the boreholes gives a area mass density over the fracture area. But what exactly should those parameters be? The overburden is completely non-applicable in the space application we're looking at.

Nonetheless, I picked some parameters just to see. My estimate was about 12 kg TNT / m

^{2}of fracture. However, if you used this to cut all of the needed rock, you will quickly find that you'll be sending more explosives than what you would otherwise be sending as a pressure vessel to cover a similar volume. This would make the proposal absurd and clearly uneconomic.

Are there other ways to fashion rocks in the way you want without explosives? I picture space probes with chainsaw-like things attached. This would have complications too, since most of their cutting would be done on the center piece, they would either need really long cutters or they would need to excavate a very wide fracture so they could fit into it. Alternatively, a more simple and primitive method may suffice. Why not just bang rocks together in order to break them up? Nothing about this proposal actually required a well-controlled shape. So sure, why not? Although this assumes movable large monolithic rocks available to begin with. Plus, how do you shatter the largest rock if it's the biggest one around?

Finding out how much area needed to be cut turned out to be disappointingly easy. This is because the 2D cross section is a straightforward multiple of the spheres that would occupy it. It's just 4 Pi r

^{2}/ 2 divided by Pi r

^{2}, you get 2. But now what is the total area that we're dealing with in the first place? This requires the advanced science of calculating the area of an equilateral triangle.

For reference, the ballpark figure I'm looking at is D=1 km. Maybe more. In order to get the area of the rocks with the slight over-sized method I mentioned previously, you'd multiply it by that over-sizing factor. Again, none of this was particularly difficult or insightful, and it's not clear what the application would be in the first place, since smashing rocks together doesn't have an easily quantifiable cost.

Inside-out Regolith, Carried by a Million Robots?

In spite of all this, we're still not lacking for an argument that voids can be sealed with a hierarchical distribution of rocks. Why? Because nature already did it. We are relatively confident that many asteroids have large voids inside, and we are also confident that they have something kind of like a contiguous (even if dusty) surface on the outside. At first these facts seem contradictory. But it makes sense when you consider that 1) first large rocks congeal into a pile and then 2) smaller rocks continue to fall onto this pile. While some small rocks will fall between the larger rocks, some will get stuck. As more and more get suck, the probability that the next rocks that fall onto it will get stuck increases, up until the point when the surface is effectively sealed. Not sealed against gases by any means, but against particles falling on the asteroid.

As a bit of a troll argument, you could simply rearrange the rocks from the outside regolith into the structure to seal a void which I have discussed here. That would require nothing more than pushing them around. But this would be a complicated low-gravity maneuver of moving many rocks around the surface and interior of an asteroid.

But we have another point to grapple with - loose material still exists within the interior of the asteroid. Hypothetically (and somewhat realistically) you could run an inventory of all the rocks around the asteroid, and chances are that you'll have a good shot at finding a sufficient size distribution to make your seal without any blasting. Some will only need to be moved a small distance, for instance, from one void into the next, but many others will need to be dragged through the byzantine path through the asteroid pores. The largest rocks will be the worst.

In fact, the very largest one, the center stone, is the problematic one here. By the very fact that it can't FIT out of the void's entrance-way, we would expect that it can't fit INSIDE in the first place if we're hauling it from somewhere else. This isn't strictly geometrically true, but it's close enough. This leads to a dilemma. Maybe you cut the center-stone, and then glue the pieces back together. Thankfully, no glue is strictly required. If the forces balance correctly, then it can just "sit" there, held by the outward force from the habitat's pressure. I'm not very concerned about that detail.

The only real problem is that you'll have to move a lot of rocks. Exactly how many depends on the convergence between the price of moving stuff and the strength of the pressure liner. I would assume that the robots would move non-propulsive in a method similar to lead climbing in the outdoor pastime of recreational climbing. By attaching anchors and ropes at various points, there should be enough degrees of freedom to get where you want to go.

A Big Dump Truck of Tiny Qualifiers

I don't think this discussion is pointless. I think there is a compelling argument, but there are far more unknowns than what there are certainties. Also, an analytical approach isn't nearly as helpful as what I had hoped.

But none of this suggests that we should toss the idea out. We don't know if this method will be viable, but our best understanding lightly suggests it will work. This is still saying nothing of the vastly complex structural engineering problem that comes along with potential re-seating of the structural rocks. Once again, this doesn't appear as a game changer to me, it just seems like a tremendously complicated analytical task.

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