Thursday, July 4, 2013

Space Colonies Inside Irregular Asteroids

Any asteroids within the size-range suitable for a early gravity balloon colony would likely be highly irregular.  That is to say:
  1. the body exhibits a non-spherical shape, like a blob
  2. its rock has some amount of compressive strength
This was established with my last post, where you can see that basically any body with a natural internal pressure of 0.3 to 10 Earth atmospheres is likely to be irregular.  Those are also the internal pressures you need to use it as a colony.  Now, I'm using the term internal pressure, but so far I'm only referring to the contribution of self-gravity to internal pressure.  It could be more or less according to how the asteroid formed and what kind of internal forces it experiences as a result.  In this post I want to talk about how you would manage the most obvious kinds of internal forces for an early gravity balloon colony.

Irregular Asteroid Stresses

Call the largest dimension the polar axis.  This is chosen because it is the axis of symmetry.  In that case of planets, the axis of symmetry is typically the smallest one because they are roughly oblate spheroids, whereas 10-km scale irregular asteroids are more like a prolate spheroid.  In other words, this is the geometric model I'm using:

My terminology:
large dimension = polar axis

This doesn't look very far off for many of the asteroids we've photographed.  Eros, in particular, among others.  With this in mind, we can consider the stresses that he object's own self-gravity will cause on its interior.  For a mental model, this is somewhat similar to a building on Earth.  In order to "stick up", it has to have some strength.  The only difference between an asteroid sticking out as a prolate spheroid and a building standing up on the surface of Earth, is that the asteroid deviates from a spherical shape, while the building deviates from the flat surface of the ground.

This protrusion will only lead to compressive stress here.  After all, it doesn't have to stand up to the wind or any other dynamic forces.  If you want a rough formula for exactly how much compressive stress, then consider the gravitational head.  For a bad approximation, imagine that the gravitational field only varies with radius.  Then it's obvious that the equatorial radius is smaller than the polar radius.  That leads to an inconsistency if you imagine the object material is fluid-like.  You obtain a different pressure if you measure the elevation drop from the equator surface to the center versus if you measured from the polar surface to the center.  It is precisely the difference between these two pressure that is the non-isotropic force, or the compressive force.  It has the same units as pressure, because these are both elements within the stess tensor - the bread and butter of civil engineering.  After all, the entire proposal basically comes down to civil engineering.

For some math, you can imagine that this stress will be approximately equal to the gravity on the surface of the asteroid (which is an average figure itself) times the elevation difference at the equator and the pole.  Without going into details here, you can get a factor of 2/5 to add onto this.  This isn't perfect, but it's pretty good.

$$\sigma_z \approx \frac{2}{5} \rho g \left( R_x - R_z \right)$$

To illustrate the occurrence of this compressive force, I've used some arrows here.  Imagine that a pressure without any material stress would entail 4 arrows from all directions in this 2D approximation.  In reality pressure acts all around you.  So instead of that, we have some preferential direction where the pressure squeezes from only two sides.  I did my best to illustrate that for the prolate spheroid shape I'm talking about here.  This is briefly representative for the natural state of the asteroids I'm talking about.

You can easily extend the idea to a hole in the center.  About the same amount of net force will be present over a cross section at the equator.  If you drill a hole in the center, that means that there's less area over which to distribute this force.  Logically, that means that the stress would be intensified by the presence of this hole.

There are some finer points to this argument - mainly that in the above model the bubble would have to have some internal pressure.  This is what we're talking about for the gravity balloon.  Specifically, the pressure would have to be exactly enough such that the top and the bottom of the hole in the above image wouldn't experience any compressive stress.  You could change that with a different pressure inside the bubble.  If you increased the pressure in the bubble enough you would induce tensile stress - stress that tears the material apart, not pushes it together.  I have assumed that the pressure set-point would be carefully managed to keep all stresses compressive.

Why would you want to do this?  Because of the failure mechanisms associated with breaking.  If you compressive force doesn't hold up, then you could see some material rearrangement - just like if you had built your sandcastle too high.  That might still not result in loss of atmosphere.  Particularly if you had added some membrane to keep the atmosphere from diffusing into the rock to begin with.  There still remains a danger that whatever material rearrangement happens would destroy some part of that membrane, but it's still a smaller danger compared to failure of tensile stress holding the atmosphere in.  If you relied on the tensile stress of the asteroid, you would risk a catastrophic loss of atmosphere.  This is the same sort of event we concern ourselves with the international space station, or any similar design.  Breaks are generally fatal.

Impact of a Deformed Central Bubble

Similar thought experiments can be used for imagining that a small bubble is deformed in the shape of a prolate spheroid.  Start with the assumption that tensile forces are unacceptable.  Then imagine that we deviate from the spherical shape that I've always talked about for the inner bubble of air.  If you do this, that is effectively adding material around the equator region and subtracting it from the polar regions.

In doing this, we introduce a quadrupole moment.  This behaves as you would expect from a quadrupole field:

generic example of a quadrupole field

But in the case of a gravity balloon, we can only allow compressive stress, by adjusting the pressure of the air bubble lower.  Start out drawing the gravitational field lines from the quadrupole gravitational moment.  Then, due to the compressive stress argument, draw a compressive stress in all places where two arrows point toward each other.  This is what I've done in the image below.  You still have to use your imagination to think of this being an prolate spheroid.

There is an obvious utility to this - because the compressive stresses are in the opposite directions to the stress from the asteroid shape itself.  That means that an odd shaped bubble may be used to less the compressive forces within the interior of the asteroid.  That is exactly what you would do for management of interior forces, building a primitive gravity balloon colony.

In short, the combination of the two above stress diagrams gives a result that is overwhelmingly balanced back to isotropic forces.  That is, just pressure, no material stresses.  Like illustrated below.

gravitational balloon designed to relieve
stress within the asteroid rock

There's no reason to believe this could be done perfectly, but I haven't done the calculations.  I imagine it would be quite an involved project to do so.  Even as we imagine there will be some residual forces, it needs to be considered what the criteria is.  There seems to be every reason to believe that you could start with an asteroid like Eros and establish a colony with breathable air, several kilometers in diameter, all the while keeping the asteroid internal forces less than its natural state.

You would not want to stress the asteroid more than its natural state, because that will give you a virtual guarantee that loss of atmosphere will not happen.  That's the type of guarantee needed to have people seriously consider moving there, and that's why a gravity balloon is a competitive concept for space colonies.

Irregularity of Asteroids by Mass

The very fact that an asteroid is non-spherical proves conclusively that it has some material strength.  It would then be tempting to use its material to hold in an atmosphere, literally with no processing at all.  This would be cheap, but it would also be potentially dangerous as well as unnecessary.  The reason lies in a mass-scale argument.

Looking at what we know about asteroids, we can find that the size cutoff at which most bodies appear highly spherical is fairly close to the point where their internal pressure approaches 1 Earth atmosphere.  Because of this, it would be tempting to imagine that an asteroid's material strength may be nearly sufficient to maintain an atmosphere.  There are a few hazards with that argument, but the main takeaway is that it's not necessary.  Basically, self-gravitation is more useful than tensile strength because we have no reason to believe that an asteroid's natural tensile strength is reliable.  There are more complicated arguments involving the role of compressive strength, but I'll get into those with a later post.

Abundance of Irregular Asteroids by Pictures

Here are a few examples of asteroids at different scales that we have pictures of.  There is a combined imgage out there which preserves the length scale, although this isn't as useful as just looking at the images side-by-side, since I'm interested in their degree of irregularity.

PictureNameM (kg)Center Pressure
due to self-gravity
(in Earth atmospheres)

1 Ceres9.4 x 1020221spherical

4 Vesta2.6 x 1020185borderline

21 Lutetia1.7 x 10186.3irregular

253 Mathilde1 x 10170.27somewhat

243 Ida4.0 x 10160.36irregular

951 Gaspra2.0 x 10160.24irregular

433 Eros7.0 x 10150.12irregular

2867 Steins1.0 x 10140.0032irregular

4179 Toutatis5.0 x 10130.0027somewhat

25143 Itokawa3.0 x 10102.0 x 10-5irregular

There is somewhat a deficit of information beyond this.  If you want to see more pictures of asteroids (like I do), you might be out of luck, because almost every one of the pictures above represents a major space exploration mission.  There's also a pronounced deficit of information for Earth-like center pressures, since the above tables skips over an order of magnitude between Mathilde and Lutetia.

Note that pressure doesn't follow directly from mass.  This is because the objects have different densities, and I used those when calculating the center pressure.  A lower density will result in a lower central pressure, even for the same total mass.  That is simply a quirk of self-gravitation.

Once we get to the extremely small bodies, they seem to commonly take on the shape of an prolate spheroid.  Beyond one Earth atmosphere, the bodies all seem to conform to a roughly spherical envelope, even though there are a lot of irregularities on the surface.

For those small bodies, it's interesting to note that the speed of rotation also puts a limit on the amount of material stress we could expect from them.  A list of fastest rotating objects shows that some have a rotation on the order of a minute, but these are all several meters in diameter.

Largest Bodies Identified Irregular

Another reference for this is the Wikipedia list of solar system objects by size.  I went through that table and grabbed the largest objects that were identified as irregular.  This is a good approach because it gives a mass figure below which irregular objects start to appear is large number.  However, its major shortcoming is that we have no idea how many corresponding regular bodies exist around their mass scale.

ObjectM (kg)Shape
Proteus (moon)5.0 x 1019 irregular
Nereid (moon)3.1 x 1019 irregular
52 Europa1.7 x 1019 irregular
Davida4.4 x 1019 irregular
Sylvia1.5 x 1019 irregular
Cybele1.8 x 1019 irregular

Reasonable accounting would put the limit for irregularity at somewhere between 1017 and 1019 kg.  However, the limit I've found for a habitable inner pressure is around 1016, and can even be lower than that.  Engineering limits are still more complicated.  Even though bodies in this size range are irregular, they're still roughly spherical in many cases, indicating that material strength doesn't hold up on the scale of self-gravitational forces.  Within the size range we're interested in, the bodies are showing that they withstand some deformation against the equipotential criteria, but exactly how much is unclear.

This evidence does give a clear message - that using an asteroid to hold breathable air would involve both self-gravitation forces, as well as some material pushback.