Transit within Space Mega-Cities

This is intended to be the first post in a series on transport issues applicable to moving between one artificial gravity cylinder and another (not necessarily specific to a gravity balloon). The growing list of topics under this umbrella, new and old, have grown so much that a full series is needed. This post acts as a summary of those topics and establishes a central claim.

In particular, I want to talk about a billion person city. That is, a 3D city in space or inside an asteroid which has on the order of a billion people. For the purposes of this topic, I will use the term colony to denote a rotating artificial gravity cylinder in such a cluster, and the word city to refer to the entire cluster. In previous posts, I outlined how 1 billion people is just on the elbow of population constraints shifting from packing density to heat removal. Even that latter constraint isn't absolute, but it's not important because transportation times will put larger sizes in a different stratum anyway. The level of cultural connection is fundamentally different between occasional travel versus daily service.

Just that word "daily" introduces all kinds of qualifiers in the first place. Any concept of a day/night cycle would be artificial. My position is that this is only slightly different from modern cities on Earth since artificial light became rife, except that there's no natural suggestion from nature. The specifics of how a society manages the biological need for a day/night cycle is one thing I don't care to take a position on. The only kind of assumption I'm interested in here is how many trips per day someone takes between seperate colonies. Ideally this wouldn't include grocery trips or other relatively fungible activities. However, the entire draw of such a city is the connectedness, so I expect the relatively routine days of living in one colony and working in another would be balanced by extremely active collaborators. If you only go between your work and home colony that will obviously result in 2 trips. To balance things out, I'm saying 4 on average per person per day, for 4 billion A->B trips each day.

How you accomplish this volume, and the specifics of the physical manifestation is what I'm interested in here.

1. The Comparison

As an alternative hypothesis, this discussion will entertain the idea of lots of conventional artificial gravity habitats packed closely together. That conventional concept is an independently pressurized colony that rotates to create gravity. The problem I'm interested in here is inter-colony transport. Inter-city transport would be very different, the demand for it isn't well established, and there's not much I could say anyway. Comparing to the gravity balloon, the one obvious difference is that the space between the colonies is vacuum.

This still accomplishes the argument that a larger number of smaller colonies will have more surface area than a large single colony with the same volume. Transit is a lot easier with this system. Transportation within a colony is not much easier than transportation on Earth since it is a terrestrial-like environment. With no convenient fossil fuels you'll find public transportation more appealing too.

It should also be noted that these colonies would need to be tethered together in some way to avoid eventual collisions and resist destruction due to any perturbation. For instance, if a transport ship lost its pressure boundary, that air will then impart an impulse to the nearby colonies. It's obviously necessary to prevent those disturbances from causing millions of people's death by slow drift and collision of their colonies.

My reason for introducing this dichotomy is to identify very stark different between the gravity balloon and the conventional scenarios. Starting out, we don't want to prejudice ourselves to one or the other. Indeed, I think there are scenarios where the conventional approach might be better. It all comes down to what it is that you value. My predictable insinuation, however, is that if you desire a hyper-connected community of a billion people, the gravity balloon is almost certainly your best option... mostly because it doesn't need airlocks for intra-colony transportation.

But first, we need to formalize why airlocks matter.

2. Transport Network Topologies

Transportation can either be on-demand or mass transit. On-demand can entail either a personal or a shared vehicle. It also doesn't have to be the case that an on-demand vehicle is for a single person, but if it's not this will impose additional constraints which I don't want.

My perspective is that whether a trip is serviced by on-demand or mass transit depends on the volume of demand for the trip, measured in people per second. While the ideal volume for mass transit is disputable, I put it roughly in the range of 1.0 people per second. However, the minimum acceptable volume also depends on the number of transfers someone has to take during their trip. Doing 5 transfers at a cost of 1 minute each is taken to be equivalent to waiting 5 minutes for direct service.

Quantifying the difficulty of a transfer isn't precise but is still obvious. Going through an airlock is going to carry a higher time cost than stepping from one platform to another.

 2.1 Spoke-hub

On one extreme, the spoke-hub network topology is heavy on transfers. If all colonies only connected to one hub by their own dedicated route (or "spoke"), then even to get to your neighbor colony you'd have to travel to the very center of the city, transfer, and travel to the destination.

This complexity buys you a reduction in the total number of routes that have to be planned and maintained. Since the number of travelers is a constant, this increases the volume that each route sees. The 1 central hub is the extreme example, because there's only 1 route per colony.

Such extremism is not necessary. If you turn this into a hierarchy then the more central routes and hubs will have even higher volume than the local routes. As such, this method lends itself nicely to mass transit. The local routes will inevitably see usage rates 8 times the population of the colony per day. Since this will be in the 1,000s at least, any literal hub-and-spoke topology could be serviced entirely with mass transit.

Relaxing the extremism a little bit more, you would likely add cross-links either between colonies or regions. Since the volume is so high, this can be done while keeping mass transit viable, but still reducing the number of transfers.

Our tradeoff for extremely high volume is a larger number of transfers. I'll call the number of hierarchy levels I, and the number of regions/colonies serviced by a hub N. Then the total number of colonies is N^I. This established, the maximum number of transfers would be 2*I-1. If you used the smallest possible value of N, N=2, then you could be facing on the order of 30 transfers.

Such transfers would have to be virtually instant in order for the transportation system to be practical.

2.2 Point-to-Point

By putting people into colonies, we are dividing up our billion people into groups which are probably 10,000s to millions of people. Even the smallest realistic number of colonies is around 1,000. Even with this number of people, mass transit could only be minimal in a point-to-point system. This is because the number of routes in a full point-to-point system goes with the number of colonies squared. You'd be looking at 2-3 people for every route every minute.

This still forms a decent alternative hypothesis. By making the colonies large (although practically so), regular flights between colonies can be possible with the ridership being a handful of people.

2.3 Penalties for Distance and Transfers

A space city might find either one of these topologies favorable depending on the type of city and available transportation technologies.

Broadly, larger colonies are associated with conventional colonies because the task of reducing air drag in a gravity balloon becomes more onerous. Larger colonies are also associated with greater propensity to point-to-point transport topology.

More transfers are also disassociated with conventional colonies. The self-evident facts are that all colonies maintain their own atmosphere, and that colonies are stationary relative to each other. That means that you will use a minimum of 2 airlocks. If find it unlikely that any system designer would want to do any more than this. Any mid-range transfers would likely avoid merging the atmospheres of the vehicles if at all possible. Airlock cycling is slow and expensive.

3. Physical Mechanisms

Routes and transfers have been addressed in the abstract sense up until this point. That is, a route connects a hub or a colony to another hub or colony, while a hub is a point where one can exit one vehicle and enter another.

Even without any statements regarding the possible or likely network topology, a great deal can be inferred about the transportation system of a future space city from the underlying technologies. Popular illustrations of space colonies with artificial gravity sometimes add one or two details of ships entering and exiting... but if you start increasing the expected volume things start to get real weird real fast.

3.1 Means of Docking

By docking, I mean mean moving in and out of a colony or a transportation hub. Docking entails getting on a shuttle, getting off, and even the process of finding your next shuttle.

3.1.1 Conventional

Ports are most commonly depicted on the axis of a rotating space colony. For a cylinder, these are the two ends. For a sphere, these are the two poles on the axis. This rotates along with the space station, but the rotation rate is slow, and the accelerations mild.

But this is inherently one-lane. You could dock off-center but this will use propellant which I find to be incompatible with a large space city because the propellant demand for billions of trips would be huge. The only alternative would be to surround the city in a barrier and recover and recycle propellant. As such, I find the only valid solution to be sending larger ships through the airlock. This will likely be a ship that carries other ships in order to satisfy the point-to-point topology.

3.1.2 Gravity Balloon

A colony inside of a gravity balloon has open ends. This means that, hypothetically, someone could just walk off the edge. For our (more serious) purposes, we need some kind of architecture that gets them somewhere else after leaving. Since most of my reference designs have ends 30 m in diameter or more, this will entail collecting the people into a smaller space.

The solution I've been leaning toward is to have people's path through the colony mirror that which the air flow takes. Come in at the edge of the inlet end. Either take an escalator, shuttle, or a slide down to the terrestrial environment. Once they decide to leave, a building which spans the full diameter of the colony rises to the center-line. Here, they enter a tube where the low gravity and air flow move them out.

Individuals still need to make their own decision about what route to take, so you would essentially need a lot of handles and rails so they can move through the hub to their transport, which will be mass transport. Other than this, there's not much else to figure out. The atmosphere is continuous, so the only reason for doors in a transit shuttle is to keep people from falling out. A similar logic will apply for the transit hubs.

3.2 Means of Moving

The primary difference between the gravity balloon and conventional cities is the existence of an atmosphere in the former. This is a detriment to power consumption and speed limits in transit. However, it also can be a nice thing to have something to push against.

My position is that whether an atmosphere is a merit or a disadvantage is a function of the total size we're talking about. Since a gravity balloon can more easily host an extremely high density, these attributes somewhat make sense in conjunction. Alternatively, some methods might be available to get extremely high speeds in less dense clusters of conventional colonies.

3.2.1 Conventional

In the conventional space city concept, we are forced to move relatively small shuttles through the vacuum between colonies. I'll start out with the assumption that these shuttles are mostly free-flying and I'll classify the issues into 3 major parts, and only briefly touch on them here.

After departing the docking facilities of a colony, the shuttle obviously has to first accelerate. Since we're not very interested in multiple km/s speeds, this can almost certainly be accomplished mechanically. But we also must keep in mind Newton's law of motion. The transit volume almost certainly rules out reaction engines, so the shuttles are necessarily pushing against either the colony or some ancillary structure. I would favor the latter. A non-rotating envelope could surround the colony to provide easy access and also transfer momentum to and from the colony through axial bearings. So this non-rotating structure could launch shuttles through mechanical tracks or even mechanical arms. However, each shuttle is going in a different direction. The shuttle departure and arrival rate is also fairly high, so you would likely have many of these systems serving a single colony.

Once you get going, there will necessarily have to be some avoidance software, if we assume a mostly crude method of moving in the direction of the destination colony. If I assume a shuttle is about a square meter in cross-section, then random walks through my reference space city would result in roughly 1 collision every week. This is probably most surprising in illustrating the enormity of the 3D space. This is even after subtracting the volume of the colonies themselves. Avoidance might be one thing for which propulsive maneuvers makes sense because they are both infrequent and random.

Not only do you have to avoid other shuttles like yourself, but the lattice of colonies needs to be navigated around, but I classify this differently in the category of steering. Because you'll be going on movingly a preplanned route, it's thinkable that your redirects can be done by stationary infrastructure along your way. I would imagine that magnetic forces would be the most ideal for this because frequent catching and relaunching would make for a very uncomfortable ride.

Lacking a good vision of implementation of a system that handles these issues, I'm tempted to say that free flying shuttles won't exactly be the architecture that a conventional space city winds up with. I think something much more akin to roads are likely, but very different space roads. I would imagine these are steel guiding rails which the shuttle only makes contact with during acceleration, braking, and steering. This road network, thus, would still somewhat resemble a hierarchical network topology, although certainly not spoke-hub. I envision something much closer to an interstate system. For the large volume routes, the shuttles might even combine themselves in something resembling a mass transit system, but still while avoiding connecting via an airlock.

3.2.2 Gravity Balloon

Moving through a continuous microgravity atmosphere is the same as planes on Earth, except without the added complication of gravity. Thus, no novel technologies are needed and we can concern ourselves only with the economics of the system. If we use an extremely hierarchical transit system, the basic unit of transportation would likely be the individual and they would make decisions about where to go by grabbing onto guiding anchors as I argued before. Starting from that point, we only need to further identify the technology options for high-volume routes. These could use several mechanisms to minimize air drag, but the logistical implications of these approaches are also important.

One approach might be to just not worry about drag. You could even use a rope tow to transfer people relatively locally between colonies of the local hub. That is essentially a simple pulley in zero gravity. Many of these trips would be around 1 km in length, but they would need to be completed quite fast and very safely. The biggest problem with a person directly holding on to a fast rope tow would be the they can't breathe. If everyone wore aviator masks, the speed could potentially be rather high. That raises another complication - of how to grab on in the first place. This could likely be solved by a series of rope tows in steadily increasing speed, but all within arm's reach to the next stage. This also sounds somewhat dangerous, so a simple alternative might be a shuttle moving along a rope. It could accelerate with its own wheels, or it could just be on a pulley system. The only drawback is the relatively inflexibility since these approaches very strictly connect either two places or a loop.

The rope tow might be the most simple system I can think of. However, if we want to go in the extreme of maximizing logistical prowess while still not worrying about energetic efficiency, even a pneumatic tube starts to look like a valid option. If a traveler's destination was obvious (as in a true spoke-hub topology), the system planner can avoid a transit center where the person chooses where to go, and simply opt to suck the people up with a current of air. Yes, like the Jetsons. Actually, this could boost a much higher throughput with much greater simplicity than any other alternative. On the other hand, it can't be very fast (without killing people), and it would be highly inefficient.

Fixing the problems with the pneumatic tube by adding greater complexity would, in fact, be possible. You could actually invent a system that borrowed the physical principle from the friction buffers that I've discussed for the rotation of the colonies themselves. This concept was obvious to me a while ago, but I found it completely ridiculous. However, after studying the nature of hierarchical transit systems, it seems that the most central transportation routes can quite possibly get used by a large fraction of all the trips taken in a day. I mean, one route might service 2 billion trips per day. This is 1,000s of people per second. Even at 100 miles per hour, you could have 100s of people per meter. Any conventional system would involve massive docks or massive ships. The logistics of getting people in and out of a shuttle would be mind-boggling at these volumes. So a pneumatic tube isn't absolutely a terrible idea.

Considering these economic pressures, let me propose a ridiculous idea which would feasibly service the ridiculous populations of a space mega-city. Take a closed tube that goes in a large loop - a torus. Now surround this tube in another tube. Do this over many tubes. Have the inner tube rotate about its perpendicular axis at the desired speed. Each enveloping tube will move slightly slower. Let's say 5 mph over 20 tubes for a total of 100 mph travel speed. Then, of course, you would have doors in each level of tube so that someone could eventually make their way to the central tube. This could handle extremely large volumes, would be 100% continuous, and the power requirements wouldn't be all that high because the flow is controlled carefully. The main problem, I would imagine, would be training people to use it.

For people desiring methods which are more familiar, borrowing from planes, both jet and propeller craft are possible. You would only exchange the wing for another control surface or two. For the biggest legs of a mass transit system, air-breathing shuttles could reduce energy consumption and increase speed by increasing their scale. This is a coherent vision, but I doubt that the air-breathing engine would be the ideal choice for acceleration at the start of the trip since a simple mechanical boost at the start would be fairly trivial. Direction changes can be handled slowly over the course of the trip by control surfaces, so any launcher wouldn't need to aim either.

We don't necessarily have to consider on-demand transportation, but that is also relatively easy. It might also be a preference under certain circumstances, or just plain fun.

4. Gravity Balloon Connectivity Argument

Here's the point:

Why build a gravity balloon space city instead of the alternative?

I've been as general and generous as I can with the conventional type of space city. Indeed, I'm not trying to argue that any particular reference design is bad, just different. The attribute where a gravity balloon is appealing is transit connectivity. One real reason for pursuing a gravity balloon is desire for an interesting, vibrant, and interconnected city in space. I did not go into numbers here, although I hope to get to those in later posts, so you'll have to take my word on matters of degree. I wouldn't be saying any of this if I didn't have some empirical basis ready.

The desire to mingle among a billion person community daily might be somewhat difficult to reconcile with conventional space colonies. Even if one favored the extreme over-sized space habitat of a McKendree cylinder, it would take a heroic technology effort to make daily transit among the entire pseudo-terrestrial area viable.

As we try to get design convergence in the reference conventional I mention, several issues pop up. Smaller colonies result in greater transit network challenges and shorter average travel distances. However, larger colonies raise some serious safety concerns due to the speeds of the rotation. Even on the large side of ideal design range, I'm absolutely sure that the space between colonies would involve a large amount of clutter in order to make the formation of the movement of shuttles workable and comprehensible.

A gravity balloon, on the other hand, has really weird possibilities for the transportation routes, and considering the size that we're talking about these might even be relevant. Either way, having air to push against is a huge benefit. Also, no separate life support system is needed for every trip and people can live without constant airlock cycling. Those issues with conventional colonies are not deal-breakers, but the gravity balloon clearly has much greater desirability in this context.

Rockfill Pressure Boundary in Asteroids

Context of Void Use

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"

Visual Conception of Rubble Pile Interiors
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 / m² 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 r² / 2 divided by Pi r², 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.

SVG Version of Fractal Rock Filling Pattern

There will be another post on this soon, but I want to avoid testing a new trinket in the middle of a larger post.  The figure below is the output from a code I developed in order to quantify things that will be needed for the cutting of rock to section off an area of a porous asteroid.

Rock Distribution to Plug a Porous Interstitial Site of Regular Lattice of Spheres

This is an svg element, which is a vector graphics format.  That means that every circle is specified by markup language saying where it is, what color it is, and so on.  When I post details, I'll restrict myself to png images.

Ederworld Analyzed (Concentric Gravity Balloons to Maximize Volume)

From the Orion's Arm website, I have been directed toward another similar design.  The original proposal seems to trace back to Dani Eder (possibly in 1995).  I believe this is the same person who wrote the Space Transport and Engineering Methods Wikibook.  In that discussion, people were trying to ask what the largest realistic structure that could be built is.  He correctly concluded that this would be a gravity balloon of Hydrogen, made even larger by rotation.  Wording used there was a "Bubbleworld", which appears to be the concept I call a "gravity balloon".  This Bubbleworld is much more relevant, but I can find very little information on it.  Worse, I had passed over the concept in the past because it shares a name with other proposals that are not relevant to this blog.  I would like to find more writing on this Bubbleworld, but I'm not sure if I will.  The wording used for Ederworlds on Orion's Arm is "Inflated, self-gravitating megastructures," which is very promising to me.  However, it seems that the title of "Ederworld" should be reserved for the goal of creating the maximum possible living space, which is the quirky design that I'll get into here.

I've repeated the calculations for the Ederworld described in the email that it apparently comes from.  It is a serious mind-twister.  The goal is to create the largest possible living area in terms of volume.  Previously, I have entertained the concept of the Virga world, which is the largest world you can create with a simplistic gravity balloon concept.  In order to go further, Ederworld introduces a bubble of Hydrogen gas in the center, which is prevented from mixing with the ordinary air on the outside.  As you increase the size of this central Hydrogen bubble, the thickness of the air shell decreases from Virga's dimensions.  That means that you're accepting a reduced thickness of the living area in favor of an increased radius.  Exactly where the optimal point will fall isn't trivial.

If you assume that the Hydrogen gas isn't compressed any, then it turns out the optimal point is infinity.  This problem is fairly sophisticated, but can be done with simple algebra.  I've done that, and taken the limit for large values of R.  In the final form, you can see that the volume scales with R² times a constant that comes from the problem parameters.

This means that a larger Hydrogen balloon will always give you a larger livable area - and it means that our assumption was wrong.  At sufficient sizes, it's obvious that Hydrogen gas compression will be significant.  In fact, it is this factor which will ultimately dominate.  That also makes the problem very very difficult to solve.  I used the numerical integrator which I described in my post on superlarge gravity balloons.  That is applied for the Hydrogen core.  After that, the "large" assumption is appropriate for the livable area and outer shell.

With that numerical integrator, we can start with some central pressure and then find the radius at which the pressure has dropped to 1 atm.  That's the condition for the interface between the Hydrogen gas and normal atmosphere.  The result of this inquiry is quite fascinating.  Beyond a point, adding mass to a ball of gas actually decreases its size.  Of course, a ball of gas with these central pressures would not be viable naturally.  They would evaporate away.  However, this is still the same basic idea as with our gas giants.  In a simple model, they just lie further to the right on this curve.  However, some phase changes happen, so it's not actually so simple...

Ederworld Inner Hydrogen Balloon Radius

Adjusted by Attempting Different Central Pressures

You can see that the radius itself hits a maximum.  So obviously adding more Hydrogen won't get you more living area, because at that point it decreases both dimensions, the thickness and the radius.  The volume peaks at a slightly smaller radius.  Going between radius, mass of the Hydrogen, and thickness is a little bit detailed, but involves the same equations that I posted above, for the case without considering Hydrogen compression.  The volume of the living space is obviously 4 Pi R² t, where R is the radius and t is the thickness.

Graph of the Living Space Volume

The Central Pressure is the Pressure at the Origin in the Hydrogen Balloon

Again, these are all done by the integrator for spherical distributions of gas that I made.  With the above graph, we pinpoint the absolute maximum living space that can be made by this method without rotation.  For the big picture, here is a table of my numbers:

Region	Thickness (km)	Mass (kg)	Volume (km3)	Average Density (kg/m3)	Surface g (m/s2)	Inner Pressure (atm)
Hydrogen	237830.5	7.50E+24	5.63E+16	0.13	0.0089	3.5
Living	2719.7	2.36E+24	2.14E+15	1.11	0.0114	1.0
Wall	0.461	4.31E+22	3.80E+09	11340.	0.0114	0.7

No doubt, this is way larger than Virga.  To illustrate, here is what I'm describing:

 

(Hydrogen gas isn't actually red, I couldn't

think of another color to differentiate it)

We have a central bubble of Hydrogen, for which the pressure shifts significantly.  Interestingly, even the density at the center of the Hydrogen bubble isn't quite up to the density of normal air.  Hydrogen spans a range of about 0.08 kg/m³ to 0.3 kg/m³, and air is closer to 1.3 kg/m³.  The parameters here actually match the previous work by Dani Eder surprisingly well.  He said:

If we assume that the bubbleworld is non-rotating, and the living space is at one atmosphere at the gas interface, the gas used is hydrogen, and the gas is at the same temperature as the living space i.e. 300K, then the answer is a sphere about 240,000 km in radius.

That's essentially what I obtained.  The major reason for the difference is probably that he assumed a temperature of 300 Kelvin, while I used 293 Kelvin.  Considering what it takes to get this number, I'm actually surprised that he was able to get this.  However, I think there is a little bit of misspeak in the proposal here:

If we assume that the living space has an average density of 10 kg/m³ (air is 1.2 kg/m³, the balance is people, houses, trees, etc.), then the living space has a limited thickness based on breath-ability. After 2400 km of thickness, the air will be at the equivalent of 3000 m above sea level on Earth, which is about the limit of ordinary breathing with no problems.

The livable portion is about 2,400 km if you use 1.3 kg/m³ as the density, not 10 kg/m³.  Using the former, the thickness comes out to only 300 meters.  However, I think that 10 is a little bit too high for this metric.  The reference tube design I described contributes a mass of only about 2.8 kg/m³, for a total of 4 kg/m³ if you include the air.  I have a hard time imagining a comfortable environment where things are packed more tightly than this.  After all, if you've made a habitat so absurdly huge, you would think that you wouldn't run out of space.  But I guess that's a subject for the sci-fi writers.

The density of the livable portion doesn't affect the calculations for the Hydrogen part, because its boundary condition is 1 atmosphere of pressure either way, and shell theorem prevents any gravitational effects.  However, the density does affect the total mass.  It is also quoted that:

The bubbleworld is too diffuse to hold the atmosphere in by gravity, so an outer shell (steel is handy) is used to keep it in.  It works out to 500 m in thickness.  The total mass of such a world is 3x the Earth's, so there should be enough raw material to build one out of a typical solar system.

You'll get 3x of Earth's mass (of almost this) if you use 10 kg/m³ for the livable area.  But this isn't compatible with the thickness of that area of 2,400 km.  Otherwise, the calculations for the "wall" are the same as mine.  I find about 500 m if you assume a density of lead, which is extremely heavy at 11.34 specific gravity.  If you change this density, it doesn't change the total mass of the construction.

If you return to the assumption of a 1.3 kg/m³ living space density, then the mass is about 1.7x Earth.

Buoyancy (Actually Differentiation)

It's also not stable, naturally.  You can say that no gravity balloon is stable.. but you would kind of be wrong, unless you're counting the liner to keep the gas inside.  A gravity balloon should have a gas on the inside and a solid as the wall.  As a rock-like substance, the rock has some strength to it.  It's not going to start tumbling due to the presence of microgravity due to a slight displacement from perfect spherical symmetry.  That's because we're dealing with a fluid underneath a solid.  For the Ederworld, we're dealing with a fluid under another fluid.

Fluid-fluid constructions are extremely unstable if not already differentiated.  By that, I mean that the Hydrogen gas is lighter than air, and because of that, it wants to rise to the top.  The air wants to fall to the bottom.  This is an issue, because you have to stabilize the Hydrogen balloon.  Any off-center movement will want to grow, and that will be a big problem.  This construction is so large that you really can't credit much material strength.  However, since the livable space is only a few 1,000 km across at most, it's reasonable that this could be spanned with stabilizing tethers.  This is obvious - stability is never absolutely limiting.  It's only a complicating factor in an engineering sense.

Aside from stability, I was also convinced that tidal forces would be a big problem for a balloon of this size.  I was very convinced of this, but it turns out that I was wrong.  You would have to be within 1 AU of the sun for the tidal forces to approach the surface gravity of this, which is the general metric I would use to set expectations.  It would appear that it could sit in many places in the solar system and still be viable at the stated pressure and size without tidal forces ripping it apart.

Qualifier: This goes into the library of physically viable constructions, but it would take a society far beyond anything that we have ever known on planet Earth.  This is something to "outdo" the Virga design - which is the extreme of a Bubbleworld, which was the starting point for the Ederworld.  The Virga design is vastly beyond what we could hope to do as a society that resembles what we have today.  Even the designs I refer to as "Anahitan" and "Sylvian" are mind-bogglingly massive constructions, and those assume all kinds of shortcuts taken to reduce the needed resources.

Thermal Engineering of Free-Floating Artificial Gravity Tubes

This blog describes a concept of rotating tubes for artificial gravity, which are free-floating in a large zero-gravity atmosphere.  Specifically, my reference tube design fits completely inside of a 1 cubic-kilometer cube and could house 22,000 people in a familiar environment.  Obviously, the zero-gravity air environment would need to be much larger than this.

With this population, an obvious question follows: where does the heat go?  My intention is that these tubes will have "windows" on both ends which are open to the environment.  At this scale, no amount of conduction can be relied on, and air flow within zero gravity is notoriously stagnant.  The tube itself isn't zero gravity, so there will naturally be air currents as the inhabitants use energy locally, that heats the air, and that drives currents.  What kind of air flow rate (through the windows) would this drive?  Obviously that depends on the window size.

I previously imagined the window being about 20 meters in diameter, which limits its relative speed to a slow walking pace.  For a test case, consider "lateral flow", where air comes in one window and moves out the other.  Assume 2 kW of power consumption per inhabitant, and allow a 5 degrees Celsius temperature change as the air travels through.  The air speed at the edge of a window will then be 21 miles per hour.  That's a little bit too much.  Because of that, I'm altering the design.  The window will need to be about 40 or 50 meters in diameter.  I use 40 if it's double-flow (air comes in both windows) and 50 if one window is inlet and the other is outlet.  I choose these figures because it minimizes the air speed.  As you increase the diameter of the window, the speed of its edge relative to the zero-gravity atmosphere increases.  However, as you reduce it, the air flowing into the habitat has to be faster.  By making these two components roughly equal, you obtain the universal minimum.

Illustration of Wind Flow into an
Artificial Gravity Tube

With the parameters I've described here, the total wind speed comes out to about 13 miles per hour.  This isn't quite what I was hoping for, but it's still fairly permissive of easy travel into and out of the tubes.  To get more practical, however, it's obvious that what I've described as "lateral flow" won't exactly work.  To get this type of flow, you'll have to power it, but how?  People will be moving in and out constantly.  If you put a large fan there, you'll have to have an air-tight door for them to travel through, all the while avoiding being sucked up by the fan.  This might not sound challenging, but this is almost entirely a zero-gravity atmosphere (the window has about 1/10th Earth gravity).  A person's motion will be dominated almost entirely by the air currents, and not by their footing.  Because of this, it's worth while to compare the general desirability of flow schemes on both a technical and qualitative basis.

Advantages and Disadvantages of different Flow Schemes

In addition to "lateral flow", I'm also considering "centrifugal flow", where valves on the floor of the habitat allow some air flow through, into the friction buffers area.  It also follows that the friction buffers would have to allow some amount of air flow.  This could have an unrelated benefit of helping to hold the friction buffers into place.  The flow paths might be easy to engineer as well, since they could be macroscopic windows, which could even allow maintenance people to go into them.  This scheme is using the rotating tube as a centrifugal pump essentially, as I've described in prior posts.

Thinking more about the flaw of the first two, I came up with a natural circulation approach that is now my favorite.  Doing the calculations, you will find that the driving force from the centrifugal flow is too much by orders of magnitude.  This can be fixed by piping the flow back to the window of the tube.  With this setup, the flow is powered by the thermal difference of 5 degrees.  Cold air flows down and hot air flows up.  The only challenge is to them corral the hot air to the exit, and make sure the cold air makes it to the surface before going anywhere else.  This would muck up the aesthetics a little bit, to be sure.  The area of flow dividers would be massive.  Nonetheless, the problems of the other schemes seem to outweigh this.

Possibly the largest issue, and the one that concerns me the most, is vortex formation as the air travels from the windows to the surface and back again.  If these are not reduced or eliminated by radial flow dividers, then it could become a fairly bumpy ride.  This is important for intake as well as outtake - since the torque from both must balance each other.  To any degree that it doesn't, that could create big problems.

Numerically, there are several flows of energy at work.  Their relative scales are illustrated below.

• Population's Consumption - multiply the number of people by the power each uses, for microwaves and computers or whatever they use power for.
• Rotational Drag - this comes from the posts on friction buffers, it is the power necessary to power the motors that keep that station spinning against the drag of the atmosphere.
• Driving Force to the Baffles - power that would be released if the necessary air flow to cool the city were directed through the floor, into the friction buffers.  If it flows through valves, then it will increase the temperature of that air.  However, since this is a lower power level than the population's consumption, it will only be about 1 degree C, compared to the 5 degrees it has already heated by.
• Natural Circulation Driving Force - this is the power you would have to drive the flow in the natural circulation scheme against the all sources of friction
• Coolant Kinetic Energy at Window - using the lateral flow scheme (also the same for natural circulation), this is the power needed to accelerate the air flow at the windows to the free jet condition.  In other words, this is the power consumption to drive the flow if the hydrodynamic forms loss coefficient is equal to 1.0.

Returning to the last two quantities, it is obvious that natural circulation flow would NOT be sufficient to drive flow at the stated temperature difference.  This is because the effective forms loss coefficient for the entire process would certainly be greater than 1.0.  That means that, in practice, the window would need to be larger than 50 meters across, people would have to consume less power, or a higher change in temperature would have to be accepted.  I have not gone into detail for these calculations, because the above number is still a good ballpark.  It tells me the design is within engineering possibility.  Interestingly, another approach to improve natural circulation would be to make the diameter of the tube larger.  Even if this increased the population by an R^2 factor, it would increase the driving force and the area at the same time, so the total cooling capacity would follow with R^3.

Through this examination, I've established the following claims.

•  Heat removal limitations are not trivial.  Increasing population density by much could push thermal limitations.  Plus, very intentional designs to exchange heat with the surrounding air are required for my habitability constraints.
• Tossing air out into the friction buffer space would hypothetically work but is economically unrealistic for a 500 meter diameter tube, and gets worse as that gets larger.
• Flow dividers inside of the habitat are necessary to transport the air from the windows to the edges and back.
• Natural circulation is viable for the reference design, but only barely.
• Air movement will have an impact on accessibility in general, and has to be engineered together.

This is an important component of the overall proposal of gravity balloons, because the artificial gravity habitats need to be both economical and desirable.  They certainly seem to be.  Nonetheless, they will be complicated.

The subject of thermal management of these tubes also deserves mention of the analog on Earth: the Stack Effect.  A rotating artificial gravity of a given radius is extremely similar to an Earth building of that height.  Buildings on Earth also have to have HVAC systems, and this resembles one of those very strongly.  The stack effect for buildings on Earth essentially does the same thing that my natural circulation flow scheme does.  But on the other hand, those buildings don't have as many people, and also have more area (of windows and doors and such) exposed to the atmosphere.  So these two cases are similar, but one can't be called easier than the other.

The Specific Stability Requirement of Shell World

Gravity's shell theorem is extremely neat, but there's some heavy critical thinking that needs to be applied when we start going into the ideas of mega-structures.  The shell theorem states that when something (the object) is placed inside a spherically symmetric distribution of matter (the shell), then neither the object or the shell will experience a net force.  However, "net" is the key.

The shell world is a proposal to surround a small body in our solar system in a shell that covers the entire planet/moon/asteroid and holds in the atmosphere.  In other cases, it may be called worldhouses, worldsheets or anything else reflecting the general concept.  If you look at this recent conception of the idea, you'll notice that the advocates are specifically advocating relying on the weight of the shell as an alternative to a pressure vessel.  That means that, in fact, this is a particular type of gravity balloon.

Mars or perhaps a moon in another solar system could be encased in a shell of dirt, steel and Kevlar fiber.

Key word here is "dirt".  This reflects a desire to "weigh down" the sheet so that it doesn't have to rely on material strength.  This is obviously possible from a physics standpoint.  The structure would be initially balanced.  Obviously, the pressure of the atmosphere pushing up would be equal to the weight of the shell pushing down.

The question of stability is "what happens when the planet moves inside the shell?"  This is actually surprisingly easily answerable, and it's a straightforward stability analysis.  But only for specific assumptions - namely if there is no rigidity.  You have two different cases you can entertain as limit cases:

1. the shell is so rigid it never experiences deformations
2. the shell is completely non-rigid

In the first case, the structure is always stable.  The reason is that the net force between the planet and the shell is always zero.  So even if the planet moves, it will not affect the shell.  Only the atmospheric pressure at different locations will change - after all, the planet still has some gravity.  The atmospheric forces are restorative, so this means the structure is always stable.

The case with no rigidity is harder to appreciate well, but it's somewhat of a fundamental assumption I want to keep in mind when writing about gravity balloons.  You would certainly prefer to place no requirement on the materials.  Then the problem is only one of gathering the materials - not doing anything with them.  So for the shell world we can apply this sort of assumption.  This means that any acceleration on one segment on the shell is not taken to influence other parts of the shell.  You still have some obvious problems.  In order for this to be useful in any way, there would have to be a flexible membrane that maintains a pressure barrier even when the shape is deformed.  And yes, this is precisely what I have in mind for the nature of the shell.

We will also have to neglect the gravitational effects from the shell itself.  This could be significant in some cases, but for something large like Mars, it's irrelevant.  It can be ignored with no major ill-effects.  In fact, if the shell starts to gain significant mass relative to the planet itself, then that starts to look a lot like a gravity balloon.

That's a lot of qualifiers, so let's get into the specifics.  I couldn't find an expression for atmosphere pressure including the full form of Newtonian gravity, so I made one.  With that, we can set the stability requirement.  To do this, we imagine that the sheet lies at some radius from the center of the planet, and at this point, the upward force from the pressure exactly equals the downward force from gravity.  Because of that, we can formulate a simple stability requirement.  To have stability, the fraction change in pressure with a change in altitude needs to be larger than the fraction change in gravity.  Following that logic, and using the form I've developed, you can get a very clear stability requirement.

Calculation of the Stability Requirement

for a Shell World

This stability requirement is formulated in terms of radius.  This is to say that below that radius, a world sheet would be stable, but above it, it would not.  What does it mean to not be stable?  It means that pressure will "outrun" gravity.  If a deformation of the sheet develops, the pressure will cause it to expand until the sheet hits the surface of the planet, or until it breaks.  The system obviously isn't differentiated to begin with, so failure is always a possibility.  Even if it's stable, a leak can destroy it - meaning that it will eventually fall to the ground.  This is to be expected, since we don't normally expect things to float in the air without additional support.  The worldsheet can, but only if it fits this criteria.

Now, the H' value needs a calculation associated with it.  Take that from my Physics Stack Exchange post.  The expression has mass of the planet in it.  This is the dimension over which we want to investigate the system.  Here is the requirement restated:

Stability Requirement in Terms of Planet's Mass

and Other Universal and Gas Parameters

This is a useful form, because it can directly be applied to find out which bodies can have a stable world sheet.  I did that for a few interesting bodies, here is the table:

Table of Viability of Worldsheets for Various

Interesting Bodies in the Solar System

	M (kg)	R0 (km)	Stable r (km)	r / R0
Earth	5.97E+24	6,371.0	2,370,438	372.1
Mars	6.42E+23	3,390.0	254,698	75.1
Moon	7.35E+22	1,737.1	29,166	16.8
Ceres	9.50E+20	476.0	377	0.79
Vesta	2.59E+20	262.7	102.8	0.39
Sylvia	1.48E+19	143.0	5.9	0.041
Phobos	1.07E+16	11.3	0.0042	0.00038

From this, we conclude that Ceres could not have a world sheet without additional stabilizing features on it.  There is physically no radius at which it could work with air at normal conditions.  Now, there are ways we could relax this.  Factors that affect the air properties will affect the conclusions here.  Refer to the mass implication equation above to see how to apply these.  My conclusions are:

• Lower temperature -> more stable
• Pressure at surface -> no effect on stability
• Higher molecular weight -> more stable

Even considering these, it would be difficult to imagine how a Ceres worldsheet could be made to be stable.  I used the temperature of 293 Kelvin, because I'm assuming that the world sheet habitat would be at room temperature.  That's not very easy to change if you're going to have humans living there.  Moving on, the pressure won't affect stability because the worldsheet's thickness is designed to exactly balance the pressure with gravity.  A higher molecular weight is an interesting proposition, but what would you use?  Nitrogen has a higher molecular weight than Oxygen gas, so maybe with a higher Nitrogen partial pressure a Ceres worldsheet would be more viable.  Otherwise, more Argon could do the trick.  But that will only buy you a factor of 2 or so.  For smaller bodies, it just could not be made stable period.

Given these results, I would say that some of the illustrations of the shell world can't really be practical.  You wouldn't just need the shell to have tensile strength.  That wouldn't be enough.  It would have to be fully rigid, meaning that it tolerates torsional stresses, and that just doesn't sound practical.  In reality, it would make far more sense to have it tethered to the surface of the planet.  Without that detail, it seems very dubious that it could ever work on a Ceres size object.  On Mars it could work without any tethers to the surface.

With some active tensioners in the tethers, it wouldn't be that big of a deal, actually.  Tethers are easy to shorten or loosen.  You could have some active control for the shell's altitude at different points over the planet, as well as ways of sensing how far it is from a balanced point.

This result comes with an interesting implication.  This means that there is a range of masses (roughly spanning from Anahita to Ceres) that are both too large to host a habitable pressure at their center as a gravity balloon, but also too small to host a stable world sheet.  Of course you could just abandon the world sheet and instead use an interconnected network of caves.   That replaces the "tensioners" that I was referring to with more-or-less a big pile of rock.  I suppose I'm still left wondering why anyone would be happy with gravity 2 or 3% of Earth's, which is about where the stability point for world sheets comes up.

Population Limits of Large Space Habitats

On the International Space Station (ISS), people are packed in like sardines.  For manned stations such as this, the constraint is generally taken to be that some bare minimum amount of volume per astronaut is need.  I'm not exactly sure what biological justification they use for this, but I'm pretty sure that not going insane is a big part of it.  My calculations put that volume in the ballpark of 100 cubic meters per astronaut.  That figure is complicated by the fact that it is not currently occupied by the intended number of astronauts.  There's also a lot of equipment.

For a distribution of artificial gravity rotating tubes in a zero-gravity atmosphere, I've made a reference design which also has a particular packing factor of people per unit volume.

In both of these cases, we're assuming some density constraint given to us from some other type of engineering (aside from the station design itself).  In the case of the ISS, human-factors engineering dictates a volumetric population density, and in the case of my reference tube design, various energetic constraints prevent sizes much larger (which I hope to write more about later).  Neither of these are very absolute, and compromises by the inhabitants would allow you to use a higher density.

At a certain size, however, there is a thermal limit based soley on how fast the station can radiate away its heat.  Assume a spherical habitat (I understand this is a bit of a physics meme, but in this blog it's literally the case).  With that, we very easily establish the maximum population that can be housed with a given sphere radius and a given radiator temperature.  This assumes the radiator covers the entire spherical surface and there is perfect heat transfer from the habitat's core to that surface.

Maximum Population Constraint Given

- Density Limitation and then

- Heat Transfer Limitation

It's obvious from the form of the equations that this concern is relevant for large sizes, but not for small ones.  The ISS still has a thermal management system, but it only needs a small radiator compared to the size of the entire station.  The density-based maximum population grows with R³ (radius cubed) and the heat radiation constraint grows with R².  That means there will be a cross-over point.

For the given parameters, these population constraints are plotted below.  I entertained two cases, one where people use 2 kW on average (which is consistent with the modern developed world electricity use), and one where they use 10 kW.  The electricity use alone doesn't account for all energy use, and then there is the issue that farming is very energy-intensive and may need to be conducted within that habitat.

Log-Log Graph of Habitat Diameter versus

Maximum Population that can be Sustained

The cross-over point is fairly obvious here.  To give real numbers, here is a table.

Radius of Crossover from

Density-Limited to Energy-Limited

and Maximum Population at that Radius

Assuming Maximum Packing Density of 22,000 people/km³

Per-capita energy use	Radius	Population
10 kW	5.7 km	17 million
2 kW	28.5 km	2.1 billion

Small asteroids-turned-habitat would not be limited by the heat removal, since they couldn't get much larger than a 10 km inner diameter.  But then again, a very large habitat size is quite desirable.

For someone designing a habitat of a size much greater than these crossover points, they'll have some decisions to make.  Imagine they're given population as a prior constraint.  They might simply choose to have a lower population density in order to avoid having to deal with the heat rejection issue.  On the other hand, to make use of air and asteroids as efficiently as possible, perhaps they'll implement some more exotic system.

The constraint that the radiator lies on the surface can be broken, and it wouldn't be particularly difficult.  A neat approach would be to use geosynchronous orbit to increase the effective radiating area.  This is logical because most asteroids rotate with relatively fast day-lengths, meaning that it wouldn't be difficult to build the needed "space elevator" from the surface to GEO, and also run large pipes over that distance.  Nonetheless, the material constraints would be much more significant in this case.  If you're only working between the inner edge of the habitat and the outer surface, there are lots of ways you can use the asteroid rock to help make building the coolant channels easier.  If you're "over-sizing" the entire thing with the ring radiator concept, then you're going to be tacking on a lot more material constraints.

So I sketched a brief image of what I'm talking about.

A Ring Style Radiator for Increased Population
in Large Gravity Balloons

Natural circulation would be desirable for this configuration, but it would not work beyond GEO.  Alternatively, your pumping force could be from natural circulation from GEO to the center, but it has to overcome the density difference pumping in the other direction from GEO to the outer edge of the radius.

Yet another problem is whether the ring approach would buy you any increased area in the first place.  For the concept to make sense, GEO would have to be much further than the radius of the gravity balloon.  With short rotation periods, this isn't quite the case.  Small asteroids rotate at a variety of rates, with some of them spinning so fast that GEO is below the surface.  Those are too small for the purposes of a gravity balloon in the first place.  In fact, most objects would have a much greater area covered in the GEO circle than by the outer surface area.  Additionally, this ratio will grow when the gravity balloon is inflated.  That process increases the moment of inertia, so the rotation rate slows and the GEO point moves outward faster than the surface itself.  If it is too far out, however, material limits will become a problem, as well as debris and other things.  Generally, the ring radiator would be a much more challenging concept to build.  But that makes sense, considering that it would never be built unless the size of the colony was gigantic to begin with.

I like to think of Virga as a world that is limited by heat removal.  The author even makes some references to ice formation on the inner side of its shell.  Heat removal limitations are actually a convenient explanation for the density of people in the sphere.  In the story, rotating cities are separated by great distances.  We can say this is because the atmosphere would get too hot if the density was much higher.  It's a neat idea.  It paints a picture of relatively empty and expansive gravity balloons when their diameter exceeds 10 km.  However, make no mistake that this is a constraint, not exactly a good detail.