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.

Questions on Physics and Space Exploration sites about Gravity Balloon Habitats

This blog is a bit of a tangent, which stemmed from questions which were being tossed around on the Q&A site Physics Stack Exchange.  Since I started writing, another site, Space Exploration Stack Exchange has started up.  I do ask questions about specific components of the gravity balloon habitat concept, but there have also been some relevant questions that other users ask, which just happen to be pertinent.  I went through and quickly grabbed a list of these questions here.  Surely there will be more in the future but this covers all that I could find today.  Some of them correspond exactly to posts on this blog, and that was my intention.

Physics (http://physics.stackexchange.com/)

• Keeping air in a giant gravitationally-bound space balloon -Addresses the basic concept of center-caves.
• Amount of material required for a pressure tank - Concluded that structural materials scale directly proportionally to the pressure x volume product with conventional pressure vessels.
• Stress in a thick-walled pressure vessel -Asks how the structural materials would change if the pressurized volume was small compared to the vessel.  It seems to be a more restrictive case.
• Attenuation mass-thickness for sunset light - I wanted to get these parameters for the transport of sunlight within large space habitats, and we found them.
• Can a minimum ultimate tensile strength (UTS) for an asteroid be established based on its diameter and rotation? - Turns out that some asteroids rotate so fast that their geosynchronous orbit is effectively below the surface.  This is rare, and exclusive to small bodies.
• Effect of water on wobble of rotating structure - Interesting query about management of rotating artificial gravity tubes.  Turns out that water will have some very spooky effects, like pushing a half-full jug of milk on the counter.  But this isn't necessarily a bad thing.
• Can structural trusses in centrifuges be helpful at all? -Compared the structural requirements of a "barbell" type of artificial gravity structure to a simple ring.  Turns out the barbell is easier to build, but for complicated reasons.

Space Exploration (http://space.stackexchange.com/)

• What is the hollow asteroid concept in "Mining the Sky" - A similar concept is purportedly mentioned in this book, although I have yet to find out what that concept is.
• Method to estimate asteroid density based on spectral type -I've started to use the method from my reference here.  You have to factor in porosity, which has potential problems, but it's better than using no density estimation routine.
• What stability issues plague long artificial gravity cylinders? -Kalpana One is a recent proposal for a rotating space station, which limits itself to a "stubby" shape.  Exactly why was discussed, and I've adopted the design thinking myself.
• How to derive asteroid mass/size estimate from JPL parameters - Absolute magnitude is a measure of brightness of an object from 1 AU, so this gives us a very good estimation of diameter of a spherical object if the albedo is know.  Albedo is often not known.
• Full list of asteroids that have been optically imaged as of 2013 - Similar to a post I made on the subject.  There turn out to be more than what I gave credit to.
• Theories of what asteroid macroporosity actually means - Some dialogue on the nature of rubble piles.  The packing factor is surprisingly low for some asteroids, and this reflects a fairly recent area of research.
• What is the range of strength for asteroidal materials? - I was somewhat thinking that asteroid materials could be used to build structures in space without any processing.  Still not sure if that works.
• Are there any theoretical size limits of man-made space stations/structures? - About self-gravitation.
• In space can the difference in temperature between the inside and outside, provide useful energy? - One of my favorite questions.  Has lot of interesting implications related to Landauer's Limit and our thermodynamic existence.
• What is the largest naturally occurring body that could be hollowed and safely lived in? - Question sort of converges on the idea of a gravity balloon, although it wasn't intended.