Wednesday, December 11, 2013

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:

RegionThickness (km)Mass (kg)Volume (km3)Average
Density (kg/m3)
Surface g (m/s2)Inner
Pressure (atm)
Hydrogen237830.57.50E+245.63E+160.130.00893.5
Living2719.72.36E+242.14E+151.110.01141.0
Wall0.4614.31E+223.80E+0911340.0.01140.7


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 linear 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.

Monday, December 9, 2013

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.

Wednesday, December 4, 2013

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
Earth5.97E+246,371.0 2,370,438 372.1
Mars6.42E+233,390.0 254,698 75.1
Moon7.35E+221,737.1 29,166 16.8
Ceres9.50E+20476.03770.79
Vesta2.59E+20262.7102.80.39
Sylvia1.48E+19143.05.90.041
Phobos1.07E+1611.30.00420.00038


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.

Tuesday, December 3, 2013

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 useRadiusPopulation
10 kW5.7 km17 million
2 kW28.5 km2.1 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/)

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

Thursday, November 21, 2013

Why Not Live in the Empty Spaces Inside Asteroids?


Rubble piles are everywhere in our solar system.  The majority of asteroids are likely to be rubble piles.  This means that they are a conglomeration of many smaller fragments which are just resting  against each other, held by their collective gravity.  Within one of those pores inside an asteroid, why not just partition off a defined volume by lining the crevices with a thin sheet that can act as a pressure barrier, and use that as a giant permanent habitat?  This turns out to be a surprisingly good thought.

The object you use for this would have to be well beyond the minimum size for a normal gravity balloon.  That's because if 1 atmosphere of pressure is greater than the force transmitted through the rocks, filling it with air would cause the rocks to unset themselves and lose their structure.  For this idea, let's just leave all of the load-bearing rocks where we find them.  There is certain to be lots of smaller rubble that is not structurally relevant.  Our solar system also gives us an upper bound on the object's mass.  Beyond a certain mass limit, the rubble no longer has sufficient strength and will collapse its pores.  Interestingly, physics then predicts that the surface gravity will then increase, and this may cause a cascading failure that compresses the entire body, beginning the process of differentiation.

To pick an asteroid to build a habitat out of, we're not constrained by mass anymore, since the goal is to use the pores, which are already at vacuum.  I will refer to the "rock pressure" as the pressure that has to be transmitted by the structural rocks.  Between those rocks, you could keep breathable air.  I illustrated the basics of the concepts for Sylvia, which is very nearly the largest body with significant porosity.  By my data, it also has the greatest void volume (which comes from a combination of mass and density parameters).  You would need to place the pressure boundary somewhere below the 1 atm line, but anywhere within that boundary would do.  So here's what I have in mind:


This would create the maximum amount of volume with breathable air (without an "inflating" process).  It is huge, and the total mass of air which can be contained comes out to nearly 18% of Earth's entire atmosphere.  If you sum up values for even a small set of asteroids for which we know the porosity of, it becomes clear the the main belt could easily hold several times as much breathable air as Earth with this method.

This also gets around the problem of "inflatability" that I've referred to several times.  There's nothing to inflate and little matter to move around.  You just erect a thin plastic-ish barrier to keep the air in, and the "supports" for this surface are already there.  Well, almost.  There remains an issue that the crevices will not be trivial to seal.  If you look at packing patterns for spheres in 3D space, the area that needs to be sealed will be on the order of the diameter of the balls themselves.  This is a problem, but one possible solution would be to use the movable rubble as a backstop.  If the rubble comes in a fairly smooth size distribution, then the large rocks will be locked in place due to the pressure from gravity and the smaller ones will be floating around, but those smaller ones will be fitting into the interstitial locations.  If you corral those smaller rocks to the edge of where you want to put the pressure boundary, they can provide a more continuous backstop for the pressure boundary.  I tried to illustrate this below.

Detail of Pressure Boundary Engineering



At the smallest end, you might end up needing to mill some rocks into basically gravel, and use that to fill in the final crevices.  Basically, it's a hierarchy problem.  You will need rocks of all sizes to compliment themselves in order to create a relatively flat surface over which to overlay the air boundary.  This concept produces surprising parameters.

There is really no denying that:
  • This is possible with drastically minimal technology as opposed to other gravity balloon concepts
  • It can scale to diameters of around 300 km - unthinkably huge

Obvious problems with the design:
  • Pore size might not be large enough for artificial gravity rotating structures
  • Movable rock sizes might not be small enough to support pressure seal
  • Could be too much rubble, difficult to clear
  • Thermal and chemical impact on structural rocks from the breathable air

Obviously, you would not have a totally continuously atmosphere, of the type portrayed in the Virga world.  However, we don't know how large these pores are.  A single pore inside Sylvia might be as large as the largest inflatable gravity balloon.  It is truly a mind-blowing concept.  Of course, you can't get something for nothing, so this is a concept assisted by the material strength of the rubble.  Compressive, not tensile strength.  Gravity pulls "in" and the rocks push "out".  Then, the pressure boundary can physically be constructed with minimal effort by pushing "out" on a layer of rocks which are pulled "in" by gravity.  I return to the concept of breaking length that I've addressed before.  The size of Sylvia corresponds to a material breaking length of about 375 meters.

Equating Breaking Length on Earth to
Side of Porous Spherical Asteroid


In other words, the rocks are stressed to the same degree that they would be if sitting on the surface of Earth and raising to that height.  This is the type of scale we're looking at.

Demonstrated Rock Strength in Porous Asteroid Structures
Compared to the Eiffel Tower
(which also holds up against gravity)

Does that sound implausible?  No, not at all.  That scale is, actually extremely common on Earth.  We see cliffs in Earth's natural geography that rise much higher than this very frequently.  We even sometimes see free-standing rocks that approach the scale that I'm talking about.  Like this:

Example of Large Earth Rock that
Approaches the Relevant Scale

If the asteroid Sylvia is made of about the same type of rock as this thing, then it makes sense that it has not collapsed into itself, and it also makes sense that bodies much larger than it have.  This is also the boulder "size" that the inhabitants would have "hanging over their heads".  Of course, this has no direct bearing on the pore size, because those were formed in zero gravity, which evolves very differently from rocks on Earth.

For more information about this reference case, I compiled this table.

Parameters of a Habitat in 87 Sylvia

ParameterValue
Natural Asteroid Radius139.2 km
Central Rock Pressure46 atm
Radius for 1 atm137.7 km (98.9% of radius)
Radius for 2 atm136.1 km (97.8% of radius)
Usable Empty Space10.6 million km^3
Containable Air18.9% of Earth's atmosphere
Tubes that could fit10.6 million of reference design
Corresponding Population116 billion people
Incident Sunlight6.5 TW
Max Surface Radiation878 TW
Rock Thermal Energy53,659 TW-years
(heatup to room temp)
Surface Gravity0.05 m/s2 (0.005 g)
Weight of Tubes13,459 tonnes of force
Geosynchronous Radius205.8 km
Delta V for:
Center-to-Surface84 m/s (188 mph)
Surface-to-Space119 m/s (266 mph)


Many interesting things lie in this information.  For some values, I've used the reference tube design in my last entry.  That is a tube where about 20,000 people can live and uses fairly conservative assumptions.  It also assumes reasonable packing density.  With the parameters of this potential habitat, we find that it could house a huge number of people by this method.  However, that seems impractical.  There are energetic concerns that come along with this.  It might make sense to reduce people-density up an order of magnitude or so, and leave more empty space.  But that depends on the type of vision you're trying to realize.

Either way, the amount of air that could be held is gigantic.  This object has a semi-major axis of about 3.5 AU, which means that the solar radiation is much less, and insufficient to supply any more than maybe 1 billion people.  But if they had an independent nuclear fuel source, they use energy at a much higher rate.  I included the black-body radiation limit for this object's size at 293 Kelvin, which is room temperature.  The idea is that if you went above this number, you would eventually have to place radiators far above the surface itself.

Then again, why would you need radiators at all?  The object of 87 Sylvia is huge, and it's far below room temperature to start out at.  At that size, internal heat from nuclear decay isn't very significant, so the inside has probably equalized with the average surface temperature, which should be somewhere around 147 Kelvin.  If you were going to increase it from that temperature to room temperature, it would consume 10s of 100s of years of the full power output of such a society.  So in the beginning, there's no need for radiators in the first place.  But this bring up another good issue - might it be too cold?  Well, it would depend on the heat transfer rate, and this will depend on the size of the pores, which again, we don't know.  However, km-scale gravity balloons are almost perfectly insulated.  If the pores are workable large to begin with, then cool-down probably isn't going to be a problem.  I suspect that I could prove this with numbers, but I haven't done it yet.

Now let's talk about the issues with gravity.  In this concept we've forfeited the attractive idea of having a fully zero-gravity habitable area.  This will change many things, and stuff can no longer float around the volume.  The most important consequence is that the artificial gravity tubes will have to be anchored.  It would still take a really really long time for them to fall any significant distance, but it still must be dealt with by mechanical bearings of some type.  Cranes on Earth can have capacities on the order of 400 tonnes or so, so it seems that this problem is of a greater scale than familiar construction activities.  Still, the mechanical bearing wouldn't have to do much, it would be a simple tether, and it would be static.  For the scale of about 14,000 tonnes of force, I imagine that would be workable.

There are also some interesting orbital properties of this object.  It does have two moons, which has already attracted the attention of astronomers, although it has no relevance to the idea I'm pitching here.  If you wanted to use it as a space transport slingshot, you could since geosynchronous orbit isn't very far from its surface.  However, it's location is not ideal.  It is a poster-child of the asteroid belt, at about 3.5 AU.  That would create difficulty in getting to it.  Even extremely quick, snappy, trips would probably take on the order of years to complete.  There are much closer objects to set up simple habitats nearer to Mars orbit, and this is not one of them.  This is the gulf that you have to cross in order to find such a favorable natural structure which can serve as a habitat for countless numbers of humans.

I can think of some other really quirky issues with using this body in this way.  Some of the rubble might be ice.  Who knows?  As you use this to hold air at room-temperature, you might cause changes in the rock.  This would be bad if it caused collapse, but it also might cause unpredicted geysers and things of that sort.  There is sure to be no lack of material, and even a great diversity of material since it may have been created by accumulating many other smaller (diverse) bodies.  There's no telling what kind of scientific unknowns you would be dealing with.

For now, even the most basic geometry of this thing's interior is a secret.  But you could, in theory at least, just walk in, put up drapes, and you would could have a massive pressurized environment.

Monday, November 18, 2013

Rotating Tube Reference Design

Without doing much new analysis, I want to add a post in which I formalize the parameters associated with the tubes that rotate for gravity in my vision.  For gravity balloons, you start out getting to a usable object (like an asteroid), sealing off the pressure boundary, and then adding air.  After that, some rotating living area will be needed.  Many layers of sheets surround this structure in my vision, which reduce the drag on the construction.  This makes it a practical proposal, and that will not be true without the friction buffers.

So, just for the sake of completeness, I will present a set of workable parameters here.  These are all highly tunable.  You could change anything, but that will affect others.  I hope the links of what affects what is somewhat obvious.  First, here are the images, and more detail is included in tables below.


If you will, take this 1 km cubed box and repeat it spatially in your head.  I wanted to give a sense of how close they would all be, so I'm trying to illustrate this with the image below.


To be perfectly clear about what can be specified, and what follows from other values, I'm presenting the "independent" variables first.

Specified ParameterValueDescription
Inner Radius500 metersThe radius of the rotating tube, where people would be standing. This is basically the same for the floor and the structural supports.
Length500 metersTaking inspiration from the Kalpana One, this is a conservative choice to eliminate possible rotational instabilities. Longer tubes can be problematic.
Friction Buffers20 sheetsThis was selected from a balance of the number of sheets and power dissipation, discussed in a previous post.
Buffer Width103 meterAlso from the prior post in the subject, it's mostly constrained by volume constraints and diminishing marginal value of wider regions.
Population21,000 peopleYou could set this to a range of values, but this demands a reasonable power consumption for rotation and affords enough floor space.
Ramp Slope45 degreesThis is the rise over run slope for the ramp to the ends, which leads to the zero gravity space.
Access Diameter20 metersDiameter of the area open to the zero-gravity atmosphere at the end. The edges would have a slow speed and acceleration for moving in and out.
Mass Density2 tons/m2Assumed mass per unit area. This includes all lifestyle-associate things.

Variables which can be calculated from the system specified so far:

Derived ParameterValueDescription
Power for Rotation5.2 MWThis is the mechanical power needed to keep the tube spinning. It could be more when corrected for motor efficiency.
Usable Area0.79 km2Area with 1 g of gravity, so this does not include the access ramp.
Total Area1.32 km2Internal area, including the ramps. Obviously a good deal of this would still be usable.
Displaced Volume0.26 km3Volume including the friction buffer space. This means that 74% of the total space would be unoccupied in the repeating lattice.
Lattice Shape1 km3 boxThe arbitrary bounding box I'm using so that a repeating pattern of these can be discussed, converting volume metrics to more tangible things.

To make this more personable, I'm also including some per-capita parameters.

Per PersonValueDescription
Rotation Power250 WattsThe power needed to keep the construction spinning for each person. This is similar to an appliance, so it wouldn’t be overly burdensome.
Area37 m2Livable area per person, which corresponds to a relatively high density city. However, roads and other things can be through the center.
Mass per Person75 tonsThis is the total mass in the area that corresponds to one person's living space. For reference, a house may weigh 60 tons, so this is still relatively low, but workable.

For further information, I have looked into the farmland necessary to sustain a human.  The National Space Society has made a variety of claims, which result in parameters you can use, but the range is large.  Anyway, it would be reasonable to assume about 50 to 200 square meters would be necessary to grow food for one person.  This is obviously a problem if the rotating tubes include farm production.

I would advocate a different method of growing crops.  For some algae (and others), it would make the most sense to just have the suspended in zero gravity.  For most crops, however, I think it would make sense to grow them in low gravity, such as 1/10th Earth gravity.  With such a selection of parameters, you could have farmland with no friction buffers at all, which would be much more economical.

Sunday, November 17, 2013

Inflation Process of a Gravity Balloon

Let's talk about the process of turning an asteroid of a decent-size (about 60 km diameter) into a gravity balloon.  Doing so means that you have to deal with something I call the "pressure droop".  In one example, we start at a pressure of 3 Earth atmospheres (atm) and infate to the benchmark suggested by the National Space Society of 1/2 atm.

Pressure droop of a gravity balloon presents several difficulties, but it's a necessity that must be dealt with if the goal is to work with a single object and avoid moving large masses around our solar system in the construction process.  The constraint of constant habitability through construction (hereafter I'll call it "inflation") also confounds things a little bit.  Humans can survive in up to 3 atmospheres of compressed air without succumbing to Oxygen toxicity, but it would not be desirable.  It would be much more preferable to maintain relatively constant Oxygen partial pressure and fill the rest with an inert gas.

In space, the most available inert gases might be Argon or others, but for simplicity of analysis I'm sticking to Nitrogen gas.  In the most simple sense, air is a combination of Oxygen and Nitrogen gas.  Because the overall pressure of the gravity balloon is set by the pressure-volume relationship (which comes from gravity), we now have two constraints that dictate the quantity of gases that must be in the balloon at all times:
  1. Set the partial pressure of Oxygen to 0.21 atm
  2. Keep the rest filled with the inert gas
While this seems relatively simple, there are some tricks.  The above two requirements mean that to maintain constant habitability, you must insert new Oxygen and Nitrogen gas at constantly varying rates.  You can inflate at any total rate you like, but the ratios between the two are set by this requirement.  I've formulated the exact forms for the gas masses using the stipulated requirements, but I should note that they require use of functions that I've written about elsewhere on this blog.  There are some other qualifiers as well.  For instance, since the temperature is set by habitability constraints it is constant, so the density is proportional to pressure.  I use Earth sea-level as a benchmark to reference this to.  Given that, these are the equations needed.  The pressure is actually the function P_RM, and the volume is implemented as 4/3 pi r^3, which puts everything in terms of radius.

Masses of O2 and N2 for the Entire Process of Inflation
Expressions are Based on Radius and Independent Variable


Here the expressions are plotted for a particular set of parameters.


other parameters of this situation include:
Pressure at start of process is 3 atm
Density of asteroid rock is 1 g/cm^3

The rationale behind the Oxygen mass is somewhat self-obvious, but the Nitrogen gets interesting.  It's not unexpected that it drops beyond a certain point.  After all, if you were going to inflate the structure to the partial pressure of Oxygen itself, then you would have to eliminate all of the Nitrogen content in the process sooner or later.  In fact, for the situation described above, the pressure gets down to around 0.8 atm before it makes sense to actually remove Nitrogen.  But if you have to remove some of the gas, that is difficult in-and-of itself, but it's also expensive and non-ideal.

A more likely outlook is that the relatively small final droop in Nitrogen mass would be avoided altogether.  This could be done if the Oxygen partial pressure is fungible to some degree, and it is.  Instead of using the above program, you would probably bump up the Oxygen content a little bit before the Nitrogen peak, and then only level it out to the desired final concentrations.  This would be fairly workable for a goal of 0.5 atm.  But of course, as good scientists, we need to work to break the model first.

I struggled a little bit to reduce the problem to a set of unit-less parameters.  The issue being described here is mostly a geometric issue (meaning it can be reduced to dimensionless constants), but with an added pressure scale due to the demands of humans to have a constant pressure of Oxygen.  Well I figured a method out.  Here are some extra terms I need to introduce in order to communicate the dimensionless parameters:
  • R_max_N2: the radius at which the Nitrogen mass reaches an absolute maximum in the inflation process
  • R_PO2: the radius you would inflate it to if you continued until the pressure dropped down to the partial pressure of Oxygen
  • Ratio of these two: a good dimensionless metric for the problem at hand, which is generally the radius relative to the ultimate inflatable radius
  • Pressure ratio: the ratio of the initial central pressure to the partial pressure of Oxygen (note that this is not the same as the ratio for the maximum Nitrogen point)

I will be referring to this point of maximum Nitrogen as the Nitrogen Tipping Point because it is, in fact, an undesirable result for someone considering building a gravity balloon.  You would not like to have to "trash" any of the valuable gas you produce.  It follows that the more swing in N2 gas is, the more difficult the entire construction would be.  It's not hard to mentally picture this problem becoming very difficult for extremely large cases.  You need to produce lots of gas to initially "prop" it up, like a car jack for changing a tire.  The cost of doing that prop might be prohibitive, so we need to know in what cases that will happen.

If the radii ratio is close to 1, then the Nitrogen tipping point can not be a problem.  You will, after all, desire to leave some inert gas in there at the end of the process, so you will not need to remove any of it.  If the radii ratio is very low, you may have a problem.


If you had your hopes set on "reasonable" size gravity balloons, then this might be a sigh of relief to you.  There is hardly any practical scenario where a pressure droop of a factor of 10 would be tolerated.  At that point, you would be working with something which was initially uninhabitable.  There are lots of really cool ideas that see pressure ratios for the inflation process span from 1.1 to about 6 or so.  Throughout all of these, the inert gas that needs to be trashed is basically small - to the point that a mild O2 partial pressure adjustment could result in nothing being thrown away.

I should note that for any futurologists with big dreams - large gravity balloons are still viable in a sense.  I took the ratio of the PV product for the Nitrogen Tipping Point relative to the PV product for the fully inflated (only Oxygen) point, and it never surpasses a small fraction of the total.  So if you resolve to "throw away" a "prop" gas in this process, it won't be large in volume compared to the total amount of atmosphere you have to make for the habitat.

But where would all this gas come from?

I don't know.  This is a subject a bit outside of what I know well.  The logical options are:
  1. Transport volatiles from somewhere they're abundant (comets, Ceres, Mars, Jupiter)
  2. Process the asteroid rock itself to extract N2 and O2
The latter option is obviously more appealing, but it's not obviously easy.  Mineral chemical forms aren't often easy to work with.  You might have the option of just putting crushed up rock in an autoclave, heating it, and then separating what comes out.  That would be the ideal scenario.  It's also vaguely more plausible with the center material rather than the outside material.  Since you'll be drilling down there anyway that's obviously not a deal breaker.  But there's no guarantee it will be this easy.

Perhaps a better question is to set a defined limit to start out with.  Is there enough O and N to begin with?  In C-type asteroids, the answer is probably "yes", but barely.  We have some good literature on the elemental compositions of meteorites, which are vaguely representative of the asteroid materials.  The result from that is:
  • Nitrogen: 0.14%
  • Oxygen: 40%
It seems that there is no shortage of Oxygen at all.  Nitrogen is the difficult one.  If we imagine an asteroid with an initial 3 atm central pressure, inflated to 0.5 atm, then that has a Nitrogen requirement of about 2e14 kg.  Using the above ratio, we can find that the elemental Nitrogen content of the original asteroid is about 3.5e14 kg.  This means that you would need to extract all the Nitrogen out of over half of the body, and this is just not reasonable.

Even looking at common minerals, like olivine, isn't particularly helpful.  I have not seen a chemical formula with N attached to it which is predicted to be common in C-type asteroids.  However, I think this is because of my own lack of thoroughness rather than anything intrinsic.

A more promising approach is to look to predicted out-gassing of asteroids.  This paper, for instance, addresses the exact thing I'm interested in.  It even mentions N2 gas itself, although in fairly low concentrations.  There are many other compounds with Nitrogen attached to it, and probably some obvious chemical processes that could lead to extraction of it.  But that doesn't guarantee this would be helpful for a gravity balloon.  It may be that the analysis that shows common Nitrogen is only applicable to bodies much larger than needed for an effective gravity balloon.

In short, it's hard to say with my level of knowledge.  Or more briefly, I don't know.  There still seems to be no reason to rule out the gas production as a deal-breaker.  It could certainly be done without the economics blowing up.  I just don't know how.

Wednesday, November 13, 2013

The Principle of Mediocrity (regarding access and size)

Or: why the ideal candidate depends on what you're after

Conceptually, this blog is mostly focused on the idea of "moving into" an asteroid by drilling a hole into the center, sealing the area off, and slowly replacing the rock with air.  Denser, more metalic, rock will have a higher central pressure for the same mass.  This effect is actually quite dramatic.  Let's consider the wall thickness needed to have a marginally small balloon of air in the center.

Thickness need to create 1 atmosphere of pressure at center
varies a great deal with the material density 



I used several density benchmarks from a useful paper for the points in that graph, which just represent different cases.  From this, you would obviously conclude that a more dense asteroid would be more easier to work with, at least at first.  This is somewhat the case with Eros.  Its material has a specific gravity of 2.67 on average, so it can hold normal atmosphere even though it is much smaller than Phobos, which struggles to do so itself.

The more complicated twist is a concept I call "Inflatability", which seems approproate given the balloon references.  Basically, as you enlarge the center cavity, the pressure due to the force on gravity on the rock walls decreases, and the air pressure experienced in the habitat (in the absence of other structural factors) droops.  This will obviously put the pressure below the habitable limit at some point, and this point will be different for different bodies.  However, the math works out in an interesting way.  Basically, the geometry can be reduced to a "dimensionless parameter", which represents a relative degree of inflation.  So start with a marginal central volume, and then increase this to a final state that looks more like a balloon.  We can show that the percentage drop in pressure follows from the fraction of the asteroids initial radius the volume is expanded to.  This is actually fairly easy to picture with a graph that I've sense updated since I revisited the governing equations last post.

Pressure Droop with Radius Increase of Habitable Area
Radius is the Inner Air Bubble Radius divided by the initial asteroid radius


Side note: This graph was corrected from the previous version.  This one correctly reflects the slight stall at low radii, due to the nature of the hole's effect.  Keep in mind that volume goes as R^3, so the function's curvature is uniformly positive when in terms of volume.  I still included the case for an initially rotating asteroid but this would be extremely uncommon in practice - it's rare to see a center pressure correction of more than 1% due to rotation.


If you think about it, this raises an interesting dilema.  Since the pressure droops according to the *fraction* of the initial radius, lower density bodies have a lower droop, but longer access tunnel.  Consider that this is a tradeoff of considerations that would matter more to a civilization depending on the stage of development they're on.  Their challenge might be:
  1. Establishing the access tunnel and pressure seal
  2. Ability to further increase the size of the habitat
A selection that skimps on the first point will go on to have trouble with the second point.  If your goal is a massive mixed gravity world, then you can't take the "easy route" of a dense asteroid, which may only need an access tunnel of 5 km.  You'll have to start with a less dense asteroid that likely has a starting radius of 20 km or more.  As a rule of thumb, the pressure starts to very significantly drop when your habitat radius approaches the original asteroid radius.

The droop that can be tolerated isn't very clear to establish.  Skylab was a space station with a cabin pressure of around 0.34 atmospheres, so this is obviously workable, but it's not clear if its desirable long term.  The NSS suggests 0.5 atmospheres for a permanent space station.  On the other end, I've looked into compressed air Oxygen toxicity limits, which tend to be a problem around 3 atmospheres.  In fact, people have spent days on-end in this kind of pressure.  However, Earth has a pretty constant pressure for the most part, and the upper limit is based on assumptions that don't have to be true.  So we conclude that both ends of this limit can be pushed one way or the other, but for now, we might as well look to a factor of 8x or so just to establish the maximum pressure fluctuation that we'll tollerate.  This isn't absolute, but by the nature of biological properties of humans, the error bars are large.  For a given pressure ratio, we look to querry how large of a habitat we can make.  I tried to illustrate this with the following graph.

Pressure droop ratio that follows from a given
Radius of Inflation



This tells a very interesting story.  The merit of a gravity balloon is large volume with no structural materials.  This puts the maximum size you can produce into a helpful context.  These only radii figures, so it follows that (for instance) a livable diameter of 80 km can be produced by using an asteroid with a specific gravity of 1.3 and tollerating a pressure drop of a factor of 8.  Now, an 80 km sphere is pretty large, but I certainly think it's a desirable *concept*, although obviously long-term.  Even if you can't go beyond a diameter of 2 or 4 km, creating such a habitat would be a mindblowing advancement of humans into space.  For every case, however, producing the air itself will remain a challenge since it scales directly proportionally with volume.