Introduction to the Gravity Balloon

Ideation of space habits rests in a strange place between fiction and science.  I believe it's a subject part of a great blue unknown that real researchers won't write in detail about for fear of being taken less serious, or perhaps for their loyalty to the practical.

To sweepingly divide all proposals into two categories, some things have a concrete path with current infrastructure... and some don't.  The tragedy of neglecting the latter is that we may hold back an inspiring vision.  Some imminently possible pictures of the future require a rainbow of imaginary technology.  If done well, this requires suspension of disbelief, not in any physical principle, but that such an infrastructure so different from today will be thought a worthy pursuit by our descendants.

There are two routes for extending human presence into space and they are both completely unattractive for anytime soon.  On one hand, space habitats in empty space don't scale well, and on the other hand, terrestrial environments are inhospitable locations at the end of an inhospitable trip.  Artificial gravity concepts are generally proposed for empty space, although there is the occasional exception of moon hotels.  Popular literature divides these rotating space stations into annular structures, such as the Bernal sphere and O'Neil cylinder, and toroidal structures such as the Stanford torus.  If you think about these from a mechanical engineering standpoint, the load-bearing materials will be found either around the circumference or directly span the diameter.  As it turns out, the material requirements for the two approaches are identical as long as you use sufficient potent materials.

Most of the common discussed ideas probably outright ignore the needs of radiation protection.  In a sense that's not wrong, since we already have people temporarily living in space with only their pressure vessel to provide shielding from radiation.  The problem is that the International Space Station habitation isn't sustainable.  The radiation dose on-board is 100s of times higher than Earth, and the material quantity needed to get parity between those is intimidating.  If you are crediting that radiation shielding is integral to the structure (which almost all proposals do), the idea of a long-term artificial gravity space station is at least somewhat self-defeating.  Either it has very high radiation levels (not so long-term anymore), or you will need a much stronger structure to hold all the shielding in place.  This particular problem is solvable by surrounding the station with shielding that doesn't rotate along with it.  In order to further simplify the structural needs for supporting artificial gravity, one could also use modularized pressurized containers, or the toroidal approach.

Approaches for artificial gravity space station cross-sections:

Now, limiting the discussion to the first of these, an annular space habitat, we can discuss the magnitude of materials needed.  The material strength needed per unit area increases with increasing radius.  This runs contrary to another constraint, minimizing the Coriolis force, which calls for as large of a structure as possible.  Keeping in mind that the requirements scale with radius, we can directly compare the contributions from the three components of load, internal pressure, and radiation shielding.  The "standard ruler" for this is meters of water equivalent, which creates an instant intuition to daily life.  The pressure above us is equivalent to the additional pressure you will experience going underneath 10 meters of water.  That is to say, the mass-thickness of the atmosphere above us is the same as 10 meters of water above us.  That is also the same value that we can thank for shielding from harmful cosmic rays, which is the reason for correspondence between shielding and pressure effective thickness.  Finally, we have the actual stuff that would exist within the space station.  The value for this is inherent subjective, but if you can imagine you and all of your possessions being melted down and coalescing to a constant fluid level on the ground, that's an appropriate interpretation of it's water-equivalent mass-thickness.  In the following image I'm illustrating this as 2 meters, which is quite high, but one needs to keep in mind that any amount of agriculture will require substantial amounts of soil and water, which put together, may be enough for you to drown in.

Wall makeup of the annular habitat:

This shows just how budensome the material requirements of the typical vision of an artifical gravity space habitat are.  In the modular approach to pressure modules, the load is decoupled from the pressure support, but it doesn't fundamentally change anything about the difficultity of these structures for very deep physical reasons.  Any pressure vessel in empty space that we speak of is inherently assumed to be made of rigid materails supporting that pressure.  That isn't completely other-worldly, since you go into a pressure vessel every time you fly commercially.  When a plane rises to 30,000 feet it maintains an internal pressure considerably higher than outside.  In order to do this, however, the plane's body has to be heavily reinforced and sealed well.  Obviously this is even more true for space habitats.  A good question that follows to ask is if we can find some way to make a pressurized volume that reduces the materials needed.  The answer is that we can't.  Consider the equation for material stress in a pressurized sphere.  The specifics aren't important other than the fact that the quantity PV, pressure times volume, is equal to an expression times mass.

Material stress for pressurized sphere

This is an unfortunate physical reality that will not budge for us and fundamentally limits the rate at which space habitats can scale.  In order to expand a space habitat, not only do we have to bring in new material proportional to the volume we want, but we have to process that material into strong structural fiber.  it doesn't matter if you change the shape or add cross-support, the above proportionality remains.  Perhaps the only "good" news is that it still scales better than building skyscrapers, which becomes MORE material intensive the higher it goes.

Breaking the mold

It is argued that space can usher in an age of abundance.  If the economics of self-replicating zero-gravity industrial infrastructure is as good as we hope it can be, then perhaps habitat mass that scales with volume is simply irrelevant, since the advanced civilization could make as much of anything as they need.  I see this as using the "brute force" thinking of the industrial age of oil.  It is better to paint a vision of the future that cleverly works with the laws of nature.  The expensive approach I've described so far for an artificial gravity station is a go-to plot device for many science fiction authors, or at least those who make an attempt at realism.  They have, themselves, perfectly illustrated the problems with the concept by invoking Carbon Nano-tubes to hold together habitats of sufficient sizes to mimic Earth.

Alternative proposals seem hard to come by, but here are two that I know of with some amount of merit.

Trent Waddington is an active space proponent online that has attempted a proposal for colonization inside an asteroid that includes artificial gravity.  He details his observations in his Living Inside An Asteroid post, which calls for tunnels and a circular train track for artificial gravity.  Although left unsaid, I would presume that the tunnels are a full vacuum and the train contains the pressure boundary.  Under the assumption that the asteroid is fully rigid this could make sense.  Most importantly, the force on the tracks would replace the tensile strength in my example of a modular rotating habitat with stationary shielding.

To attack his idea, the friction with the tracks seems to be an unresolved issue.  The absence of gravity and a vacuum environment doesn't change anything about rolling friction.  If anything, it makes it harder because lubrication is no longer a simple thing.  There are ways around this, certainly.  The larger issue is that nothing has been done about the scaling of pressure vessel materials.  Given that's not solved, there's not really an benefit to eliminating the structural materials for the load, remember, tensile strength to create artificial gravity pales in comparison to what's needed to hold internal pressure.  It's hard to see how any track-based approach would be superior to a simple tether, save for the unique case that there is an extremely high premium on reducing Coriolis forces.  Even in that case, the largest spinning radii that are practical to support with a tether is also close to the limit of perfect asteroid rigidity.

Karl Schroeder is a science fiction author that has written several fiction books based in a fantasy world called Virgra, which is mostly stagnant atmosphere contained in a "space balloon".  The volume is extremely large at 5000 miles in diameter and is littered with human presence in the form of rotating habitats.  It is a good president of someone exploring the immensity of an Earth-sized three-dimensional world enabled by zero gravity.  It also highlights how easy certain things become (like transportation) when you eliminate the need for pressure vessels to cope with the environment of space.

Nonetheless, I absolutely can not understand how he possibly thought it was a good idea to claim the balloon was made of carbon nano-tubes, in spite of apparently being directly told by a physicist that self-gravity alone could hold the air in.  When science fiction authors use unrealistic technology to get around a problem, then that's just how things go.  But here we have a case where the author invoked extremely exotic technology when completely ordinary stuff would have worked just fine.  Carbon nano-tubes are cool, but one has to consider the fact that we're specifically talking about employing them for >1,000 years in a high radiation environment with huge thermal stresses as well as ice constantly forms and breaks off the inside.  The world also has a completely unnecessary and unphysical giant sun located in the middle of the thing.  Neither of these things are the biggest problems with Virgra.  One could just imagine replacing them and call what's left physically accurate, if not for....

The spinning structures, for which:

"After all, any one of the smaller rotating structures could still be hundreds of kilometers in diameter."

No, they couldn't.  Even the smallest acceptable radius for Coriolis forces (about 500 meter diameter, 2 rpm) would be subject to hurricane gale force winds in this environment.  Given that the atmosphere is the same that we know on Earth, with plenty of N2 diluting the Oxygen, the speed of sound is also the same, and beyond a diameter of roughly 10 km you find yourself at fully supersonic speeds.  Now you may say "well the air will be dragged along with the tube", and it won't be.  This is a well understood problem if you take it to be a cylinder rotating in open air.  It is in a highly turbulent regime, and the type and amount of turbulence you will see under particular flow conditions is well understood and well studied.  I looked into some papers on this, and with some simple calculations got a general idea of the sheer force that the tube would see, and it's on the order of 80 Watts per square meter.  Multiply that by the entire surface are of the tube, and even for the slender habitats, you're looking at massive power dissipation by the air, which is basically what a hurricane is.

There are plenty of others that seriously do not have any real merit.  Some are mind-blowing by just how inefficient and round-about they are.

The gravity balloon

I argue for an approach of using an asteroid's self-gravity to obtain livable pressure.  This has several challenges associated with it, but I argue that once those can be overcome, extremely favorable scaling is obtained.  Most importantly, a volume can be built that satisfies all the requirements for long-term habitation while keeping all of the flexibility of a small gravity well.  This can all be done with a comparatively small material investment.

In short - start with at an asteroid of a diameter on the order of 20 km to 100 km.  Establish a transport tunnel to the center of the asteroid.  Begin production of useful gases (Oxygen would be nice), and pump it into the center.  Both rigid candidates and rubble piles can be used, but depending on which it is, material will either have to be cut out of the center and transported out in place of the gas, or the sides will simply be allowed to buckle and grow, forced by the pressure of the gas itself.  In that latter case, there is still concern of the Raleigh-Taylor instability, and this would have to be dealt with with basically a liner.

For our efforts, we get a large volume of air with no need for structural materials.

Ultimately, this results in an uphill battle against asteroid differentiation.  Nature "wants" to sort the materials in order of density, and we are going in the other direction by using an outer shell of heavy matter to hold an internal pressure that is amendable to life.  Why do I think this is a solvable problem?  There are several benefits the particular geometry affords us, but one should also keep in mind that many asteroids of this size already are not differentiated.  Also, since gas is so incredibly light compared to gas, the gravitational field at the wall is practically nothing.  Keeping giant boulders from falling off the inner wall would be accomplished by the force of a fly's weight.  More importantly, any membrane that separates gases at all can keep the inner boundary stable.  There are still global concerns against differentiation, you can imagine the danger of the central bubble floating to the top like in a bath tub.  Like the other concerns, however, nature is kind to us in this case.  The field at the center of any body is already zero, and provided you don't deviate far from this position, any small imbalances can be corrected very slowly (by pushing boulders around) without fear of catastrophic failure.  Finally, it's a very quantifiable problem.  The only big unknown missing is the nature of the asteroid material, which we already hope to be gaining a great amount of information about.

The starting point for the minimum asteroid size is illustrated below.  We consider the minimum size the smallest asteroid mass that will have nearly an atmosphere at its center.  In real life, density matters in addition to the mass.  I've assumed 1.3 specific gravity for the purposes of illustration, but the calculations can be done with other densities as well.  As a general rule, higher density buys you higher pressure.

For the reference "large" case, I sought the smallest size that initially has a tolerable internal pressure.  Since the gravity is maintained by self-gravity, the pressure decreases as you add more gas.

The end goal could then be for a very stable habitat for a very large number of people for a very long time.  Alternatively, this can (and would) be used for simple storage of large quantities of gas in deep space, irrelevant of habitation.  But wait, I was just badmouthing the problems with spinning for artifical gravity in an atmosphere.  What about that problem?  Like other problems, it's solvable.  It just needs careful thought.

To begin working at the problem, let's ask the question: what fundamental energetic limits does nature impose on friction?  Well, viscosity turns out to be pretty fundamental.  Imagine the simple situation of Taylor-Couette flow.  In that case, we are speaking about a plane moving at a certain velocity relative to another plane.  In the large radius limit (which is all we're concerned with), the sheer force for laminar flow is:

$$ \tau = \frac{ \mu U }{d} $$

• U - relative speed between plates
• d - distance between plates
• mu - air dynamic viscosity

If you plug in some reasonable-sounding value for the distance between sheets, you wind up with a laminar value for energy loss per unit area of milliwatts per square meter, which is entirely tolerable as long as you have at least a somewhat economically competitive energy source.  But that's not a real thing - this flow isn't laminar so the above equation doesn't apply.  But let's put it another way, can we make the above equation apply?  Signs point to "yes".  You should notice the appearance of U divided by d.  If you insert a sheet between the two boundary sheets that moves at a velocity half way between them, you decrease both variables by the same factor... and at the same time get closer to the laminar cutoff point.
http://physics.stackexchange.com/questions/55387/can-a-divider-laminarize-turbulent-flow-and-thus-reduce-friction

So here's the picture we're building, we can "tame" the flow around a rotating cylinder for artificial gravity by inserting intermediate sheets between the rotating structure and the external air.  For the sake of mathematics, we can simply imagine the final sheet to be fully stationary.

Illustration of laminarization

Scaled up to a practical application around rotating habitats, we're looking at something like the following illustration.  You can't just apply this "friction buffer" around the outside edge, you would have to taper the ends to a pinch so that the relative speed of the sheets is small and can be managed.  One could also imagine the ability to walk "up" this end ramp, which provides access to the bulk zero-gravity gas and back again.

Illustration of rotating habitats with friction buffer

Some questions do remain.  For instance, how do you keep the sheets in place?  That's not entirely clear, but it's possible that a pressure differential could help.  This could be maintained around the end seals or by using sheets that are semi-permeable and allowing air to slowly leak out from the habitat - causing the rotating structure to operate partially like a centrifugal pump.  There's another difficult reality to deal with - if the sheets are no more than a few centimeters apart even a small wobble could cause them to bump and possibly be destroyed.  As it would turn out, water itself seems to offer great promise as a ballast to minimize these wobbles.  Aside from this, one can also image the possibility of intelligently placed counterweights.  These could be in the form of interconnected water towers.

The number of stages needed for the friction buffer would depend on the amount of space allocated to the friction buffer.  For a simple calculation, I find around 200 are needed for a 500 meter diameter tube.  I imagine this would almost certainly be reduced in reality, accepting some losses to turbulence for greater simplicity and less effort.

In the end, it seems that energetic limits might be more challenging than anything related to providing air, gravity, and radiation shielding.  The asteroid belt doesn't even start until further than 2 astronomical units from the sun.  In terms of candidates within the inner solar system, we seem to be limited to Phobos and 433 Eros.  Within the asteroid belt, there are abundant candidates within the size parameters I've entertained - I estimate on the order of 530.

The problem with starting at a distance of 2 AU, however, is that light is attenuated by a factor of 4.  A large array of mirrors would be needed to gather a significant amount of light.  The size where this light can be focused, however, can be pushed pretty small (though not infinitely small).  If sunlight was the primary energy resource, then maybe this isn't the right track to take.  Of course, nuclear power of sorts is still a possibility, which would add a distinct heat source to the volume.  Active removal of heat would become a necessity at some early point, since the 10 to 20 km of asteroid material would be an almost perfect insulator.  The challenge of energy cycling would grow with the size of the volume if we assume some relatively constant number of people per unit volume.  Using the method and the asteroid size range I've discussed so far I calculated the total amount of volume that one could obtain by inflating all of them, and found a number on the order of 1% of the total volume of Earth's atmosphere (would have if it were all at sea level density).  Then, I scaled this to 7 billion people just for fun to get a person-volume density.  The the previously discussed "large" case resolves to roughly 93 million people, which can imply energy needs on the order of 200 GW.  Putting all this together, this is my illustration of the situation.  The rotating habitats are really just provided for scale.  I don't think it would be practical to make any much bigger than the 500 meter diameter.

A few things I thought about including but didn't include pipes for heat removal, transportation facilities, and tethering of the rotating habitats themselves.  It's thinkable that they could just float around.  Interestingly, the period of oscillation only depends on the density of air in there and comes out to something like 2 days.  But drag force would constrain those oscillations to near the center for this size.  I think, however, that with even very simple fan propulsion the habitats could steer themselves.