## 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.

Parameters of a Habitat in 87 Sylvia

ParameterValue
Central Rock Pressure46 atm
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
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
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

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

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.

## Monday, November 4, 2013

### Basic Gravity Equations for a Sphere with a Hole in it

For a solid sphere, expressions for the gravity inside and outside of this body are fairly easy to find online.  That is (4/3) G Pi rho r within it, and G M / r^2 outside of it.  Now cut out a sphere in the center of this sphere, and replace it with empty space.  The gravity in this case will not be linearly increasing, and this result took some work for me to figure out.  The most simple correct approach is to subtract gravity from the removed sphere, which is using the principle of superposition.  This is straightforward algebra, so I've put that in the following equation, and a comprehensive roundup of the gravitational field forms inside the hole, in the shell, and outside it all.
• solid: the entire sphere without the hole cut out
• hole: the spherical region in the center that is cut out
• shell: the points within the solid sphere that is not in the cut out region
• R: the radius of the inner sphere cut out
• t: the difference between the original sphere's radius and the hole radius

Gravitational Field for Solid Sphere with Hole in Center

Since it's what I'm interested in, I derive the pressure of the center cavity due to hydrostatic force from the shell's weight.  The principle is that height time field equals gravitational potential.  Since the field is constantly changing here, we integrate that, but only do it in the shell region because it's the only area that has a non-zero density.

Getting Pressure from the Field

This is the most important equation related to a gravity balloon, since it quantifies the pressure with respect to the other physical parameters.  However, this form isn't necessarily the most useful possible combination.  For any given body, we would like to presume that we're starting out with the mass value because there's only so much matter in that place to work with.  That makes the task a little more computationally difficult.  To do this, we introduce a mass balance equation.  This is nothing more than a geometric statement, with calculation of the volume of the shell.

Pressure and Mass Equations

These give two equations with 5 variables.  In any case, we're probably going to know the density.  That leaves 4 variables and 2 equations, for 2 degrees of freedom.  For instance, these could be satisfied by specifying what mass you start with and then how much of an inner radius you excavate.  That will then return a pressure.  Alternatively, maybe we query what radius we can get with a given asteroid, assuming that we want Earth sea level pressure.  That case can present some difficulty since there is no answer when the mass is below a certain value.

I've addressed several combinations of values.  The governing equations can be rearranged quite simply for several cases to get an explicit function.  This is done in a simple form for the pressure and mass equations individually, and then in combination between them.  Two functions are then left which can't be found except for with a cubic formula application (or worse), so these are left as implicit functions.

Function Set

All of the above about directly implimentable, except for the fact that the implicit functions need a solver in order to work.  I found Newton's method to be the ideal option for this.  The challenge is keeping the convergence in check, but that's workable since these functions are relatively simple.  Implicit solves are often somewhat of an art form.

The thickness of the shell (variable t) demonstrates a behavior where it remains in a definable range for the small and large cases.  This is useful to set a guess at the start of Newton's method, as a simple average of those two limit cases are used.  For the guess of the radius (looked up by pressure and mass), a similar type of trick is used.  A limit case of thickness is put into the equation for radius, and that gives a value.  We want to over-estimate the radius because we don't want it to ever go negative.

Guesses for Variables in the Implicit Solves

In my formulation, the functions don't start out as a root find, so the objective value (for instance, of pressure) is subtracted from the listed function, which doesn't affect the derivative.  The derivates can be found algebraically fairly easily, but there's a complication in the case of R_MP.  That is a root find on a function which takes the output of another function as an argument.  That requires application of the chain rule for differentiation, but the exact way it plays out is a little bit complicated.

Multivariate Chain Rule with Composite Function

That pretty well covers everything needed to get correct and efficient implimentation of lookups for a gravity balloon.  The total list of functions are:
• dPdR_RM (R , M , rho )
• dPdR_Rt (R , t , rho )
• dPdt_Rt (R , t , rho )
• dtdR_RM (R , M , rho )
• M_RP (R , P , rho )
• M_Rt (R , t , rho )
• P_RM (R , M , rho )
• P_Rt (R , t , rho )
• R_MP (M , P , rho )
• R_Mt (M , t , rho )
• R_Pt (P , t , rho )
• t_RM (R , M , rho )
• t_RP (R , P , rho )

As a side note, you need functions for the gravitational constant, G.  Also pi.  I implimented all of this, and put the code here:

The Visual Basic (for Excel) code link on Pastebin

This post covers the basic mathematics that underpin the entire idea.  I wanted to make this one technical post in extreme detail because I had these equations wrong before, so this is a correction of some of the things posted in this blog, although not everything.  I had made the wrong assumption that gravity within the shell would be linear, and this is demonstrably false by a couple of arguments.  I have good confidence in the equations here, and the functions seem to perform rather well to the extent of my testing so far.  It's good to have this documented well, because it forms the starting point for all of the rest of the discussions.