Friday, October 25, 2013

Inclusion of Air Pressure Effects for Super Large Gravity Balloons

Although impractically huge, the math of a gravity balloon changes a great deal when they get beyond a certain size, in large part due to the pressure of the air itself.  There are quite a few related discussions that stem from this.  On Earth, for instance, we need our large gravity well to hold in air, but it's not so much because the upper atmosphere extends to escape-able distances at thermal velocities - far from it.  In terms of sheer value of density, our atmosphere is basically gone when you reach 40 km.

That altitude represents a Delta V value (the speed you would accelerate to if you fell that height in a vacuum) of 0.6 km/s, whereas the escape velocity for Earth itself is around 11 km/s.  The role our gravity-well plays is, instead, to fight against the "boiling" off our our upper atmosphere by higher energy particles from the sun and the cosmos.

To put this in different terms, if you wanted to blanket the Earth in a sheet to hold in its atmosphere, that sheet wouldn't need to be very thick.  A simple mathematical treatment of this is to say that the pressure drops off exponentially.  You need a constant to use in that equation, which is simple, and given by this reference to be 7 km.  You can very easily see that at 40 km, you're left with less than 1% of the air pressure, according to [P = P0 exp(- h/h0 )].

One might correctly notice that this has implications for a gravity balloon.  This raises the possibility of a super large type of gravity balloon, which perhaps looks more like a gas giant planet than a space station.  Formalize the problem by requiring a habitable pressure in the center.  Then that pressure decreases as you move away from the center.  There are a few possible non-ideal consequences of this, which are quite interesting from a physics standpoint.  Most obvious, the outer regions might not be entirely habitable due to a low barometric pressure.  This would only happen for extremely large sizes, and represents the ultimate upper limit.  Second, you forfeit the zero-gravity environment for a large part of the volume.  While the outer regions aren't zero gravity, they can still be very low gravity.  Going from 1 atmosphere of pressure to near-zero pressure requires a certain potential difference, but unlike Earth, that difference may be spread over a large difference for a super large gravity balloon, resulting is a very low field at any point.  It's possible that any residual gravitational field could be counteracted by wind turbines which have relatively low energy requirements.

The world I am describing comes awfully close to the conception of the Virga world.  Coincidentally, they just came out with a new comic series set in the Virga world, which I hope to write more about later.  In addition to the changing pressures, this world has convection currents (which might be physically accurate).  The author also goes on to write in a great deal of politics and dynamics to this world, which is tangential to this blog.

I have noted (as have others) that the Virga world doesn't need to have a wall of carbon nanotubes, but I haven't given very detailed consideration to how thick the walls might need to be, or even if these walls would be stable.  Wall thickness is, after all, a function of the pressure near those walls.  The technical guidance we have for Virga is that it is 5,000 miles in diameter.  I don't think we are given a pressure at any point inside this, but I will happily assume 1 Earth atmosphere in the center.

The Mathematical System

Governing equations are straightforward, but I found them deceptively subtle to get correct.  There are number of components, so I'll outline them in a list.  The goal here is to obtain a set of 2 differential equations that describe the atmosphere pressure and gravitational field within this structure as a function of the radius.  In order to get there, we have to formalize several physics equations.

Technical steps:
  1. I used the ideal gas assumption and constant temperature.  This relates pressure and density in a linear relationship.  Use the specific gas constant to write this explicitly.
  2. Mass is taken to be a function of radius, and includes all air mass below that radius.  This is then related to the field, making use of the shell theorem.
  3. Change in mass for a differential increase in radius is the sphere surface at that radius times the air density at that radius.  This is a geometrical statement.  Additionally, density is replaced with pressure from the ideal gas equation.
  4. Initial conditions are obvious.  The gravitational field is zero at the center, and this formalization specifies the pressure at the center to be the habitable goal.
  5. The differential equation for mass is then changed to be in terms of gravitational field, which comes from making a substitution from the shell theorem.  Some algebra is done, using the chain rule of calculus, and then rearranged.
  6. Pressure falls according to the density of air and the gravitational field, in the same way that Earth's atmosphere does, which I outlined in the introduction here.  This is written in a form usable as a differential equation.
For such a large volume, transport of heat out from the center to the outer regions becomes non-trivial.  This makes the assumption of constant temperature a little dubious.  We would probably expect some falling temperature with radius.  This is still pure presumption as to how this thing would actually be built, so I'll keep T constant for now.  The equations are complicated enough as they are.

This math is a little bit intimidating, and it turns out, the system can't be directly solved easily.  In the following equations, look at the last 3 lines.  Those fully specify the differential equation system.  With the constants filled in, you can put this directly into a mathematical software package.  Unfortunately, I have not yet found such a package that will give an algebraic answer, so I will have to satisfy myself noting that it can't be done without extremely exotic functions.

Governing Equations of the system
Last three constitute complete differential equations


The constants need definition.  Since this is a technical blog, I will list all of the physical constants employed here.  The gas constant is for air at sea level.  Generally, the values are sought to provide a normal Earth room temperature atmosphere, consistent with most of this blog and the idea of Virga.  It's possible to make an extension of this math to do primitive analysis of gas giants or stars.  Hopefully I can do that as another post some other time.
  • R_{specific} = 287.058  J / (kg K)
  • T = 293 K
  • G is Newton's gravitational constant = 6.67384e-11 m3/(kg s)
With this, the system is fully specified mathematically.  You have the ability to input the above equations, with the above constants, into a numerical integrator and obtain a spatial picture of the pressure and field within one of these bodies.  I have made my own code to do these calculations available on Pastebin here.

When talking about a gravity balloon, we are terminating the gas by adding a wall (in literal terms).  By the shell theorem, we should be comfortable ignoring everything beyond whatever radius we're looking at, because the gravitational field contributions all cancel out.  That is why these mathematics are relevant for large gravity balloons.

I was also interested in the effect that setting different pressures would have.  I tested two cases, where the central pressure was 1 atm and 3 atm.  A surprising result came out of this - that the pressure at large radii was lower when starting at a higher central pressure.  Actually, this makes complete sense.  This is why gases consolidate into gas giants instead of always hanging out in a large volume at low density.  This is telling the story of gravitational collapse of gases.

Pressure versus Radius graph


This has interesting consequences for gravity balloons.  You would think that containing more gas in the same space would require more container material... but that's just not the case here.  This is the strange nature of self-gravitation.  The air holds itself in (to a limited extent).  Now, there's also the valid question of whether 3 atmospheres of pressure is actually usable, and it's likely not because of Oxygen toxicity.  Because of that, it's not at all clear how the usable volumes between both of these compare.  But for sake of argument, let's take the pressure range for the "habitable" volume to be 0.8 atmospheres to 1.0.  With that specifier, I can compare the habitable volume between these two cases.  We're imagining that the center area of the 3 atm case will be treated as uninhabitable, but people could live beyond that radius.  Honestly, I think this looks closer to the sketches of Virga.

Table of Radii that certain Pressures occur at
and corresponding volumes with given range
Radiifor thePressure
(given in km)
1 atm3 atm
0.8 atm018,384
1 atm10,98718,590
Habitablevolume
caseV (km3)
1 atm5.5562E+12
3 atm8.83007E+11


Here we see that the habitable pressure would not be increased by adding more air to the system.  That's not entirely surprising, for the same reason that Jupiter doesn't have much volume at "habitable" (again, just 0.8 to 1 atm) pressures.

Properties of the Structure

There are two mass values of interest - the mass of the wall required to hold the air in, and the mass of the air itself.  For the wall requirements, the formula I have used so far for the "large case" needs to be revised.  Going back to my original question on physics stack exchange about this question, the large case has fit the equation of [ P = 2 G pi mu2 ].  That equation takes into account the self-gravitation from the wall itself, but not the gravity from the air.  So I've wrote another equation that does take it into account.  To solve this equation, it needs to be solved for the mass-thickness of the wall, and then simply multiply by the sphere surface area at that radius and that's the wall mass.


The calculation of air mass is trivial because it follows the same equation used in setting up the differential equations to begin with.  That equation is just recycled.  Now, here are the equations.  These are in terms of P(r) and g(r), which are the pressure and gravitational field throughout the air.  These are outputs of the code that I have on pastebin.

Expressions for mass of the air and wall of super large gravity balloon
(require numerical solution of previous set of equations)



We can now look at how the two independent variables (center pressure and the radius) affect the mass needed to construct the wall of this gravity balloon and fill it with air.  With my code output and these equations, I produced the following graph to illustrate this, and it gives a good picture of the general mass scales involved for different cases.

Graph of calculated masses of wall and air
given different radii of structure

Keep in mind that this graph is still using linear scales.  In terms of general observations:
  • The material requirements for the wall never reaches an absolute maximum.  This was one of the primary questions that was motivating me.  The wall mass requirements grow at a rate below even the surface area of the volume, but it continues to grow.
  • At super large radii, the gravitational field from the air itself dominates, which isn't very surprising.
  • The cross-over point is around 20,000 km (40,000 km diameter), which has an atmospheric pressure of 0.5 to 0.7 atm.  In other words, the air gravity starts to dominate while the outer regions still remain disputably habitable.

There's now a need for better reference values.  Earth's moon has a mass of 7.3 x 1022 kg.  This is a facinating reference point, because it establishes the the maximum practical size of a gravity balloon is right around the mass of the moon.  In terms of length scales, I just want to quickly note some other bodies for comparision.

radii for comparison
  • The Moon 1,738 km
  • Virga 4,023 km
  • Earth 6,378 km
  • Saturn 60,268 km
  • Jupiter 71,492 km

Compared to what's possible, Virga is somewhat small.  My expectation was that the pressure would varry significantly between different regions in it, but that expectation has proved wrong.  Virga's outer regions would only be about 4% lower pressure compared to its center.  However, I also need to volunteer the fact that if Virga was made as a gravity balloon, the walls would have more mass than all the air (and all other stuff) on the inside.

I also wanted to write a little more on the stability of such a massive construction, but I find myself at a loss on the subject.  This analysis included effects from changing pressure over the volume and gravitational effects from the air.  These will affect the stability of the walls, but I'm not entirely sure how or to what extent.  The gravity of the air is at least partially destabilizing, just how tidal forces are.  Come a little closer to center, and the gravity increases, and vice-versa.  This is characteristically unstable, but it probably isn't a game changer.  There's also the fact that the pressure increases as you go in further, and as far as I can tell, this effect will be more significant (in any case) than the tidal forces from the air.  All of this is concerning the particular deformation of one part of the wall falling in a little bit.  It seems that the dominating factor for that contingency is what it's always been - the change in the self-gravitation of the wall.  My expectation was that the wall-self gravitation would become irrelevant on large scales because the mass-thickness of the wall declines.  That happens, but probably not to the point of irrelevancy.  Even if you go so big that wall self-gravitation didn't dominate the stability discussion, the air pressure would be the dominant mechanism for these global deformations.  Of course, there's still the matter of "local" instabilities - which consist of leaks and wall Rayleigh-Taylor instabilities.

As for other observations, I want to quickly hit the escape velocity and the wall thickness.  At around 100,000 km radius, I find the wall thickness to be around 240 meters, if I assume a density of 3.5 grams per cubic centimeter.  Crowlspace was looking at about the same thing and came up with 1,345 meters at a 200,000 radius.  Of course, with larger radius the wall thickness will decrease.  This shows that Crowlspace's number is truly quite different from mine.  I believe this is because of the nature of the calculation he was trying to do, which wasn't considering any hetrogenous spatial distribution of the gas.

For the 1 atm central pressure case, I find an absolute maximum escape velocity (from the surface, including air and wall) to be 761 m/s.  This is a facinating result, because as long as you don't change the parameters like the central pressure or density, it is the maximum escape velocity that a gravity balloon can ever have.  It's also baffling because in Newtonian gravity, the gravitational potential of an infinite sheet of matter is infinite, and in large cases of this the wall starts to look a lot like an infinite sheet.  But that doesn't happen because the wall's thickness decreases with increasing radius.  Even more surprising is the magnitude of this number.  It's just not very fast, and even a bullet from a conventional gun can meet it.

Of course, the real question is where you would get all of that air from.  Wall materials for a gravity balloon can be anything, so the moon itself would literally suffice for this large limit gravity balloon's wall.  Earth's atmosphere is made out of fairly common elements, so that's not a constraint, but they would have to be processed in some sense.  I agree with the sense that such a large habitat would be in the outer edges of a solar system, but possibly they would be a complete interstellar space.  If near some clouds of gas of plentary nebula, perhaps the gases would be easier to collect.  I don't doubt that some dumb rock for the wall materials would be hard to find either.  But there's still the matter of turning whatever gas you have into molecular Nitrogen and Oxygen.  It's certainly a reasonable idea for speculation.  The scale is just so impossible for a humble human to consider.  A habitable area could literally exist accross a region 2 times the diameter of Earth.  The math tells us that is easily possible in terms of pressure alone.  But what would anyone do with all that space?

3 comments:

  1. Quite right. I wasn't trying to account for heterogeneity of the air mass when I did that computation - I assumed an isotropic pressure through-out. As I realised later this wasn't going to happen, as self-gravity would set up a significant pressure gradient. Instead - and this is for a future post still being head-edited - I worked out the maximum radius for a balloon of constant density air to have a central "gravity pressure" the same as the kinetic pressure of the air. At 1 atm and Earth-like composition, the maximum radius is 22,000 km. Is it stable? I think it'd need to be actively stable, with a nearly constant density and temperature through-out. Thus internal heat sources are mandatory. If the whole thing is at ~288 K on average, then we'll need the surface to be radiating at that temperature - at least. However the skin will need to actively transfer the heat from its inner surface to the outside. Passive heat conduction will cause the interior to rise a temperature sufficient to cause the skin to radiate at a luminosity equivalent to the internal energy sources. Too much thermal inertia of the skin and the interior will COOK.

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    1. The stability question is something I might revisit later. My post on shell world is a different kind of method, which could hypothetically generalized to really really large Virga-type worlds. The issue is whether you're balanced by the shell self-gravity or the air gravity. You would think that if you get far enough out, it will be 100% the air gravity, which would make it solvable.

      But of course you were correct in your assumption. Just assume the air density is constant and you can predict the fraction by which the pressure drops. If that is much less than 100%, you know you're right. If not, then you're not. Virga's radius is about 8,000 km, I believe. So I think that puts it in the 10% range, so you were correct. You didn't need this analysis to tell you that either.

      I might or might not understand your intention of the central "gravity pressure". I think there was discussion of the Hadley cell type circulation, which is basically natural circulation from the center to outside. This relies on the density difference and the integrated gravity field. Obviously, anything solid over 1 km thick won't be very good at transferring heat. If you get super super large, the air is so thin that even that can't transfer heat well. So, engineered heat removal channels are always required.

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