# How big can the Parachute cities be?

Nobody knows where the big Blue is. Some people say it's a dream. Some people say it's the space between the worlds, where all the things that fall through the cracks end up. The occasional rains of odd socks and pencil sharpeners make me inclined to believe the latter though.

Either way: I know we're falling. Everything, forever, falling through an endless blue sky. The only real stable places are the four Parachute City states: Maelstrom, Charybdis, Freefall and Slip. Four agglomerations of rock and dirt, wooden ships and aluminium planes and some metal parts that nobody left here recognises.

Inside we fall along with whichever city we were born on, protected from the winds below by the lee of the city (though the crosswinds can be brutal). The careful or clever can practically jump from one city to the next, helped along by wing suits or windskiffs.

The cities keep an even keel and can slightly control where they are in relation to each other with strategically placed hot air balloons and vanes around the rim of the city, where the vicious turbulence as they plummet through the endless sky can dash careless airship captains or wing suit swimmers into pieces.

But how large can one of the cities get before they start to tumble or can't hold themselves together any longer? That's what I've been directed to find out.

How large (in terms of flat square meters on the top, assuming it's held steady) can an object falling at its terminal velocity get before the various stresses tear it apart?

Please note the hard science tag: I'm not looking for hard science criticism of the world, just a hard science analysis of the large falling object. If it helps the atmosphere is equivalent to Earth's at sea level and there is 1g of gravity.

Time to break out the fluid flow equations! (Or not. ;-)

A quick note: I don't care if the answer to this question is 'very small', I can work with that. I just want to see some kind of maths or formalised rationale (with citations or equations, preferably) behind the maximum size.

• So the cities are in perpetual free-fall through an infinite sea-level-equivalent atmosphere, but the atmosphere itself doesn't fall except where something like a city pushes it downward? Mar 23, 2016 at 21:58
• I'm not enough of a scientist to work the math, but it seams to me that as long as the bottom of each city presents a much lower drag than any of its other surfaces, then it won't tumble. So your cities should be designed somewhat like a water droplet or a dart with the sharp end pointing down. Mar 23, 2016 at 21:59
• @robwatts: yep. Pretty much nailed it there. Mar 23, 2016 at 22:33
• @user16295: as high tech as you like, if you can back it up with hard science. Neither answer so far actually matches the tag. :-( Mar 24, 2016 at 11:43
• Mar 24, 2016 at 12:46

It's been too many years since I tried to do a calculation like this so I can't fulfill the requirements of the "hard-science" tag but you'll need to compare it to ship design.

For your city to be stable: The centre of mass has to lie below the centre of buoyancy.

Now of course there's no such thing as a centre of buoyancy in a falling object but there will be a calculated centre of aerodynamic resistance defined as the point in the structure where you could say "air resistance applies here in this direction" which could be used as an equivalent. (I didn't do anything much with turbulent flows so don't bite my head off if this ceases to be valid outside laminar flow)

I my head this requires your city to look something like a more flared golf tee, point downwards with most of the mass in the stem and the population living on top, trailing behind like a parachute over a weight.

In terms of the available surface area, assuming you have a fairly compact lower length on the stem, you could go happily multistory on the city as the density of housing will be lower than the density of the city base and would help to keep your city stable. The shear stresses caused by the wind on the flare of the Tee are your problem and unless this has been deliberately constructed with metal reinforcement the whole thing will tear itself apart remarkably quickly.

Balloons round the edge of the city would be in turbulent flow and as such would have no effect.

Other amusing problem: The city is going to have a higher terminal velocity than a person, wingsuits aren't your problem when jumping between cities, the problem is not being able to catch up with the city again as it falls away beneath you.

A long object's resistance is based on its length, a short object's resistance is based on its cross sectional area. To build a city that falls more slowly than a person you need to keep it in the "short object" range. However this will result is a much less stable profile and if a "natural accumulation of stuff" design is followed, it will probably be highly unstable in turbulent flow.

Preventing the city from spinning is a life or death issue. It's that or build a hollow cylindrical city with high spin (foils on the outside could do this) and the people live down inside the tube.

Shortening and stabilising the structure - The parachute spill hole.

Sticking with the golf tee but shortening it somewhat to reduce streamlining effects, we're now going to have to stabilise the turbulent flow around the edge. The simplest way to do that is to inject air into the flow just inside the edge of the city, sourcing the air from the highest pressure point, namely the downward tip. To get this right does require some "hard-science" grade calculations that I'm not going to do but we're going to hollow out the leg of our tee as much as we dare so as not to upset the mass balance and run those tunnels to as many points as we can just inside the outer edge of the city. I think we want a slight negative pressure at the point, so the total possible airflow through these smaller tubes should be slightly higher than that through the main leg. We also want to have as many outlet points as possible round the rim.

This will (should) have the effect of stabilising the turbulence around the edge of the city and hence allowing a shorter structure to be more stable.

It will also allow you to surround the city with balloons and small parachutes to allow better control as you no longer have (have much reduced) the turbulent zone.

• Thanks: That's shape and mass distribution worked out at least! Now the harder part of stresses on the flare and dealing with the edge turbulence... Mar 24, 2016 at 10:11
• The aim of the city is for it to have a lower terminal velocity than a person, ideally, hence the name! Mar 24, 2016 at 10:12
• Unfortunately the object described here, while stable falling in air, is also really quite streamlined... Mar 24, 2016 at 10:35
• Note: actual parachutes are a stable falling object with a lower terminal velocity than a human Mar 24, 2016 at 10:57
• @Rugnir, I should also say, a parachute is not a stable object, a parachute of correct size for the mass below and a hole in it is a stable object. The hole is an interesting concept to consider here though. A hole through the city could do wonders for stability. Mar 24, 2016 at 11:10

The problem here is that wind erosion is going to be pretty terrific. Those "clumps of dirt" won't last long against hurricane force winds. Remember that a human body falling at terminal velocity falls at about 120 MPH, and dirt plus rocks certainly won't fall slower; in fact, with higher density, they'll fall faster.

So either the city inhabitants have coated the bottom with something to keep things from being blown away, or else they have to constantly replenish what gets eroded away. I can see a story idea here, with workers having to constantly work on the canvas (or whatever) covered by netting, which is needed to keep the soil from being blown away.

For larger structures, the question of fluid dynamics does come up. But equally important is whatever provides the structure keeping these falling "islands" together. Steel, aluminum etc. simply can't hold an extremely large structure together. The cube-square law places severe limits on the ability of large structures to support their own weight. (And these islands are not actually in "free fall" as the term is used in physics. They are falling at terminal velocity, which means they do have weight. The people walking around on these island will not float off; they'll feel weight.)

Frankly, unless you're talking about tiny islands no bigger than a skyscraper -- which would be the approximate limit of size using concrete and steel, or possibly even smaller if constantly subjected to hurricane force winds -- you're looking in the wrong place for a hard science answer here. If you want something that far removed from everyday reality, either assume the laws of physics are radically different (but then, that means anything living has to be far different from life as we know it, so nothing like human beings in the story), or else magic is being used to hold the islands together.

Another possibility is to use "exotic matter" with an arbitrarily high tensile strength, like /scrith/ in the RINGWORLD series. But that's near-magical science fiction, not real science. /Scrith/ has a structural strength on the order of the strong nuclear force. With a material that strong holding things together, you can make the islands as big as you want... or limited only by how much of the exotic matter the inhabitants can get or make. If you go that route, I suggest a honeycomb structure of the exotic matter (since it's that strong, the walls of the honeycomb can be extremely thin... even thinner than aluminum foil) with the voids filled with normal matter, or even empty space to make the structure lighter. The exotic matter could also be used for (amazingly thin) structural beams, arches etc. to support buildings on the islands.

Of course, if it's exotic matter, you can choose any arbitrarily high structural strength you want. It doesn't have to be as strong as /scrith/, but it should be stronger than carbon nanotubes... which in theory can form structures 30 to 100 times stronger than steel, but in practice don't seem to form structures with long-term stability. In other words, carbon nanotubes break down at the molecular level after a fairly short period of time, so apparently aren't suitable for building things.

"The aim of the city is for it to have a lower terminal velocity than a person, ideally, hence the name!"

Well then, you need to rethink the idea of the cities being composed of a random collection of dirt clumps, airplanes, and random metal junk. They will have to either be structures built with a density less than the human body -- a lighter overall weight per unit of volume -- or else they'll have to have something like giant literal parachutes slowing their passage through the air... and how long would those last if made of ordinary silk and nylon ropes? Or even kevlar? Again, this is suggesting exotic materials.

Even if they are held up by literal parachutes, there's still the problem of holding the entire structure together, and the severe limitations to size of using ordinary materials (for example, steel or aluminum I-beams).

The simplest solution would be to use the "golf tee" overall shape for the city suggested by user16295, constructed of a lightweight honeycomb of some material stronger than what we can make with current technology. Assume much of the "tee" is empty space (or even vacuum) surrounded by honeycomb, with only the surface inhabited, furnished with soil to grow crops for food, etc. (I hope they get plenty of rain to provide water?)

You might consider investigating the theoretical properties of carbon nanotubes, but assume that some method of producing large structures made out of them, plus some method of stabilizing them on the molecular level has been discovered... which I suppose isn't impossible in theory.

"Actually: just thought some more about my last point: they'll be up at a full g, won't they?"

EDIT: I see I made an error in what I posted below. I'll leave it there for purposes of illustration.

No, you were right: A fraction of a gee, assuming the entire space is in a 1-gee field. That's 32 feet per second per second of acceleration for something on the ground. If, for example, the parachute city is falling at 10 feet per second, then that's (32-10=) 22 feet per second, which means 22/32 = 0.6875 g for those standing on the ground of a parachute city.

EDIT: My error above was confusing velocity with acceleration. No, those standing on the ground on the parachute city will be subject to 1 gee, just as those standing on the surface of the plant. They would only experience lower gravity if the parachute city was under constant acceleration. Since it's falling at a fixed (terminal) velocity, it's not accelerating. [end of edit]

But why would this endless column of air have a one-gee gravity field? This is obviously not Earth! Since you're world-building, you can arbitrarily choose any gravity level you like. And if the outside gravity field is lower, well then you can make the structure larger without using exotic matter. Lower weight means you can use weaker materials to hold the structure together, altho if the winds are as terrific as you suggest, the inhabitants would still have a severe problem holding things together for long.

Groups of objects held together by something flexible? Well now, that's something that might actually be possible given current building materials. Kevlar is your best bet, but your original question suggested that all that's available is some random debris. If they have to rely on ropes woven out of grasses or reeds, those structures are gonna have to be pretty small!

"If you can expand your answer up with any numbers to back up the 'no bigger than a skyscraper' assertion..."

Well, it depends on how flexible the overall structure is. Bridges can be bigger than skyscrapers, partly because they can flex and partly because they're more horizontal than vertical... so they can be supported by multiple piers rather than having the entire weight of the structure supported by the bottom floor. (If the weight is hanging from parachutes, the entire weight has to be supported by the framework the parachutes are attached to. But even so, a bridge might be a better analogy.) However, think of the behavior of a flexible bridge in hurricane conditions. It won't last long, will it? So that's why I think the skyscraper analogy is closer to what would actually survive under such conditions.

Sorry, I'm not gonna do math here. I don't do math for fun, and you haven't provided enough specifications to be able to give any hard numbers to it anyway. We don't know what the turbulence is, and we don't know how fast the parachute city is falling. And altho you've specified a one-gee gravity field, there doesn't seem to be any good reason to impose that limitation. Choose a lower gravity, things become easier for the inhabitants to build stable structures, probably the turbulence is reduced, and it becomes easier to fly from one structure to another. If it really was a one-gee field with everybody falling along with some random debris, it's almost impossible to believe that they could actually organize and build anything before starving to death.

BTW -- I suggest you read Larry Niven's THE INTEGRAL TREES for some ideas about humans living in free-fall... and on structures which are "falling through the sky" but nevertheless have portions of the structure where you feel weight.

• The point about weight is accurate, but we'd be measuring it in fractions of a g unless we can really get the parachute cities to act like parachutes. As far as the sizes go: This is exactly the kind of answer I need for this question as I'm trying to work out how big I can make any one piece of city, or if it would just be better to have a series of loosely interconnected pieces. If you can expand your answer up with any numbers to back up the 'no bigger than a skyscraper' assertion then it's pretty much right on the money. Mar 24, 2016 at 10:46
• Actually: just thought some more about my last point: they'll be up at a full g, won't they? That works even better for the purposes of this world! Mar 24, 2016 at 10:52
• Re coating or erosion: it could be catching material, esp. Living matter. That might have xriven the overall design in the first place. Btw, I picture a pocket of gas at breatable composition and pressure in a constant updraft in a gas giant. The cities are essentially keeping station. Mar 24, 2016 at 17:18

I'm no good at the hard science, but if your goal is floating cities in the air, could you use a buoyant gas bubble in the crust of the big rock? I bring this up because I remember a wedding that took place on top of a hot air balloon envelope, not in the basket. It was years ago, pre-internet and I can't find any images. lots of sales brochures though. I just remember the happy couple jumped with parachutes after. That kind of image sticks in a kids mind.

A very dense load at the bottom of the teardrop (or hot air balloon) shaped rock, buoyant pocket in the middle, and thriving civilization on top. Augment with other gas balloons. It seems to me that the city would float at the equilibrium point in the atmosphere. That should give you standard gravity and pressure above even a gas giant.

Sorry I don;t have the math to back any of this up. I just think this may be a logical approach, something that could possibly be proved out by someone smarter than me.