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When I was a boy I used to daydream about building a tower so tall that the top of it would stick out of the top of Earth's atmosphere project into near space.

There would perhaps be a zero gravity area in the penthouse where my friends and I could bounce around and play space versions of various earth-based games and sports in most excellent zero-g conditions.

Much to my continued disappointment and despite all the technological advances of the last thirty or so years, no one has built such a structure.

Can anyone explain the physical limitations/constraints that are preventing someone from realising my fantasy of a 'Space Tower'?

UPDATE: This Kickstarter Project seems to be pretty confident ...

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you'd have to go a lot higher than LEO to achieve your zero-g penthouse. –  TheSheepMan Aug 30 '11 at 11:38
    
Indeed, in order to acheive the zero-G environment, your tower would have to be thousands of kilometers tall and built on the equator. en.wikipedia.org/wiki/Geosynchronous_orbit –  AdamRedwine Aug 29 '12 at 14:35
    
This is a stupid question. Wind load alone will not make this possible, among other things –  mcodesmart Jan 14 at 22:17
    
There are no stupid questions, @Programmer13. Feel free to elaborate on your terse response if you wish to add to the discussion. –  5arx Jan 15 at 11:46

7 Answers 7

up vote 6 down vote accepted

First off, the limitation is a material that would not collapse under the weight - earth crust is not quite hard enough. Buckling and other instabilities, nope. Generally, forget a tower built on earth. Not a chance, no such material.

Start building from geostationary orbit and extend the "rope" both inside and outside the orbit. The outside may be just heavy counterweights, as the inside will begin to pull towards earth. Make the orbital part thicker to support extra weight, as you extend the lower part, until it reaches earth surface below.

Now the problem is the material. The only material in existence with sufficient weight-strength ratio is buckytubes. These are currently centimeters long at most, extremely expensive and you'd not only need thousands of kilometers of them... the rope to sustain its own weight would have to be about 1km thick in the thickest place (near the geostationary orbit).

Now consider:

  • earth carbon supply, I don't think all coal mines combined could mine that much carbon
  • construction craft fuel. This all would have to be lifted high enough. A LEO rocket takes many times more fuel than its payload weight. A geostationary orbit rocket - much more. The good news is the fuel can be hydrogen+oxygen which is water, and we have that aplenty. The bad news is you need at least as much energy to separate them as you gain from burning them, so the power consumption for fuel production would exceed whole world's power production.
  • environmental impact of that much steam released into atmosphere
  • account for micrometeorites that can really rain over your parade. And this thing being that big, collisions WILL happen. Also account for space junk.
  • account for winds and storms once you reach the atmosphere. Also, upper atmosphere is pretty hot... not nice work environment, also nanotubes aren't extremely fire-proof.
  • cost and impact on economy. Coal becomes super-expensive and we look for alternate sources of carbon.

And when you finally build it, calculate how long a lift travelling some 300km/h would take to reach 37,000km of the 0-gravity orbit...

EDIT:

I can't currently find the article that listed 1km thickness, but let us try to calculate parameters of the tower merely strong enough to sustain itself.

The nanotube tensile strength is $UTS=6422kg/mm^2$ (1)

The density is $\rho=1.4g/cm^3$

The ribbon is said to be 1m wide.

The thickness will vary. For the needed $M_0=20t=20000kg$ capacity it needs $A_1=0.31mm^2$ cross-section at the bottom. At 1000mm width that's 0.00031mm thick.

Now I'm really not in the mood to solve a differential equation of thickness - mass - tensile strength - gravity so let me try a discretization, approximating with $h=1km$ long wedges. At 35000 samples that should give us a decent approximation.

$$ V_n= {A_n+A_{n+1} \over 2}h \\ M_n=\rho V = \rho {A_n+A_{n+1} \over 2}h $$

Now we can't happily assume weight not to vary with altitude. After all, near the orbit it will be zero. It varies with distance from center of Earth. At surface, $r_0=6378km; M_{earth}=5.97 10^{24}kg; G =6.67300 × 10^{-11} {m^3 \over kg s^2}$;

So, the weight function of each segment will be

$$ Fw_n=G{M_n M_{earth} \over r_n^2} \\ r_n=r_0+n[km] $$

And the tensile strength surface $A_{n+1}$ must overcome is

$$ F_{n+1}=F_n + Fw_n \\ F_{n+1} = A_{n+1} UTS $$

We seek $A_{35000}$ which will trivially yield thickness by dividing by 1000mm.

$$ A_{n+1} UTS = F_n + Fw_n \\ A_{n+1} UTS = F_n + G{M_n M_{earth} \over r_n^2} \\ A_{n+1} UTS = F_n + G{\rho {A_n+A_{n+1} \over 2} h M_{earth} \over r_n^2} \\ A_{n+1} = F_n + (A_n+A_{n+1}){ G \rho h M_{earth} \over 2r_n^2 UTS } \\ X := { G \rho h M_{earth} \over 2r_n^2 UTS }\\ A_{n+1} - = F_n + (A_n+A_{n+1})X \\ A_{n+1} = F_n + X A_n + X A_{n+1} \\ A_{n+1} - X A_{n+1} = F_n + X A_n \\ (1-X)A_{n+1} = F_n + X A_n \\ $$ We get our two fundamental equations for numeric computation: (with helper X, which I'm really not in the mood to transform into something nicer.) $$ X = { G \rho h M_{earth} \over 2r_n^2 UTS }\\ A_{n+1} = { F_n + X A_n \over (1-X) } \\ F_{n+1} = A_{n+1} UTS $$

Now excuse me, it's 3AM and I'll finish the calculations at a different time.

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Great debunking of the space elevator ideas. It's a sad reality that most people who advocate space elevators simply don't understand the simple math behind it. What assumption is the 1 km thickness based off of btw? Doesn't the depend on the weight of the payload? If you're assuming that it only has to have the ability to lift a small mass, then would a much smaller mass be required? Pragmatically, I wonder if a space pipeline for propellant would be the most likely use for such a thing since batch lifting would present great difficulty. –  AlanSE May 24 '11 at 15:34
    
In the kinds of SF where they simple assume the beanstalk, the cars pick up speed in a hurry after they leave the atmosphere: there is no need to limit them to a few hundred m/s. Depending on the implied technology there may be a change of cars. The real stinker is that it makes non geo-stationary orbits unsafe. –  dmckee May 25 '11 at 3:08
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@dmckee: On one hand, yes. On the other, there is still friction against the tower, friction of the mechanisms, energy transport losses and so on. Even if it was built as maglev, dynamic tensile strains would create a whole lot of heat. Not impossible to do, but still travel time would be of order of days. –  SF. May 25 '11 at 7:21
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Yes, your calculation shows that the free breaking length of a constant-cross-section carbon nanotube rope in constant 1g gravity is not quite long enough to reach from geostationary orbit all the way to the surface. However, all modern space elevator designs do not have constant 1g gravity, and are not constant-cross-section -- they are tapered. –  David Cary Jan 7 '13 at 20:53
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@AlanSE: I am also mystified by the "1 km thickness". All the references listed at the Wikipedia: space elevator article that I've seen so far seem to assume ribbon of 1 meter wide or less. –  David Cary Jan 7 '13 at 21:03

First of all, it is an elementary misconception that there would be a "zero gravity" environment in a tower that would only reach the top of the atmosphere.

Most of the air molecules exist at a height smaller than 10 kilometers - and above 100 kilometers from the Earth's surface, the air is so diluted that it becomes undetectable.

At the height of 10 kilometers - where the atmospheric pressure is almost zero - the gravitational acceleration is just 0.3% weaker than it is on the surface and even at 100 kilometers, it is just 3% weaker. So forget about "lunar games". The gravitational forces over there are pretty much indistinguishable by humans from those we know on the surface. At 100 kilometers, a 75-kilogram person may feel 2 kg lighter but it may be compensated by the suit he needs to avoid suffocation. ;-)

The absence of air has nothing to do with the absence of the gravitational force. The air tries to be at low attitudes in order to minimize its potential energy; how much it wants to minimize the energy is given by the molecular mass and the temperature. However, the air density is something totally different than the gravitational acceleration - they're surely not proportional to each other in any sense.

The air density is proportional to $\exp(-\Phi m / kT)$ where $\Phi$ is the gravitational potential, $m$ is the molecule's mass, $k$ is Boltzmann's constant, and $T$ is temperature in Kelvin's degrees. However, the gravitational acceleration is $d\Phi / dh$. These two functions depend totally differently on the height $h$.

Tall buildings

The tallest building in the world is Burj Khalifa in Dubai - it has 828 meters. It's about 10% of the thickness of the atmosphere in the "narrow sense". It's hard to build tall buildings - one must guarantee that they're stable despite the immense weight of the material above each floor and despite the wind and vibrations of the Earth's surface.

But there are no "strictly physical" limitations that would prevent one from building a tower that reaches 10 kilometers above the surface. One may say that all such limitations are of engineering character. Tall mountains such as Mount Everest may be viewed as "natural tall buildings" and their height isn't far from the top of the atmosphere (in the narrow sense). The design of very tall buildings would probably have to be a bit hierarchical - with a solid base made out of a heavier material and lighter floors near the top, just like in the case of mountains.

One would surely start to face problems to find reasonable materials if he wanted buildings that substantially reduce the gravity on the roof - buildings that are thousands of kilometers tall. For example, one of my kindergarten visions was to build an elevator that could take one to the Moon and that could convert the kinetic energy of the Moon's motion around the Earth. That's a really challenging task for engineers. One will run into problems with conventional materials etc. - but one may still say that the limitations are of an engineering (and budgetary) character rather than fundamental physics limitations.

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Thanks. Have reworded my question in light of your answers about the reach of Earth's gravity. –  5arx Mar 23 '11 at 9:56
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You can estimate how high a mountain can be if it's simply a pile of rocks, it depend sonly on gravity and youngs modulus for the rock ias.ac.in/jarch/jaa/2/165-169.pdf –  Martin Beckett Mar 23 '11 at 17:21
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@Martin, No, Youngs modulus does not rule that! The relation of force to elastic deformation is irrelevant. The ruling factors are maximum pressure/shear/etc when the material starts to flow, rupture, crunch etc, (which is where Hook's law ends, and Youngs modulus is no more existent) –  Georg Mar 24 '11 at 10:29

The level at which this question is being asked is uncertain. I thought I would mention the idea of the space elevator, which some people take seriously. However, Lubos is correct in saying that the edge of the atmosphere does not mean the end of gravity. A spacecraft orbits the Earth because it is falling towards the Earth. However, it is just moving fast enough so that it keeps missing the Earth which curves away under the spacecraft’s trajectory. The apparent loss of gravity, as seen with space shuttle and ISS astronauts floating around, is due to the fact the astronauts and everything in the spacecraft is falling and moving with the rest of the spacecraft. Remember that Galileo demonstrated that different masses fall with the same acceleration, and so everything in a spacecraft falls with the same acceleration. However, the whole thing is moving fast enough to keep missing the Earth, and this acceleration of gravity provides the centripetal force which maintains a circular orbit.

There is this “Jack and the Beanstalk” idea of the space elevator. I seriously question whether this will ever be built, but the idea is possible in principle.

http://en.wikipedia.org/wiki/Space_elevator

The idea has some problems of course. In particular it is tough to stack up mass elements without it falling over. If the gravity force at the center of mass deviates from its foundation the stack fall over. So I think the idea of building the tower from the ground up is probably wrong.

The prospect for this lies in building from the top down. There are ideas about manipulating the orbits of asteroids. The Russians want to change the orbit of Apophis asteroid, which will come close to the Earth in 2029. Suppose we get good at doing this, and we manipulate the orbit of an asteroid into geosynchronous orbit around the Earth. A geosynchronous orbits is at a radius of 37,000 km where the orbital period is equal to the rotational period of the Earth. As the Wiki page shows one must then have a counter weight beyond geosynchronous orbital radius. So if one had an asteroid of sufficient mass and with the proper material constitution one could then build the tower downwards from this point. This would be accompanied by building upwards with an amount of mass so the center of mass of the emerging structure remains at geosynchronous orbit. Eventually this would then be constructed into this tower. The gravity gradient on this emerging structure would have to be carefully monitored and the vibrations on this controlled. It would not be at all trivial to do this.

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Wasn't it in an Arthur C. Clark science fiction story? –  anna v Mar 23 '11 at 13:32
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To do zero gravity frolicing in the penthouse, the tower had to have that 37 000 km (geostationary) height (I dont know, whether this is measured from center of earth or from sea level, but the difference will not matter with respect to "feasibility". –  Georg Mar 23 '11 at 15:06
    
actually it would have to be well over the geostationary height to balance out (in average) the weight of the cord below geostationary height. –  lurscher Mar 23 '11 at 18:31
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AC Clarke did write a novel about a space elevator about 20 years ago. I don’t remember how they built the thing in the novel. I did try to keep the point that the center of mass of the whole thing has to be at geosynchronous orbit. So you do need a counter weight further out. My suspicion is this is pretty pie in the sky stuff. I suspect it is very unlikely we will ever build this. The further back in history you go you find commitments to building large things, pyramids, cathedrals etc over long periods of time. The modern world is “fast-food,” where we “want it now.” –  Lawrence B. Crowell Mar 23 '11 at 18:51
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The longevity of policies, programs and the preeminence of nations have become more and more time compressed. Egypt was a major power for 1000 years, Rome for 500, the British for 250, and now the US primary position is about to expire in less than 100 years. A mark of progress is time compression --- and impatience. Future superpowers may have tenures measured in decades. Andy Warhol has cursed us all with 15 minutes of fame. –  Lawrence B. Crowell Mar 23 '11 at 18:52

Re: the original question:

If you wanted "zero gravity" at the top of the tower, you'd have to build a tower tall enough to reach the height of geostationary orbit: a point at which the orbital period of an object in freefall matches the time it takes the earth to rotate once. As other commenters have pointed out, that'll occur at a height of roughly 35000 km above ground. Good luck!

Re: The claim that "but one may still say that the limitations are of an engineering (and budgetary) character rather than fundamental physics limitations." I disagree. One fundamental physical limitation is the fact that matter is held together by chemical bonds. This limits the ultimate strength of any material (although AFAIK for all macroscopic materials mankind currently produces, the ultimate strength is far shy of what one would get from a "perfect" material). The ultimate strength will limit how tall a tower one can build. This is, for example, the reason why small asteroids can be quite aspherical, but large asteroids (and planets) cannot: an highly aspherical planet would collapse under its own gravitational force and become near-spherical due to the finite strength of the materials it's made of. The ultimate strength limit also means that you can't build the "space elevator" other folks mention using any currently available materials that I know of.

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There's an interesting review article on the subject:

Review of New Concepts, Ideas and Innovations in Space Towers
Mark Krinker, (2010)

A lot of new concepts, ideas and innovation in space towers were offered, developed and researched in last years especially after 2000. For example: optimal solid space towers, inflatable space towers (include optimal space tower), circle and centrifugal space towers, kinetic space towers, electrostatic space towers, electromagnetic space towers, and so on. Given review shortly summarizes the[s]e researches and gives a brief description [of] them, note some [of] their main advantages, shortcomings, defects and limitations.

http://arxiv.org/abs/1002.2405

The above is sufficiently interesting that I would recommend it as a place to look for interesting problems for your undergraduate classes in mechanics.

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Towers supported from the bottom are a bit tricky. Buckling limits how tall a column can be. One needs to additional lateral stiffness to overcome this, usually by putting up guy wires. Even so there are going to be real limits, as Anonymous Coward has mentioned above, solids obtain their stiffness from chemical interactions between molecules and atoms, and the strength to weight ratio is limited. There are some plans for some structures up to about a kilometer, but the cost per unit volume of building goes up for tall buildings. We could probably go a lot higher by the use of carbon nanotubes, but we are years away from being about to construct practical guy wires from them.

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Rather than using material, perhaps magnetic fields configured in stages. Imagine a stack of plates separated at a distance on the order of a meter. Magnetic fields, from superconducting magnetics repeal the plates above or below. Sensors and an electronic system dynamically adjust the fields. I wonder if the fields would need to become attractive at some point due to centifigual forces from the earths rotation.

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