Space Elevator/Lift Physics

Recently I gave a lab talk on a topic that's always been dear to my heart - space elevators!

Here I will attempt to summarize my findings. Fortunately there are minimal numbers of equations!

The basic idea: A satellite in geostationary orbit completes one orbit every 24 hours, so thus remains at the same point above the Earth. If you unspooled a counter-balanced cable that was long and strong enough, it could touch the surface. Then you could climb up into space without using a rocket, like Jack and the beanstalk.

Question: How far above the earth must the center of gravity of the satellite be in order to be geostationary?

Answer: About 36,000 km.

Maths: In the geostationary point in a rotating frame the gravitational force is balanced by the centrifugal force. GM/r^2 = r w^2. The period of Earth's rotation is 86400 (-240*) seconds, giving a radius of 42,100 km. The Earth's radius is about 6380km, giving the above answer. *The sidereal day is only 23 hours and 56 minutes long.

Discussion: This is a long way - about 3 times the diameter of the earth. From the top, the earth would look smaller than the field seen inside a stadium. The cable length would nearly wrap all the way around the Earth. By comparison, a train line around the equator would be easier to make, but nowhere near as useful, and about the same length. Similarly, aiming for a travel time of 5 days implies a speed of 300km/h, about the same as a bullet train.

Question: How strong would the cable have to be?

Answer: Specific strength of 50,000 kNm/kg, which is about 100 times stronger than steel.

Maths: g(h) = GM/(r+h)^2 - (r+h)w^2. The acceleration is zero at the geostationary radius. 

dT = rho A g(h) dh, where rho is density and A is cross sectional area. Integrate from the surface up to any height H.

The peak tension is at the geostationary point, and is about 50,000 kNm/kg. This has the units of the sound speed in the material squared.

Discussion: One way of comparing the tensile strengths of materials is by their breaking length. Eg A-36 steel has a breaking length of 3.2km. That is, a cable 3.2km long would break under its own weight. Unfortunately this breaks down for really strong materials, because the Earth's gravity is not the same if you go up high enough! So we'll stick to specific strength as a unit of strength. Divide by 10 to get the breaking length in km. 

50,000 kNm/kg is a lot stronger than Nylon, the polymer from which climbing ropes are made. A standard climbing rope is about as thick as your pinky and can lift a car, or (more importantly) absorb a large fall from a big man. If climbing ropes were made from something strong enough to make a space elevator, not only would they be even more expensive, they need only be as thick as a human hair. Of course, heat dissipation in descenders is already a problem with narrow climbing ropes...

Question: How much energy do you need to escape from Earth?

Answer: 62MJ/kg. This corresponds to an escape velocity of 11km/s. (REALLY FAST!)

Maths: U = Gm/r. The earth weighs 5.94*10^24kg. G is Newton's constant or 6.67*10^-11 in SI units.

Discussion: This is the MINIMUM energy requirement, assuming 100% efficiency. If you use a rocket with chemical energy (the current best method) typically you need at least 100 times as much mass just in propellant, and that is about 1/1000th of the cost of the rocket structure, as a rough proxy of how much energy is needed to build the metal parts of the rocket. So the attraction of a space elevator is a million fold increase in energy efficiency for operation. In comparison aeroplanes, bridges, and trains can consist of more than half cargo or payload, not less than 1%. Note that a space elevator does not get you to escape velocity, only to geostationary orbit. However, extending the counter weight with a tail out further into space can allow people to steal momentum from the rotation of the earth to be flung into space. Escaping the sun requires about the same delta-v again (30 km/s to 42 km/s). Such space elevator extensions could be readily added onto an existing space elevator as it is already in equilibrium, and would require an extension of 109,000 km and 260,000 km respectively. The tensional forces in the cable continue to decrease out until Earth escape velocity, then reverse direction. To obtain cable-flinging escape velocity from the solar system, the cable would extend 5/6 of the way to the Moon and need to be somewhat stronger: 200,000 kNm/kg if it were uniform, somewhat less if it tapered as the inverse of distance from the Earth. Tapering can also work on the Earth side, though practically speaking you still need stuff two orders of magnitude better than steel anyway.

Question: Material strength. What is strong enough to do this?

Discussion: 50000 kNm/kg is a really big number. Some other materials commonly regarded as strong, and their specific strengths (in kNm/kg) are:

A-36 steel: 32

A1S1 A11 steel: 694

Titanium: 209

Tungsten: 750

Glass fibre: 1900

Carbon fibre: 3540

Spider silk: 1270

Kevlar: 2500

Spectra: 3800

Bone: 80

Silicon (crystal): 3000

Diamond: 800

This looks a bit bleak. To get there, we're going to have to have to resort to using the words nano and meta. 

Carbon nanotube: 46000

This is the right ballpark. It's hard to imagine a stronger material than certain allotropes of carbon. 

Question: Just how strong can carbon structures be?

Answer: Graphene (the subject of the 2010 Physics Nobel Prize) is a remarkable material with a specific strength of about 450,000 kNm/kg. It is hard to imagine a material that could be stronger.

Maths: Graphene is a single layer of graphite, and consists of layered hexagonal lattices bound together by SP2 hybridized orbitals. The spacing between atoms is 0.142 nm and between layers is 0.335 nm, which is about a thousand times smaller than a wavelength of light. From this the density can be estimated at 2.272g/cm^3. The tensile strength can be ball-parked by dividing the first ionisation energy by a typical deformation energy, giving 7.5*10^-8 J/bond, which is a really big number considering these are just atoms! In a lattice, this gives a yield strength of 1 TPa, which is (not coincidentally) the laboratory measured value.

Discussion: It is hard to overstate just how miraculous graphene is. A one square meter sheet of graphene strong enough to support a cat would weigh less than one whisker! This is a far cry from 99% of a rocket's mass being not-cargo, which illustrates nicely the difference between the two regimes. However it is worth remembering that the longest nanotube ever made is 18.7cm. So there is a long way to go before we are able to bind a bunch of these together to make a strand that can wrap all the way around the earth. It is also worth noting that if the day were much longer or the Earth much fatter, a space elevator would be impossible.

Question: Architecture? How might you build one?

Discussion: Consider for reference value a cable strand 1mm across and 36,000km long. A very special piece of string! When spooled up it has a volume of 36 m^3, or roughly equivalent to my office. Its mass is 75 T, which is somewhat more than the mass of the occupants of my office. With a specific strength of 60,000, which has been demonstrated in the lab, it would be able to support a weight of 100 T, most of which is its own weight. The loaded space shuttle also weighed about 100 T, although it only barely got to low earth orbit.

Question: How might a space elevator system work?

Discussion: With departures every 15 minutes, a 5 day transit time and 2 directions of traffic, there are 1000 climbers in total. Each is structurally and functionally analogous to a train car or a mid-size commuter jet such as the A320, weighing about 50 T when loaded. Additionally, the cable has to support the mass of attitude rockets, shielding, power, escape pods, tracks, damping, and so on. Estimating a mass of 1.5 T/km, the cable has to support a mass of 100,000T. Estimating the structural ratio at 10%, or each kg of stuff requires 9 kg of cable, the total mass of the cable is a billion tonnes, or 1 Tg. Only in computing can you use giga, tera, and exa with a straight face! This would require something like 15,000 of the 1 mm model string-like strands, which all bundled together would be 15 cm across. So as of today, we can make a carbon nanotube long enough to span a space elevator cable, only in the wrong direction! Obviously in an actual cable, the fibres would be separated to avoid bulk failure and aid weight distribution. Note that this billion tonne cable needs a counter balance or counter cable above the geostationary point to prevent it from falling down!

Question: How do you build it?

Discussion: Good question! Wave a magic wand! In all seriousness, there are two possible approaches.
Send a nuclear powered robot to the asteroid belt, find an appropriate sized carbonaceous chondrite asteroid. We're talking ~300 m long. Have it mine water and blast it out to move its orbit to the Earth, inserting it in geostationary orbit. Construct a cable by extrusion, probably pointing away from the earth. When it's finished, rotate it into place above a suitable point on the Earth. Profit. Problems include possibly crashing said asteroid into Australia, and a slow rate of construction. Even if it built a cable at 1 km/day, it would still take 100 years to reach the required length.

Launch a seed cable to geostationary orbit. Currently the largest rocket available can launch 6 tonnes to geostationary orbit. This is more than enough for a clever communications satellite, so we'll have to make do. The satellite consists of a spool and other gizmos, and the cable can be paid out at the appropriate angle to compensate for Coriolis forces rather rapidly. The cable in question would be about a quarter of a mm thick, but capable of supporting a large car, in addition to its own considerable weight. Part of that extra mass could consist of attitude control rockets at the LEO altitude to help it avoid space junk, and possible laser systems for micrometeorites. Once the feeder cable was secured, additional strands could climb and fix themselves into place at an exponential rate. Finally, shielding, tracks, dampers, giant space lasers and other equipment could be fixed in place and the cable completed in as little as 5 years. Note that construction on a tail or counterweight could proceed at the same time. As multiple strands could be build on the earth at the same time, there is no bottleneck in this approach. Disadvantages include the cost of the initial launch, weight balance issues, and the vulnerability of the early cable to severing.

Question: What about vibrational modes?

Discussion: The math of this is left as an exercise for the reader. The sound velocity is given by the square root of the tension divided by the linear mass density, just like a guitar string. The sound velocity goes to zero near the ground, implying that the wavelength also goes to zero, and the amplitude necessarily increases. Fortunately the atmosphere provides some viscous dissipation, but ultimately avoiding harmful vibrations is a matter of clever design. Some transverse vibrations are probably an excellent way to transmit power along the cable and to enable it to avoid collisions with satellites at lower orbits.

Question: What might other hazards be?

Discussion: Corrosion, micrometeorites, lightning, attacks, etc. Most of these issues can be dealt with by covering the structure in a shield like the shield on the ISS. It consists of a ceramic layer which absorbs the impact by shattering, and a metallic shield layer which can absorb lots of small impacts. There are other valid approaches for absorbing smaller collisions as well. The tracks would also be covered, possibly by moveable hatches that open as the climber passes and close after it leaves. Tracks would be multiply redundant and would be able to be switched in case of damage or maintenance such that the overall system could continue to operate without breaks. Although the cable could be as narrow as 15 cm, it makes more sense to separate the strands in a weight-sharing structure to which the tracks and other stuff can be attached. That way the breakage of compromise of any particular strand is less likely to affect those around it.

Question: What would it look like?
Discussion: Not much. At any point along its length, perspective would disappear it out of sight within a few km. On the earth's surface, it would appear as a rope going up and disappear from sight before reaching the clouds. From the top looking down the earth would be the size of a basketball at arms' length. On the way down, the earth would appear to dominate the view with a horizon for the whole day, but only in the last half hour would you enter the atmosphere. For comparison, it has similar length-width ratio with a railway rail that stretches across a continent.

Question: So how much does it cost?

Discussion: This is impossible to estimate! The cost of a launch to geostationary orbit is about $150m, but this would be a trivial expense compared to the materials. If each climber cost the same as an A320, then they would cost $100b, including a few spares. If the cable cost the same as steel, the materials would cost $300b. Cable would likely be made from coal or oil, or possibly atmospheric CO2. What you lose in material construction costs you might make back because of the lack of refinement needed. At a billion tonnes, the cable represents about 10% of the current annual carbon output, so could be viewed as positive carbon sequestration! Including development costs and peripherals, a cost in the trillion dollar range seems possible.

Another way of estimating the cost is the cost necessary to be competitive. If there was a demand for 4000T or 600 launches a day, which is the projected capacity of the cable in the above discussion, the cost of doing that over 30 years using rockets is about $2*10^15. However there is an excellent chance that rockets will gradually become fully and rapidly reusable, bringing the cost down by a factor of a thousand to only $2t. By comparison, this is the cost of 10 years of war and a national reputation. This is also roughly equivalent to the cost estimate above, indicating that with enough demand, space elevators could be competitive. While 600 launches a day might seem ambitious now, especially considering that each has a payload of 6 T, consider the history of aviation.

One hundred years ago, a plane that could carry 6 tonnes, let alone the demand for 600 flights a day seemed impossible. Today, there are more than 5000 A320s (just one type of aircraft) built. The US has an average of 28,000 commercial passenger flights a day, each carrying an average of 20T of cargo. Similar narratives apply to the history of mass rail. If today a million tonnes of people and stuff gets moved by air every day in the US, a country with a mobile population of about 200m, then 4000 T/day into space implies an off-world population of less than a million. As crazy as that seems today, it's only a tiny fraction of humanity.

The purpose of this blog post was not to predict when, but to demonstrate feasibility. Jules Verne and Leonardo da Vinci both knew that flight was possible, just not yet technologically available. There is no physical law that prevents space elevators from being possible. I cannot say for sure that by the time they become competitive, there won't be a better or more awesome way to get to space. Hopefully involving warp drives. But what I've shown is that it's nearly economically feasible too. 

If I were to hazard a guess for when it would be likely to build one? 15 years to put people on Mars. 5 more to set up a permanent base. 5 more to begin mass migration. 20 more to move half a million people there. At this point continued migration would be cheaper by space elevator, even if people were only going one way. So 2055, which is in my lifetime, hopefully. A space elevator on Mars could be built much more easily due to its reduced gravity.