__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.

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.

__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:

__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.

__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.