The ribbon must be made of a material with a huge tensile strength/density ratio (the stress a material can be subjected to without breaking, divided by its density). A space elevator can be made relatively economically feasible if a cable with a density similar to graphite and a tensile strength of ~65-120 GPa can be mass-produced at a reasonable price.
By comparison, most steel has a tensile strength of under 2 GPa, and the strongest steel resists no more than 5.5 GPa, but steel is dense. The much lighter material Kevlar has a tensile strength of 2.6-4.1 GPa, while quartz fiber can reach upwards of 20 GPa; the tensile strength of diamond filaments would theoretically be minimally higher.
Carbon nanotubes (a material that was first discovered in the 1991) appear to have a theoretical tensile strength and density that is well above the desired minimum for space elevator structures. The technology to manufacture bulk quantities of this material and fabricate them into a cable is in early stages of development. While theoretically carbon nanotubes can have tensile strengths beyond 120 GPa, in practice the highest tensile strength ever observed in a single-walled tube is 52 GPa, and such tubes averaged breaking between 30 and 50 GPa1).
Even the strongest fiber made of nanotubes is likely to have notably less strength than its components. Improving tensile strength depends on further research on purity and different types of nanotubes.
The most likely designs to succeed call for single-walled carbon nanotubes (SWNT). While multi-walled nanotubes (MWNT) may attain higher tensile strengths, they have disproportionately higher mass and are consequently poor choices for building the cable.
Note that as of 2006, carbon nanotubes have an approximate price of $25/gram, and 20,000 kg - twenty million times that much - would be necessary to form even a seed elevator. This price is decreasing rapidly, and large-scale production would reduce it further, but the price of suitable carbon nanotube cable is anyone’s guess at this time.
Carbon nanotube fiber is an area of energetic worldwide research because the applications go much further than space elevators. Other suggested application areas include suspension bridges, new composite materials, lighter aircraft and rockets, and computer processor interconnects. This is good news for the space elevator because it is likely to push down the price of the cable material further.
Due to its enormous length - a space elevator cable must be carefully designed to carry its own weight as well as the smaller weight of lifters. The required strength of the cable will vary along its length, since at various points it has to carry the weight of the cable below, or retain the cable and counterweight above. In an ideal cable, the actual strength of the cable at any given point would be no greater than the required strength at that point (plus a safety margin). This implies a tapered design.
Using a model that takes into account the Earth’s gravitational and centrifugal forces (and neglecting the smaller solar and lunar effects), it is possible to show that the cross-sectional area of the cable as a function of height is given by:
Where <math> A® </math> is the cross-sectional area as a function of distance <math> r </math> from the Earth’s center.
The constants in the equation are:
* <math> A_{0} </math> is the cross-sectional area of the cable on the earth’s surface. * <math> \rho </math> is the density of the material the cable is made out of. * <math> s </math> is the tensile strength of the material. * <math> \omega </math> is the rotational frequency of the Earth about its axis, 7.292 × 10-5 rad·s<sup>-1</sup>. * <math> r_{0} </math> is the distance between the Earth’s center and the base of the cable. It is approximately the Earth’s equatorial radius, 6378 km. * <math> g_{0} </math> is the acceleration due to gravity at the cable’s base, 9.780 m·s-2.
This equation gives a shape where the cable thickness initially increases rapidly in an exponential fashion, but slows at an altitude a few times the Earth’s radius, and then gradually becomes parallel when it finally reaches maximum thickness at geostationary orbit. The cable thickness then decreases again out from geosynchronous orbit. The relative thickness at all points is determined by the strength density ratio. This is shown in the figure to the right.
Thus the taper of the cable from base to GEO (r = 42,164 km), :<math> \frac{A(r_{\mathrm{GEO}})}{A_0} = \exp \left[ \frac{\rho}{s} \times 4.832 \times 10^{7} \, \mathrm{ {m^2}\!\!\cdot\!{s^{-2}} } \right] </math> Using the density and tensile strength of steel, and assuming a diameter of 1 cm at ground level, yields a diameter of several hundred kilometers at geostationary orbit height, showing that steel, and indeed most materials used in present day engineering, are unsuitable for building a space elevator.
The equation shows us that there are four ways of achieving a more reasonable thickness at geostationary orbit:
* Using a lower density material. Not much scope for improvement as the range of densities of most solids that come into question is rather narrow, somewhere between 1000 kg·m-3 and 5000 kg·m-3. * Using a higher strength material. This is the area where most of the research is focused. Carbon nanotubes are tens of times stronger than the strongest types of steel, hugely reducing the cable’s cross-sectional area at geostationary orbit. * Increasing the height of a tip of the base station, where the base of cable is attached. The exponential relationship means a small increase in base height results in a large decrease in thickness at geostationary level. Towers of up to 100 km high have been proposed. Not only would a tower of such height reduce the cable mass, it would also avoid exposure of the cable to atmospheric processes. * Making the cable as thin as possible at its base. It still has to be thick enough to carry a payload however, so the minimum thickness at base level also depends on tensile strength. A cable made of carbon nanotubes (a type of fullerene), would typically be just a millimeter wide at the base.