Space elevator: Encyclopedia II - Space elevator - Physics and structure

Space elevator - Physics and structure

There are a variety of tether designs. Almost every design includes a base station, a cable, climbers, and a counterweight.

Space elevator - Base station

The base station designs typically fall into two categories—mobile and stationary. Mobile stations are typically large oceangoing vessels, though airborne stations have been proposed as well. Stationary platforms are generally located in high-altitude locations, such as on top of high towers.

Mobile platforms have the advantage of being able to maneuver to avoid high winds, storms, and space debris. While stationary platforms don't have this, they typically have access to cheaper and more reliable power sources, and require a shorter cable. While the decrease in cable length may seem minimal (typically no more than a few kilometers), that can significantly reduce the minimal width of the cable at the center, and reduce the minimal length of cable reaching beyond geostationary orbit significantly.

Space elevator - Cable

The cable must be made of a material with an extremely high tensile strength/density ratio (the limit to which a material can be stretched without irreversibly deforming divided by its density). A space elevator can be made relatively economically if a cable with a density similar to graphite, with a tensile strength of ~65–120 GPa can be produced in bulk at a reasonable price.

By comparison, most steel has a tensile strength of under 1 GPa, and the strongest steels no more than 5 GPa, but steel is heavy. 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 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 [4] 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 63 GPa, and such tubes averaged breaking between 30 and 50 GPa. 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.

Most designs call for single-walled carbon nanotubes. While multi-walled nanotubes may attain higher tensile strengths, they have disproportionately higher mass and are consequently poor choices for building the cable. One potential material possibility is to take advantage of the high pressure interlinking properties of carbon nanotubes of a single variety. [5]. While this would cause the tubes to lose some tensile strength by the trading of sp² bond (graphite, nanotubes) for sp³ (diamond), it will enable them to be held together in a single fiber by more than the usual, weak Van der Waals force (VdW), and allow manufacturing of a fiber of any length.

The technology to spin regular VdW-bonded yarn from carbon nanotubes is just in its infancy: the first success to spin a long yarn as opposed to pieces of only a few centimeters has been reported only very recently (March 2004); but the strength/weight ratio was not as good as Kevlar due to the inconsistent quality and short length of the tubes being held together by VdW.

Note that as of 2005, carbon nanotubes have an approximate price of $50/gram, and 20 million grams 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 so on. This is good for space elevators 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 climbers. 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 resist the outward centrifugal force of 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 Sun and Lunar effects), it is possible to show that the cross-sectional area of the cable as a function of height looks like this:

Where A(r) is the cross-sectional area as a function of distance r from the Earth's center.

The constants in the equation are:

• A₀ is the cross-sectional area of the cable on the earth's surface.
• ρ is the density of the material the cable is made out of.
• s is the tensile strength of the material.
• ω is the rotational frequency of the earth about its axis, 7.292 × 10^-5 radian per second).
• r₀ is the distance between the earth's center and the base of the cable. It is approximately the earth's equatorial radius, 6378 km.
• g₀ is the acceleration due to gravity at the cable's base, 9.780 m/s².

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.

Thus the taper of the cable from base to GEO (r = 42,164 km),

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 and 5000 kg/m³

• 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 nanotube would typically be just a millimeter wide at the base.

Space elevator - Climbers

A space elevator cannot be an elevator in the typical sense (with moving cables) due to the need for the cable to be significantly wider at the center than the tips. While designs employing smaller, segmented moving cables along the length of the main cable have been proposed, most cable designs call for the "elevator" to climb up a stationary cable.

Climbers cover a wide range of designs. On elevator designs whose cables are planar ribbons, some have proposed to use pairs of rollers to hold the cable with friction. Other climber designs involve moving arms containing pads of hooks, rollers with retracting hooks, magnetic levitation (unlikely due to the bulky track required on the cable), and numerous other possibilities.

Power is a significant obstacle for climbers. Energy storage densities, barring significant advances in compact nuclear power, are unlikely to ever be able to store the energy for an entire climb in a single climber without making it weigh too much. Some potential solutions have involved laser or microwave power beaming. Other possible designs use energy from regenerative braking of down-climbers passing energy to up-climbers as they pass, magnetospheric braking of the cable to dampen oscillations, tropospheric heat differentials in the cable, ionospheric discharge through the cable, and other concepts. The primary power methods (laser and microwave power beaming) have significant problems with both efficiency and heat dissipation on both sides, although with optimistic numbers for future technologies, they are feasible.

Climbers must be paced at optimal timings so as to minimize cable stress, oscillations, and maximize throughput. The weakest point of the cable is near its planetary connection; new climbers can typically be launched so long as there are not multiple climbers in this area at once. An only-up elevator can handle a higher throughput, but has the disadvantage of not allowing energy recapture through regenerative down-climbers. Additionally, an up-only elevator would require some other method to return people to Earth. Finally, only-up climbers that don't return to earth must be disposable; if used, they should be modular so that their components can be used for other purposes in space. In any case, smaller climbers have the advantage over larger climbers of giving better options for how to pace trips up the cable, but may impose technological limitations.

Space elevator - Counterweight

There have been two dominant methods proposed for dealing with the counterweight need: a heavy object, such as a captured asteroid, positioned past geosynchronous orbit, or extending the cable itself well past geosynchronous orbit. The latter idea has gained more support in recent years due to the relative simplicity of the task and the fact that a payload that went to the end of the counterweight-cable would acquire considerable velocity relative to the Earth, allowing it to be launched into interplanetary space.

Space elevator - Launching into outer space

As a payload is lifted up a space elevator, it gains not only altitude but angular momentum as well. This angular momentum is taken from Earth's own rotation. As the payload climbs it "drags" on the cable, causing it to tilt very slightly to the west (lagging behind slightly on the Earth's rotation). The horizontal component of the tension in the cable applies a tangential pull on the payload, accelerating it eastward. Conversely, the cable pulls westward on Earth's surface, insignificantly slowing it. The opposite process occurs for payloads descending the elevator, tilting the cable eastwards and very slightly increasing Earth's rotation speed. In both cases the centrifugal force acting on the cable's counterweight causes it to return to a vertical orientation, transferring momentum between Earth and payload in the process.

We can determine the velocities that might be attained at the end of Pearson's 144,000 km tower (or cable). At the end of the tower, the tangential velocity is 10.93 kilometers per second which is more than enough to escape Earth's gravitational field and send probes as far out as Saturn. If an object were allowed to slide freely along the upper part of the tower, a velocity high enough to escape the solar system entirely would be attained. This is accomplished by trading off overall angular momentum of the tower (and the Earth) for velocity of the launched object, in much the same way one snaps a towel or throws a lacrosse ball.

For higher velocities, the cargo can be electromagnetically accelerated, or the cable could be extended, although that would require additional strength in the cable.

Space elevator - Extraterrestrial elevators

A space elevator could also be constructed on some of the other planets, asteroids and moons.

A Martian tether could be much shorter than one on Earth. Mars' surface gravity is 38% of Earth's, while it rotates around its axis in about the same time as Earth. Because of this, Martian areostationary orbit is much closer to the surface, and hence the elevator would be much shorter. Exotic materials might not be required to construct such an elevator. However, building a Martian elevator would be a unique challenge because the Martian moon Phobos is in a low orbit, and intersects the equator regularly (twice every orbital period of 11 h 6 min). A collision between the elevator and the 22.2 km diameter moon would have to be avoided through active steering of the elevator, or perhaps by moving the moon itself out of the area.

Conversely, a Venusian space elevator would need to be much longer. Although a tether placed at the stationary orbit of the slowly rotating Venus would intersect the sun, one could be constructed that rotated with the fast-moving cloud decks of the planet which take only four earth days to make a complete cycle. The cable would need to exceed 100 thousand kilometers long but, counter-intuitively, would experience less stress due to the slightly smaller gravity exerted on the cable. Such an elevator could service aerostats or floating cities in the benign regions of the atmosphere.

A lunar space elevator would need to be very long—more than twice the length of an Earth elevator, but due to the low gravity of the moon, can be made of existing engineering materials. Alternatively, due to the lack of atmosphere on the Moon, a rotating tether could be used with its center of mass in orbit around the Moon with a counterweight at the short end and a payload at the long end. The path of the payload would be an epicycloid around the Moon, touching down at some integer number of times per orbit. Thus, payloads are lifted off the surface of the Moon, and flung away at the high point of the orbit.

Rapidly spinning asteroids or moons could use cables to eject materials in order to move the materials to convenient points, such as Earth orbits; or conversely, to eject materials in order to send the bulk of the mass of the asteroid or moon to Earth orbit or a Lagrangian point. This was suggested by Russell Johnston in the 1980s. Freeman Dyson, a physicist and mathematician, has suggested using such smaller systems as power generators at points distant from the Sun where solar power is uneconomical.

It may also be possible to construct space elevators at the three smaller gas giants, Saturn, Uranus and Neptune. These would all involve tapering several times greater than those of the inner solar system, and would need to be approximately 50-60 thousand kilometers long, yet are still within the limits of advanced nano-tubes. These outer space elevators could facilitate the exchange of supplies and helium-3 between floating mining colonies in the atmospheres and local moon settlements. However, difficulties such as the equatorially orbiting lower rings and moons of these giant planets would first need to be overcome.

Adapted from the Wikipedia article "Physics and structure", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki