One possible event that would destroy the elevator cable would be a lightning strike. Lightning has sufficient current and voltage potential in its arc to heat and destroy any composite that we have been considering. One could argue that the carbon nanotubes (melting point ~6000°) would survive a lightning strike and that there may be a similar hightemperature epoxy that could be used for the lower section of the cable. However, we consider this a higher-risk option and believe that there may be a better solution to the problem.
The electrical properties of Earth's atmosphere are impressive as can be seen in figure 10.1.1. Potential differences of 400 kV/m can be produced with 40 C of charge stored in thunderstorm cells. The cells of potential and charge are isolated so any part of a thundercloud could be charged even if another part is shorted to ground (lightning). If we look at our cable, this means the cable will appear to be the least resistance path to ground. In other words the cable will be the path lightning will take between cloud and ground. It also means that the cable will not be able to discharge the cloud sufficiently to retard lightning, the cells could be too isolated and any individual cell could damage the cable. If we decide to try to make the cable with a higher resistance than air (a possibility) we might avoid a lightning strike by being the most resistive path to ground. However, rain often accompanies lightning storms and if the cable were to become wet the water may form a conductive path to ground. The lightning may take the water to ground and in the process enough of the lightning's energy may be imparted to the cable (through the EMF field or heat or explosive pressure or ...) to destroy the cable.
Possibly the best solution to the lightning problem is to locate the cable anchor in a "lightning-free" zone like the one off Ecuador (figures 10.1.2 and 10.1.3). The anchor station would move the lower end of the cable out of the path of the few storms that do occur in these regions. In such regions less than several lightning strikes will occur over the course of a year in a 100,000 km2 area. These strikes will also be concentrated in only several storms of limited spatial extent. This location and anchor movement scenario may also be required for avoiding high winds that may damage the cable.
A second alternative location for the cable anchor could be above 6 kilometers near the peak of a mountain. At least in one study it was found that there was a greatly reduced occurrence of lightning at these altitudes [Dissing, 1999]. This effect can also be seen in lightning frequency maps (figure 10.1.2). However, there are difficulties with locating the cable anchor on a mountain peak (Chapter 6: Anchor).
Meteors are a serious concern for the survivability of the space elevator cable. Meteor fluxes have been measured from Earth, and their impact characteristics have been studied in-situ (LDEF) and in high velocity impact facilities. In the thick plate regime meteors will destroy an volume 50 times that of the impactor and to depths of several times the impactor's diameter. Much of this destruction is due to the energy shock that is created in the bulk material by conversion of kinetic to thermal energy. In a thin plate scenario this changes. The shock is more intense in a solid thin sheet because the reflection off the back face combines with the initial shock. However, much of the energy also escapes out the back side of the thin plate without destroying more of the plate so the total volume destroyed in a thin plate is less than in a thick plate. In our case we have a more unique situation, we have a sheet composed of independent fibers in a very thin plane. We will see how this affects our situation. But first we will examine our environment.
Published micrometeor fluxes from several sources give roughly the same distribution (see figure 10.2.1). Below about 1 cm radius natural micrometeors dominate the population of material near Earth, above about 1 cm radius man-made space debris is the major constituent.
From the published fluxes we can calculate the impact rate we would expect and the resulting damage. If we are to assume a micrometeor will survive long enough to go straight through the cable and destroy a section (worst case), we will have catastrophic damage for large meteors at any angle of impact and for small meteors at grazing angles across the ribbon face (figures 10.2.2 and 10.2.3). For objects larger than 1 cm diameter we see the impact rate on our initial, vulnerable cable is once in several decades (figure 10.2.4). Grazing impacts by small meteors nearly parallel to the cable's long axis will not cause catastrophic damage. The criteria we find for which meteors will damage the cable is:
where r is the radius of the meteor, w is the width of the cable, f is the fraction of cable that must be destroyed to sever it, f is angle between the ribbon face and the incoming trajectory of the meteor and q is the angle in the plane of the ribbon face between the meteor trajectory and the cable's long axis (figures 10.2.2 and 10.2.3). Integrating over the relevant angles we can find the fraction of the meteors that can sever the cable as a function of meteor radius. Combining this with data on meteor fluxes (figure 10.2.1) we find how often we can expect the cable to be severed by a specific sized meteor (see figure 10.2.4). Examining figure 10.2.4 we find that the small, grazing-incident meteors will destroy our cable quickly. Initially, this does not sound good, but we need to understand the situation better to really determine how much of a problem we have. First, we assumed the meteor would pass straight through the cable destroying everything in its path. Second, we assumed the cable was perfectly flat.
When we look at our scenario, for example, we realize the 100 micron particles are coming in at angles of less than 0.25 degrees to the ribbon face and continuing straight into the cable plane (one to ten microns thick on edge) for over two centimeters without being deflected. In many cases it will run into alternating regions of bare carbon nanotubes and epoxy/nanotube composite. This implies that the cable is flat to better than 100 microns across large sections of its face. What we have is a grazing impact on a thin sheet. It turns out that studies of this situation have been done for composite sheets [Lamontage, 1999: Taylor, 1999]. These experiments used impactors with roughly the same diameter as the target thickness and examined the affect of incident angle on the penetration, damage and ejecta. What was found was that the impact and debris were deflected normal to the plane of the target and did not continue on their original path. It was also found that the damaged area and penetration depth dropped dramatically with increasing incident angles and with target thickness (the thinner the target the less area damaged). These experiments strongly argue that all impacts where the impactor and cable thickness are roughly of the same dimension (even grazing impacts) will damage no more cable area than several times that of our impactor. Now when we talk about impactors of 2 mm radius on our cable we are no longer in the same situation as in the published experiments because our impactor is now much larger than our cable is thick. In these cases further studies are required but from the data it looks likely that we will find that grazing incident impacts on our cable by meteors even several millimeters in radius will not stay in the plane of the cable and cause serious damage.
Ignoring the discussion in the previous paragraph, let's assume we still have a problem with 2 mm radius meteors (figure 10.2.4). One way to eliminate this hazard is to give the flat face of the ribbon a curvature that has a displacement out of the plane of the ribbon of more than 8 mm over a length of 2.5 cm. This would eliminate the possibility of a 2 mm radius meteor being able to damage more than 2.5 cm of our cable (and does not substantially increase our problems for larger meteors). This curvature essentially makes a tube (radius of 19 mm) of our 10 cm wide ribbon. With a curvature of this magnitude we will eliminate the problem of small, grazing incident meteors entirely. These calculations are also only for our initial 10 cm wide ribbon. If we can place a wider initial ribbon in orbit then the meteors of concern drop in frequency linear with the width and a larger area needs to be destroyed before our cable has problems. The outcome is a flatter cable can be used. Curving the cable at the levels we are discussing has little impact elsewhere in our program.
Currently space debris larger than 10 cm diameter is tracked by U. S. Space Command. This accounts for roughly 8000 objects (satellites and space debris). An additional 100,000 objects with diameters between 1 and 10 cm are in Earth orbit. Of these objects most are in LEO (500 - 1700 km) which has the highest and most deadly relative velocity to the space elevator cable. With this density of debris we can expect the cable to be hit and possibly severed once every 250 days. One possible solution to this problem is to track all of the space debris between 1 and 10 cm diameter and move the cable out of the path of any that are on a collision course (Chapter 6:Anchor).
Haystack observatory is beginning to study and track objects in Earth orbit down to 1 cm. Optical tracking systems are also coming on-line at this time. Tracking space debris down to 1 cm has been a concern of NASA because of its affect on the space station. A study was done at Johnson Space Center [Loftus, 1993] on the construction of a new debris tracking network and came up with a design that would monitor objects down to 1 cm with 100 m accuracy using essentially current technology. This is very close to the tracking network we would need for the space elevator. An alternative system could be a set of five facilities located on the equator based on the Berkeley's One Hectare Telescope. This would be an easily implemented and inexpensive solution.
Initially, we want to avoid all impacts on the cable from objects larger than 1 cm (this becomes less stringent as the cable grows). Based on the system proposed by Johnson Space Center the space elevator would need to avoid a piece of space debris every fourteen hours on average (see table 10.3.1). With an understanding of cable dynamics, a good computer system and the proposed anchor facility (Chapter 6: Anchor) this level of active avoidance is feasible.
One additional design modification that could be implemented is to widen the initial cable slightly at orbits where the debris is highest (figure 10.3.1). The vast majority of the critical debris is located between 500 and 1700 km altitude (Interagency Report on Space Debris). If we were to design the cable to be twice as wide for these 1200 km we would reduce the risk of serious damage by roughly 30% pushing the critical meteor size up to roughly 3 cm which is more easily tracked. The increase in the cable mass would be 0.65% including the greater cable to support this extra mass.
Let's assume we have the initial and weakest cable deployed and a wind blowing across 1 km of its length. For a first example, we will also assume it is acceptable for the cable to be displaced such that we have a 10° deviation from a nominal position. The question is: what wind velocity will break our cable in this scenario.
The force from the wind perpendicular to the ribbon face required to break the initial and weakest cable is:
To do the calculation correctly we need to calculate the aerodynamic drag on a ribbon or set of strings or rods (the individual fibers in the ribbon) with regularly spaced plates connecting the rods (composite sections). In addition, the cable will rotate in the wind to some extent. However, we will start with a slightly simpler and worse case where the ribbon is face on to the wind. In this case, all of the fibers and composite plates see the full force of the wind. In reality there would probably be some shadowing of the wind as the ribbon turned in the wind as well as some turbulence and fluttering of the cable.
The drag of an object can be expressed as:
where D is the drag, r is the air density, A is the frontal area, v is the air velocity and Cd is the drag coefficient (1.28 for a flat plate, ~0.07 for a cylinder in low velocities)
For a large number of individual fibers we then have:
where N is the number of fibers. For a flat plate like the composite sections we have:
The total drag will be:
This can be solved for v:
With the force required to break the cable from above (2110N), 1200 fibers of 10 micron diameter (one possible configuration for the first, smallest and most vulnerable cable), 5% of the length in composite and wind effective over a 1 km vertical extent we get:
It is easily seen that almost the entire drag comes from the composite sections. The reason for this is they fill the area between the fibers making for a much larger effective area per unit length and the drag coefficient for a flat plate is almost twenty times that of the cylinder.
Looking at the wind speeds near the proposed anchor location [Chelton, 1981], (figure 10.4.1) we find the velocity distribution is actually considerably below the 32 m/s breaking velocity. However, a larger margin of safety, if we can easily get it, is always better. A second paper [Sandwell, 1984] gives a global map of the seasonal average wind speed. In the maps it can be seen the spatial distribution of high and low wind regions. The proposed anchor location (~1500 kilometers west of the Galapagos Islands) is found to have yearly low wind speeds and is in the area of very few lightning strikes.
|Type of Storm||Category||Winds (mph)|
Areas where we can further reduce the risk of damage by wind include:
If the design modifications push the critical wind velocity to roughly 154 mph then we are discussing destruction by a cyclonic storm. If we are looking at an ocean platform anchor then the storms we are concerned with are specifically category 4 hurricanes. We then have a possible solution by considering the anchor location in terms of the spatial distribution of hurricanes. Figure 10.4.3 shows the general and historical spatial locations of hurricanes. As can be seen in the global maps, hurricanes tend to exist at low latitudes in both the northern and southern hemispheres but do not occur at or cross the equator. It can also be seen that the eastern pacific off the coast of Ecuador has essentially no hurricane activity. This area is our current first choice for an anchor location based on the spatial distribution of lightning and may now solve our wind loading problems as well.
Atomic oxygen exists in the upper atmosphere between about 60 and 800 km with the highest density near 100 km altitude. It is extremely corrosive and will etch the epoxy in our cable and possibly the carbon nanotubes. On NASA's Long Duration Exposure Facility (LDEF) mission atomic oxygen etched carbon fiber/epoxy composites at rates up to 1 mm/month , preferentially etching the epoxy in some cases. This high etch rate was only seen on the leading face of the spacecraft where the atomic oxygen is being swept up. On the trailing edge and in the shadowed regions the etching by atomic oxygen was found to be zero in many cases. The reason for this is the thermal velocity of atomic oxygen is about 1 km/s whereas the LDEF satellite velocity was over 7 km/s. For our stationary cable we would expect the etch rate would be down by about two orders of magnitude from what LDEF experienced on its leading edge. On the flip side, LDEF was at about 400 km which places it at an altitude with an atomic oxygen density two times less than the maximum our cable will experience. The bottom line is we should expect to see etching rates of 1 mm/month. This etch rate would be sufficient to destroy our cable in a few weeks in the current design (see figure 10.5.1). There are two possible solutions to this particular problem.
The first and probably best solution is to coat the affected segment of cable with a material that is resistant to atomic oxygen as suggested by many of the LDEF experiments. In addition to carbon/epoxy composites, LDEF also had bulk and thin film metal experiments, and metal-coated composites. During the 5.8 year life of the LDEF mission, gold and platinum were unaffected by atomic oxygen, while aluminum and several other metals were found to have minimal degradation. In the metalcoated composite tests it was found that coatings (nickel plus SiO2) as thin as 0.16 microns could protect the composite from the affects of atomic oxygen. If we coat the affected length of the cable with a metal such as gold or aluminum we will need to make a trade-off between durability of the metal coating under the passage of climbers and minimizing mass so as to not weigh down the cable. Coating thicknesses between 0.02 and 25 microns may be acceptable depending on durability and the density of the coating. What needs to be completed are a set of tests that determine: 1) if the carbon nanotubes as well as the epoxy will be etched, 2) will a metal layer adhere to the epoxy and nanotubes of our cable, and 3) what is the minimum layer of metal that will survive the passage of several hundred climbers.
The second solution is to modify the cable geometry. The baseline, pre-modification cable has an average thickness of 1.5 microns and a width varying from 5 to 11.5 centimeters. This ribbon can be a uniform 1 micron sheet or spaced fibers of 5, 10, 20, 30, up to 400 micron diameter. The larger diameter round fibers would survive much longer in the atomic oxygen environment. As can be seen in figure 10.5.1 unprotected fibers less than about 100 microns would be insufficient to survive in the atomic oxygen environment long enough for the cable to be strengthened. These large diameter fibers may cause problems for the climbers, be more difficult to add fibers to and have a higher risk of damage by meteors.
Heating of the cable can be produced by passage through the local magnetic fields. The potential induced along the cable can be expressed as: E = B(r)v(r) where E is in volts/meter, B(r) is the magnetic field, and v(r) is the velocity of the cable relative to the magnetic field. For radii (r) <10rEarth, B(r) ~0.35_10-4rEarth 3/r3 and v(r) is approximately zero. However, if we assume the worst possible case where the magnetic field is fixed and the cable is rotating with the Earth (v(r) = 463 r/rEarth m/s) we get potentials from 0.00026 V/m at 10rEarth to 0.016 V/m at Earth's surface. At distances of greater than 10rE, the cable is in the interplanetary magnetic field during the day (Bave ~6 nT and Bmax ~80 nT) and is in the Earth's magnetosphere at night. This corresponds to a maximum potential of 0.00068 V/m at the far end of the cable. With a minimum resistance of 0.4 W/m we have a maximum of 0.0064 W/m of heating occurring near the Earth end of the cable and 1 mW at the far end. The cable would quickly radiate this level of heating away into space.
The segment of the cable in Earth's radiation belts will experience less than 3 Mrad per year (energetic electrons and protons) [Daly, 1996]. Studies of epoxy/carbon fiber composites (epoxy/nanotube composites would be expected to be comparable or better) have found them to be radiation hard to greater than 104 Mrad [Egusa, 1990: Bouquet, 1979]. This would allow them to survive more than 1000 years in the expected environment.
The other radiation damage that must be considered is that of solar UV. Often specific materials can be corroded by ultraviolet radiation and this must be considered when selecting an epoxy for the cable construction.
Initial work by Pearson, 1975, on oscillations induced by the moon, sun and motion of climbers found the problems avoidable. However, since we have a fairly different system scenario the calculations need to be repeated. In Pearson's work, he had a cable of 144,000 km long with no counterweight on the end and examined only taper ratios above three. Our shorter cable will increase our characteristic frequency, the counterweight will essentially fix the upper end of the cable (doubling our frequency from the case Pearson calculated) and our smaller taper ratio will decrease the frequency.
If we examine the standard oscillation of a string under tension [Nagle, 1996] with the ends fixed, we find our system is very close to the ideal case. The initial-boundary value problem for the standard problem is:
where u is our equation of motion, L is the length of the string, f (x) is the initial location of the string, g(x) is the initial velocity of the string and a2 is equal to the ratio of the tension to the linear density of the string. This assumes: no gravity, the string is perfectly flexible, the string is a constant linear density, the tension is constant and no other forces are acting on the string. This doesn't sound like our situation at all. Well, let's look at it a little closer. We have gravity, but that is what is giving us the tension and it is along the cable in our case. For any individual segment of the cable, the force applied by gravity is small compared to the tension. Our cable is pretty much perfectly flexible, the width is much less than the length. Our cable does not have a constant linear density or constant tension. But let's look at where these two come into our calculations. Equation (1) is equivalent to F=ma for each segment of the cable. The only place where the design of the cable comes in is in a2. And a2 is also where the restriction on constant tension and linear density comes into play. The one unique thing about our cable design is it is designed specifically such that the tension for any segment is exactly proportional to its crosssectional area (or linear density). To be precise, to correct these equations we would need to put in a function of x in equation (1). However, because of our specific relation between tension and linear density this function is defined as 1 at all locations and has no affect on our problem. The initial boundary value problem we have stated above is a very close match to our situation. The last constraint was that there were no other forces acting on the string. To a large extent this is true, only the moon, sun and climbers will act on our cable. These are small forces that can pump oscillations but not dramatically change the characteristic frequency of the cable.
The solution to this problem is given in many differential equations textbooks and gives a characteristic frequency for our cable of
where L is 9.1x107 m, a is 7.1 x103 m/s and n is 1, 2, 3, 4, .... The first mode has a period of 7.1 hours. This period is sufficiently far from 12 hours (the sun), 12.5 (the moon) or integral fractions of these so our cable should not be pumped by either the sun or moon. A cable 15% shorter than the one we propose (76,000 km) could have serious problems. Small oscillations or traveling waves that may be induced by wind or meteors can also be actively damped out at the base of the cable if a cable displacement monitoring system is implemented to detect any movements in the cable.
One oscillation that Pearson investigated was that of transverse waves induced by climbers. The bottom line on this oscillation is that large oscillations can be induced when the climber transverses the length of the cable in one period of the cable's characteristic frequency. (Pearson assumed no counterweight so had the climber traveling twice the length of the cable during one period.) Since we just calculated our cable's characteristic period to be 7.1 hours we will only need to worry about this particular affect when we plan to have climbers traveling at close to 10,000 km/hr.
When considering the construction of a space elevator the possible environmental impacts must be examined. Two of those environmental impacts will be considered here. The first is the possibility of discharging the ionosphere and the second is the impact if a space elevator were to be severed and fell back to Earth.
The charge production rate in the ionosphere ranges between 2000 and 6000 q/cm3/s. For an area around the cable of 1km x 1km and 500km in vertical extent this relates to 1x1025 q/s or 625,000 C/s. With a resistivity 10-4Wm for carbon nanotubes, a 20-ton capacity cable (2 mm2 cross section) would have a minimum resistance of roughly 5MW. For the cable to discharge the ionosphere at the same rate as charge is being produced would require a current of 625,000 Amps to flow through the cable. To produce this current a voltage difference of ~3 x 1011 Volts would be required between Earth and the ionosphere. The measured electric field under thunderclouds just before a lightning strike is 10 - 20 kV/m. If we extend this electric field up to the ionosphere (which does not occur but should be a worst case) we find the static voltage potential would be less than 2 x 109 Volts. At this voltage difference with no redistribution of charge in the ionosphere we could discharge an area 100m around the cable. Since we have assumed the most conducting cable possible (in reality it would probably be down by orders of magnitude due the epoxy sections) and the highest potential difference conceivable it is more likely that only a small volume of centimeters radius would show any affect from the cable's presence.
If a cable is severed the lower segment will fall back to Earth while the upper portion floats outward. The worst case would be if the countermass breaks off the far end of the cable and the entire 91,000 km of cable falls back to Earth.
Depending on the location of the break, the epoxy used, the dynamics of the fall, etc. the cable will re-enter the Earth's atmosphere at a velocity sufficient to heat the cable above several hundred degrees Celsius (figure 10.9.1). If the cable is designed properly, the epoxy in the cable composite will disintegrate at this temperature. This means the cable above a certain point will re-enter Earth's atmosphere in small segments or carbon nanotube / epoxy dust. About 3000 kg of 2 square millimeter crosssection cable (20 ton capacity) may fall to Earth intact and east of the anchor. Detailed simulations will be required to determine the possible sizes of segments that will survive and the health risks associated with carbon nanotube and epoxy dust. In terms of the mass of dust and debris that will be deposited, we can compare what will happen to what naturally happens now. Each year 10,000 tons of dust accrete onto Earth from space, the additional 750 tons of the first cable will increase that year's infall by 7.5%. A larger 1000-ton capacity cable would have a mass of 30,000 tons or roughly equivalent to 3 years of normal global dust accretion. Further investigations are required to determine the environmental impact of depositing this much dust along the Earth's equator.
In the opposite case where the break is at the bottom of the cable, the entire structure would float up away from Earth. The cable would remain in orbit with the lower end hovering above the Earth at some low altitude. Re-connecting the lower end may be a possibility, but we will not speculate on that here.
In all cases, there will be some amount of time (hours to days) between the initial break and any substantial change in the configuration of the cable. Various scenarios can be derived for saving the cable or stopping it from re-entering during this delay but these will all greatly depend on many aspects of the design and current state of the cable at the time of the break.
In any analysis of the environmental impact the possibility of a falling cable and the damage it will cause must be compared to the alternative which is continued use of rockets. During rocket use both pollutants from the burning fuel and from the re-entry of the spent rockets must be considered. For example, each Titan IVB has a dry mass of 65,000 kg, much of which ends up re-entering and burning up in Earth's atmosphere. The Titan IVB also burns roughly 500,000 kg of propellant. Our proposed 20 ton capacity cable has a mass of 750,000 kg. A strictly mass comparison is far from the proper comparison to make but it gives a rough idea of scales of the environmental impacts we need to compare.
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