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We have built a space elevator using carbon nanotube. The cable is just strong enough to sustain the counterweight, with a narrow margin to survive storms.
PROBLEM: Terrorists of the Spaghettic State have vowed to destroy the gluten-free appendage.
All using current or near-future technologies.
There are three potential avenues of attack, some of which have multiple ways to cause major damage to your (very fragile) space elevator.
1: Surface
For a direct attack the ground is the easiest to deal with. A large exclusion zone with very good surveillance will deter or flat out stop a terrorist (by terrorist I'm assuming not a major military incursion) attack.
The issue with the ground would come with the payloads being launched. Even a pipe-bomb could have massive destructive potential, as the space elevator would have to have exquisitely balanced and orchestrated payloads going up and down the ribbon in order to avoid destructive resonance etc. Stop one of the payloads for a few minutes: tower comes down. The security procedures for getting to the space elevator would have to make the Pentagon look a bit tame. Personally I'd be tempted to stop people travelling with their own possessions and undergo a week long quarantine to flush out any ahem foreign bodies.
Against a major military incursion: you've got nothing. As strong as the ribbon might be it's also under a lot of tension. A bullet striking the ribbon might be enough to tear, shatter or otherwise ruin the ribbon.
2: Air
There are 2 ways that the tower can reasonably be attacked from the air by a terrorist group. Hijacking and missiles.
Hijacking is easy to deal with, but would carry some major political flak. It's basically the same principle currently used to defend high profile buildings: Blow up the compromised aircraft. Brutal but would get the job done. Have a few radar stations and a couple of nearby fighter jets ready to scramble and hijacking is no longer a threat
Missiles are more of an issue. Even an indirect strike can carry major damage risk (shrapnel and concussive forces don't play well with the ribbon). The easiest way to prevent this being a risk is to make sure your ground based exclusion zone is wide enough that the terrorist missile launchers haven't got sufficient range to hit you. This might be an exorbitantly huge area. As a second line of defence you might consider point defence cannons similar to those used in naval warfare, or even explosively launched interceptor missiles. It all depends on how well funded your terrorist cell is.
Again: Against a military attack you've got nothing. Long range missiles, drone launches, stealth or high altitude bombers, or even good old 'throw planes at it until one of them gets lucky' means that your tower won't survive.
3: Space
The least likely for a terrorist attack, but also the hardest to defend. Space is Big. The capital letter is deserved there. Compared to the amount of ribbon you have to defend against aerial attack there's a ridiculous length of ribbon utterly exposed between the earth and the terminal point (the terminal point being 2x higher than geostationary orbit and a reasonable distance towards the moon). An impact here will almost certainly destroy the ribbon, and be almost impossible to repair even if it doesn't. Luckily your terrorists are terrorists. If space is in any way monitored, patrolled or governed you can restrict who has access to what, and deal with it from there.
If there is a determined attacker in space then high quality lenses are your friend. You'll need an awesome 360 degree sensor package (which is eminently doable for distances less than the moon) with as many bells and whistles as you can think of to ensure redundancy and spot anything on even a vague intercept course well before it gets to you. This sensor grid would have to extend around the world in order to watch for threats from the other side of the planet. Then you'll need several fancy missiles/ space fighters/ whatever you like in order to defend the ribbon in a similar manner to the air defence scenario. If it's on a path that will, at some point in the future, be able to intercept using current engine technologies, then you warn it off and blow it up if it doesn't immediately start a corrective burn. Finally you'll need an array of high power lasers on satellites in geostationary orbit to deflect/ deorbit any smaller pieces of junk/wreckage, as a bolt moving at orbital velocities will carry enough energy to punch a hole through the ribbon or at the very least destabilise the tower.
Preventing/ stopping space based attacks would be a real pain, but also be the foundation for the kind of orbital defence grid you'd need in order to stop people simply dropping rocks and letting kinetic energy do all the hard work. If the terrorists in space can drop a 2 tonne ceramic cone onto your space elevator's base station they also have the power to turn any city in the world into a kilometre wide crater.
Again: If the military gets involved you're boned. A few hundred light rockets with a grudge will tear down the tower like it's made of tissue paper. They don't even have to worry about re-entry criteria like an ICBM or getting up to orbital velocity like a spaceship.
Summary:
It's doable if you're sensible and already have infrastructure in place to prevent space terrorists from ending the world (you've got that, right?)
My suggestion: Find a nice tropical island somewhere (or build your own, if you've got the resources for a space elevator). Outfit it with it's own airstrip and enough defences to cover a half dozen Carrier groups, and let everyone know that there's a twenty mile wide exclusion zone with a 'trespassers will be killed to death' policy that has no exceptions. Create a security port somewhere nearby that specialises in stripping people of everything (like a Duty-Free Guantanamo Bay) before loading them onto high-security boats/helicopters for transfer to the elevator. In space: Kill anything that looks at you funny and then deorbit it using high powered lasers.
Oh, and pray that the terrorists aren't secretly backed by the military, or you're doomed.
I think the key defense that hasn't been mentioned yet:
Build more than one.
Basically, the core costs of building a space elevator are greatly reduced when you build a second one. No more R&D, and the ability to use the existing elevator to lift material into space helps enormously. Losing the world's only space elevator hurts, but losing only one of hundreds?
Good disaster planning would help a great deal.
Most likely any attack would take place low, near the anchor point. This means not much weight will fall down and cause damage, while the majority of the cable would instead want to rise upwards.
When the attack happens, well designed safety mechanisms would kick in. Motors at the top of the elevator would wind up the cut cabling. Climbers would detach and use emergency rocket motors or parachutes to land safely, or make their way to collection points near the other elevator terminuses. After the immediate damage near the base anchor is cleaned up, a backup cable could be lowered, the climbers refurbished and reattached, and the whole thing brought back to operation. The broken cable could be assessed for possible repair.
In the best case scenario, nobody dies, and everything is back to normal in about a week. Take that, space-terrorists!
http://www.wikipedia.org/wiki/Space_elevator_safety
Attacks near the top of the cable would suck much more. Maybe the best you can do is try to control the descent of the falling cable, again by using emergency rocket motors in the climbers.
The attack vectors have been covered by others, so i will concentrate on how to defend it.
Since a determined attacker will eventually find a way to attack, damage or even destroy the cable, or the fixture at the bottom that holds it, i think the best way to defend against an attack is:
don't be a target
While of course you cannot conceal this structure, you can avoid being an interesting target for an attack.
Terrorists attack symbols of whatever group they consider as enemies.
Most of the time, the alleged reasons for the attack are just mumbo-jumbo. Terrorist attacks most of the time are in fact publicity actions, because they help the terrorist group that performed the attack in getting or increasing "street credibility", and as a result get more followers and more money.
After all, terrorist groups are mostly criminal organizations.
To gain street credibility, the target they choose should have high idealistic value to their enemies (so everybody can see that they managed to hit them where it hurts), but it should be of no (great, at least) value to the group they recruit members and money from.
That means: Whichever group is operating the space elevator should spend quite some energy into not making enemies. They should generously reimburse anybody whose land they used to build the elevator and the perimeter, they should be generous employers, benefactors to the surroundings and the whole world, and they should make really sure that at least something like 80% of the world's population has at least mildly positive feelings about the elevator and its operators.
That way, you minimize the risk of being attacked.
As a bonus, it most likely means you are being a nice person.
One way to destroy it would be to wind it in. This would pull the geostationary space station out of orbit.
The cable(s) are designed with enormous tensile strength but resistance to shearing may not be good. Repeated missile attacks would weaken or even melt or snap it.
You haven't specified the thickness and number of the nanotubes. If there are only a few and they are narrow, a good pair of bolt-cutters would do the job.
The terrorists could use 9/11 tactics and simply fly a plane into the cables.
Prevention would be a real problem. A large portion of the Air Force would have to be on constant standby. So would air traffic control in the area - both to avoid civilian planes hitting it by accident and to detect intruders.
Personally I think it would be so unstable that simply sending a heavy load up would throw the whole thing out of balance and cause a catastrophe - imagine the collateral damage!
The cables would act as a lightning conductor so there would have to be a way of preventing thunderstorms.
On the other hand, if the cables are non-conducting then the process of going up and down would create a huge static charge.
Considering that you said 'near future', I am assuming that a lot of the currently in-development technologies will be easily available and that technologies that would build off of them would be in development at the time the space elevator attacks would happen. As other answers have stated, defending against a ground attack would be quite easy. The air is probably the place that needs defending. Currently, swarming robots are in development, and at the time of a space elevator construction, it will probably be quite possible that there are automated swarming missiles that can easily identify and dispose of a threat. Warheads have also gotten much smaller and more powerful, so these missiles may very well be able to handle a large aircraft without being very large. Swarm drones and swarm missiles could surround and patrol the area around the cable and stations, and anything unauthorized that comes close to the elevator gets blown into oblivion. If even one of the swarm bots detects any incoming aircraft, all of the nearby bots will join in the attack. Because each of the swarm bots is rather simple, they could be manufactured and replaced quickly. A small terrorist organization would likely be using missiles or hijacked/stolen air or spacecraft, all of which would be easily intercepted by swarm missiles. The problem of preventing shrapnel from hitting the ribbon would also be easily solved with swarm bots, as it there could be specialized shield bots that would encircle a blast zone and shield the surrounding area with heavy, magnetized armor plates that would prevent shrapnel from crossing them and redirect compression waves. Of course force fields would make this a lot easier, but I don't think that the 'near future' would have those yet.
Several considerations can be made:
The elevator may be capable of self defense. Most practical designs for the space elevator have the climbers remotely powered by lasers on the ground energizing specialized solar cells. With enough laser energy and assuming there are going to be several lasers always on standby as backup for the currently engaged lasers, you can form a pretty formidable laser shield around the tower to deal with air and space threats (one of the things the lasers could do on normal duty is act as "laser brooms" to deorbit small pieces of space junk that might impact the tower).
The elevator is not anchored to the ground. Many variations of the space elevator are actually tethers that orbit the Earth and only extend partially into the atmosphere. One variation is a "bolo" or Rotovator which turns on its own axis much like two spokes on a bicycle wheel and the "lower" end can reach down as far as you like (theoretically even to the ground) to pick up cargo. Normal precautions to ensure the cargo is inert or safe are made on the ground terminal, and payloads can be launched via suborbital rockets or even supersonic jets for pickup. Since the bulk of the cable is in orbit, only military or paramilitary groups with a high degree of expertise and funding will be able to attack the structure itself.
The elevator is so valuable that the local population will come out to defend it. You can assume the immediate area around the elevator is a "company town" with the employees of the corporation which built it. The economic benefits of the elevator are going to be vast, so various mechanisms should be made to tie the people for a late area around to the economy of the elevator, and a clear understanding made that the economic benefits are a result of the elevator's operation (an employee share holding corporation, for example). Terrorists try to hide among the population and exploit grievances within the population, but if the majority of the people are tied economically and symbolically to the elevator, they will be a thousand times better than any police or counter terrorist force in ferreting out any threats to their way of life. (This is only possible so long as the corporation itself remains true to the charter, and the host government resists becoming corrupt and becoming a major rent seeker of the elevator's economic activity, which will invert the process and provide the terrorists with a safe haven instead).
I'll go with security of the space elevator.
of course the easiest way for damage to be done is get close, and walking a bomb into the facility, and setting it off, would be the easiest way. It is also the easiest way to prevent. Everything and everyone going into and onto the elevator is searched.
That would leave the vector of attacking the elevator/cable itself which would be either an air to air strike or a missile. In this case I think having automatic defense systems in place to shoot out incoming items. The cable would need a several mile no fly zone around it. With system targeting anything that could be a threat.
The problem of course is that should the elevator fall down, it will be an ecological disaster and could kill untold numbers of people. An elevator will be over 35,800km long and the earths circumference is 40,000km. So it would wrap most of the way around the Earth if cut near the top. Granted if it was cut near the bottom I think it would move away from Earth and every thing and everyone on it would be in severe danger.
If there is an extensive constellation of other satellites on a variety of missions around this planet, to a terrorist's eye, there are already thousands of high velocity missiles just barely missing the tether hundreds of times a day. Keeping a space elevator safe requires sophisticated coordination of literally all the other satellites orbiting the planet. Every day, several satellites will have to make corrective maneuvers to avoid hitting the tether. A terrorist group could take control of a satellite and steer it into the tether, or they could introduce an error into the mission control system that coordinates satellites.
Hacking an individual satellite
A satellite is programmed to execute certain commands that it recognizes. Sometimes the security on satellites is astonishingly poor. (https://motherboard.vice.com/en_us/article/its-surprisingly-simple-to-hack-a-satellite) If you want to take control of a satellite, you need to know what signals to send. If the satellite has no security features, this is as simple as finding out what commands it responds to and sending them. You could find these out by stealing a manual or some other documentation from the manufacturer, or just guessing, if you have to. Defend against satellite takeover by encrypting transmissions and introducing passwords.
Hackers might be able to defeat the encryption through brute force, especially if they have a quantum computer or access to a massive bot net for parallel computing, depending on the type of encryption. More likely, they will try to trick someone into revealing the encryption and passwords, or steal this information from the manufacturer or the satellite operator on the ground, possibly by hacking related computer systems. The satellite codes should be carefully guarded, so the terrorist group might need to go to the trouble of planting a mole or interrogating former employees. We're talking real cloak and dagger stuff.
Messing with the system
If there is a centralized facility that coordinates all the satellite traffic, hacking or infiltrating that and moving a decimal point somewhere could be enough to take down (or up, actually) a space elevator. Feed slightly inaccurate orbital parameters to the maneuver scheduling software and the space elevator will be dead in a day or two. If there are multiple command centers - possibly owned by different governments - that need to coordinate with one another, interfering with those communications could accomplish the same thing.
Defend against this by having redundant schedulers that double-check each other. Terrorists would probably have to get a mole inside the company that makes the scheduling software and introduce a tiny bug in the next update. The obvious defense to this is to never update the software (which is typical of engineering operations, in my experience).
Terrorists could try and feed bad tracking data to the entire command and control network. Live tracking data would be collected from ground tracking stations around the globe and perhaps from some satellites and the space elevator station itself. But the maneuver calculations are probably not being computed at the ground stations or in the satellites. Maybe they are done at the mission control building, in which case tapping into their connection to the tracking system and feeding bad data might be enough.
The defense here is to do scheduler calculations in multiple places and compare results.
And then there's Kessler Syndrome
Hack a satellite, crash it into another satellite. Now nobody has control of those satellites - which are now in many pieces in many orbits - and the pieces will almost inevitably hit the tether. If they don't hit the tether, they might hit other satellites, possibly head-on, which will also break up and hit other satellites. If you can't specifically target the space elevator, just ruin space for everyone.
Although it violates one of the stipulations of the question, I'd come at this quite differently. Anything can be destroyed (especially things that require heroic engineering even to exist under favorable circumstances); and yet most things are not in danger of being destroyed most of the time.
Why are planes targeted by terrorists? It's not like Islamic fundamentalists are anti-aviation activists; largely the reason is that our governments have invested heavily in promoting fear of this specific scenario. Planes are a focus because they are surrounded with theatrics proclaiming "you'll never blow up this plane! Just try it!" ...and so that's what terrorists (very, very rarely) do.
I don't think a space elevator could even be built if one of the design requirements was "must be immune to hostile acts", because it couldn't be. You'd have to take it more or less on faith that people wouldn't try to destroy it. So, if I were imagining a space elevator I might not imagine extraordinary defensive measures. The priority would be having a backup plan if it was broken-- more than one elevator, certainly, but perhaps also ways of repairing it rapidly.
One strategic advantage a space elevator has is that while any one point is vulnerable, it's very hard to attack its whole length at once. Perhaps there are repair stations spaced along the length, ready to launch rockets that unspool a replacement tether and catch the broken end as it falls (the top station would probably need powerful station-keeping rockets, but they'd only need to run for a brief period).
Attacks:
Defenses:
Although it isn't so much of a precedent, some games have touched on this before. Front Mission Evolved (2010) features the world (in 2171) have 3 (and 1 apparently unfinished) orbital elevators spread over the main factions of the world. The three main operational ones are attacked by the fictional mercenary group (working as terrorists), Apollo's Chariot. The cinematics in-game (of which a quick YouTube search will probably yield some videos) show each elevator being attacked in turn and a view of the systems used to defend them.
The initial scenes of the game feature New York City's orbital elevator (nicknamed Percival) coming under attack by Apollo's Chariot using a mixture of missiles and automated guns. The scene starts with complex security software scanning all the boats in NYC harbour and finding some which can't be identified (and then appear to be standard mercantile vessels after beacon spoofing software is used). When the attack begins you see the mercenaries leave these ships and commence attacking the elevator. The elevator itself is equipped with many automated turrets which can rotate a full 360° around the structure. Although these initially shoot down the incoming missiles, they eventually prove ineffective against a full scale barrage which renders the defence mechanisms useless and thus the destruction of the elevator (by falling on NYC) ensures.
Although it isn't a carbon nanotube (it is a lot largely than your planned tube presumably) it is still destroyed at a lot greater cost (to lives, weaponry and infrastructure).
To answer your questions.
What are the most probable attacks terrorists will launch to cut the cable?
What is the best way to prevent against these attacks?
