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The idea of an elevator reaching to the near cosmos is very tantalizing when trying to think of ways off planet. A major draw back to this idea is that a long, thin tower into the atmosphere seem like a very likely candidate to be damaged by Space junk, asteroids, etc.

My question is, is there a truly effective way to protect a space elevator from the elements of space, and whatever other damage it could come across?

My mind drifted to ideas like orbiting shields, or possibly automated turrets on the structures exterior set to target and destroy projectiles. I would love to here other ideas.

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  • $\begingroup$ I suppose it is pretty much same thing. I was just focused on the one structure because it reaches from the ground to space, which seems more vulnerable. $\endgroup$ – Renzler Dec 13 '17 at 6:50
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    $\begingroup$ Hi Renzler. I see you have accepted an answer only about ten minutes after it was posted, and only about three hours after you posted your question. It is usually recommended to wait at least a day before accepting an answer, to allow for people in different time zones to read, research, and answer your question. Questions with accepted answers typically receive less attention from the community, which may deprive you of even better answers. See Please wait at least 24 hours before Accepting an answer for a more in-depth discussion. $\endgroup$ – a CVn Dec 13 '17 at 9:46
  • $\begingroup$ I think the best way would be to not protect it from the results of a collision (though I wonder if it had any effect, given how strong a space elevator tether would need to be) but to remove the debris from the orbits. If we can do elevators, we can surely have space junk collection. $\endgroup$ – PlasmaHH Dec 13 '17 at 12:46
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    $\begingroup$ Surround it with space goats? $\endgroup$ – RDFozz Dec 13 '17 at 16:52
  • $\begingroup$ @PlasmaHH I hope so. We've already had attempts at it. $\endgroup$ – John Dec 14 '17 at 1:51

12 Answers 12

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Lasers.

You don't need to destroy the debris, just push it out of the way by the slightest amount. The vast majority of space junk is tracked, as others have mentioned, so you'll know what's a threat long before it actually hits. Hit the debris with a laser to alter it's path slightly and you can avoid collisions.

Because you can track most of the debris, you can even start this process well in advance. Debris will be speeding past very quickly, but you should be able to hit anything dangerous multiple times as it passes by with each orbit.

You can even use this system whenever it's not directly protecting the elevator to slowly reenter space debris as it passes. Just slow down the speed of whatever passes by and eventually stuff will start to fall out of orbit. It cleans up space and protects the elevator all at once!

As for the stuff that isn't tracked, you'll need to be able to detect it in advance, so some sort of sensing equipment will be necessary, but you'll need that already for laser targeting. Luckily the stuff that's not tracked is all very small stuff, so it will be easier to push out of the way, and you won't need as much advance notice.


Delving deeper!

Luckily for anyone who wants a thorough, scientific analysis of this option, I'm not highly original and laser reentry of space junk has been studied! One particular paper DOI: 10.1016/j.asr.2012.02.003 goes quite in-depth, and I'll cite some of its more important findings here.

First, how big of a problem is space junk?

There are about $N_1 = 2,200$ large objects (diameter ≥ 100cm, mass of order 1 ton) in LEO, and $N_2 = 190k$ small objects (diameter ≥ 1cm). The flux for the small ones in the peak density region is about $R_2 = 1.4E-4$ $m^{-2}year^{-1}$ ... [Applying these figures], the chance that a big object will impact a big object is once in $T_{11} = 134$ years, whereas the chance a small object will impact a big object is once in $T_{21} = 3$ years.

One variable seems to change names at some point in their calculations, so I'm not certain about how this translates to a stationary object, but it does give us a time scale.

So if we do nothing, we'll have a small strike every few years. But let's not do nothing! Let's use some lasers!

First of all, how do lasers work to deorbit debris? Basically, the laser hits the surface and rapidly superheats the material, vaporizing it. This vapor is still in the path of the laser however, so it continues to be struck by the laser beam, where it superheats a bit more and becomes plasma. The plasma rapidly decompresses, pushing the object away, and essentially forming a small jet on the object's surface. After some time, adding more energy to the system is counterproductive so the authors of this paper suggest a pulsed laser, rather than a continuous one.

The authors assume a period of $\tau = 5ns$ for the laser pulse, and using this number, find that they need to apply $53\ kJ/m^2$ to the object to optimally accelerate an aluminum target.

In a practical case where $D_{eff} = 10m$, if $T = 80\%$, $T_{eff} = 0.5$. In order to deliver $53 kJ/m^2$ to a target at $1000km$ range, the 
product $
WD_{eff}^2
$ must 
be 
at 
least
 $993
kJm^2$, 
laser
 pulse
 energy 
must 
be $
7.3kJ$, 
and the
 mirror 
diameter $
D$ 
must 
be 
$13m$.

This means that the lasers don't even have to be in space! With pretty much current tech, you can deploy a system of them on the surface and deorbit from there!

The authors find that any object with mass less than 1kg can be reentered in a single pass. Larger objects (their example gave mass of 1-ton) would take several years to reenter, but, if you recall the frequency of interactions between large objects, the chance of a catastrophic collision is very small, and we can very easily nudge these objects out of the way in a single pass.

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    $\begingroup$ That about only slightest amount is correct, but only when planning multiple orbits ahead of any potential collision. For small stuff that's not tracked, merely changing the velocity by fractions of a percent just seconds before an impact you'd like to avoid would not be sufficient. $\endgroup$ – leftaroundabout Dec 13 '17 at 18:28
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    $\begingroup$ Space junk (small debris in the region of paint flakes) are very much a problem for any spacecraft. Currently we have no way of tracking this kind of debris, yet when it collides with a satellite it is capable of completely destroying a surface, even though it hardly weights anything. - The problem is so large that when discovered many organizations (both the sovject union and the US defense) worked on preventing space debris. (Without that decision in the 70s we wouldn't be able to go into space nowadays). $\endgroup$ – paul23 Dec 13 '17 at 20:44
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    $\begingroup$ @paul23 I think that's a bit of an overstatement. A 1mm^3 piece of steel at low earth orbit produces only 250 J of energy and 0.03144 Newton Seconds so nothing huge. The tether should be able to withstand such a hit (maybe not repeatedly I'll admit). The protection against such a hit would be very thin plating around the tether. At these speeds both surfaces behave a lot like liquids and the debris will be destroyed $\endgroup$ – bendl Dec 13 '17 at 22:35
  • $\begingroup$ Accidentally forgot to remove the 1/2 from the momentum calc so it's off by a factor of 2! Still very small though $\endgroup$ – bendl Dec 13 '17 at 22:41
  • $\begingroup$ also mirrors would achieve the same thing at a lower level of tech (but since OP mentioned shields and stuff I'm sure lasers wouldn't be an issue.) $\endgroup$ – Aequitas Dec 14 '17 at 0:28
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This won't likely be an issue:

  1. Space elevators aren't really towers, they are tethers. A strong but relatively thin series of wires made of a super strong material extending way beyond geostationary orbit (To have some tension in it).
  2. Space Junk isn't just random pieces of metal floating around everywhere. We have zones and orbits that are safe and ones that are not safe at all. We track these pieces. And by the time we have the tech and willingness to make a space elevator, we would probably have a fleet of space garbage trucks collecting all that trash, shredding it and dumping it into the atmostphere
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    $\begingroup$ "We have zones and orbits that are safe", but a space elevator isn't really in orbit, the cable is stationary. A highway is pretty safe when you ride it in a car going in the same speed and direction as everyone else, but planting a big pole right in the middle is another story. IMHO that is a huge issue that is almost always overlooked when talking about space elevators. $\endgroup$ – Keelhaul Dec 13 '17 at 11:07
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    $\begingroup$ Coincidentally I just saw a tv documentary about various "get into space" ideas. They said there that within 90 days the tether would suffer a lethal collision because it is stretching such a wide area of space that there are no debris free zones. Also it can not course correct like a lot of other space vehicles that get out of the way of tracked space junk. $\endgroup$ – PlasmaHH Dec 13 '17 at 12:45
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    $\begingroup$ @user3770822 space elevators ARE stationary poles (more like tether, but still stationary relative to the Earth's surface), save for the occasional wiggle. Furthermore, it would have to be placed on the equator, where the concentration of space junk is at the maximum. And again, the comparison with the ISS and its occasional course correction is moot : the ISS is "flying with the flow". Your tether is standing still in the middle of it. Think of it as standing in the center of a battlefield : the density of flying bullets is ridiculously low, but you're almost sure to get hit pretty quickly. $\endgroup$ – Keelhaul Dec 13 '17 at 13:09
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    $\begingroup$ @Keelhaul "Furthermore, it would have to be placed on the equator, where the concentration of space junk is at the maximum" not true... Since most rockets were launched from the likes of Florida and Russia most space junk is in an oblique orbit that crosses the equator once every 45 minutes or so. $\endgroup$ – Trevor_G Dec 13 '17 at 18:06
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    $\begingroup$ @Trevor assuming that they never perform a course correction to return to an equitorial orbit... I honestly don't know how often this is done, but it's probably often enough. Either way, the density of space junk at the equator will be higher than anywhere else because any orbit is guaranteed to cross it twice, not so the farther away from the equator you go $\endgroup$ – bendl Dec 13 '17 at 18:26
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A magnetic shield would be the most effective way to shield a space elevator from pieces of space debris and cosmic radiation.

A DC magnetic field would cause high energy particles to curve and bend around the field source, shielding occupants and cargo alike.

A AC magnetic field would "energize" inductive materials and "repel" from the field source. (See Eddy Currents).

A large superconducting magnet would provide cheap way to produce the magnetic field. It would also serve a second purpose, the ferrying of power to and from the space elevator's ground station and orbital platform.

A power station located at the base of the Space Elevator would provide sufficient power for orbital craft, Space Elevator Elevator's and the superconducting magnet cooling system. In fact, the field itself could be used to increase the structural integrity of the Space Tether.

The tether itself would need to be "hinged" to allow sway, or else structural fatigue would result in the material composition of the tether. Because of Newton's law of opposing forces, the tether's magnetic field would act on a piece of space junk pushing itself away, and the space junk aswell. However, a piece of space junk having a mass larger than the tether itself, could cause an emergency situation, especially if the tether is unable to sway out of the way in time.

The Sci-Fi book "Web between Worlds" talks a little bit about this subject.

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  • $\begingroup$ I thought it was settled long ago that if the "cables" were conductive they would create power themselves acting as a giant thermocouple? And wouldn't an EM field around the station severely interfere with communication and instruments? I do thing magnets have best potential but perhaps in a leading/trailing configuration? Collected materials can add mass to ballast/end of tether and provide some small bit of building material. $\endgroup$ – Hebekiah Dec 13 '17 at 11:08
  • $\begingroup$ Too lazy to go into the math myself, but I wonder how using a space fountain would affect this. Possibly it would already have a magnetic field, and would already deflect the high energy particles. $\endgroup$ – bendl Dec 13 '17 at 15:07
  • $\begingroup$ I think this is much more impractical than you would imagine. The tether would need to be made of an ultra light and strong material which super conducting material isn't. You would required a giant cooling system and a huge amount of power to keep it running 24/7 along a large portion of the elevator. Also, space debris travels about 10x faster than a bullet, the amount of energy you would need to invest to deflect it would be insane, and that's assuming the material doesn't melt and lose its conductivity the movement it hits the field. $\endgroup$ – Shadowzee Dec 20 '17 at 0:06
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Yes, the simplest option is to use some form of physical shielding in areas of space-junk concentration. This adds to the total mass of the tether, but is surely worthwhile.

Some sort of composite armour would be ideal, a combination of stand-off shielding to break up large pieces of debris, a shock absorbing layer to slow them down then a carbon nano-mesh to mop up anything that's still moving.

It would need replacing regularly, and would add a large amount of weight to the tether, but if you can build a space elevator you should be capable of building some high-strength, low mass composite armour.

The basic principle would be that the armour needs to be capable of protecting the tether only from space junk that you are not capable of tracking, space junk that you can track could be removed or redirected using other means.

You would also want to build redundancy into the design, e.g. having 4 tethers instead of 1, with the loss of any 1 tether not causing a catastrophic failure

Also bear in mind, that in the event the tether breaks, nothing 'above' the break will fall to earth, so all the expensive lifting equipment, space port etc at the top of the lift would be safe from an unplanned re-entry. However simulations show that the counterweight would probably leave the Earth's orbit.. which would be inconvenient for anyone not wanting to go into outer-space. As such, another sensible precaution would be to fit the 'counterweight' with enough fuel to thrust itself back into a stable orbit. I haven't calculated whether this would be feasible, but getting the fuel there wouldn't be a problem, you have a space elevator after all! In all likelihood the space port at the top is used to fuel ships for onward travel anyway.

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The current method popular amongst researchers for protecting a space elevator is to 'waggle' it.

The base station needs to be able to move the base of the tether in order to dampen oscillations of the tether (and to pre-stretch it before adding a payload) - the same mechanism would be used to set up a wave that travels up the tether so it isn't where the debris is when it passes.

Specialist tether repair vehicles would be used to repair damage from undetectable debris.

The JBIS vol. 69, no. 6/7 issue was dedicated to space elevators, and might be a good place to start research, should you be interested. £15.00 if you can't find it at a local library

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  • $\begingroup$ Wouldn't waggling create a whiplash effect at the end of the elevator? $\endgroup$ – Miguel Bartelsman Dec 21 '17 at 9:09
  • $\begingroup$ @MiguelBartelsman if there wasn't a relatively massive counterweight on it, possibly. $\endgroup$ – JCRM Dec 21 '17 at 9:31
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Something not mentioned in any of the other answer, which I think should be on this topic is the difference between collisions between space junk and something orbiting vs. a fixed target like a space elevator.

If you've ever seen a plot of an orbit over the ground, the ground track looks like a sinusoid centered on the equator. Some ground tracks are different for polar or geostationary orbits, but the one thing they all share is they cross the equator at some point and due to precession, eventually their orbit will cross every point on the equator.

enter image description here

So why do we care? For orbital collisions, you're concerned when the orbit of whatever you care about intersects orbits of debris, many orbits will never cross the orbit of our target and can be ignored and others will only infrequently cross, and many of these objects are moving in the same direction so the relative impact speeds will be lessened.

A traditional space elevator needs to be located on or near the equator, so given sufficient time every item in an orbit lower than geostationary will at some point intersect our space elevator, it's not a matter of finding the rare items on collision course, literally everything will hit it, eventually. As for relative velocity, our space elevator is stationary, so every impact is at orbital velocity, no glancing blows here (on the positive side there are no head on collisions of 2x orbital speed)

So given that everything is a concern, having a space elevator is going to require a very intensive program to track everything in orbit (within certain size limits). Not just mapping it but actively tracking changes and calculating the upcoming potential collisions with the elevator and take the appropriate action. There you have three options; move the item into a different orbit that won't collide with the elevator for some time, move the elevator out of the way, or absorb the collision.

Move the Collider

There are a number of ways this could be done:

  • If it's a functioning satellite, station, or vehicle with maneuvering control it would need to actively adjust it's orbit slightly to avoid the collision, and because all orbits eventually will intersect the cable, everything in low to medium orbit now has additional station keeping thruster needs to avoid the periodic collisions.

  • For larger non functioning items and debris you would want junk removing or refueling satellite tugs to dock with the item and remove it from orbit or put I back into the first category. This will be expensive, but it should be a one time cost as you eventually clear out most of the large junk (if you start actively deorbiting end of life satellites).

  • For smaller debris, Laser ablation causing some of the item to boil off to push the item to a different orbit that will miss your elevator or deorbit. This would provide very small amounts of delta-V and therefore will need to target items well in advance of the potential collision.

  • Blowing the item up completely is not likely to be very effective, you will just leave smaller bits of debris now moving in lots of slightly different orbits, now requiring more tracking and collision avoidance.

  • And if it wasn't hard enough, deorbiting items (our normal junk cleanup method) has some new hazards. These deorbits are inherently unpredictable due to the complicated interactions with the atmosphere. You usually end up with a large predicted path, which you should make sure doesn't ever cross you elevator.

Move the Elevator

So you know where the debris is going to be, just make sure the elevator is somewhere else:

  • Space elevators are not monolithic rigid structures, they are more like really long ropes. As you move material up and down the cable you will develop oscillations in the cable, which you could control based on how much and how fast you move material up and down the cable. Minimally, you would want to monitor these vibrations to prevent a dangerous build up of harmonic motion. You could also control the motion to move the area of concern a few kilometers out of the way of a potential collision.

  • Some proposals have advocated a movable ground site anchored to a floating oil rig type structure (Most of the equator is ocean, and in a space elevator the last few kilometers of a mountain don't actually help that much). You could also potentially have the upper end or intermediary sections provided with thrusters to allow active movement of sections of the cable. These motions would have limits and they would cause vibrations along the cable, but could push a section enough to avoid a collision (just make sure your vibration doesn't move another section into a collision).

Take the Hit

Sometimes you can't move them, you can't dodge, and you have to take the hit, but there are ways to ensure it's not a catastrophic hit.

  • Redundancy is your friend. A rope isn't made of one big string, it's made of many, similarly your space elevator should be made up of multiple redundant cables interconnected with a separation distance between them, so it would take a very large avoidable collision to sever all of the cables, and multiple cable breaks can be supported by adjacent sections.

  • Design thickness and safety factors, these are standard engineering design methods. If something is going to degrade at x rate and it needs to last y time you make it sufficiently thick to still survive until it can have maintenance or be replaced. Similarly when you have calculated how thick something must be to not break, you make it thicker by some safety factor.

  • Shielding, provide the cable with an outer sheathing, similar to a Whipple Shield. These shields would break up and absorb the impacts from the smaller micrometeoroids and debris.

  • Active repair, the cable should have active robotic climbers going up and down constantly repairing and replacing damaged sections. You will likely need this type of technology to even construct the space elevator in the first place, it only makes sense to keep it in place for maintenance. On the more theoretical level you could active materials using nanotechnology which sense and repair damage.

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It's not a question of can you, it is a question of you must if you want to keep your elevator working for any length of time.

Now how to actually do that?...

Anything we attach directly to the structure like large shields or anything would likely make our structure totally infeasible, adding so much weight and bulk would really hinder our ability to build, maintain, and use our elevator.

We don't want to use any guns or missiles to shoot debris out of the way, because the resulting collisions/explosions would create even more debris for us to have to deal with.

We could use magnetic fields as already suggested, but those would probably cause massive interference with communications and computers aboard vessels launching off the elevator...

Our tether will probably have to be very flexible for two reasons:

  1. Our anchor at the space side could attempt to maneuver the tether to bend it out of the way to avoid large collisions.
  2. But not all objects are traceable/avoidable. They are too small. So for these unavoidable collisions, our tether must be able to survive. The best way is to absorb as much energy as possible, like a soccer net absorbing a sweet goal.

But even then we will suffer some damage from time to time. So our tether must be built in a way to be repairable/healable from these little nicks without completely rebuilding.

As the neighborhood gets more crowded (as will definitely happen with our easy access to space on our elevator!) we will need something more than just avoidance and absorption. We need satellites capable of intercepting space debris and deorbiting them safely. Currently, such a satellite would be illegal, since it could be easily turned into a weapon. It would be trivial to target another countries operating satellites and deorbit them, thus denying them space access. Having weapons (or anything that could be weaponized) in space is a big no no for international law right now. But by the point we are capable of building space elevators, the need for garbage collectors will be so great that I'm sure some sort of agreement for their operation could be reached.

So, in short, protect your elevator in three ways:

  1. Bend out of the way to avoid the big stuff
  2. Be flexible, strong, and repairable enough to handle minor collisions
  3. Use garbage collection satellites to clear the neighborhood of as much debris as possible.
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  • $\begingroup$ being flexible enough to absorb the impact like a football net means it would stretch under tension, which is a bad thing for a tether, which is constantly under tension. $\endgroup$ – JCRM Dec 14 '17 at 9:29
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I suggest that the biggest problem with a space tether is that every single inch of it is a potential point of catastrophic failure.

The solution to which is...make the structure of it have redundancies. The orbital altitude sections should be a much wider diameter mesh-like structure, or lots of thinner cables rather than one big one. These could be bound together at regular intervals to allow damaged sections to be detached and replaced without replacing literally hundreds of miles of cable.

The first rule of engineering safety is to assume that all preventative measures will eventually fail and the system will need to stand against disaster itself.

In the event that a piece of debris hits the tether at several kilometers a second or more, it's better to have it sever part of the cable than the whole thing.

To cope with weight issues, you could make the lower sections physically have fewer strands.

The additional upper weight would actually be a good thing, allowing the overall length of the tether to be made substantially shorter as it acts as a counterweight (a role usually filled by a captured asteroid or simply extending the cable much further than necessary)

Pro - Segmented space elevator is easier to repair and easier to construct as well as more resilient to disasters

Cons - Segments are likely to be weaker than a contiguous cable.

You'd want to combine this with other measures such as pre-emptive deorbiting/capture/destruction of debris, or nudging the elevator itself out of the way. but being able to replace partial-segments of cable in-situ would lower the stakes of potential disasters a lot.

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  • $\begingroup$ Welcome to WorldBuilding! If you have a moment please take the tour and visit the help center to learn more about the site. Have fun! $\endgroup$ – Secespitus Dec 14 '17 at 14:34
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    $\begingroup$ @Ruadhan2300 - good thinking; don't know why we usually assume only one main cable. A complication is that the sections would have to permit the climber capsule thingy to soar past the joints as the elevator would take a week to climb if limited in speed (150kph to geosynchronous orbit 35,786 km is 9.5 days), so regular ski lift type sections aren't going to work. If the climber is able to switch cables this could work if alternating half circle sections. But each ring or half ring is going to have high wind forces on it so stress and twisting becomes a problem. $\endgroup$ – Hebekiah Dec 15 '17 at 7:26
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You can protect it with a turret based auto targeting system quite well. However I am guessing you want something more complex.
SO! How about a secondary, independent, structure? One that is like a honeycomb-bell and absorbs all the shock and allows for you to harvest said debris.

The base contains the engines and command parts allowing it to remain in orbit. I think the structure might serve as a base of sorts for the elevator

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  • $\begingroup$ The disvantage with this approach is it requires constant maintenance replacing already damaged structures, also the harvesting process can easily exceds costs and become ecnomic infeasible $\endgroup$ – jean Dec 13 '17 at 16:16
  • $\begingroup$ It does require constant maintenance however the materials will be cheap. You need to remember that the setting is in the future. The tech level is beyond us. $\endgroup$ – Cbm.cbm Dec 14 '17 at 6:29
  • $\begingroup$ Costs are nos material but personnel you do need a Swarm of drones, thecies, enginners, sensors moving partes for hundreds square Miles. $\endgroup$ – jean Dec 14 '17 at 7:18
  • $\begingroup$ In your first comment you said "the harvesting process can easily exceeds costs and becomes economically infeasible" Now you switched to " Costs are not material" Alright. However it does answer what was asked: "is there a truly effective way to protect a space elevator from the elements of space, and whatever other damage it could come across?" $\endgroup$ – Cbm.cbm Dec 14 '17 at 8:08
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I'm thinking this is a non-issue.

If you have the technology to build the super-light super-strong materials you would require to build a practical space-elevator, making said materials strong enough to survive, or self-heal, after a hit by space junk, or duck out of the way, might just be a secondary and trivial issue.

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  • $\begingroup$ materials strong enough to support their own weight are nearly in reach. being able to survive strikes at orbital velocities is orders of magnitude harder. $\endgroup$ – JCRM Dec 14 '17 at 9:30
  • $\begingroup$ @JCRM There is a lot more to it that just being able to support their own weight when something is greater than 35,786 km high or long... and spinning. Not to mention the almost impossible task to get it into position in the first place. $\endgroup$ – Trevor_G Dec 14 '17 at 11:49
  • $\begingroup$ Being able to support it's own weight is most of the problem. Getting it into place isn't trivial, but getting things into geostationary orbit is something we've been doing for a long time, and lowering the tether from there needs a little delta v on the tip and a counter-weight $\endgroup$ – JCRM Dec 14 '17 at 12:07
  • $\begingroup$ @JCRM you cant just lower something from orbit. It does not work that way, As the lowered end gets closer to earth it also has to be slowed down so it is at the right speed for its new height. In fact ..every point in the cable is at a different speed. If you do not do that the cable will proceed out ahead of the satellite in a curve, loop back out into space never reaching the ground. $\endgroup$ – Trevor_G Dec 14 '17 at 12:15
  • $\begingroup$ @JCRM also do not forget geosynchronous satellites also spin on two axis. They are set up in a controlled once per day spin so they always face the planet. What happens when you extend your arms when you are spinning... the spin decreases. Good luck controlling that and balancing it while keeping all the forces in line when you are trying to extend a 35,000 km long "cable". Especially when you have nothing solid to hang on to. $\endgroup$ – Trevor_G Dec 14 '17 at 13:34
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Let's start by admitting defeat, and assume there is no practical way to prevent the accidental severance of a Space Elevator's tether by space junk. Once the tether is in its operating position, we can expect a catastrophic failure of the tether once every N days, on average, for whatever value of N our analysis provides us with.

So are we doomed? No, because we can always just put up multiple tethers near each other. There's a chance that any single one of them might get severed, but the chances of them all getting severed at the same time are much smaller -- presumably any space junk big enough to cut all of them at once is also big enough to be easily tracked and avoided.

To make repairing a severed tether easier, you might have the tethers cross-connect at intervals, so that a severed tether will remain mostly in place rather than falling to Earth (or out into space).

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Clean up space and armor the cable

Space particles hit too hard

The biggest problem with the space elevator's cable is that it is not going at orbital velocity. That means anything that hits it will be going orbital velocity relative to the cable.

For some basic kinetic energy comparisons, a 200g object will strike the cable with the force of an battle tank's main cannon and a 20g object will strike with the force of a 2-ton car at 70 mph. Every gram of object is equivalent to about 10g of TNT.

Obviously, the accumulated damage from even the tiniest strikes will make the elevator non-viable.

Big particles must be removed

You simply can't take too many hits from highway speed cars. Fortunately, particles such as these are well cataloged. The standard for detected space particles is 10 cm. A 10 cm sphere with the density of plastic is about 4g; with the density of steel is about 35g. Particles of this size must be removed. According to an article in the Vanderbilt Journal of Law (related to space liability claims, I was surprised to find info in a law journal too!), the Air Force is tracking 21,000 objects of 10 cm or larger, and more than ten times as many over 1 cm.

Active debris removal is a technique that will work over the long term. A 2007 NASA working paper calculates that the 200 year effective reduction factor of removing one piece of space debris is 36; that means removing one piece of debris now removes 36 pieces of future debris (due to collisions causing debris to proliferate, ie Kessler Syndrome). The study also showed that instead of an exponential increase in space debris over time (again, Kessler Syndrome), debris could be limited to a linear increase by removing only 5 pieces of space junk per year, as long as these were the biggest 5 pieces.

Since in-space collision of large objects simply does not happen right now, with a linear growth of space debris, a space elevator would be safe from large impacts for a long time. Since the 10 cm and greater particles are tracked, in a space elevator-having future they would be tracked with greater accuracy, and advanced warning could allow countermeasures to prevent collisions.

Removal methods

There are generally two classes of removal: deorbiting and absorption.

Larger objects make more sense to de-orbit. If you are only going to remove 5 objects a year, you can build a space tug to gently nudge these objects into an orbit that sends them into the atmosphere to burn up. If you are going to remove smaller objects, then a laser to push them into the atmosphere would work well too. A laser would not work well on something the size of a full-on satellite, there would be too much danger that the laser would spawn its own debris.

For smaller objects in the 10 cm and below range, you could catch them with aerogels. A space tug would drag a large aerogel block through space. Small objects would impact and be absorbed by the gel. This is not very cost effective, and requires a lot of sweeping over time, but when the risks are loss of a space elevator, which surely cost hundreds of billions if not more, then it would be worth it.

You still need to armor the cable

You will never be able to sweep all of the smallest objects up, even with an army of aerogel dragging tugs. Space is just so big. So the last line of defense would be to armor the cable itself.

The entire structure, including the space that the elevator cars travel up and and down in, should be protected by a whipple shield. This is just a bumper to absorb impacts and break impacting particles up into plasma. Since you have a space elevator, the costs of bringing materials into space to repair the shield are not that great, so it really doesn't have to be much more than aluminum foil. More advanced materials like ceramic fiber might be available and used for such this shield.

Conclusion

  • Strictly control the number of satellites in space, removing defunct ones with space tugs by pushing them into the atmosphere
  • Blast smaller particles into the atmosphere with lasers
  • Sweep up the smallest particles with aerogels
  • Armor the cable with a whipple shield

Congratulations, you now have a safe and working space elevator!

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