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= Space Elevator Displacement =

The lower portion of the space elevator, below 2000km altitude, must be able to move to avoid satellites and space debris in LEO orbit.  According to the ESA Meteoroid and Space Debris Terrestrial Environment Reference ( [[ http://www.master-2005.de/ | MASTER-2005 ]]) there are more than 600,000 objects larger than 1 cm (0.4 in) in orbit.  Assuming a 1000 kilometer altitude circular orbit, the average orbital period is 105 minutes, and these objects cross the equator twice in that time.  Note that the MASTER data do not give ephemerides, just counts based on limited sampling.

The current NORAD tracking accuracy (for 2%, or 13,000 of those objects) is 1 kilometer.  NORAD is not especially concerned with collisions between small debris and gossamer square-kilometer objects.  NORAD's main mission is protecting more robust assets like the space station from collisions with satellite-sized debris, and watching for incoming ICBMs.  Given that radar return power goes down as the 4th power of the distance, we can expect the "1km accuracy" is much better at 200km altitude, and much worse at 2000 km altitude.  The NORAD database may fall off with altitude steeply because we can't see the smaller objects higher up.  As we will see, the higher altitude objects are challenging in many ways.

Using the MASTER estimates, the average debris flux passing through a 1 km wide (east-west) vertical band is about 15 per hour.  The number passing through a 1km wide cylindrical column will be higher, as the flux will be divided by the average of the sine of the inclination.  Assuming an average inclination of 45 degrees, the flux passing through our cylindrical column will be about 21 per hour.  The average time interval between objects passing through our cylinder is 172 seconds.

But that is an average interval.  If we assume an average rate of 0.35 objects per minute, the probability of 0, 1, 2, 3 ... per minute will follow a Poisson distribution:

==== Table 1:  Objects per minute, 1 km cylinder ====

|| Number || Probability || Occurrences<<BR>>per 10 years ||
||   0    ||  0.704688   ||  3706377.48  ||
||   1    ||  0.246641   ||  1297232.12  ||
||   2    ||  0.043162   ||   227015.62  ||
||   3    ||  0.005036   ||    26485.16  ||
||   4    ||  0.000441   ||     2317.45  ||
||   5    ||  0.000031   ||      162.22  ||
||   6    ||  0.000002   ||        9.46  ||
||   7    ||  0.000000   ||        0.47  ||

Over the course of 10 years, we have an even chance of having to dodge 7 objects, along a 2000 kilometer high vertical column, in one minute.  That will be difficult, the ribbon will have multiple jigs and jogs propagating up it.  Probably impossible, especially as climbers on the ribbon will prevent rapid movement for the ribbon above them.  So some of those objects will pass within our cylinder - if the ribbon is 20 cm wide, each of those objects has a 1/5000 chance of impacting the ribbon.

Further, this assumes accurate and complete debris catalogs and ephemerides.  As we enter the "Kessler Syndrome Era", many debris objects will be created by over-the-horizon collisions, with new orbital parameters, and will travel for many orbits before they are encountered by the space debris tracking network.   The only reliable way of characterizing and avoiding debris is to [[ Radar | track the oncoming objects ourselves ]].

So, make two bold assumptions:
 * We do our own tracking, with '''complete''' coverage to 2000 km altitude and 4000 km range, and 
 * We can characterize incoming centimeter-sized object trajectories accurately, to within '''20 meters'''

That changes our table above.  Assume we need 5 minutes to move the ribbon 20 meters at 2000 kilometers altitude (difficult, as we will see below!).  Our collision table now becomes

==== Table 2:  Objects per five minutes, 20 meter cylinder ====

|| Number || Probability || Occurrences<<BR>>per 10 years ||
||   0    ||  0.965605   ||  1015739.65  ||
||   1    ||  0.033796   ||    35550.89  ||
||   2    ||  0.000591   ||      622.14  ||
||   3    ||  0.000007   ||        7.26  ||
||   4    ||  0.000000   ||        0.06  ||

Over the course of 10 years, there will be about 7 times when we will need to avoid 3 objects at once over a 5 minute period.  This time, the chance of an impact is 0.2/20, or 1 in 100, so if we can only avoid 2 at once, we will have a 7% chance of collision and possible failure from these events.  Even very good tracking does not keep us completely out of trouble!


The speed of propagation of a transverse jog of the ribbon is the proportional to the square root of the specific loading.  A ribbon loaded to 30 [[ http://nextbigfuture.com/2009/01/understanding-strength-of-materials-and.html | Mega Yuri ]] will have a transverse (back and forth, not up and down) propagation speed of 5500 m/s .  

Since the ribbon is tapered, we will get mechanical reflections.  The result is that the heavier ribbon above will move less than the ribbon below.

The easy case is if there are no climbers below the top end of the debris zone.  In that case, an 81 meter horizontal movement of the anchor will propagate at 5.5 km/s up the ribbon, with the amplitude diminishing slowly as we encounter wider-tapered ribbon.  If the wave encounters a 4X step increase in $ \mu $, the ribbon mass per length, then the mechanical transverse stiffness, proportional to $ \sqrt{ \mu T } $ doubles. In that case, 2/3 of the wave (54 meters) will continue to propagate upwards, while a "-1/3" wave (from 81 to 54 meters) will propagate back down to the anchor point, where it will reflect and send a 1/3 wave (54 back to 81 meters) back up.  If the discontinuity is 1100 km up, then the step occurs at 200 seconds up the ribbon, and the next reflection arrives 600 seconds after the initial wave.  That sends 2/9 additional displacement (54 to 72 meters) up the ribbon, and -1/9 (81 to 72 meters) down.  The position of the discontinuity moves like so:

||   time  ||<-3> displacement, meters   ||
|| seconds || anchor || middle || top    ||
||    -1   ||    0   ||  0     ||  0     ||
||     0   ||   81   ||  0     ||  0     ||
||   100   ||   81   || 81     ||  0     ||
||   200   ||   81   || 81     || 54     ||
||   300   ||   81   || 54     || 54     ||
||   400   ||   81   || 54     || 54     ||
||   500   ||   81   || 81     || 54     ||
||   600   ||   81   || 81     || 72     ||
||   700   ||   81   || 72     || 72     ||
||   800   ||   81   || 72     || 72     ||
||   900   ||   81   || 81     || 72     ||
||  1000   ||   81   || 81     || 78     ||
||  1100   ||   81   || 78     || 78     ||
||  1000   ||   81   || 78     || 78     ||
||  1000   ||   81   || 81     || 78     ||
||  1000   ||   81   || 81     || 80     ||
||  1000   ||   81   || 80     || 80     ||
||  1000   ||   81   || 80     || 80     ||
||  1000   ||   81   || 80     || 80     ||
||  1000   ||   81   || 80     || 80.67  ||

The tapered case behaves somewhat like that, but the reflections are distributed.  In this simple case the displacements make it past the discontinuity attenuated but continuously, and after a few reflections step their way up to full displacement.

== The climber filtering problem ==

However, the worst case is if there is a climber just below a section of ribbon we need to move, slackening the ribbon below it to a lighter tension.  That slows down speed of propagation, and lowers the stiffness, which is unfortunate in three ways, as we will soon see.  Assume that the ribbon below a climber is slackened to 16 MYuri, for a propagation speed of 4000 m/s.

Even worse, the climber may be between the moving anchor point, and a section of ribbon we have to move.  The mass of the climber will resist lateral movements.  Assume the climber is at 1600km altitude, and we need to move the ribbon just above it.

||{{attachment:SEdisplacement.png}}||The tension below the climber is T = 160 KN, the mass per meter is $ \lambda $ = 0.01 kg/m.  While the ribbon is actually tapered and stretched (changing density), ignore those effects for now. This means the transverse propagation velocity is $ V_T = \sqrt{ T / \lambda } = 4000 m/s $ <<BR>><<BR>> Assume we move the bottom of the ribbon by $ \Delta x $ = 500 meters in $ \Delta t $ = 25 seconds, resulting in lateral velocity of $ v_x \equiv \Delta x / \Delta t $ = 20 m/s.  This results in a 500 meter wide moving edge (100km high) propagating at 4000m/s . Assume the climber is at 1600 kilometers altitude (effective gee = 6.2 m/s), and weighs 20 tons, and the the 1600km ribbon weighs 116 KN, putting an additional 240 KN load on the ribbon above.  The total loading on the ribbon above is approximately 160KN+124KN+116KN, or 400KN, so for a 30 MYuri design the ribbon above the climber weighs 13 g/m and the propagation speed above the climber is 5500 m/s. <<BR>><<BR>> The edge travels from the ground to the climber in 400 seconds.  The edge is moving laterally at 20m/s, and involves 100km or 1000kg of tether, so the total momentum in the forward wave is 20,000 kg-m/s.  When the wave hits the climber, it will reflect, dumping not only the momentum of the forward wave but that of the reverse wave as well, minus the momentum launched by the now-starting-to-move-sideways climber.  40,000 kg-m/s will accelerate the 20,000 kg climber to 2 m/s, so the reverse wave is diminished from 20 m/s to perhaps 18 m/s.||

The reverse wave bounces off the anchor point and returns in another 800 seconds.  Meanwhile, the moving climber is itself launching transverse energy up and down the ribbon, so its velocity will decay exponentially:

$ \Large { dv \over dt } \approx { { \sqrt{ \mu_1 T_1 } + \sqrt{ \mu_2 T_2 } } \over { M_{climber} } } v ~ ~ ~ ~ ~ ~ ~  v = v_0 + v_1 e^{ t / \tau } ~ ~ ~ ~ ~ ~ \tau = { { M_{climber} } \over { \sqrt{ \mu_1 T_1 } + \sqrt{ \mu_2 T_2 } } } $

... with an exponential decay time $ \tau $ of about 150 seconds.  So it will decelerate, stopping and perhaps swinging back long before the next reflection arrives.

The climber (and some the ribbon above it, with a slope of about 1km vertical per two meters horizontal) will not start moving until 400 seconds after we start moving the anchor.  It will move 40 meters 430 seconds after we start moving the anchor point, and will move about 290 meters before the next reflection at 1200 seconds re-accelerates it (more slowly the next time). This stepwise process will continue for hours until we move the anchor point again, or the whole ribbon finishes moving 500 meters.  The climber may need a day or two to climb out of the way.  

The actual behavior of the tapered ribbon and the moving climber will be more complex.  Keep in mind that the whole system, including the tapered ribbon above, will propagate waves of displacement energy both above and below.  Because of reflections at mass discontinuities like climbers, the anchor, and the taper, there will be resonances, and the resonant energy will be difficult to remove by launching upwards waves from the anchor point.  A complete and accurate measurement of ribbon position will be needed by the control system, which will feed some big numerical simulations.  That will help us choose the how we control some big motors, making big displacements.  

Damping out 90% of the unwanted transverse energy in the space elevator may require (WAG) 10 round trip times from anchor to counterweight;  at an average of 5km/sec prop delay, that would be almost 5 days.  Remember that all this resonant energy is adding to the transverse position of the ribbon at potential collision points, and we must know that position within some fraction of 20 meters in order to accurately dodge impactors.

Because of the large propagation times, we will need to anticipate upper-tether movements many minutes before we need to move them.  With debris objects at 2000km altitude moving at 7000 m/s, 7 minutes of warning means a 3600 km radar range, very difficult for tracking centimeter-scale objects.  See [[Radar]] for some possibilities.

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