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← Revision 66 as of 20210620 03:27:51 ⇥
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## page was renamed from GEOrail  
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= GEO Rail = == "Landing" Launch Loop payloads in a destination without rockets == 
= Capture Rail = === "Landing" Launch Loop payloads on tethers in circular orbit, without apogee kick motors === 
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Imagine a long and heavy orbiting tether, running vertically from 29900 km radius, through GEO (42164 km), to a counterweight above. The tether has a thin, passive conductive rail around it. It will be used to magnetically capture vehicles and move them to GEO, using the sliding rail to push payloads transversely while they magnetically slide upwards.  The launch loop provides low cost deltaV from the earth's surface, injecting vehicles into highly elliptical transfer orbits, or earthescape orbits. What a launch loop cannot do directly is put a payload into a circular orbit. The perigee of a launchloop launch will always be at the launch track height, even if the apogee reaches the moon. To put a vehicle into a higher circular orbit requires additional angular momentum, usually supplied by a rocket. Tethers provide an alternative. A zerovelocity capture on a vertical tether means that the launch loop launches payloads with an angular velocity matching that of the tether, at a lower altitude ( or radius ) than the center of the tether's orbit. == Classic Tether Capture == '''Yellow is a vehicle in a transfer orbit from launch. Green is a capture, red is a missed capture.''' {{ attachment:cap12.png   width=960, height=320 }} [[ attachment:cap12.c  Here is the source ]] , and you will need [[ http://www.libgd.org/  libGD ]] and [[ http://apngasm.sourceforge.net/apngasm  apngasm ]]. Capturing on a tether is not a new idea. The transfer orbit is chosen such that the vehicle arrives at the tether with zero velocity, and clamps on. It will have a small relative acceleration downwards, perhaps 2.5% of one gee, a fairly light load, assuming the tether is much more massive than the vehicle. There are problems with classic tether capture. For a tether in geostationary orbit, capturing a vehicle in a transfer orbit with a perigee of 80km, the capture radius (the apogee of the transfer orbit) is at 29870 km, 12294 km below the geostationary altitude of 42164 km. Somehow, the vehicle must climb up the tether for more than 12000 kilometers. With the very small gee loading, this is not nearly as difficult as climbing up a space elevator from the Earth's surface, and a tractor vehicle can be waiting on the tether to pull up the launch vehicle. However, with limited power and wheel velocity, this will still take a while. At 200 kilometers per hour, the climb will take 61 hours. The climb will probably be slower on the higher gee lower section. We will be stealing angular momentum from the tether, so we will need something like a VASIMR electric rocket to add the momentum again. Worse still, what if we miss the capture? The period of the circular transfer orbit is 6.768 hours; the next time the vehicle reaches apogee, the tether will be a long way away, and we may need many days and hundreds of meters per second of delta V to catch it again. == Climbing Tether Capture == {{ attachment:cap11.png   width=960, height=320 }} [[ attachment:cap11.c  Here is the source]], and you will need [[ http://www.libgd.org/  libGD ]] and [[ http://apngasm.sourceforge.net/apngasm  apngasm ]]. But who says the vehicle is constrained to zero vertical/radial velocity? Vehicles leave the launch loop with a magnetic rail attached to the bottom, which magnetically drags on the rotor and propels them to orbital speeds. The magnetic rail is passive  it can only produce drag  but it can drag on anything conductive, such as a very thin metal tube around a tether. If we arrive at the tether with zero transverse/horizontal velocity, and a high radial/vertical velocity, we can ride the magnetic rail, just like the launch loop track that we left a few hours before. As we ride up the rail, we slow down against it. In a matter of minutes, we brake to a stop at geostationary radius. This requires a few percent more deltaV at launch, but launch velocity is cheap with a launch loop. Transit time from start of launch to arrival at GEO will be less than 4 hours. The transit time will be slightly faster than a classic Hohmann transfer orbit with an apogee kick motor. Other benefits include a somewhat shorter hanging tether, which also reduces the amount of angular momentum the vehicle removes from the tether. For a geostationary target orbit, we can choose a transfer orbit with a 12 hour period and a semimajor axis of 26562 km (0.62996 smaller semimajor axis than the 42164 km GEO orbit). If apogee remains at launch loop exit altitude (6458 km), then drag can help the vehicle reenter 12 hours later, on the other side of the earth from the launch loop. It we instead choose to make another try, we can add 10m/s deltaV at apogee, and raise perigee by 45 km, a 200x reduction in drag. That will give us one or two more passes at capture, 24 and 48 hours later. The unmodified orbit is the "12 hour low" orbit, while the boosted perigee orbit is the "12 hour high" orbit. Actually, we must also perform a second burn at perigee, to lower apogee, if we wish to maintain the same semimajor axis and the same 12 hour orbit period. We can also choose a transfer orbit with a 24 hour complete period, and perform the same perigeeraising maneuver if we wish to reduce drag and make a second pass. That higher orbit will have higher angular momentum, and draw less angular momentum from the tether.  Transfer Orbit Perigee ApogeeV,,P,,V,,A,, AngM R,,capture,,   km  km  m/s  m/s  m^2^/s km   12 hour low  6458  46665       12 hour high  6503  46620   24 hour low  6458  77870   24 hour high  6503  77825  A 24 hour orbit, geostationary capture tether, fed from a fixed point on earth, is perhaps the easiest capture tether to imagine. However, a nonsynchronous capture tether offers more flexibility, as we will see below.  THE REST OF THIS PAGE WILL BE REWORKED  Imagine a long and heavy orbiting tether, running vertically from 29000 km radius, through GEO (42164 km), to a counterweight above. The tether has a thin, passive conductive rail around it. It will be used to magnetically capture vehicles and move them to GEO, using the sliding rail to push payloads transversely while they magnetically slide upwards. 
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=== VASIMR and "Zero Orbiting Exhaust" momentum restoration === 

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A [[ http://en.wikipedia.org/wiki/VASIMR  Variable Specific Impulse Magnetoplasma Rocket ]] (VASIMR) engine operates optimally at 50 km/sec, so operation at 80 km/sec is close to optimum. The reaction mass is argon, which can be frozen at 200C with a density of 1600 kg/m3 and a vapor pressure of 0.1 atmosphere for transit from earth. We will need about 1 kg of argon reaction mass shipped up for every 50 kg of uncompensated mass shipped up from Earth. Argon is 0.93% of the atmosphere, and costs about $1/kg (liquid) in high quantities. It is a byproduct of the production of liquid oxygen. We will expend about 40 kilowatt hours per "uncompensated vehicle kilogram" operating the plasma rockets, including coolers and radiator operation. Typical solar cell weights for existing satellites are 65 kg/kilowatt, so 1 kilogram of solar cells provides enough energy in a year to bring aboard 3.5 kilograms of mass  a power doubling time of 5 months. However, if we can use "server sky" style ultrathin InP cells, closer to 1000 kg/kW, we can approach 50 kilograms of mass per kilogram of solar cell. In the very long term, solar cells made from lunar materials will bring both momentum and cheaper launch to the GEO rail, but solar cells are a hightech undertaking, and it will be a long time before that level of manufacturing technology can operate practically in space. In the shorter term, we may be forced to get by with lower ISP engines than 80 km/sec VASIMR, perhaps operating only around noon with an exhaust velocity of 20km/sec. That increases our argon needs 4X, but reduces our power needs 16X. We will only be operating our engines with about a 30 percent duty cycle, so our solar panel needs only drop 5.3X. Still, our power doubling time drops to a month or so. Probably worth it during the building phase. We may even be rascals, and run the VASIMR engines 24 hours for a while, dumping a few tons of midnight argon into orbits in the inner solar system. == Operation of an M288 rail base == A similar rail system can supply the M288 [[ http://serversky.com  server sky orbits ]] MORE LATER == M288 rail bases for orbital debris capture == Server sky thinsats are cheaper to launch if they are lighter, but below a minimum masstoarea ratio, the orbits are unstable under light pressure. However, we can add mass ballast to a very light satellite, using gramweight chunks of space debris. The processing is simple, just let the debris shatter as it is collisioncaptured inside a cornucopiashaped funnel. Larger chunks can be cut apart at a simple processing plant at the M288 rail base, smaller chunks can be welded together. Capture vehicles can drop below the tether, in intercept orbits where the debris crosses the equatorial plane. Depending on their orbit, they can return to the same rail base, but more likely a different rail base will be more convenient. MORE LATER 
A [[ http://en.wikipedia.org/wiki/VASIMR  Variable Specific Impulse Magnetoplasma Rocket ]] (VASIMR) engine operates optimally at 50 km/sec, so operation at 80 km/sec is close to optimum. The reaction mass is argon, which can be frozen at 200C with a density of 1600 kg/m3 and a vapor pressure of 0.1 atmosphere for transit from earth. We will need about 1 kg of argon reaction mass shipped up for every 50 kg of uncompensated mass shipped up from Earth. Argon is 0.93% of the atmosphere, and costs about $1/kg (liquid) in large quantities. It is a byproduct of liquid oxygen production. With 80km/s exhaust, we will expend about 40 kilowatt hours per "uncompensated vehicle kilogram" operating the plasma rockets, including coolers and radiator operation. Typical solar cell weights for existing satellites are 65 kg/kilowatt, or 16 watts per kilogram, so 1 kilogram of solar cells provides enough energy in a year to bring aboard 3.5 kilograms of mass  a power doubling time of 5 months. However, if we can use "server sky" style ultrathin InP cells, closer to 1000 kg/kW, we can approach 50 kilograms of mass per kilogramyear of solar cell. In the very long term, solar cells made from lunar materials will bring both momentum and cheaper launch to the GEO rail, but solar cells are a hightech undertaking, and it will be a long time before that level of manufacturing technology exists in space. In the shorter term, perhaps there is some variant of VASIMR that can operate at lower exhaust velocities, expending more argon but using less energy. If we fire the argon out at 4500 m/s, and don't have too much thermal spread, the argon will be in a ballistic orbit that falls into the atmosphere (returning the argon where we found it). This will require 1 kilogram of argon for every 3 kilograms of payload shipped, which may mean that every other payload is argon during construction. However, at 50% efficiency, and 50% mass fraction we will need only 3.8 kWhours of energy per kilogram of incoming nonargon mass, resulting in 1 kg per 10 kgdays of solar cell. That will lead to very fast growth rates of the power and engine systems, rapidly evolving towards the high velocity exhaust scenario. The above analysis assumes that the argon plasma neutralizes  if it stays ionized, it will gyrate in the Earth's magnetic field and not go much of anywhere, because the pitch angle will be high. That will be interesting, perhaps the trapped particles will act as shielding mass against higher energy particles trapped on the same field lines, but that analysis must wait for another time. Also  do not assume that we can fire the argon at night at just the right velocity so that the exhaust falls into the sun. The sun is a small target, and distant, so with thermal spread much of the exhaust will end up in highly elliptical orbits that reach earth's orbit. The gas will be dilute, and most of it that returns will collect in the morning sky, but it will add some pollution in the long run. How much argon will we use? The Earth's atmosphere weighs 5E15 tons, and about 1% is argon, so 5E13 tons of argon are available. With 80km/s exhaust resulting in a 50:1 payloadtoargon ratio, we can ship up 2.5E15 tons before we run out of argon. Chances are, we will be shipping down material and momentum from the moon long before we make a dent in the argon supply. == Conductor Track Weight == Assume a 5 ton vehicle moving at 2700m/s with 1.0 m/s^2^ transverse acceleration (lots of vibration!), and 0.3 m/s^2^ radial deceleration from the magnetic field. The total transverse force is 5000N, the deceleration force is 1500N. The deceleration power is 4MW . Assume a 10 meter magnet rail with 0.2T undulating fields and 50% fill, and an average of 4 cm of "track width" on a 6cm diameter aluminum tube (bonded to a Kevlar core tube), so the "bearing surface" is 0.2 m^2^. The field pressure is 5000N/0.2m^2^ or 25000N/m^2^, which is 0.2T times 125,000 amps per meter in the conductor. Assume the current path is effectively 10cm long and 5 meters wide for 625KA total. For a deceleration power of 4MW, the resistance of the current path must be 4MW/(625KA^2^) or 10 microohms. The resistivity per square of the aluminum will be (5.0/0.1) times that, or 500 microohms per square. Aluminum is about 30 nanoohmmeter, so a thickness of 60 microns will have that resistivity. The cross section is $pi$ * 0.06m * 6E5m or 1.13E5 m^2^, for a mass per meter of 30 grams (aluminum density is 2.7). That is 30kg/km, or 360 tons of aluminum for the whole 12000 km track. 4MW/2700m/s, or 1500 Joules per meter of heat energy will be deposited into the 30g/m track. Aluminum heat capacity is 0.9 J/gmK, so it could heat by about 56K. However, it is bonded to a much larger mass of Kevlar (1.4 J/gmK), and the thermal diffusion time through 60 microns of aluminum is about 40 microseconds. The 10 meter magnet rail passes by in 3.7 milliseconds, so we can assume that most of the heat has passed into the Kevlar core, and the aluminum heats much less. MORE LATER  
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$ a ~ = ~ { \Large { r_p \over { 2 ~  ~ \LARGE { { r_p {v_p}^2 } \over { \huge \mu } } } } } $ ` ` semimajor axis  $ a ~ = ~ { \Large { 1 \over { \LARGE { 2 \over r_p } ~  ~ \LARGE { { {v_p}^2 } \over { \huge \mu } } } } } $ ` ` semimajor axis 
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For the classical ( zero vertical velocity ) tether capture, $ v_R $ = 0 and $ \theta $ = $ \pi $ . So: $ {\omega_T} ~ = ~ { \Large { { \mu^2 ( 1  e )^2 } \over { {r_p}^3 {v_p}^3 } } } $ $ ~ ~ ~ ~ ~ ~ = ~ { \Large { { \mu^2 ( 2  r_p {v_p}^2 / \mu )^2 } \over { {r_p}^3 {v_p}^3 } } } $ $ ~ ~ ~ ~ ~ ~ = ~ { \Large { { ( 2 \mu  r_p {v_p}^2 )^2 } \over { {r_p}^3 {v_p}^3 } } } $ The perigee velocity $ v_p $ can be computed by iterating on this equation: $ v_p ~ = ~ \sqrt{ 2 \mu / r_p  \sqrt{ \omega_T r_p {v_p}^3 } } $ Given $ v_p $, we can compute the capture radius, which is $ r_a $ as above. But we aren't done! We need to do two more things  climb up the tether, and restore momentum to the tether. Assume the tether is massive compared to a vehicle, so it does not deflect much. The will slow down by a tiny amount, and we must restore the momentum, perhaps with rocket engines. We must also supply energy to lift the vehicle up the tether against the small gravity (minus centrifugal force) at that high altitude. <4:> '''Classical Tether Capture'''   $ \mu $  gravitational parameter  3.986004418e14  m^3^/s^2^   $ \omega_T $  GEO angular frequency  7.29211515e5  radians/s   $ r_{GEO} $  GEO radius  42164172.4  m   $ v_{GEO} $  GEO velocity  3074.66  m/s   $ E_{GEO} $  GEO energy / kg  14180301.17  J/kg   $ H_{GEO} $  GEO angular momentum/kg  1.296e11  m^2^/s   $ r_p $  perigee radius  6458137.0  m   $ v_p $  perigee velocity  10074.5754  m/s   $ e $  eccentricity  0.64446    $ a $  semimajor axis  18164241.20  m   $ r_a $  apogee radius  29870345.40  m   $ v_a $  apogee velocity  2178.1800  m   $ acc_a $  apogee radial accel.  0.2879  m/s^2   $ E_a $  apogee energy / kg  15716587.30  J/kg   $ H_a $  apogee ang momentum/kg  6.506e10  m^2^/s   $ \Delta r $  radius increase  12293826.96  m   $ \Delta v $  velocity increase  896.4800  m/s   $ \Delta E $ radial energy increase/kg 1536286.13  J/kg  
For the classical ( zero vertical velocity ) tether capture, $ v_R $ = 0 and $ \theta $ = $ \pi $ . We are at apogee. We know three things about the transfer orbit: $ \Large { 1 \over a } ~ = ~ { 2 \over r_p }  { {v_p}^2 \over \mu } ~ = ~ { 2 \over r_a }  { {v_a}^2 \over \mu } $ Conservation of energy $ v_a r_a ~ = ~ v_p r_p $ ` ` Conservation of angular momentum $ v_a ~ = ~ \omega_{GEO} r_a $ ` ` Definition of zero relative transverse velocity, vehicle to tether That is enough to solve for $ r_a $ : $ \Large { { 2 \mu } \over r_p }  \left( { { \omega_{GEO} {r_a}^2 } \over r_p } \right)^2 ~ = ~ { { 2 \mu } \over r_a }  { \omega_{GEO} {r_a}^2 } $ Which can be reduced to the following iterator: $ r_a \Large = \left( { { 2 \mu } \over { {\omega_{GEO}}^2 } } \left( { r_p \over { r_a + r_p } } \right) \right)^{ 1 \over 3 } $ Iterate on that for a while. It converges in about 20 steps to high accuracy starting anywhere between 0 and $ r_{GEO} $, but slightly faster starting at $ 0.7*r_{GEO} $ If we rendezvous with a hanging tether at the radius $ r_a $, we will have zero velocity vertically and horizontally with respect to the tether, and a small acceleration downwards. We can clamp on and hang at the end of the tether. Our small residual weight will accelerate the tether down, slightly, and launch strain waves up it.v But we aren't done! We need to do two more things  climb up the tether, and restore momentum to the tether. Assume the tether is massive compared to a vehicle, so it does not deflect much. The tether will slow down by a tiny amount, and we must restore the momentum, perhaps with rocket engines. We must also supply energy to lift the vehicle up the tether against the small gravity (minus centrifugal force) at that high altitude. <9:> '''Classical Tether Capture'''   $ \mu $  gravitational parameter  3.986004418e14  m^3^/s^2^   $ \omega_T $  GEO angular frequency  7.29211515e5  radians/s   $ r_{GEO} $  GEO radius  42164172.4  m   $ v_{GEO} $  GEO velocity  3074.66  m/s   $ E_{GEO} $  GEO energy / kg  14180301.17  J/kg   $ H_{GEO} $  GEO angular momentum/kg  1.296e11  m^2^/s   $ r_p $  perigee radius  6458137.0  m   $ v_p $  perigee velocity  10074.5754  m/s   $ e $  eccentricity  0.64446    $ a $  semimajor axis  18164241.20  m   $ r_a $  apogee radius  29870345.40  m   $ v_a $  apogee velocity  2178.1800  m   $ acc_a $  apogee radial accel.  0.2879  m/s^2   $ E_a $  apogee energy / kg  15716587.30  J/kg   $ H_a $  apogee ang momentum/kg  6.506e10  m^2^/s   $ \Delta r $  radius increase  12293826.96  m   $ \Delta v $  velocity increase  896.4800  m/s   $ \Delta E $ radial energy increase/kg 1536286.13  J/kg  
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If we haul the payload up with a solar cell power source weighing as much as the payload, then our "hauling power" will be about (16/2) = 8 watts per kilogram using current satellite technology. Against a 0.2879 m/s^2^ acceleration field, we can move at (8/0.2879) = 28 m/s near the bottom, faster near the top. At 8 watts per kilogram, and 1536286 joules per kilogram, climbing the tether will require at least 53 hours. However, at some point we are velocity limited  perhaps 200m/s for our motor. That occurs when the acceleration force is ( 8/200 ) = 0.04 m/s^2^, about 39800 km high. The velocity limit increases climber time to 55 hours .  If we haul the vehicle up with a solar cell power source weighing as much as the payload, then our "hauling power" will be about (16/2) = 8 watts per kilogram using current satellite technology. Against a 0.2879 m/s^2^ acceleration field, we can move at (8/0.2879) = 28 m/s near the bottom, faster near the top. At 8 watts per kilogram, and 1536286 joules per kilogram, climbing the tether will require at least 53 hours. However, at some point we are velocity limited  perhaps 200m/s for our motor. That occurs when the acceleration force is ( 8/200 ) = 0.04 m/s^2^, about 39800 km high. The velocity limit increases climber time to 55 hours . 
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The capture radius $ r_c $ is below orbit apogee by some asyet undefined amount. The angular velocity will be $ \omega_T $ as before, so the transverse velocity will be $ \omega_T r_c $. The total velocity squared will be the sum of the squares: $ {v_{vc}}^2 + ( \omega_T r_c )^2 $. $ \omega_T r_c ~ = ~ v_0 ( 1 + e \cos( \theta ) ) $ $ v_{vc} ~ = ~ v_0 e \sin( \theta ) $ $ r_c ~ = ~ r_0 / ( 1 + e \cos( \theta ) ) $ So: $ \sin( \theta ) ~ = ~ { \Large { v_{vc} \over { v_0 e } } } $ $ ~ ~ ~ ~ ~ ~ ~ ~ = ~ { \Large { v_{vc} \over { v_p  \mu / r_p v_p } } } $ $ \cos( \theta ) ~ = ~  \sqrt{ 1  \sin( \theta )^2 } $ ` ` . . . Note the quadrant! $\theta$ is near $\pi$, not 0! $ ~ ~ ~ ~ ~ ~ ~ ~ = ~  \sqrt{ 1  { \Large \left( { v_{vc} \over { v_p  \mu / r_p v_p } } \right) } ^2 } $ $ \omega_T ~ = ~ ( v_0 / r_0 ) ( 1 + e \cos( \theta ) )^2 $ $ ~ ~ ~ ~ ~ ~ = ~ { \Large { \mu^2 \over { {r_p}^3 {v_p}^3 } } } ~ \left( 1  \left( { \Large { { r_p {v_p}^2 } \over \mu } }  1 \right) \sqrt{ 1  { \Large \left( { v_{vc} \over { v_p  \mu / r_p v_p } } \right) } ^2 } \right)^2 $ Let's solve this mess for $ v_p $. We know all the other terms. It will be iterative, as before: $ v_p ~ = ~ \LARGE \sqrt{ { \Large { \mu \over r_p } + } { { { \mu \over r_p }  \sqrt{ \Large \omega_T r_p {v_p}^3 } } \over \sqrt{ 1  { \left( { v_{vc} \over { v_p  \mu / r_p v_p } } \right) } ^2 } } } $ If $ v_{vc} $ = 0, this reduces to the classical case above. The capture rail is a long way out of the gravity well, so "natural" vertical deceleration of a vehicle is small. However, there will be some drag from eddy currents in the thin and rather resistive rail. In a production system, the drag profile will be optimized and rather complex. For this analysis, we will just compute the kinetic energy lost by moving up the rail, multiply that by a drag factor, and use that to compute the kinetic energy and velocity left as we approach the dock at GEO. 
The capture radius $ r_c $ is below orbit apogee, and below the GEO altitude . This is a good starting point. The angular velocity will be $ \omega_T $ as before, so the transverse velocity will be: $ v_{tc} = \omega_T r_c $ and the angular momentum is $ h = \omega_T {r_c}^2 $ We know that at launch, at perigee, the angular momentum is the same, so: $ v_p = h / r_p $ We can define the orbit from there, computing $ a $, $ e $, $ v_0 $, $ r_0 $, and $ \theta $ from the equations above. Most interesting is computing the vertical velocity at capture, $ v_{rc} $: $ v_{rc} ~ = ~ \sqrt{ 2 \mu { \Large \left( { 1 \over r_p }  { 1 \over r_c } \right) } + {v_p}^2  {v_{tc}}^2 } $ Also the vertical energy per mass needed to climb up the tether,$ E_{cc} $: $ E_{cc} ~ = ~ { 1 \over 2 } {\omega_T}^2 \left( {r_c}^2  {r_{GEO}}^2 \right) + \mu { \Large \left( { 1 \over r_c }  { 1 \over r_{GEO} } \right) } $ If the eddy current losses are negligible ( which requires a thick and heavy conductor track ), then the vehicle will reach (or pass) GEO if $ E_{cc} < { 1 \over 2 } {v_{rc}}^2 $ If we define the capture parameter $ c \equiv r_c / r_{GEO} $ and the perigee parameter $ p \equiv r_p / r_{GEO} \approx $ 0.1531665 then the inequality can be massaged into: $ c^5  2 p^2 c^3 + ( 2 p^2 + p ) c  2 p^2 > 0 $ MORE LATER  that does not look right, yet. The next few bits may be invalid, too The capture rail is a long way out of the gravity well, so "natural" vertical deceleration of a vehicle is small. However, there will be some drag from eddy currents in the thin and rather resistive rail. In a production system, the drag profile will be optimized and rather complex. For this analysis, we will just compute the kinetic energy lost by moving up the rail, multiply that by a drag factor, and use that to compute the kinetic energy and velocity remaining as we approach the dock at GEO. 
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<4:> '''GEOrail Tether Capture'''   $ v_{vc} $  Vertical V  2700.0000  m/s   $ acc_s $  slowdown deceleration  5.0000  m/s^2^   $ D $  drag factor  2.0000    $ \mu $  gravitational parameter  3.9860E+14  m^3^/s^2^   $ \omega_T $  GEO angular frequency  7.2921151E05  radians/s   $ r_{GEO} $  GEO radius  42164172.37  m   $ v_{GEO} $  GEO velocity  3074.6600  m/s   $ E_{GEO} $  GEO energy / kg  14180301  J/Kg   $ H_{GEO} $  GEO angular momentum/kg  1.2964E+11  m^2^/s   $ r_p $  perigee radius  6458137.0  m   $ v_p $  perigee velocity  10462.7761  m/s   $ e $  eccentricity  0.77363    $ a $  semimajor axis  28529281.45  m   $ r_a $  apogee radius  50600425.91  m   $ v_a $  apogee velocity  1335.3651  m/s   $ \theta $  orbit angle at capture  143.73  degrees   $ r_c $  capture radius  30440398.47  m   $ v_{tc} $  capture tranverse velo.  2219.7489  m/s   $ E_c $  capture energy  15558098  J/kg   $ acc_c $  capture vert. accel.  0.2683  m/s^2^   $H_{capture}$ capture ang. momentum/kg 6.7570E+10  m^2^/s   $ y_s $  slowdown run  177982.60  m   $ r_s $  slowdown radius  41986189.77  m   $ v_s $  slowdown velocity  1334.10  m/s   $ t_s $  slowdown time  266.82  s    tether climb time  6604.70  s    Partial orbit time  18533.41  s    Launch time  348.76  s    Total time  7.15  hours   $ \Delta v $  velocity increase  854.9111  m/s   $ \Delta E $ radial energy increase/kg 1377796.89  J/kg   $ \Delta H $ radial ang. mom. incr./kg 6.2070E+10  m^2^/s  
<9:> '''GEOrail Tether Capture'''   $ v_{vc} $  Vertical V  2700.0000  m/s   $ acc_s $  slowdown deceleration  5.0000  m/s^2^   $ D $  drag factor  2.0000    $ \mu $  gravitational parameter  3.9860E+14  m^3^/s^2^   $ \omega_T $  GEO angular frequency  7.2921151E05  radians/s   $ r_{GEO} $  GEO radius  42164172.37  m   $ v_{GEO} $  GEO velocity  3074.6600  m/s   $ E_{GEO} $  GEO energy / kg  14180301  J/Kg   $ H_{GEO} $  GEO angular momentum/kg  1.2964E+11  m^2^/s   $ r_p $  perigee radius  6458137.0  m   $ v_p $  perigee velocity  10462.7761  m/s   $ e $  eccentricity  0.77363    $ a $  semimajor axis  28529281.45  m   $ r_a $  apogee radius  50600425.91  m   $ v_a $  apogee velocity  1335.3651  m/s   $ \theta $  orbit angle at capture  143.73  degrees   $ r_c $  capture radius  30440398.47  m   $ v_{tc} $  capture tranverse velo.  2219.7489  m/s   $ E_c $  capture energy  15558098  J/kg   $ acc_c $  capture vert. accel.  0.2683  m/s^2^   $H_{capture}$ capture ang. momentum/kg 6.7570E+10  m^2^/s   $ y_s $  slowdown run  177982.60  m   $ r_s $  slowdown radius  41986189.77  m   $ v_s $  slowdown velocity  1334.10  m/s   $ t_s $  slowdown time  266.82  s    tether climb time  6604.70  s    Partial orbit time  18533.41  s    Launch time  348.76  s    Total time  7.15  hours   $ \Delta v $  velocity increase  854.9111  m/s   $ \Delta E $ radial energy increase/kg 1377796.89  J/kg   $ \Delta H $ radial ang. mom. incr./kg 6.2070E+10  m^2^/s  
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MORE LATER 
MORE LATER == M288 capture rail == A similar rail system can supply the M288 [[ http://serversky.com  server sky orbits ]]. {{ attachment:cap14.png   width=960, height=320 }} [[ attachment:cap14.c  Here is the source ]] , and you will need [[ http://www.libgd.org/  libGD ]] and [[ http://apngasm.sourceforge.net/apngasm  apngasm ]]. <9:> '''M288 rail Tether Capture'''   $ v_{vc} $  Vertical V  2933.86  m/s   $ acc_s $  slowdown deceleration  4.9836  m/s^2^   $ \omega_T $  M288 angular frequency  4.37527e04  radians/s   $ r_{M288} $  M288 radius  12769564  m   $ v_{M288} $  M288 velocity  5587.028  m/s   $ E_{M288} $  M288 energy / kg  15.607e+06  J/Kg   $ H_{M288} $  M288 angular momentum/kg 7.1344E+10  m^2^/s   $ r_p $  perigee radius  6458137.0  m   $ v_p $  perigee velocity  9603.4591  m/s   $ e $  eccentricity  0.49426    $ a $  semimajor axis  1276564  m   $ r_a $  apogee radius  19080992  m   $ v_a $  apogee velocity  3250.3790  m/s   $ \theta $  orbit angle at capture  112.54  degrees   $ r_c $  capture radius  11905978.35  m   $ v_{tc} $  capture tranverse velo.  5209.1859  m/s   $ E_c $  transfer orbit energy  15.607e+07  J/kg   $ acc_c $  capture vert. accel.  0.5328  m/s^2^   $H_{capture}$ capture ang. momentum/kg 6.2020E+10  m^2^/s   $ y_s $  slowdown run  863.59  km    Partial orbit time  2157.42  s    Launch time  304.42  s    Total time  50.84  minutes   $\Delta H$  ang. momentum change/kg  6.2020E+10  m^2^/s  With a capture radius of 12769km, and an elliptical orbit with the same semimajor axis the same as the M288 orbit, a miss will result in another capture opportunity one orbit (about 4 hours) later, plus or minus drag in the upper atmosphere. We may choose to add some apogee velocity, and otherwise adjust the orbit so perigee is above the high drag portion of the atmosphere. The M288 orbit is low enough and the magnetosphere particle density is high enough that we can use electrodynamic tether acceleration to restore angular momentum after a capture. 9.3235e9 m^2/s times the angular velocity of 4.375E4 rad/sec is the energy we need to add back to the tether, about 5 MJ/kg at 80% thrust efficiency. With 16 W/kg solar cells at 62% availability (tilting to the terminator to avoid night sky illumination), 1 kg of solar cells can generate 5MJ in less than 6 days Vertical deltaV could be a problem, but this velocity is tangential to the orbit, doesn't change the capture rail semimajor axis much, and is as likely to add as subtract from radial velocity, depending on where on the capture rail orbit it is added. By timing our launches, we can manage the average towards zero.          Angular   slowdown    retry  retry    slowdown  Restore   distance  total  energy  orbit  sidereal  launch  capture  accel  Energy   km  minutes  ratio  hours  orbits  m/s  m/s  m/s2  J/kg   863.59  50.84  19.1703  3.99  1.0000  9132.52  2933.86  4.98  4.08E+6   631.83  42.86  57.5671  11.97  1.5000  9510.04  3695.37  10.81  3.01E+6   507.13  39.81  109.132  7.98  2.0000  9716.17  4069.73  16.33  2.43E+6   371.80  37.09  245.410  11.97  3.0000  9942.27  4457.78  26.72  1.79E+6  MORE LATER == M288 rail bases for orbital debris capture == Server sky thinsats are cheaper to launch if they are lighter, but below a minimum masstoarea ratio, the orbits are unstable under light pressure. However, we can add mass ballast to a very light satellite, using gramweight chunks of space debris. The processing is simple, just let the debris shatter as it is collisioncaptured inside a cornucopiashaped funnel. Larger chunks can be cut apart at a simple processing plant at the M288 rail base, smaller chunks can be welded together. Capture vehicles can drop below the tether, in intercept orbits where the debris crosses the equatorial plane. Depending on their orbit, they can return to the same rail base, but more likely a different rail base will be more convenient. MORE LATER 
Capture Rail
"Landing" Launch Loop payloads on tethers in circular orbit, without apogee kick motors
More rockets bite the dust ...
The launch loop provides low cost deltaV from the earth's surface, injecting vehicles into highly elliptical transfer orbits, or earthescape orbits. What a launch loop cannot do directly is put a payload into a circular orbit. The perigee of a launchloop launch will always be at the launch track height, even if the apogee reaches the moon. To put a vehicle into a higher circular orbit requires additional angular momentum, usually supplied by a rocket.
Tethers provide an alternative. A zerovelocity capture on a vertical tether means that the launch loop launches payloads with an angular velocity matching that of the tether, at a lower altitude ( or radius ) than the center of the tether's orbit.
Classic Tether Capture
Yellow is a vehicle in a transfer orbit from launch. Green is a capture, red is a missed capture.
Here is the source , and you will need libGD and apngasm.
Capturing on a tether is not a new idea. The transfer orbit is chosen such that the vehicle arrives at the tether with zero velocity, and clamps on. It will have a small relative acceleration downwards, perhaps 2.5% of one gee, a fairly light load, assuming the tether is much more massive than the vehicle.
There are problems with classic tether capture. For a tether in geostationary orbit, capturing a vehicle in a transfer orbit with a perigee of 80km, the capture radius (the apogee of the transfer orbit) is at 29870 km, 12294 km below the geostationary altitude of 42164 km. Somehow, the vehicle must climb up the tether for more than 12000 kilometers. With the very small gee loading, this is not nearly as difficult as climbing up a space elevator from the Earth's surface, and a tractor vehicle can be waiting on the tether to pull up the launch vehicle. However, with limited power and wheel velocity, this will still take a while. At 200 kilometers per hour, the climb will take 61 hours. The climb will probably be slower on the higher gee lower section.
We will be stealing angular momentum from the tether, so we will need something like a VASIMR electric rocket to add the momentum again.
Worse still, what if we miss the capture? The period of the circular transfer orbit is 6.768 hours; the next time the vehicle reaches apogee, the tether will be a long way away, and we may need many days and hundreds of meters per second of delta V to catch it again.
Climbing Tether Capture
Here is the source, and you will need libGD and apngasm.
But who says the vehicle is constrained to zero vertical/radial velocity?
Vehicles leave the launch loop with a magnetic rail attached to the bottom, which magnetically drags on the rotor and propels them to orbital speeds. The magnetic rail is passive  it can only produce drag  but it can drag on anything conductive, such as a very thin metal tube around a tether. If we arrive at the tether with zero transverse/horizontal velocity, and a high radial/vertical velocity, we can ride the magnetic rail, just like the launch loop track that we left a few hours before. As we ride up the rail, we slow down against it. In a matter of minutes, we brake to a stop at geostationary radius. This requires a few percent more deltaV at launch, but launch velocity is cheap with a launch loop.
Transit time from start of launch to arrival at GEO will be less than 4 hours. The transit time will be slightly faster than a classic Hohmann transfer orbit with an apogee kick motor.
Other benefits include a somewhat shorter hanging tether, which also reduces the amount of angular momentum the vehicle removes from the tether.
For a geostationary target orbit, we can choose a transfer orbit with a 12 hour period and a semimajor axis of 26562 km (0.62996 smaller semimajor axis than the 42164 km GEO orbit). If apogee remains at launch loop exit altitude (6458 km), then drag can help the vehicle reenter 12 hours later, on the other side of the earth from the launch loop. It we instead choose to make another try, we can add 10m/s deltaV at apogee, and raise perigee by 45 km, a 200x reduction in drag. That will give us one or two more passes at capture, 24 and 48 hours later. The unmodified orbit is the "12 hour low" orbit, while the boosted perigee orbit is the "12 hour high" orbit. Actually, we must also perform a second burn at perigee, to lower apogee, if we wish to maintain the same semimajor axis and the same 12 hour orbit period.
We can also choose a transfer orbit with a 24 hour complete period, and perform the same perigeeraising maneuver if we wish to reduce drag and make a second pass. That higher orbit will have higher angular momentum, and draw less angular momentum from the tether.
Transfer Orbit 
Perigee 
Apogee 
V_{P} 
V_{A} 
AngM 
R_{capture} 

km 
km 
m/s 
m/s 
m^{2}/s 
km 
12 hour low 
6458 
46665 




12 hour high 
6503 
46620 

24 hour low 
6458 
77870 

24 hour high 
6503 
77825 
A 24 hour orbit, geostationary capture tether, fed from a fixed point on earth, is perhaps the easiest capture tether to imagine. However, a nonsynchronous capture tether offers more flexibility, as we will see below.
THE REST OF THIS PAGE WILL BE REWORKED
Imagine a long and heavy orbiting tether, running vertically from 29000 km radius, through GEO (42164 km), to a counterweight above. The tether has a thin, passive conductive rail around it. It will be used to magnetically capture vehicles and move them to GEO, using the sliding rail to push payloads transversely while they magnetically slide upwards.
This is not the normal apogee capture system, where we launch into an orbit whose perigee transverse velocity matches a hanging tether at the right altitude. Instead, we launch the vehicle a little faster and a little later. That puts the vehicle in an orbit with an apogee trailing the tether. The vehicle will never get to that apogee, because we will slide in front of the tether below apogee, where we still have a high (2700 m/s) radial velocity. We will still match the tether's transverse velocity and altitude, though.
In the rotating tether frame of reference, the vehicle will approach from below at high speed, from the rear, and decelerate in the traverse ("horizontal") direction as it approaches the tether. The vehicle is rising radially, conserving angular momentum, and losing angular velocity as the radius increases. In the tether's rotating frame, this looks like Coriolis acceleration, equal to twice the velocity times the tether's angular velocity. The angular orbital velocity for a GEO tether is 7.2921E5 radians per second, and the radial ("vertical") velocity will be around 2700 meters per second, so the Coriolis acceleration is 0.27 m/s^{2}. If the vehicle misses the tether rendezvous, it will continue to accelerate retrograde in the rotating frame, with the backwards velocity turning into downwards Coriolis acceleration in the rotating frame, reaching apogee above the intended attach point but well below GEO.
The tether is made with notexcessivelytapered Kevlar. We will need many tons of it, especially at GEO, as an angular momentum bank. More mass is better for this system. THIS is where we put the hotels, the radiation shielding, and the heavy buffet tables for obese tourists.
We launch a vehicle off the launch loop at 10463 m/s, and with very good radar and some trim thrusters, we "land" on the bottom of the rail at 30440 km altitude. At that point, we have the same angular frequency and circular orbital velocity, and a large vertical velocity component, 2700 m/s .
As the vehicle slides up the tether, the weak gravity and eddy current drag slows it down radially, as it accelerates transversely (forwards) in orbit to remain on the "faster at higher altitude" tether.
As the vehicle slides upwards, it magnetically levitates/pushes against the east side of the rail with Coriolis acceleration. That is 39 centimeters per second squared at 2700 m/s vertically, and 19 centimeters per second squared at 1334 m/s, our speed as we approach GEO altitude. The Coriolis acceleration and the eddy currents both reduce with relative velocity  most convenient! The vertical deceleration (gravity minus rotational centrifugal acceleration) decreases to zero as we approach GEO, still with significant velocity, though with less than the velocity of a frictionless ballistic because of the eddy currents.
As the vehicle approaches GEO, it is still moving fast, which is good, because it has 12000 kilometers to travel. The trip will take about 5 hours from earth, and 2 hours riding up the tether. It will be slower than a Hohmann with an apogee kick motor, but much cheaper and less polluting. At "slowdown height", about 180 kilometers below GEO station, the rail surface changes to increase eddy current drag, slowing down the vehicle. The vehicle reaches the drag section at 1334 meters per second, and deceleration increases from a few millimeters per second to 5 m/s^{2}, a bit more than half a gee (with passengers facing backwards, backs into chairs). The vehicle slows to 25 meters per second about 40 meters below the station, and with active track control, reduces deceleration and velocity to zero over the next 5 seconds. The payload or passenger compartment is plucked off the magnetrail and wing, and pushed into the station through an airlock.
Since the tether rail is not magnetic or active (that would be far too heavy for a 12000 kilometer gossamer structure) the vehicle magnetrail will need to wrap partly around it in some way. 39 cm/s^{2} Coriolis acceleration is far too weak to hold the vehicle against a rippling tether at high speed. For example, a ripple with a 20 kilometer wavelength and a 20 meter peaktopeak amplitude will shake the vehicle back and forth at 0.8 gees ( 12x Coriolis ) and a period of 7 seconds  a frisky roller coaster. Such a transverse ripple will not be a standing wave, but traveling up or down the tether at around 1 km per second. It will be a challenge to remove it, perhaps by periodic transverse shaking at GEO by the large momentum mass. It would be more difficult to remove by shaking the bottom counterweight vertically, inducing Coriolis accelerations, because a lot of power is required.
We can reverse the process. Using a magnetic accelerator to launch a vehicle down a rail on the west side of the tether, we falling off the end in a transfer orbit back to the upper atmosphere. This restores most of the GEO rail momentum.
The rocket thrust needed by GEO rail vehicles will be pure velocity correction, centimeters per second if we've done our radar and orbital mechanics calculations correctly. If we miss, we just reenter normally  a delay, but not a disaster. Besides a little correction exhaust, and whatever we need to add to for momentum restoration, the system is mass conservative and mostly energyrecycling.
Restoring GEO Rail orbital momentum
If we deorbit GEO station trash much faster than we receive payloads (aiming for empty and lifeless portions of the ocean) we can add more momentum by exploiting Coriolis force acceleration. We can also receive mass (and momentum) from the moon, or launch mass from the loop in slingshot orbits around the moon. However, the moon will not always be conveniently located for this, and vehicles may make many 20 day orbits before arriving with a suitable orbital position.
A slightly stronger tether can swing like a pendulum. That allows vehicles to arrive from the launch loop with higher angular velocity, and leave with lower angular velocity. This variation needs further study.
VASIMR and "Zero Orbiting Exhaust" momentum restoration
We will probably need to make up some energy and momentum losses with plasma rocket engines at GEO station, especially if we accumulate more upward vehicles than downward ones. But overall, the energy will be tiny compared to the the launch loop energies, and we can float a lot of solar cell around GEO as part of our "momentum anchor".
As part of an overall "space antilitter" ethic, we should make sure the plasma rocket engines are offset so they are firing exhaust away from the rest of the geosynchronous ring, and that the rocket exhaust is traveling fast enough to escape the solar system ( >16 km/sec at noon when the rockets are pushing retrograde to the Earth's orbit around the sun, >76 km/sec (!!) at midnight, when the rockets are pushing retrograde ).
A Variable Specific Impulse Magnetoplasma Rocket (VASIMR) engine operates optimally at 50 km/sec, so operation at 80 km/sec is close to optimum. The reaction mass is argon, which can be frozen at 200C with a density of 1600 kg/m3 and a vapor pressure of 0.1 atmosphere for transit from earth. We will need about 1 kg of argon reaction mass shipped up for every 50 kg of uncompensated mass shipped up from Earth. Argon is 0.93% of the atmosphere, and costs about $1/kg (liquid) in large quantities. It is a byproduct of liquid oxygen production.
With 80km/s exhaust, we will expend about 40 kilowatt hours per "uncompensated vehicle kilogram" operating the plasma rockets, including coolers and radiator operation. Typical solar cell weights for existing satellites are 65 kg/kilowatt, or 16 watts per kilogram, so 1 kilogram of solar cells provides enough energy in a year to bring aboard 3.5 kilograms of mass  a power doubling time of 5 months. However, if we can use "server sky" style ultrathin InP cells, closer to 1000 kg/kW, we can approach 50 kilograms of mass per kilogramyear of solar cell. In the very long term, solar cells made from lunar materials will bring both momentum and cheaper launch to the GEO rail, but solar cells are a hightech undertaking, and it will be a long time before that level of manufacturing technology exists in space.
In the shorter term, perhaps there is some variant of VASIMR that can operate at lower exhaust velocities, expending more argon but using less energy. If we fire the argon out at 4500 m/s, and don't have too much thermal spread, the argon will be in a ballistic orbit that falls into the atmosphere (returning the argon where we found it). This will require 1 kilogram of argon for every 3 kilograms of payload shipped, which may mean that every other payload is argon during construction. However, at 50% efficiency, and 50% mass fraction we will need only 3.8 kWhours of energy per kilogram of incoming nonargon mass, resulting in 1 kg per 10 kgdays of solar cell. That will lead to very fast growth rates of the power and engine systems, rapidly evolving towards the high velocity exhaust scenario.
The above analysis assumes that the argon plasma neutralizes  if it stays ionized, it will gyrate in the Earth's magnetic field and not go much of anywhere, because the pitch angle will be high. That will be interesting, perhaps the trapped particles will act as shielding mass against higher energy particles trapped on the same field lines, but that analysis must wait for another time.
Also  do not assume that we can fire the argon at night at just the right velocity so that the exhaust falls into the sun. The sun is a small target, and distant, so with thermal spread much of the exhaust will end up in highly elliptical orbits that reach earth's orbit. The gas will be dilute, and most of it that returns will collect in the morning sky, but it will add some pollution in the long run.
How much argon will we use? The Earth's atmosphere weighs 5E15 tons, and about 1% is argon, so 5E13 tons of argon are available. With 80km/s exhaust resulting in a 50:1 payloadtoargon ratio, we can ship up 2.5E15 tons before we run out of argon. Chances are, we will be shipping down material and momentum from the moon long before we make a dent in the argon supply.
Conductor Track Weight
Assume a 5 ton vehicle moving at 2700m/s with 1.0 m/s^{2} transverse acceleration (lots of vibration!), and 0.3 m/s^{2} radial deceleration from the magnetic field. The total transverse force is 5000N, the deceleration force is 1500N. The deceleration power is 4MW .
Assume a 10 meter magnet rail with 0.2T undulating fields and 50% fill, and an average of 4 cm of "track width" on a 6cm diameter aluminum tube (bonded to a Kevlar core tube), so the "bearing surface" is 0.2 m^{2}. The field pressure is 5000N/0.2m^{2} or 25000N/m^{2}, which is 0.2T times 125,000 amps per meter in the conductor. Assume the current path is effectively 10cm long and 5 meters wide for 625KA total. For a deceleration power of 4MW, the resistance of the current path must be 4MW/(625KA^{2}) or 10 microohms. The resistivity per square of the aluminum will be (5.0/0.1) times that, or 500 microohms per square. Aluminum is about 30 nanoohmmeter, so a thickness of 60 microns will have that resistivity. The cross section is pi * 0.06m * 6E5m or 1.13E5 m^{2}, for a mass per meter of 30 grams (aluminum density is 2.7). That is 30kg/km, or 360 tons of aluminum for the whole 12000 km track.
4MW/2700m/s, or 1500 Joules per meter of heat energy will be deposited into the 30g/m track. Aluminum heat capacity is 0.9 J/gmK, so it could heat by about 56K. However, it is bonded to a much larger mass of Kevlar (1.4 J/gmK), and the thermal diffusion time through 60 microns of aluminum is about 40 microseconds. The 10 meter magnet rail passes by in 3.7 milliseconds, so we can assume that most of the heat has passed into the Kevlar core, and the aluminum heats much less.
MORE LATER
The Math
We will neglect oblateness, light pressure, and the gravity of other bodies for now. Simple Kepler two body orbits around the earth.
The vehicle leaves the launch loop at 80km altitude, the apogee of an elliptical orbit with a perigee of r_p = 6458.137 km and velocity v_a . The earth's gravitational parameter \mu = 398600.4418 km^{3}/s^{2} = 3.986004418e14 m^{3}/s^{2}. Given those three values, we define the orbit:
h ~ = ~ r_p v_p angular momentum
a ~ = ~ { \Large { 1 \over { \LARGE { 2 \over r_p } ~  ~ \LARGE { { {v_p}^2 } \over { \huge \mu } } } } } semimajor axis
e ~ = ~ 1  { \Large { { r_p } \over a } } ~ = ~ { \Large { { r_p {v_p}^2 } \over \mu } }  1 eccentricity
v_0 ~ = ~ { \Large { { v_p } \over { 1 + e } } } ~ = ~ { \Large { { \mu } \over { r_p v_p } } } characteristic velocity
r_0 ~ = ~ a ( 1  e^2 ) ~ = ~ \Large { { {r_p}^2 {v_p}^2 } \over \mu } characteristic radius
r( \theta ) ~ = ~ \Large { r_0 \over { 1 + e \cos( \theta ) } } orbit radius as a function of \theta , the angle from perigee or true anomaly
{v_T}( \theta ) ~ = ~ v_0 ( 1 + e \cos( \theta ) ) orbit transverse velocity
{v_R}( \theta ) ~ = ~ v_0 e \sin( \theta ) orbit radial velocity
{\omega } ( \theta ) ~ = ~ { \Large { h \over{ r^2 } } } ~ = ~{ \Large { { \mu^2 ( 1 + e \cos( \theta ) )^2 } \over { {r_p}^3 {v_p}^3 } } } orbit angular velocity
r_a ~ = ~ ( 1 + e ) a ~ = ~ { \Large 1 \over { \LARGE { { \huge 2 \mu } \over { r_p {v_p}^2 } } ~  ~1 } } apogee
The angular velocity of the tether in geostationary orbit (defined by the stellar day ) is \omega_T = 2 \pi / 86164.098903691 = 7.29211515E5 radians per second.
At the capture point, the orbit has the same angular velocity as the tether, \omega = \omega_T .
Classical Tether Capture
For the classical ( zero vertical velocity ) tether capture, v_R = 0 and \theta = \pi . We are at apogee.
We know three things about the transfer orbit:
\Large { 1 \over a } ~ = ~ { 2 \over r_p }  { {v_p}^2 \over \mu } ~ = ~ { 2 \over r_a }  { {v_a}^2 \over \mu } Conservation of energy
v_a r_a ~ = ~ v_p r_p Conservation of angular momentum
v_a ~ = ~ \omega_{GEO} r_a Definition of zero relative transverse velocity, vehicle to tether
That is enough to solve for r_a :
\Large { { 2 \mu } \over r_p }  \left( { { \omega_{GEO} {r_a}^2 } \over r_p } \right)^2 ~ = ~ { { 2 \mu } \over r_a }  { \omega_{GEO} {r_a}^2 }
Which can be reduced to the following iterator:
r_a \Large = \left( { { 2 \mu } \over { {\omega_{GEO}}^2 } } \left( { r_p \over { r_a + r_p } } \right) \right)^{ 1 \over 3 }
Iterate on that for a while. It converges in about 20 steps to high accuracy starting anywhere between 0 and r_{GEO} , but slightly faster starting at 0.7*r_{GEO}
If we rendezvous with a hanging tether at the radius r_a , we will have zero velocity vertically and horizontally with respect to the tether, and a small acceleration downwards. We can clamp on and hang at the end of the tether. Our small residual weight will accelerate the tether down, slightly, and launch strain waves up it.v
But we aren't done! We need to do two more things  climb up the tether, and restore momentum to the tether. Assume the tether is massive compared to a vehicle, so it does not deflect much. The tether will slow down by a tiny amount, and we must restore the momentum, perhaps with rocket engines. We must also supply energy to lift the vehicle up the tether against the small gravity (minus centrifugal force) at that high altitude.
Classical Tether Capture 

\mu 
gravitational parameter 
3.986004418e14 
m^{3}/s^{2} 

\omega_T 
GEO angular frequency 
7.29211515e5 
radians/s 
r_{GEO} 
GEO radius 
42164172.4 
m 

v_{GEO} 
GEO velocity 
3074.66 
m/s 
E_{GEO} 
GEO energy / kg 
14180301.17 
J/kg 

H_{GEO} 
GEO angular momentum/kg 
1.296e11 
m^{2}/s 
r_p 
perigee radius 
6458137.0 
m 

v_p 
perigee velocity 
10074.5754 
m/s 
e 
eccentricity 
0.64446 


a 
semimajor axis 
18164241.20 
m 
r_a 
apogee radius 
29870345.40 
m 

v_a 
apogee velocity 
2178.1800 
m 
acc_a 
apogee radial accel. 
0.2879 
m/s^2 

E_a 
apogee energy / kg 
15716587.30 
J/kg 
H_a 
apogee ang momentum/kg 
6.506e10 
m^{2}/s 

\Delta r 
radius increase 
12293826.96 
m 
\Delta v 
velocity increase 
896.4800 
m/s 

\Delta E 
radial energy increase/kg 
1536286.13 
J/kg 
\Delta H 
radial ang. mom. incr./kg 
6.458e10 
m^{2}/s 
If we haul the vehicle up with a solar cell power source weighing as much as the payload, then our "hauling power" will be about (16/2) = 8 watts per kilogram using current satellite technology. Against a 0.2879 m/s^{2} acceleration field, we can move at (8/0.2879) = 28 m/s near the bottom, faster near the top. At 8 watts per kilogram, and 1536286 joules per kilogram, climbing the tether will require at least 53 hours. However, at some point we are velocity limited  perhaps 200m/s for our motor. That occurs when the acceleration force is ( 8/200 ) = 0.04 m/s^{2}, about 39800 km high. The velocity limit increases climber time to 55 hours .
If we can use a pulley system without tangling cables, running at 400m/s, climber time is 8.5 hours. We can go a lot faster if we are not moving the motor and the solar cells.
MORE LATER
Rail Tether Capture, with vertical velocity
This is a little trickier. The system moves faster if we have a lousy "lift to drag" ratio, because we can travel the tether faster. It also makes the math a little simpler if we pick a fixed starting speed, as long as it is more than enough to make the climb. Lets start with the energy increase for the classic case, 1.54 MJ/kg , double it, and turn the result into a velocity: about 2500 meters per second. We will arbitrarily choose that as v_{vc} , the vertical speed the vehicle will be moving up the tether when it is captured. Since we are launching from the loop with about 52 MJ/kg anyway, an extra 3% energy loss will not be a show stopper, and we will get to the GEO rail station faster. Time is money!
The capture radius r_c is below orbit apogee, and below the GEO altitude . This is a good starting point. The angular velocity will be \omega_T as before, so the transverse velocity will be:
v_{tc} = \omega_T r_c
and the angular momentum is
h = \omega_T {r_c}^2
We know that at launch, at perigee, the angular momentum is the same, so:
v_p = h / r_p
We can define the orbit from there, computing a , e , v_0 , r_0 , and \theta from the equations above.
Most interesting is computing the vertical velocity at capture, v_{rc} :
v_{rc} ~ = ~ \sqrt{ 2 \mu { \Large \left( { 1 \over r_p }  { 1 \over r_c } \right) } + {v_p}^2  {v_{tc}}^2 }
Also the vertical energy per mass needed to climb up the tether, E_{cc} :
E_{cc} ~ = ~ { 1 \over 2 } {\omega_T}^2 \left( {r_c}^2  {r_{GEO}}^2 \right) + \mu { \Large \left( { 1 \over r_c }  { 1 \over r_{GEO} } \right) }
If the eddy current losses are negligible ( which requires a thick and heavy conductor track ), then the vehicle will reach (or pass) GEO if
E_{cc} < { 1 \over 2 } {v_{rc}}^2
If we define the capture parameter c \equiv r_c / r_{GEO} and the perigee parameter p \equiv r_p / r_{GEO} \approx 0.1531665 then the inequality can be massaged into:
c^5  2 p^2 c^3 + ( 2 p^2 + p ) c  2 p^2 > 0
MORE LATER  that does not look right, yet. The next few bits may be invalid, too
The capture rail is a long way out of the gravity well, so "natural" vertical deceleration of a vehicle is small. However, there will be some drag from eddy currents in the thin and rather resistive rail. In a production system, the drag profile will be optimized and rather complex. For this analysis, we will just compute the kinetic energy lost by moving up the rail, multiply that by a drag factor, and use that to compute the kinetic energy and velocity remaining as we approach the dock at GEO.
The vertical deceleration is given by:
acceleration = \omega^2 r  \mu / r^2 . . . zero at GEO, of course!
and the specific energy ( J/kg ) can be computed by integrating that:
E(r) = 0.5 \omega^2 r^2  \mu / r
The drag factor is D , and will always be greater than 1, so the velocity nearing GEO will be:
v_{GEO} \approx \sqrt{ {v_{vc}}^2  2 D ( E_{GEO}  E_{capture} ) }
The velocity at radius r will be:
v(r) \approx \sqrt{ {v_{vc}}^2 + D \omega^2 ( r^2  {r_c}^2 )  2 D \mu ( 1 / r_c  1 / r ) }
The transit time from the capture point to GEO will be:
time = {\huge\int} _{r_c}^{GEO} { \Large { dr \over {v(r)} } } ... which probably requires numerical integration
Since we want to keep the transit time small, we would like to keep the rail very straight and travel up it quickly. So we keep the drag and deceleration relatively low for most of the trip. As we approach GEO, however, we "hit the brakes", decelerating at a much higher rate, so we come quickly and smoothly to a stop at the dock. The transition from low to high deceleration occurs at the slowdown radius r_s .
Here's a closetominimum transit time capture:
GEOrail Tether Capture 

v_{vc} 
Vertical V 
2700.0000 
m/s 

acc_s 
slowdown deceleration 
5.0000 
m/s^{2} 
D 
drag factor 
2.0000 


\mu 
gravitational parameter 
3.9860E+14 
m^{3}/s^{2} 
\omega_T 
GEO angular frequency 
7.2921151E05 
radians/s 

r_{GEO} 
GEO radius 
42164172.37 
m 
v_{GEO} 
GEO velocity 
3074.6600 
m/s 

E_{GEO} 
GEO energy / kg 
14180301 
J/Kg 
H_{GEO} 
GEO angular momentum/kg 
1.2964E+11 
m^{2}/s 

r_p 
perigee radius 
6458137.0 
m 
v_p 
perigee velocity 
10462.7761 
m/s 

e 
eccentricity 
0.77363 

a 
semimajor axis 
28529281.45 
m 

r_a 
apogee radius 
50600425.91 
m 
v_a 
apogee velocity 
1335.3651 
m/s 

\theta 
orbit angle at capture 
143.73 
degrees 
r_c 
capture radius 
30440398.47 
m 

v_{tc} 
capture tranverse velo. 
2219.7489 
m/s 
E_c 
capture energy 
15558098 
J/kg 

acc_c 
capture vert. accel. 
0.2683 
m/s^{2} 
H_{capture} 
capture ang. momentum/kg 
6.7570E+10 
m^{2}/s 

y_s 
slowdown run 
177982.60 
m 
r_s 
slowdown radius 
41986189.77 
m 

v_s 
slowdown velocity 
1334.10 
m/s 
t_s 
slowdown time 
266.82 
s 


tether climb time 
6604.70 
s 

Partial orbit time 
18533.41 
s 


Launch time 
348.76 
s 

Total time 
7.15 
hours 

\Delta v 
velocity increase 
854.9111 
m/s 
\Delta E 
radial energy increase/kg 
1377796.89 
J/kg 

\Delta H 
radial ang. mom. incr./kg 
6.2070E+10 
m^{2}/s 
\Delta r 
radius increase 
11723773.89 
m 
Here's a table of transit time versus versus vertical capture velocity. The capture velocity is quite sensitive to launch velocity, which must be controlled to mm/sec. Note that if we launch very fast, we spend a long time heading out and coming back to the tether. These are all "drag=2" times; if we reduce drag, the optimum occurs at a lower launch velocity.
Transit time versus vertical tether velocity 
time, seconds 
total 

vertical 
vp 
a 
rc 
launch 
orbit 
climb 
slow 
hours 
3200.00 
10615.08 
37940037 
30661158 
354 
24481 
4330 
446 
8.23 
3000.00 
10551.38 
32915772 
30569017 
352 
21591 
4991 
381 
7.59 
2800.00 
10491.37 
29807595 
30481967 
350 
19422 
5934 
308 
7.23 
2750.00 
10476.96 
29148708 
30461018 
349 
18964 
6245 
288 
7.18 
2700.00 
10462.78 
28529281 
30440398 
349 
18533 
6605 
267 
7.15 
2650.00 
10488.83 
27946160 
30420112 
348 
18129 
7028 
244 
7.15 
2600.00 
10435.13 
27396521 
30400159 
348 
17749 
7542 
220 
7.18 
2550.00 
10421.67 
26877836 
30380544 
347 
17390 
8193 
193 
7.26 
2500.00 
10408.45 
26387830 
30361268 
347 
17053 
9075 
162 
7.40 
2400.00 
10382.74 
25485852 
30323743 
346 
16433 
13467 
70 
8.42 
There may be another set of optima at much higher launch velocities. If we fire the vehicle off the launch loop very fast, we are shooting it almost straight up the tether. The simulations did not capture that because \theta is less than 90 degrees, the cosine changes sign, and the orbit is probably parabolic with an eccentricity greater than 1. While that might shave off some transit time, if the vehicle misses a capture, it will be lost in interplanetary or even interstellar space.
MORE LATER
M288 capture rail
A similar rail system can supply the M288 server sky orbits.
Here is the source , and you will need libGD and apngasm.
M288 rail Tether Capture 

v_{vc} 
Vertical V 
2933.86 
m/s 

acc_s 
slowdown deceleration 
4.9836 
m/s^{2} 
\omega_T 
M288 angular frequency 
4.37527e04 
radians/s 

r_{M288} 
M288 radius 
12769564 
m 
v_{M288} 
M288 velocity 
5587.028 
m/s 

E_{M288} 
M288 energy / kg 
15.607e+06 
J/Kg 
H_{M288} 
M288 angular momentum/kg 
7.1344E+10 
m^{2}/s 

r_p 
perigee radius 
6458137.0 
m 
v_p 
perigee velocity 
9603.4591 
m/s 

e 
eccentricity 
0.49426 

a 
semimajor axis 
1276564 
m 

r_a 
apogee radius 
19080992 
m 
v_a 
apogee velocity 
3250.3790 
m/s 

\theta 
orbit angle at capture 
112.54 
degrees 
r_c 
capture radius 
11905978.35 
m 

v_{tc} 
capture tranverse velo. 
5209.1859 
m/s 
E_c 
transfer orbit energy 
15.607e+07 
J/kg 

acc_c 
capture vert. accel. 
0.5328 
m/s^{2} 
H_{capture} 
capture ang. momentum/kg 
6.2020E+10 
m^{2}/s 

y_s 
slowdown run 
863.59 
km 

Partial orbit time 
2157.42 
s 


Launch time 
304.42 
s 

Total time 
50.84 
minutes 

\Delta H 
ang. momentum change/kg 
6.2020E+10 
m^{2}/s 
With a capture radius of 12769km, and an elliptical orbit with the same semimajor axis the same as the M288 orbit, a miss will result in another capture opportunity one orbit (about 4 hours) later, plus or minus drag in the upper atmosphere. We may choose to add some apogee velocity, and otherwise adjust the orbit so perigee is above the high drag portion of the atmosphere.
The M288 orbit is low enough and the magnetosphere particle density is high enough that we can use electrodynamic tether acceleration to restore angular momentum after a capture. 9.3235e9 m^2/s times the angular velocity of 4.375E4 rad/sec is the energy we need to add back to the tether, about 5 MJ/kg at 80% thrust efficiency. With 16 W/kg solar cells at 62% availability (tilting to the terminator to avoid night sky illumination), 1 kg of solar cells can generate 5MJ in less than 6 days
Vertical deltaV could be a problem, but this velocity is tangential to the orbit, doesn't change the capture rail semimajor axis much, and is as likely to add as subtract from radial velocity, depending on where on the capture rail orbit it is added. By timing our launches, we can manage the average towards zero.








Angular 
slowdown 


retry 
retry 


slowdown 
Restore 
distance 
total 
energy 
orbit 
sidereal 
launch 
capture 
accel 
Energy 
km 
minutes 
ratio 
hours 
orbits 
m/s 
m/s 
m/s2 
J/kg 
863.59 
50.84 
19.1703 
3.99 
1.0000 
9132.52 
2933.86 
4.98 
4.08E+6 
631.83 
42.86 
57.5671 
11.97 
1.5000 
9510.04 
3695.37 
10.81 
3.01E+6 
507.13 
39.81 
109.132 
7.98 
2.0000 
9716.17 
4069.73 
16.33 
2.43E+6 
371.80 
37.09 
245.410 
11.97 
3.0000 
9942.27 
4457.78 
26.72 
1.79E+6 
MORE LATER
M288 rail bases for orbital debris capture
Server sky thinsats are cheaper to launch if they are lighter, but below a minimum masstoarea ratio, the orbits are unstable under light pressure. However, we can add mass ballast to a very light satellite, using gramweight chunks of space debris. The processing is simple, just let the debris shatter as it is collisioncaptured inside a cornucopiashaped funnel. Larger chunks can be cut apart at a simple processing plant at the M288 rail base, smaller chunks can be welded together.
Capture vehicles can drop below the tether, in intercept orbits where the debris crosses the equatorial plane. Depending on their orbit, they can return to the same rail base, but more likely a different rail base will be more convenient.
MORE LATER