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Deletions are marked like this.  Additions are marked like this. 
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$ a ~ = ~ { \Large 1 \over \LARGE { { {v_p}^2 } \over \mu }  { 2 \over { r_p } } } ~ ~ ~ ~ ~ ~ $ semimajor axis  $ h ~ = ~ r_p v_p $ ` ` angular momentum 
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$ e ~ = ~ 1  { \Large { { r_p } \over a } } ~ = ~ { \Large { { r_p {v_p}^2 } \over \mu } }  1 ~ ~ ~ ~ ~ ~ $ eccentricity  $ a ~ = ~ { \Large 1 \over \LARGE { { {v_p}^2 } \over { \huge \mu } }  { { \huge 2 } \over { r_p } } } $ ` ` semimajor axis 
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$ r_a ~ = ~ ( 1 + e ) a ~ = ~ { \Large 1 \over { \LARGE { { \huge 2 \mu } \over { r_p {v_p}^2 } } ~  ~1 } } ~ ~ ~ ~ ~ $ apogee  $ 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 { { v_p } \over { 2 ~  ~ { \LARGE { r_p {v_p}^2 } \over { \huge \mu } } } } } $ ` ` nominal 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 is $ \omega $ = 2 $ \pi $ / 86164.0905 = 7.2921159E5 radians per second MORE LATER 
GEO Rail
"Landing" Launch Loop payloads in a destination without rockets
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.
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 (1800 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 1800 meters per second, so the Coriolis acceleration is 0.26 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 10148.7 m/s, and with very good radar and some trim thrusters, we "land" on the bottom of the rail at 29979.7 km altitude. At that point, we have the same angular frequency and circular orbital velocity, and a large vertical velocity component, 1746.8 m/s .
As the vehicle slides up the tether, the weak gravity slows it down radially, as it accelerates 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 26 centimeters per second squared at 1800 m/s vertically, and 7 centimeters per second squared at 400 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 4 hours. About 10000 meters below GEO station, the rail surface changes to increase eddy current drag, slowing down the vehicle. The vehicle reaches the drag section at 400 meters per second, and deceleration increases from a few millimeters per second to 10 m/s^{2}, a bit more than 1 gee (with passengers facing backwards, backs into chairs). The vehicle slows to 25 meters per second about 40 meters below the station, and with the aid of some linear motors, 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. 26 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 0.3 gees ( 12x Coriolis ) and a period of 11 seconds  a mild 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.
More rockets bite the dust ...
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.
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 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 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
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 = 6458km and velocity v_a . Given those two values, we define the orbit:
h ~ = ~ r_p v_p angular momentum
a ~ = ~ { \Large 1 \over \LARGE { { {v_p}^2 } \over { \huge \mu } }  { { \huge 2 } \over { r_p } } } 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 { { v_p } \over { 2 ~  ~ { \LARGE { r_p {v_p}^2 } \over { \huge \mu } } } } } nominal 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 is \omega = 2 \pi / 86164.0905 = 7.2921159E5 radians per second
MORE LATER