For a vertical tether in circular orbit, there's a point where the net acceleration is zero. Above that point, so called centrifugal force exceeds gravity. Below that point, gravity exceeds so-called centrifugal force. If a payload is released on this point of on the tether, it will follow a circular orbit alongside the tether. This point I call the Tether Center.
It will be a conic section. Call the conic's eccentricity e. Call the distance from tether point r.
If dropped from below center, r = (1-e)1/3.
If released from above center, r = (1+e)1/3.
Here's my derivation. Mark Adler also gives a nice demonstration in the comments on that post.
This is true of any vertical tether in a circular orbit.
If there are two prograde, coplanar vertical tethers at different altitudes, there's an elliptical path between them where the perigee velocity matches a point on the lower tether and apogee velocity matches a point on the upper tether.
If a payload is released from the lower tether at the correct time, it will rise to the upper tether which will be moving the same velocity as the payload at apoapsis. Rendezvous can be accomplished with almost no delta V. Cargo can be exchanged between tethers with almost no reaction mass.
Let r for the release point above the tether be (1+e)1/3 and release point below the tether be (1-e)1/3. Then both the larger and smaller ellipse will be the same shape.
Center of the above tether is 8000 km. I tried to place it above the dense orbital debris regions of low earth orbit. The tether is 461.6 kilometers long. Dropping from the foot will send a payload to a 150 km attitude perigee. Throwing a payload from the tether top will send a payload to a 9780 km apogee.
From a 150 km altitude orbit, it takes about .33 km/s to send a payload to the tether foot.
Both ellipses have the same eccentricity, about .0864
I repeatedly clone, scale by 126% and rotate 180º:
But there's a problem with this scheme. A tether loses orbital momentum each time it catches a payload from below. Ascending and throwing to a higher orbit also saps orbital momentum. How do we keep these tethers from sinking?
Imagine resources parked in lunar orbit. Maybe propellent mined from the lunar poles. Or perhaps platinum from an asteroid parked in a lunar DRO. To send cargo to earth's surface or low earth orbit would entail catching from a higher orbit, descending and dropping to a lower orbit:
If cargo is moved down as well as up, momentum boosting maneuvers can be balanced with momentum sapping maneuvers.
Thus mass in high orbits are sources of up momentum. This itself could be a commodity, a way to preserve orbits of momentum exchange tethers.
This tether spiral scheme cuts tether length, especially in regions of high debris density and the Van Allen Belts.
In this illustration successive ellipses vary by a factor of 21/3. Other rates of expansion are possible. Let k be the ratio of one ellipse apogee to the apogee below. k = (1+e)4/3/(1-e)4/3. Thus we can wind the spiral tighter or loosen it by choice of ellipse eccentricity.