## A spiral of tethers

3 Apr 2015, 19:00 UTCFirst off let's look at the great granddaddy of vertical tethers, the Clarke tower.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.In this case, the tether center is at geosynch height, about 42,000 km from earth's center. I set 42,000 km to be 1. What path does a payload follow if released from the tether below the 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 ...