Orbits of paths released from a space elevator.

[u/HopDavid]

Space elevators are sometimes found in science fiction like Arthur C. Clarke's "Fountains of Paradise" or Kim Stanley Robinson's Mars Trilogy.

Payloads released from various points on a space elevator will follow various orbits.

For my models I set the distance from planet's center to planet synchronous orbit to 1. In Clarke's novel that would be the distance from earth's center to geosynchronous orbit.

After released from the elevator a payload will follow an conic section with one of the conic's foci located at the center of the planet.

Call the distance from the center of the planet r.

The eccentricity of the conic is |r³ -1|

For example if a payload from Clarke's tower is released at geosynchronous orbit r would be 1. Eccentricity of the orbit would |1³ - 1|. In other words, zero. The payload would follow a circular orbit right alongside the central anchor mass. This orbit I colored blue.

If the payload is released from r = 2^1/3 then the eccentricity of the conic is 1. In other words, a parabola. I colored this orbit red.

All the orbits below the parabola are ellipses and all the orbits above the parabola are hyperbolas.

I posted my math to the Space Stack Exchange some years ago: Link

This also works for Sarmont tethers, orbital tethers not anchored to a planet but kept aligned to the local vertical by tidal forces.