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 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

**. This itself could be a commodity, a way to preserve orbits of momentum exchange tethers.**

*up momentum*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 2

^{1/3}. 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.

^{}

## 6 comments:

Hi Hop,

One thought I had is that your tethers are rotating at a rate of one rotation per revolution. Each tether had a winch that could pull in tether mass or play it out. Adjusting the mass in this way would cause the tether to rotate due to conservation of angular momentum. This might be a more convenient way to move the payload to the other end of the tether than having the payload climb the tether.

By the way, what do you estimate the payload mass to structural mass ratio to be for the whole system?

I have a few papers about space tethers in my library which you might find interesting: https://drive.google.com/drive/#folders/0B0uK2L6eQG0UYU5DejV2Y0Y3YWM/0B0uK2L6eQG0UYUVMSk1CVGk5RG8/0B0uK2L6eQG0UaUZ2bnM3U29oY2M

Interesting notion -- winching a vertical tether to a rotovator and vice versa. If memory serves, Jon Goff had also talked about that on Selenian Boondocks.

Regarding tether mass...

The short vertical tethers I describe have a taper ratio of 1-- even with ordinary materials like Kevlar. So that's a mass advantage over long tethers and/or rotovators that have high acceleration at the tips.

However I believe we'd need an anchor mass at each tether center. If the total tether mass were close to payload mass, catches and throws would have too much impact on tether orbit. The tether center would be a good place for solar array wings.

The first game of tether catch I imagined was between Phobos and Deimos elevators. Those two moons would be nice momentum banks.

A taper ratio of 1 is very promising. Boeing did a study for the HASTOL tether and concluded that structural-mass-to-payload-mass ratio of 200:1 or better is required for the economics of their particular architecture. With a more extreme ratio their system would not have been economically competitive with conventional rocket transport. A different ratio would apply in your case for destinations beyond LEO but the underlying economic principles are the same. This ratio determines the amount of time it would take for the system to "pay for itself."

If I'm reading your sixth image correctly, there are a total of 16 tethers in your architecture. However, each tether could simultaneously be holding it's own payload (or some up-payloads and some down-payloads). I would not be surprised if a ratio near 10:1 is possible.

If you wish to flesh out this architecture more you might consider using the spent upper stages as counterweight when constructing the system.

I'd also like to see Hall ion engines and xenon as part of the anchor mass.

Then the tethers might be a way to quickly impart momentum that has gradually been built up over time.

"Ascending and throwing to a higher orbit also saps orbital momentum. How do we keep these tethers from sinking?"

The most obvious method is to use the Earth's magnetic field. As the tether rotates through the magnetic field, it will generate a current via induction; this was demonstrated (partially) by NASA's Tethered Satellite System-1 and 1R program (among others). The physics here are reversible. By running an electric current through the tether, thrust can be generated, allowing momentum to be created and keeping the tether from sinking.

Here's a link to a similar earth-moon tether transport system using only two tethers: http://www.tethers.com/papers/cislunaraiaapaper.pdf

The structural-mass-to-payload-mass ratio that was arrived at in this study is 28:1.

Not mentioned in the paper are the orbital inclination limits of the lunar tether. The lunar tether in this study is in circular low lunar orbit. Due to the moon's lumpy gravity, such orbits will only be stable at specific inclinations or "frozen orbits" of 27º, 50º, 76º, and 86º.

My opinion regarding the purpose of these systems is that it is more worth wile to provide lunar material to LEO and to EML2 than to provide terrestrial payloads to the lunar surface.

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