(Note: the following is one of various posts that were copied here from the Moonwards.com forum for the sake of preserving the early days of the project. It was originally posted there on Feb 28th, 2016.)
From Sigvart Brendberg:
Lunavators is an approaching topic, so I just threw up the basics here if anyone is interested
A major concern in space travel is not distances, but change in velocity, commonly called delta v. A spacecraft is limited by its delta-v budget, governed by the rocket equation that has a nasty exponential nature. For example, a spacecraft taking off or landing on the Moon typically needs to be half propellant, with regular chemical engines. Thus, spacecraft are normally mostly fuel, and less of everything else.
Tethers, when completed, provides an efficient way of cheating the troublesome equation, what if you could just climb down to the Moon? The bottom part of the elevator is anchored to the surface, so the momentum of the spacecraft is simply transferred to that enormous chunk of rock.
The problem: Why does not the Lunavator simply fall down? That is after all our everyday experience. The classical approach to this problem is to use the spin of the planet to counteract gravity. (Try to spin around with a rope). However, that does not work for the Moon, as it spins way to slowly. Instead, the elevator extends to a point between the Earth and the Moon called EML1, a place where Earth's and the Moon's gravity cancels out, and together with the inertia creates a place where a thing would remain fixed relative to both the Moon and Earth. A nice place for a space station. From there, a counterweight can be lowered down into the gravity well of Earth. The lower end is also of interest because it moves slower than orbital velocity, making it possible to just "jump off" and re-enter the atmosphere. In other words, a free return from the Moon. Of course, it also works in reverse, making it possible to climb from a transfer orbit to the surface of the Moon. The total one-way delta-v saved is about 2500 m/s.
The math is not that horrifying:
The cable is stretched, so the cable strength we look for is ultimate tensile strength. Also, the density of it is important, the lighter the better. The combined metric for how good a material is as a tether, strength divided by density, gets the unit m^2/s^2, sometimes called Yuri.
This is the same unit that we use for the requirements for the tether, acceleration over a distance. Simply put, you take the acceleration profile over the whole tether span, and finds the area under the graph by integration. This is also a quantity in m^2/s^2.
This can be put into the tether equation, R = e^(requirement in Yuris / tether strength in Yuris). R is here the ratio of the cross sections at the start and at the end of the tether. If this ratio is very large, the material is too weak.
One promising material is Zylon. At a tensile strength of 5800 MPa and a density of only 1560 kg/m^3, it ends up at 3.7 mega-Yuri. That is good compared to the potential between EML1 and the lunar surface, 2.7 mega-Yuri. The end result is a taper ratio of 2.1, perfectly acceptable.
That anchored lunar elevator concept is especially exciting, as it would basically be a highway into space. However it would weigh so much that a lot of infrastructure would need to exist in order to build it. Maybe it should be the thing in the Phase 3 Mission 1 section, instead of taking over the world. I was reading a bit about this the other day and it was mentioned that because the Moon's orbit isn't exactly circular, and other variants such as the tug of the sun, Jupiter, and Venus, and irregularities in the gravitational field of the Moon, the EML1 point moved around a fair bit. I went and found where - it was this Orbiter forum thread (where Hop David shows up, surprise, surprise). So the station will require significant orbital maintenance and will need to move continuously to stay in the right relationship to all the elements. But it still is a much easier project than an equivalent elevator from the surface of the Earth.
The rotating 'skyhook' version, credited to Hans Moravic, allows for a system with much smaller mass that does something similar on a much smaller scale. It can be in a low orbit around the Moon and by rotating the whole structure as it orbits, one end can touch the lunar surface with no relative velocity.
The clever additions by Robert Hoyt to this design allows for a tether 200 km long with a counterweight on one end and a travelling station or hub that moves up and down the tether to transfer momentum from the Moon to the Lunavator. (That's what he called his specific design, though i agree it's so catchy we should just use it for all lunar tether systems.)
Not that i have yet read his paper properly, but i can't find what material he was using in his calculations. I imagine it was Zylon. At any rate, the tether mass in his design is listed as only 4706 kg. (I like it that he didn't round that to 4700 kg.) It is so light that just one Falcon Heavy could launch not only the tether but also the rest of the system, considering that there is no need to land on the Moon. So maybe the first Lunavator should indeed be done that way. Still - there is soooo much basalt to be had on the Moon, and it may be up to the job, if not as good as Zylon. Sure, it is twice as dense. But getting it where you want it takes about a fifth of the delta v.
There are few sources on basalt tensile strength and what they say is highly variable. The particular source of basalt makes a big difference, from what i've read. This page by Dr Alexander Novyskyi seems to give a fair treatment of the subject. There is a chart on that page shown below that makes clear how much the particular basalt used and the process technology affect final strength.
That product 2nd from bottom has a pretty good tensile strength, the two above it aren't too shabby either. And note this quote from the accompanying article:
'An important step in obtaining quality is melt degassing process which involves the removal of gas held in the melt. This process takes place upon exposure of the melt at temperatures above 1720 K. The duration of exposure is about two hours.'
Degassing you say? Well, the Moon will have no problem with that at all. In fact, this brings up again the speculation that lunar glasses may be significantly stronger than Earth glasses in general because they are completely anhydrous. This article by James Blacic rather gushes about that. (There is a lot of other fascinating stuff in that booklet too.)
Which development route to take... much to ponder...Sigvart:
The whole rotating tether set-up have momentum, and we pick up something that stands still on the surface. Afterwards, the combined mass of the two must have orbital velocity, in order to not crash into the ground. We want the resulting orbit to be circular. We can manage to do that by either drop mass when picking up the payload, or initially having an elliptic orbit. Of course, we can chose to only lift payloads that are much smaller than the tether mass to approximate the orbit as a circle. The core principle does however remain, we balance cargo up and down from the surface.
If the tether orbits at an altitude of 100 km, the end of the tether has to rotate at 1720 m/s in order to be able to catch the payload. Let us see what the acceleration is then:
(1720 m/s * 1720 m/s)/100000 m = 29.6 m/s^2
That is over 3 g. Ouch.
And when we pick up the payload, we have lunar gravity too. (In total 31.2 m/s^2)
This is great opportunity to present another fundamental property of tethers:
Tether mass is proportional to tether-end acceleration. (for the same payload)
So even if a skyhook is shorter than an anchored tether, it has to be much ticker.
Next: Taper ratio.
The potential (in m^2/s^2, Yuri) caused by the rotation is simply the tether length times half the tether-end acceleration. For the low lunar orbit tether:
29.6 m/s^2 * 1/2 * 100000 m = 1.48 mega-Yuri
Quite substantial. In comparison, the potential caused by the lunar gravity over the 100 km is just 0.155 mega-Yuri.
Although a smaller item, we must consider that too. Taper ratio with Zylon fibre:
R = e^(1.64 MY / 3.7 MY) = 1.56
An improvement over the anchored tether, but the almost 20 times larger acceleration causes a 20 times larger cross section.
A longer tether, for example orbiting at 200 km, has a tether-end acceleration of 15.65 m/s^2 and a taper ratio of 1.58 (very close to the ratio for 100 km).
500 km: acceleration 6.48 m/s^2 and R of 1.65
In conclusion, the mass of the skyhook tether is almost independent of the orbital altitude (unless very high or low), but a higher altitude reduces the acceleration giving more time to make the catch.