Vertical Lunar Tether In A Polar Orbit
This sky hook is a gravity gradient stabilized vertical tether. It's in a polar orbit so it will pass over the poles as well as the lower lunar latitudes.
Unlike an equatorial orbit, there are only two occasions during a lunar orbit where a tether's Vinf velocity vector is anti-parallel to the moon's velocity vector. So launch windows to earth would only occur each two weeks. That's still pretty often. These occasions are also good times to rendezvous with the tether.
Playing with earth moon three body simulations, polar orbits seem to remain stable up to a radius of around 20,000 kilometers. That is where I will set the anchor mass at the balance point of this sky hook. I believe this is far enough above the lunar surface that the mascons won't damage this tether's orbit.
Asteroid Anchor Mass Via a Keck vehicle
What to use for the anchor mass? With the asteroid retrieval vehicle proposed in the Keck Report, it is possible for a vehicle of moderate mass to retrieve a much larger mass to the earth moon neighborhood. The Keck authors believe a rock could be placed in high lunar orbit for around .17 km/s. A lunar orbit with a 20,000 km radius has a speed of around .5 km/s. I believe it would take around .7 km/s to park a rock in the orbit we want.
The Keck vehicle includes solar panel arrays and Hall ion thrusters. These would be great to have on a vertical tether. It takes awhile for ion engines to impart momentum, but given time they're about ten times as efficient as the best chemical rockets. A tether can build up momentum over time but release it suddenly. Thus they are a good way to enjoy an ion engine's great ISP and an Oberth benefit.
As well as adjusting the tether's orbit the Keck vehicle's solar arrays might also power elevator cars moving up and down the tether. If water is exported from from the lunar cold traps to the tether, the arrays might also crack water into oxygen and hydrogen bipropellent. There are a number of possible uses for this power source.
The tether length above the anchor mass can be built in increments. I imagine the tether growing longer and more able with time. Here are three possible stages:
To EML2 or EML1
EML1 and 2 are about 65,000 km from the moon. To reach this apolune, we'd need an upper tether length of about 2700 kilometers. Using Wolfe's spread sheet, this tether length has a taper ratio of 1. With a safety factor to 3, tether mass to payload ratio is about .02.
This is pretty good. I believe this low stress tether length could accommodate copper wires to transmit power to the elevator cars.
Once at apolune, I believe it would take about .3 km/s to park the payload at EML2 or EML1.
EML2 is a good staging location should we want to travel to and from destinations beyond the earth-moon neighborhood.
To a Perigee at Geosynchronous Orbit
Transfer orbit from GEO to the moon is about a an ~36,000 x 378,000 ellipse. Apogee speed is about .45 km/s. The moon's speed is about 1.02 km/s. So the tether needs to hurl a payload to a Vinf of (1.02-.45) km/s or about .57 km/s.
To achieve this Vinf our tether needs to be 12,200 km. Zylon taper ratio is 1.09. With a safety factor of three, Tether to payload mass ratio is about .167. So a ten tonne tether could accommodate a sixty tonne payload. This is still pretty good. A power cable along this length is also doable.
Perigee velocity of our transfer orbit is ~4.13 km/s. Geosynch orbit velocity is ~3.07 km/s. If the transfer orbit and destination geosynch orbit are coplanar, geosynch circularization would be about 1.06 km/s. But I expect that would be the exception rather than the rule. If the orbit inclinations differ by 20º, 1.6 km/s would be needed to park in geosynch.
To a Perigee at Low Earth Orbit.
A 300 x 378,000 km orbit has apogee velocity of ~.19 km/s. (1.02 - .19) km/s = .83 km/s.
To throw a payload to a trans earth orbit, our tether needs to impart a Vinf of .83 km/s. This takes a tether length of 19,200 kilometers. With a safety factor of three, Zylon taper ratio is 1.2. Tether to payload mass ratio is .38.
If perigee is through earth's upper atmosphere, aerobraking can provide a large part of the 3.1 km/s delta V for circularizing at LEO.
Again, the tether length below the anchor mass can be built in increments. Incremental growth with time is more doable than trying to do the whole length in fell swoop. Here are some possible steps along the way.
To a Perilune at Low Lunar Orbit.
To drop a payload to a 90 km altitude perilune, length needs to be 7360 km. Given a safety factor of 3, Zylon taper ratio is 1.06. Tether to payload mass ratio is .15.
Velocity of transfer orbit's perilune is about 2.2 km/s. Low lunar orbit is about 1.6 km/s. It'd take about .6 km/s to circularize at low lunar orbit.
To the Moon's Surface, Impact Velocity 1 km/s.
If the tether is extended to a length of 17890 km, tether foot altitude is about 370 km. Dropping a payload from this tether foot would result in a 1 km/s impact.
Given a safety factor of two, Zylon taper ratio is 2.88. Tether to payload mass ratio is 26.87.
Note the safety factor is less than in the other scenarios. As we descend further into the moon's gravity well, stress climbs more rapidly. It would be more difficult to include copper wires for power along the lower parts of the tether.
To a Tether Foot Just Above the Moon's Surface.
Dropping the tether foot to an altitude of 10 kilometers gives us a length of 18,252 km. Safety factor of 2 and Zylon taper ratio is 3.72. Tether to payload mass ratio is about 51.
Dropping from this tether foot, a payload would impact the lunar surface at .184 km/s.
A .2 km/s payload delta V budget for soft landing seems quite doable. Likewise it would take about .2 km/s to launch a payload from the lunar to rendezvous with the tether foot.
However dropping the tether foot this far is considerably more ambitious than the other scenarios described above.
Travel About The Moon
Kim Holder noted such a tether might serve as transportation between locations on the moon.
Without a tether, going from pole to pole would take about 3.4 km/s: 1.7 km/s to launch and another 1.7 for soft landing. Going from equator to pole would take 1.53 km/s to launch and another 1.53 km/s for a soft landing, totaling 3.06 km/s.
So a 18,000 km lower lunar tether length would make travel about the moon easier.
A Location to Process Asteroid Ore
It takes about .6 km/s to park ore from some of the more accessible asteroids in 20,000 km lunar orbit. If rendezvous with the tether top is doable, it could take considerably less.
I envision infrastructure accreting about the tether anchor mass 18,262 km above the lunar surface. Water, platinum, gold, rare earth metals, and other materials could be extracted at the anchor. Refined commodities could climb to the top of the tether and then tossed earthward.
A Synergy Between The Moon and Near Earth Asteroids
Moon and asteroid enthusiasts are often at odds with one another. They should be allies. In terms of delta V, it's a lot easier to park asteroids in lunar orbit than lower earth orbits. And given growing infrastructure in lunar orbit, the moon's surface becomes more accessible.