It is the skyhooks that allow ships to move around the solar system using a tiny fraction of the fuel that would otherwise be needed. The skyhooks that greatly reduce the time needed to move between worlds, and extend the launch windows when launches can happen so much, that if the spatial relationship between worlds is really unfavorable some fuel can be burned and they can be reached anyhow, making it possible to launch almost any time. The Earths's skyhook turns launching to orbit from here into a job that, compared to rockets today, can be done by a ship both giant and robust that can make the trip thousands of times, without needing to take off with such fury it is like carpet-bombing the launch pad, and shaking anyone in it to near unconsciousness[?]. The Moon's skyhooks allow shuttles to move between any two points on its surface for about a quarter of the fuel otherwise needed. Each of the worlds where skyhooks are built receive this same benefit. Skyhooks change everything. They are the future, and the sooner we really understand that, the sooner we can move towards that future.
The sort of skyhooks we are talking about here are vertical skyhooks. Think of them as junior space elevators. They have a lower tether hanging down towards the surface of the body they orbit from a large anchor mass, the same as space elevators do. They just aren't attached to its surface. The platform at the bottom of their cable (the foot) instead moves above the surface at some suitably low altitude. Ships launched from the surface need to catch up to the foot, and to brake to return to the surface. The advantage is the delta v required is much less than what is needed to get to orbit, because the foot is moving much slower than orbital speed. For the first skyhook planned for the virtual colonies, it takes 1/5th of the normal delta v. That is the case if the foot hangs to 20 km above the surface and the anchor mass at the center of gravity orbits 5000 km up.
Skyhooks also have an upper tether extending upwards from the anchor mass deeper into space. The tip of this tether experiences the opposite effect from the foot - it is moving faster than orbital speed at its altitude. The effect is that objects holding onto the tip feel a force pulling outwards, like if you twirled a ball on a string above your head. The amount of that force depends on how much faster they are moving than orbital speed. As things climb the tether from the anchor towards the tip, that force appears and slowly grows. Again the opposite effect is felt climbing from the foot towards the anchor. At the foot, the pull of gravity is almost as much as on the surface. At the anchor, gravity is imperceptible because it is moving at orbital velocity. The closer you are to the anchor as you climb the tether, the closer your velocity is to orbital velocity, and you feel gravity less and less.
Essentially there are two things that have to be managed for skyhooks to work - angular momentum and cable strength. Angular momentum is how fast a skyhook is moving in its orbit. As payloads arrive and move along the tethers, the momentum of the whole skyhook is changed. It will slow down, which will cause its orbit to drop, or it will speed up, causing its orbit to rise.
There are several options for correcting this, explained in the expanding section below. It's pretty detailed. The short version is that to minimize the amount of fuel engines on the anchor and at the tip of the skyhook use, payloads travelling down the tethers need to be balanced by payloads travelling up them. When this isn't enough, dummy masses can be moved up or down from the anchor and released such that they enter elliptical orbits that accompany the skyhook. Later they can be retrieved and returned to the anchor. Both activities will affect the skyhook's angular momentum and can be carried out when convenient. The essential thing is to increase the mass of the anchor over time until it is thousands of times as massive as any payload that moves along the tethers. Then its angular momentum is affected so little by those movements that it takes a long time for changes to accumulate to the point where it affects operations. Thus there is lots of time to make changes with passive techniques instead of using engines.
Maintaining Skyhook Orbits
The speed of a skyhook's foot is determined by the altitude of its orbit, which is located near its anchor mass. The pull of gravity decreases as something gets farther from a world, so orbital velocity is slower for the anchor mass than for something 20 km above the surface. Plus, the foot is always aligned vertically under the anchor mass - it's hanging from it. So, it takes the same amount of time to orbit the Moon once as the anchor mass does, but the distance it covers is far less, because it is travelling along a much smaller circular path. The animation in the sidebar gets this across.
So now think about this - as something moves down from the tip, it feels less and less of a force pulling up, and as something moves up from the foot, it feels less and less of a force pulling down. Where does that energy go? Energy doesn't just disappear, any more than mass does. The motion of the whole skyhook changes so the energy remains the same. When a car climbs to the anchor, as it feels less downward pull, a pull to the side is experienced by the tether. That pull is in the direction opposite its orbital motion around the Moon (which is called the retrograde direction, and the direction of orbit is called prograde). That's because the car started out moving much more slowly prograde, when it was at the foot. So, the tether is dragging it prograde as the tether's prograde speed increases. The end result is that the anchor mass experiences a net pull retrograde, which slows it down. The amount it is slowed down depends on how much more it masses than the car does.
Managing a skyhook is far easier if you have a nice, heavy anchor mass so big that the momentum it has dwarfs the momentum changes caused by cars moving up and down the tethers.
The nice circular orbit it needs to have then moves very little as cars ferry things along the tethers. The feet of the skyhooks in the lunar constellation travel only 20 km above the surface, and there is 5000 km between them and the anchor station. If the anchor mass slows by only a few meters per second, in a few hours the foot platform will slam into the ground and be destroyed. If the situation is not quickly corrected the whole skyhook could be dragged down and crash.
The anchor of a skyhook must have engines on it (or even better, the skyhook tip must have engines) to speed up the anchor to compensate for the cars slowing it down (or speeding it up, as happens when a car descends from the skyhook's tip - that's a problem too). But, there are other ways to manage a skyhook's momentum budget that use no fuel at all. The first step is to try to balance incoming and outgoing payloads.
Just as a car climbing from the foot to the anchor slows the skyhook down, a car descending from the tip to the anchor speeds it up. And each of these processes in reverse does the opposite thing - when a car goes from the anchor to the foot, or the anchor to the tip. So, the best time to have a car descending from the tip, is when there is a car climbing from the foot, and that way they balance each other out.
The ships coming in from other worlds are far larger than the shuttles coming to the foot platform. So, to keep things balanced, you need to ferry the cargo on the interplanetary ship downwards bit by bit, while many shuttles bring cargo to the foot and bit by bit that gets ferried up. With the really huge ships at the end of the timeline, it could take weeks to complete this operation. It takes so long, it's best to slowly ferry the interplanetary ships that dock with the skyhook tip down to the anchor mass, even as they are also being unloaded, so that other ships will be able to dock with the tip or launch from it sooner.
This reduces a lot the need to use rocket engines to correct the skyhooks' orbits, but it won't do it all. There are lots of reasons why the traffic won't always balance in a timely manner. Also, in Phase 4 of the timeline the skyhooks are turned into specialized pairs of skyhooks, one the launch skyhook that mostly handles interplanetary ships, and the other the surface skyhook that mostly handles traffic to and from the surface. Loads can still be balanced as the cargo moves between the two skyhooks in each of these pairs on ships called hoppers, but it adds another layer of complexity to the job. It is a good thing that the cars on the tethers and the hoppers are completely robotic and take direction only from the anchor's navigation computers, short of some human override. Those computers constantly monitor orbit closely and calculate when and where all the cars and hoppers need to be based on that, and the schedule of ships arriving and departing.
Those computers have other tricks up their sleeves when all this orchestration isn't enough. If there isn't a ship or a car in a good place to move in some way to balance things, then a simple rock will do. A suitable supply of really big boulders massing many tons is stored on the anchors of the skyhooks, which have to be as heavy as possible anyhow, and don't have a direct effect on the loads being borne by the tethers themselves, as they are at orbital altitude and aren't being pulled in any direction by the forces we're talking about.
The rocks do need to be ferried by cars to where they do their work, but at least there doesn't need to be any actual payload with a destination involved. And they don't need to move all the way to the tip or the foot - that would complicate traffic of actual goods and be much slower. They are just ferried up or down by up to a fifth of the length of a given tether (which can mean up to 7000 km on the longest tether), and then released. A rock released from below the anchor has been pulling the tether prograde as it descends, causing a small net increase in the orbital speed of its skyhook. At the altitude it is released, it enters an orbit around the skyhook's world. That orbit will be an ellipse that comes closest to its world a quarter orbit after it was released. As it gets closer, it speeds up, and as it gets farther away, it slows down. Eventually it comes back to the point at which it was released, and can be easily captured. It will come back with the same speed at which it was released, which is to say, no speed relative to the skyhook tether at that point, and it will brush by it nice and close. If it isn't convenient to grab it at that moment, it will continue to return like that, and it can be caught on another pass[?]. Then it is ferried back to the anchor to be reused another time. The same effect happens when a rock is released from above the anchor station, except that makes the skyhook slow down.
What this would look like from the perspective of the skyhook, is that the rock first drops downwards, because it isn't going fast enough for orbit where it is. As it drops it gains speed. Long before it hits the ground, it has acquired orbital speed. To the skyhook, it is moving faster than the tether at its altitude, and it pulls out in front of it. It is moving fastest when it is 90° beyond the point in its orbit where it was dropped, and is also at its lowest altitude. The orbital dynamics add up to the rock moving in a circle aligned edge-on to the tether, a few thousand kilometers across, to observers on the skyhook. Its orbit around the Moon is an ellipse. It takes the same time for it to go around the Moon as the skyhook does, but it moves fastest when closest to the Moon, slowest when furthest away, in accordance with Kepler's 2nd Law.
There could often be a small swarm of such rocks accompanying a skyhook, gently looping around and almost kissing it each time they pass by. That means they are navigation hazards. They will need lights and radio beacons. Also thrusters sufficient for emergency collision avoidance maneuvers (rarely necessary), and to line them up properly with the skyhook (which might be necessary if they have been looping around for weeks - there are always little factors that perturb orbits).
A variant on this trick is occasionally tossing rocks between skyhooks, especially between the skyhooks of the Moon and Earth. The skyhook that throws one loses some momentum, and the one that catches it gains some. Really, the main thing is to keep building up the mass of the anchors over time, whenever possible. When asteroid mining grows to a major industry, this is easy. The more momentum is stored up in a giant anchor mass, the less a skyhook's speed is affected by the puny ships travelling up and down.