There have been many suggestions about a space tether being used as a slingshot to sling one satellite away from another, suggestions about a space tether being used as a docking device to bring one satellite toward another, and some proposals to use a tether line to hoist materials into space from the surface of a planet. These proposals are impractical because of Coriolis forces that result from the fact that the space tether is actually spinning because the tether is in orbit, going around and around the planet. If a space tether draws two satellites together, the satellites wind up spinning around each other, rather than simply becoming docked together. If the tether goes slack for any reason, then it will become un-manageable and will probably break when the tether is pulled into tension again. If the two tethered satellites are drifting apart and then the tether suddenly becomes taught, the tether will snap and break into two or more pieces.
The Frankenstein satellite has a motorized reel that will take up any slack that develops in the space tether, and the reel will also let out some more tether length if the tether tension forces become too large. The tether management system is a heavy and complex part of the buoy itself. In the simplest model, the tether goes through a Lagrange point called L3. This model supposes that the Earth and the Moon are stationary. Each space buoy acts an anchor for the other space buoy. The two buoy / anchors are connected by a space tether. In this model, the Moon-side buoy has more mass, so that the gravity force on the Moon-side buoy is equal to, and opposite from, the gravity force on the Earth-side buoy.
The Moon-side buoy hangs down one-half kilometer from L1 toward the moon, and Earth-side buoy hangs down one-half kilometer from L1 toward the Earth. The two buoys and the L1 Lagrange point have the same orbital period as the Moon. As long as the tether remains intact, the tether and the two space buoys will hang up there in space, always between the Earth and the Moon, neither rising or falling even though the space buoy's velocity isn't enough to keep a normal satellite in orbit! If one end falls down toward the Earth, the space tether lifts the other end up from the Moon more quickly, and we all know that things don't fall up as easily as they fall down . The motorized reel in Frankenstein's Brain (the Earth side buoy / anchor) will cleverly let out some tether length or reel some in, based on calculations to make the space buoy's orbit more stable.
There are a few reasons why we can't use L1 as a location for the space tether buoys. The space in L1 is already taken, or claimed, and exists as a theoretical position for a K13 orbit that goes around the L1 Lagrange Point. L1 itself is unstable, but it is possible to orbit around L1. The K13 orbit (not shown) is a stable orbit that is always in Earth's shadow, always in darkness hence it is an ideal place to put a space telescope. Such an orbiting telescope would collide with and pair of space buoys placed at L1.
Lagrange point L5 is inherently unstable. L5 exists in the direction that the Moon is orbiting, in other words L5 is out ahead of the Moon and L5 jumps around a little bit, something like sea foam on the bow wave of a ship moving through the water ocean. Just as the bow wave of a ship can go higher, lower, left or right, so does the "gravitational bow wave" jump around ahead of the Moon as the Moon moves along in Lunar orbit, and the exact location of Lagrange point L5 walks around the theoretical location of L5.
Lagrange point L4 trails along behind the Moon, hence is inherently more stable. We will place our space buoys and space tether near Lagrange Point L4. The placement is "near L4" rather than "at L4" for two reasons. One reason is that the tether runs through a saddle point in space, and the notion of which-way-is-down is kind of complex. If we were to go aboard Buoy 2, we would be able to stand up in microgravity, and we would notice that Buoy 2 could fall down toward the Moon if the tether would break or become cut apart. Because the gravity field lines (not shown) are curvy and squiggly near the four dimensional saddle point, gravitation forces cause the Moon-side buoy / anchor to hang "down" in a direction that is not directly toward the Moon. To a lesser extent the Earth-side buoy / anchor to hangs "down" in a direction that is not directly toward the Earth. The tether itself represents a straight line in the fourth dimension.
The second reason that the placement is "near L4" rather than "at L4 is: Adding mass to a Lagrange (or saddle) point will move the Lagrange point in space. A massive Lagrange point location is slightly different than the theoretical massless location of a Lagrange point.
Most astronomers won't look at a star through L4 or L5 with a telescope, because starlight becomes refracted when the starlight passes through a Lagrange point hence the image of the star seems to dance around a little. The notion of a straight line, such as a light ray, can be changed if the light ray passes through a Lagrange point. For some reason, astronomers have been ignoring this effect for nearly 400 years, ever since the telescope was invented. Perhaps the older astronomers among us are frightened of observing the effect that there could be two or more straight line path ways from a distant star to a localized telescope or observatory. We are modern enough to recognize that a Lagrange point is actually a little bit of warped space. This warped space occurs naturally and this warped space (also known as a Lagrange point or saddle point) has been there as long as the Moon has been there which is more-or-less forever. Warped space is nothing new and has existed as long as there have been planets orbiting stars. When we look through a Lagrange point and see the light from a distant star, we understand that looking at or through a Lagrange point can result in some kinds of optical illusions. These optical illusions are harmless.
If you want to be "Searching For Frankenstein's Brain" and you don't have access to the Deep Space Radar Station, it is possible to use a high-powered telescope to search for a man-made object that follows the moon, near L4 in the Lunar ecliptic plane. A ten inch reflector telescope might be big enough to see a hint of it passing in front of the distant stars. After you launch into space, your space capsule radar (yes it has one) will easily see the radar reflector on the Earth side buoy / anchor known as "Frankenstein's Brain". Pilot your space capsule to the space buoy, then along the space tether, which represents a shortcut through space; Escape velocity on a path along the tether is approximately 44 KPH slower than escape velocity on a path that doesn't pass through a Lagrange point.
Fin.
