The wider an L2's shield, the more space the cold colony behind it has to play with. For Venus, the shield is ice; a rocky world might end up with various regoliths and slags. The wider the shield, the more energy it delivers to the ion drives and, also, the drives are further out to provide more balance against L2 instability.
For Venus I've got polar stations to beam power at its outer rim and whatever is behind it. From Venus (and Jupiter) also-also, more ions can be caught from the wind. (None of this matters to the polar stations which can just shift their beams appropriately.) Earth will enforce strict mass limits on what is floated in all its relevant Lagranges excepting STL3-5. Mars nearly already has a SML1-2 in high-flying Deimos. Jupiter is more likely to get a SJL1 if we ever get around to that. So: another Venus post.
As the shield is widened, a wider colony torus can be inserted closest to it. This pushes the older, smaller toroids back - as segments on a tapering worm. Back there they are all in position to take the polar relays' energy.
The L in "L2" is for "libration"; the front is always moving. The back segments are connected by cables and padded with foam lest they scrape up against one another. Once the back segments leave this halo their tail shall curl behind L2's orbit which is its planet's orbit, moving less. Eventually the tail shall leave the shield's shade. Segments once outside the shade may post energy panels facing the annular solar disc.
Against the tail's drag the ion drive up front must, as well as pushing toward the planet, also push toward its orbital direction.
All sections must depend on the kindness of another station, and further back they must fear (or plan!) that the cables will be cut. The Coriolis starts in for Earth/Venus natives forced in more-cramped toroids. At least they won't rock about as much as the middle segments might. I think the lower classes will cluster back there...
By "lower classes" I mainly mean Martians, then Lunars and Belters. If the labour force is a mile or more from the ice, the central zero-G spine will be a pneumatic tube to get the workers to and from the ice. That's easiest if the links all spin at the same rate. As the toroids toward the ice get larger, they spin slower. If all the legacy links are spinning slower, less Coriolis... but also less G. Beyond a certain point I expect to detach the last link(s): promoted to the front as a hub-spoke with a new rim, or sent elsewhere in orbit or even the Solar System.
Eventually I'd sequester some of these links for non-industry, non-habitat purpose. The water-work is seasonal, for Venus mostly on the Earth/Luna timetable. These planets do have amenities to house retirees, aerostats for Venus. L2's middle links, so depopulated, can include a university and all its labs.
[MOVED: I did most of this Sunday noonish under the "Building" section. I'm separating the act of building this station from the nature thereof, and abstracting it for other planets.]
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