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A colony structure like this would be easier to build if one could eliminate the mirrors and windows. This could be done by replacing the windows with a big optical diffuser running down the core of the cylinder. This would be made from ultra-light holographic membrane. A relatively small window in one end cap could provide entry for light collected by a much larger Sun-facing, non-rotating solar collector mirror in the manner of a Cassegranian teoptics telescope

6.1 Rotating Space Habitats
Suitable locations for space colonies include orbits around the Earth-Moon Lagrange points. Consider a large
fat cylinder rotating on its axis (Fig 6.2); a landscape of seas, plains and mountains girdles it within, while
sunlight, reflected from external mirrors, enters through the end caps.
Paper read to BIS/CEMS Syposium "Bringing Worlds to Life" at Birkbe... http://www.paulbirch.net/CustomPlanets.html
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The total mass of the habitat works out at around 40 tonnes per square metre of habitable land, more or less
equally divided amongst atmosphere, crust, geosphere base and toposphere (Fig 6.3).
The geosphere base uses a low-density fractal architecture to keep the structural mass low, slotted together
out of strong but readily available fractal blocks of fused rock (Fig 6.4). This is a constructional technique
equally suitable for supramundane planets and rotating space habitats; it provides strength in tension as well
as compression whereas normal masonry construction only has strength in compression.
The maximum size of conventional rotating space colonies is determined largely by the strength of available
structural materials. The limit is about 250 km radius for quartz, 1000 km for sapphire, and 2500 km for
diamond. That last case is actually pretty big, a colony equalling the land area of the Earth. Even so, most
conventional space colonies will be considerably smaller than natural planets.
To some extent we can get around the size limit by making colonies longer, because the constraint is on the
radius not the length. An extreme example is the macaroni habitat (Fig 6.5), an ultra-long habitat with a
sunshine tube down the middle, if we loop it all the way round a star the total habitable area may be around
2000 times that of the Earth. And if we stretch a macaroni tube along tramlines between one star and another,
the habitat area becomes something like 100 million Earths. There is sufficient material in the solar system to
construct macaroni habitats thousands of light years long.
Even the largest diamond colony or the longest macaroni habitat is not a planet, though, because planets have
horizons and you live on the outside, whereas space colonies have the outside on the inside!

I think the hill sphere of Titan would be greater than the 24-hour orbit radius, so you can just have the orb orbit the moon. Anyway if placed at L1 with a moon that is tidally locked would create the same situation that supposedly exists on Proxima b.

lily-pad city floats inside a gigantic and nearly-invisible sphere in Saturn's upper atmosphere; a balloon kilometers across with a shell of fullerene-reinforced diamond below and a hot hydrogen gas bag above. It's one of several hundred multimegaton soap bubbles floating in the sea of turbulent hydrogen and helium that is the upper atmosphere of Saturn, seeded there by the Society for Creative Terraforming, subcontractors for the 2074 Worlds' Fair.

The cities are elegant, grown from a conceptual seed a few megawords long. Their replication rate is slow (it takes months to build a bubble), but in only a couple of decades, exponential growth will have paved the stratosphere with human-friendly terrain. Of course, the growth rate will slow toward the end, as it takes longer to fractionate the metal isotopes out of the gas giant's turbid depths, but before that happens, the first fruits of the robot factories on Ganymede will be pouring hydrocarbons down into the mix. Eventually Saturn — cloud-top gravity a human-friendly 11 meters per second squared — will have a planet wide biosphere with nearly a hundred times the surface area of Earth. And a bloody good thing indeed this will be, for otherwise, Saturn is no use to anyone except as a fusion fuel bunker for the deep future when the sun's burned down.

The lily-pad habitats have proliferated, joining edge to edge in continent-sized slabs, drifting in the Saturnine cloud tops: ... The green and pleasant plain stretches toward a horizon a thousand kilometers away, beneath a lemon-yellow sky. ...

if you go outside and you die, then the world is not habitable.

I like having an outdoors, other people, they like living in tunnels.

I define terraforming as not needing a wall to keep inside your habitable environment.

You can talk about domes all you want, but that is not really terraforming.

Being that far out from the Sun, its rotation rate makes no difference. Since its diameter is about 2200 km and its maximum distance from the Sun is 97 AU, if we build a mirror that is 220,000 km in diameter, we can focus enough sunlight on it to give it an Earth like level of solar intensity. if you want to know how big this is, it is bigger than Jupiter. I think Eris would be a candidate for an artificial Sun, probably fusion powered. Building a mirror this big is probably a waste of time and resources.

The density of liquid Tungstein is 17.6 g/cm3 or 17.6 tons/meter^3
Volume of the sphere is 628143703822703 meters cubed times 17.6 tons = 11,055,329,187,279,572.8 tons of Tungstein or 1.1e16 tons.
Now the question is what's the best way to heat this molten ball of Tungstein to the temperature of the surface of the Sun, the great thing about liquids is they hold their volume, rather than expand as a gas, in a condition of microgravity, surface tension will keep it to a ball of white hot liquid. So what's the best way to heat it? Perhaps a laser, a very powerful solar powered x-ray laser, target this ball with it, but avoid hitting Titan and this ball of molten metal and illuminate one hemisphere of Titan heating up its surface to Earth like temperatures. This leads to another problem. Much of the surface is ice, not only water ice, by dry ice, its nitrogen atmosphere will probably get a lot of carbon-dioxide as the dry ice on its surface sublimates, the water ice will eventually melt and form an ocean. The good news is there are a lot of carbon compounds in Titan's atmosphere and crust, through chemical rearrangement, they can be formed into artificial rock, to form a shell in the shape of the topography of this Moon.
http://images.nationalgeographic.com/wp … 00x450.jpg

In total, at the distance of 15,000 km, it would have to be 338.24 kilometers in diameter, this will take into account the fact that the mirror will have to be angled at 45 degrees to reflect incident light rays from the Sun, so this reduces its apparent diameter to 0.7 of its actual diameter. At a distance of 15,000 kilometers the orbital period is 8 hours and change, if we want, we could increase the orbit to 24 hours, but then we'd have to increase the size of the mirrors too. the size of the mirror would have to be proportional to the mirror's distance from the Earth. If we increase to orbit to 35,870 km we would get a 24-hour orbit, if we did this however...
35,870/15,000 km = 2.39133.
2.39133 * 338.24 = 808.84 km
We'd have to increase the mirror diameter to 808.84 kilometers the Sun would then rise on a 24-hour schedule, it could be synchronized with the appearance of the real Sun to produce twice as much sunlight, the mirror doesn't have to be 100% reflective of course, and the real Sun when it appears is at a low angle, heavily filtered by the atmosphere, so it won't be quite twice as much as we see the Sun at high noon in the New York area for example. You probably do want to warm it up more to compensate for the cooling effect of the surrounding glaciers.