Life on the Broken Ring - an issue of size
One of my ongoing projects is what I think of as the "constructed worlds gallery", a series of Megastructures as settings for stories and games, including things like the "Flying Pie-plate" a world sized dish as suggested by Larry Niven as the starting point for the construction of the habitat in Ringworld, an Alderson Disk galactic lifeboat, and a trefoil mobius knot magicked up by a group of Dragons on the run from Cthulhu. Mainly I like playing with the implied geophysical issues that would otherwise make such structures unhabitable relatively quickly, the strange environments that result from solving them, and the everyday life of their dwellers.
My latest project is a piece, or rather pieces, of a broken Ringworld but I'm having trouble working out how large the habitable zone on such objects will actually be so given the assumptions below...
My understanding is that under the conditions listed below the side walls of the original ring continue to perform their role and the curvature of the ring segment will eventually be sufficient to have a similar effect.
Question: How long do the chunks of a broken Ringworld, per the original design specifications, need to be for the maximum sized pocket of atmosphere to remain in the construct and how big will it be?
Assume that:
Apart from being separated from their neighbouring ring sections the pieces are otherwise intact.
The pieces in question have broken across the width not along the length of the ring so both sidewalls are intact.
The sections are under the standard 0.992 gee acceleration. This is thrust induced, necessary to hold them in an orbit closer to their primary than it should be at its orbital velocity.
They are the in the same Goldilocks orbit they were built for, or something similar.
I know that the scenario creates appreciable engineering challenges in set up, those are entirely out of scope for this question.
Also please note that while the ultimate result of this question may well be a matter of relatively simple math it is primarily concerned with getting the right math to work from (thus the hard-science designation) as I have tried this twice, using different approaches, and gotten consistent results that differ by several orders of magnitude and neither of which looks right when drawn to scale.
It appears that certain assumptions are being made that are not valid let me clarify:
This is not a ringworld in its original state.
This is not even necessarily a ringworld in its original setting.
Think of broken pieces as salvaged objects that have been repurposed as mega-habitats.
This question is not at all concerned with the fact that this scenario falls outside our current understanding of physics this is a matter of whether the construct as described can hold atmosphere and if so how much.
This post was sourced from https://worldbuilding.stackexchange.com/q/149706. It is licensed under CC BY-SA 4.0.
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I'm going to take a bit different approach to this based on the revisions... unfortunately I don't have math to back this up, but it should at least get you started thinking in a useful direction. (So far, I haven't seen anyone else accounting for the chunk's own gravity, and my own calculus is too rusty, so I asked P.SE to help us.)
Assuming you are talking about a relatively small chunk of ring (say, 45° or less; bigger than that, I expect the orbital mechanics just get... interesting¹), you are essentially dealing with a really oddly-shaped planetoid. In particular, this chunk has to be in orbit around the star, because otherwise it is almost by definition not going to stick around very long.
(¹ The whole ring is easy to reason about because it is balanced and many forces will cancel out. This is also why a whole ring can mostly ignore gravity and spin as fast or as slow as you like, subject only to torsional stresses.)
This being the case, unless your chunk is really close, relatively speaking, to the star, or really big in relation to its orbital distance, you mostly only have to worry about the chunk's own gravity, because the other forces are going to be comparatively weak. (Just as the Earth doesn't lose its atmosphere due to centrifugal force.)
What you have, essentially, is a tidally-locked planet (i.e. in a fairly "normal" orbit, with zero axial tilt and a rotation period of exactly one revolution per orbit). Note that this has to be the case, because, unlike a complete ring which is gravitationally stable at any rotational speed, a fragment on its own is either in a regular orbit, or, by definition of not being in an orbit, isn't going to stick around your star for very long.
At a large enough size (and we're almost certainly talking about such a size), we are mostly looking at atmosphere retention being primarily a function of the segment's gravity. A 1/300 chunk has roughly Earth-mass, and that's slim; Ringworld's width is 1/625 it's circumference, so we're talking about a chunk only twice as "long" as wide (and you said we aren't cutting it along the width).
This is where things get weird and hard. Because your segment is curved, your point of maximum gravity is going to be above the surface... but because of the distances involved, they aren't going to exert much effect. The strongest gravity is going to tend to 'hug' the surface to a fair degree.
Now... I haven't developed my model far enough to actually prove this, but I think you ought to be able to simply park an atmosphere on the thing, and it will mostly stay put just via gravity. Doing things like bending the walls in may actually make things worse, since, as others have noted, that shifts the point of maximum gravity up away from the surface. Moreover, it's going to exacerbate a problem you're going to have anyway, which is that dirt (and rock) near the edges is going to want to slide toward the center... which is going to tend to turn your system even more into a "regular" planet. In any case, you're going to have thinner air toward the edges, but I think you'll have plenty of air towards the center and even spread out a bunch.
For bonus points, this means you can also spin your chunk to give it a day/night cycle. In fact, you can probably give it an atmosphere on both sides. (This might be a very good idea, because it will eliminate atmosphere loss in case you get a hole through it.)
Your real problem, of course, is keeping this monstrosity from collapsing under its own weight. Although if the intact ring managed this, you're probably okay. (Also, you mentioned that the structural problems are out of scope...) On the other hand, if you let it collapse into a ball (this would end up looking like two hemispheres in the middle of a big honking plate), well then, you've got yourself a nice little planetoid that will have no problem retaining atmosphere. (The weather might be interesting!)
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Frame Challenge
One of the criteria you gave is:
The sections are under the standard 0.992 gee acceleration. This is thrust induced, necessary to hold them in an orbit closer to their primary than it should be at its orbital velocity.
I propose that this doesn't make sense and should be discarded.
First off... the energy output of such a drive is ludicrous. If your "chunk" is merely "squarish" (about as long as it is wide), we're talking about 5e25 Newtons of thrust. You didn't supply enough information to convert this to energy output, but without some extreme hand-waving, there's a good chance we're talking about stellar energy levels. (In fact, IIUC, this drive produces about 1 Solar output if the chunk is moving at a piddling "” by the astronomical standards we are talking about "” 10 m/s. By comparison, Earth's orbital velocity is 3e4 m/s.) No matter how much hand-waving you want to employ, that drive is probably going to produce some heat. Time for even more hand-waving to explain how you can dissipate all that without cooking your habitat. (Maybe forget the star and just have it accelerating through empty space?)
Second... if your thrust is really continuous, and not varying in something like a sinusoidal cycle, then, to be useful, it has to be at a constant angle relative to your star, which means your chunk of Ringworld is (effectively) tidally locked. However, once this is true, I can see no benefit to mucking about with your orbital velocity in the first place, other than "because we can". When you're tidally locked, you don't have seasons, and "year" doesn't mean much unless you're practicing astrology.
If you still want to play with your orbital velocity... then I think the question is unanswerable without additional information. Namely, the answer will depend on your actual velocity and how your actual orbital mechanics modifies the effects of your star's gravity.
Let's assume, instead, that your chunk is tidally locked facing the star, with your thrust directly toward the star such that the perceived gravity in the center of your chunk is 0.992G. Let's also assume that you've chosen an orbit such that the combination of stellar illumination and waste heat from your drive makes your chunk "comfortable". This feels like a much more plausible scenario, and fortunately, it has an easy answer:
Since we're talking about near-Earth "effective" gravity, we can assume that the atmospheric "depth" will be comparable. Atmosphere, as elsewhere noted, doesn't "stop" at any particular point, but Earth's atmosphere is anyway regarded as being about 500km deep, so Ringworld's 1000km walls should retain this pretty well. The answer to your question, then, is that the curvature should be such that the ends of the arc are about 1000km "higher" than the center (i.e. the distance between the midpoint of the arc and the midpoint of the line between the ends is about 1000km). Calculating the required angle for this is left as an exercise for the reader. In fact, because your thrust accounts for only part of the perceived "gravity" towards the ends, actual gravity from the mass of the chunk itself will have the effect of "flattening" the arc somewhat in terms of its apparent gravity. (The actual calculations for this effect requires moderately complicated calculus or approximation via FEA.) On the one hand, this will increase the length of the arc needed to match the side walls. On the other hand, 1000km may be more than necessary for atmospheric retention.
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