Earth is overpopulated and needs somewhere to create a settlement. If they decided to settle on Saturn's asteroid ring and If the people who are making the settlement have the technology we currently have now and had a trillion dollar budget to build a settlement, could it actually work properly and will it be self-sustained? It is a city kind of settlement so no mining would be done for earth's economical purposes but only to help the settlement build so it can't buy. It has to be self-sustaining so it can't buy or bring anything else from earth. Pretty much no contact to earth after the settlement would be built.
With today's technology the answer is most likely No".
There's two important factors at play here:
Going into space is a very specialized niche which very few companies deal with. There's not a lot of manufacturers for the sort of tech you're going to need to build a large habitat. It's debatable if we even posses the technology to build a large-ish habitat (we already have the ISS, but that is not adequate for long term habitation, and is protected by Earth's magnetosphere, which won't apply out in the depths of the solar system).
But even if we did have the theoretical knowledge to build such a construct, we lack the industry, and skilled labor to do so.
Politics and Economy
The second major aspect of this undertaking is politics and economy.
A trillion dollars sounds like a lot of money, but when it comes to setting up several new industries, and developing extremely advanced and expensive materials en masse, it just isn't enough. Furthermore, some of those materials can't be found in the quantities you're going to need - other industries need them, and they're eventually going to block you from buying out all the available stocks.
Even trickier are the politics of it all. If the Earth is already over-populated, and taking the current geo-political, and global economy into consideration, you are most likely looking at a planet in turmoil. There's already many conflicts being fought, and we're looking at a possible world war within our lifetimes. When you announce a major undertaking to leave the planet you may spark accusations of trying to build weapons in space, racism based on your colonist selection process, etc. etc. etc.
More than likely one govenrment or another would simply shut you down because they wouldn't want a massive space station to be in anyone else's control other than themselves.
However, assuming that the situation on Earth is very similar to what it is today, you wouldn't have to go to Saturn. Mars has a big "For Rent" sign on its front lawn, as it were, and no one has made it over there yet.
And while it's debatable whether we have the tech to allow colonists to survive the decade long journey to Saturn, we're pretty sure that we could get someone to Mars in one relatively healthy piece.
Add to that that other companies are already designing missions to Mars, and so some industry is already geared in that direction, and you may just have a working concept.
On the more practical side of 'if it is possible?'...politics and sustainability aside.
The ring is not a solid structure and there would be no atmosphere on any of the ring debris that makes up the ring, ranging from dust particles up to fractured small moons. As far as I recall, if you were to compress the entire ring system you would only have an atmosphere a micron thick?! So pretty much you will be building a giant spaceship-city like stargate atlantis. No walking on the ring itself...unless you hopping from debris to debris in a spacesuit.
You would have to provide some sort of CONSTANT shielding/protection from the ring debris, from the larger city killers down to the tiny specks of dust. Look what happened to the ISS window when it hit a ?paintfleck? (or more accurately, the paintfleck hit the ISS)!
You would be safer on the smaller shepherd moons (but even these will have debris raining down on them occasionally) or better yet one of the larger moons. If you really insistent on being 'in' the rings, you could try for a settlement on the leeward side of the shepherd moons away from any bombardment. Then you get a view of the rings side-on.
Edit: that last suggestion about the leeward side of the shepherd moons would only work if your moon was tidally locked to Saturn. If it isn't tidally locked you could end up building your settlement in a sheltered area only for the moon to rotate your settlement into the 'highway to hell!'.
There is one very clear factor which speaks in favour of building in Saturn's rings; all the materials are already processed in bite sized pieces and it does not take a very great deal of energy to get around.
Going from planet to planet, or even from moon to moon in Saturn's system requires using a fair amount of energy to change orbits between the various starting and stopping points. In interplanetary space this can be measured in kilometres per second, in the rings, since everything is quite close and in similar orbits, you might only need to metres per second deltaV to move around.
So the ices and dust needed to feed your civilization are there for the taking with little more than a net and a very low powered rocket (or a tether if you like to save on reaction mass).
For the more advanced civilizations, the rings make a great place to set up a massive tether to reach down into the atmosphere of Saturn and draw 3He up as nuclear fusion fuel. Since the amount of sunlight is very feeble in Saturn's orbit, a solar powered civilization would have to build platoons of mirrors in orbits free of the rings and beam sunlight or microwave energy to the rings, since collisions between ring debris and fragile mirrors would be catastrophic. (The colonies will be well armoured anyway in order to withstand the intense radiation environment).
There will be a distinct lack of metals and many of the elements needed for advanced civilization, but the availability of cheap ice and volatiles would give Saturnian ring dwellers an advantage when trading with other space "nations".
Sure - why not? Here is some information about the rings from NASA:
The ring particle sizes range from tiny, dust-sized icy grains to a few particles as large as mountains. Two tiny moons orbit in gaps (Encke and Keeler gaps) in the rings and keep the gaps open. Other particles (10s to 100s of meters) are too tiny to see, but create propeller-shaped objects in the rings that let us know they are there. The rings are believed to be pieces of comets, asteroids or shattered moons that broke up before they reached the planet. Each ring orbits at a different speed around the planet.
The main rings are, working outward from the planet, known as C, B and A. The Cassini Division is the largest gap in the rings and separates Rings B and A. In addition a number of fainter rings have been discovered more recently. The D Ring is exceedingly faint and closest to the planet. The F Ring is a narrow feature just outside the A Ring. Beyond that are two far fainter rings named G and E. The rings show a tremendous amount of structure on all scales; some of this structure is related to gravitational perturbations by Saturn's many moons, but much of it remains unexplained.
The fun the about science fiction is you have some measure of freedom to invent new technologies or scientific discoveries. They can mine the ice and other chemicals, establish a base on the larger rocks, or have a station orbiting in the Encke and Keppler gaps (maybe there are two groups and they are so named?)
a trillion dollar budget to build a settlement
Well this alone would be your shortfall.
$100 billion for six astronauts?
A manned Mars mission would be incredibly expensive. NASA estimates peg the overall expenditures at about $100 billion over 30 or 40 years, Sherwood said, but those numbers may be too low.
The International Space Station (ISS), after all, was initially anticipated to cost $10 billion over 10 years. But it ended up costing 10 times that, and took nearly three decades to assemble. http://www.space.com/16918-nasa-mars-human-spaceflight-goals.html
An estimated trip to mars is estimated to cost 100 billion dollars. However, the ISS was estimated at 10 billion and cost ten times that. So if this project followed suit you would have 100 billion x 10 = $1 Trillion.
...Well, looks like you're out of money and your 6 people (a few shy of a city) are 2 planets short and have no money to build a sustainable environment.
I would have to say no, mostly due to the fact that with today's technology, we wouldn't be able to get anyone out there to live on it let alone build a habitat. Also, I don't think a sustainable settlement of any real significant size would be able to sustain itself out there without food supplies from earth, assuming they could terraform an asteroid to grow things I still don't think it would be enough. On top of that I don't think that whatever small settlement they could make out there would have a very big impact at all on the Earth's overpopulation problem.
With enough money, many MANY things can be made possible while still being completely impractical.
You provided a budget of \$1 Trillion US. According to this stack exchange answer, the cost per KG to launch something into geo-synchronous orbit is approximately $56,000 with an Atlas V rocket. That means that, within your budget, you could put about 1.5 billion KG (or 153,467 tons) of free material into geostationary orbit if you blew it all on launches. That is just getting all of your materials/modules/fuel/food/oxygen/water into a place where it could be assembled together before then heading out towards the outer planets. It doesn't account for their individual costs.
I don't have a calculation handy for figuring out how much fuel you'd need, but it would be significant (especially if you plan on landing your craft instead of crashing it into a planetary body at tens of thousands of miles per hour. Oh, and you are also going to have to carry some type of fuel to support life-support operations on your ship unless you are just planning on transporting corpses. Fuel costs need to be subtracted from your available budget.
Maybe your ship consists of habitat modules that can be disassembled when you get to your destination and used as permanent living space for the settlers. For that to be the case, they would have to be durable enough to survive the journey, land, be disassembled, and have a reasonable expectation of lasting long enough under daily living conditions to support your goal. This probably means that they are both HEAVY and EXPENSIVE, so subtract that from your budget.
You are also go to have to purchase all of the supplies/food/oxygen/water/recreational equipment/exercise equipment/medical supplies (anyone priced an MRI machine lately?)/medicine/tools/manufacturing equipment (because you can't just run down to the parts store)/redundancies/computer equipment/educational tools/nuclear fuel for a permanent power source/etc. so subtract all those from your budget.
If, after purchasing all of your supplies you still have enough left to fly it into space and take some people along, then your journey has become economically possible.
Assuming that, after all the math is done, we still have enough carrying capacity to put a few hundred people (still not "city-sized") on the ship to go along with the cargo, there are several logistical challenges to consider.
First, we know how to put things in space, how to fly something to Saturn's neck of the solar system, how to land "Earth things" on "space things," and so on. What we haven't done is do all those things at once. Yep, we've put people on the moon, and we've put landers on other planets, and we've done long space journeys with unmanned craft, and we're starting to understand the impact of long-term space flight on human health. Now suddenly all of those things need to be put together for an extremely complex operation involving lots of people for an extremely long mission with no hope of rescue if something goes wrong. Since we've never done it all at the same time, we don't know exactly what challenges we'll face.
Take for example the variety of passengers. Astronauts are carefully vetted and extensively trained. It might not be practical to do that with all of your residents, so, while best efforts might be made, you are taking risks just by putting your crew and passengers on the ship. What if there are high instances of severe depression due to leaving an entire planet behind forever? What if people reproduce onboard? What if someone commits a criminal act? What about terrorism? What if we find that space causes strange health issues in people that have a rare genetic defect? What if the module holding the vitamins takes a meteor to the hull and everyone gets scurvy? All of these contingencies have to be planned for with procedures in place to handle them. Forget just one critical thing that pops up (like a flu-bug that mutates into something really nasty due to interaction with cosmic radiation), and everyone is dead.
Then there is the question of sheer time and distance. It took Voyager 2 about 4 years to reach Saturn. Assuming we could travel at that speed, there are a lot of things that could go really wrong in that amount of time while out in space. It becomes a logistical problem because who wants to spend a trillion dollars on a mission, throw their hands up in the air, and hope it succeeds? Plans need to be in place for rescue, repair, and resupply missions in the event that a problem occurs. The further away and the longer it takes to get there, the more complicated and expensive (or impossible) these missions become.
According to NASA:
Saturn, the "Ringed Planet," is so far away from the Sun that it receives only about 1/80th the amount of sunlight that we receive here on Earth.
So, there is your figure for growing crops or producing solar power. You are probably going to have to bring nuclear power sources to run lights, grow food, create heat, power factories, and process waste back into usable compounds. Nuclear power, while lasting a long time, is still going to need expensive replenishment and maintenance. If enough fuel isn't brought along, can it be readily obtained from the environment? Will it weigh too much to take the quantities needed? Uranium and Plutonium are pretty DENSE stuff if I remember right.
How about basic things needed to maintain the habitat and systems? To be completely self-sufficient, all of those will need to be manufactured onsite. Otherwise you will need to plan supply missions with necessary components, and those missions will take 4 years to get there. Forget to bring along a spare "main air scrubbing unit?" Hope you can hold your breath a long, long time.
While (perhaps) technically possible, it wouldn't be at all feasible simply due to the extreme distance and the availability of far, far better choices. The moon is a few days away and plenty big enough for small settlements. You could have access to solar power 24/7, and the chances of something bad happening on the way are minuscule in comparison. If something bad did happen, Earth is right around the corner to lend a hand.
If the moon wasn't available (maybe due to international politics), then Mars (or maybe one if its moons) would be a far better choice. Or maybe Venus would be better? The abundance of heat energy and an atmosphere from which to extract some raw material has to be advantageous.
I'd generally tend towards a natural disaster or world war instead of overpopulation as the reasoning for off-world settlements - for a few reasons.
I agree with the sentiment expressed in earlier comments that given essentially unlimited resources, it's always going to be more economically feasible to build on earth than offworld. Unless earth is suffering from some global disaster.
Additionally, there's significant evidence to suggest that there's no such thing as global over-population. As currently accessible resources get used up and therefore get more expensive, there is a greater economic incentive to find new deposits of the same resources or find alternative methods of achieving the same goal.
For example, it took rising petroleum prices in the tried-and-true middle eastern markets for comparatively expensive oil shale and oil sand extraction in Canada to become economically viable, and for comparatively expensive hybrid and electric car development to become viable.
The same logic applies to livable space.
One option - a trope similar to the one used in Sunshine or Interstellar, where the world is slowly dying, and it makes economic sense to fly as many people and resources offworld as possible. And if Saturn's rings are still a bridge too far (considering places like Mars or Europa might be more livable sooner, you could say that those places are dying too.)
Another option - human civilization is vast and wealthy, and lots of people are eager to be frontiersman on the edge of civilization and deal all the inherent challenges.
I'm gonna say no, due to Kessler Syndrome. The relevant part of the syndrome states that once you have a sufficient number of debris you will be hit. Given that Saturn's ring is a massive amount of debris, staying there for a long time will guarantee your eventual obliteration. At which point we will have more debris than before, which is the famous part of Kessler syndrome.