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I'm planning on sending a colony ship on an extended, multi-generational voyage to a distant star. My colony ship is an oblate spheroid, 2km long on its long axis and 1km across on its shortest access. It's built using near future technology, with power generation provided by fusion plants and thrust via a large array of solar sails, which are mounted on external control pods and don't take up much of the 'internal' space on the ship. The sails can expand to an enormous area, but primarily use albedo changes to move around based on solar pressure, so their control doesn't heavily tax the power supply systems on the ship.

Internally, I've got my power generation systems located along the central axis of the ship, with all technical areas and industry clustered around those. Following my industrial section, which must produce and recycle all goods for the populace of my ship, I've got habitation and commerce modules wrapping around the core in cylindrical layers, with farms and natural areas forming the outermost layers. I'm expecting the natural areas to be responsible for the majority of organic recycling and air filtration on the ship, though there is a backup system of atmospheric scrubbers to maintain a breathable atmosphere in the event of a biological breakdown in the nature modules. The entire ship is close to a zero-g environment, with the exception of several exercise modules, which generate artificial gravity by rotating around the rest of the ship. These gently rotate the rest of the ship, in accords with conservation of angular momentum, resulting in very gentle gravity acting on the rest of the ship, which leads to objects slowly settling towards the 'up' direction over the course of several minutes.

The voyage is expected to take at least several centuries, so most systems on the ship will need to be replaced/maintained via onboard facilities multiple times. Technology has progressed to the point where the most space efficient way to handle this is by recycling all materials, rather than by carrying spare parts, though manufacturing methods are still broadly similar to what we have today, rather than using nanobots or the like.

If I pack all of my essential systems in as efficiently as possible, about how many colonists can I expect to fit on board my ship?

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You presumably mean how many can live aboard the interstellar space potato at any one time? Not over the course of the trip. – Samuel Feb 22 at 17:59
    
Yep! Edited the title to make it a bit clearer. – ckersch Feb 22 at 18:04
    
How long is your voyage planned to be? do they need to be able to fit in factories for replacing failed microchips and other high-end failed parts(long long voyage) or can they carry spares? how much of the ship is taken up by the drive and power plants? – Murphy Feb 22 at 18:50
    
@Murphy The voyage is a several-century long trip, at the very least, so the colonists will need to be able to manufacture replacements. The drive system is based on solar sails, which are mounted externally and won't take up much of the main hull, but I'm not sure how big the power plants would need to be. Big enough to supply energy needs for all of the colonists using fusion power. – ckersch Feb 22 at 18:59
    
Is cryo/stasis an option? – aroth Feb 23 at 7:59

Food

It may take 1 acre on earth to feed a single person, but that's relying on soil, sun, and animals. Cattle would be fairly inefficient to take in space, and you can't grow plants "naturally", so hydroponics makes much more sense. Since you have fusion power available, I assume hydroponic greenhouses can feed a human with much more efficiency than 1 acre per person. One source says it is about 10x more water and land efficient.

Also note that cows eat a lot of grain. One pound of meat requires about 10x times the water to produce compared to a pound of soybeans. It would be best if most of your population were vegetarian, unless population size isn't critical and you wanted the cattle for populating the destination (or as a luxury reward).

Shielding

Also, it would be best to put as much stuff between your humans and the outside as possible. Hard radiation will make your ship pointless if the crew isn't adequately shielded. You can use water as a shield, or lead. But probably best to just put as much mechanical towards the outside as possible (but keep sensitive electronics near the interior). If your fusion reactor runs on D/T, then you can use heavy water as the fuel and keep that near the outermost layer.

Capacity

One acre is about $4000 m^2$. Hydroponics can reduce that to maybe $400 m^2 * 2 m$, so let's say $10^3 m^3$. Thus, you should allocate at least this much space per colonist. As long as their "personal space" is much less than this, that becomes mostly a rounding error, and the rest of the space can be used for power, life support, manufacturing/recycling, transportation, etc. So, absent a significant improvement in farming density, I'd say that $10^6$ is a hard upper bound on your population, and $10^5$ is more realistic (though anything in that range is passable).

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+1, This is just like something I would have done. – Quiquȅ Feb 23 at 3:43
    
I assume the "on Earth" comment was in response to my answer below. Thanks for going the extra step! – ChronoD Feb 23 at 4:44
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Mealworms are quite efficient and allow for animal proteins without the need for complement. – MakorDal Feb 23 at 5:31
    
Since it's clarified that parts/replacements will need to be manufactured in flight you'd probably need to allocate a reasonable chunk of space to manufacturing and maintenance. Having machines and backup machines to replace all the parts that might fail (and things needed by a new colony as it's starting up) is going to take up a fair amount of space. – Murphy Feb 23 at 10:29
    
Yes, I presume that readers will see answers by o.m. and Chris G as well. Sorry for the lack of attribution! – Lawnmower Man Feb 23 at 22:22

The ship is approximately $10^9$ cubic metres.

People can easily live in personal quarters with $10^2$ cubic metres per person. That would be roughly 33 square metres and 3 metres high. Of course people on Earth get out from time to time, walk in the park, so let me increase the volume by a factor of ten for corridors, gardens, etc. -- $10^3$ cubic metres per person.

As a wild guess, assume that 90% of the ship are engines, storage, whatever.

That would allow $10^5$ colonists. If you allow them less generous spaces, it could be $10^6$ or $10^7$ colonists.

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Does $10^2$ cubic meters include food production, or is that in the 'whatever' category? – ckersch Feb 22 at 20:53
    
Are you assuming the people live just on the inside surface, or are you assuming a sponge like 3D structure with people inhabiting bubbles of space all the way through? – Thucydides Feb 23 at 1:23
    
@Thucydides, a sponge as specified by the question. – o.m. Feb 23 at 5:53
    
Is '10^2' really more convenient than '100'? – aroth Feb 23 at 8:00
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@aroth when the rest of the numbers mentioned follow the same format, yes. – Pimgd Feb 23 at 8:26

(Adding onto o.m.'s answer and to try and answer ckersch's comment on it.)

It takes about 1 acre to feed a single person based upon this website which equals 4000 square meters. I assume this can be reduced because of two reason:

  1. Advanced technology in comparison to the modern technology the website is basing their calculations on.

  2. Reduced need for energy consumption (since they no longer will likely have reduced muscle mass based on this question about the same universe.

I assume that $10^4$ cubic meters per person, INCLUDING food generation, would be more accurate for approximating the capacity, which allows for $10^4$ colonists instead, using the same space assumptions of o.m. (90% unusable space for engines/storage etc.)

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Hmm, but that one acre is an acre on Earth, with all of the benefits of the surrounding planet, its water cycle, and ecosystems, and the light of the Sun at Earth distance. – Dronz Feb 23 at 7:51
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Agreed. My answer was just to give a basic idea going in that direction. Lawnmower Man (currently #1 answer) expanded more on it if you haven't taken a look yet. – ChronoD Feb 24 at 19:10

The most efficient packing method would be if the colonists were liquified pre-flight, in which case each colonist would occupy roughly 60 liters of volume. An oblique spheroid as specified would have a volume of roughly 8.4 x 10^12 liters, so the answer to the question "How many colonists could fit on a 2km long multi-generational colony ship" is roughly 1.7 trillion.

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Welcome to the site. I can honestly say I didn't expect this. However, 60 liters of liquid human does not count as a colonist. I seriously doubt murdering trillions of people and sending the remains across the galaxy is what the OP desires. – Frostfyre Feb 23 at 5:07
    
Or do you have a plan to get them back to some more solid form later? – bilbo_pingouin Feb 23 at 7:18
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If they're like the Kelvan Andromedans in Star Trek, they could store them in dried form without the need for liquid. :-) hexnet.org/files/images/hexnet/content_cuboctahedron-kelvan.jpg – Dronz Feb 23 at 7:55
    
@Frostfyre but this should indeed be a solid upper limit to the transport of humans (or so to speak human biomass) ^_^ – Falco Feb 23 at 12:47

To maximise the number of colonists and ensure that they arrive at their destinations alive, there is no better technique than...

Suspended animation.

By freezing the colonists into liquid nitrogen and later reviving them, there can be massive savings in space. Since there are no ambient heat sources in deep space, loss of coolant is not a significant factor during the vast majority of the trip, and therefore there is no need to carry excess coolant to replace natural boiloff due to heating.

Assuming that the volume required to store an average colonist is equal to that of a modern-day funeral casket with volume ~=$\text{0.9m}^3$, you should be able to comfortably fit a billion colonists into the ~$1\text{E9m}^3$ of volume. The remaining ~10% or so of extra space would be dedicated to shielding, engineering and support for non-suspended crew members, who will treat it as a generational job to maintain the ship during the trip. Since the billion or so colonists are effectively maintenance-free cargo, the manpower required for running the entire ship would be minimal. In fact, it is possible for the entire thawing system to be automated, and in that case no thawed humans need be present on the ship for the vast majority of the journey.

The technology to freeze a human into liquid nitrogen with minimal damage has already been pioneered by Alcor Life Extension Foundation, and as long as the technology to safely thaw out these frozen humans is perfected, this would be by far the most space-efficient way to transport colonists on multicentennial trips.

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I have a book here somewhere written by Yuri Gagarin (the first cosmonaut) it says humans need 2 pounds of oxygen per day. I don't remember from the book anything about space required. But I do remember a debate on the news where they discussed prison overcrowding, and they determined that 17 cubic feet of air was required for each inmate for it not to be considered a health hazard. (0.481386 cubic meters.) (Round it off to .5 for easy math, and say you chose short people.) If the ship was only one meter wide and one meter tall.. then, roughly 4,000 people would fit front to back.

Making the ship wider or taller would increase this number. But you would subtract for walls, fuel, engines, batteries, food, etc.

4,000 people at 2 lbs oxygen = 8,000 lbs per day. life support should have AT LEAST one day to spare!

Liquid oxygen takes up less space, and would be the easiest way to transport it. The coldness of space makes refrigeration a little bit easier. 1 lb of liquid oxygen equals 337.56 liters (337,560 cubic centimeters). .. so 675,120 cc per person @ 2 lbs. x 4k people = 2,700,480,000 CCs. = 2700.48 cubic meters. So, if the oxygen tank(s) were one meter wide and tall they would would be 2.7 kilometers long (you wanted the ship to be 2km) Making them 2x high and wide (4x total) makes them 675.12 meters long. I would HIGHLY suggest a series of smaller tanks to keep liquid from splashing around during turns that much weight as a liquid could rupture the tanks from inertial force alone! and pressure at the end of a 2km tank during acceleration or deceleration would equal the force of a waterjet cutter! Imagine a cold-laser.

Because you are using a sphereoid, I will cut it in half to do calculations of 2 domes (front and back) 1km wide 1km front to back. and 1km tall, the other domes height will make it 2 km tall and keep the same 1km dimensions on the other 2 axis.

calculator tool @: http://www.onlineconversion.com/object_volume_ellipsoid.htm You have 1,047,197,551.1965976 cubic meters of space to work with.

3.7 liters of water/day (wikipedia), or 1.9 l/day according to the mayo clinic? call it 2 and recycle.

Each crew/colonist would need: .5 cubic meters of space to live (cramped prison-style). .67512 cubic meters for oxygen .002 cubic meters of water (2 liters) - food? 2 liters as well. Imagine eating steak and potatoes from a cola bottle. .002 cubic meters of food. 1.17912 / call it 1.2 cubic meters per colonist.

You could cram in 872,664,626 colonists if you did not carry fuel. (1/8 Earth's population?) But this is somewhat unrealistic. No batteries, electronics or machinery either. Average weight of a human adult...180? but their are children, being a generational ship.

unless you want everyone in suspended animation, not needing a generational ship. Any more info you can give?

I gave storage space for food, water, and oxygen for one day only, instead of for the entire trip, as food can be grown, and air and water recycled.

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