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So first off I will assume that the generation ship has a reasonable capacity of 1 billion and 45,000 people are the first generation on it. Also assuming an 8/1000 per year death rate(the average death rate around the world) and an annual birth count of around 6,979 babies per 45,000(this takes into consideration miscarriage, stillbirth, fertility rate(and the fact that infertile couples make a minority on the generation ship), the 2 years between the birth of 1 baby and the conception of another and thus 11 pregnancies on average per generation between 20 years(youngest ideal age for pregnancy) and 50(average age of menopause), and multiple births from twins to quads(which has the lowest prevalence, like only 5 per 30 years per 45,000 if you take the natural conception rates into consideration only(like I did)), but not infant mortality and preterm birth because of these 2 assumptions:

  1. Technology to keep baby in the uterus until full term(like artificial amniotic sacks and stopping labor entirely until full term)

and

  1. Infant mortality is so low, even in those with severe defects, due to medical technology, that it simply isn't significant enough to affect the annual surviving birth count by much

that a linear graph would be very close to the actual population curve until 1 billion is reached, at which point people will have to die before anyone else is born so realistically, the curve would zigzag with the maximum always being 1 billion at most.

Now if we take the population density of Bangladesh, this means that at most, there will need to be 798,722 sq km just to fit 1 billion people in the generation ship, not to mention the complex water treatment plants with 100% of water being recycled from sewage and sodium hydroxide being used because, well, thousands of pregnant women in 1 cycle = a lot of HCl getting into the water = water approaching 2 on the pH scale = a lot of sodium hydroxide needed to form water in the least toxic and least likely to explode way(the only byproduct is sodium chloride which is soluble and can easily be drawn out of the water via evaporation. Or the tons of plants and animals including bees that would be required for the least gross space diet(seriously, eating algae all the time, um no thanks, that sounds like an unstable source and it just sounds gross) That would require so much fuel, it would only work if it was built in interstellar space, otherwise, gravity would prevent the generation ship from going into interstellar space.

So let's take the population density that is the highest of any state in the world, that of Macau, a city that is technically part of China but also technically its own state under One China 2 Systems. This brings the area needed for the humans way down to 46,704 sq km, an order of magnitude smaller than with Bangladesh's population density. But there are cities with an even higher population density than Macau. The most densely populated city in the world is Manila in the Philippines.

This doesn't bring the area needed down by an order of magnitude but it does go down by more than 10,000 sq km. If 1 billion people were in a place with the population density of Manila, Philippines, then the needed area would be 24,087 sq km. But this is only if you are talking about "city proper". According to the UN's Habitat Data, the most densely populated city is Dhaka, Bangladesh at 44,500 people/sq km which is even more dense than the "city proper" of Manila, Philippines. This would bring the needed area down to 22,471 sq km.

You see a pattern here? Yeah, as the density increases by a fixed amount, the amount of area needed gets smaller at a slower rate.

22,471 sq km though is still a lot and I think that is way too big for a generation ship, even though this does leave more added room for water treatment, farming, etc.

But just how dense can it get? I mean, I have heard of the theoretical maximum density of 5 people per sq ft but that seems way, way too dense for anything really, especially considering that that is a theoretical maximum assuming that people can't move because they are all contained in 6 cb ft chambers(1 sq ft * average height of 6 feet). That just would be impractical on so many levels, that even thinking that it could be possible is an overstatement.

So how dense would the population have to be in order for the generation ship to be small enough that it can relatively easily get out into interstellar space, even if it takes years to get to the sun's escape velocity due to the high mass? I'm assuming some very strong metal is used because metals can more easily block gamma rays(the most concerning of all radiation, considering that neutrinos don't cause damage and other subatomic particles at average energy would easily be blocked at a much lower thickness than is needed for gamma ray protection and very high energy particles like the OMG particle are very rare) than any other everyday material and I don't know if carbon fibers, which are way stronger than even the strongest metal, could block gamma rays with a reasonable thickness, much less that less thick than any metal that would be used.

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    $\begingroup$ I think you should start from how this ship would be powered and propelled, otherwise calculating human density is an important, but essentially pointless exercise. $\endgroup$ – Alexander May 26 '18 at 4:52
  • $\begingroup$ Well I plan on having the ship have nuclear fusion powered engines which would have much more energy than even the ion engines that have been used in some spacecraft. I plan on having nuclear fusion also be the main source of electricity because magnets close to the fusion engines could attract the electrons into giant batteries while the nuclei, being much more massive, would stay in the fusion tanks to propel the ship. $\endgroup$ – Caters May 26 '18 at 5:01
  • $\begingroup$ Human density had to be adjusted for farmland / factories. How many square miles of land does it take to feed Macau? $\endgroup$ – pojo-guy May 26 '18 at 5:52
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    $\begingroup$ Damn this is a huge bland wall of text. Consider learning about markdown to make it easier on the eye... $\endgroup$ – dot_Sp0T May 26 '18 at 7:25
  • $\begingroup$ Do you want a population density that includes necessities such as food, hygiene, common areas, etc...or one that disregards all that. Your city examples dont seem to include all the necessary imports to maintain that. $\endgroup$ – spade May 26 '18 at 11:25
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Lots of guesstimates ahead, no offense intended.

People will want to get comfortable: it's a ship for life, not just travel. Let's pick 9 m^2 per person: that's 9e9 m^2 = 9000 km^2 for a billion people. Cut the area in pieces, arrange 10m height separation between them (to make space for infrastructure), and the volume becomes 90 km^3 - a cube with sides smaller than 5 km.

But that's just the residential sector. There will be need for factories, food production, recycling, commerce... I will guess that all of these occupy 20 times the volume of the residential sector, so 1800 km^3.

But there's more! Such a giant station will need fuel, nuclear or otherwise, and propulsion systems. Given that most of a rocket's mass is fuel, the ship's volume will grow even more; let's say that 4/5 of the mass is fuel. This will get the ship to about (1800 + 90) * 5 = 9450 km^3.

Add radiation shielding, against these pesky cosmic rays, and the volume gets to a nice round 10,000 km^3, a cube of sides about 21.5 km.

So, population density isn't a big factor on ship's size.

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  • $\begingroup$ I like this answer. Its thinking in 3 dimensions rather than 2. However, it does conflate volume with mass. Volume overall isnt really a problem, there are over 500 asteroids in the asteroid belt that you could fit your 1800 km^3 city in (wwu.edu/skywise/a101_asteroids.html) However, the fuel would like be 99/100 of the ships mass, though the fuel is likely denser than the city section (which would have a lot of air in it. This reddit post (reddit.com/r/estimation/comments/3jpy0i/…) estimates the weight of New York would be 26 trillion pounds $\endgroup$ – Richard Hansen May 26 '18 at 23:33
  • $\begingroup$ So the fuel weight would be around 3 quadrillion pounds. $\endgroup$ – Richard Hansen May 26 '18 at 23:34
  • $\begingroup$ Good thinking. Another point to consider: How do you get rid of the heat? Even if the engines are perfectly efficient, a billion people and their infrastructure will produce a lot of heat and you have only the small area of the surface of the ship to get rid of it through. This is a huge engineering problem. $\endgroup$ – Mark Olson May 27 '18 at 1:15
  • $\begingroup$ So is getting such a high mass to escape velocity. I would think that for a generation ship to receive gravity assist, it would have to be miles away from the astronomical object and slowly orbit it enough times. Even if I was using nonrelativistic speeds, surely general relativity would affect the gravity there so much that it is way more likely to crash into something than get gravitational assist from it. Also, I would think that as long as there is enough material to protect from gamma rays, conduction would spread the heat out enough that the increase in temperature would be insignificant $\endgroup$ – Caters May 27 '18 at 4:52
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Any ship is going to require reaction mass. A LOT of it. Assuming your usual scifi staple fusion torch drive we're talking anywhere from 2/3rds to 90% of the vessel's mass needing to be just fuel and engines (dependent upon % of C you are accelerating to, what fuel your fusion method uses, and how efficient it is at reacting it). A true generation ship (not a sleeper, icebox, or Methuselah style ship) Is going to be MASSIVE! Kilometers and kilometers, engines the size of mountains, and small oceans of whatever fuel you use.

Let's err on the upper end of efficiency and say that you are traveling at 5 percent C. The rear 2/3rds of the ship are basically fuel tanks and radiation shielding. The habitat module is in the middle. Building such a device that can simply withstand a trip that might last a few thousand years just inst feasible or explainable with current science. So instead of simply lasting that long it's gotta have factories, ideally highly automated and highly versatile constructors which can build anything the ship needs to repair itself on its journey from raw materials. Having the factory in front of the habitat module but behind the frontal armor makes sense for a reason I'll discuss further on. It can also take a hit instead of your colonists hab module if something penetrates the frontal armor. Ideally this factory could also gather some resources as it travels in the form of asteroids and such it can capture. Maybe even try to snag the odd comet to top off on fuel. (fuel is a thing you might have gigatons of and still not have enough).

Next is its frontal section, which is meters and meters of armor. Doesn't matter what (within reason) there just has to be lots of it. It's going to ablate over time, quite a bit in fact. So I'm imagining the armor is in giant overlapping slabs that can move slowly on tracks kind of like a giant tank tread. As one wears out it cycles to the rear where it is repaired then cycled to the front again (again, like a giant tank tread). The factory right behind the armor section spends a significant portion of its operating capacity not needed for other functions repairing these armor slabs as they are almost imperceptibly cycled through the rotation.

But, your ship is practically packing a small star for its reactor(s?) so why just stand by and let your factory burn up all of its resources building new armor plates? Why not tap into a bit of its engines' fusion power and have a highly adept suite of automated sensors, quick reaction targeting computers, and obscenely powerful lasers. These sensors constantly scan ahead and if they detect even a speck of dust the ravening beams of death vaporize it. This will minimize impact on the frontal armor slab. Now you can conserve some of that raw material for manufacturing manufacturing for other stuff. Maybe a bigger habitat module or make modifications to the ship. (maybe you receive sluggish but much awaited updates from earth on new tech or develop some of your own along the way and decide to fix/modify/retrofit something.) Always good to possess at least some surplus.

In any case, by this point we're talking about a vessel at least tens of kilometers long burning gigatons of fuel all to support a mere 10,000 or so passengers. It's a gargantuan laser blasting heavily armored behemoth lumbering along at a mere 5% the speed of light.

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    $\begingroup$ Regarding your forward ablative armor - why not use a gel reservoir, like an enormous run-flat tire? As the surface is pitted away, the gel seeps out and hardens, and it can be replenished (if it needs replenishing) from the inside. It's the same effect as modular armor but fewer moving parts. You'd need to formulate it carefully to get the right mix of physical characteristics but heck, if you're mega-engineering already... $\endgroup$ – Cadence May 26 '18 at 12:34
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What about a Bussard Ramjet for power and propulsion? I don't know if this would work in reality, but it's been used in enough SF novels that I'd accept it as a reader.

Quick definition: When the ship is going fast enough, a magnetic shield/funnel takes hydrogen ions from "empty" space and compresses them until they fuse. In some stories they use excess to refill their fuel tanks. And you do need fuel, since the ramjet doesn't work until you're going fast enough.

Examples of stories that use it: "Tau Zero" by Poul Anderson, and many of the early Known Space stories by Larry Nivens (the Pak used them also). How could I forget this one, "Mayflies" by Kevin O'Donnel, a story about a generation ship. Mayflies follows the generation ship from the start to the finish of the journey, though the POV jumps in time so you get to see major societal changes that occur during their extremely long trip (the ramjet got turned off early and couldn't be turned back on again until near the end of the book). The ship was huge, and like yours started from a relatively small population and it grew. I don't remember what the target population was though.

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