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Assuming the ship has access to fusion power, a large volume of solar panels, and crude antimatter power generators (say maybe 1 gram a day per generator, or whatever is reasonable for technological development a few hundred or a thousand years in the future), how big could a generation ship designed to travel between .1-.99c be? If it's too big, its mass will prohibit it from achieving the desired speeds. If it's too small, it may not have enough living space leftover to accommodate millions of people in at least relative comfort, and it would require less effort to destroy in the event of an incursion by hostile forces (plus it wouldn't look as cool as it would if it were bigger).

Keeping in mind that the ship can be allowed a few weeks, months or even years to reach top speed given the combined power output of the sources listed above, what would be a plausible size for a generation ship that's meant to carry millions as well as defend itself?

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    $\begingroup$ Possible duplicate of I don't want to live on this planet anymore $\endgroup$ – TrEs-2b Oct 11 '16 at 0:53
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    $\begingroup$ @TrEs-2b It's not a duplicate. That question didn't involve faster relativistic travel & the numbers were much smaller (only 500) not millions. It is relevant, but it's not a duplicate. $\endgroup$ – a4android Oct 11 '16 at 3:33
  • $\begingroup$ Why would its mass prevent it from reaching the desired speed? It will just take more time to reach the desired speed because the acceleration is lower, but you will be able to. Even if it is only after a couple hundred years (and you should have started braking a long time before that...) $\endgroup$ – subrunner Oct 11 '16 at 12:59
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    $\begingroup$ Carrying solar panels along on an interstellar journey is wasted weight, because you spend almost all your time in interstellar space, where the energy available from starlight is minute. Making antimatter requires putting in the all the energy you can get from reacting the antimatter, plus inefficiencies. Antimatter is potentially useful if it is made at the start point in an industrial facility and carried on the ship, because it's a low-weight way to carry a lot of energy. Making it on board uses energy that you should be able to apply more directly to creating thrust. $\endgroup$ – John Dallman Oct 12 '16 at 6:38
  • $\begingroup$ What are you actually asking for here? I don't really understand the question. $\endgroup$ – Tim B Oct 19 '16 at 10:40
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You can get an initial benchmark for the size of the ship from the largest cruise ship on Earth today:

Royal Caribbean's newest ship, Harmony of the Seas, debuts Friday on its pre-inaugural sailing out of Southampton, England. Weighing 226,963 gross registered tons with a passenger capacity of 5,479 guests at double occupancy (it fits 6,780 guests total) and 2,100 crew members, Harmony is now the world's largest cruise ship.

Thus, a cruise ship has about 26 tons displacement per person on board with quite a bit of crowding and only a week or so's supply of food, but also with fuel and engine space in addition to areas intended to be occupied (we will need to account for fuel and engine space separately in this case because the fuel and engine requirements of an interstellar ship are very different from those of a cruise ship).

By comparison, the Nimitz Class aircraft carriers are the largest warships ever built at 102,000 tons with over 6,000 personnel, which is about 17 tons per person for tours that are typically six months at a time, with less fuel (since it is nuclear powered - an aircraft carrier can run itself for about 30 years before refueling, but not no fuel because it has fuel for aircraft), but with space devoted to aircraft that would not be as necessary on an interstellar craft (although some landing craft would surely be needed).

But, you wouldn't go wrong in estimating that for a many year long trip as opposed to a weak long holiday outing or a six month tour for someone use to the hardship of a soldier, that you'd need at least 45 tons per person of living space per person.

It takes roughly 10,600 square miles of arable land with crops growing on them to feed the population of Seattle (with a population of about 652,000), while the city itself has less than 84 square miles of land area and not all of that is arable land (i.e. land that it is possible to grow crops upon). This is about 10 acres per person. And, any interstellar trip is going to need to grow most of its own food (with artificial light because starlight is too dim). Even if you could be significantly more efficient than terrestrial farming on Earth which isn't optimized for land being extremely scarce, by an order of magnitude, you'd probably need at least an acre per person of space for food production.

State of the art terrestrial farming techniques and a vegan diet leave you at about 2 acres per person. The most optimistic estimates I've seen are as little as 1/4 to 1/8th acres per person, but some of the assumptions that go into that aren't well proven or demonstrated in practice. So, an estimate of 1 acre per person is fairly reasonable middle ground.

Still, if you were really going to build a ship like this, you'd want to invest a lot in improving agricultural productivity per square meter because each percent of reduction in this reduces you ship size and cost by 1%. If you can produce enough food per person with 1/2 an acre instead of 1 acre, you can cut the size of the ship in half.

A 102,000 ton aircraft carrier has about 4.5 acres of deck space. So, you'd need about 22,700 tons of food production area per person in addition to the 45 tons of living space.

So, in round numbers you'd be talking 23,000 tons per head of living space and food production space and nuclear power production for ship life support operations from which to feed and house them assuming an order of magnitude improvement in food production per square foot relative to Earth would be possible (e.g. by reducing less efficient animal food relative to more efficient plant food proportionately).

I'll assume that "millions of people" means 2 million, for sake of having a number to work with here, so you'd need at least 46 trillion tons of living and food production space (which would still be quite cramped) before considering fuel and engines.

You can make your ship as big as you desire, because for very large interstellar space ships, the amount of fuel and engine required per ton of living space is going to be nearly constant.

There is another moving part here. More fuel can allow you to travel faster, less fuel limits your speed (engine size is pretty indifferent as the key question is how long you run them at an acceleration of 1 G, not the amount of peak acceleration you can generate). For example, if you decided that 95% of the ship would be devoted to fuel and engines for a near maximal peak speed given the available technology (whatever that ends up being), then you have a ship carrying 2 million people that is at launch 1 quadrillion (i.e. 10^18 kg) of displacement. An aircraft carrier by comparison is 10^8 kg of displacement. So, this ship would need to be on the same size as about 10 billion aircraft carriers.

A new aircraft carrier (which is a reasonable comparable given its size and technological sophistication) costs about 13 billion dollars (aircraft not included). So, this ship would cost about 1.3 * 10^20 dollars which is 130 quintillion dollars (i.e. 130 million times a trillion dollars). By comparison, the entire U.S. national debt is about 19 trillion dollars. So, this would cost about six and a half million times as much as the entire U.S. national debt.

And honestly, both the ship size and cost are pretty stingy estimates.

UPDATE: A new academic article addresses this in detail with greater precision. It thinks that a generation ship can be 224 meters in radius and 320 meters in length with a population of 500 people that can be stable for centuries.

Numerical constraints on the size of generation ships from total energy expenditure on board, annual food production and space farming techniques

F. Marin, C. Beluffi, R. Taylor, L. Grau

(Submitted on 28 Jan 2019)

In the first papers of our series on interstellar generation ships we have demonstrated that the numerical code HERITAGE is able to calculate the success rate of multi-generational space missions. Thanks to the social and breeding constraints we examined, a multi-generational crew can safely reach an exoplanet after centuries of deep space travel without risks of consanguinity or genetic disorders. We now turn to addressing an equally important question : how to feed the crew? Dried food stocks are not a viable option due to the deterioration of vitamins with time and the tremendous quantities that would be required for long-term storage. The best option relies on farming aboard the spaceship. Using an updated version of HERITAGE that now accounts for age-dependent biological characteristics such as height and weight, and features related to the varying number of colonists, such as infertility, pregnancy and miscarriage rates, we can estimate the annual caloric requirements aboard using the Harris-Benedict principle. By comparing those numbers with conventional and modern farming techniques we are able to predict the size of artificial land to be allocated in the vessel for agricultural purposes. We find that, for an heterogeneous crew of 500 people living on an omnivorous, balanced diet, 0.45 km2 of artificial land would suffice in order to grow all the necessary food using a combination of aeroponics (for fruits, vegetables, starch, sugar, and oil) and conventional farming (for meat, fish, dairy, and honey).

Comments: 12 pages, 14 figures, 3 tables, accepted for publication in JBIS

Subjects: Popular Physics (physics.pop-ph); Instrumentation and Methods for

Astrophysics (astro-ph.IM)

MSC classes: 85-04, 91C99

ACM classes: J.2; K.4

Cite as: arXiv:1901.09542 [physics.pop-ph]

(or arXiv:1901.09542v1 [physics.pop-ph] for this version)

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  • $\begingroup$ Apologies for the bad link on the Seattle farming source. I'll try to find it again. $\endgroup$ – ohwilleke Oct 28 '16 at 3:30
  • $\begingroup$ +1 for an excellent answer. Unfortunately, you used national debt as your unit of measure for describing the scale of this economic venture. As the last few presidencies have demonstrated, our national debt is growing much quicker than our scientific capabilities. I fear that by the time we are technically able to build this ship, its' construction will only take a fraction of that current day's national debt to pay for it. Assuming there is a nation around to even consider picking up the tab. $\endgroup$ – Henry Taylor Oct 28 '16 at 4:49
  • $\begingroup$ As additional consideration, the ISS is currently 370 metric tons, with 6 crew members. That's 111 tons per person, which is 2-3 times your answer, but in the ballpark. On the other hand, I don't think you need nearly as much room as you're suggesting for food. We only need "acres" of land because the sunlight is effectively 2D. By using artificial lights in a 3D environment, you could substantially reduce the space requirements. $\endgroup$ – MichaelS Oct 28 '16 at 5:53
  • $\begingroup$ @HenryTaylor If you'd prefer (and honestly it is a more robust measure), global GDP is about $76 trillion a year, so the cost would be about 2 million years of global GDP. $\endgroup$ – ohwilleke Oct 28 '16 at 17:23
  • $\begingroup$ Aircraft carrier as measure of weight seems shady, specially considering that they bring armor, that the structural loads in an aircraft carrier are way more unpredictable, and that they have so much empty internal space that maybe even some of the weight is "overlooked" should it topple under mild wind. As crude example, a tank the size of a truck will easily weigh 5 times more. And don't even compare it to airborne machines. The weight is probably orders of magnitude overestimated. Through considering the increased cost of lighter mats probably the cost ends up around the same. $\endgroup$ – Oxy Jan 18 at 19:36
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@ohwilleke's excellent answer clearly describes the scaling issues inherent in moving millions of full grown and wide awake humans.

If all you want from the majority of your travelers is genetic diversity, consider shipping all but a few thousand of them as frozen fertilized embryos. Then cryogenically suspend everyone except the few dozen who are needed to operate the ship. Make sure that these living crew are all female and in each generation, so they can gestate, birth and subsequently educate their own crew replacements, using some of the ship's supply of fertilized female embryos.

Now when your ship eventually reaches its destination, the current generation of crew can wake up the thousands of frozen grown-ups and help them build the initial settlement. Then everyone can get busy turning the the embryo banks into babies. With in a few generations, all of your millions of colonists will be living on a wonderful new world.

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My answer is going to focus on the energy requirements of your ship which touch on the relative amount of space you need for getting the ship to its destination and supplying power to your people.

Summary

Solar power is useless, so don't bother. Either antimatter or fusion reactors would be perfectly useful for personnel energy demands, and would consist of a very negligible amount of the overall size and mass of the ship.

Your 1 gram of antimatter per day will support at least 2 million people. It will also require total hand-waving to acquire that midflight.

However, any kind of nuclear fusion, and a paltry 1 gram of antimatter per day, will be completely worthless for powering the ship, assuming we're hitting top speed within 10 years. If you want it to take much longer time frames, it might be possible, but I didn't calculate that.

You'll need to bring about 10% of your ship's mass worth of antimatter to hit 0.1 c in 10 years, or about 350% of your ship's mass worth of antimatter to hit 0.9 c in 10 years. I don't think 0.99 c is remotely doable without extreme advances in propulsion technology.

That much antimatter would take a few billion times the current age of the universe to produce at current rates, so you'll need way better tech. Still, the Sun outputs plenty of energy to accomplish it if you can build enough generators.

Fusion Power

At this point, we're outside the range of hard science. We know generally how fusion works, but we've never done it in a lab in a sustainable form. (We've done fusion, but it takes more power than the fusion produces, so it's an awesome experiment, but utterly worthless as a power source.)

That said, this MIT experiment is estimated to produce a lot of power if they ever make it work.

A working ARC fusion reactor would use 50 megawatts (MW) of power to produce 500MW of fusion power, 200MW of which could be delivered to the grid. That's enough to provide 200,000 people with electricity.

The reactor itself is about 1 meter across, so we don't need to worry about its mass too much. The infrastructure for ITER's reactor is about three stories tall, but a few dozens rooms worth of space for every 200k people is negligible.

The Culham Center for Fusion Energy estimates

A large power station generating 1,500 megawatts of electricity would consume approximately 600 grammes of tritium and 400 grammes of deuterium each day.

That equates to about 0.243 kg of fuel per megawatt per year. Given the 1 MW per thousand people figure on the MIT article, that's 243 kg fuel per million people per year, which is pretty negligible.

Antimatter Power

As I pointed out in the answer to this other question of yours (and pointed out in John Dallman's comment above), creating antimatter to use as a power source doesn't really make any sense. The power used to create the antimatter is millions to billions of times higher than what you finally get out of the antimatter annihilation.

You could use some kind of hypothetical device that collects antimatter with total handwavium (say, there's enough antimatter hanging out in interstellar space that you can just grab it on the way by, or zero-point energy). In that case we can calculate the energy from 1 gram per day (about 50 gigawatt hours per day which converts to about 2 GW total output). But none of that is remotely hard science.

From the section on Fusion Power, humans living in modern Boston use about 1 MW per thousand people, 2 GW would provide for 2 million people. That number would likely be far lower in an actual generational ship as people would learn how to do more with less. Still, it's a good upper bound.

Importantly, one gram per year per 2 million people means the normal matter mass you'd need to annihilate the matter is negligible. Presumably, you'd have some kind of reactor that takes space and mass, but since we don't have antimatter collectors and/or generators it's hard to say exactly how much. I'll assume it's about the same as a fusion reactor.

Without any handwavium, you'd need to bring the antimatter with you. The amount of 0.5 g per year per 2 million people is going to be totally negligible in terms of size and mass, but will require some very advanced means of actually producing that much antimatter.

Solar Power

Also, as pointed out in John Dallman's comment, solar panels are probably a huge waste. At 93 million miles from the Sun, we're seeing about 1.3 kW per square meter. At 0.1 c over a single generation (about 28 years), you'd travel about 16 trillion miles, which happens to be about halfway to the nearest star. Power output will fall off with the square of distance, so you're looking at about 44 nanowatts per m² at that distance, and an average of 7.6 milliwatts per m² across the trip.

Even if you could somehow fly in a line that gets you really close to each star you pass, your best case is going to be about 1.6 watts per m², assuming you're literally touching the surface of each star on the way past.

To be fair, not all stars are like ours in power output, but the Sun is actually the top 10% by mass, so your realistic solar influx will be even lower than the calculations above. Further, your realistic path will probably stay substantially farther from nearby stars than the calculations above, further lowering the average power.

From Sunmetrix, typical solar panels are 10-20 kg per m². At the lower value, you're looking at about 6 million kg per megawatt) in the best case of zooming right up to sun-like stars.

In addition to the mass issues, you have to have some way to spread them over an enormous area without shearing or folding from the torque. One megawatt is 600 thousand square meters in our best case scenario. That fits into a circle with a 437 meter radius.

If the ship is accelerating at 0.01 gees, a 1 m² section at the edge, with its mass of 10 kg, requires about 1 N force to keep it in place. At 437 meters from the center, that's 437 N-m of torque per m². There are about $2\pi r$ of these 1 m² sections around the outer radius. Then $2\pi (r-1)$ sections around a slightly smaller section. Turning that into an integral gives us about 600k N-m torque on the center of the disc.

You could probably solve the torque issues for a 437 meter disc by using supporting structures and so forth. But you need a thousand such discs for each million people on your ship. And realistically, you're looking at something closer to the 44 nW per m² figure. That requires some 23 trillion m² of panels per MW, or 23 quadrillion m² of panels per million people. Which ends up with a 151k km radius array, which has about 16% of the area between Earth and the Moon's orbit. The total torque is about $72\cdot10^{15}N-m$ and you're really not getting around that with extra supports, unless your entire ship is about that large.

As a side note, solar panels have a best-case efficiency of about 86%, and are realistically sitting around 50%. Your advanced people could likely hit 70-80%, but this is pretty trivial when there's so little sunlight available in the first place.

Acceleration Energy Requirements

Ok, so we need a negligible amount of extra space for the fusion and antimatter reactors compared to personnel energy usage. But we still need to accelerate the ship.

To hit 0.1 c in ten years, we need about 0.1 g acceleration.

To get one megaton of mass to 0.1 c, we need about $4.49\cdot10^{23} J$. That requires about 5 million kg, or five kilotons, worth of energy.

For the anti-matter propulsion system, the extra mass for propulsion is pretty negligible at about 0.5%.

For nuclear fusion, we're getting about 1.5 GJ per kg of fuel. That means about $3\cdot10^{14}kg$ of fuel. Which means about 3 parts per million of the spaceship's mass is payload; the rest is fuel. So really, antimatter rockets are the only way we're getting this ship to 0.1 c.

If we up the cruise velocity to 0.9 c, we'll need 0.4 megatons worth of energy. That's huge, but doable, in the sense that your rocket will still be 71% payload.

On the other hand, getting 0.2 megatons of antimatter is insane. With current antimatter production methods that use 15 billion times the antimatter's mass energy to create it, at a rate of 1 billion years per gram, you'd need about 22 times the Sun's annual output of energy, and way more particle accelerators than we currently have to accomplish it before the Sun dies. That is extremely advanced technology, but it seems plausible to an advanced enough society.

From Wikipedia, chemical rockets have an energy efficiency of about 60%. Anti-matter rockets have between 10 and 85% efficiency. But it really doesn't matter; anti-matter will have negligible mass, fusion will be way too high.

Reaction Mass

Now, you'll need reaction mass dependent on how much energy you can impart into each particle, and is given by $M=P\left(e^{\frac{\Delta v}{v_e}}-1\right)$, where $M$ is reaction mass, $P$ is payload mass, $\Delta v$ is the change in spaceship velocity, and $v_e$ is the exhaust velocity.

We've set $\Delta v=0.1c$. From this page, antimatter rockets have a specific impulse of 0.6c. From what I can tell, they're using "specific impulse" to mean "effective exhaust velocity", so $v_e=0.6c$. Plugging this into the equation, we get 18% P. Calculating a final velocity of 0.9 c requires about 78% of the ship to be reaction mass.

If we use dense reaction mass, this means the actual size of the spaceship isn't largely affected by the reaction mass. And adding 20% mass to the ship isn't particularly substantial in the grand scheme of material requirements.

From here, fusion rockets have exhaust velocities up to 700 km/s, or about 0.0023 c. This brings our required reaction mass up a lot (about 7 billion billion P). No way we're doing anything with that.

So if your guys are using fusion reactors to power the spaceship, you'll have to assume they achieve much higher exhaust velocities. Around 0.14 c is required to keep the reaction mass lower than the payload mass. This seems reasonable enough for an advanced society, but we don't have any current means of achieving it. Not that fusion was ever a real choice considering how much mass we need to power it.

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