THE LIMITING FACTOR IN ONE'S ABILITY TO GO INTERSTELLAR much probably...
PACKING UP BREAD AND DIRT BLOCKS
We have so far contemplated how rare phosphorus is, how it arrived here, and why it is so important for Life. Now let’s figure out how important that is for space travel.
Let's start with rations, my proxy for vegetables will be lettuce and cabbage. While milk and poultry will serve as protein proxies. So by mass, a kilogram of meat contains about 1% phosphorus while dairy is around 0.08%. While vegetables contain only 0.04%. I’ll be estimating the needs of the crew based on a 2006 study on the Finnish consumption of natural resources.
Around 25% of the consumption will consist of dairy, while 10% is protein, 17% vegetables, and the rest is mostly water, alcohol, and sugar based drinks. Which provides 170g, 850g, and 58g of phosphorus, respectively, with trace amounts in other sources, throughout the year (~3g/day), which is compatible with the daily flux of phosphorus for a healthy human.
So not only you'd have to be able to store 840 kg of rations per person per year, but also produce it during the trip, after all, cabbage and chicken are also eating.
For crops, the requirements for phosphate fertilization may vary, both corn and alfalfa need about 5.9 kg per ton of crop (5 and 2 tons per acre), and crops only use about 20% per year, so a single fertilization may yield up to 5 crops before it is needed again. Monoammonium Phosphate, a common granular fertilizer, is only 13% P by mass.
So if our ship has to sustain 1000 people for a year, it has to provide 160-170 tons of vegetables a year-round, and 84 tons of meat. And we include at least 991 kg of fertilizer consumed per year.
Because tissue printing and meat culture is already a developing technology I'm initially assuming this meat can be produced without the animal necessarily on-board, nowadays this technology can produce 10kg per hour (per unit I assume), once you have a culture of cells and their nutrients. So those 84 tons are of raw material, including 840 kg of phosphorus.
The conclusion here is that, not only you need 2.04 tons of raw phosphorus per 1000 people, but you also need all of the water, air, but also other elements such as carbon, nitrogen, and calcium to circulate properly around this system, and add a slight surplus there because bacteria will grow in these conditions want you or not, you won't like to wake up one day and find that the nutrients are being drained by the soil itself.
The International Space Station has a pressurized volume of 915.6m³ at 1.0 atm, and a crew of 7 people, though about 13 astronauts have been at the ISS at one time in 2009, several sources cite the ISS optimal capacity as something in between 6 and 10 people, so let’s go with 5 people, since we also have to share those resources with our growing food and unwanted bacteria.
Using that pressurized volume and air density at 1 atm, we get 1,163 kg of air per 5 people onboard, or 232.6 tons per 1000 people. While CO² is disposed of into space by the ISS. Chemical and electrical equipment exists to break CO² into carbon and oxygen, the carbon reacts within the system to create amine compounds, in that case, 40-50% of atmospheric carbon gets trapped in those amine reservoirs.
Active air production can take a few routes, either by burning oxygenic fuel, or by electrolysis of water, the latter which averages at 1 liter of water per person per day, while the released hydrogen is reacted in another unit with carbon dioxide, to produce water and methane.
For water, the ISS is able to recycle 90% of its water — with around 1,514 liters per 2-3 months. While each astronaut is limited to 4.4 liters of water a day (drinking, shower, etc), being conservative we can use between 4 liters astronaut-style, or 384 liters for a more Earth-based routine.
For comfort and leeway into near-future advancements, I’m using:
Drinking, 4 liters a day
Shower, 3 minutes, 4 liters a minute, 12 liters per shower, 2 showers a day.
Food water, about 0.5 liters a day.
Total: ~28.5 liters a day.
The ISS also has the capacity to recycle between 10 and 20 liters of water a day (apparently), and so we need a daily recycling capacity at least 5-6x that of the ISS, which approaches that many times more equipment since we are at 90% efficiency, we are also carrying enough water to operate 90 days before touching recycled water. So 2,565 liters per person on our trip.
Still on water usage, vegetables need 40-80% soil humidity to grow, and so we are looking for around 60% water by volume of soil — and that’s another unknown variable, many crops can be cultivated through hydroponics, but some need soil, assuming we can generate artificial gravity, a more hands-on approaching to vegetable growth will be my route here, we had already established that we can get between 2 and 5 tons of crop per acre (~4,047 m²), let’s settle at 4 tons per 4,047m² for these calculations, and I’m also using ~1 meter deep of soil, so the ship literally carries a farming ground with it.
Humus has an average density of 2.65 g/cm³, clays is 1.7 g/cm³, and clay+gravel stands at 2.7 g/cm³. At thicknesses 12.7cm, 63.5cm, and 24cm for the soil layers, we have a total density of 2.06 tons/m³, which means 8,336.8 tons of soil per acre.
For the vegetables (165 tons) we need an average of 50 acres, for a total soil mass of 416.84 thousand tons, and with 60% humidity, that’s 121.4 thousand tons of water…
Because we need to account for active bateria around, literature shows that up to 5-10% of soil organics are microbes, so basically we have to account for extra “crew” onboard, and thus extra phosphorus, air, and water we need to carry accordingly, bacteria are between 1-2% phosphorus, while I’m assuming their body mass mostly water as well, and so I’m using our humus mass as our reference for organic matter, so about 16.83 kg of bacteria per cubic meter. A single-person ship with a 417 ton garden hoards as much bacterial biomass as extra 30-40 people, and these “people” also consume oxygen and emit carbon dioxide. And so two solutions exist for that:
- Add more resources on board.
- Use methanogenic bacteria to even out CO² and O² emissions.
Because the 1st option is too complicated, I’ll opt for the 2nd way.
AEROBIC:
C6H12O6 + 6 O2 + 38 ADP + 38 phosphate → 6 CO2 + 44 H2O + 38 ATP
METHANOGENIC:
CO2 + 4 H2 → CH4 + 2 H2O
The aerobic:methanogenic respirations processes need to happen in a 1:6 proportion to balance out, in perfect conditions. In reality, the biogas conversion rate is about 60%, the unprocessed CO2 could be used to produce carbonate water, but we can’t take that many risks of leakage.
Cultivated soils exhale 7-8 tons of CO2 per hectare per year, so 3.04 tons per acre, and by our counts, a single human would need 0.05 acres of land, so 0.42 kg/day.
Exhaling air from a human contains 4.4% CO2, which amounts to about 0.7-1kg of carbon dioxide per day depending on activity level, because the only function of these methanogens is to process respiration waste, and not biomass, we can get away with ignoring the fermenting time it takes for them to process solids.
With leeway, I calculated the methanogen-to-aerobic respiration ratio to get rid of 1.5 kg of CO2 per crew per day, as 9-10x.
I haven't found anywhere relating microbial mass to methane yield, so I'm using a cow's rumen capacity and methane yield as my proxy biodigestor, with 100 liters rumen populated by methanogens and other bacteria it can produce up to 500 liters (357g) of methane per day, so for 0.5kg of CO2 → 0.182 kg of CH4, we need at most 50-150 L biodigestor per person and their cubes of dirt, which I'm assuming has a sludge consistency 1.3 g/cm³, that's extra 65kg of mass per person, minimum. Or 200 kg per person if you're neutralizing atmospheric carbon through this process.
So here is the thing, a single person in our expedition here needs for a year: be able to produce and farm 840 kg of food.
0.05 acres of land to farm vegetables, 1m thick, 416.8 tons of soil.
3.41 tons of bacteria within the soil, plus ~70 kg of methanogens.
1 kg of fertilizer.
2.66 tons of human usage water.
121.4 tons of water for plants.
62.58 tons of water dedicated to microbes.
Total phosphorus onboard 53kg.
TOTAL MASS (so far) 707.7 tons, or 1.6x International Space Station.
In that regard, I now understand why the ISS crew don't farm for sustenance, instead having small crops for greens and research, but they can have new rations and water be brought to them every 3 months, our interstellar expedition can't.
ARCHITECTURE AND DRY MASS
I'll be using both ISS and MIR as my references/proxies for equipment and infrastructure. I see two ways to approach this estimate, monolithic and modular constructions, monolithic stations are ones composed of a single vehicle or part.
Examples of this design are Salyut 7 and Skylab:
While modular constructions are well modular, like ISS and MIR:
While modular designs improve a lot on the amount of continuous structure and division of space, I’m actually unsure if the modular architecture can withstand thrust or significant acceleration rates required for space travel due inertia of modules outside of the main axis.
I also assume, given the crew size of both stations, that the total space per crewmember is between 100-200 m³, and from that volume will be calculated the air mass. Because there are a variety of pressurized modules in each station, with different functions and equipment, the exact mass of components vary as well, but within that mass power law, solving for the ISS and MIR pressurized volumes gives their total mass as expected.
A single habitat with 90m³ would weigh on the order of 22 tons ±5 t, while science modules with 30m³ would weigh 5.4 tons ±1.5 t.
Unless we develop FTL drives for such small ships, we won’t need to worry about generating gravity for a trip of a few months. But permanent stay will require some sort of system for gravity, a centrifuge or tether is the logical solution.
Centrifuges are more self-contained than tethers, we can put several habitats along the circumference of the centrifuge and have the crew climb up to the main axl of the ship to move to other parts.
There are a few sites that can calculate the centrifuge size for you, I’m being conservative with materials and putting up 0.5g of acceleration on it, for a radius of 112 meters rotating 2x per minute. Luckily for them, the 112 meter climb won't be a burden, because the gravity will decline the closer one gets to the main axl.
With a module length of 13m and diameter 4.2m (space station sized), with 90m³ of pressurized volume. About 54 of the modules fit tightly end to end on the centrifuge wheel, totaling 4,860 m³, and the rim weighting over 4,000 tons. Because here we have way more space than a few space stations combined, I'm considering it safe to assume we could move part or the operation to the centrifuge.
For an active 5 person crew, we have 0.25 acres of land that goes on another centrifuge with 156 habitat-sized modules side to side, and 69 modules in double line and triaxial symmetry (for balance), used for farming. The whole structure weighs 5,000 tons while dry, plus 2,084 tons of soil and 607 tons of crop water, 7,691 tons total. The farm ring needs lighting as well, let's position a 5kg 280 W full spectrum lamp on top of the plants, one per square meter, extra 5.06 tons.
The whole crew will be using at least 750-1000 m³ combined. Leaving 3,860m³ free for more equipment, recreational areas, more resources, and tool storage. Like proper bedrooms, dining areas, etc. Assuming a few of these modules lead or are designed as aircraft to be used for decent once on their destination, while others serve as genetic banks. Because both centrifuges spin at the same rate, that is, have the same gravity, we can couple them close together.
We need a place to store 315 tons of water destined to nurture microbes and the crew onboard. Large spherical tanks along the main axl can do the job, less pumping work on the way down into the centrifuge areas. 76 spheres with 10m radius can do the job, plus 2-4 extra (just in case), still using our volume-mass relationship we get 3,020 dry tons for the tanks. The best way to arrange them is along the main axl, with about 2m in between them, and arranged in four opposing lines of 20, the length of the water tank section is about 440 meters.
The air mass calculated from the total pressurized volume (~25,110 m³) here is around 31.89 tons of air. Plus our expedition needs 6kg of raw fertilizer per 5 years for the trip.
The sum of all our calculations here yield approximately 15,063 tons, not counting thrusters, fuel, and the whole rig that holds this together. While input of the raw volume gives 33,572 tons. Which I find interesting because the sum of the ISS 915.6 m³ in 22 ton modules give 223,8 tons, about half of the total mass in structure and unpressurized life support and scientific modules. So between 30 and 60 thousand tons is the mass of a 5 crew spaceship.
PROPULSION PROBLEM
We need rockets to make this thing go interstellar, or to go anywhere for that matter. I'll be using the Falcon Heavy first stage as my proxy. At 8.2 MN thrust in a vacuum, exhaust of 3050 m/s, provides 3 km/s of delta-v.
Since we need to escape the solar system, our final cruise delta-v should be above 16 km/s, and so I'm calculated fuel initial mass for 20 km/s delta-v… 20 rockets, each pushing this ship contributing for 1 km/s of delta-v, 69,400 tons of fuel per rocket. And double that amount because we gotta decelerate on our destination.
This is just out of question unless we can fancy ourselves some really good non-chemical fuel thrust source, like those ion or nuclear fission plasma thrusters, at 5-10% the speed of light, it is feasible to reach nearby star systems in a few decades to centuries.
MISSION LAYOUT PROBLEM
OK, so there is no way we can arrive at the destination with the same crew we've put in, basically.
We can't put out a sleeper mission, and what's the point of uncrewed missions in this post? We want humans. And so we don't need humans for like 99% of the time for this to work. Granted that until we solve the thrust source problem, we will also get to the artificial uterus technology, and also good enough AI that can raise our little crewmembers into colonists once they arrive. And so, depending on how complete our mission layout is (raise them on orbit or ground), we can basically discard all our work until here about human sustenance in space…
So one way we can solve the ship mass problem is to simply diss the entirety of the human sustenance modules and keep only the essential, in a way, Avatar’s Interstellar Vehicle Venture Star does this job well, with its payload necessitating little space aboard, since the crew is asleep for most of the trip awakening prior to the descent procedure. Except what is more realistically likely to happen within the next century is to solve the artificial intelligence problem, and the ethics of raising humans in space without parents or choice of career.
REMINDER
Your destination MUST HAVE enough phosphorus in acessible form to sustain a growing population, else you're stuck.
-M.O. Valent, 05/06/2022
