One thing that necessarily happens on a generation ship is that people die of old age. Now that poses a tension: On one hand, you need to handle the dead with dignity. On the other hand, you probably cannot afford losing the materials the dead are composed of.
So in short: How would a generation ship responsibly handle the dead?
To complement @Ville Niemi's suggestion Thermal hydrolysis, there is already an equivalent in the context of death customs. It's called Alkaline hydrolysis. Although alkaline hydrolysis is rather controversial in that it "does not show sufficient respect for the teaching of the intrinsic dignity of the human body" as the New York State Catholic Conference put it – link in wikipedia – whether it is acceptable or not depends largely on your culture.
Note there are also alternatives to burial, e.g. "The Yanomami have the practice of cremating the remains and then eating the ashes with banana paste." However, this so called Endocannibalism can – in it's traditional form – transmit the incurable degenerative neurological disorder termed Kuru if your population contains infected individuals.
Regarding prions the CDC states on cremation
[...] cremated remains can be considered sterile, as the infectious agent does not survive incineration-range temperatures.
The alkaline hydrolysis process, using a pressurized vessel that exposes the carcass or tissues to 1 N NaOH or KOH heated to 150°C, can be used as an alternative to incineration for the disposal of carcasses and tissue.
The process has been shown to completely inactive TSEs (301v agent used) when used for the recommended period
@user3082: Thanks for your contribution. I edited my answer according to your objection.
Soylent Green are people.
It's the new recycling.
On a more serious level, I'd suggest looking into what are now known as woodland burials. The body returns to the forest as nutrients. You could do the same in the biosphere pod.
One variation of the woodland burial involves freeze drying people in liquid nitrogen then breaking the frozen remains up into little chunks that are a lot more biodigestible.
Organ transplants and Cannibalism
Humans are resources, they have:
Once organs are harvested, the remaining biomass can be pumped dry of water, solid mass like bones then taken out, skin removed, and the rest used as fertiliser. Anything not used there and then can be stored for future use.
There's also a multitude of other things to be done:
Thermal hydrolysis combined with a bioreactor of some kind. Turns organic waste and dead bodies to fertilizer and nutrients. There probably would be separate systems for managing sewage and the dead, though. This would not really be that different from cremation, but should waste less energy. The methane generated by the bioreactor could be used to power the hydrolysis producing carbon dioxide and water. Which together with the fertilizer produced would maintain food production.
Dignity is a society-invented concept. In one society, it's dignified to bury a corpse under ground in a designated graveyard. In other, it's dignified to leave the corpse in high ground to have vultures eat it, called sky burial. Yet another used small ships or boats to dispose of the body with honors. Ferengi from Star Trek broke the body in collectible pieces to be sold as "last profit". Very dignifying procedure - for a profit-centered society.
A space-faring society would inevitably work out their own ways, and it will be those ways they'll see as "dignified". Most likely consider our own ceremonies shocking, despicable, barbaric and wasteful.
Bottom line: make it any way you see fit. It's not for our society to judge it.
Approximately the same way we do it here: Composting, by way of burial.
Depending on the time frame and capacity, it might be necessary to carefully manage the compost pile. We might have to give up on embalming and persistent spacious coffins, and put in the dirt microbes that will digest humans quickly. Bones might need special handling. But if the capacity is high enough and the time frame is long enough, the need for careful management could be minimal.
Depending on how the ship's engines run, one could use the concept of the fusion torch. Essentially, the body is fed into the ship's reactor (or whatever else powers it, provided that its energy output is similar or greater). The conflagration blasts it into its component atoms, which can then be collected and sorted by a device akin to a mass spectrometer. Once you have these stores of raw elemental material, you can use them as you would any other store of that same material: it is, in many ways, the ultimate form of recycling.
Research on the fusion torch didn't work out so well on Earth, but for something like a generation ship it could be ideal. There is no sanitation risk, because any pathogens on the body are blasted just as the body is. There is very little wasted matter, because every component of the body can be recycled in this way. There is very little wasted energy, beyond what is necessary for the ship to run, because you already needed to run the engines (or at least a power plant), and the sorter can also be used to recycle other matter.
In a spiritual sense, this need not be much different from how things are done now. Many religions include a concept of "returning the body to the Earth" in a symbolic sense. The process I outline above returns the body to the ship, both symbolically and, to a degree, literally.
Cremation. The body is converted into Carbon dioxide and water vapour, which rejoin the atmosphere, plus carbon ash mixed with a few other nutrients which can be sprinkled on the soil and will fertilise the plants. Nothing is lost.
You will presumably have machinery in place to balance the levels of carbon dioxide and oxygen if your vegetation isn't up to the job.
Assuming a population of 10,000 and an average lifespan of 80 years, you can expect one death every 2.9 days or so. An average North American adult human is 80kg. That's 1.1kg per hour, most of which will be water. This is a very moderate amount of carbon dioxide for a good size biosphere to soak up.
I can see a few methods:
1. Cremation
This is already a popular method of dealing with your body when you've passed on (at least here in the UK). A furnace doesn't take up much space, and especially on a ship where you have huge amounts of power, it certainly won't take up much of it. You could then drop the ashes out into space - one astronomer is doing this with Pluto - but you have to decide whether that's ethical or not: is it polluting the Universe?
2. Cryogenic freezing
We do already have the technology to do this one. When someone dies, you can give them a coffin, a funeral, and then put the coffin in a cryogenic freezing unit until you get to your destination. The big disadvantage of this is that it does take up a lot of space - with that many people on board, you're going to need a big room to keep all these dead people in. And of course, it'll be a lot of work to get them all off the ship when you get to your chosen planet.
3. Sanitary Vacuum Disposal System
Also known as "throwing bodies out into space", this method is the simplest: after the funeral, you stick the body in an airlock and open the outside, thus letting them fly off into space. This might actually appeal to some people because of the almost romantic ideology that your dead relatives are out there somewhere, watching over you - in this case they really are out there. However, this does again have the problem of are you polluting space?
While I side with @RoboKaren (my version is a woodchipper in the forest), and I like the other unique answer, @VilleNiemi - I might be persuaded to argue with @TonyEnnis.
Depending on the length, and losses (and surplus material) available on your ship/trip, it might be possible to save some parts of people for burial on the destination planet.
I'd think it'd be something like 'save the skull' or 'save a bone chip', while recycling the rest. But, only if the biosphere doesn't need the materials.
I anticipate that any generation ship is going to have losses; guy who suicides out the airlock. Guy in the collection/maintenance probe whose fuel tank exploded and jettisoned him into deep space... Etc.
Hugh Howey's "Silo Series" touches on your exact problem. The inhabitants bury the dead in the growing levels and the bodies decompose and are used for food crop growing. It even mentions how there is a different smell in the growing levels compared to the rest of the habitat.
Social conditioning over a couple of generations could make it so that people honour the food grown, literally, by their forebears which has the added benefit of stigmatising food waste on a generation ship.
Begin Flowchart:
Step 1: Did the deceased die of natural causes: Yes - Step 2, No - Step 4
Step 2: Does the deceased have any healthy organs that the organ bank is missing: Yes, remove and store - Step 3, No - Step 4
Step 3: Are there enough bodies in storage for medical training: Yes, store body - End, No - Step 4
Step 4: Reclaim water
Step 5: Reclaim bone - for use in fertiliser
Step 6: Cremate and process remains, remove carbon, phosphorus, remaining calcium, iron, potassium, sulfur, sodium, copper, selenium and magnesium
Step 7: Inject remainder into space/intern/return to family
End
They'd probably have a ceremony, similar to a funeral today. The funeral ceremony would depend on where the generation ship originated, as each culture of Earth seems to have their own traditions.
The ceremony could begin with kind words, or shooting a flaming arrow, placing coins on eyelids, or letting them sail with the river. Whatever does happen will be performed by a close loved one.
The family of the deceased need to grieve but still get a sense of closure. As long as the body isn't ripped apart in a chamber in front of everyone, there is a lot of inspiration with Earth's cultures as to how to handle this.
I recently read Artur C. Clarke's 2001: A Space Odyssey.
(Forgive me if my knowledge of the story differs from the movie; I haven't watched it yet)
At one point in the book, when Hal kills the three hibernating astronauts, the surviving crew-member (Dave Bowman) wraps their bodies in white, and sends them out of the airlock to make their own travels through space.
I kinda thought that this was the space version of burial at sea (out of necessity, not due to preference).
Admittedly, the craft that he was in was not a generation ship, but I thought that this incident might help explain some of the questions at hand vis a vis the question on ships -- assuming there's nowhere to place the body, and nothing to do with it (either truthfully, as in the case of Dave Bowman; or because of "dignity" issues), then perhaps this might serve as a model for the procedure on generation ships.
I believe that in a large enough spaceship, there is no reason the same "services" being provided on earth could not be provided in space. This takes care of the different "dignified" methods of handling the dead. Where the difference comes, is in the handling of the bodies after the ceremonies. If the body is to be buried and maintained buried for say 100 years, there should be a cemetery to accomplish this. After the 100 years any remains are exhumed and used as fertilizer or burned in the reactor. If the body is to be cremated, then it is burned in the reactor. If body parts are donated (to be reused), obviously, they are removed from the bodies prior to either of the two previous procedures.
As in most of the answers I believe it all comes down to culture. I think a good analogue of this is within the Frank Herbert's Dune Universe.
On Dune (A mostly barren desert planet with very limited water resources) the native inhabitants Freemen recycle and reuse all water including capturing exhaled breath, bodily waste and even the processing of corpses. Such is the value of water, that spitting is considered a very high form of respect (An activity usually frowned upon in other cultures). I believe the key sentiment in the processing corpeses is that the dead are dead and require their water (or other consitutants) less than the living.
To Quote:
“We will treat your comrade with the same reverence we treat our own,” the Fremen said. “This is the bond of water. We know the rites. A man’s flesh is his own; the water belongs to the tribe.”
The other half of the solution is that in a ship that travels for generations with a small group of people with presumably little outside contact it becomes much easier for social norms to be transformed over time.
