So you have a scenario in which some number of people are stranded, in some context, on a distant world. What happens to them? This and variations on it are a common theme in several categories of fiction.
My context is that a world is under some kind of terraforming effort, and there are a few dozen to a few thousand people involved and resident on it. That world’s backstory is that it orbits a sun that is similiar to Sol, they having been born in the same stellar nursery billions of years ago. This sister star is now far away, but with a similiar start produced some similiar planets. One is Earthlike, and life started there as well. That life evolved to multicellular forms. The world had it’s own ‘great oxygenation event’ followed by a long period of generally increasing oxygen.
BOOM! A nearby star goes supernova, wiping out all life, bathing the world in radiation that penetrated deep, resulting in total extinction, perhaps even destroying organic molecules of more than a dozen atoms. This for the background. This event came and went several thousand years ago. So now we have this sterilized earthlike world with a nitrogen – oxygen atmosphere, oceans, continents, islands, and a climate similiar to Earth. An axial tilt results in seasons. Maybe there are some moons, but none as big as our own. The soil, though sterilized, is fertile. Maybe the day length is a little or a lot different. You can do a lot with this.
Humans arrive, and after some investigation, decide “let’s terraform this place.” There is no other life to conflict with. You can do this. No alien-predator situation. No hidden, lurking, monsters. Just a planet you can seed with Earth biota. Assuming you can get it there. You would be picky about what you brought to seed it with. We can imagine a space travel FTL technology that makes it possible. And a logic in doing so. Earth has many times had extinction events. Some came close to being total. And now, it’s possible to garantee that never can happen again because there will be samples of Earth life on this new world, expanding their habitat, and evolving. Maybe even humans.
And then…
The ships no longer arrive for whatever reason. No resupply, no new people, no going back to Earth.
Parameters
How many people make a viable population? This is a multi-modal question as there are different frames.
Genetic
Sufficient genetic variability to not end up so inbred as to become super vulnerable to some disease or hereditary condition that extinction is a near future certainty. Different sources give different numbers, but 100 is a start. Even so, it’s a challenging start.
Better Genetics
Have sufficient genetic variability that there is a capacity to adapt to a changing environment, not merely survive in the existing one. How many people is an open question as you need to consider how much change might occur, and that will affect the numbers.
There are many interesting genetic variations on Earth. There are people who have adapted to high altitudes, people with sufficient melanin to gain some resistance to skin cancer, the ability to tolerate lactose, malaria resistance, and others. There are also genetic weaknesses and vulnerabilities, but that is more a concern for the first case.
If you start with, for example, 3,000 people, they may still be genetically not very diverse. Your future people can possibly work around it by ‘managed reproduction’, requiring genetic tracking and assigned mates. Not a happy situation and likely unstable as people just might rebel in the near future.
Technological
How much tech could be preserved or reimplemented to sustain more than a paleolithic society? And how many people would know how to identify a suitable rock and shape it into something useful?
Your colony has a bunch of technology. If they are terraforming, they are going to have computers, machinery, possibly boats and aircraft. An array of communication, climate, and observation satellites might exist, and the gorund station to support them. And also simpler items such as rechargeable flashlights, knives, hand tools and farm implements, not to mention a variety of clothing and communications devices.
The problem is that it all eventually wears out or breaks down. Even if you have some people to do repairs, eventually there are no more spare parts. All those things were shipped in. There is no manufacturing capability. There are no mines, and no smelters, no factories.
You might have a machine shop, but your only really complete capability is likely farming corn, vegetables, grains, fruits and so on. Why? Because if you have a few hundred or few thousand people and you have to bring them all their food, that is all you will be doing. It pays off to make your colony completely self sufficient in food production as early as possible.
Now much of it might be mechanized, but the gardens and farms can be manually managed with hand tools. This is a good starting point, and sets a base capability.
More Tech Needed
So your abandoned colonists have a farm and vegetable gardens. Maybe they even have some sheep and chickens. Overall, survival is assured. And as the population grows, so will the farms, gardens, and animal population. You could stop at this point, job well done, and they live happily ever after.
But that’s not what happens. Eventually even the hand tools wear out, break or are lost in one or another small disaster. You need to be able to sustain the inputs to agricultural operations.
One thing thats not too challenging if you know how or can figure it out from whatever library you are left with is finding, mining and smelting metals. A little copper, a little iron, and soon you are able to make hammers, shovels, knives, mesh metal nets, and even chains. You of course need people to do this. Not a lot of people, and those have to be fed, but a few. And over time, practice to get usably good at it.
Because you can mine and smelt copper, you can draw copper and make wire. If you can insulate it, you can preserve electrical technology. You are not going to make LEDs, computer chips, or anything close. But its not a stretch to assume a small colony could support things like power generation and transmission on a small scale, and telegraphy. In addition to communication over a distance, they may be able to build such things as motors and pumps. Given the colony size and how many people can be spared for such things, some things will be prioritized, some possibly forgotten for centuries. You can hand pump water for irrigation and other uses, but an electrically powered pump can free up people for other tasks.
Other potential tech is heating and cooling. Air conditioning is a simple tech, but your colonists have to know how to create copper pipes and valves. Chemicals for cooling are another matter, also somewhat easy, but again, you need the numbers to support this and everything else.
Some other tech is desirable, especially medical technology. And that’s a hard one. But you can, with other capabilities, have decent sewage and water purification systems. At the least, you can prevent a lot of diseases with those.
How do you make a pen, pencil or paper? Everything is a challenge.
Sociological and Political
So you have a thousand people in a town of some sort that is the center of the terraforming project. Now what?
Humans have developed a lot of political and social systems. What would a particular group do best with and how would you make that possible and sustainable in the face of competing interests?
Government is not necessary. In the past, many societies, even some decent sized cities, have been entirely self organized. Maybe your colony could start out this way. Later, or sooner, maybe a coalition of aggressive ‘we are the anointed’ forcibly take over through violence and threats of violence. There are situational defences against such things, but not all situations are equal. The best defence is to be able to go someplace else. If your terraforming was sufficiently well along, this is possibly viable.
You could have almost any kind of pre-existing administration and this would persist for a while. But the people and the system would need to adapt to the new reality. It is possible the terraforming project persists for a while. But eventually, the colony will be looking inward exclusively.
Will they apply a selective breeding (of humans) campaign to sustain genetic diversity? It’s popular as an idea, not so popular in reality. Arranged marriages are a long and persistent part of human history, but this adds a new dimension. An alternaitve is the Genghis Khan system where one, or a few, men “own” the right to reproduce and father all children. Up to 1 in 200 people alive today can trace their genetic ancestry to Khan, so this is not a far fetched idea. Humans have a disturbing history of the extermination of every male in cities but the enslavement of the women as sex slaves.
You might get some kind of direct democracy. This has happened quite a bit too. From ancient Greece to the buccaneers of the Carribean. What we recognize as ‘the state’ today is a relatively recent invention. There are very many ways to organize a society, and many will work to a greater or lesser extent. What you are unlikely to get unless you set up for in advance is anything theocratic. But humans can surprise you.
Driven by Beginnings
My way of doing this is kind of world building is to set up the inputs, then work out plausible outcomes. Pick something and run with it. Which leads to yet other possibly plausible scenarios. The story unfolds in my mind, going where it needs to go.
Others might set up the starting point and end game, then build a story path that connects them. This can work. But it seems that will force a lot of contrived and occasionally obviously contrived situations to get there.
For me, what happens next is based on what the various people in the story are like, what is the context of their situation, what information and resources do they have available, and what possibilities do they consider. Being in character matters.