Just like everything else on-board, the human population itself (as well as for all livestock and pets) needs to be self-sustaining. There need to be enough crew to keep the systems running, enough farmers to crew everyone's food. Even if you automate, you just end up replacing a workforce of laborers with an equally large workforce of repair technicians.
"But Sean," you say, "in 1910 there were 14 million farmers supplying 100 million people. Today (2021) there are only 3 million farmers supplying 300 million. Surely automation will affect your assumptions."
To which I will reply, those figures (14 % the population in 1910 vs 1% today) don't account for the fact that a lot of work that goes into farming has simply been shifted into other industries. Yes there are only 3 million people with "Farm Worker" on their business cards. But those workers operate farm machinery. An industry which employs hundreds of thousands of people. Those farm machines operate on petroleum fuel, an industry which employs millions of people. And even the farm machinery companies nowadays are just the top of a pyramid of sub-contractors and suppliers which employ millions more. Caterpillar may make tractors, but they don't make their own steel, plastic, or computer chips.
Both modern and historical numbers are also questionable because a lot of farms use labor that is off-the-books. In the past: children. Today: migrant workers. And before you give me the side-eye about the children remark, need I remind you that the reason most school districts up until recently have a summer break is so that the students could work the fields at home.
So whatever population you decide that you are going to need for the actual services provided by the settlement, you need to factor in the people who are going to support the people doing the actual work. Farmers. Technicians. Physicians. Teachers and Day Care providers.
After a while this space settlement ends up sounding like just a small town. And basically, that's the idea. It is far enough out that you need to provide all of the creature comforts of home. And that includes inhabitants of the opposite sex, and all of the facilities for caring for little inhabitants. With allowances for family leave, etc.
The concept of a minimum viable population is often floated in space exploration. I've seen figures as small as 180 people. This is bonkers. You don't see any viable small town with a population that small.
The size of the population effects the services offered locally. A model known as Settlement Hierarchy charts this:
|Miniscule||< 150||Limited to none|
|Village||150-1,000||Church, Grocery Store, Post Office|
|Town||1,000 to 10,000||Basic medical facilities, banks, supermarket, emergency services, schools, commercial zones|
|City||10,000 to 1,000,000||Advanced services (doctors, expert mechanics, etc.)|
|Metropolis||> 1,000,000||Public infrastructure, political influence, luxury services, universities|
As I stated, a viable population center is a growing population center. But how much growth is sustainable?
I am going to assume that most space settlements are going to have some sort of core industry that requires a certain sized workforce to operate. Well, people are going to move into your town. But if they really like the town, they will stay even after they retire. The younger members of your work force are also going to want a family. So women in the workforce are going to be taking a few years out to raise children, and those children need to factored into your logistics just like the retirees. They don't contribute to the workforce, but you can't have a society without them.
For an example of a frontier port at the edge of the Solar System see: Population Case Study: Intersteller Generation Ship
For an example of growth in action in a sealed system see: Population Case Study: Interplanetary Logistics Platform
Settlements are built for a reason. Is it a mining outpost? Is it a shipyard? Does it serve as the Solar System's major vacation resort? And for every person who is working towards the Settlement's stated mission, there are a number of people who work behind the scenes to support them.
We have a lot of data on the expected number of service workers for a given population of people, going back to the middle ages. In a stable settlement 44% of the population will be non-laborers. 10% will be operating in a support capacity. That means for every worker devoted to your settlement's core industry you need to allow for 1.14 inhabitants that are not devoted to that industry. A settlement with 100,000 workers will have:
|Services||Craftsman - Misc||1419|
|n/a||Women on Maternity Leave||1400|
|Services||Craftsmen - Artisan||568|
|Services||Craftsmen - Food||508|