How many people would it take to to run a self sufficient colony? I'd say it would take from 20-1,000,000 to live comfortably and still be a colony. But I want to know if it's possible with less than ten. Also, famous engineer and inventor Elon Musk says it's nearly impossible with less than a million people including children.
One problem with 10 colonists is you do not have a very deep bench. If a couple of them croak you may not have enough people left to continue farming. Consider Jamestown, one of the early American colonies.
The following winter, disaster once again struck Jamestown. Only 60 of 500 colonists survived the period, now known as “the starving time.” Historians have never determined exactly why so many perished, although disease, famine (spurred by the worst drought in 800 years, as climate records indicate), and Indian attacks took their toll. On June 7, 1610, Jamestown’s residents abandoned the hapless town, but the next day their ships were met by a convoy led by the new governor of Virginia, Thomas West, Lord De La Ware, who ordered the settlers back to the colony.
That is 88% mortality, which would leave 1.2 of your colonists alive. I bet the 0.2 guy would not be great company either. Unless your tech could keep his head alive in a jar, in which case he could tell jokes.
Where was I? Your colony's mortality rate will depend on the circumstances of your colony. Jamestown suffered from cold weather and famine. The missing colonists at Roanoake were probably killed by natives. Disease can be a factor. 10 people with advanced tech can work a farm and sidestep disease but if you lose some people, you may come to a tipping point where even with tractors you cannot produce your own food anymore.
In the short term the size of your colony will depend on how much food your colonists can produce for themselves (which depends on your environment, climate, crops and tech level) and causes for mortality / mortality rate. Jamestown seems pretty miserable. But if you are a boatload of shipwrecked slaves in the tropical Caicos, with no disease and the natives long gone, you can probably live off the ocean and your gardens very nicely.
The other issue with a colony is reproduction and sustainability over generations. This will be tricky in the short term with 10 adults. Colony implies self-sustaining, which means a birthrate and babies and kids. Someone has got to bear these kids and someone has got to keep these kids from falling down wells. You need kids if your colony is not going to die with the original colonists but child care (and no old folks or older kids to do it) will cut into your workforce as well.
The real long term deal killer for your 10 colonists colony: your gene pool is too small. When these kids grow up and have kids, who with? With 10 starting colonists, everyone in the colony will be related within a few generations. You are going to have serious inbreeding problems. If this is science fiction you could assert that these colonists have been made genetically pure, or perhaps there is a vat of frozen sperm to use which artificially increases the size of the gene pool. But if it is just 10 ordinary people their descendants are going to struggle.
20,000 over 50 years
This is pretty hard to answer in a rational way, since the factors governming a space colony are many. Instead of reasoning out a number, lets see what a historical colony saw in immigration to establish it as a successful colony.
Plymouth colony (the Pilgrims, in Massachusetts) is a pretty close approximation to a self-contained colony for about 100 years. The Pilgrims left Europe to form their own society, and had little interest in intermingling either with the natives of Massachusetts or non-Puritan Europeans. 103 pilgrims landed from the Mayflower in 1620 (including one born on the way!), and 58 survived the first winter. Only four of the original 18 adult women made it to the first Thanksgiving in 1621; not an auspicious start.
37 new settlers arrived in 1621, 96 more in 1623. All in all, about 20,000 colonists went to the Plymouth colony and the total population in 1690 was about 7,000. This could probably be considered a stable colony at that point; it had repaid the debt that it owed for the land settled so it was reasonably economically prosperous. After all, the first child born on the Mayflower would have been 70 by this time; time enough for three generations on the colony. After 1690, the colony was annexed to Massachusetts and ceased being so isolated.
A Use Case - might be able to work out some rules.
I am assuming a non-garden of Eden (not Earth like ) environment so requires significant technology to survive for example Mars or in Space.
100,000 general purpose population (they can do any job with minimal training). 1% population expansion and as it takes at least 20 year to mature 20% or 20,000 children extra.
Assume 1 Specialist(Doctor, Mechanical Engineer etc) for each 200 people, therefore 500 people in each specialty. If there are as many as 50 sub-categories, then at least 10 are available in each sub-category.
For 20 specialties (500 * 20) => 10,000 ;
The total is at least 100,000 + 20,000 + 10,000 => 130,000.
If we assume that 3 or 5 in each specialty is the minimum needed then 39,000 or 65,000 respectively would be a minimum sustainable population.
Conclusion: The minimum is at least 10s of thousands.
Note: The more people the better, certainly 1 Million will significantly reduce the risk of a colony dying compared to 130,000.