I'm going to ignore genetic diversity because the necessary population to sustain a technological environment will always demand enough people to satisfy that requirement.
Assumptions:
Maintenance of the basic technology that brought them to the new planet. Not necessarily rocketry... but vehicles, automation, manufacturing, etc. In other words, I'm assuming we didn't spend trillions of dollars to move a population to a new world so they could wear skins and fish with spears sharpened by broken rocks.
Self-sustaining colony. All the mining, farming, logging, maintenance, manufacturing, etc. must exist within the colony.
The colonization process brought a single set of all the tools needed to get through, say, a year, but after that the colonists must provide for themselves 100%.
Automation is commensurate with what's available in 2017. Automation changes these numbers RADICALLY, but the OP doesn't really tell us anything other than it's a intrasystem space-faring society in the near future.
Family Size
Your average family in the United States is 2.53 people. I'm not even going to speculate on what 0.53 people looks like. Worse, even if you round it to 3, that's two parents and one child, which is a decreasing population. You need 4 to break even and 4.1 (better known as 5) to grow. Let's favor 5.
Out of that 5 people let's assume you have two adults, one sub-adult, and two children. That's 2 laborers, one laborer/secondary education, and 2 dependents requiring child care and education.
Your biggest problem is children. They can't be ignored. If you arrive with no children and expect to start making babies you need adults for child care immediately and both education and medical by age 5-6. Therefore, you really must account for children at the outset or your fooling yourself. You could squeeze my final number by minimizing children based on age distribution so that you're getting replacement adults "just in time," but that's much more complicated.
Farming
From here we learn that on average one farmer can feed 155 people. This assumes an established farm in 2010. If we assume the farmer has a family and that he can draw on a "basic labor pool" to get him up and running, then our farmer +4 (family) can feed 150 people or 5:155.
farming = 5T/155 where T = Total population (keep this in mind).
Logging
I thought I could assume if we can get this group to another planet then we can construct with plastic and metal and therefore there is no need for logging. However, it's likely you need crates, paper, rubber, composting, and who knows what other non-foodstuff organics. I can't find statistics for how many people are involved in this. Let's use the farming number and assume one family can serve an additional 150 people with non-food organics. So, another 5:155.
Logging = 5T/155
Mining
This is one of the more problematic issues. Different materials appear in different locations on a planet. This means our colony has many labor centers, all requiring administration, law enforcement, etc. I'm going to define mining as "anything we take from the ground that we can't eat," so it includes petrochemicals.
According to here the average citizen of the U.S. needs 40,000 pounds (20 tons) annually of materials of over 15 types. (I'm being optimistic about the "and other" categories and counting them as one each.)
Mining requires blasting/digging/drilling, hauling, and processing. I have friends who work the hard-rock mines in northern Idaho, and even with automation, they have hundreds of laborers. Granted, it's production mining rather than subsistence colony mining, but still.... Let's assume you need 2 blast/dig/drill, 1 haul, 2 processing for 5 people (25 with families). I must assume a massive ratio or this simply doesn't make sense, so let's assume 25:3000 per item on average.
Remember, that's 2 blasters/diggers/drillers producing their share of 60,000 tons per day. It can be done. I must be done, because...
Total: 825:3000 (see my problem?)
mining = 825T/3000
We need to sub-calculate the number of labor centers for later calculations. That would be T/150 + T/150 + T/2175 = 30T/2175.
Transportation
We need to get things from one place (e.g., farm or mining location) to someplace else (at least another farm or minining location). If we only assume two drivers + families (trucks or trains... pray we're not dealing with trains...) per labor center.
trans = 10(30T/2175)
Maintenance
Keeping the equipment running is a very difficult variable to define. Let's assume two full-time mechanics per labor center.
maint = 10(30T/2175)
Education
The United States has 5.49 teachers per thousand students. We have some austerity going on, so 5:1000. Secondary eduction is 5.5, but this is more important, so 6:1000. Plus families.
education = 55T/1000.
Medical
Keeping all these people alive will be a problem. Keeping them healthy an even bigger one. You've gotta love the CIA, who suggest 2.55 physicians per 1,000 people in the U.S. I doubt this includes specialists and it certainly doesn't account for multiple labor centers. So, 2:1000 MDs and 1 MD per 3 labor centers and 5:1000 specialists. Plus families.
medical = 2T/1000 + 10T/2175 + 5T/1000 ≅ 25T/2175 + families = 125T/2175
Administration
This includes everything from paper-pushing bureaucrats to police, firemen, code enforcement, phone answerers, etc. My right-big-toe tells me we need 3 per labor center + families.
admin = 3(30T/2175) = 90T/2175 + families = 450T/2175
Manufacturing
According to here, manufacturing jobs are 8.5% of the U.S. workforce. Let's assume that's only one parent in a family (like everything else).
manufacturing = (0.085)T/5 + family = 0.085T
I'm going to assume this number includes drivers, administrators, maintenance, etc. It's going to be a bit low because of the compounding affect of unassigned workers, but all that would happen is fewer unassigned workers increased by the same amount of additional manufacturing jobs. So the total population estimate should remain "accurate." (ahem).
Entertainment
I'm going to ignore entertainment of any and all kinds. Colonists should plan on bringing kazoos.
Unassigned Workers
I've intentionally not tried to compound the spouses or sub-adults into the workforce. This is because there will be jobs ranging from street sweeper to technical assistant that I'm not even going to try to estimate. All those jobs must draw from the unassigned labor force (and sub-adults, as necessary).
What am I missing? I'm missing retail outlets, which would presume some kind of town or village. For an initial colony, central distribution or coordinated transportation could solve this until the community begins to grow. I'm sure I'm overlooking/ignoring a lot of other things, but this is a long enough post.
That's a mess... how do you calculate the minimum population?
We're going to iterate through the equation.
We'll use T=1 to find our starting point and ignore the divisors so we have one of each primary laborer plus their family, which means our "initial population" is 2,070. We know this isn't right because we don't have enough raw goods to feed/supply all those people.
We want to assume that we never increase a number unless the population has actually exceeded the amount required for the increase. No fractional doctors, please, otherwise all the fractions add up to wholes that run out of control. However, this means people are a bit overworked... but that might be expected on a new colony.
With the exception of our raw materials. We must have enough farmers, loggers, and miners. So we'll round those numbers rather than keeping them at the floor.
Total minimum population: 2,260
- Children: 904
- Sub-adults: 452
- Unassigned Workers: 452
- Farmers: 14
- Loggers: 14
- Miners: 123
- Transport: 61
- Maintenance: 61
- Education: 24
- Medical: 25
- Administration: 92
- Manufacturing: 38
Is this realistic? To be honest, in real life you probably need 10X this number of people... but I can't prove it without spending 10X the time to analyze the situation.
My Program (PHP)
$a = 100;
$t = 1;
$ifar = 5;
$ilog = 5;
$imin = 825;
$itra = 300;
$imai = 300;
$iedu = 55;
$imed = 125;
$iadm = 450;
$iman = 5;
$t = $ifar + $ilog + $imin + $itra + $imai + $iedu + $imed + $iadm + $iman;
$count_check = 0;
$count_max = 1000;
while($a > 0.01){
$far = 5 * round($t/155);
$log = 5 * round($t/155);
$min = 5 * round(165*$t/3000);
$tra = 5 * floor(60*$t/2175);
$mai = 5 * floor(60*$t/2175);
$edu = 5 * floor(11*$t/1000);
$med = 5 * floor(25*$t/2175);
$adm = 5 * floor(90*$t/2175);
$man = 5 * floor(0.085*$t/5);
$pop = $far + $log + $min + $tra + $mai + $edu + $med + $adm + $man;
$a = abs(($pop - $t)/$t);
$t = $pop;
if($count_check >= $count_max){echo "\n\nFAILED TO CONVERGE!\n\n"; exit;}
$count_check++;
}
echo "\n\n";
echo "Total Population:\t".$pop."\n";
$children = 2*$pop/5; echo "Children:\t\t".$children."\n";
$subad = $pop/5; echo "Sub-adults:\t\t".$subad."\n";
$spouses = $subad; echo "Unassigned Workers:\t".$spouses."\n";
$far /= 5; echo "Farmers:\t\t".$far."\n";
$log /= 5; echo "Loggers:\t\t".$log."\n";
$min /= 5; echo "Miners:\t\t\t".$min."\n";
$tra /= 5; echo "Transport:\t\t".$tra."\n";
$mai /= 5; echo "Maintenance:\t\t".$mai."\n";
$edu /= 5; echo "Education:\t\t".$edu."\n";
$med /= 5; echo "Medical:\t\t".$med."\n";
$adm /= 5; echo "Administration:\t\t".$adm."\n";
$man /= 5; echo "Manufacturing:\t\t".$man."\n";
echo "\n";
