How much will this civ expand?

Question

The general thrust of this question is how big should I expect the territory of colonization to be?

The factors that I know...

You have 5 colony ships with 2 million colonists each that set up colonies on 5 planets. They are cut off from Earth and have to start from scratch.

Edit: Sorry - 2 million is just the initial numbers of the first colonies. Colony ships after that hold 1600 to 2200. Also they live to around 120 years old.

I currently assume that there is 245 years between this initial colonization and these colonies start to send out ne colonies.

When they set up their colony they have the ships and can build a "jump gate". These ships can travel at roughly 30c normally, and 300c assuming their is a "jump gate" at both the start and destination, which means they need to get to any place first and then they can travel 10x quicker.

Let's say this speed holds true from year 0 (initial colonization date) to year 182. After that they can travel at 40c or 400c.

Each new colony they set up, they will start building a new colony ship and launch it when they reach a population of 2 billion. Also, these colony ships serve as a space elevator/space station when they arrive.

What i need to know is, how by this sphere of colonization will be in year 513. Assuming a continuous travel, they reach ~10,000 light years out, but that doesn't seem right that they'd colonize all that,

Also assume they have infinite energy and infinite computer processing, but need to get the resources from the planets. The stations are used as habitats too, but each person has a plot of land that they exclusively own.

So, how small/big can I arguably make this sphere of territory given these "facts"

Answer 1

I'm going to tackle the population half the problem, and come back later for the travel time half.

Making a few assumptions:

• People pair off and have four kids when they're 20 (each generation is twice the size of the last)
• People die when they're 60
• Once a colony hits 2 billion people, it makes a new colony ship every 2 million thereafter

Your scenario has two types of colony: growing and established.

Growing Colonies

A growing colony has, roughly, three generations. By the time the first generation (2Million) is 60, their kids are 40 (4m) and grand-kids are 20 (8m). In 200 years from the start, the active generation has doubled 10 times, giving a population of 2048M (plus 1024M 40 year olds and 512M 60 year olds). So a colony switches from growing to established around 185 years after being established. I'll call it 200 years given some internal wars, disasters, and resource constraints.

Established Colonies

Assuming the colony (or Earth itself) can sustain a rate of production on colony ships, and it sends one out every time the population hits 2002M, it would send out 1024 every 20 years.

Net Result

With no travel time, you've got ~520 years of Earth at full production, giving 26624 colonies. The original five colonies, with 220 production years, make a total of 81920 colonies. Every 20 years, the number of established colonies increases by 1024 for the first 200 years, then a self-compounding number after that. Suffice to say a million colonies pretty easily.