On the mathematics of interstellar colonization

[Emspak]

I've been turning something over in my head for a bit and I wanted to check some math. It has to do with how long it would take a hypothetical alien civilization to colonize a big chunk of nearby space.

Over at the https://sites.google.com/a/upr.edu/planetary-habitability-laboratory-upra/projects/habuniverse they estimate there are some 160 habitable planets in a volume of space 33.6 light years in radius. How long would it take to fill that up?

Let's further posit a doubling process. That is, we send out one ship, and that sends out two, which sends out four, et cetera.

In a volume of space 33.6 light years in radius you have a total of 158,893 cubic light years, using V = \frac{4}{3} \pi r^3 . Assuming the 160 systems you want are evenly distributed each one is in a region of 993 cubic light years, or if it is a cube a space that is ~9.97 light years on a side. A sphere of 993 cubic light years would be ~6.1 light years in radius. So the stars we can say are an average of about 9 to 12 light years apart from each other. (Anyone who can please tell me if you think this calculation is way off).

To reach the 160 habitable worlds in this volume you would (assuming they are evenly distributed) go through just over seven doubling steps -- 1, 2, 4, 8, 16, 32, 64, 128, 256 -- so between step seven and eight you hit your 160 systems.

Now, the worlds you want are separated by 9 to 12 years at light speed. Call it an average of 10. So you'd hit your target after 80-90 years, filling that volume and assuming that as soon as a ship lands they send out two more. (Obviously this isn't realistic).

But what if my volume were a sphere with 10 times the initial radius? 336 light years in this case. Now my volume is 158,893,479 cubic light years. That gets me to 160,000 planets to go to and colonize (assuming the distribution of worlds I want remains about the same). In that case the number of doublings goes from between seven and eight to 17 and 18. (I actually get 17.2) So here, assuming the same average distance, I end up with a total colonization time of around 180 years at the outside.

BUT. The volume of space I want to fill is 336 light years in radius. In 180 years even at light speed you couldn't get from one side to the other. So even though I have filled (mathematically) all the planets in the region I run into a seeming contradiction.

There's a resolution to this and I suspect it's a simple mathematical principle I am forgetting, but any help would be appreciated here. I further suspect it has to do with how I calculated the number of planets in a 10x larger radius.

Any help in seeing what might be blindingly obvious? :)

 

[jerromyjon]

To reach 160 habitable worlds

But what if my volume were 10 times

That gets me to 160,000 places

??

 

[Emspak]

??

i edited a bit for clarity, tell me if it makes more sense now.

 

[jerromyjon]

How do you get 1000 times the planets in 10 times the space?

 

[Emspak]

because the RADIUS I mentioned is 10x. (10x 33.6 = 336). Volume is proportional to the cube of radius. If the number of planets per unit volume is n, then for any n*V you get n * (4/3)*πr³ and that means n is proportional to r³. Increse r by a factor of 10 and n goes up by a factor of 1000.

 

[DrStupid]

To reach the 160 habitable worlds in this volume you would (assuming they are evenly distributed) go through just over seven doubling steps -- 1, 2, 4, 8, 16, 32, 64, 128, 256 -- so between step seven and eight you hit your 160 systems.

Now, the worlds you want are separated by 9 to 12 years at light speed. Call it an average of 10. So you'd hit your target after 80-90 years, filling that volume and assuming that as soon as a ship lands they send out two more. (Obviously this isn't realistic).

That requires that every new colony finds two additional nearby habitable planets.

But what if my volume were a sphere with 10 times the initial radius?

If the volume gets larger you will get closer to an average of 1 nearby habitable planet per colony. That means that the rate of expansion turns from exponential to linear.

 

[Emspak]

That requires that every new colony finds two additional nearby habitable planets.
If the volume gets larger you will get closer to an average of 1 nearby habitable planet per colony. That means that the rate of expansion turns from exponential to linear.

OK, so is there a good way to work out the relation for nearby planets in relation to volume? Like I noted, mathematically speaking the doubling process just means you full the requisite number of planet "slots" pretty fast, but I was trying to work out how fast the colonization wave would move in anyone dimension. Your answer makes sense in that a circle increases r and the circumference of that circle increases proportionally to r (linearly). But if I read you right a colonization wave would be limited by the speed of travel (obviously) so that once you get to a large sphere it would basically increase at our rate of ~10 light years per colonization doubling even if the saw two nearby planets each time, because a) the direction is random and b) the distribution of planets is a constant. Is that about right?

 

[M Quack]

Try to look at shells around the planet of origin. At each step you increase the radius by 10 light years, such that the n-th shell has radius 10*n ly.

Then calculate the number of new, uncolonized worlds in each shell. If the density of habitable planets is constant at 1 per 1000 ly^3 that would be

volume of the hollow shell: 4pi/3 (n^3-(n-1)^3) (10 ly)^3
number of new planets: 4pi/3 (n^3-(n-1)^3) = 4pi/3 (3n^2-3n+1)
total explored/colonized volume: 4pi/3 n^3 (10 ly)^3.
total number of planets: 4pi/3 n^3.

So, if for convenience you assume that all the habitable planets are on the outer edge of each shell, and new planets can be reached in 10 years (i.e. by flying radially outward), then the total number of colonized planets cannot grow faster than n^3. For large n that is a far cry slower than the 2^n you postulated in the beginning!

So for the first few generations you will be limited by the number of ships available, after that you will be limited by the volume of unexplored space that can be reached.