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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 [itex]V = \frac{4}{3} \pi r^3 [/itex] . 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? :)

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 [itex]V = \frac{4}{3} \pi r^3 [/itex] . 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? :)

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