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I found a really cool sounding order-of-magnitude modeling question but am a bit at a loss on how to approach this:

The Milky Way contains 100 billion stars and has a radius of 250,000 light years. Suppose that humans set a long term goal to colonize every inhabitable star system in the Milky Way and that 10% of star systems are suitable. This colonization might be accomplished by sending out a ship of settlers to a nearby solar system, establishing a colony, and then waiting long enough - 500 years? - for the colony to develop enough that it can send its own settler ships. Do an order of magnitude estimate of how long it would take to colonize the galaxy, explaining any assumptions you make.

My first assumption is to put our solar system into the center of the galaxy with the argument that this makes calculations more symmetric, assuming that it will change the results by a factor of order 1. (I'm shorter away from the "ends" of the solar system but I benefit more from the hopefully exponential growth).

Now the question is how to model things further. A crude estimate of the average distance between inhabitable star systems tells me that they are about 5 light years apart from each other. I guess since 5 is much smaller than 250,000 I could move to some sort of continuum description of colonization progress, but then I'm not sure on how to proceed from there.

Another important question would be the speed of my colony ships. I guess it should be at least on the order of the escape velocity of the earth-sun-system, so again using crude estimates I'll get something like $5\cdot 10^4\, \mathrm{m}/\mathrm{s}$ which means that a ship needs about 5000 years for a journey.

If the time per journey was significantly larger than the time to prepare the next ship (500 years), I guess I could ignore the 500 year delay and assume instant readiness, but that doesn't seem to be appropriate here. And if I don't do that, then the process seems a bit tricky: During the 5000 year journey of the first ship, I can produce 10 additional ships and send them to a journey, i.e., growth of colonized space is not just proportional to the number of fringe systems but depends in a non-obvious complicated way on the number of colonized planets AND when they were colonized. (Similar to "broadcasting in networks" scenarious?)

Maybe someone immediately recognizes a commonly used underlying model that is appropriate here?

PS: I'm open for suggestions re tags.

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2 Answers

This is stupidly simple, so perhaps not what you're looking for, but here's my thought: given an average journey distance of 5 ly and time of 5000 years plus 500 year "stops," the furthest distance from Earth that can be colonized grows at an average rate of no more than (5 ly)/(5500 yr). So just multiply this velocity by the distance from Earth to the furthest planet in the galaxy.

Of course that's an exceedingly rough estimate, but think about the possible correction factors that you could apply to refine that estimate, and see whether they would add up to multiple orders of magnitude. I suspect they will not.

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Well, I guess the biggest correction would come from the fact that the number of suitable stars/planets a distance $r$ from earth grows linearly with $r$, so the time to cross the galaxy once, as you calculate it, is probably much lower than the time to reach each of the suitable planets. –  Lagerbaer Dec 2 '11 at 1:03
    
Yeah, I was kind of just thinking about the colonization "front." I'd just leave the rest of the galaxy to take care of itself :-P –  David Z Dec 2 '11 at 1:05
    
Addition: I would estimate the distance of the habitable stars from their density: $L \approx \sqrt{\rho} = \sqrt{\frac{Area}{Stars}} = \sqrt{\frac{\pi(250000ly)^2}{100\cdot10^{9}*10\%}}\approx 4.4ly$ @Lagerbaer: corrected –  troyaner Dec 2 '11 at 2:36
    
@troyaner The thickness of the galaxy will also matter a great deal. Using your general method with the volume I get about $17 ly$. But we should be in agreement that the travel time between stars will dominate either way. –  AlanSE Dec 2 '11 at 3:23
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a. Assume ave distance between any star system at 5 l.y. You can colonize anything and an asteroid or airless moon is a lot simpler than a habital planet in which decades may be needed to understand the bio-chemistry before popping any neck-seals of a helmet... Colonists may be building orbitals interim. Colonists, in fact, may be 'grown' from embryo from robotic Orion starship. See c.

b. Assume 10% speed of light using Orion nuclear pulse engine propulsion. Ave time to any system in any direction then = 50 yr

c. Assume two, or maybe three generations before original Orion starship is refurbished, or enhanced using technology acquired from research or communication from homeworld, or new Orion starship is built. Will not assume significant velocity upgrade, however. Need Bussard Ramjet for this - constant acceleration. So, 100yr minimum in-system turn-around before next ship leaves outward.

d. Assume 3,000 yr colonization period without taking into effects of Great Filter or Copernicus Effect.

Can you make a 3D map of this?

duanepeed@yahoo.com

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