Time take to colonize the galaxy with interstellar arks

[notnewton96]

Homework Statement

Calculate the time taken to colonize the galaxy using an interstellar ark, making very rough approximations and broad assumptions. There are two rules:
1) The propulsion system must be something that already exists, no warp drives etc
2) Once the ark reaches a planet, the colonists forget which direction they came from

Homework Equations

See below

The Attempt at a Solution

This was a question given out a while ago. Our group didn't get very good grades on it so I'm giving it another go. The marks are awarded mostly on the theory of the answer rather than its accuracy.

I've already redone part of the question. I used an average stellar distance of 5ly. Using the rocket equation based on an ion drive with exhaust velocity 50,000ms^-1 and the equations of motion I came up with an average travel time of approximately 20,000 years which I've been told by the lecturer is ok.

But I'm a bit stumped by how the colonists would decide on which direction to head out in after colonizing a planet. Seeing as they have no idea which direction they came from the only option must be heading in a random direction. But how can I represent this mathematically?

I guess an analogy would be a board of switches (the same number as stars in the galaxy) that are all in the off position. The goal is to turn every switch on. The time taken to turn every switch on is approximately 20,000 years. The switches are turned on in a random order. Therefore every now and then the same switch is reached.

My random statistics knowledge isn't too great. So if anyone could point me in the right direction (just noticed how funny that is :P) it would be a great help. Thanks :)

 

[mfb]

But I'm a bit stumped by how the colonists would decide on which direction to head out in after colonizing a planet. Seeing as they have no idea which direction they came from the only option must be heading in a random direction. But how can I represent this mathematically?

Some reasonable models:
1) head to the nearest star only. This will quickly end in a double star system.
2) head to a single random star nearby. This gives a random walk.
3) head to all stars nearby.
4) head to a few stars nearby.

 

[notnewton96]

What I managed to do in the end was this: http://www.wolframalpha.com/input/?i=sum+from+n=1+to+1e9+of+(30000*(1e9-n)/(1e9-1))

Is this correct? I'm actually trying to do this in python now and struggling to recreate that as code. Although I have created random walk programs in python before, how would this apply to this case though? Because random walks apply to distances and random turns. I've never seen on where an action is taken at every distance step. Would this not fail to take into account the already colonized star systems?

 

[Dick]

What I managed to do in the end was this: http://www.wolframalpha.com/input/?i=sum+from+n=1+to+1e9+of+(30000*(1e9-n)/(1e9-1))

Is this correct? I'm actually trying to do this in python now and struggling to recreate that as code. Although I have created random walk programs in python before, how would this apply to this case though? Because random walks apply to distances and random turns. I've never seen on where an action is taken at every distance step. Would this not fail to take into account the already colonized star systems?

I doubt a question like this has a single correct answer. And I have no idea what assumptions you made to get an answer from Wolfram. What were they? You aren't going to have much of a chance of colonizing the galaxy in a single ark before the stars start dying. You should be able to calculate that. If you want a strategy that could actually win then once you colonize a planet build two new arks and send them out. Look up fission chain reaction. That's the way to get the job done. You'll still have to make a lot of assumptions. Just make them. Assume anything you want that's physically plausible. I think that's how this game is played.

 

[notnewton96]

I assumed that there are 1e9 stars in the milky way and that each had one colonizable star. I also assumed that the travel time and the time taken to colonize a planet was 30,000 years. Giving me an answer for one ark colonizing the galaxy of 1e13. Which seems like a sufficiently large number...

This obviously assumes that no stars would go supernova etc during this time and ignores expansion effects.

 

[notnewton96]

I've been searching for nuclear fission chain reactions to see any similar mathematics. But I don't see how I could apply this to the colonization case. I should be able to modify the function I created in wolfram but I just don't see how. Instead of the probability of finding an uncolonized planet remaining linear as with 1 ark, it would become exponential with each colonized planet sending out 2 arks.

 

[mfb]

Well, as Dick mentioned, you probably don't want to use a single ark for 1e13 years. You don't have to, if one ark can be built, it is possible to build more.

With many arks, it is sufficient to model the volume where planets are colonized - once the first colonies in some region exist, it does not take long until all planets there have colonies.

 

[DEvens]

So you need some assumptions. You've got a transit time, that's good.

How long does it take from ship getting to a new star till colonists are crowded enough to want to move on? Suppose they break up their ship to make colony material, and have to advance far enough to have enough extra to afford a new ship. Might be 10,000 years. (Think of the opening sequence of the first season of the cartoon Futurama.)

How many ships does any particular planet build? If it's more than one then the full colonization time goes way down.

Do colonies die out? If the time is billions of years you have to expect some will if for no other reason than stars going pfft.

What does a ship do when it gets to a planet and the planet is already colonized? Hopefully the colonists don't forget their path in this case. They won't repeatedly come back to a particular already colonized planet.

So, suppose a planet takes 10,000 years to build up after colonization. And then it builds ten ships. And suppose that on average 5 colony ships out of 10 fail utterly. Suppose that if a ship comes to a world and it's already populated, it moves on without stopping, so has a doubled transit time.

When there are a bunch of colonized planets in a cluster, the inner ones will have to random-walk-with-memory till they find a non-colonized. The ones on the edge will have roughly a 50/50 chance of finding a planet. So their net hop time will be 20,000 (transit) plus 10,000 (half of another on average) plus 10,000 to get to enough strength to make new ships. So something like 40,000 years per generation. And the generation produces 5 times as many as before. These are also reinforced to some extent from the "inner" planets. That gets you a billion colony ships in only 13 generations or so. And 10 billion in 14 generations. So be generous and clal it 25 generations. So 25 times 40,000 years, or roughly 1E6 years. If only 2 ships are successful it changes the 14 generations to about 38 genrations. And again to be generous call it 50 generations. So I'd say that 1 million to 2 million years is in the ballpark.

If the transit time is 10 times as long it's onlly 20 million years.

Which brings up Fermi's question: Since the galaxy is much much MUCH older than that, where is everybody?
Dan

 

[mfb]

If 5 light years take 20000 years and the journey is velocity-limited, you cannot colonize the whole galaxy in 20 million years - even if you send ships to all stars at once, they won't get more than 5000 ly away from the origin.
I think your scenario would give regions of 0% and regions with nearly 100% colonization rate, with a very thin boundary region (and many colony ships there).