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I was told this sort-of riddle by someone, having to do with the proof for the finite age of the universe, and I'm not sure how to approach the answer.

Assuming that the entire universe is uniformly filled with sun-like stars (let the sun-radius be $R$) with a density of $N$ stars per cubic Mpc, how far out into space one would have to look, on average, before the line of sight intersects a star

It seems like a simple calculation, but I can't seem to wrap my head around it.

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This is the topic of Olbers' paradox, and might more properly be addressed at Physics.SE. Cf. Is observable universe an explanation against Olbers' paradox? – hardmath May 7 '13 at 16:49
I don't think this belongs on Physics.SE. It's a perfectly well formed math question. – Alex L May 7 '13 at 18:57
@hardmath: Related to Olber's paradox, but not the same thing. – bob.sacamento May 7 '13 at 19:03

Imagine your line of site as a cylinder of radius R. It stretches to the point where, on average, you would expect there to be one star in the cylinder, i.e.

$\pi R^2 l N \approx 1$


$l \approx \displaystyle\frac{1}{\pi R^2 N}$

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Why a cylinder of radius R? – Alex L May 7 '13 at 18:51
Think of a line going through space filled with stars of radius $R$. Now, alternatively, think of the stars as points, and the "line" as having a half-width of $R$. The probability of intersection should be the same, but it's easier to think about it as a line having a width. Or, try this: Instead of thinking about how far out your line of sight has to go, think of how far do you have to pull all of space toward you before you get whacked by a star? – bob.sacamento May 7 '13 at 19:05

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