# Monte Carlo problem - mean free path to a star

1. The problem statement, all variables and given/known data
(We are to solve this with Monte Carlo programming. Based on the universe from Olber's paradox)
Suppose you are in an infinitely large, infinitely old universe in which the average
density of stars is n = 10^9 Mpc^−3 and the average stellar radius is equal to the Sun’s
radius R = 7 × 10^8 m. How far, on average, could you see in any direction before
your line of sight struck a star? (Assume standard Euclidean geometry holds true in
this universe.)

We are allowed to assume:
-The universe is static
-The stars are roughly homogeneously distributed

2. Relevant equations
My thoughts:
l = 1/(n * sigma)
where sigma is the cross sectional area of interaction, or Pi R_sun^2

3. The attempt at a solution
I have working code and get a decent answer (with a LOT of waiting...) but I wanted to verify my answer. We had a similar 2D problem where we calculated the MFP of an arrow shot in a forest with average tree density of 0.005 trees per m^2, and tree radius of 1m.

I was easily able to verify my results by using the formula from section 2 (above), which was 100m. With a large enough forest, and enough runs, I was able to get ~100m from my Mathematica program.

For this 3D problem, I wanted to verify the answer (by hand) to confirm my program's results.

If the formula from part 2 can be applied here, where n= 10^9 Mpc^-3, and sigma = Pi*R_sun^2, I get:
MFP = 6.26 x 10^23 pc

(Keep in mind this is a homogeneously distributed universe, meaning no galaxies, no clusters, etc)

Anyway, my program gets around 5 x 10^24, so I'm wondering if my calculation is wrong, or if my program just needs higher precision or something...

Any help is appreciated!

Last edited:
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#### Greg Bernhardt

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

Thanks for the bump. All the relevant information is there. I think MC problems are generally notoriously difficult, but maybe I should have posted this in a math section, since what I really wanted was just a confirmation of a formula to use for 3D mean free path...

#### haruspex

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Thanks for the bump. All the relevant information is there. I think MC problems are generally notoriously difficult, but maybe I should have posted this in a math section, since what I really wanted was just a confirmation of a formula to use for 3D mean free path...
I got the same formula from first principles. Numerically I get slightly less: 6.19E23. Maybe you need to post your algorithm.

"Monte Carlo problem - mean free path to a star"

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