Monte Carlo problem - mean free path to a star

[Adoniram]

Homework Statement

(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
-Every star has radius = solar radius

Homework Equations

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

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!

 

[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?

 

[Adoniram]

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]

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.

 

[paranormal]

Your calculation seems to be correct. The mean free path (MFP) in this scenario is the average distance a photon can travel before encountering a star. In this case, we can use the formula l = 1/(n * sigma) as you mentioned, where n is the number density of stars and sigma is the cross-sectional area of interaction.

Using the given values of n = 10^9 Mpc^-3 and R = 7 × 10^8 m, we can calculate the cross-sectional area of a star as sigma = Pi * R^2 = Pi * (7 × 10^8)^2 = 1.54 x 10^18 m^2.

Plugging this into the MFP formula, we get l = 1/(10^9 * 1.54 x 10^18) = 6.49 x 10^-28 Mpc. This is equivalent to 6.49 x 10^-19 pc, which is close to your calculated value of 6.26 x 10^23 pc.

It is possible that your program may need higher precision or more runs to get a more accurate result. Additionally, since this is a Monte Carlo simulation, there may be some random fluctuations in the results. Overall, your approach and calculation seem to be correct.