Lost in a thicket

Folk tales are filled with cautionary tales about getting lost in the middle of a dark forest. This trope is not just a convenience for story tellers, it is a lesson about light scattering. As one goes farther into a forest, photons have had more opportunities to hit a tree and bounce backward out of the forest.

Suppose you find yourself 10 m\text{10 m} into a dark forest which is illuminated from a (one) side. The canopy is extremely thick so that no light penetrates from above. Moreover, the forest consists of cylindrical trees with average radius rˉ\bar{r} which are randomly distributed with number density ρ\rho (i.e. 2 trees per square meter).

What fraction of the incoming light the can you see, on average?

Assumptions

  • Ignore second scattering events, if a photon hits a tree once, it exits the forest forever.
  • rˉ=6 cm\bar{r}=6\text{ cm}
  • ρ=2 tree m2\rho = 2\text{ tree m}^{-2}
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