Forest for the trees

Problem 1

If the Universe was infinitely old and infinitely extended, and stars could shine eternally, then in whatever direction you look the line of your sight should cross the surface of a star, and therefore all the sky should be as bright as the Sun's surface. This notion is known under the name of Olbers' paradox. Formulate the Olbers' paradox quantitatively L.~Anchordoqui, arXiv:0706.1988, physics.ed-ph.

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Problem 1

There are $n$ trees per hectare in a forest. Diameter of each one is $D$. Starting from what distance will we be unable to see the forest for the trees? How is this question connected to the Olbers' paradox\footnote{Ryden, Introduction to cosmology, ADDISON-Wesley.}?

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Problem 1

In a more romantic formulation this problem looks as following. Suppose that in the Sherwood Forest, the average radius of a tree is $30cm$ and average number of trees per unit area is $n=0.005m^{-2}$. If Robin Hood shoots an arrow in a random direction, how far, on average, will it travel before it strikes a tree?

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Problem 1

The same problem in the cosmological formulation looks this way. Suppose you are in a infinitely large, infinitely old Universe in which the average density of stars is $n_{st}=10^11 Mpc^{-3}$ and the average stellar radius is equal to the Sun's radius: $R_{st}=R_{\bigodot}=7\cdot10^8m$. How far, on average, could you see in any direction before your line of sight strucks a star?

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Problem 1

Demonstrate, that stars in galaxies can be considered as collisionless medium.

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