# Forest for the trees

### Problem 1: the Olbers' paradox

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:physics.ed-ph/0706.1988.

### Problem 2: a down to earth setting

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$^*$?

$^*$ Ryden, Introduction to cosmology, ADDISON-Wesley.

### Problem 3: a Robin Hood setting

In a more romantic formulation this problem looks as follows. Suppose that in the Sherwood Forest the average radius of a tree is $30cm$ and the 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?

### Problem 4: the cosmological setting

The same problem in the cosmological formulation looks this way. Suppose we 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 we see in any direction before your line of sight hits a star?

### Problem 5: stars collisions

Demonstrate that stars in galaxies can be considered a collisionless medium.