Olber's Paradox: Why is the Night Sky Dark?

Suppose that the universe is (1) static, (2) infinite, (3) eternal and (4) uniformly filled with stars (or galaxies, which are made of stars). If we look in any direction, our line of sight must eventually run into a star (galaxy), just as a frictionless arrow shot in the middle of an infinite forest will eventually stick itself into a tree. Therefore, the night sky should be as bright as the average star (galaxy) and certainly should not be dark.

Of course, most distant stars will appear to be fainter than nearby ones; their brightness will fall off as 1/d2, where d is their distance. But, at the same time, there will be many more stars at larger distances than smaller ones, and the number at the larger distance will increase with d2 so that the two effects should cancel.

Basically, the flux arising from all the stars in a given shell, is equal to the product of the flux arising from an average star at that distance times the number of stars per shell. If the density of stars in constant (assumption 4 above), then the number of stars in a shell is just the number of stars in the shell times the volume of the shell:

where t is the thickness of successive shells, and n is the number density of stars per unit volume. Therefore the flux arising from each shell is constant, that is, stars at any given distance from us contribute the same amount of flux as stars at any other distance. In every direction our line of sight should intercept the surface of a star and thus the sky should be bright (and the universe should be warm!).

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