   Chapter 34, Problem 6CQ

Chapter
Section
Textbook Problem

Olbers’s paradox poses an interesting question: If the universe is infinite, then any line of sight should eventually fall on a stars surface. Why then is the sky dark at night? Discuss the commonly accepted evolution of the universe as a solution to this paradox.

To determine

The reason for the sky to be dark at night and the commonly accepted evolution of the universe as a solution to this paradox.

Explanation

Consider following conditions for the universe: -

1. Filled uniformly with stars (which are part of galaxies).
2. Eternal
3. Infinite
4. Static

If we consider that a person is looking at any particular direction in the sky, then the line of the individual’s sight should be towards a star (or galaxy), like an arrow which is frictionless in nature and shot.

Therefore, the sky at night must be as bright as similar to the normal star (galaxy) and must not be dark.

Although most stars which are at far distance will appear to be less bright than the stars which are present nearer, and the brightness of the stars at distant location reduces as 1/d2

Where ‘d’ is their distance.

At the same time duration, there are many stars at far distances than the stars at near distances, and the total number of stars at the larger distance will keep on increasing with d2, so that both the effects cancel out each other.

Generally, the total flux which is present in the number of stars present in a particular given shell and is also same to the multiple of flux arising from an average star at that times of distance and the total number of stars per shell.

If the total amount of density (in number) of stars is constant (assumption 1 above), then the total number of stars in a particular shell is the total number of stars in the particular shell multiplied with the total volume that the shell has:

fshell=fstar×numberofstars

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