# A little math problem

1. Jan 13, 2005

### homology

Okay, let's say that stars have an average radius and an average density in an infinite universe (infinitely old as well). How far on average could you look before "seeing" a star in your line of sight?

Its a homework problem, all I want is a little nudge. I've blanked out and can't even begin to figure out how to work it (I know... I'm ashamed of myself). So if one of your dudes could get me started I'd be pretty stoked.

Thanks,

Kevin

2. Jan 14, 2005

### Chronos

Look up Olber's paradox. It's a good starting point. Newton also suggested the universe could not be both static and infinitely old because it would collapse under its own gravitation.

3. Jan 14, 2005

### homology

Hmm, I should have mentioned that we did study Olber's paradox. Figured out that something was wrong with the cosmology of Newton's time. These exercises follow that section. I'm positive I've done something like this before but for the life of me I can't remember.

4. Jan 14, 2005

### hellfire

Very interesting problem. I will propose some steps for a simplified solution in two dimensions.

Assuming euclidean space, the angular size of the stars as a function of the distance D and their radius R is $$\theta = \frac{2R}{D}$$.

Then, knowing the star density (which gives an average number of stars in a “volume” $$N_V = \rho \pi D^2$$), one can calculate the average number of stars on a “surface” $$N_S = \rho 2 \pi D dD$$.

The angular size covered by stars on this “surface” will be $$\phi_s = \frac{(2R) (\rho 2 \pi D dD)}{D}$$. The total covered angular size throughout the whole “volume” will be $$\phi_T = \int \phi_s = \rho 4 \pi R D$$.

To cover the whole sky, the condition $$\phi_T = 2 \pi$$ holds and therefore $$D = \frac{1}{2 R \rho}$$

If you assume that the number of stars N in a given volume follows a Poisson distribution, there will be a probability to see farther away. Note that none of both assumptions are acceptable: the distribution is not a Poisson distribution (correlation functions are used) and the space is not euclidean due to expansion (which has an influence on the angular size, as discussed here: https://www.physicsforums.com/showthread.php?t=59595).

Last edited: Jan 14, 2005
5. Jan 14, 2005

### hellfire

this solution is wrong, sorry. It would be right if one could simply add the $$\inline \phi_s$$ from different "surfaces" in the "volume" to impose the condition $$\inline \phi_T = 2 \pi$$, but one cannot, because the area covered by stars in one surface may overlap de area from the previous or from others. At the moment I have no better idea how to proceed.

6. Jan 14, 2005