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© Copyright 1997, Jim Loy
How much light do we get from the stars? Let's assume that the stars are more or less evenly distributed throughout space. Now, let's draw a sphere around us:
The number of stars in the sphere is roughly proportional to the volume of the sphere. A fairly simple integral (which I won't bother to derive) gives the amount of light energy that we receive from the stars in our sphere. I will call this amount of light energy x. It turns out that it doesn't matter what x is. So, we don't even have to fill in the average density of stars, and the radius of the sphere, and the average brightness of the stars, and calculate what x is.
Let's draw a second sphere, twice as big as the first one, creating a shell of stars. This shell has more volume, and therefore more stars, than the original sphere. But it is farther away from us. The same integral that I mentioned above shows that we receive the same amount of light energy from this shell as we do from our original sphere. We get the same x amount of light energy from this shell.
We continue drawing shells, each one as thick as that first shell (a thickness that is the same as the radius of our original sphere). The same integral, every time, yields x as the light energy from each and every shell.
Well, in an infinite, relatively uniform universe, there are infinitely many shells of stars. So, we get an infinite amount of light energy from these infinitely many shells. And we burn to a crisp.
But, we don't burn to a crisp. And this situation is called Olbers' Paradox.
Blocking of light:
Stars are not evenly distributed. Stars are clustered into galaxies. But, if we just substitute the word "galaxy" for "star" (and make our shells much thicker), in my arguments above, then we are much closer to the true situation. But, galaxies are clustered into clusters of galaxies. And clusters of galaxies are clustered into super clusters.
But, even if there is no end to the clustering, the reasoning still holds. There is just more variation in the stellar density of each shell. We should still burn to a crisp.
The books claim that a flaw in our reasoning is that the light of some stars never gets to us. It is blocked by nearer stars and dust clouds.
Supposedly, the integrals can be adjusted for this. And it is then deduced that we don't get an infinite amount of light energy from the stars (super clusters?). We are told that the sky would be roughly as bright as if we were at the surface of an average star.
We would still burn to a crisp.
Again, we don't burn to a crisp. Is there anything else wrong with our reasoning?
Well, we have made two assumptions which must be examined. We assumed that the universe was infinite (and of infinite age). It may not be infinite. And we assumed (without mentioning it) that the universe was not expanding. There is strong evidence that the universe is expanding.
Let's assume that the universe is infinite, but is expanding. The light of receding galaxies is shifted to the red, and we get less light energy from these galaxies, than we would if they were not receding. And it turns out that we would get a finite amount of light from infinitely many galaxies. This solves the paradox. The universe is expanding. There is also reason to believe that the universe is also finite in size.
My adjustment to Olbers' Paradox:
Let's back-track to the earlier situation, where we assumed an infinite, non-expanding universe. I am dissatisfied with the arguments about the light being blocked by stars and dust, giving us less than an infinite amount of light energy.
Let's pretend that nothing blocks the light from the stars, for a moment. Then let's imagine that there is a huge shell around us, made of a material that reflects half the light that hits it, and absorbs the other half. No light, from the stars, reaches us directly.
Each point on this shell receives about one half the light from the stars that we would receive without the shell. The shell itself is blocking half the sky from these points on the shell. Only half of that light is absorbed by each point on the shell. Well, Olbers' Paradox showed that each point would receive an infinite amount of light energy. Here each point would absorb one-fourth of that, which is still infinite. So each point on our shell would radiate, as a black body, an infinite amount of thermal light energy. We at the center would get an infinite amount of light energy from the shell.
Now, let's imagine a second such absorbing shell, outside the other shell. The inner shell gets an infinite amount of light energy from the outer shell. And, again, we get an infinite amount of light energy from the inner shell.
Now, imagine an infinite number of concentric shells, around us. We still get an infinite amount of energy at the center. This is merely a result of the fact that none of the energy is lost. It is just re-radiated by each shell.
Well, we don't have these absorbing shells. Instead, we have absorbing shells (containing galaxies) with holes in them (between the galaxies). So the galaxies block light, just as our imaginary shells would, but much less efficiently. We still get an infinite amount of light energy at the center. And we burn to a crisp.
But, again, we don't burn to a crisp (if you didn't notice). So, again, the universe is expanding, or finite, or both. The above argument also deals with light striking dust particles, which absorb and then emit the same amount of energy, probably at a longer wave length.
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