Olbers' paradox: why is the night sky dark?

Started Nov 12, 2012 | Discussions thread
Jonny Boyd
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Re: Olbers' paradox: why is the night sky dark?
In reply to trisd, Dec 1, 2012

trisd wrote:

Jonny Boyd wrote:

I'm afraid that I'm going to have to disagree with you here! To a certain extent at least. Stars occluding one another isn't quite the problem it at first appears to be for exactly the reason that dust clouds don't explain the paradox – the blocked light will heat up the blocking stars, causing them to radiate more light.

That is not a part of the original treatment which discarded inverse square law as the explanation for the "dark night sky". Once you realize the mistake, realize it actually is inverse square law that is responsible for the appearance of the night sky as we see it, then you will see there is no any infinity of light heating everything to glow with the same brightness.

The inverse square law is used in the calculations for the brightness of each shell.

I also used the inverse square law with a slightly different method and got the same result. Perhaps you can tell me where I've gone wrong and show me how the inverse square law should have been sued properly:

  1. You suggested that a 100x100 pixel detector would be enough to judge the brightness of the sky. Let's assume that this detector has a lens that allows it to see an area of sky subtended by a solid angle of 1.024x10^-10 steradians. Each pixel therefore covers 1.024x10^-14 steradians. Happy with that?
  2. Let's also imagine a shell of stars at a distance 1r. These stars are close enough to occupy 1.024x10^-14 steradians, so they are exactly imaged by 1 pixel. There are so many stars at this distance that our detector can pick up 5000 of them. This means that there is a star every other pixel. And those stars are measured like point sources. Happy with that?
  3. Photons arrive from one of these stars at a rate of 1024 photons per second. (That gives a flux of 1024 photons per second per pixel.) If our detector takes a 10 second exposure, that means that a pixel illuminated by a star from the 1r shell will have a brightness of 10,240. Happy with that?
  4. hat happens with the rest of the pixels? Well let's look at the shell of stars at 1.41r. These stars are 1.41x further away, so there will be twice as many i.e. 10,000. This means that each pixel will have one of these stars and they are measured like point sources. Happy with that?
  5. These stars will only occupy 5.12x10^15 radians (half a pixel) because they're 1.41x further away and photons will come in at a rate of 512 photons per second. (That gives a flux of 1024 photons per second per pixel). If our detector takes a 10 second exposure then one of these stars will contribute 5120 photons to the brightness of the pixel. Happy with that?
  6. We're not done yet though, because Olbers' paradox states that every sight line ends in a star. At the moment we've still got 5.12x10^15 steradians to deal with for this pixel. Happy with that?
  7. So let's go to the shell at a distance of 2r. This will have 20,000 stars, which is 2 per pixel. Happy with that?
  8. Each of these stars occupies 2.56x10^-15 steradians (quarter of a pixel) and photons will come in at a rate of 256 per second. (That gives a flux of 1024 photons per second per pixel). If our detector takes a 10 second exposure then each of these stars will contribute 2560 photons to the brightness of the pixel. Happy with that?
  9. At this point, the whole pixel is covered, meaning that every line of sight ends in a star. We can therefore turn to working out the brightness of the pixel. Happy with that?
  10. There is a total of 5120 photons from the one star in the 1.41r shell and 2*2560=5120 photons from the two stars in the 2r shell, giving a total value for the pixel of 10,240 photons. Happy with that?
  11. This is the same value as the pixels which are illuminated by the stars from the 1r shell, so every pixel has the same brightness value, meaning that the sky is uniformly bright. Happy with that?
  12. This result isn't surprising because the flux from every star is the same. So if each pixel sees the same flux, has the same area, and is exposed for the same length of time, then of course each pixel will have the same brightness. Happy with that?
  13. You can change the numbers, but as long as you ensure at every line of sight ends in a star, you'll find that every pixel ends up with the same brightness. What that brightness is will depend on the values you put in, but the brightness will be uniform. Happy with that?
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