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Am I the only one for whom the paradox isn't intuitive? It's not clear to me why many distant dim stars should add up to a bright sky. It depends on how dista

Am I the only one for whom the paradox isn't intuitive? It's not clear to me why many distant dim stars should add up to a bright sky. It depends on how distant and how dense they are, doesn't it? Even assuming that the universe is infinite and eternal and approximately equally dense everywhere. The embedded video describes the paradox by saying that the night sky "should be as bright as the sun". Imagine if that were the case. Then move yourself twice as close to the sun, making the sun four times brighter. Then the night sky is 1/4 as bright as the sun. So clearly, distance and (lack of) density could make the night sky less bright, presumably to below the threshold of human perception. Right? Even aside from their explanations of distant light not having had time to reach us, and/or being red-shifted out of our visible range.

splat 14y ago
The reason is that the flux from any particular star goes as 1 / distance^2, but the number of stars observed in a patch of sky increases like the distance^2. The two cancel each other out. An easier way of thinking of it is to think about looking along a single direction. No matter which direction you look, this ray will always be pointed towards the surface of a star. (Incidentally, when thinking about these sorts of problems, it helps to distinguish flux from specific intensity: http://en.wikipedia.org/wiki/Specific_radiative_intensity http://en.wikipedia.org/wiki/Specific_radiative_intensity)
- IsTom 14y ago
  But wouldn't the ratio of these two matter? It could be that density of stars in space is very low and you'd get shade of almost-black?
drcube 14y ago
It's only a paradox if you think the universe is infinite in size and age, and the stars are evenly distributed and have been shining the whole time. Basically, the fact that the sky is dark tells us that at least one of the above assumptions is wrong.
- Someone 14y ago
  Not even that. One can postulate that there are things that shield out starlight. Let's call it "ether" or "dark matter". Alternatively, most of the stars really far away could be covered in a Dyson sphere.
  - drcube 14y ago
    That doesn't matter. Eventually the dust will radiate too.
    - Someone 14y ago
      Yeah, you are right. It could be in frequencies we cannot observe with the naked eye, but eventually, we would have to tweak some other postulates, too. Let's start over and postulate a gigantic turtle that carries the world…
  - lutusp 14y ago
    > Alternatively, most of the stars really far away could be covered in a Dyson sphere. The principle of energy conservation dictates that all the energy inside the sphere gets out, in one form or another. So the existence of Dyson spheres, dust clouds, dark matter etc. only delays the appearance of the energy in the universe at large.
- drcube 14y ago
  I should correct myself that historically, the paradox didn't hinge on the universe being infinitely old, but on the speed of light being infinitely fast.
jbri 14y ago
How does distance affect brightness? Distant stars are exactly as bright as close stars - however, the "brightness" is just spread over a greater area of space. In particular, the perceived brightness of a star falls off according to the square of distance - you can work this out for yourself by comparing the surface area of a sphere of radius d, with the surface area of a sphere of radius 2d. But how many stars are at that distance? If you assume uniform density, then the answer is exactly the opposite of what we found above! The number of stars at a distance of d is proportional to the square of the distance. This means that the total amount of light perceived from stars at a distance d is exactly the same, irrespective of the value of d. Then if the universe was also both eternal and infinite, then there would be an infinite amount of light reaching the earth!
- surrealize 14y ago
  Okay, if the universe were perfectly homogenous, i.e., uniformly dense at all scales, then this argument would be easy to buy. If you allow density variation at some scales, like the difference in density between the interior of a star and interstellar space, then it changes a bit, right? If you're really close to a star, then obviously that star is brighter than the average brightness of the universe. Right? If so, then the night sky wouldn't be "as bright as the sun" (like the video says), as observed from earth. But would the night sky be uniformly bright? Doesn't it depend on the average density of the universe? I.e., if the average density gets low enough, then don't those small-scale density variations start to matter at some point?
  - jbri 14y ago
    It's generally hard to speculate on how things "would" be if the universe were markedly different than it how it actually is. Or rather, it's easy to speculate, but it's nigh-impossible to come up with one solution to the exclusion of others. I can come up with many hypothetical universes where the night sky would be uniformly bright, and many hypothetical universes where there would be substantial amounts of variation. The general statement of the paradox is that the total amount of light coming from distant stars massively outweighs the light coming from a local star, so while the local star might create a slightly brighter patch of sky, that difference is small enough that even in its absence the sky would look "bright".
gurkendoktor 14y ago
It reminded me of the 'paradox' that 1 + 0.5 + 0.25 + 0.125... is finite and not even very large. You could have an infinite universe with infinitely many stars and still end up with a certain percentage of the sky being dark. Or maybe I'm mathematically wrong, but the above summation has broken any intuition for me that "the sum of infinitely many things must certainly surpass everything" :(
- thaumasiotes 14y ago
  That sum is even easier to process as finite if you represent it in binary: 1 + 0.1 + 0.01 + 0.001 + 0.0001 +... clearly, after any number of terms have gone by, you're still not at two (though of course, once they all have, you are).
  - gurkendoktor 14y ago
    Oh, that's a nice explanation. I like to think that I am not adding something, just always going half the way to 2.0. Still, when this was introduced to me in school, I remember my head hurting a little. :)
mikeash 14y ago
The sun's brightness per area is constant with distance. If you move twice as close to the sun, the brightness per area is exactly the same as it is on Earth, it just covers four times as much of the sky. If every sight line ended on a star, you'd expect the entire sky's brightness per area to be that of a star.
deleted 14y ago
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splat 14y ago

The reason is that the flux from any particular star goes as 1 / distance^2, but the number of stars observed in a patch of sky increases like the distance^2. The two cancel each other out. An easier way of thinking of it is to think about looking along a single direction. No matter which direction you look, this ray will always be pointed towards the surface of a star. (Incidentally, when thinking about these sorts of problems, it helps to distinguish flux from specific intensity: http://en.wikipedia.org/wiki/Specific_radiative_intensity http://en.wikipedia.org/wiki/Specific_radiative_intensity)

IsTom 14y ago
But wouldn't the ratio of these two matter? It could be that density of stars in space is very low and you'd get shade of almost-black?

drcube 14y ago

It's only a paradox if you think the universe is infinite in size and age, and the stars are evenly distributed and have been shining the whole time. Basically, the fact that the sky is dark tells us that at least one of the above assumptions is wrong.

Someone 14y ago
Not even that. One can postulate that there are things that shield out starlight. Let's call it "ether" or "dark matter". Alternatively, most of the stars really far away could be covered in a Dyson sphere.
- drcube 14y ago
  That doesn't matter. Eventually the dust will radiate too.
  - Someone 14y ago
    Yeah, you are right. It could be in frequencies we cannot observe with the naked eye, but eventually, we would have to tweak some other postulates, too. Let's start over and postulate a gigantic turtle that carries the world…
- lutusp 14y ago
  > Alternatively, most of the stars really far away could be covered in a Dyson sphere. The principle of energy conservation dictates that all the energy inside the sphere gets out, in one form or another. So the existence of Dyson spheres, dust clouds, dark matter etc. only delays the appearance of the energy in the universe at large.
drcube 14y ago
I should correct myself that historically, the paradox didn't hinge on the universe being infinitely old, but on the speed of light being infinitely fast.

jbri 14y ago

How does distance affect brightness? Distant stars are exactly as bright as close stars - however, the "brightness" is just spread over a greater area of space. In particular, the perceived brightness of a star falls off according to the square of distance - you can work this out for yourself by comparing the surface area of a sphere of radius d, with the surface area of a sphere of radius 2d. But how many stars are at that distance? If you assume uniform density, then the answer is exactly the opposite of what we found above! The number of stars at a distance of d is proportional to the square of the distance. This means that the total amount of light perceived from stars at a distance d is exactly the same, irrespective of the value of d. Then if the universe was also both eternal and infinite, then there would be an infinite amount of light reaching the earth!

surrealize 14y ago
Okay, if the universe were perfectly homogenous, i.e., uniformly dense at all scales, then this argument would be easy to buy. If you allow density variation at some scales, like the difference in density between the interior of a star and interstellar space, then it changes a bit, right? If you're really close to a star, then obviously that star is brighter than the average brightness of the universe. Right? If so, then the night sky wouldn't be "as bright as the sun" (like the video says), as observed from earth. But would the night sky be uniformly bright? Doesn't it depend on the average density of the universe? I.e., if the average density gets low enough, then don't those small-scale density variations start to matter at some point?
- jbri 14y ago
  It's generally hard to speculate on how things "would" be if the universe were markedly different than it how it actually is. Or rather, it's easy to speculate, but it's nigh-impossible to come up with one solution to the exclusion of others. I can come up with many hypothetical universes where the night sky would be uniformly bright, and many hypothetical universes where there would be substantial amounts of variation. The general statement of the paradox is that the total amount of light coming from distant stars massively outweighs the light coming from a local star, so while the local star might create a slightly brighter patch of sky, that difference is small enough that even in its absence the sky would look "bright".

gurkendoktor 14y ago

It reminded me of the 'paradox' that 1 + 0.5 + 0.25 + 0.125... is finite and not even very large. You could have an infinite universe with infinitely many stars and still end up with a certain percentage of the sky being dark. Or maybe I'm mathematically wrong, but the above summation has broken any intuition for me that "the sum of infinitely many things must certainly surpass everything" :(

thaumasiotes 14y ago
That sum is even easier to process as finite if you represent it in binary: 1 + 0.1 + 0.01 + 0.001 + 0.0001 +... clearly, after any number of terms have gone by, you're still not at two (though of course, once they all have, you are).
- gurkendoktor 14y ago
  Oh, that's a nice explanation. I like to think that I am not adding something, just always going half the way to 2.0. Still, when this was introduced to me in school, I remember my head hurting a little. :)

mikeash 14y ago

The sun's brightness per area is constant with distance. If you move twice as close to the sun, the brightness per area is exactly the same as it is on Earth, it just covers four times as much of the sky. If every sight line ended on a star, you'd expect the entire sky's brightness per area to be that of a star.

deleted 14y ago

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