Exploring Olber's Paradox: Is the Wikipedia Explanation Wrong?

[HowardTheDuck]

I'm having a bit of a problem getting my head around Olber's paradox. The explanations haven't convinced me (I'm sure the fault is with me). According to Wikipedia:

"To show this, we divide the universe into a series of concentric shells, 1 light year thick (say). Thus, a certain number of stars will be in the shell 1,000,000,000 to 1,000,000,001 light years away, say. If the universe is homogeneous at a large scale, then there would be four times as many stars in a second shell between 2,000,000,000 to 2,000,000,001 light years away. However, the second shell is twice as far away, so each star in it would appear four times dimmer than the first shell. Thus the total light received from the second shell is the same as the total light received from the first shell."

Yes, I can agree that the second shell has more stars, but those stars are spread over the greater surface area of the second shell, and they don't seem to take that into account. When you're looking up from the earth, surely the only factor that matters is the density of stars in the shell, not the total number (it is the surface density which decides how bright a region of sky is). It seems to me that the density (stars per surface area) is going to be the same for all shells (more stars, but spread over more area). Therefore, the deciding factor is the distance - and that means that the further shells will be less bright.

So, yes, as Wikipedia suggests, the total light received from the second shell will be the same, but the total light received is surely irrelevant. The surface density is the same in all shells, and the distance is further, so the shell will appear less bright the further away it is.

At the very least, surely the Wikipedia explanation is wrong.

Am I wrong? Thanks.

 

[Bandersnatch]

When you're looking up from the earth, surely the only factor that matters is the density of stars in the shell, not the total number (it is the surface density which decides how bright a region of sky is)

What matters is the density - which is constant, the total number - which raises with the square of the distance, and the brightness of each individual star, which falls with the square of the distance.
You end up with a constant flux from all shells.

the total light received from the second shell will be the same, but the total light received is surely irrelevant.

It's not only relevant, it's the definition of what brightness means - unless you want to change the angular size of the patch of the sky you're looking at. Which you shouldn't, and which is most likely the source of your confusion.

The surface density is the same in all shells, and the distance is further, so the shell will appear less bright the further away it is.

The distance is further, so the area of the shell encompassed by the same solid angle(the same patch of the sky) increases at the same ratio that the brightness of any single star falls.

 

[HowardTheDuck]

Thanks very much for your help.

I really don't like this Olbers paradox thing. I think I'm just going to ignore it!

 

[Bandersnatch]

Perhaps not the healthiest attitude when learning science, but whatever floats your boat.

I'd say sleep on it, roll it around in your mind for a while, and see if it clicks. Maybe come back later and ask some more quesitons.

As far as I can tell, you were just skipping one of the steps in visualising the brightness as a function of distance, due to misunderstanding of what brightness is - i.e.the amount of light coming from a given solid angle, not from some given area of space.

 

[skeptic2]

Let me suggest a more down-to-earth example. Have you ever looked at distant objects through the rain. The farther away they are the more they fade into the rain. At some point you can't see them at all even with a telescope. If the raindrops were stars, your whole view would be nothing but stars.

The brightness of the disk of the star at a distance isn't really dimmer. It appears dimmer because the size of the disk is smaller. In fact as the ratio of distance to diameter increases, the angular size of the disk approaches D/d in radians where D = diameter and d = distance. As you can see if you double the distance, the number of stars increases by a factor of 4 but the angle of their diameter decreases by a factor of 2. The star's brightness decreases by a factor of 4 because the diameter is reduced by 2 in the horizontal direction and also by 2 in the vertical direction. Consequently just as at some point you can't see through the raindrops, at some point the whole sky would be filled with stars.

 

[HowardTheDuck]

I do kind of get it now, thanks.

I think the Wikipedia explanation is very poor.

 

[Drakkith]

Yes, I can agree that the second shell has more stars, but those stars are spread over the greater surface area of the second shell, and they don't seem to take that into account.

Of course they do. That's why there are 4 times as many stars in the second shell.

When you're looking up from the earth, surely the only factor that matters is the density of stars in the shell, not the total number (it is the surface density which decides how bright a region of sky is).

The total number determines the density of stars in each shell.

It seems to me that the density (stars per surface area) is going to be the same for all shells (more stars, but spread over more area). Therefore, the deciding factor is the distance - and that means that the further shells will be less bright.

The density is the same for all shells, but the brightness stays the same because you have more light. 4 times as many stars as the 1st shell, but each one is 4 times dimmer, so the total light received is the same.

So, yes, as Wikipedia suggests, the total light received from the second shell will be the same, but the total light received is surely irrelevant. The surface density is the same in all shells, and the distance is further, so the shell will appear less bright the further away it is.

Total light is what determines how bright each shell would be. If the total light is the same, the shell is just as bright. Remember, we are talking about the angular area here. The amount of light coming from a square arcsecond of sky will have equal amounts of light from each shell.

 

[Chronos]

Olbers paradox was motivated by the obvious - the night sky is dark. If, as believed in those days, the universe is infinitely old and infinitely populated with stars, the night sky should be exceedingly bright. Philosophers tried many ways to wriggle out from under the paradox, without much success. Fortunately, as usual, progress marches on. New and better observations of the heavens have convincingly revealed the universe is not neither infinitely ancient, nor infinitely voluminous. We can only see back to the time of recombination [about 13.8 billion years ago] and the volume of space we can see [i.e., is within our particle horizon] is clearly finite. So, it turns out that neither premise used in Olbers paradox [the universe is infinitely old and large] is valid.

 

[ImaLooser]

Olbers paradox was motivated by the obvious - the night sky is dark. If, as believed in those days, the universe is infinitely old and infinitely populated with stars, the night sky should be exceedingly bright. Philosophers tried many ways to wriggle out from under the paradox, without much success. Fortunately, as usual, progress marches on. New and better observations of the heavens have convincingly revealed the universe is not neither infinitely ancient, nor infinitely voluminous. We can only see back to the time of recombination [about 13.8 billion years ago] and the volume of space we can see [i.e., is within our particle horizon] is clearly finite. So, it turns out that neither premise used in Olbers paradox [the universe is infinitely old and large] is valid.

It seems to me that you need the finite speed of light as well. If the speed of light were infinite then we could see the entire Universe, finite or infinite. The age wouldn't matter because we wouldn't be able to see anything in the past.

 

[skydivephil]

Isnt the last scattering surface a wall in the sky preventing any light from before it getting to us? Hence if the unvierse were infintiley old the night sky still would in fact not be infinitley bright ? I find it hard to believe that those proposing a possibly infintley old unvierses such as Roger Penrose and Sean Caroll arent aware of Olbert paradox.

 

[Drakkith]

Isnt the last scattering surface a wall in the sky preventing any light from before it getting to us? Hence if the unvierse were infintiley old the night sky still would in fact not be infinitley bright ? I find it hard to believe that those proposing a possibly infintley old unvierses such as Roger Penrose and Sean Caroll arent aware of Olbert paradox.

That surface exists precisely because the universe (as we know it) is not infinitely old.

 

[skydivephil]

That surface exists precisely because the universe (as we know it) is not infinitely old.

The surface exists becuase of the hot big bang, but if there was a pre big bang universe no matter how long that pre big bang universe existed for, as long as it led to our big bang then the last scattering surface will still exist.

 

[yenchin]

There's a nice video of this on Minute Physics:

 

[Chronos]

Since Olbers paradox is phenomenologically based, unobservables are not an issue. In that sense, it true the surface of last scattering is the practical limit on what we can see.

 

[Drakkith]

The surface exists becuase of the hot big bang, but if there was a pre big bang universe no matter how long that pre big bang universe existed for, as long as it led to our big bang then the last scattering surface will still exist.

Of course! That's why I put 'as we know it' in my post.