Inverse square law explains Olbers' paradox?

[humbleteleskop]

The most correct way to do it I can think of would be to make the first set of stars 4 pixels and the second set 1 pixel, but 4x as many. All would be the exact same brightness.

I agree. But then we have to draw the third shell, so we are back looking at the same pickle and we can't just keep increasing the size to compensate. After all the first shell is according to the paradox thousand million light years away, those stars really shouldn't appear much bigger.

My main objection against scaling the size however would be because the paradox explicitly states the stars in the second shell are actually dimmer.

 

[humbleteleskop]

Time to draw some diagrams. Define a sphere of diameter D, at a distance d from a point U. In this scenario, D is a star's Photosphere, and U is your eye. Assume that all light given off by the disc is radial (meaning on a straight line between the center of the sphere and each point on the surface of the sphere). With this model, how many lines intersect at U?
One.
If that point is giving off 1,000,000 photons steadily every second, point U will receive 1,000,000 photons steadily every second. Right? With me so far? Ok, given this simplified model, how does distance, d, reduce the number of photons reaching point U? (hint: think of the words:"it don't").

I'm afraid that's incorrect. We can talk about it in terms of numbers of photons. Light intensity is indeed a measure of amount of photons per unit time per unit area, but that's exactly where and how inverse square law applies.

http://www.astronomynotes.com/starprop/s3.htm

 

[russ_watters]

I agree. But then we have to draw the third shell, so we are back looking at the same pickle and we can't just keep increasing the size to compensate. After all the first shell is according to the paradox thousand million light years away, those stars really shouldn't appear much bigger.

Yes, it is difficult to make an accurate diagram.

My main objection against scaling the size however would be because the paradox explicitly states the stars in the second shell are actually dimmer.

I don't think that's true; where are you seeing it?

Is it based on the assumption that stars are point sources? It just isn't true, no matter how many times you say it. It is an approximation that sometimes works, but doesn't here.

Perhaps it would be useful if you considered what it would look like if you captured two or four or an infinite number of stars in one pixel.

 

[russ_watters]

I'm afraid that's incorrect. We can talk about it in terms of numbers of photons. Light intensity is indeed a measure of amount of photons per unit time per unit area, but that's exactly where and how inverse square law applies.

http://www.astronomynotes.com/starprop/s3.htm

Your diagram doesn't show the areas. Again, you are misusing the inverse square law and need to start paying closer attention to how it actually works.

 

[humbleteleskop]

I don't think that's true; where are you seeing it?

http://en.wikipedia.org/wiki/Olbers'_paradox
To show this, we divide the universe into a series of concentric shells, 1 light year thick. Thus, a certain number of stars will be in the shell 1,000,000,000 to 1,000,000,001 light years away. 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.

Is it based on the assumption that stars are point sources? It just isn't true, no matter how many times you say it. It is an approximation that sometimes works, but doesn't here.

I don't like assumptions. I simply see no other way to visually represent that sentence I quoted above.

Perhaps it would be useful if you considered what it would look like if you captured two or four or an infinite number of stars in one pixel.

I want to draw what the paradox postulates and I don't see any such bunching effect has relevance, but if you have some idea how it might actually come in play just tell me about it and I'll incorporate it in the picture so we can see how it fits.

 

[humbleteleskop]

Your diagram doesn't show the areas. Again, you are misusing the inverse square law and need to start paying closer attention to how it actually works.

I posted a link where that diagram came from which contains explanation. I didn't think it was necessary to copy it here.

http://www.astronomynotes.com/starprop/s3.htm

How am I misusing the inverse square law? If you believe something I said or referred to is incorrect please tell me about it.

 

[russ_watters]

I don't like assumptions. I simply see no other way to visually represent that sentence I quoted above.

Nobody does, but you do need them - and more importantly, you need to understand the assumptions others are making. The wiki quote does indeed include the unspoken assumption that the stars are point sources.

I want to draw what the paradox postulates and I don't see any such bunching effect has relevance, but if you have some idea how it might actually come in play just tell me about it and I'll incorporate it in the picture so we can see how it fits.

Eventually if you have enough shells, you will start getting more than one star per pixel, right?

The problem here is simply that you want to draw something that can't be accurately drawn. So you make assumptions and draw the scenario accurately per the assumptions, but without forgetting that you made assumptions.

 

[Bandersnatch]

If you want to draw the paradox on a computer screen, and you really want to start with stars whose diametre is below the pixel resolution, do what you did with your 10 and 40 stars, only don't stop there.

Say, the picture has got 9 pixels(3x3). Draw a one-pixel star of set X brightness(at distance R=1). Draw another shell 2 times farther, so 4 stars of 1/4 X brightness each. Then another, 3 times farther, with 9 stars of 1/9th brightness. At this point, you're already adding brightness value to all pixels, or marking some pixels more than once.

Even though stars from each consecutive shell will add a 1/R^2 fraction of the initial X brightness to the pixel they're drawn in, the number of pixels to be marked will be equal to R^2. So, with each shell, you're adding the equivalent of the initial brightness X, spread over 9 pixels. Reapeat infinite amount of times, and you get each pixel to be infinitely bright(which is what you get if you treat stars as ideal point sources that don't obscure each other).

 

[russ_watters]

If you want to draw the paradox on a computer screen, and you really want to start with stars whose diametre is below the pixel resolution, do what you did with your 10 and 40 stars, only don't stop there.

Say, the picture has got 9 pixels(3x3). Draw a one-pixel star of set X brightness(at distance R=1). Draw another shell 2 times farther, so 4 stars of 1/4 X brightness each. Then another, 3 times farther, with 9 stars of 1/9th brightness. At this point, you're already adding brightness value to all pixels, or marking some pixels more than once.

Even though stars from each consecutive shell will add a 1/R^2 fraction of the initial X brightness to the pixel they're drawn in, the number of pixels to be marked will be equal to R^2. So, with each shell, you're adding the equivalent of the initial brightness X, spread over 9 pixels. Reapeat infinite amount of times, and you get each pixel to be infinitely bright(which is what you get if you treat stars as ideal point sources that don't obscure each other).

By the way, this phenomena is easy enough to demonstrate when taking pictures of stars or galaxies: the higher the desired resolution, the dimmer the picture (with the same telescope) because each pixel contains fewer stars.

(Assuming the telescope resolution isn't exceeded)

 

[humbleteleskop]

Let's get one thing straight here. the inverse-square law is the reason that individual objects get dimmer as distance increases. No one's arguing against that.

The reason you can't see those galaxies when you look towards the Hubble Deep Field is because they are too dim for your eyes to detect them.

We agree then the reason galaxies in the Hubble Deep Field are so dim is due to inverse square law.

The reason the sky is mostly black is because there is a very large distance between most visible objects in space and light from more distant objects that would "fill in the gaps" has not yet had time to reach us.

Suppose there was enough time, those galaxies behind would still not make the galaxies in the Hubble Deep Field any brighter or more visible, nor would they themselves be visible to the human eye.

The inverse-square law explains why objects get dimmer as the distance increases. That's all. The law itself does not explain Olber's paradox.

I don't understand the issue here. Even the wikipedia article on Olber's Paradox gives the answer right in the opening paragraph.

If the universe is static and populated by an infinite number of stars, any sight line from Earth must end at the (very bright) surface of a star, so the night sky should be completely bright. This contradicts the observed darkness of the night.

That contradicts our observation of the Hubble Deep Field. We can look in the direction of anyone of those billions of stars, and yet we see nothing but black. To make the night sky in Olbers' paradox universe completely bright we would need eyes with exposure time of about several months.

 

[humbleteleskop]

Nobody does, but you do need them - and more importantly, you need to understand the assumptions others are making. The wiki quote does indeed include the unspoken assumption that the stars are point sources.

At least we cleared up that one. I wish you jumped in sooner.

Eventually if you have enough shells, you will start getting more than one star per pixel, right?

The problem here is simply that you want to draw something that can't be accurately drawn. So you make assumptions and draw the scenario accurately per the assumptions, but without forgetting that you made assumptions.

Yes, there are obviously some resolution limits which can impact the brightness. If our resolution was only one pixel, for example, then even a single star would make the whole night sky appear uniformly bright.

However, I believe our image have enough resolution to represent at least four shells before any such effects come into play, given we start we only 10 stars in the first shell. And then, whatever visual peculiarities happen behind, will not change how those first four shells look like, I suppose.

 

[humbleteleskop]

If you want to draw the paradox on a computer screen, and you really want to start with stars whose diametre is below the pixel resolution, do what you did with your 10 and 40 stars, only don't stop there.

Say, the picture has got 9 pixels(3x3). Draw a one-pixel star of set X brightness(at distance R=1). Draw another shell 2 times farther, so 4 stars of 1/4 X brightness each. Then another, 3 times farther, with 9 stars of 1/9th brightness. At this point, you're already adding brightness value to all pixels, or marking some pixels more than once.

Even though stars from each consecutive shell will add a 1/R^2 fraction of the initial X brightness to the pixel they're drawn in, the number of pixels to be marked will be equal to R^2. So, with each shell, you're adding the equivalent of the initial brightness X, spread over 9 pixels. Reapeat infinite amount of times, and you get each pixel to be infinitely bright(which is what you get if you treat stars as ideal point sources that don't obscure each other).

Hmmm. Well, this is definitively the turning point. If you are right, you win, and I lose. It seems my only hope is that somehow occlusion would at some point prevent that from happening all the way. Let me think...Actually, exposure time!

If we are talking about taking a photo of Olbers' paradox night sky, then yes, each pixel would eventually become completely bright, but it would not happen at once, it would take some time, possibly very long time. Would it not?

 

[russ_watters]

Hmmm. Well, this is definitively the turning point. If you are right, you win, and I lose. It seems my only hope is that somehow occlusion would at some point prevent that from happening all the way. Let me think...

I thought you already agreed with me when I described this before? I said four pixels for the first and one for the second; if you want three shells, you just need to start with 16 pixels. But again, all of the pixels will be the same brightness.

Exposure time!

Yes, each pixel would eventually become completely bright, but it would not happen at once, it would take some time, possibly very long time. Wouldn't it?

These diagrams are computer generated drawings. They aren't photographs.

 

[Drakkith]

We agree then the reason galaxies in the Hubble Deep Field are so dim is due to inverse square law.

Of course.

Suppose there was enough time, those galaxies behind would still not make the galaxies in the Hubble Deep Field any brighter or more visible, nor would they themselves be visible to the human eye.

The key your missing is that the light from the distant galaxies adds up with the light from the nearer galaxies. And the light from the even more distant galaxies adds up with all that light. So that area of the sky that the galaxy occupies would be MUCH brighter than it is now. You are correct in that each individual galaxy wouldn't be visible to the human eye, but the light from all the galaxies would add up and cause a diffuse "glow".

That contradicts our observation of the Hubble Deep Field. We can look in the direction of anyone of those billions of stars, and yet we see nothing but black. To make the night sky in Olbers' paradox universe completely bright we would need eyes with exposure time of about several months.

No, because the light adds up to make that part of the sky much brighter.

 

[Bandersnatch]

Actually, exposure time!

If we are talking about taking a photo of Olbers' paradox night sky, then yes, each pixel would eventually become completely bright, but it would not happen at once, it would take some time, possibly very long time. Would it not?

If it were a CCD camera matrix and not a drawing, the camera would record maximum brightness instantly, as each of its 9 pixels would receive infinite number of photons per unit time(no matter how short the exposure).

You can make every pixel of the camera maximally bright even with finite amount of stars, as long as you've got enough stars to shine at every pixel(i.e., camera resolution is low enough), and you take long enough exposure.

The point of Olber's paradox is, that it would happen instantly, which is most certainly not what we observe.

Even if you allow for non-pointlike sources, the sky would still be blindingly bright, as the stars obscuring the light from farther away would need to absorb and then reemit all that incident energy.

 

[humbleteleskop]

I thought you already agreed with me when I described this before? I said four pixels for the first and one for the second; if you want three shells, you just need to start with 16 pixels. But again, all of the pixels will be the same brightness.

I agreed for 4 pixel size in the first shell, but to scale it further to include more shells the stars in the first shell would grow to the size of the Sun and larger, which does not correspond to reality. On the other hand Bandersnatch talks about stars of equal apparent size in every shell, consequently having different brightness.

These diagrams are computer generated drawings. They aren't photographs.

Yes, but ultimately it is supposed to represent what the human eye would see, or mimic how a photograph of Olbers' paradox night sky would be formed.

 

[humbleteleskop]

The key your missing is that the light from the distant galaxies adds up with the light from the nearer galaxies. And the light from the even more distant galaxies adds up with all that light. So that area of the sky that the galaxy occupies would be MUCH brighter than it is now. You are correct in that each individual galaxy wouldn't be visible to the human eye, but the light from all the galaxies would add up and cause a diffuse "glow".

Ok, we are down to the last point left to discuss. I see what you are saying, but that doesn't strike me as logical or intuitive, so I must ask for proof or explanation. How do you arrive to that conclusion, is there some theory about that phenomena or experiment which can demonstrate it?

 

[russ_watters]

I agreed for 4 pixel size in the first shell, but to scale it further to include more shells the stars in the first shell would grow to the size of the Sun and larger, which does not correspond to reality.

This is just a diagram, not reality, but instead of considering the pixels growing, couldn't you just consider the resolution increasing? More pixels in the same area?

On the other hand Bandersnatch talks about stars of equal apparent size in every shell, consequently having different brightness.

No he doesn't. He's talking about the pixels having different brightness, not the stars. He states clearly that the stars are much smaller than one pixel, so in addition to each pixel showing a star, it also averages-in the brightness (none) of empty space.

Yes, but ultimately it is supposed to represent what the human eye would see, or mimic how a photograph of Olbers' paradox night sky would be formed.

In that case, you need two models, just one, because if you start with stars less than 1 pixel (or even points), the pixels get dimmer for a while, then start getting brighter again as the star density becomes greater than the pixel density. As I said before, you can actually take pictures of this phenomena (I've taken a bunch).

Ok, we are down to the last point left to discuss. I see what you are saying, but that doesn't strike me as logical or intuitive, so I must ask for proof or explanation. How do you arrive to that conclusion, is there some theory about that phenomena or experiment which can demonstrate it?

That's the rest of the statement of the paradox: you add the shells together to get the total brightness observed:

Thus each shell of a given thickness will produce the same net amount of light regardless of how far away it is. That is, the light of each shell adds to the total amount. Thus the more shells, the more light. And with infinitely many shells there would be a bright night sky.

 

[humbleteleskop]

If it were a CCD camera matrix and not a drawing, the camera would record maximum brightness instantly, as each of its 9 pixels would receive infinite number of photons per unit time(no matter how short the exposure).

1.) Instant maximum brightness, how do you arrive to that conclusion?

Consider a patch of sky similar to the Hubble Deep Field. In reality we can not see any brightness there unless we increase exposure time, why would that be any different with Olbers' paradox universe?


2.) Receive infinite number of photons per unit time, how is that possible?

This reminds me of Zeno's paradox and the problem of infinite divisibility. It seems your claim is that infinite number of stars can fit in finite field of view arc. Can you elaborate?

 

[russ_watters]

1.) Instant maximum brightness, how do you arrive to that conclusion?

2.) Receive infinite number of photons per unit time, how is that possible?

If the stars are assumed to be point sources, then they have infinite surface brightness and since there are an infinite number of them, the sky is therefore infinitely bright.

Consider a patch of sky similar to the Hubble Deep Field. In reality we can not see any brightness there unless we increase exposure time, why would that be any different with Olbers' paradox universe?

The whole point is that his universe is infinite, infinitely old and static. So it has no horizons and no redshift: nothing to keep light from traveling forever to reach you.

This reminds me of Zeno's paradox and the problem of infinite divisibility.

Your misunderstanding is vaguely similar to Zeon's, yes.

It seems your claim is that infinite number of stars can fit in finite field of view arc. Can you elaborate?

If they have zero size, you can fit an infinite number in any area. Just like you can say any space, surface or line/curve contains and infinite number of points.

 

[Drakkith]

Ok, we are down to the last point left to discuss. I see what you are saying, but that doesn't strike me as logical or intuitive, so I must ask for proof or explanation. How do you arrive to that conclusion, is there some theory about that phenomena or experiment which can demonstrate it?

I don't know what you don't understand about it. Galaxies are not large, opaque objects. They have a lot of empty space, so light from objects behind the galaxy can shine through unless it is blocked by large dust clouds.

Take a look at the following picture (Warning: Large File): http://upload.wikimedia.org/wikipedia/commons/c/c5/M101_hires_STScI-PRC2006-10a.jpg

Zoom in and you can literally see more distant galaxies through the Pinwheel galaxy.

 

[humbleteleskop]

This is just a diagram, not reality, but instead of considering the pixels growing, couldn't you just consider the resolution increasing? More pixels in the same area?

I could, but the paradox states they are actually dimmer. If increased resolution was true substitute for the lack of brightness we could make Hubble Deep Filed galaxies visible by increasing resolution instead of exposure time, and I don't think that's true.

No he doesn't. He's talking about the pixels having different brightness, not the stars. He states clearly that the stars are much smaller than one pixel, so in addition to each pixel showing a star, it also averages-in the brightness (none) of empty space.

Brightness is a property of pixels, it describes appearances. If something appears to be grey, you can't say it's actually white only smaller. Although both are functions of the same actual or objective properties, as subjective properties apparent size and apparent brightness are separate and independent.

In that case, you need two models, just one, because if you start with stars less than 1 pixel (or even points), the pixels get dimmer for a while, then start getting brighter again as the star density becomes greater than the pixel density. As I said before, you can actually take pictures of this phenomena (I've taken a bunch).

That's maybe straight forward and intuitive to you, but not to me. I think I should reserve my comments until I'm more familiar with it. I'll search the internet now. In the meantime please feel free to point some links concerning this relation between brightness and resolution.

 

[humbleteleskop]

If the stars are assumed to be point sources, then they have infinite surface brightness and since there are an infinite number of them, the sky is therefore infinitely bright.

The whole point is that his universe is infinite, infinitely old and static. So it has no horizons and no redshift: nothing to keep light from traveling forever to reach you.

Your misunderstanding is vaguely similar to Zeon's, yes.

If they have zero size, you can fit an infinite number in any area. Just like you can say any space, surface or line/curve contains and infinite number of points.

I realize now all of my arguments actually boil down to this one question: can infinite number of stars indeed fit into finite field of view arc? You say these stars have, or appear to have, zero size, but we know in reality they actually do have certain size greater than zero. Let's forget about the paradox and diagrams for a moment, are you saying your answer is actually a fact of reality?

 

[russ_watters]

I realize now all of my arguments actually boil down to this one question: can infinite number of stars indeed fit into finite field of view arc? You say these stars have, or appear to have, zero size, but we know in reality they actually do have certain size greater than zero. Let's forget about the paradox and diagrams for a moment, are you saying your answer is actually a fact of reality?

No, that's your demand (and it's implication) - we've been telling you for the entire thread that it isn't true in reality or even in Olbers' Paradox!

In reality and in Olbers', stars have size. They aren't point sources even though we are unable to detect their size with our eyes or a pixel on a camera. So Olbers' universe would be as bright as the surface of the sun (minus the secondary effects Bandersnatch mentions, which are tough to include and aren't part of the thought experiment). You received this answer first in post #6.

 

[humbleteleskop]

I don't know what you don't understand about it. Galaxies are not large, opaque objects. They have a lot of empty space, so light from objects behind the galaxy can shine through unless it is blocked by large dust clouds.

The part where their brightness add up is unconvincing to me, as well as notion that it would manifest instantaneously. The only way brightness of individual light sources can add up is if their light converges to impact the same pixels. I don't have a problem with that actually, apart from it happening instantaneously. What really boggles me is implication that brightness of Olbers' paradox night sky would then simply be a sort of visual artifact caused by resolution limits, because an eye or a photo resolution is digital rather than analog. And again it really boils down to question whether infinite number of stars can indeed fit in finite field of view arc.

 

[russ_watters]

The part where their brightness add up is unconvincing to me, as well as notion that it would manifest instantaneously.

Note, that instant infinite brightness thing was only in your version of the paradox. It isn't true in Olbers' or in reality.

The only way brightness of individual light sources can add up is if their light converges to impact the same pixels.

It doesn't have to converge. Here's a picture I took of a globular cluster:

The center of the image is bright because there are so many stars - more than one per pixel - in it. Note that the apparent size of the stars is an artefact of the imaging process; in reality, all stars are much smaller than a pixel.

What really boggles me is implication that brightness of Olbers' paradox night sky would then simply be a sort of visual artifact caused by resolution limits, because an eye or a photo resolution is digital rather than analog. And again it really boils down to question whether infinite number of stars can indeed fit in finite field of view arc.

No, it doesn't: again, Olbers' paradox doesn't claim that, you do.

[we posted at the same time, so please make sure you don't miss my previous post, # 54]

 

[Drakkith]

Russ is correct. Even with an optical system capable of resolving any arbitrary detail, every sight line would still end on the surface of a star in the eternal, static, infinite universe of Olber's paradox and the night sky would be blindingly bright instead of near-pitch black.

 

[russ_watters]

Russ is correct. Even with an optical system capable of resolving any arbitrary detail, every sight line would still end on the surface of a star in the eternal, static, infinite universe of Olber's paradox and the night sky would be blindingly bright instead of near-pitch black.

Indeed, ultra-high resolution (impossibly high) is what we would need to resolve individual stars and it would produce an image akin to the animation in post #3. Instead, based on our technological limitations, we'd actually just see a relatively smooth/evenly lit sky with little detail.

 

[humbleteleskop]

The center of the image is bright because there are so many stars - more than one per pixel - in it. Note that the apparent size of the stars is an artefact of the imaging process; in reality, all stars are much smaller than a pixel.

As a side question, if you have looked in the direction of the Hubble Deep Field with that telescope of yours, what did you see... something or nothing at all?

 

[russ_watters]

As a side question, if you have looked in the direction of the Hubble Deep Field with that telescope of yours, what did you see... something or nothing at all?

Not much; my telescope is much smaller and is located on earth, so it is more limited in capabilities. However, amateurs with better equipment and locations often take pictures with many background galaxies.