Inverse square law explains Olbers' paradox?

[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.

 

[humbleteleskop]

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

I demanded so to reflect what the Wikipedia article says, I didn't think it would yield answers that do not correspond to reality.

In reality and in Olbers', stars have size. They aren't point sources. 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.

That may be the answer, but to me it's a long jump to conclusion. The paradox talks about stars that get dimmer and dimmer in every subsequent shell. I think it's too much for you to expect it should be obvious how those dim, dimmer and very dim stars actually combine to become bright. To me that's not obvious at all, sounds more like a paradox of its own.

On the bright side, a lot of questions were answered and I only have a few more left. I hope everyone participating is enjoying this as much as I do, and I thank you all for your time.

 

[humbleteleskop]

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

Ok, we are talking now about Oblers' paradox as if it was real so that our conclusions correspond to reality. If necessary let us suppose all the stars are equal to our Sun.

It's past midnight 12:25 am, we take a camera with ISO 100 film, aperture size f/256 and shutter speed 1/1000 of a second. We point the camera towards the sky and snap a photo, which after we develop looks:

a) uniformly maximally bright (completely white/overexposed)

b) uniformly bright, but less than maximally bright

c) non-uniformly bright

d) rather dark but we can see some of the closest/brightest stars

e) something else

 

[humbleteleskop]

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.

But if we look at very distant star which appears very dim due to inverse-square law, and if we have enough resolution so no other star adds up its brightness to this star we are looking at, then shouldn't we see it as dim as it is?

 

[russ_watters]

I demanded so to reflect what the Wikipedia article says, I didn't think it would yield answers that do not correspond to reality.

I may have confused things by a previous answer -- and the wiki may not be worded the best it could either. The wiki for Olbers' paradox doesn't say that the stars are assumed to be point sources (it just invokes the inverse square law) and in the diagram they show and in reality, they clearly are not. That glosses over the complication of how the inverse square law applies. As the wiki for the inverse square law shows, in most cases the error in that wrong assumption is small:

The intensity (or illuminance or irradiance) of light or other linear waves radiating from a point source (energy per unit of area perpendicular to the source) is inversely proportional to the square of the distance from the source; so an object (of the same size) twice as far away, receives only one-quarter the energy (in the same time period)...

In photography and theatrical lighting, the inverse-square law is used to determine the "fall off" or the difference in illumination on a subject as it moves closer to or further from the light source. For quick approximations, it is enough to remember that doubling the distance reduces illumination to one quarter;[7] or similarly, to halve the illumination increase the distance by a factor of 1.4 (the square root of 2), and to double illumination, reduce the distance to 0.7 (square root of 1/2). When the illuminant is not a point source, the inverse square rule is often still a useful approximation; when the size of the light source is less than one-fifth of the distance to the subject, the calculation error is less than 1%.

http://en.wikipedia.org/wiki/Inverse-square_law#Light_and_other_electromagnetic_radiation

In this case, I think the difference between assuming they are point sources or not is that if you assume they are point sources, the sky should be infinitely bright and if you assume they are not, it should "merely" as bright as the surface of the sun. But of course, neither assumption produces the view we actually see or the view you think we should see.

You appear to be confused about the inverse square law; thinking it applies to the surface brightness of an object. It can't: if the object is twice as far away, it appears 1/4 as big, so in order to shine 1/4 as bright in total, the surface brightness must be the same. If their surface brightness were cut by 1/4 as well, then they'd look 1/16th as bright to our eyes.

That may be the answer, but to me it's a long jump to conclusion. The paradox talks about stars that get dimmer and dimmer in every subsequent shell. I think it's too much for you to expect it should be obvious how those dim, dimmer and very dim stars actually combine to become bright. To me that's not obvious at all, sounds more like a paradox of its own.

See the bold part above: they appear dimmer because they send to you about 1/4 as much light when the distance doubles. But that's their total light sent to your eye, not their surface brightness (intensity). I think you are confusing the total light received with the surface brightness; they are not and cannot be the same.

Here's another source that addresses this specific issue:

Why isn't the night sky uniformly at least as bright as the surface of the Sun? If the Universe has infinitely many stars, then presumably it should be. After all, if you move the Sun twice as far away from us, we will intercept one quarter as many photons, but the Sun's angular area against the sky background will also have now dropped to a quarter of what it was. So its areal intensity remains constant. With infinitely many stars, every element of the sky background should have a star, and the entire heavens should be at least as bright as an average star like the Sun.

http://math.ucr.edu/home/baez/physics/Relativity/GR/olbers.html

 

[russ_watters]

Ok, we are talking now about Oblers' paradox as if it was real so that our conclusions correspond to reality. If necessary let us suppose all the stars are equal to our Sun.

It's past midnight 12:25 am, we take a camera with ISO 100 film, aperture size f/256 and shutter speed 1/1000 of a second. We point the camera towards the sky and snap a photo, which after we develop looks:

a) uniformly maximally bright (completely white/overexposed)

b) uniformly bright, but less than maximally bright

c) non-uniformly bright

d) rather dark but we can see some of the closest/brightest stars

e) something else

In Olbers' universe, the entire sky would be as bright as the surface of the sun. That would probably be a, but could be b; that isn't something I know offhand (I haven't tried to take unfiltered pictures of the sun - I don't want to damage my camera!).

 

[humbleteleskop]

You appear to be confused about the inverse square law; thinking it applies to the surface brightness of an object. It can't: if the object is twice as far away, it appears 1/4 as big, so in order to shine 1/4 as bright in total, the surface brightness must be the same. If their surface brightness were cut by 1/4 as well, then they'd look 1/16th as bright to our eyes.

See the bold part above: they appear dimmer because they send to you about 1/4 as much light when the distance doubles. But that's their total light sent to your eye, not their surface brightness (intensity). I think you are confusing the total light received with the surface brightness; they are not and cannot be the same.

Yes, I am aware of that and I agree. What I don't agree with is when they say "dimmer" that they actually mean "smaller". Here is why:

http://en.wikipedia.org/wiki/Apparent_brightness

Is "apparent brightness" about differences in size or color brightness?

 

[humbleteleskop]

In Olbers' universe, the entire sky would be as bright as the surface of the sun. That would probably be a, but could be b; that isn't something I know offhand (I haven't tried to take unfiltered pictures of the sun - I don't want to damage my camera!).

I think photographing the Sun with those parameters would actually produce very dark photo, that's what I was aiming for anyway. I found parameters for photographing the Sun and I cranked them up to allow for much more brightness, here:

http://www.astronomy.no/sol310503/ekspo.html


I couldn't think of how to formulate it at the time, but what I meant to ask really is this: if we set camera parameters so that we get almost completely dark photo of Olbers' paradox night sky, then a few bright spots on it would be images of the closest stars. But you seem to say it would be all or nothing, that is it would be uniform regardless of how dark or bright the resulting photo is. To me it makes more sense that photons from the closest stars would have higher chance to hit the camera in sufficient number to make an impression than photons from further away stars.

 

[Bandersnatch]

Yes, I am aware of that and I agree. What I don't agree with is when they say "dimmer" that they actually mean "smaller". Here is why:

http://en.wikipedia.org/wiki/Apparent_brightness

Is "apparent brightness" about differences in size or color brightness?

Yes, humbleteleskop, they do, we do, everybody does. Usually that's what it means for a faraway star to be dimmer - it's just smaller.

The disc of a star sends photons towards your detector(eye, ccd, whatever). The less photons reach it, the dimmer the star appears. There are various processes that could obstruct photons on their way(like scattering, absorption by interstellar dust; there could be redshifting making them less energetic, and leading to failure to trigger the detector), and make the resulting image dimmer, but the inverse square law is specifically, and only, about the geometric reduction of the area of the stellar disc. Stars two times farther away are four times dimmer exactly, and only, because their apparent discs are four times smaller.

The end result on the side of the detector is just less photons impinging on it, so as far as it is concerned, there's no difference between calling the source four times smaller and four times dimmer - there will be the same amount of photons hitting it in both cases. But the physical reason for the dimming remains the reducion in apparent size, and the distinction becomes important once you deal with objects that are larger than the maximum resolution of the detector.

In other words, you can use the point source approximation in many cases, but you need to keep in mind the real reason for the dimming, so as to know when the approximation doesn't apply anymore.

I couldn't think of how to formulate it at the time, but what I meant to ask really is this: if we set camera parameters so that we get almost completely dark photo of Olbers' paradox night sky, then a few bright spots on it would be images of the closest stars. But you seem to say it would be all or nothing, that is it would be uniform regardless of how dark or bright the resulting photo is. To me it makes more sense that photons from the closest stars would have higher chance to hit the camera in sufficient number to make an impression than photons from further away stars.

The flux of photons would be constant over the whole sky, so it wouldn't matter how far a star is(as long as all the stars have the same surface luminosity). The time needed to travel from the star wouldn't matter, as the universe is supposed to be eternal. None would stand out.

 

[humbleteleskop]

Yes, humbleteleskop, they do, we do, everybody does. Usually that's what it means for a faraway star to be dimmer - it's just smaller.

Are you kidding me?!? What's next, "wet" actually means "tall"? I can't possibly be the only one who thinks "brightness" is something that describes color. So many articles about it and no one cares to point at that semantic nonsense. Why in the world is it not called "apparent size" then? Unbelievable!

You win, I lose. Rrrrhh!

 

[humbleteleskop]

The flux of photons would be constant over the whole sky, so it wouldn't matter how far a star is(as long as all the stars have the same surface luminosity). The time needed to travel from the star wouldn't matter, as the universe is supposed to be eternal. None would stand out.

Wait a second, are you saying this is wrong:

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