Inverse square law explains Olbers' paradox?

[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

 

[humbleteleskop]

One other thing...

Hubble telescope gazed at those galaxies for 23 days to obtain this photo. At the beginning it was all dark and eventually got brighter, right? It didn't grow larger, the actual color got brighter. Doesn't that mean "apparent brightness" and inverse-square law is actually about color brightness and not the size, in this case at least?

 

[Bandersnatch]

Why in the world is it not called "apparent size" then?

Because size is the reason behind brightness difference. You can't resolve most stars as anything bigger than just a point, so all you can measure is the brightness. You call it apparent brightness, because brightness is what you measure. The brightness is what it is, because the size is what it is. It makes little sense to call it apparent size, as size is something you do not observe, even if it directly influences brightness.

Is it in any way becoming clearer now?

Wait a second, are you saying this is wrong:

No. We're talking about Olber's paradox, remember? The whole sky packed with stars with no empty spaces left between them, so that it looks like one big surface of the sun on the firmanent.

 

[russ_watters]

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?

I don't know what "color brightness" is, but the article doesn't provide the details relevant to the question, so it is better to use a source that does. However in this case i don't think it really matters which assumption you pick. The one thing you may not do, however, is use both at the same time, which appears to be what you want to do. So please answer clearly:

Do you recognize that geometrically an object that is twice as far away covers 1/4 as much area in your field of view?

 

[russ_watters]

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!

Because it isn't just size, it is size AND surface brightness.

 

[russ_watters]

Wait a second, are you saying this is wrong:

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

No, it isn't wrong, you are wrong. Repeating it over and over again isn't going to change that.

Please go back and reread the first page of the thread. You are making the same wrong claims as you made before and chasing your tail. You should already know the things that you are saying are wrong.

 

[russ_watters]

One other thing...

Hubble telescope gazed at those galaxies for 23 days to obtain this photo. At the beginning it was all dark and eventually got brighter, right? It didn't grow larger, the actual color got brighter. Doesn't that mean "apparent brightness" and inverse-square law is actually about color brightness and not the size, in this case at least?

No, this has nothing whatsoever to do with photographic exposure time. You are just adding to your confusion by searching for other ways around this. Focus on the specific cases at hand.

 

[Drakkith]

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?

You are still thinking of stars as point sources and are ignoring what we've said about surface brightness. As I explained earlier, a 1 arcsecond x 1 arcsecond section of the Sun is exactly the same brightness whether you're at 1 au or 2 au. In other words, if you were to measure number of photons emitted from this 1x1 arcsecond square, it would be equal in both cases. You need to forget everything else in this thread until you understand why this is so.

 

[humbleteleskop]

No, this has nothing whatsoever to do with photographic exposure time.

What do you believe was the purpose for 23 days exposure time?

 

[russ_watters]

What do you believe was the purpose for 23 days exposure time?

It makes the image bright enough to see. But again, this has nothing to do with Olbers paradox, since the HDF was not completely filled with star.

Now please: if a star's surface brightness is dropped to 1/4 and size is dropped to 1/4, how much less light is received?

 

[humbleteleskop]

You are still thinking of stars as point sources and are ignoring what we've said about surface brightness.

- "Generally a source of light can be considered a point source if the resolution of the imaging instrument is too low to resolve its apparent size. Examples: Light from a distant star seen through a small telescope"
http://en.wikipedia.org/wiki/Point_source

Where do you get your information from?

As I explained earlier, a 1 arcsecond x 1 arcsecond section of the Sun is exactly the same brightness whether you're at 1 au or 2 au. In other words, if you were to measure number of photons emitted from this 1x1 arcsecond square, it would be equal in both cases.

Photons emitted have nothing do with the distance it's measured from. Brightness, which is a function of photons received, does vary with the distance. For example, apparent brightness of the Sun as seen from Venus is -27.4, as seen from Jupiter is -23, and as seen from Neptune is -19.3.

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

 

[russ_watters]

- "Generally a source of light can be considered a point source if the resolution of the imaging instrument is too low to resolve its apparent size. Examples: Light from a distant star seen through a small telescope"
http://en.wikipedia.org/wiki/Point_source

"If". For Olber's paradox, they are not considered point sources.

Again, if you want to make up your own different thought experiment that is different from Olber's paradox by using point sources, that's fine, but you have to recognize it is different and analyze accordingly...which we've already done and explained that it does not provide the result you desire.

 

[Drakkith]

- "Generally a source of light can be considered a point source if the resolution of the imaging instrument is too low to resolve its apparent size. Examples: Light from a distant star seen through a small telescope"
http://en.wikipedia.org/wiki/Point_source

Where do you get your information from?

The key phrase here is "can be considered". This means that real light sources are NOT point sources. We can "consider" real light sources to be point sources because we have limitations to our optics, and until the object's apparent size is larger than its airy disk we generraly don't have to worry about it, allowing us to simplify certain models and calculations. However, Olber's paradox is one of those situations where considering stars to be point sources will NOT help you understand.

Photons emitted have nothing do with the distance it's measured from.

Okay, change "emitted" to "received".

Brightness, which is a function of photons received, does vary with the distance. For example, apparent brightness of the Sun as seen from Venus is -27.4, as seen from Jupiter is -23, and as seen from Neptune is -19.3.

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

That's measuring the brightness of the Sun as a whole as seen from those planets. A small 1x1 arcsecond section of the Sun has the same brightness at every planet. And by that I mean the number of photons received from this section will be the same. But since the Sun shrinks in apparent size as you move further away, there are fewer and fewer 1x1 arcsecond squares, so total brightness does go down.

 

[humbleteleskop]

It makes the image bright enough to see. But again, this has nothing to do with Olbers paradox, since the HDF was not completely filled with star.

I didn't say it has anything to do with Olbers' paradox. I said it has to do with apparent brightness and inverse-square law, and I pointed out how nothing grew in size, but only increased in brightness.

Now please: if a star's surface brightness is dropped to 1/4 and size is dropped to 1/4, how much less light is received?

I guess 8 times less, who knows. I thought the lesson you wanted to teach me was that amount of light received would be equal in either case.

Do you recognize that geometrically an object that is twice as far away covers 1/4 as much area in your field of view?

Yes. And it would have the same apparent brightness if it was at that same distance but 4 times bigger and with 4 times less of surface luminosity.

 

[humbleteleskop]

The key phrase here is "can be considered". This means that real light sources are NOT point sources. We can "consider" real light sources to be point sources because we have limitations to our optics, and until the object's apparent size is larger than its airy disk we generraly don't have to worry about it, allowing us to simplify certain models and calculations. However, Olber's paradox is one of those situations where considering stars to be point sources will NOT help you understand.

I disagree. If the size can not be resolved and the distance is increased it can not get any smaller only its color can get dimmer.

That's measuring the brightness of the Sun as a whole as seen from those planets. A small 1x1 arcsecond section of the Sun has the same brightness at every planet. And by that I mean the number of photons received from this section will be the same. But since the Sun shrinks in apparent size as you move further away, there are fewer and fewer 1x1 arcsecond squares, so total brightness does go down.

That arc-second will not correspond to the same surface area if the distance is increased, but larger area, so yes. I guess that example is supposed to represent a "wall of stars" relating to Olbers' paradox, but it's misleading as those stars are not in the same plane perpendicular to the line of sight.

 

[humbleteleskop]

"If". For Olber's paradox, they are not considered point sources.

Again, if you want to make up your own different thought experiment that is different from Olber's paradox by using point sources, that's fine, but you have to recognize it is different and analyze accordingly...which we've already done and explained that it does not provide the result you desire.

As Boris the Animal would say: let's agree to disagree.

 

[Vanadium 50]

I disagree. If the size can not be resolved and the distance is increased it can not get any smaller only its color can get dimmer.

Well that's just plumb wrong. The angular size of an object is what it is, independent of whether or not we can resolve an object of this size or not.

 

[russ_watters]

Separate post because of how important this is:

I guess 8 times less, who knows. I thought the lesson you wanted to teach me was that amount of light received would be equal in either case.

This raises a bunch of big, red flags:

1. I gave you the answer (in bold, no less!), so the fact that you answered wrong means you aren't trying hard enough. Our help here is not free: it comes with the requirement that you make an effort to learn what we are trying to teach you.

2. Who knows? Everyone who is posting in this thread and making a claim must know. That includes you: you can't claim to explain a principle in science if you can't do even the simplest calculations that describe it.

3. You didn't just guess wrong, you were doubly wrong: You contradicted your own claim (4x brightness reduction) with your wrong answer. You need to grasp that the math does not support your claim and listen to us when we explain why. Which makes:

4. You don't even recognize your own scenario when it is recited back to you! You need to organize your thoughts better: again, you need to try harder here.

 

[russ_watters]

I didn't say it has anything to do with Olbers' paradox. I said it has to do with apparent brightness and inverse-square law, and I pointed out how nothing grew in size, but only increased in brightness.

This is your thread on how the inverse square law relates to Olbers' paradox. If it doesn't relate to Olbers' paradox, then it isn't relevant to the thread and we shouldn't be discussing it.

Yes. [twice as far away = 1/4 the size]

So how can you claim that if you have 1/4 the size and 1/4 the surface intensity, you get 1/4 the total brightness? ...or, for that matter, 1/8th the total brightness (your two claims). It should be obvious to you that you are contradicting yourself.

As Boris the Animal would say: let's agree to disagree.

That's really not an option here. This is a pretty simple issue and there is a straightforward right and wrong answer. You can choose to be wrong if you want, but we won't indulge your insistence that your wrong answer is right for much longer.

 

[humbleteleskop]

Well that's just plumb wrong. The angular size of an object is what it is, independent of whether or not we can resolve an object of this size or not.

I was of course referring to apparent size. Let me try again. If the angular diameter of a star can not be resolved and the distance from the star is increased, then its apparent size can not get any smaller, only its apparent color can get dimmer. True?

 

[Vanadium 50]

False.

Let me say it again: The angular size of an object is what it is, independent of whether or not we can resolve an object of this size or not. The size of an object is not determined by our ability to measure.

This thread has gone on quite a while, largely because you post one incorrect statement after another. Are you really asking a question? Or are you trying to promote a position.