Illustrating Olbers' paradox

I agree.


EDIT: I could argue "photon deterioration" I pointed to in my previous post, but I think I have solid argument even without it, so I'll leave that out and put in my sleeve.

Brightness is not only a function of number of photons per unit time, but also depends on the sensor area they are projected or spread over. Say certain number of photons distributed over 10 pixels will make those pixels white, but the same amount of photons distributed over 40 pixels will make those pixels 4 times less bright. Do you agree?

 

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Which immediately tells us that the assumptions of the paradox can't all be true. Our universe cannot be infinitely old, infinitely large and static.

Fortunately, we have model to hand which does a much (much!) better job of accounting for what we see (although there are still a few issues).

What proportion fail to arrive in comparison with the expectations of the standard (Lambda-CDM) cosmological model?

 

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The light from the stars in the Hubble Deep Field obviously reached us and is in visible spectrum, so I don't think those explanations apply.

I'm not aware of that which you are talking about, but it sounds as if some kind of "photon deterioration" indeed exist as a proven fact. Can you give us the formula?

 

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They do all arrive. The calculations match the observations. There is no such discrepancy.

He's asking you, not telling you. Clearly, the claim you made above is just handwaving: you don't actually know, you are just guessing.

Yes, but that isn't the situation being described so it isn't relevant: both of your pictures have the same area....and if we're talking about observing it with our eyes, it's the entire sky in both cases.

 

The detector I have in mind only has one "pixel", and has no lenses or mirrors. It simply counts the photons which arrive each second from within the solid angle defining its field of view, and ignores any that don't, so the notion of "brightness" you describe is not as relevant to my argument as the idea of photons per unit time.

I'll rephrase the paragraph you quoted to avoid the troublesome word:

Thus, in an infinite large, infinitely old, static universe filled uniformly with stars, every shell sends the same total number of photons towards Earth per unit time. Because there are infinitely many shells, if it were not for the fact that stars can intercept photons from other stars, the assumptions would actually lead to the conclusion that infinitely many photons per unit time arrive at the Earth from every non-zero-sized patch of the sky. As it is, due to the intervening stars we only get a finite number of photons per unit time from each patch of sky in this imaginary universe, and the number of photons per unit time per unit solid angle is equal to that which we receive from Sun, assuming the Sun is a typical star in this scenario.
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We may be talking at cross purposes. What I mean is, the night sky does not look like the surface of a star, so the assumptions of Olber's paradox cannot all be true.

The Lambda-CDM model is the standard cosmological model (big bang + inflation + cold dark matter + dark energy).

No, that's not what I was implying. You said:

Take any star in the Hubble Deep Field, it is supposed to emit large quantities of photons in our direction, as if there was a continuous line between us, but apparently they don't all arrive. Due to relative translation and rotation this supposed continuous ray doesn't connect to the same point on the star's surface, but this shouldn't matter much. So then beside dispersion and inverse-square law, these photons also seem to deteriorate somehow on the way.​

I am asking you to clarify and support the statements I bolded, if possible.

 

And that is where the paradox is flawed. It's like measuring light intensity without taking into account "per unit time" part of its definition. If your eyes had only one pixel resolution and you gazed at our real world night sky, wouldn't you conclude it's uniformly bright, and at least as bright as the brightest star in your field of view?

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Would your sensor conclude these two patches of sky are equally bright? Would its conclusion be correct?

 

A neat version of Olber's paradox is this one (simplified slightly):

If the universe were uniform, static and infinitely old, it would be in a dynamic equilibrium such that the energy content of a large volume of space (large enough to contain at least one star) remains stable over time. Therefore the amount of light energy arriving at a particular star per unit time must equal the amount leaving. Therefore the sky must have the same radiant intensity as the surface of a star.

 

I'd be counting the total number of photons entering via my adjustable field of view, which is exactly what I want my detector to do. Remember that my detector had an adjustable field of view? For every direction, I was able to reduce its field of view so that only part of the surface of a single star was monitored, and thereby showed that the number of photons received per unit time per solid angle was the same wherever I pointed it.

(Applying that argument to a series of narrow wavelength bands, we actually find that not only must the night sky have the same radiant intensity as a star's surface, but the same spectrum too.)

I wrote about your pictures in a previous post, which you might have missed:

http://www.sciforums.com/showthread...bers-paradox&p=3195507&viewfull=1#post3195507

 

We are back at your 1st argument. You are measuring brightness of the sky and all you do is measure brightness of a single star in each shell? Why not use wide angle field of view?

I'm on it, but we are getting there with this current conversation anyway.

 

I was referring to long exposure time required to capture light from Hubble Deep Field, but I realize now you will say it's due to angular size even though it can not be resolved. So let me ask you something instead...

Does there exist any such sensor in the real word that can confirm what you said here?

 

Since Wikipedia says the first shell is at thousand million light years, I think even those stars in the first shell would not be resolved to a size greater than a pixel, so I'll go strait to the alternative below.

Ok. The question is, would not stars occlusion and/or limited time exposure prevent the image for getting uniformly white?

 

If stars have no size? The problem with your logic is that your second statement contradicts your first statement. First you understand objects only appear smaller the further they are, and then you forget about it and use cartoon physics to draw your second conclusion.

 

This is YOUR logic we are discussing. I was explaining how your logic doesn't work regardless of which assumption you use. And in fact YOU are the one trying to have it both ways, saying in the previous post that you think stars with zero size can occult each other. Or, perhaps, that even though the stars are much smaller than a pixel, only one star's light can be seen on each pixel - which you must know is wrong.

 

The sensor would conclude they are equally bright because the same number of photons is captured by each.

You can measure this: zoom in on a star in each photo and measure with a photo editor the luminosity value of each pixel, then add all them all together. If they were made correctly, you should get the same total.

 

Before making an attack you really ought to put some effort first into making sure that we understand each other and are talking about the same thing, or we both are going to waste more time than necessary. I of course do not think stars have zero size, so let's not go there ever again, it's just plain insane.

I actually think that's the only way to make the paradox valid. I am only questioning will that really lead to full and uniform saturation of all the pixels, and how quickly would it happen in terms of some everyday camera with normal setting, say a smartphone camera on auto-settings as that's approximately similar to what the naked human eye would see.

 

Yes, single pixel sensor would measure total intensity and arrive to wrong conclusion in respect to how humans perceive brightness. Regardless of what your definition of "brightness" is, surely you have to agree if we ask 100 ordinary people whether those two images are equally bright all of them would say - no. Just like all of them would say that in this image below the white square is brighter than the grey square regardless of their size.

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I agree the total intensity is the same, but total intensity does not describe what people consider is "brightness" in normal everyday language. It obviously fails to recognize there is a contrast between the two images. It fails to account for the fact the human eyes have more than one pixel resolution.

It really is unfortunate brightness is not defined quantitatively so we are having this silly semantic argument where you can say a grey square is equally bright as a white square. What normal everyday people call brightness should have units in power per unit solid angle per unit projected area, or something among those lines. I was looking at the definition of "radiance" which kind of fits the description, but Wikipedia says although "radiance" and "luminance" were previously synonymous with "brightness" that usage is now discouraged in favor to make "brightness" a non-quantitative or subjective property. To make things worse some of these terms related to light measurement, like "intensity", have different meaning in physics and photography. And if we don't use the same language we can hardly ever agree.

 

OK, let's do that since it's reasonably trivial.

First, though, I stress that I am not measuring brightness, because brightness is ill-defined. I am counting photons, which is something objective. If you want to know how "bright" something looks, you can just count how many photons with optical wavelengths hit your retina each second and figure it out from there.

Consider one specific, narrow range of optical wavelengths to begin with, say 499 nm -- 501 nm. Look up at the sky. Now consider a single atom on your retina, which is exposed to the light entering your pupil. The solid angle subtended by your pupil at that atom's location is S steradians, say. Now, to find the total number of photons hitting that atom each second, we must integrate some function over the solid angle S, to get the contributions from all visible directions in the sky. What sort of function? Well, it will be some f(d) telling us the number of photons hitting the atom per steradian of sky per second for the direction d (the quantity d is just a unit vector pointing from the atom into space, in whatever direction we're considering). The total hitting the atom per second is N, where

\( \displaystyle N = \int f(\mathbf{d}) \textrm{d}\Omega , \)
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where dΩ is the element of solid angle, and the integral is over all directions d within the field of view. The next question is, how do we determine the function f(d)? Well, that's exactly what my previous argument was about; the result tells us that f(d) is the same for every direction in the sky, and so the quantity N above is just fS, with f now a constant. In other words, N is directly proportional to the solid angular size of the field of view, regardless of how wide our pupil is or which bit of sky we are looking at. Every patch of sky looks like what we see when we look at the same-apparent-sized patch of the Sun's surface, within the 499 nm -- 501 nm range of wavelengths.

You can repeat the argument for all other wavelengths, and the result is that the spectrum also matches what we see when we look at the Sun.

 

The inverse square law, plus some basic geometry, already confirms what I said there.

Edited to add:

You can postulate new laws of physics to circumvent the paradox if you want, but then you should be careful to check that they don't contradict anything else we have observed. It's easier said than done!

 

The opacity of intervening stars has the effect of preventing the detector from becoming exposed to infinitely many photons per unit time, but does not affect the image's uniformity. There will be some small non-uniformity due to deviations from perfect uniformity in the spatial distribution of stars, but I think that is not in the spirit of what you're asking about.

As for time exposure, apeture size, and how quickly the sensors saturate, you may be better able to see some interesting features like sunspots by playing with such things, I suppose. However, effects like that don't materially affect the conclusion that every patch of sky looks like an equally-sized patch of the Sun's surface. The oceans will still boil, I'm afraid.

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Edited to add:

On the general strategy of drawing the stars 1 px in size but with varying greyscale, I have some cautionary remarks about combining layers. It would actually be pretty tricky to implement that in such a way that you didn't introduce errors. For example, if you plot a grey pixel in some layer, it should not necessarily replace a grey pixel plotted when rendering a further-away layer (because the new star may not actually be obscuring the old one). I'm not even sure it is possible to do this in a way which doesn't mess things up, without doing something really fancy like proper raytracing.

I think the only way to really be sure is to begin with a very large bitmap, so that you don't need to plot stars as single pixels, and plot n[sup]2[/sup] white circles of radius K/n for values of n going from 1 up to about K/2 (i.e. we cut off the rendering after a finite number of layers). You'd draw the circles using simple overplotting of the previous pixels (this models the occlusion effect). Once you're done, scale the bitmap size down to something manageable and see how it looks. Then try it again with more layers (which may require increasing the initial bitmap size).