Inverse square law resolves Olbers' paradox

tris_d

Treatment originally used to discard inverse square law as solution to Olbers' paradox was not set up correctly. If we include sensor (camera) in the treatment and model light as photons the result describes what we actually see.

Barakn

Not really.

tris_d

You see I got the red card for posting this, we are not allowed to discuss it here.

Ah well, I guess this is the point where I get banned.


http://www.asterism.org/tutorials/tut09-1.htm



Since the area of a sphere of radius r is

A = 4p r2 (1)

the volume of such a shell is

V = 4p r2t (2)

If the density of each of the luminous objects within the shell is "n", then the total number of these objects in the shell must be

N = 4p r2nt (3)

Now let us ask just what amount of energy such a shell will send to the Earth. Since the shell's thickness is small, it is reasonable to assume that the entire shell is at a distance "r" from the earth. The energy, E, emitted by any source at distance r, produces an intensity, "I", over a given area, A, on the Earth of (inverse square law)

I = E/4p r2 (4)

The total intensity received on the Earth from all the sources in the shell r units away must then be the intensity produced by each source times the total number of sources or

T = IN (5)

Substituting the value of N previously calculated into the above, we find that

T = tnE (6)

We notice at once that the total energy received from any chosen shell does not depend upon its distance from us (no r in the above equation). The total energy received from all the shells is the sum of the contributions of each shell. If there are M shells this total is

S = tnEM (7)

But there is an infinite number of shells and so the total intensity on the earth must be infinite. Therefore, the nighttime sky should be blindingly bright!


--//--

They completely ignored sensor surface area, that is some 2-dimensional image receiving this light, like a photo or human eyes, and by ignoring that they get result as if the image has only one pixel. So instead of to "see" many dots, some bright some less bright, they practically sum all the received intensity in only one pixel and thus result wrongly indicates the sky is bright.

They also ignored exposure time. The rate of incoming photons is proportional to distance, due to inverse square law, which is known and accepted fact, that's why very distant stars do not produce any dots on a photo-plate unless we wait long enough. Just by looking at this fact makes it clear to me inverse square law explains it all.

Let me explain with an example. Two stars at distance r would impact photo-plate with intensity I, and four stars at double the distance will also impact photo-plate with the same intensity I. That's what they are saying, and that's true. However, what they are not considering is that two closer stars will produce two dots each with brightens I/2, but four further stars will produce four dots each with brightness I/4.

There is difference between two bright dots and four less bright dots of course, and there is difference between two dots on 10x10 resolution image and 1x1 resolution image. So when they ignore this sensor surface area they practically work with 1x1 resolution image where all the intensity gets summed up at one pixel, and of course all they see is "bright sky". To summarize I draw this conclusion: at infinite distance there will be infinite number of stars and if we had infinite resolution they would produce infinite number of dots, but the brightness of each dot would be I/infinity, which is pretty much nothing but black.

Bandersnatch

Is there a difference between two halves of the Sun and four quarters of the Sun? They all add up to one Sun.

tris_d

Two or more stars do not add up to one star, there is spatial separation between them.

Drakkith

The Inverse Square Law does not resolve Olbers' Paradox. Instead it is the state of the universe that does so. Consider the following. IF the universe was not expanding, and IF it was infinitely old we would be swamped with visible radiation, since an infinite number of stars lie at every line drawn away from the Earth, leading to an infinite number of photons coming in. However this isn't the case because both of those things are not true. It is the combination of an expanding universe that has a finite age that resolves the paradox.

But even if the inverse square law did happen to "solve" the paradox, it still wouldn't mean anything. Olbers' Paradox is about an eternal static universe. In such a universe the known laws of stellar evolution wouldn't apply, as there would need to be some sort of "recycling machine" to produce new hydrogen from old, heavier matter that stars produce. Otherwise you wouldn't get an infinite number of stars for an infinite amount of time. The fact is that a great many things resolve Olbers' Paradox, from the expansion of the universe, to the basics of nucleosynthesis and more.

Barakn

Huh? Maybe you better explain to us what the difference between brightness and intensity is. Most of us view them as being the same, so you've just produced two mutually exclusive statements.
Now you've definitely gone off the deep end. You've just assumed that all stars are infinitely distant. The ones that count are much, much closer. But even at that infinite distance, you've made a bad assumption. The "outermost" shell (in quotes because if there's a shell at infinity, there's another one at infinity+1) has an infinite surface area and therefore has an infinite volume and infinite number of stars. You've essentially divided that by r² where r = ∞ to estimate the intensity here, and you've arbitrarily assumed that the result of dividing one infinite number by another is zero. It doesn't have to be - it could be zero, infinity, or any number in between. The proper approach would involve using limits as r → ∞. Until you show some math, your argument remains just so much handwaving.

tris_d

The rate of incoming photons from very large or infinite distance would proportionally go down, so the chance for even a single photon to hit us from there would be very or infinitely small.


I've been e-mailing this to hundreds of professors and researchers at famous universities around the world, and I just got my first reply from one Oxford professor. He said:

- You seem to ignore the key aspect, which is that the further away a star is, the faster it is moving away from us. This motion dims its light in two ways: it shifts the photons to lower frequencies where they have less energy, and it leads to longer gaps between the arrival times of successive photons than there were between the emission of the photons. So the brightness of a star decreases with distance faster than 1/r^2, while the # of stars per unit angle of the beam increases only as r^2.


To which I replied:
- "Thank you for your time. I don't see what you said contradicts what I said, rather just adds up to the effect of making light even dimmer than what would manifest only due to inverse square law.

Can you tell me are we actually able to measure any difference in the arrival time of successive photons of certain galaxies if we compare measurements of today with measurements from say several years ago?"



I see people are interpreting the paradox in different ways. For me it's just about answering the question: why is the night sky dark? And as you say there could be many factors, one of them certainly 'expanding universe', as Oxford professor explained, but what I'm saying is that inverse square law by itself would be enough to account for the night sky not being completely bright as the original treatment concluded.

tris_d

Good point.

http://en.wikipedia.org/wiki/Brightness
- "Brightness" was formerly used as a synonym for the photometric term luminance and (incorrectly) for the radiometric term radiance. As defined by the US Federal Glossary of Telecommunication Terms (FS-1037C), "brightness" should now be used only for non-quantitative references to physiological sensations and perceptions of light


http://www.its.bldrdoc.gov/fs-1037/dir-005/_0719.htm
- brightness: An attribute of visual perception in which a source appears to emit a given amount of light.

Note 1: "Brightness" should be used only for nonquantitative references to physiological sensations and perceptions of light.

Note 2: "Brightness" was formerly used as a synonym for the photometric term "luminance" and (incorrectly) for the radiometric term "radiance."


http://en.wikipedia.org/wiki/Apparent_magnitude
- The apparent magnitude (m) of a celestial body is a measure of its brightness as seen by an observer on Earth


It's scales proportionally. The further the star the longer is the time interval between successive photons to reach us from there. Basically what I said is that you would need very long or infinitely long exposure time to see any of those stars.

I'm not sure what more math there is. It's pretty simple. The original treatment does not take image resolution into account, so it can not differentiate between two stars with brightness I/2 and four starts with brightness I/4, thus it practically sums up all the intensity from any shell into an image with only one pixel resolution, ignoring all the spatial separation, and therefore the result is not complete. They get correct intensity, but that's not what we see or what camera captures, what we see is 2-dimensional image where each star has its own spatial location, so we need to divide this intensity across all the "dots" to get result indicating what we actually see.

Drakkith

Then I recommend you stop using Olbers' paradox as your basis for this question. Olbers' paradox is about a static and eternal universe. When I searched for Olbers' Paradox I immediately found this:

In astrophysics and physical cosmology, Olbers' paradox, named after the German astronomer Heinrich Wilhelm Olbers and also called the "dark night sky paradox", is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe.

If you are not specifically talking about Olbers' Paradox in an infinite and eternal universe, then simply ask why the night sky is dark.

Of course, as there hasn't been time for much light to reach us from distance sources. This of course may not be an issue in an eternal and static universe. It really depends on a great many factors.

Not really. The change in velocity is far too small over such a short period of time. Plus, the arrival of photons at a detector is subject to Poisson Noise, and arrive in a semi-random fashion, which would make it even more difficult to find a change, especially for extremely dim galaxies.

tris_d

I see what you mean, and finally now I understand why some people thought what I am saying does not fit in mainstream theory. They thought I was trying to argue the universe is infinite, eternal and static, but what I am really saying is that inverse square law would work just the same regardless.

I think it's when they found inverse square law does not answer the question when they started calling it a paradox, and since then it seem like inverse square law was completely ignored and not considered to be even a part of the solution due to conclusions of the original treatment, which I think is a mistake and should be reconsidered.


Do you know where I could find some actual numbers? I'd like to see how many photons per second we get from two similar stars that are at different distance, and also from two different stars that are about the same distance away from us.

Drakkith

It's called a paradox because for most of history the Universe has been thought of as static and unchanging. It wasn't until the early-mid 1900's that we began to even realize the actual size of the universe and that it was expanding. During this time arguments were thrown back and forth for various models, including one that claimed the universe was both eternal and static. For such a model the darkness of the night sky introduces a "paradox" in that IF the universe is static and eternal, the night sky should not be dark. Or at least not as dark as it is.


I'm not actually sure how to work it all out. Try this article and see if you can figure it out from there: http://en.wikipedia.org/wiki/Apparen...e#Calculations

Chronos

Actually, the night sky should be about as bright as the surface of an average star under Olber's reasoning - since every possible line of sight falls upon the disc of a star. We are forced to conclude one or more of Olber's premises - the universe is static, spatially and temporally infinite, and infinitely populated with stars - are invalid. Modern observational evidence strongly suggest all these premises are invalid.

H2Bro

There actually isn't a difference between two dots each I/2 bright, and 4 dots each I/4 bright, so long as each group is seen as a pointsource by the observer.

You keep mentioning Olbers paradox, but you explicitly say your not arguing under the premise that the universe is eternal and static. I think what your getting at is that even IF the premises of Olbers paradox - static and eternal universe - held, the night sky would be black because the inverse square law.

If thats the case then picture this. Imagine concentric shells around an observer, but not filled with stars, instead the surface area of the shell is covered with 50,000K plasma. At distance D from the observer, observer gets a total amount of energy S. At distance 2D you get still get sunlight S, even though the source is twice as far the surface area shining light at you has also doubled, so you get the same.

Now picture half of the shell is at 2D and the other half at D. Still equals S light. Now imagine the shell fractured into billions of points each at different distances, but still forming a continuous shell from the point of the observer. You still get S, and this is why Olbers paradox is a paradox. Also this is independent of what size of sensor your using. So long as your pointing the sensor up at the sphere, it will receive S/(focal arc of the sensor) amount of energy.

If your not arguing within the hypothetical situation of of Olber's premises, then your just saying that the night sky is black because really distant stars aren't as bright as closer stars. Which isn't much of a statement at all, akin to saying you've discovered circles are round.

Have you actually done this?

tris_d

It does not matter if every line of site ends up at some star. Light is not some continuous instantaneous rays, as they obviously thought it is at the time they made that conclusion, but there are gaps between incoming photons that increases as the distance increases, so we would not see far away stars without making long exposure time regardless of how many stars there are and regardless of the size of the universe.

http://en.wikipedia.org/wiki/Apparent_magnitude
- Note that brightness varies with distance; an extremely bright object may appear quite dim, if it is far away. Brightness varies inversely with the square of the distance.


You gonna get me banned. I don't want to argue the universe is static or infinite, I just want to say the original treatment is incomplete, and even with the result I'm suggesting it does not lead to either conclusion.

tris_d

I'm not sure what do you mean. If you see four stars as a single point source than that would be just one star for all it matters.


Yes, that's what I'm saying. Inverse square law would make it dark in either case, due to rate of incoming photons that drops with the distance. Expanding universe just makes it even darker.


That's interesting, but I don't think it compares, as in that case it seems even just the 1st shell would be sufficient to make the night sky bright, and would occlude further shells so they wouldn't even matter.


Well, that's certainly nice example, it's getting me confused. Let me try the same argument as before: emitted amount of light would be the same, but received amount of light, per unit time, would not be the same as the rate of incoming photons would be slower from more distant shell.


Then you should be able to point which one of my sentences is false or does not follow. Like when you said two dots with brightness of I/2 is the same thing as four dots with brightness of I/4. If you can prove/explain that, then I will have to agree with you about everything else.


Yeah. Haha! And it's not the first time I did that either. Every university has very accessible list with all the professors, researchers, alumni, emeritus and all other kinds of associates, friends and visitors. They don't really respond to random e-mails like that so you have to send hundreds of e-mails to get some response. That's my experience anyway. Funny!

Drakkith

I think that is exactly what he's saying. If you can't resolve the individual stars, then they can be considered to be a single point source and not multiple ones.