Olbers' paradox: why is the night sky dark?

[tris_d]

Olbers' paradox - Wikipedia, the free encyclopedia

I think treatment originally used to discard inverse square law as the solution to this question was not set up correctly. I would like to confirm my findings, but people working in Astronomy and Cosmology field are not receptive to even discuss it as it seems that would somehow contradict mainstream theory, which is what brings me here. So this is what I think about it, please let me know if you see any mistakes:


The original treatment ignores 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 would be bright. They also ignore exposure time. The rate of incoming photons is proportional to distance, due to inverse square law, which I find is well documented and accepted fact, and just by looking at that makes it clear to me inverse square law explains it all.

Let me explain with an example (all the stars are the same). Two stars at distance r would impact photo-plate with intensity I, and eight stars at double the distance will also impact photo-plate with the same intensity I. That's what they are saying, and that's fine. However, what they are not considering is that two closer stars will produce two dots each with brightens proportional to I/2, but eight further stars will produce eight dots each with brightness proportional to I/8.

There is difference between two bright dots and eight less bright dots, 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 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.

 

[fjrabon]

If you want this to be taken seriously you need much more than back of the envelope 'logic'. Individual photons traveling from a far distance aren't less bright. Why the light traveling froma distance is less bright is because the photons have dispersed and few of them are hitting 'the sensor'. The problem with your logic is that when you have infinite light sources, you will still have an infinite steam of a very small number of photons hitting the sensor. Each photon will be as bright as a photon is, which would mean white sky. Your treatment basically assumes that the inverse square law derives from light being dimmed at the photon level by distance, which isn't true.

 

[tris_d]

If you want this to be taken seriously you need much more than back of the envelope 'logic'. Individual photons traveling from a far distance aren't less bright. Why the light traveling froma distance is less bright is because the photons have dispersed and few of them are hitting 'the sensor'. The problem with your logic is that when you have infinite light sources, you will still have an infinite steam of a very small number of photons hitting the sensor. Each photon will be as bright as a photon is, which would mean white sky. Your treatment basically assumes that the inverse square law derives from light being dimmed at the photon level by distance, which isn't true.

As you said photons get dispersed, so the amount of photons impacting image per surface area per unit time is less coming from more distant stars. It is this amount of photons and the time of exposure that will determine the brightness imprinted on the image, and that amount of photons is proportional to 1/r^2, correct? So, based on that I made these four pictures where exposure time is the same and the stars are the same, only distance doubles:

Now, this series of pictures is either true or false representation of how inverse square law influences image brightness relative to distance. And if you think it is incorrect, then please point out why, where is the mistake? How do you think it should look like?

 

[fjrabon]

the problem is that if ANY number of photons are reaching the viewer from the source, the fact that there are infinite source stars would equal pure brightness. Think about it this way, the photons aren't going anywhere when the inverse square law takes effect, they're just getting more dispersed, however, as they become more dispersed, their dispersion fields begin to overlap with nearby stars in the visual frame, which would then cause areas of overlap to brighten. That's what your pictures don't take into account. In your pictures, none of the 'stars' points of emission ever start to overlap, and thus total brightness only gets dimmer. But, when you have an infinite number of point light surfaces, the inverse square law no longer holds, because then you no longer have a point surface of light, but a flat plane. The inverse square law applies to point sources of light only. The fact that every star is immediately adjacent to another in the visual plane means that the sky should act like a complete plane of light that completely envelops the universe.

And additionally, the fact that the universe is expanding has been confirmed pretty heavily, and is sufficient to explain olber's paradox anyway, since we now know the universe isn't in fact static.

 

[tris_d]

the problem is that if ANY number of photons are reaching the viewer from the source, the fact that there are infinite source stars would equal pure brightness. Think about it this way, the photons aren't going anywhere when the inverse square law takes effect, they're just getting more dispersed, however, as they become more dispersed, their dispersion fields begin to overlap with nearby stars in the visual frame, which would then cause areas of overlap to brighten. That's what your pictures don't take into account. In your pictures, none of the 'stars' points of emission ever start to overlap, and thus total brightness only gets dimmer. But, when you have an infinite number of point light surfaces, the inverse square law no longer holds, because then you no longer have a point surface of light, but a flat plane. The inverse square law applies to point sources of light only. The fact that every star is immediately adjacent to another in the visual plane means that the sky should act like a complete plane of light that completely envelops the universe.

And additionally, the fact that the universe is expanding has been confirmed pretty heavily, and is sufficient to explain olber's paradox anyway, since we now know the universe isn't in fact static.

I would like to confirm the basics first, so please forget about infinity for the moment. In my example there is only 30 stars in the whole universe, and if you prefer let it be 30 light bulbs in a dark room. The question is whether the series of four pictures I posted above is correct representation of how inverse square law influences image brightness relative to distance. Can you confirm? Or even better, can you point me to some actual photos taken of the same light bulb with the same exposure time at double, triple and quadruple distance, or something among those lines?

 

[fjrabon]

the problem is that if ANY number of photons are reaching the viewer from the source, the fact that there are infinite source stars would equal pure brightness. Think about it this way, the photons aren't going anywhere when the inverse square law takes effect, they're just getting more dispersed, however, as they become more dispersed, their dispersion fields begin to overlap with nearby stars in the visual frame, which would then cause areas of overlap to brighten. That's what your pictures don't take into account. In your pictures, none of the 'stars' points of emission ever start to overlap, and thus total brightness only gets dimmer. But, when you have an infinite number of point light surfaces, the inverse square law no longer holds, because then you no longer have a point surface of light, but a flat plane. The inverse square law applies to point sources of light only. The fact that every star is immediately adjacent to another in the visual plane means that the sky should act like a complete plane of light that completely envelops the universe.

And additionally, the fact that the universe is expanding has been confirmed pretty heavily, and is sufficient to explain olber's paradox anyway, since we now know the universe isn't in fact static.

I would like to confirm the basics first, so please forget about infinity for the moment. In my example there is only 30 stars in the whole universe, and if you prefer let it be 30 light bulbs in a dark room. The question is whether the series of four pictures I posted above is correct representation of how inverse square law influences image brightness relative to distance. Can you confirm? Or even better, can you point me to some actual photos taken of the same light bulb with the same exposure time at double, triple and quadruple distance, or something among those lines?

I have no idea what you're asking. My point is that when two stars are close by, and their light overlaps, you get a brightening effect. This can be seen with all sorts of pairs of stars that are close, and to the human eye look like one bright star, brighter than either individual would be. That's what your examples above aren't taking into account, since none of your 'stars' overlap. Once you start getting overlap on a large scale, you no longer have point sources of light. The inverse square law ONLY APPLIES FOR POINT SOURCES of light, and thus, if any of the stars in the universe are close enough in the visual 2-D plane to overlap, you can no longer use your method of reasoning.

 

[fjrabon]

look at the following picture:

bokeh-photography-9.jpg

notice how the areas where the bokeh overlaps are brighter? That's the piece of this puzzle that you're missing.

 

[Buckster]

As previously mentioned, the fact that we don't live in a static universe is the biggest factor/key to the answer.

Stars are born and die at different intervals and at different distances from Earth, requiring different amount of time for the photons to reach Earth. In other words, they turn on and turn off, so there's not a steady stream of photons from any part of the sky, let alone from all parts of the sky all at once and forever.

Also, as the universe expands, many stars (and more as the universe expands) are beyond the event horizon, so they will never be seen from Earth because they (the ones beyond the event horizon) are, in fact, moving away from Earth at faster than the speed of light (relatively), and as the expansion of the universe accelerates, that is more of a factor every day.

In addition, not all of 'empty' space is empty to allow the photons to make it to Earth unimpeded. There is a lot of dark matter and dust out there that blocks a lot of photons along the way. Photons zipping too near a black hole get sucked in and never make it to us either, and it turns out there are a lot more black holes out there than first thought as well.

Because of those factors, every possible part of the night sky is not populated with an observable star from our point of view and, in fact, most of it isn't.

Eventually, because of the expansion of the universe, especially because that expansion is accelerating, an observer anywhere in it will see no stars at all, because they will ALL be beyond the event horizon, no matter where in the universe the observer is.

Lawrence Krauss has a very interesting lecture that dives into this as a part of it here:



Jump to about the 49 minute mark of the video for the punchline, but I DO recommend watching the whole thing to best understand how he gets to that point.

 

[amolitor]

There's no problem at all having an infinite universe with an infinite number of stars scattered around randomly in it, and NONETHELESS having whatever fraction of sight lines you like fail to terminate on the surface of a star. As always when dealing with infinities the details are quite subtle. This is loosely waved at in the "Fractal star distribution" section of the wikipedia article.

This is separate from the issues raised above, but still important.

 

[tris_d]

I have no idea what you're asking.

I'm asking about the basics first. -- There are two light bulbs in a dark room. Closer one (bottom-left) is 5 meters away, and further one (top-right) is 10 meters away from the camera. The exposure time and aperture size is such that when you take that photo and open it in Photoshop pixel brightness of the closer bulb is equal to 99. The question is, what would be the pixel brightness of the further light bulb?

I set it up in Photoshop for pixels of the further (top-right) bulb to have brightness 99/4 (~25). Is that correct, is that approximately how the photo would actually look like?

 

[TCampbell]

You might have more fun posting this question on an astronomy forum.

The inverse square law (which photographers know well), says that if the amount of light emitted from a source remains the same and only the distance is changed, then the amount of light arriving at a subject will change based based on the inverse of the square of the change in distance.

In other words.... if I have a light bulb located 10' away and now I move that light bulb 14' away (it's distance has changed by 1.4x the original distance) then the amount of light hitting me will be cut in half because 1.4 squared is 2. The inverse of 2 is 1/2. If instead of moving the light farther away, I bring it closer (so I change it from 10' to 7' or .7x) then the amount of light will double because .7 squared is 1/2. The inverse of 1/2 is 2. (Yes, I did round off the math to keep it simple.)

In your example, you double the distance... and that will cut the amount of light to the inverse square of 2... or 1/4.

Keep in mind that we can't assume all stars emit uniform brightness levels. Only certain classes of stars can be used as "standard candles". Those stars don't actually put out the same amount of brightness either... rather there's a correlation that lines up quite neatly on a graph when you graph the period of variability compared to the amount of light the star emits. Which means if you know the period of variability for the cepheid you can make an accurate assumption about the amount of light being emitted -- not that all cepheids emit the same amount of light.

If your 30 "stars" or 30 light bulbs were known to be exactly the same brightness, then yes... the inverse square law would work and you could measure the distance of each bulb based on how bright that individual bulb appears to be. This assumes the air in your "room" is clear.

If we ignore the "infinite universe" assumption and the "steady state" assumption we should still get to apply this theory not to the whole sky... but at least to some limited areas of sky. The center of the Milky Way galaxy is incredibly dense with stars. This means that if we look to the center of the galaxy what we should see (based on this theory) is a large white blotch of sky. Instead, what we really see is an awful lot of dust and we can't actually see the center of our galaxy in visible light at all. Astronomers have to use IR to peer into the center.

The area of space in which we live has a star density of roughly 1 star for every 5 cubic light years of space. In a globular cluster (globulars are VERY old) the density can be much much higher. A typical globular has a star density which as about 60,000 times more dense than the area of space we live in. So while the nearest star (Proxima Centauri) is 4.2 light years away, imagine having about 60,000 stars in the the roughly "cube" shaped area of space between our Sun and Proxima Centauri. And yet... even if we DID live inside such a cluster, the amount of light from all those stars during the night time would only provide the amount of light comparable to a room with a few candles in it. It wouldn't be particularly bright.

We do see "energy" everywhere... we've got the "cosmic microwave background radiation" which seems to be "everywhere". But that's not visible light.

My other major hobby is astronomy.

 

[tris_d]

In your example, you double the distance... and that will cut the amount of light to the inverse square of 2... or 1/4.

If your 30 "stars" or 30 light bulbs were known to be exactly the same brightness, then yes... the inverse square law would work and you could measure the distance of each bulb based on how bright that individual bulb appears to be. This assumes the air in your "room" is clear.

The light bulbs are the same, further one is at double the distance. And so I take it you confirm my image representing such two light bulbs is correct and the further light bulb would indeed appear dark gray. Ok? -- Now, before I get into argument about stars and the universe, I would like to clear this conclusion a bit more. Consider the photo below, those lights do not seem to shift their brightens to more darker shades with the distance like on my picture, they actually seem to keep their brightness and only shrink in size. Is this because the pixels get over exposed, and if we change exposure time we would actually see those lights do indeed become dimmer (darker shades) as my image would suggets?

 

[spacefuzz]

Reading stuff about light cones may help you as well better understand how the photons get dispersed after they leave the star.

Nice explanations Fjrabon.

 

[christop]

I have no idea what you're asking.

I'm asking about the basics first. -- There are two light bulbs in a dark room. Closer one (bottom-left) is 5 meters away, and further one (top-right) is 10 meters away from the camera. The exposure time and aperture size is such that when you take that photo and open it in Photoshop pixel brightness of the closer bulb is equal to 99. The question is, what would be the pixel brightness of the further light bulb?

I set it up in Photoshop for pixels of the further (top-right) bulb to have brightness 99/4 (~25). Is that correct, is that approximately how the photo would actually look like?

No, that's not right. The far object appears to have one fourth of the area of the near object, so the sensor receives one fourth as much light from it than from the near object. The two objects appear to be the same brightness because the ratio of light being received by the sensor from the object to the apparent size of the object is constant regardless of distance (ignoring the normally negligible light loss due to contaminants in the air etc).

If what you are saying were true, mountains would look black far away, or blindingly bright up close. That is obviously not the case.

Besides, this part of the experiment is easy enough to test by yourself. Use fixed settings in your camera and take several shots of a constantly-lit object or light source, varying only the distance between your camera and subject. It would probably be best to move the camera rather than the subject to avoid inadvertently changing the lighting conditions.

 

[fjrabon]

Furthermore, another error in your explanation is that you made your 'stars' both dimmer and smaller, essentially doubling your problem. The reason the inverse square law works is because of the 'smaller' effect. The light doesn't actually get dimmer itself. If you take a picture of two lights, and you cut out a piece of the picture that just covers the actual light itself, the two will appear equally bright on their surface. The inverse square law is saying that the intensity of the light from that point source that falls on a particular point in space is inversely proportional to the distance, not that the light itself seems brighter or darker on its surface from the vantage point of a distant observer.

Now, a star can get so far away that we have difficulty seeing it, not because its dim, but because its really small. But it is still adding some amount of light. And since the intensity of light is additive (ie if star x is 2 brightness and star y is 1 brightness, together they provide 3 light on the subject) any infinitesimally small amount if light, summed over infinity is still infinity.