Inverse square law explains Olbers' paradox?

  1. Hello,

    This is the thread I originally wanted to respond to, but it's closed:

    I also found this on Wiki-talk page, which seems to be the same argument:


    a.) left image representing first shell contains 10 bright stars

    b.) right image representing second shell has 40 stars each 4x less bright

    c.) total light received is the same, but does that make them equally bright?

    Now, my question is, are those two images indeed correct representation of the paradox? And if so, are those two images really supposed to be equally bright? I mean, are they? They certainly don't look equally bright. So what's the trick, is this some kind of optical illusion, or something?
    Last edited: May 25, 2014
  2. jcsd
  3. Drakkith

    Staff: Mentor

    The trick is that the light is spread out more in the 2nd picture, which leads to it looking dimmer overall. If you concentrated all the light into one spot the image would "appear" to us to be brighter because it is more noticeable. In other words, the difference in brightness between different areas of the image is greater in the first picture than in the second picture, and our eyes and brains have an easier time seeing this greater difference.
  4. Ok. But what I'm trying to point out is that those two images are not equally bright in the same way Olbers' paradox would have us believe, from Wikipedia:


    So if we receive the same amount of light from each shell, but individual stars from further away appear dimmer than closer stars, then I see no paradox since that's what we actually observe in reality and is explained by the inverse square law.
  5. Jonathan Scott

    Jonathan Scott 1,360
    Gold Member

    You're confusing the issue with the use of the word "dimmer" here. The point is that the brightness of any part of the surface of an individual star is approximately independent of distance. If it's further away, the angular size is smaller, but the brightness over the smaller angular area is essentially the same, so it's only dimmer in the sense that it covers a smaller area of the sky.
  6. I suppose that Wikipedia animation is misleading in more than one way. We are talking about human eyes here, all those stars (galaxies) appear as point light sources and their "apparent magnitude" does indeed get dimmer with the distance, according to inverse-square law.

    Besides, Olbers' paradox is taking it into account, so even if it was not really true the two pictures in my first post would still correctly represent what the paradox actually states, and my primary objective is to establish just that. I believe Drakkith confirmed those two pictures do represent the paradox correctly. I take it you disagree, so I'm asking you to reconsider.
  7. Jonathan Scott

    Jonathan Scott 1,360
    Gold Member

    Calculate what the sun looks like from twice as far away. It appears half the linear size and a quarter of the area, but the amount of the surface area of the sun you can see in a given angular area has multiplied by 4 at the same time. This factor cancels out with the inverse square effect of the distance so that the surface is emitting the same amount of power per angular area (normally known as solid angle) as seen from any distance. So overall it is a quarter of the total brightness, but each part of the surface has the same brightness (in terms of power per angular area - usually known as solid angle - at the observer) regardless of the distance.

    For a distant star, we cannot necessarily resolve it as anything other than a point, but Olbers' paradox does not require us to be able to do so.

    The limitations of human eyesight are not relevant here (except that at least our lack of resolution prevents distant stars from damaging our retina!) If you replace a single star by four stars which are twice as far away (making the same total light) then human eyes may not be able to resolve each one but their contribution to the overall level of light is unaffected by the method of observation, and the angular area they cover is the same.

    Olber's paradox doesn't even require any particular radial distribution of stars to work, except that there should be sufficient stars that every line outwards from the observer should eventually encounter a star. If that happened, the sky would be as bright as the sun in all directions.

    As far as I can see, the Wikipedia animation is accurate and helpful.
  8. I don't disagree with that, but we are talking about point light sources, thus inverse-square law applies. Do you agree?

    Therefore, if we represent the first shell with 10 bright stars, we must represent the second shell with 40 stars where they are 4 times less bright than stars in the first shell, like this:

    [​IMG] the total amount of light received from both shells is the same. Right?
  9. Drakkith

    Staff: Mentor

    Don't get caught up in "point sources". Whether we can resolve an object or not doesn't matter, the amount of light received is still the same. If it helps, just imagine that we have a perfect optical system capable of resolving any object, no matter how small/distant.
  10. I agree we need not to worry about much detail for our hypothetical scenario, it is supposed to be generalization. And I agree the amount of received light is the same from each shell regardless of any such detail. The point where we disagree is when you say 40 grey dots are as bright as 10 white dots.

    Imagine we arrange those dots into one little white square on the left and one big grey square on the right. Yes, they both "received" the same amount of light, but they are not equally bright. The white box/dots will always stand in contrast against the grey box/dots regardless of their size or quantity.

    More importantly however, I'd like if we could first establish this: the stars in Olbers' paradox are considered as point light source and thus inverse-square law applies, so that stars in every shell have the same apparent size as stars in a previous shell and are four times less bright.
  11. Jonathan Scott

    Jonathan Scott 1,360
    Gold Member

    40 dots of 1/4 the brightness do add up to the same total luminosity as the 10 dots. If you move and adjust the shape of the dots (without changing the area) until they formed a contiguous area of the same shape, there should be no visible difference between the two. Obviously the human eye can detect contrast more readily when the power is concentrated into fewer larger dots, but contrast is not relevant here, only the total power and the total solid angle from which it is being emitted.

    Olbers' paradox would not work if stars were indeed points, as it is necessary for each star to subtend a finite solid angle, however small. The "apparent size" is not relevant.
  12. There's a different issue at work here. Normally the light emitted by a computer screen is not a linear function of the RGB values.
    Try setting your screen's gamma correction to 1. You will see how your two pictures have more or less the same brightness.
  13. Drakkith

    Staff: Mentor

    We need to be specific here. The total amount of light emitted by the dots is the same. However, they are not equally bright because brightness is the visual perception of the luminance of a surface, which itself is a measure of how much light is emitted per unit area. In other words, with 40 dots the same light is spread out over a larger surface area and the image is dimmer. Note that in reality you aren't looking at the picture as a whole and deciding how bright it is. You are looking at the picture and seeing that the dots are brighter in one picture than the other. Our eyes and visual system simply aren't designed to accurately measure overall light intensity in this manner. It's much easier to see and compare distinct light sources than an overall scene.

    One thing to understand here is that if we look at both images from far enough away so that we can't resolve each dot individually, the two images will appear equally bright. Think of each dot as a flashlight. If the flashlights are so far away that we can't resolve each one as an individual light source, then the combined light from the 10 brighter flashlights looks equally as bright as the 40 dimmer flashlights.

    It doesn't matter if they are considered point sources or not, the inverse square law still works just fine. And I don't understand what you're saying about the apparent size of the stars. Point sources are not resolved and don't really have an apparent size.
  14. Luminosity is a measure of emitted light, brightness is a measure of received, or better to say perceived, light. Luminosity refers to an actual object and is objective property, brightness refers to an image of an object and is subjective property. As Drakkith points out we have to be careful about various types of light measurement which often have subtle but important differences.

    Luckily, in this case, we can avoid having any kind of semantic argument or disagreement due to differences in our definitions and interpretations. Instead of talking about words, we shall talk about pictures, and instead of talking about image brightness, we shall talk about whether our images are simply true or false. The goal is of course to establish correct visual representation of Olbers' paradox and thus conclude if it is indeed different from what we actually observe, or not.

    Therefore I state, if we represent the first shell with 10 bright stars, we must represent the second shell with 40 stars 4 times less bright than stars in the first shell, like this:

    [​IMG] the total amount of light received from both shells is the same.

    True or false?
  15. Drakkith

    Staff: Mentor

    That's right. But note that while each star in the 2nd shell is 4 times dimmer, it is also 1/4 the apparent size that the stars in the 1st shell are (assuming we can resolve them). Understand that Olber's paradox isn't about point sources. Point sources are the result of our inability to resolve far away objects. That just means that the the size of the airy disk of the focused light is larger than the image of the object at the focal plane.
    Last edited: May 25, 2014
  16. I agree it was a mistake to talk about brightness of an image as a whole, we are really only concerned about brightness of the stars. The stars will eventually completely fill our image and then "image brightness" will make more sense in relation to whether it is uniformly bright or not, but until then it's misleading.

    Yeah, but that's like photographing a photograph. I consider the image we are talking about to be the final image formed in our brain when we look at Olbers' paradox night sky. I'm not sure we are talking about the same thing, or at least not from the same perspective, so I'd rather concentrate on things we absolutely agree on and I think we should proceed from there.

    If we consider them all as point light sources it will simplify our visualization. Surely with naked eye we can not resolve the size of any star or galaxy. Maybe a few, I don't know, but for our generalized hypothetical scenario I believe considering them all as point sources is a very sensible thing to do. With "apparent size" I was trying to take into account visual artifacts where a point light source may appear smudged over some area as opposed to being illumination of a single pixel.
  17. Jonathan Scott

    Jonathan Scott 1,360
    Gold Member

    Ever looked at the Milky Way on a dark night? You will see that there is an impression of light, forming a milky band, but you will not be able to resolve many of the stars. Being able to resolve them does not affect the total light received.

    Note that if the light from an image is blurred, making it larger, this does not affect the overall balance, because it is the amount of light received compared with the amount of background hidden behind the star itself which determines the average brightness per angular area. That is why the assumption of point sources does not work.
  18. How am I supposed to draw that? They are already point light sources, they can not get any smaller, only dimmer. astronomy, stars are routinely treated as point sources for stars and other point sources of light

    ...images of point sources (such as stars)
  19. I didn't say total light received is affected by anything. I said total light received is the same for both of those two images. astronomy, stars are routinely treated as point sources

    Where do you get your information from? What are you suggesting how those two pictures should look like? Are you saying I should make the stars in the second shell brighter?
  20. Jonathan Scott

    Jonathan Scott 1,360
    Gold Member

    The two shells have the same total light. If the images of the shells were represented correctly, and were then reduced in size until you couldn't resolve the points, they should look similar (as for the Milky Way analogy).

    For purposes of Olber's paradox, the effective surface brightness of the sky is determined by the light being received from each visible star divided by the angular area (solid angle) that the star occupies against the background. This is the same for similar stars at all distances, regardless of what the individual star looks like to the human eye.

    This is like spray-painting something. It doesn't matter whether the drops are big or small; by the time the surface is completely covered, they have all joined up and overlapped, and you only see the final surface.

    When we are dealing with light from individual stars, they are of course approximately point sources. However, for Olber's paradox, one also needs to take into account the angular area (solid angle) of the source. The suggested assumption that a star is actually a point source but emitting a finite amount of energy would be equivalent to infinite brightness per angular area, which isn't going to give a sensible result for Olber's paradox.

    And it really doesn't matter if the observed image is out of focus, so light from different stars gets mixed up, as the total amount of light is unchanged and the fraction of the background which each star "covers" is unchanged, even if it cannot be resolved.
  21. Drakkith

    Staff: Mentor

    Considering the stars as point sources may simplify things in many cases, but it doesn't get at the heart of Olber's paradox. The key lies in the fact that stars have a finite apparent size and in an eternal infinite universe they would fill the sky completely, so that no matter where you look it would be like looking at the surface of a star.

    Note that nothing of what I just said involves point sources or resolving power. I've only talked about how in an eternal infinite universe your line of sight would fall on the surface of a star somewhere. The inverse-square law still applies here, but it's kind of a red herring that takes the focus away from the real explanation, which is that the luminosity per solid angle doesn't decrease with distance. For example, the Sun is about 0.5 degrees (30 arcminutes) across as seen from Earth. If we move out to 2 au the Sun is now 15 arcminutes across, which means the apparent area has dropped to 1/4 of what it was at 1 au. However, a section of the Sun 1x1 arcminute in area has the exact same luminosity at both 1 au and 2 au.

    So, knowing that the luminosity per solid angle doesn't decrease with distance, and that in an eternal infinite universe our line of sight would always fall on the surface of a star, we can say that the sky would be extremely bright if our universe were infinite and eternal.

    The explanation involving concentric shells and point-sources just muddles up everything, as it takes the focus away from the above.
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