Ask Marilyn ® by Marilyn vos Savant is a column in Parade Magazine, published by PARADE, 711 Third Avenue, New York, NY 10017, USA. According to Parade, Marilyn vos Savant is listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ."
In her Parade Magazine column of December 10, 1995, Marilyn claims that Olbers' paradox remains unexplained.
Distance is more important than you realize. Since light from a star propogates in three dimensions, the amount of light reaching an observer is inversely proportional to the distance to the third power.
Jim Thompson <firstname.lastname@example.org> and Craig Gentry <email@example.com> both pointed out that in fact, the amount of light reaching an observer is inversely proportional to the square of the distance rather than the cube of the distance. This is because the intensity of the light is inversely proportional to the area of the sphere over which it is distributed rather than the volume of the sphere through which it travels.
My original error allowed me to over-simplify my analysis, by placing all the stars in the Milky Way Galaxy at the same distance as the closest star in the Galaxy, and by placing all the Galaxies outside of the Milky Way at the same distance as the closest Galaxy. I have now revised my analysis but still assert that Marilyn is Wrong.
If stars are infinite in number and evenly distributed in space, the sky should be solidly bright with starlight in every direction; there should be no night or darkness on the surface of the Earth.
As both Jim and Craig pointed out, Olbers' Paradox is based upon the assumption that stars are infinite in number and evenly distributed in space. If one does not accept these assumptions, then no paradox exists. Marilyn was quite careless in her estimates of the number of stars and her interpretation of Olbers' Paradox when she wrote "if there are truly millions of stars in the universe, then our every line of sight into the heavens should surely end at a star."
In the following discussion, I'll use the notation 10**n to represent ten to the power n, so 5 x 10**4 should be interpreted as five times ten to the fourth power, and 5 x 10**-4 should be interpreted as five times ten to the negative fourth power.
According to Carl Sagan (Cosmos, Copyright © 1980), the Milky Way Galaxy (our galaxy) contains some 400 billion stars. The closest, Proxima Centauri, is approximately 4 light years (2.35 x 10**13 miles) from Earth. Since the Sun is only 93,000,000 miles from earth, Proxima Centauri is 252,688 times as distant. Thus, if Proxima Centauri were the same brightness as the Sun, the amount of light reaching the Earth from this star would be only 1.57 x 10**-11 as much as that from the Sun. In order to simplify the analysis, we'll assume that the stars in the galaxy are uniformly distributed and equally bright. Well assume that there are no obstructions (cosmic dust, black holes, or other stars), and that the stars are stationary (no red shift). We'll also assume that within each distance grouping (e.g. 4-10 light years from Earth), all stars within that group are at the closest distance (4 light years in this case), and therefore produces the same fraction (1.57 x 10**-11 in this case) of the Sun's light. These assumptions should, on the average, result in giving Marilyn the benefit of the doubt. In other words, our calculations should produce a result which is brighter than reality.
The total amount of light reaching Earth from all the Stars in the Milky Way Galaxy is therefore 3.79 x 10**-6, or 0.000379% of the Sun's light.
We could perform a similar calculation for stars in other Galaxies. Sagan estimates that there are 10**11 galaxies. The closest, M31, is 2,000,000 light years away, or 500,000 times as distant as Proxima Centauri, or 1.26 x 10**11 times as distant as the Sun. Assuming this galaxy consists of 4 x 10**11 stars, each as bright as the Sun, the total amount of light reaching Earth from this galaxy is 2.51 x 10**-11 as much light as the Sun. Even if all 10**11 galaxies were as close as M31, the total light reaching Earth from all of them would be only 2.51 times as bright as the Sun. If these are all uniformly distributed between 2,000,000 and 8,000,000,000 (eight billion!) light years away, the amount of light reaching us from these galaxies would be many orders of magnitude smaller.
That's why it's dark at night.
Svein Olav G. Nyberg <firstname.lastname@example.org> suggested
Why not take a look at Mandelbrot's solution to Olbers' paradox, namely that even though stars are infinitely many, the density of them is not uniform, but rather follow a kind of fractal pattern. This is the most elegant solution I've seen.
Mandelbrot's solution to Olbers' paradox can be illustrated like this: Assume the stars of the first 10**1 light years occupy 10**-1 of the sky, the stars of the next 10**2 light years occupy 10**-2 of the sky, the stars of the next 10**3 light years occupy 10**-3 of the sky, ..., and the stars of the next 10**n light years occupy 10**-n of the sky.
This is to say that the distribution of stars in the sky is not uniform, but rather follows a fractal pattern. If we sum the numbers above, we get that an infinite number of stars will occupy only 1/9th of the sky.
These would be scattered about in spots, of course. Like, one star at a time, or sometimes a cloud, or galaxy, or a supergalaxy.
In fact, the actual distributions are far less dense. I dont know the actual numbers, but a more probable estimate would be 1/1000 + 1/1000000 + 1/1000000000 ...
This would result in a distribution that occupies less than 1/9 of the sky.
Try this experiment and see if it works: Take a Hamamatsu Photon Counting Photomultiplier Tube with you anywhere on the surface of our planet and take a reading. There are photons everywhere in many different spectrums that are called light!
"Just 'cause you can't see it don't make it so" (old Southern USA saying)
I'm not an astrophysicist, and so my understanding of the answer to Olbers' paradox is based on commentaries by those with astronomically more expertise than I have, extracted from encyclopedias as well as numerous sources on the Web. Here is my impression:
The consensus seems to be that the major explanations for the darkness are:
Based on this analysis, I'm advising my friends to sell investments in companies that make automobile headlights and invest instead in a startup company that will be selling startanning lotion.
P.S. I'm not a financial expert either.
PPS. Nighttime darkness is not the only phenomenon of everyday life that Doppler effects on light have been invoked to explain. Another is the propensity of Boston area drivers to ignore traffic lights. While light sources that are retreating are "red-shifted", the opposite occurs when an observer is moving toward the light source. Thus, if one approaches a traffic light fast enough, red appears yellow, and yellow appears green.
I have always accepted the theory that the sky is dark because of the red-shifting of light due to the expanding universe. But your web page says that because of red-shifting, "light [stars] emit in a unit of time reaches us over a more extended duration, thus lessening its intensity". I think it would be more correct to say that the visible wavelengths are shifted to "invisible" wavelengths.
Here are some of the key conclusions from the translated excerpt in Clayton's book:
From that conclusion it follows that if the star filled universe is infinite, or only greater than the distance to the first magnitude shell by a factor of 760,000,000,000,000, each point of the sky would appear as luminous to us as one point of the sun... The enormous difference between this conclusion and experience lets one see that the sphere of fixed stars not only is not infinite but also that it is uncomparably less than the vastness I have supposed for it, or that the strength of light decreases in greater proportion than the inverse ratio of the squares of the distances.
Without doubt one will judge that the numbers that I set up are to be taken as conjectures; this is true, but these conjectures are not arbitrary -- least of all the distance to the stars of first magnitude that I have placed about 240,000 times farther than the sun.
As I understand it, the currently accepted explanantion for Olbers' paradox is the finite lifetime of the stars. They don't last long enough for us to see light coming from all directions.
You state that the closest galaxy to the Milky Way is M31, more commonly referred to as the Andromeda Galaxy. While it is true that M31 is the closest major galaxy, it is by no means the closest grouping of stars and gas to earn the classification of galaxy.
The Large and Small Magellanic Clouds are irregularly shaped galaxies, both of which orbit our own galaxy at a distance much closer than 2.2 million light years, the distance to M31. Both of these are easily visible from the Southern Hemisphere.
Also, astronomers have identified in recent years other smaller galaxies, some of which weave in and out of the Milky Way in their orbits.
While these galaxies are no where near as large as our own galaxy or M31, they are still galaxies, and should be considered when speaking of the closest galaxies to our own.