The Olbers’ Paradox: Why is the Sky Dark At Night? – PhotoNick
By CoperNick
Alessandro Santoni
October 1, 2018
It’s a warm summer night and you are lying on a grass field, looking at the wonderful starry sky above you. You keep staring at it until you suddenly find yourself wondering: “Why is the sky dark at night? Where is all the light from the stars out there? Is it because there’s no Sun at night?” The answer is actually not that trivial. This apparently simple question has aroused men’s curiosity since a very long time. In fact, it was first addressed in the “Topographia Christiana”, one of the earliest essays in Scientific Geography written by Cosmas Indicopleustes, a Greek monk of the 6th century.
“The crystal-made sky sustains the heat of the Sun the moon and the infinite number of stars otherwise it would have been full of fire and it could melt or set on fire”
“Cosmas Indicopleustès. Topographie chrétienne, 3 vols.”, Ed. Wolska–Conus, W.Paris
The question was tackled again during the Renaissance by some astronomers such as Kepler himself. However, the real paradox took its mature form in the 18th century in the work of Halley and Cheseaux, even if it is commonly attributed to the German astronomer Heinrich Wilhelm Olbers, who described it in 1823.
Anyway, let’s get to the point! What is this paradox really about?
Assuming that the universe is static and infinitely large, that it has a constant density D of stars – or galaxies, the distinction is here unnecessary – and no beginning, we imagine to divide it in concentric spherical shells with the Earth at the center, so that a shell of radius r will hold a number of stars equal to
![\[ \text{total surface of shell} \times \text{density of stars} = 4\pi r^2 \times D \]](https://www.teknoscienze.com/wp-content/ql-cache/quicklatex.com-ba8b8548b2e7d057c2fe783097a4bb02_l3.png "Rendered by QuickLaTeX.com")
Each of the stars in any of the shells will be shining and emitting light. Let L be the average quantity of energy irradiated per time unit by a single star at a distance r from us, then the intensity I of the radiation that reaches us will be
![\[ I = \frac{L}{4 \pi r^2} \]](https://www.teknoscienze.com/wp-content/ql-cache/quicklatex.com-72e63b7796ae0f9844540e4b99b5feef_l3.png "Rendered by QuickLaTeX.com")
because the total amount of light distributes equally on a spherical surface of radius r centered on the star itself. When we multiply this by the number of stars in each shell, the two trends balance each other out, so that the total intensity coming from a shell is independent from its distance from us
![\[ I_{TOT} = \frac{L}{ \cancel {4 \pi r^2}} \times \cancel{4 \pi r^2} D = L \times D \]](https://www.teknoscienze.com/wp-content/ql-cache/quicklatex.com-fa57c091e3a450ca47854a6c866987d7_l3.png "Rendered by QuickLaTeX.com")
A view of a square section of four concentric shells
Wikipedia, May 2009 Htkym
Therefore, since I TOT is constant and we can imagine an infinite number of shells at different distances around the Earth, the total intensity of the radiation that reaches us should be always infinite, both day and night. But this is clearly not the case: hence the paradox!
One could argue that in these calculations we have not considered that, from the Earth’s point of view, some stars could overlap in such a way that only the light of the nearest one reaches us. However, considering the same hypothesis as before, in every direction one looks at, he will eventually see a star: which implies that the entire sky should be completely covered with stars and, again, be shining day and night!
Nowadays we know that this paradox arises because some of its assumptions are wrong! The universe did have a beginning (the Big Bang) and it isn’t static at all! In 1929 Hubble demonstrated that the universe is expanding: the more an object is far from us the more quickly its distance increases. These considerations and the finite speed of light involve the existence of a finite “observable universe”, composed only by the few stars whose light has been able to arrive to Earth. Then, the sky is dark at night because the photons from lots of stars out there haven’t reached us yet; and probably never will if they are so far away that their apparent distancing speed is greater than the one of light! Curiously, the famous writer Edgar Allan Poe anticipated in some way this idea in his essay “Eureka”.
But wait a minute, that’s not the end of the story!
In our current most accepted cosmological model, the “ΛCDM model”, is hidden another problem: 378000 year after the Big Bang, the universe went through the so called “recombination era”, a transitional phase in which it became actually transparent to light thanks to the appearance of the first atoms, which are neutral entities. Before that time, all the space was filled with ionized matter that interacting continuously with photons didn’t let them propagate freely. Furthermore, at that time the Universe was so hot that the brightness of each point of the space was similar to that of the surface of the Sun. Now, since in a certain sense looking away in space is looking back in time, due to the finiteness of light speed, we should be able to see that 13 billion years old glare coming from every direction!
Fortunately the model itself actually clears our minds: indeed, in the same way intergalactic distances increases with time, also the wavelength of the photons that are traveling to us gets bigger. The Universe’s expansion therefore causes a decrease in the frequency of light, as in a Doppler-like effect. That phenomenon is called “cosmological redshift”. Then, the only reason for which we don’t see the above-mentioned radiation is that, having traveled for several billion years to reach us, it has been extremely redshifted, to the point that is only visible in the microwave range. And those are not only mental speculations! That “cosmic microwave background radiation” (CMBR), as it is now called, has been studied for more than fifty years by the time it was found accidentally in 1964 by two astronomers. This discovery earned them the 1978 Nobel prize in Physics!
All-sky map of the CMBR, create from data of the spacecraft WMAP
Wikipedia, Dicember 2012, NASA/WMAP Science Team
In conclusion, even if is pretty much uniform across the sky, CMBR cannot be seen but with the right telescopes, because its frequency is much lower than that of visible light. Problem solved! Now the next time you will look at a mesmerizing starry sky, remember that it’s not dark at all: instead it is bright and shines in every direction, but of a light invisible to our eyes!
For further reading on the history of this important question have a look at the book “Darkness at Night: A Riddle of the Universe” by Edward Robert Harrison.
Foto di Andrea Mauro
