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No 64 - December 2010

Backyard astrophysics:

Can't see the universe for the stars

If you can't see the wood for the trees, people mean to say you are obsessed with detail and fail to grasp the bigger picture. However, there is also the literal case that deep in the forest you can see only the trees nearby. The nearby trees obstruct the rest of the forest. As you enter the forest from a neighbouring field and look back, you can initially see the field and sky beyond the forest. However, at some depth L0 into the woods, the field and sky disappear and only trees can be seen. Even deeper, most of the forest itself is invisible and there is no telling how far in we have gone.

No, you have not picked up a copy of Forestry for Philosophers by mistake. The forest can be an analogy of the universe. The trees are stars; the end of the woodland is the edge of our island of stars, the field beyond is empty space reaching to infinity. The Sun is the tree we just bumped into, and the dark night sky is the bright field beyond the forest.

Night and day

It has been recognised a very long time ago that the night sky is dark [1,2], and we can confirm this each night even under the light-polluted skies of the city. After the Copernican revolution moved the centre of the universe from the Earth to the Sun, the path was open to re-interpret the starry firmament. It no longer had to be a thin sphere with bright points stuck on it. Thomas Digges [3] in his appendix to his father's Everlasting Prognostications drew it as stars distributed throughout space.

Once we also recognise that the Sun is a star like the others, the forest analogy becomes useful. We make two observations:

  1. We see very many stars.
  2. The sky as a whole is dark.

With the forest analogy, we can immediately draw two conclusions:

  1. The distribution of stars is not infinite. Beyond the island of stars is darkness - empty space or possibly no space at all.
  2. Within the island of stars, they fill only a small part of the volume. Stars are very small compared to the distance between stars. Only if the trees are planted far apart can we see through a forest.

How dense the stars?

distances to the nearest stars
Fig. 1. Distances to the nearest stars.

Can we actually be more specific about how dark the night sky is? Can that tell us something about the Milky Way or the universe? In what follows I use what some would regard as simple mathematics to illustrate how simple, but quantitative, amateur observations can generate quantitative statements about the Milky Way and the universe. If you think the maths is not simple, just skip over the equations and read the text.

Our second conclusion - stars are small compared to the space between them - we can quantify from literature data. We assume all stars are like our Sun, same diameter and same brightness. Although this assumption is wrong, the Sun is an average star, and statistically speaking, the assumption is a valid approximation. In three-dimensional space, a star has about 12 nearest neighbours. We can look up the 12 brightest stars in a catalogue to find that their magnitudes range from -1.4 (Sirius) to +0.8 mag (Altair) with a median of +0.1 mag. We can also look up that the Sun is 150,000,000,000 m away and has a brightness of -26.7 mag. The magnitude difference of 26.8 translates into a distance ratio of 230,000. Multiplying with the distance of the Sun, the distance to the nearest stars is approximately

r0 = 3.5 · 1016 m = 3.7 ly = 1.1 pc

Fig. 1 illustrates this as a map of the two-dimensional forest analogy. The inner circle is the distance to the nearest star; the outer circle encompasses 6 trees (the 2D equivalent of 12 stars in 3D). The intermediate circle encompasses three trees and marks the median distance of the 6 neighbour trees.

The density of stars can be calculated from this. Each star notionally occupies a roughly spherical space with a radius half the distance to the neighbours. Accounting also for space between spheres the density of stars is roughly 3/4 of the inverse of the sphere volume:

n = r0-3 √2 = 3.3 · 10-50 m-3 = 0.03 ly-3 = 1 pc-3

Plugging in the volume of the Sun (R = 700,000,000 m), we can calculate what fraction of space is filled by stars and how much of space is empty. The fill factor is the volume of a star multiplied with the density of stars in space:

f = (4π/3) n R3 = 4.7 · 10-23

For each star filled with hydrogen and other gas there is 1/f = 21,000,000,000,000,000,000,000 as much interstellar space with hardly any content at all. As forests go, the universe's sprinkling with stars is exceedingly thin.

The looking distance

a shell of space with stars
Fig. 2. A shell of space with stars. Areas of bright sky due to those and nearer stars.

Assuming a vast space occupied by stars at this density, we should now calculate how our sky brightness builds up from the light these stars send toward us. We imagine a thin shell of space at distance r around us and with small thickness dr. The number of stars in the shell is its volume multiplied by their density:

dN = 4 π n r2 dr

Each star has the same radius R as our Sun and fills a tiny piece of our sky with starlight. This solid angle for any one star in steradian is

s = π (R/r)2

and added up for the whole shell of stars

dS = s dN = 4 π2 n R2 dr

Observe that this is independent of the distance to the shell. In the forest, each distance-circle of trees blocks the same amount of vision beyond. Nearby trees are few, but appear large. Trees further away appear smaller, but there are more of them.

Each shell of stars of thickness dr will fill yet another piece dS of our sky with light. If we add up all the shells from zero to some distance L the area of sky filled with light is

S = 4 π2 R2 n L

Plugging in the numbers, including the radius R of the Sun we get

S = L / (1.6 · 1030 m)

The whole sky has 4π steradian. It would be filled if there were shells of stars out to a distance of

L0 = 2 · 1031 m = 2.1 · 1015 ly = 6.4 · 1014 pc

For the moment, we will just take this as a theoretical number. If we can measure by what factor the night sky is darker than the surface of the Sun, then the distance L to which we actually see stars is L0 multiplied by that fraction.

Some may argue that the looking distance is greater than calculated above, because we have not accounted for the fact that some nearby stars obstruct stars further away. This is correct, but it is significant only if a large fraction of the sky is filled with stars. In the actual universe it is dark at night, therefore mutual obstruction does not affect the calculation for the actual universe.

Measuring the looking distance

the Milky Way
Fig. 3. A negative image of the Milky Way. Taken in April 2007, Jupiter is near the dark Pipe Nebula.

I have a few reasonably deep wide field shots of the night sky that I took from a very dark site, the Paranal Observatory in Chile. I also have images of the Sun taken from the same place. In the 16-bit FITS images, the Sun registers as about 25,000 ADU. (ADU are analogue-digital-units, arbitrary brightness units. These are specific to the detector, but proportional to the brightness and comparable between images from the same detector.) To compare this with the nighttime image of Fig. 3, I have to correct for the solar filter (factor 100,000), the ISO rating (factor 0.5), the exposure time (factor 7500) and the f ratio squared (factor 13). Fig. 3 shows complete darkness as white and 150 ADU as black. In this image the Sun would have a brightness of 1.2 · 1014 ADU.

In Fig. 3, I pick one area in the brightest part and another area of mostly dark nebulosity. From another picture, I also pick an area around the south celestial pole - some distance below the plane of the Milky Way. In these three areas, the total brightness is simply averaged, sky background mixed with star light and all. The results are 100 ADU in the brightest part of the Milky Way, 45 ADU in a dark patch of the Milky Way, and 30 ADU when we are looking out above the plane of the Milky way at about -30° galactic latitude.

The ratio between the ADU values of night versus Sun is the same as the distance ratio between how far we see stars and the distance L0.

Lbright = 1.5 · 1019 m = 1500 ly = 500 pc
Ldark = 7.5 · 1018 m = 800 ly = 250 pc
Lhigh = 5 · 1018 m = 500 ly = 150 pc

The last number is most meaningful in our model of the universe as empty space sprinkled with stars to some distance L and empty beyond that: At the high galactic latitude, we look out of the disc of the Milky Way into the empty space beyond. We are looking 150 pc far at latitude -30°. At latitude -90° it should be half that. This implies a half thickness of the disc - in the solar neighbourhood - of 75 pc and a full thickness of 150 pc. This is quite an accurate estimate, the real thickness of the disc of the Milky Way overall is about 300 pc.

Interstellar absorption

the Milky Way by Herschel
Fig. 4. A historical illustration of the looking distance in various directions of the Milky Way. Courtesy of Wikipedia and F.W. Herschel [4].

William Herschel made similar observations of the Milky Way [4]. He did not have a digital camera to hand, but instead counted the density of stars on the sky. While he thought that we see more stars in the plane of the Milky Way because the distribution of stars reaches to further distances than toward the galactic poles, de Chéseaux had already raised the issue of starlight going missing between its source and the observer [5]. Today we know that both were right. The Milky Way is in fact a disc of stars, and toward the poles we see empty space beyond it. However, looking along the disc the view is actually limited by absorption, and the Milky Way is much bigger than Herschel concluded. In the plane of the Milky Way, the interstellar absorption is 1 to 2 mag/kpc [6], which compares quite well with our looking distance.

In the forest analogy, interstellar absorption is like a fog in the forest. Even though the forest might be infinite, the fog doesn't allow you to see much of it. The Milky Way in that image is a long thin strip of woodland. Look north or south and you can see the fields beyond the forest. Look east or west and you see more trees, a darker forest, but also fog hiding even more trees.

In 1823, Wilhelm Olbers wrote about the transparency of the universe [7], inadvertently repeating some of the earlier work of de Chéseaux. Both favoured a universe that is filled with stars to infinite distances. They realised that, without light going missing somewhere, our night sky should be very bright. Both had worked on comets and had seen their tails disappearing into the space between planets and stars. Interstellar space then was probably not empty and had to absorb light. If there is just a little interstellar absorption then the infinity of the stellar population of space becomes mostly invisible. At the time, this made sense and was not a paradox; indeed Olbers did not use the word.

Like trees, stars die

In the latter half of the 19th century, thermodynamics came about, and absorption in an infinite sea of stars was discredited: With all this starlight about, the interstellar matter would heat up until it radiated as much as it absorbed. The energy cannot disappear and the dark night sky cannot be explained by absorption. Absorption may exist, but it cannot turn a bright sky dark.

It was Kelvin [8], who deconstructed the problem of the dark night sky. He saw a fundamental problem with the infinite sea of stars, because he argued that any star would shine for only 50 or 100 million years. His paper was omitted from his bibliographies and so was lost until Harrison re-discovered it [9]. Well before then Harrison himself [10] had put forward the same argument with a more realistic stellar life time of 1010 years. The argument goes like this. Unless you construct a very strange conspiracy of all stars to shine on our little place in the universe at this precise moment of its history, we can expect to see light only from the equivalent of a thick shell of stars with thickness 1010 light years (3 · 109 pc). This is about 200,000 times too little starlight to brighten up our night sky.

There may or may not be stars out to infinity, and the universe may or may not be infinitely old. However, any given star lives only so long and cannot shine forever. Most of these hypothetical stars would be dead and dark. The few that are alive now can give only little brightness to our night sky. (By "alive now", I mean shining at a time such that Earth receives starlight from them now.)

The Olbers paradox

As we have seen, de Chéseaux and Olbers concluded from the dark night sky that there was interstellar absorption, and Kelvin deconstructed the notion of a universe with infinite and eternal luminous content. It might be infinite and eternal, but even if it were, at any observer's location only a finite part of it would appear luminous at any given time.

The term "Olbers paradox" was probably first used by Bondi in [11], who also presented it very much in the modern form we recognise. Bondi was a proponent of the Steady State theory of the universe as an alternative to the Big Bang theory. In the former, the perfect cosmological principle holds, whereby the universe - at very large scales - looks the same from anywhere (the plain old cosmological principle) and also looks the same at any time. If that were true, space would have to be infinite or curved back on itself like the surface of a sphere. And the universe would have neither beginning nor end, but continue forever looking the same.

To reconcile the Steady State universe with the observed expansion of the universe, matter has to be created in the space vacated by expansion. This new matter then forms stars and galaxies to fill the gaps left by the older galaxies drifting apart. This is needed to restore the old look and keep it looking the same forever. Bondi thought that the Steady State universe had a problem with the dark night sky, and showed that redshift could come to the rescue by removing energy from the starlight before much of it arrived in a telescope on another planet.

Science had lost Kelvin's solution for the dark night sky, which was and still is valid. And so, out of the argument between Big Bang and Steady State, comes about the association of Olbers with cosmology, the suppression of de Chéseaux's earlier equivalent work, and the continued fascination of the public with the alleged paradox of the dark night sky.

Jaki [12] has written a whole book proposing that the true paradox is why the night-sky paradox was not recognised sooner. To me the paradox is that we are still fascinated by it as something cosmology has to explain. Big Bang or Steady State - the night sky has to be dark simply because stars don't shine forever. The real universe provides additional mechanisms to darken the night sky, such as redshift, a limited time since stars began to shine, and a limited region of the universe being visible to us. Nevertheless, those mechanisms are not really required.

References and notes

  1. Martin Luther (translator) (1545). Biblia - das ist, die gantze Heilige Schrifft Deudsch. Wittemberg.
    Das Erste Buch Mose, Capitel 1, Vers 3, 4 und 5: "3 VND Gott sprach / Es werde Liecht / Vnd es ward Liecht. 4 Vnd Gott sahe / das das Liecht gut war / Da scheidet Gott das Liecht vom Finsternis / 5 vnd nennet das liecht / Tag / vnd die finsternis / Nacht. Da ward aus abend vnd morgen der erste Tag."
  2. Robert Baker et al. (translators) (1611). The Holy Bible, conteyning the Old Testament, and the New. London.
    The first booke of Moses, called Genesis. Chap. 1, Verses 3, 4 and 5: "3 And God said Let there be light: and there was light. 4 And God saw the light, that it was good: and God divided the light from the darkenesse. 5 And God called the light, Day, and the darknesse he called Night: and the euening and the morning were the first day."
  3. Thomas Digges (1576). "A perfit description of the cælestiall orbes according to the most aunciente doctrine of the Pythagoreans, latelye reuiued by Copernicvs and by geometricall demonstratons aproued". Prognostication euerlastinge. offers a 1595 version for browsing (JavaScript required). Start reading at "Plate and Page M3".
  4. William Herschel (1785). "On the construction of the Heavens". Phil. Trans. R. Soc., 75, p. 213.
  5. Jean-Philippe Loys de Chéseaux (1744). "Sur la force de la lumière et sa propagation dans l'ether, et sur la distance des etoiles fixes". Traité de la comète qui a paru en décembre 1743 et en janvier, février et mars 1744. Marc-Michel Bousquet et Compagnie, Lausanne, Genève.
  6. Albrecht Unsöld, Bodo Baschek (1999). Der neue Kosmos - Einführung in die Astronomie und Astrophysik - Sechste, völlig neubearbeitete Auflage. Springer, Berlin, Heidelberg, New York, etc.
  7. Wilhelm Olbers (1823). "Über die Durchsichtigkeit des Weltraums". Astronomisches Jahrbuch für das Jahr 1826, 51, p. 110.
    Olbers had three given names, Heinrich Wilhelm Matthias. In Bremen und umzu he is known as Wilhelm, which was his rufname. His paper appeared in a year book for 1826, but it was submitted and and the book published in 1823. In modern writing he is often called Heinrich, and the year of publication is often stated as 1826. Translations of the paper did appear in 1826 in the Edinburgh New Philosophical Journal and in the Geneva Bibliothèque universelle des sciences, belles-lettres, et arts. On occasion, Olbers has been called a Hamburg astronomer, which is an insult to Bremen. Hamburg has, of course, had great astronomers, but Olbers was not one of them.
  8. William Thomson (Lord Kelvin) (1901). "On ether and gravitational matter through infinite space". Philosophical Magazine, Ser. 6, Vol. 2, p. 161.
  9. Edward R. Harrison (1986). "Kelvin on and old and celebrated hypothesis". Nature, 322, p. 417.
  10. Edward R. Harrison (1965). "Olbers' paradox and the background radiation density in an isotropic homogeneous universe". Mon. Not. R. Astron. Soc., 131, p. 1.
  11. Hermann Bondi (1952). Cosmology. Cambridge University Press.
  12. Stanley L. Jaki (1969). The paradox of Olbers' paradox - A case history of scientific thought. Herder and Herder, New York.

Horst Meyerdierks


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