20 December 2007

Hey! The night sky shouldn't be dark! Everywhere you look there's a star! Olbers' Paradox and its Solution!

What the night sky SHOULD look like. But in 1823, the German astronomer Olbers noticed that the night sky doesn't look like this, and asked why.

The solution is all about the SPEED at which the light from a faraway star reaches Earth. Standing on the Earth at night, we're not just seeing the light from faraway stars. We're looking at the light the stars began sending us thousands, millions, billions of years ago. (By the time we see the starlight, the star that radiated it might not even exist anymore.) We're looking deeply into Space, but we're seeing deeply backwards into Time.

At the top is the nicely written equation which you'll find very messily written down below. Blogger just won't let me put images wherever I want.

PS, sorry I misspelled Olbers in the Comments.

Notice down in the guts of this wiki who first solved Olbers' Paradox! Vleeptron has danced with Edgar Allan Poe before -- not for his thrilling fiction and fantasy, but for his pioneering work in codebreaking. So look what else the dude was up to when he wasn't writing "The Cask of Amantillado."

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from Wikipedia:
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Olbers' paradox, described by the German astronomer Heinrich Wilhelm Olbers in 1823 (but not published until 1826 by Bode) and earlier by Johannes Kepler in 1610 and Halley and Cheseaux in the 18th century, is the argument that the darkness of the night sky conflicts with the supposition of an infinite and eternal static universe. It is one of the pieces of evidence for a non-static Universe such as the current Big Bang model. This "paradox" is sometimes also known as the "dark night sky paradox" (see physical paradox).

Assumptions

What if every line of sight ended in a star? (Infinite universe assumption #2)

If the universe is assumed to contain an infinite number of uniformly distributed luminous stars, then:

1. The collective brightness received from a set of stars at a given distance is independent of that distance;

2. Every line of sight should terminate eventually on the surface of a star;

3. Every point in the sky should be as bright as the surface of a star.

Looking at trees within a big flat wood in direction of the horizon shows the effect: The mass of dark trees will hide the horizon (imagine the trees now as lights).

The further away one looks, the older the image viewed by the observer. For stars to appear "uniformly distributed" in space, the light from the stars must have been emitted from places where the stellar density of the region at the time of emission was the same as the current local stellar density. Simple interpretation of Olber's paradox assumes that there were no dramatic changes in the homogeneous distribution of stars in that time. This implies that if the universe is infinitely old and infinitely large, the flux received by stars would be infinite.

Kepler saw this as an argument for a finite observable universe, or at least for a finite number of stars. The work of Ray D'Inverno [1] suggests that if just the assumption that the universe is infinite is dropped the paradox still holds. Though the sky would not be infinitely bright, every point in the sky would still be like the surface of a star.

In a universe of three dimensions with stars distributed evenly, the number of stars would be proportional to volume. If the surface of concentric sphere shells were considered, the number of stars on each shell would be proportional to the square of the radius of the shell. In the picture above, the shells are reduced to rings in two dimensions with all of the stars on them.

In a universe of three dimensions with stars distributed evenly, the number of stars would be proportional to volume. If the surface of concentric sphere shells were considered, the number of stars on each shell would be proportional to the square of the radius of the shell. In the picture above, the shells are reduced to rings in two dimensions with all of the stars on them.

A more precise way to look at this is to place earth in the centre of a "sphere". If the universe were homogeneous and infinite, then at a distance, r, away from the earth, the shell of the sphere would have a certain flux (viewed from earth) due to the individual flux of the stars on the shell (brightness) and also the number of stars in the shell (cumulative flux). When an observer from earth looks to a farther distance to another shell, r + x, the number of stars increases by the square of the distance, while the flux decreases by the inverse squared. Comparing the total brightness of the first shell to the second shell, one notices that both shells have equal flux, since the flux of each individual star decreases due to distance but is equally made up for by the number of stars. This means that no matter how far away an observer on earth views the sky, the brightness of each consecutive shell would not diminish, rather they would be equal. If the universe were infinite (age and volume) and had a regular distribution of stars, then there will be an infinite number of such shells and infinite amount of time for the light to reach earth (infinite flux) as long as the earth remains, effectively meaning that there would never be night on earth.

The mainstream explanation

In order to explain Olbers' paradox, one would need to account for the relatively low brightness of the night sky in relation to the circle of our sun.

Finite speed of light

The greater the distance of a star from an observer on earth, the longer it takes the star's light to reach the observer. Thus, the farther we look into in space, the farther we see into the past. This fact is a key ingredient in the mainstream explanation of Olbers' paradox, although it cannot alone explain the paradox, since the speed of light has no direct connection to the energy density and broadness of light received at any given point.

Finite age of the Universe; the origin of all light is a finite distance away

Edgar Allan Poe was the first to solve Olbers' paradox when he observed in his essay Eureka (1848):

"Were the succession of stars endless, then the background of the sky would present us a uniform luminosity, like that displayed by the Galaxy –- since there could be absolutely no point, in all that background, at which would not exist a star. The only mode, therefore, in which, under such a state of affairs, we could comprehend the voids which our telescopes find in innumerable directions, would be by supposing the distance of the invisible background so immense that no ray from it has yet been able to reach us at all."[1]

The Universe, according to the mainstream theory of the universe, called the Big Bang Theory, is only finitely old; stars have existed only for part of that time. So, as Poe suggested, the earth receives no starlight from beyond a certain distance. In fact, since the universe expands at its outskirts faster than the speed of light, the number of visible stars will actually decrease over time, causing the night sky to appear darker. According to the Big Bang theory, the sky was much brighter in the past, especially in the first few seconds of the universe. All points of the local sky at that era were therefore brighter than the circle of the sun, despite the finite and even more limited range that light could travel in that prehistoric era.; this implies that most light rays will terminate not in a star but in the relic of the Big Bang.

Expanding space

While the finite distance for the origin of any received light does not by itself solve the paradox, the Big Bang Theory also involves an expanding space that can cause the energy of emitted light to be reduced via redshift. More specifically, the extreme levels of radiation from the Big Bang have been redshifted to microwave wavelengths as a result of the cosmic expansion, and thus form the cosmic microwave background radiation. This explains the relatively low light densities present in most of our sky despite the assumed bright nature of the Big Bang. The redshift also affects light from distant stars and quasars, but the diminution is only an order of magnitude or so, since the most distant galaxies and quasars have redshifts of only around 5. Thus, the mainstream explanation of Olber's paradox requires a universe that is both finitely old and expanding.

Alternative explanations

The redshift and expanding space hypothesised in the Big Bang model would by itself explain the darkness of the night sky, even if the Universe were infinitely old. The Steady State cosmological model assumes that the Universe is indeed infinitely old, and uniform in time as well as space. It is also expanding exponentially, producing a redshift. There is no Big Bang in this model, but there are stars and quasars at arbitrarily great distances. The light from these distant stars and quasars will be redshifted accordingly, so that the total light flux from the sky remains finite, and dominated by the nearest light sources. However, the Steady State model cannot explain the detailed behavior of distant starlight and the Microwave background, since it requires a continuous transformation of the former into the latter at decreasing frequencies; this transformation is not observed.

Finite age of stars

Stars have a finite age and a finite power, thereby implying that each star has a finite impact on a sky's light field density. But the finity of the influence from any given star does not imply that our sky will be darker than the circle of the sun in most areas of our sky. Only those stars whose worldlines intersect the light cone of a point would contribute to the luminosity there in any event, so the age of any given star is largely irrelevant. Despite being neither a sufficient nor a necessary explanation of the darkness of the sky, the finite age of stars is considered by some to be a reason for the dark sky, and accordingly, is seen as a solution to Olbers' paradox.

Absorption

An alternative explanation which is sometimes suggested by non-scientists is that the universe is not transparent, and the light from distant stars is blocked by intermediate dark stars or absorbed by dust or gas, so that there is a bound on the distance from which light can reach the observer.

However, this reasoning would not resolve the paradox given the following argument: According to the second law of thermodynamics there can be no material hotter than its surroundings that does not give off radiation and at the same time be uniformly distributed through space. Energy must be conserved, per the first law of thermodynamics. Therefore, the intermediate matter would heat up and soon reradiate the energy (possibly at different wavelengths). This would again result in intense uniform radiation as bright as the collective of stars themselves, which is not observed.

Fractal Star Distribution

A different resolution, which does not rely on the Big Bang theory, was offered by Carl Charlier in 1908, later rediscovered in 1974 by Benoît Mandelbrot. They both postulated that if the stars in the universe were fractally distributed in a hierarchical cosmology (e.g. like a Cantor dust) -- the average density of any region diminishes as the region considered increases -- it would not be necessary to rely on the Big Bang theory to explain Olbers' Paradox. This model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred. This is merely a demonstration of the consequences of fractal theory as a sufficient, but not necessary, resolution of the paradox.

Mathematically, the light received from stars as a function of distance from stars in a hypothetical fractal cosmos can be described via the following function of integration:

SEE THIS EQUATION

NICELY WRITTEN

light = \int_{r_0}^\infty L(r) N(r)\,dr

AT TOP OF THIS POST

Where:

r0 = the minimum distance from which light is received > 0

r = the variable of distance

L(r) = average luminosity per star at r

N(r) = number of stars at r

The function of luminosity from a given distance L(r)N(r) determines whether the light received is finite or infinite. For any luminosity from a given distance L(r)N(r) proportional to ra, light is infinite for a \ge -1 but finite for a < − 1. So if L(r) is proportional to r − 2, then for light to be finite, N(r) must be proportional to rb, where b <>

For b = 1, the numbers of stars at a given radius is proportional to that radius. When integrated over radius, this implies that for b = 1, the total number of stars is proportional to r2. This implies that light is infinite if the minimum requirement of N(r) \propto r^2 is met, but finite if it is not.

Mainstream cosmologists reject this fractal cosmology, on the grounds that studies of large-scale structure in combination with the timeline of the universe have not produced any evidence for it.

References

1. ^ D'Inverno, Ray. Introducing Einstein's Relativity, Oxford, 1992.

* Paul Wesson, "Olbers' paradox and the spectral intensity of the extragalactic background light", The Astrophysical Journal 367, pp. 399-406 (1991).

* Edward Harrison, Darkness at Night: A Riddle of the Universe, Harvard University Press, 1987

* Scott, Douglas, and Martin White, "The Cosmic Microwave Background

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from Wikipedia:
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Heinrich Wilhelm Matthäus Olbers

Asteroids discovered:
Pallas, 28 March 1802
Vesta, 29 March 1807

Heinrich Wilhelm Matthäus Olbers (October 11, 1758 – March 2, 1840) was a German astronomer, physician and physicist.

Career

He was born in Arbergen, near Bremen, and studied to be a physician at Göttingen. After his graduation in 1780, he began practicing medicine in Bremen, Germany. At night he dedicated his time to astronomical observation, making the upper story of his home into an observatory. He also devised the first satisfactory method of calculating cometary orbits.

In 1802, Olbers discovered and named the asteroid Pallas. In 1807 he discovered the asteroid Vesta, which he allowed Carl Friedrich Gauss to name. As the word "asteroid" was not yet coined, the literature of the time referred to these minor planets as planets in their own right. He proposed that the asteroid belt, where these objects lay, were the remnants of a planet that had been destroyed. This theory is now discarded by most of the scientific community.

On March 6, 1815, Olbers also discovered a periodic comet named after him (formally designated 13P/Olbers).

Olbers was deputed by his fellow-citizens to assist at the baptism of Napoleon II of France on June 9, 1811, and he was a member of the corps legislatif in Paris 1812-1813. He died in Bremen at the age of eighty-one. He was twice married, and one son survived him.

Olbers' paradox, described by him in 1823 (and then reformulated in 1826), is the argument that the darkness of the night sky conflicts with the supposition of an infinite and eternal static universe.

Honors

The following celestial features are named for him:

* 13P/Olbers is a periodic comet.
* Asteroid 1002 Olbersia.
* Olbers crater on the Moon.
* Olbers, a 200km-diameter dark albedo feature on Vesta's surface

Mike said...

Perhaps it's arrogant of me to disagree, but I disagree. Not with you Bob, as this is a very interesting theory, but with the theory in general.

The theory assumes that there's a bunch of lights at a distance, and they're shining right at us with nothing in the way. That's just not the way the Universe is designed. Given the vast (understatement) size of the Universe, you have to assume that even if there's a light source out there, the direct line of sight between it and us is broken a huge amount of times by any number of objects. Dead stars, planets, gas clouds, comets, meteors, etc. All of these objects break down the light and send it off into other directions.

I read the entry on Wikipedia on this, and I did note the Absorption portion of the article, and it's ideas. I disagree with it too (I guess I'm one of those non-scientist people). The objects that "absorb" the light would re-radiate that light. This is true. But it re-radiates it uniformly (I'll get to this in a second). The small portion of the light that should have gotten to Earth is broken down into even smaller portions, and this happens an unknowable number of times between here and there. Also, the Universe is not infinite in size. Some of this light (I'd argue a majority of it) gets pointed away from all other objects. Even though the energy is conserved, it's not light that we'll see. Ever. What this comes down to is even though the light is out there, it's not coming at us, or it's to faint to see. As for the uniform radiation theory, I'm not sure I buy that either. Light reflects. it bounces off of things and goes in different directions.

The human eye can only see so much. The light of a candle is lost at 43 miles. Sure, that's a long way for so little light, but it's not that the light isn't reaching you. How long does it take light to go 43 miles? I'm not going to figure it out, but it's not very long.

I was reading something about Galaxy clusters the other day. There's a pretty cool picture at http://www.astrographics.com/GalleryPrintsIndex/GP0017.html. This picture is taken with a pretty amazing telescope. Can humans see it with no aid? Nope. Is the light getting here? Yep. But if we were to look right at it, we wouldn't register a thing.

If you look at the night sky on a night with no moon, the sky isn't black. It doesn't even look black. It's a dark blue. Very, very dark. Why? Because there's light there. Not much, but some.

Another thing to consider, even at as lame a distance as 4.3 light years away, our Sun is just a bright star. Dimmer than Serius. What would our Sun be after 10 light years? 50 light years? 1000 light years? That's just 1000 years out of 4.5 billion years old. It's not that the light isn't getting there. It's there. It's just not going to be bright enough to be seen.

OK, now that I've typed a lot, and proven to the world how stupid and stubborn I am, I'll stop typing.

Friday, 21 December, 2007
posicionamiento web said...

This will not work as a matter of fact, that is what I believe.

Thursday, 26 April, 2012