Olbers' paradox

From Wikipedia, the free encyclopedia

Even on clear nights, far from other light sources, the night sky is largely dark.

Olbers' paradox, described by the German astronomer Heinrich Wilhelm Olbers in 1823 (and then reformulated in 1826) and earlier by Johannes Kepler in 1610 and Halley and Cheseaux in the 18th century, is the paradoxical observation that the night sky is dark, when in a static infinite universe the night sky ought to be bright. 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".

1 Assumptions
2 Explanations
- 2.1 Accepted explanations
3 See also
- 3.1 Myths and alternative explanations
4 References

[edit] 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.

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 current scientific consensus is that effects of general relativity relating to the Big Bang and the finite age of the Universe do indeed give a finite size for the observable universe, but that it is the astronomical redshift relationship which really explains the dark sky at night.

The works of experts^[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.

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.

[edit] Explanations

[edit] Accepted explanations

Three effects contribute to the resolution of Olbers' paradox: the finite age of the universe, the redshift, and the finite radation life of stars. The first and third effects combined dominate. (Even in steady state theory models, which supposes the universe is infinitely old and spatially unbounded, the night sky would still be dark.)

[edit] Finite age of the universe

This explanation of the paradox points to the finite speed at which light travels through space. As we look further out in space we see further back in history; eventually we would pass beyond the finite age of the universe. A similar explanation was first offered, ironically, by poet and writer Edgar Allan Poe in his prose poem Eureka (1848), in which he observed:

"Were the succession of stars endless, then the background of the sky would present us an 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]

[edit] Redshift

While the finite age of the universe deals with the contribution of stars to the brightness of the night sky, it leads to the question of why the Big Bang itself does not visibly contribute instead. The reason is that the radiation from the Big Bang has been redshifted to microwave wavelengths as a result of the cosmic expansion, and forms the cosmic microwave background radiation. The expansion of the universe also limits the size of the observable universe, which means that light beyond this does not reach us, which creates the optical effect of living in a finite universe (see finite age argument).

[edit] Finite radiation lifetime of stars

The radiation lifetime of a star is about 10 billion years or so: less for bright stars, more for dim ones. Many stars have been born, lived, and their radiation has died away. Only a large amount of light from a shell of rather small radial dimension would arrive on Earth at any given time, unless the ages of stars were exactly proportional to distance (something that is the opposite of what is currently thought). Basically at any one time there are not enough stars active to fill the space of the Universe with enough radiation to light the night sky.

[edit] See also

Redshift
The Lambda-CDM model for the Universe

[edit] Myths and alternative explanations

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. There is no material which can be uniformly distributed through space and yet able to absorb more radiation than it emits without increasing in temperature. 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. Although, if there existed a particle which immediately converted electromagnetic energy into visible matter, these pockets in space of the material could continue to grow and go unnoticed.

A different resolution, which does not rely on the Big Bang theory, was offered in 1974 by Benoît Mandelbrot. He postulated that if the stars in the universe were fractally distributed (e.g. like a Cantor dust), 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, rather than a serious resolution of the paradox, for the Big Bang Theory is a more widely accepted explanation for Olbers' Paradox - among other things. In correspondence with the cosmological principle and timeline of the universe, astronomers in general have found no evidence to support a fractal distribution of the stars at the largest scale distances.

The idea of a hierarchical cosmology - what would now be called a fractal cosmology, is not Mandelbrot's invention - notably, it had been proposed in 1908, decades earlier, by Carl Charlier.

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:

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

Where:

$r 0$ = the mininum 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 $r a$ , $l i g h t$ is infinite for $a \ge -1$ but finite for $a < - 1$ . So if $L (r)$ is proportional to $r - 2$ , then for $l i g h t$ to be finite, $N (r)$ must be proportional to $r b$ , where $b < 1$ .

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 $r 2$ . This implies that $l i g h t$ is infinite if the minimum requirement of $N(r) \propto r^2$ is met, but finite if it is not.

The Milky Way is not necessarily a place where a radius of length of the largest scales would be extended from as it is not in anyway a center of a universe nor is it a region of greatest baryonic matter density.