Olbers' Paradox is based on a misconception about the nature of light and its relationship to the universe.

The validity of the assumptions.

OP  "For the sake of this paradox, we will assume that the universe is homogeneous and isotropic. This assumption means that where we are in the universe is in no way special - the same substances can be found everywhere else as well as here, and the same laws of physics apply everywhere else as well as here. This also means that in any direction, when averaged over a large enough amount of space, there is around the same number of stars."

When looked at as a whole the universe is pretty much the same in all directions and anywhere in an infinite universe is its centre, so can we accept this or does our position in relation to our Galaxy raise a special case. If we were sitting in a space station half way between the Milky Way and the Andromeda galaxy, then the universe around us might look the same in all directions but we are living on a planet belonging to a star that is 30,000 light years from the centre of a galaxy that has a diameter of 100,000 light years and contains 100,000,000,000 stars.

It wasn't untill the 1920's that a Harvard astronomer named Harlow Shaply proved that the Sun was not the centre of the universe, as had been previously thought, and was a pretty minor star as stars go, slightly north of the galactic ring we call The Milky Way.
The central hub lies in the direction of the conselation of Sagittarius but can only be seen in the infrared part of the spectrum (outside the visible range).

Around our Galaxy there are about 100 globular clusters. Our Galaxy has two small companion galaxies called the Magellanic Clouds - The Large Magellanic Cloud contains about 10,000 million stars and is about 180,000 l.y. from us. The Small Magellanic Cloud is about a fifth of this size and is about 230,000 l.y. away.

Most galaxies are members of galaxy clusters, some containing thousands of galaxies. Our own Galaxy is second largest of a group known as the Local Group containing 30 galaxies (the largest is Andromeda). Most of the galaxies in the local group are more than 2,700,000 l.y. from us. The nearest large cluster (3000 or so) is in the constellation of Virgo.

OPT   " I'm a bit bothered about the integral! "

OP   "The total flux we would receive from all stars, f_total is the flux from one star f, multiplied by the number density of stars n integrated over all space."

But this is not possible. If there are stars in all directions and eventually we reach one then that is the radius for that point. You cannot add the flux from stars beyond that point.What happens when you intergrate (i.e. vector addition) in opposite directions?

Some definitions of flux.

The flux density , or apparent brightness, or energy flux, of an object is the number of ergs per second you receive from it per square centimeter of collecting area. We normally use S for flux density, and the cgs units are ergs/cm2sec.

Suppose you are at a distance d [cm] from an object of luminosity L [ergs/sec]. Suppose further that the object radiates isotropically
(i.e. the same in all directions) and that between you and the object
there is only empty space -- no absorption. Then the flux density S is given by the Inverse Square Law.

The Inverse Square Law.

Let's start with sunlight as an example. At 1 AU, Earth receives 1 unit of sunlight; what we generally might associate with a bright sunny day at noon. How much sunlight would a spacecraft receive if it were twice as far from the Sun as Earth? Your first thought might be that, since it is twice as far it will only receive half as much light.


At a distance of 2 AUs the spacecraft is getting only one quarter of the amount of sunlight that would reach it if it were near Earth. This is because the light is being radiated from the Sun in a sphere. As the distance from the Sun increases the surface area of the sphere grows by the square of the distance. That means that there is only 1/d^2 energy falling on any similar area on the expanding sphere.




Examples:-

Sunlight on Body	Distance Miles 106	Distance A.U.	1/d^2	%	Max Mag.
Earth	93	1	1/1	100	-
Mars	141.5	1.52	1/2.31	43.28	-2.8
Jupiter	483	5.2	1/27.04	3.7	-2.6
Saturn	886	9.54	1/91	1.1	-0.3
Pluto	3666	39.46	1/1557	0.06	+14.

There is less than half as much sunlight falling on the
surface of Mars as on Earth!

Source	Distance Light Years	Distance (d) A.U.	1/d^2	%	Mag.
Sun	-	1	11	100	-26.8
α Centauri	4.3	271,929	173,945,381,041	1.352e-9	-0.27
M31 Andromeda	2,200,000	139,126,348,484	119,356,140,842,491,409,098,256	5.16632e-21	4.8
M104 (The Sombrero)	40,000,000	2,529,569,972,436	16,398,724,245,449,865,799,774,096	1.5628e-23	8.3

M31 Andromeda¹

M104 (the Sombrero)²

Back to flux.

Photography

Flux is measured in lumens which represents how much light is being transmitted over a spherical solid angle (called a steradian) with the center at the light source. A steradian is a unit area of a spherical surface with a unit radius. Thus there are four π steradians to a complete sphere. [This is the solid equivalent of a circle in a plane and measuring plane angles in radians with two π radians to a full circle.]

• The intensity of the source is candelas,
• the flux transmitted through space is lumens,
• the light illuminating a surface is foot-candles, and lux (or meter-candle),
• the light reflected from an area, the luminance is foot-Lamberts.

One can model a reflective surface just as if it were emitting light and it becomes an area source of light. Thus, the difference between a candela and a foot-Lambert: a candela is the intensity from a point source of light, and a foot-Lambert is the intensity (per unit area) from an area source of light. ("areal intensity of an extended source" is the intensity of light from an area, not a point).

From the external link in OP  "....if you move the Sun twice as far away from us, we will intercept one-fourth as many photons, but the Sun will subtend one-fourth of the angular area. So the areal intensity remains constant."
No it does not!

Luminous intensity - Candela

Luminous intensity, in a given direction, of a source that emits monochromatic radiation of 540·1012 Hz (hertz) that has a radiant intensity in that direction of 1/683 W/sr (watt per steradian)

We compare luminosity of an object to the solar luminosity, the total energy given off per second by the sun. One solar luminosity is 4 × 10³³ ergs per second. Luminosity has the same units as Power, e.g. energy per second. The Watt is the familiar unit of power. For comparison, a 400 Watt light bulb is 10^-24 solar luminosities.

(The light reflected by (or emitted by) a surface is its luminance.

A one square foot surface on a sphere with a one foot radius with a one candela source at the center of the sphere receives one foot-candle of illumination.

A perfectly reflective surface receiving one foot-candle of illumination has a luminance of one foot-Lambert.)

Resolving power.

The power to distguish between two points of light (to resolve them), depends on the diameter of a telescope measured in wavelengths.

Our eyes can resolve about 5000 wavelengths of light (i.e. we can see a bee at up to 50 yards) coresponding to an angle of 1 arc minute. In comparison, the 200 inch Mount Palomar telescope can resolve about 10 million wavelengths but due to the atmosphere is only about 60 times better than the eye (a bee at almost two miles).
The theoretical resolving power of a six inch aperture telescope is 0.8 seconds of arc (1 second is 1/36000 of a degree) and the limiting magnitude is 13.6.

In 1838 three astronomers succeded in measuring the distance of three stars, alpha Centauri (alpha Centauri is two yellow stars when seen through a small telescope), 61 Cygni and alpha Lyrae and from this and their spectra, they were able to determine the true magnitude (brightness) of the stars.

Spectral Types

First introduced by the American astronomer Pickering in 1890. The
classes were given 17 letters from A to Q but were in the wrong order of
temperature. The order below was produced by a Miss Cannon of Harvard
University. We now use the sequence from O to M subdivided in 10 steps
from 0 to 9. The Sun is a G2 Star.

1. Q
2. P
3. W
4. O - Blue - 40,000 - 25,000 °C
5. B - Blue - 25,000 - 11,000 °C
6. A - Blue-white - 11,000 - 7,500 °C
7. F - White - 7,500 - 6,000 °C
8. G - Yellow-white - 6,000 - 5,000 °C
9. K - Orange - 5,000 - 3,500 °C
10. M - Red - 3,500 - 3,000 °C
11. N
12. R
13. S

OPT   "Gas does 'glow'. There are lots of emissions from nebulae. Like, for example, the 21cm line of hydrogen which allows us to map where the hydrogen is in ours and other galaxies. This is only present due to energy entering the gas, and then leaving it again."

A wavelength of 21.11cm or a frequency of 1420MHz (radio),the 21cm hydrogen line, is a long way outside the visible spectrum.

Looking at the spectra of stars you will see black lines. These black lines show where atoms have absorbed energy from the object. If the atoms were hot enough to radiate, there would be no black line. Cool RED STARS show many lines, hot BLUE STARS do not. The emission spectra from nebulae can be seen because the nebulae are not producing a continuous spectrum as does a star.

Examples of Invisible Radiation

1. Radio Waves - 10⁸ - 10^-3.5 metres
   Outer layers of the sun,
   Remnants of exploded stars,
   Embryonic stars,
   Flare stars,
   Spiral arms,
   Exploding galaxies,
   Quasars,
   Microwave background.
   Microwaves make molecules spin.
   
   
2. Infra-Red - 10^-3.5 - 10^-6 metres
   Embryonic stars,
   Cool stars,
   Centre of the galaxy,
   Seyfert galaxies,
   Quasars.
   Infra-red makes molecules vibrate.
   
   
3. Ultra-Violet - 10^-6.6 - 10^-9.5 metres
   Solar flares,
   Young hot stars,
   Quasars,
   UV background.
   UV and x-rays ionize atoms.
   
   
4. X-Rays and Gamma-Rays - 10^-9.5 - 10^-15 metres
   Disturbed regions of the sun,
   X-ray stars,
   Gamma-rays from the galaxy,
   Exploding galaxies,
   X-ray background.
   Gamma-rays disturb nuclei.
   

The Ionosphere stops most of the electro-magnetic radiation we get from space. From earth we have two windows, one optical, the other in the radio frequencies.

• The visible spectrum - wave lengths from 400 Å to 700Å
• Radio waves from about 1cm. to 30metres

(The white noise you see on your television when there's no signal and the noise between stations on radio comes from space).

Magnitude, Apparent and Absolute.

A Greek astronomer named Hipparchus first came up with a scale of brightness for the stars he could see. He chose 6 for the faintest group and 1 for the brightest. This scale has been standardised so that first magnitude stars are 100 times brighter than sixth magnitude. Objects brighter than 1st mag. are given negative values. For example the Sun appears as a -26.8 magnitude star. Some of the faintest objects seen are 23rd. magnitude.

Each magnitude is about 2.512 times as bright as the next so a 5th. magnitude star is just over 2 and a half times brighter than a sixth magnitude star.
The absolute magnitude is the apparent brightness at a standard distance of 10 parsecs (32.6 light years). At this distance the Sun would have an apparent magnitude of 4.79. The star Sirius has an absolute magnitude of 1.5 and an apparent magnitude of -1.43. Deneb appears as a 1.23 mag. but its absolute mag. is -7. Subtracting the absolute from the apparent mag. gives the distance which, in the case of Deneb is 1,300 light years.

Stars in all directions.

OP   "So, because the universe is infinite, and there should be stars in all directions, which ever direction we choose to look in we should be able to see stars. Any line of sight should eventually intersect with a star, so all the sky should be bright." This sounds reasonable untill you look at the facts.

The proposition is that in infinite space there will be infinite stars in all directions and that if we add up all the flux from these stars we should get infinite flux. When we look at the night sky, we can't see stars less than 6th. mag. even though there are stars and galaxies in every direction. Let us look at a small area of space to see if this is true.

Looking at a ten degree square (10 x 10)°
, centred on Sirius, the total number of stars that can be seen without the aid of a telescope is 20. If we say that each pinpoint of light covers 1 square minute and that Sirius covers 10 square minutes then we have a total of almost 30 square minutes. 1 square degree equals 3,600 square minutes so in the square we are looking at, there are 360,000 square minutes. (30 / 360,000) x 100 = 0.008% is light and 99.992% is dark.

OPT   "Your argument, when you've weeded out the tons and tons of irrelevant stuff, is circular. You say we can't see stars when they're less than sixth magnitude (which is true) so we can ignore anything that is far away. In other words, we can use the fact that the sky is dark at night to prove that we can't see things far away, and then we use this fact to explain why the sky is dark at night."

The Hubble Deep Field Survey.

The Hubble Space Telescope's Wide Field Planetary Camera could see the light from a firefly at about 1,600 km. Its sensitivity range covers infrared to ultraviolet frequencies. A range much wider than we can see with our eyes.

For this survey, 342 exposures were made over a period of 150 orbits pointing the Hubble Space Telescope at a region above the north galactic pole. This direction points the HST straight up out of the disc of the Milky Way into intergalactic space with the minimum possible amount of gas and dust to obscure the view. To obtain as much light as possible, long exposure times were used (15 to 40 minutes). The images were made in four colours from infrared to blue light.

Some of the galaxies in the picture are 4,000,000,000 times fainter than the human eye can see. There are some 1500 separate galaxies, of all types,
in the picture.

The Microwave Background.

OP   "By simply looking at the night sky and seeing that it is dark, you can work out that the universe cannot be infinite in both space and time, supporting things like Big Bang theory."

The microwave background radiation (MBR), that is received uniformly from all directions of space, considered by many to be the most important evidence in support of the Big Bang (BB), may be inconsistent with that theory.
It would be expected that the observed gigantic galactic formations (The Great Wall etc.) cause irregularities in the isotropy of MBR reception, the observed spectrum of the MBR, corresponding to a near perfect black body temperature of 2.7 K, doesn't agree very well with temperatures predicted by various BB theorists. Those predictions had varied over a range of 5 to 50 K. History also shows that some BB cosmologists' "predictions" of MBR temperature have been "adjusted" after-the-fact to agree with observed temperatures.

The prediction of 5 K (by Ralph Alpher and Robert Herman in 1948), which has been selected as a basis for agreement with the observed temperature, was made by those who had accepted a BB scenario that included concepts that were incorrect. Those included the idea that all of the elements of the universe were produced in the BB, which was later determined to be erroneous

The cosmic microwave background radiation emanating from the universe could only have the observed fuzzy pattern if it contained clear amounts of dark matter and dark energy. The conclusion, based on a detailed analysis of the temperature and spacing of the bumps, was a surprise to those who felt that previous evidence for such a strange universe, based on observations of distant supernovae, was somehow inaccurate. The measurements were made with a novel group of microwave telescopes in Tenerife, Spain called the Very Small Array. The bumps are some of the oldest objects ever seen.

Space Dust - Gas, Ions, Rocks, Burnt out stars

OP   "In a black body, the dust will heat up too. It does act like a radiation shield, exponentially damping the distant starlight. But you can't put enough dust into the universe to get rid of enough starlight without also obscuring our own Sun. So this idea is bad."

Study of the gaseous filaments of the Crab nebula have been found to be much more complex, and interesting, than believed previously. They are far more chaotic, with cold and hot regions, and contain much more dust than believed before. Moreover, it was found that the filaments act as a barrier preventing the synchrtron nebula from moving outward into the interstellar space, and is in heavy interaction with this component of the nebula.
X-ray images show a dim glow of the galactic nucleus and gamma rays show molecular cloud complexs.

OP   "Absorbing dust would eventually come to equilibrium, and emit as much radiation as it absorbed. Even if it was at a different wavelength, we would still receive the same amount of light as before."

Thermal equilibrium depends on the temerature of the dust, the size of the particles and the strength of the illuminating energy. Some of the energy will be turned to heat and radiated at infrared frequencies. As the I.R. radiation is beyond the visible spectrum, it cannot add to the brightness of the night sky.

The HII regions (emission nebulae) are so named because they are composed mostly of a plasma of ionized hydrogen (HII) and free electrons. The hydrogen atoms of the interstellar medium are ionized by the ultraviolet radiation from a nearby star or stars. Only very hot stars, typically young stars, have enough radiation in the ultraviolet region at wavelengths necessary to ionize the hydrogen. The excess energy beyond that needed to ionize the hydrogen goes to kinetic energy of the ejected electrons. Eventually, by collision, this energy is shared by other particles in the gas. An equilibrium is established in a typical emission nebula when the temperature equivalent of this kinetic motion is between 7000 K and 20,000 K. For a typical emission nebulae, the density of ions (and electrons) is 1.0E8 to 1.0E10 particles per m^3.

As the ions de-excite to lower energy levels, in most cases after recombination of ions with electrons, they emit their characteristic spectral lines. The most prominent of these in the visible spectrum is the red line of hydrogen, giving most emission nebulae a characteristic red glow. There also exist "forbidden lines" (ones not normally seen in earth-bound laboratories) in the spectra from nebulae. The most prominent are green lines from doubly ionized oxygen, giving some nebula a green shading.³
These excited states may last for hours before the ion drops to a lower state and emits a photon. Only if the ion is left undisturbed by collisions (which could change the state of the ion) will the transition actually occur. In earth-bound laboratories we are not able to reduce the density low enough to see these transitions, but at the low densities of the nebulae they do occur and in fact these "forbidden" spectra may contribute significantly to the light observed from a nebula.

Rocks in space

OPT   "Don't you feel hotter if you sunbathe?"

That rocky planet we call Mercury has sunbathed for a long time just 36 million miles (on average) from the Sun. It has a rotation period of almost 59 days and a sidereal period of 88 days so the time between one sunrise and the next is 176 earth days. During its long day the temperature rises to 427 degrees C. In the night it drops to -173 degrees C. The sun's surface temperature is 5,500 degrees C. Looking at the sun from Mercury it would be 2 to 2 1/2 times as big as we see it. The surface reflects only 6% of the light shining on it.

How big does an object have to be before it becomes a star?

It would have to be 60 times the mass of Jupiter or 19,000 times the mass of the Earth. Any rocks smaller than this are going to be dark matter and therefore reduce the amount of light reaching us.

Some quotes about the Big Bang

Like Evolution and Relativity, the Big Bang is usually paraded as a proven, undeniable fact.

It isn't.


Static-universe models fit the data better than expanding-universe models.


The microwave "background" makes more sense as the limiting temperature of space heated by starlight than as the remnant of a fireball.


Element-abundance predictions using the Big Bang require too many adjustable parameters to make them work.


The universe has too much largescale structure (interspersed "walls" and voids) to form in a time as short as 10-20 billion years.


The average luminosity of quasars must decrease in just the right way so that their mean apparent brightness is the same at all redshifts, which is exceedingly unlikely.


The ages of globular clusters appear older than the universe.
The local streaming motions of galaxies are too high for a finite universe that is supposed to be everywhere uniform.


Invisible dark matter of an unknown but non-baryonic nature must be the dominant ingredient of the entire universe.


The most distant galaxies in the Hubble Deep Field show insufficient evidence of evolution, with some of them apparently having higher redshifts (z = 6-7) than the faintest quasars.


If the open universe we see today is extrapolated back near the beginning, the ratio of the actual density of matter in the universe to the critical density must differ from unity by just one part in 1059. Any larger deviation would result in a universe already collapsed on itself or already dissipated.

External links

An Australian-led team of astronomers has challenged conventional Big Bang theory by finding that large numbers of stars may be living unseen in the space between the galaxies.



http://exosci.com/news/147.html

Education establishments involved in the fields of astronomy, astrophysics, theoretical physics and cosmology are dominated by those who have accepted BB as the theory to be pursued. Scientists who seriously question the BB are generally considered disruptive, ridiculed and derogatorily referred to as big bang bashers.



http://nowscape.com/big-ban2.htm

Astronomers believe some of these galaxies could be over 12 billion light-years away (depending on cosmological models) – making them the farthest objects ever seen. A powerful new generation of telescopes will be needed to confirm the suspected distances.



http://oposite.stsci.edu/pubinfo/pr/1998/32/content/9832ay.jpg

Emil Mottola of the Los Alamos National Laboratory and Pawel Mazur of the University of South Carolina suggest that instead of a star collapsing into a pinpoint of space with virtually infinite gravity, its matter is transformed into a spherical void surrounded by "an extremely durable form of matter never before experienced on Earth."


http://www.space.com/scienceastronomy/astronomy/gravastars_020423.html

ALBUQUERQUE, N.M. - The most massive black holes share much in common with their puniest cousins, according to a new study that found the first direct evidence for the source of energetic jets shooting out in two opposite directions from one of Nature's most impressive gravity wells.


http://www.space.com/scienceastronomy/astronomy/blackhole_jets_020606.html

9-Jul-2002 An analysis of 13.5 thousand million-year-old X-rays, captured by ESA's XMM-Newton satellite, has shown that either the Universe may be older than astronomers had thought or that mysterious, undiscovered 'iron factories' litter the early Universe.



http://sci.esa.int/content/news/index.cfm?aid=1&cid=1&oid=30255