Olbers Paradox

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Olbers Paradox

• Mike Collicutt and Ryan Marciniak

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Whos Paradox?

• Heinrich Wilhelm Matthäus Olbers lived from 1758
  to 1840.
• Discovered Pallas (1802) and Vesta (1807)
• Pluralist, firm believer of moon life

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The line-of-sight argument

• Requires infinite, homogeneous universe
• Called Olbers paradox by Hermann Bondi in 1952
• Every sight line from every point intercepts a
  star
• Entire sky would be as bright as 180,000 suns

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The Forest Analogy
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Intensity

• If we look at the flux of photons from the sun at
  radius d

• If we double the distance d to the sun, then the
  solar flux decreases by a factor of 4.

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Area Intensity

• Solid angle subtended by the sun is given by

• If we double the distance to the sun as before,
  the solid angle subtended by the sun also
  decreases by 4

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• This means that the intensity per unit area per
  unit solid angle is constant

• Meaning that the Brightness B of a star is
  constant, so faraway stars should be as bright as
  nearby stars like the sun.

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Is it Dust Extinction?

• Dust particles in thick clouds absorb light
• Does dust account for why we see a dark night
  sky?

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Nope!

• Light from stars behind the dust would heat it up

Good old Blackbody Radiation
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Hubble to the Rescue

• Hubbles discovery of an expanding universe
  allowed for red shift of light
• Stars farther away from an observer have more
  expanding space in between, and hence more
  redshift
• So the brightness of the will decrease as we look
  at stars farther away

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But is that enough???
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Solutions Finite visible universe

• First proposed (poetically) by Edgar Allen Poe in
  1848, and Lord Kelvin Mathematically in early
  1900s
• "Were the succession of stars endless, then the
  background of the sky would present us a uniform
  luminosity 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."

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Finite Visible Universe Contd

• Even if our universe is indeed infinite with
  infinitely many stars, he argued that our visible
  universe is only finite in size the light from
  stars outside of this sphere hasnt had enough
  time to reach us yet
• Kelvin calculated that in order to have a night
  sky as bright as the sun, wed need to include
  all starlight up to 3x1015 ly away
• Were on the right track, but is there anything
  else?

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Finite lifetime of stars

• Typical lifetime of a star 10 billion years
• Universe 13.7 billion years old
• Stars in the early universe have since died out
• At any given time we only see a certain amount of
  luminous matter
• Not enough starlight has been emitted to make the
  night sky bright

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Age vs. Expansion

• Which factor contributes more to lack of observed
  radiation we see Finite age of galaxies or the
  expansion of the universe causing redshift?
• Start by finding the intensity of visible light
  from all of the galaxies

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All-sky intensity

• no number density of galaxies now that formed at
  tf (big bang) usually
• F spectrum of a galaxy as a function of
  wavelength time
• RR(t) scale factor for the expansion of the
  universe Hubbles law
• Ro radius of the universe in present epoch
• Actually express ? as ?oR/Ro ?o is the observed
  wavelength, ? is the actual wavelength emitted by
  the galaxy
• I(?o) is the intensity at wavelength ?o of all
  the radiation from galaxies within some imaginary
  sphere corresponding to the time of the galaxies
  formation
• Units of I power/unit area/unit wavelength
  interval

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Galaxy Spectrum

• Simplest model assume the galaxies act as black
  bodies
• As galaxies age, the luminosity of a galaxy
  decreases as spectrum becomes redder incorporate
  this by letting TT(t)
• Whats C? Blackbody gt Use Stefans Law

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Almost there

• Fixes C as
• Now can sub F into the expression for intensity
  to obtain

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Now to the point Affect of Age of Galaxies

• We can compare the intensity found under an
  expanding universe with that of a completely
  static one RoR, TTo
• So the intensity directly proportional to the age
  of galaxies in the universe by to-tf

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The point Effect of the expanding universe

• From here we want to know I/Is. The best we can
  do is use a numerical integration technique to
  gather any numerical information
• BORING
• So Lets look at the results

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The Affect of the Expanding Universe

• Can vary different parameters such as density of
  galaxies, and redshift of galaxy formation (time
  of galaxy formation)
• BUT Every time get roughly the same decrease in
  intensity I/Is 1/3
• SO the expansion of the universe only reduces the
  observed intensity of light by 1/3, whereas the
  age of the galaxies determine the observed
  intensity by orders of magnitude to-tf

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Summary/Conclusions

• Olbers paradox states that in an infinite,
  static universe with infinitely many uniformly
  distributed stars (galaxies) the night sky would
  appear as bright as the surface of the sun
• Resolutions
• Redshift due to expansion
• Finite visible universe
• Finite age of the galaxies limits light produced
• Most important effect that resolves the paradox

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