Wolfram Alpha:
Search by keyword:
Astronomy
Chemistry
Classical Physics
Climate Change
Cosmology
Finance and Accounting
Game Theory
General Relativity
Lagrangian and Hamiltonian Mechanics
Macroeconomics
Mathematics
Microeconomics
Particle Physics
Probability and Statistics
Programming and Computer Science
Quantum Computing
Quantum Field Theory
Quantum Mechanics
Semiconductor Reliability
Solid State Electronics
Special Relativity
Statistical Mechanics
String Theory
Superconductivity
Supersymmetry (SUSY) and Grand Unified Theory (GUT)
The Standard Model
Topology
Units, Constants and Useful Formulas
Olbers' Paradox
---------------
Olber thought the sky should be light at night based on the belief at the
time that the universe was infinitely big and infinitely old. In every
direction you should eventually see a star and the sum of all these sources
would cause the sky to by light. Consider a spherical shell.
volume = 4πr^{2}dr where r is radius and dr is the thickness of the shell.
Let N = number of stars per unit volume of space.
# of stars in shell = 4πr^{2}Ndr
Total intensity from all stars in shell ∝ 4πr^{2}Ndr/r^{2} = 4πNdr
Olbers' interpretation was that when r is small, there are fewer stars
but their intensities are larger. When r is large there are more stars,
but their intensities are smaller. These 2 effects cancel out, and all
shells produce the same intensity. Now, since the number of shells is
infinite in an infinite universe, I_{Total} should equal ∞ and lead to a
light sky (and a frazzled earth!). Noting that the night sky is
obviously not that bright and that we are still here, begs the
question "what is going on?". There are several options.
1. The universe is not infinite and there are only a finite number of stars.
2. The universe is of finite age so that light from stars at an great
distances hasn't reached us yet.
3. Space is expanding, so distant stars are red-shifted into obscurity.
Hubble's law states that the recessional velocity of a star, v is
related to its distance, D, by:
v = H_{0}D
So, the further the star is away, the faster it is moving. The outermost
stars can be travelling at > c, so we will never see them.
1. This is possible, but there still should be enough to light up the sky.
2. Possible, but suggests the universe was instantly created - not an
immediate answer to this question!
3. This is the generally accepted argument. The implication is that
there was a point in time when the universe was small enough to
observe constant light. In the present day, the wavelength of light
from this time has been stretched as space itself expands. We now
see this doppler shifted light in the form of cosmic background
radiation which is not visible.