Olbers' Paradox: Sky Brightness Calculator

Your details

Average number of stars per cubic parsec. Near the Sun this is about 0.14 stars/pc³. The Milky Way average is lower (~0.003), dense clusters can reach >1000.
stars/pc³
Mean luminosity of stars in solar units (1 L☉ = 3.828 × 10²⁶ W). The Sun-like average is about 0.5 L☉ because faint red dwarfs greatly outnumber bright stars.
L☉
Lookback time that limits how far light can have reached Earth. The current best estimate from the CMB is 13.8 billion years.
Gyr
Rate of cosmic expansion. Planck 2018 gives H₀ ≈ 67.4; local-distance-ladder measurements give ~73. Default 70 is a round mid-value.
km/s/Mpc
Distance to the spherical shell you want to examine. The shell flux is independent of radius in the static model, illustrating the paradox directly.
ly
Radial depth of the shell. Each 1-ly-thick shell contributes the same flux as every other, regardless of distance.
ly
Static infinite-universe sky brightnessTwilight-level brightness
Diverges to infinity (paradox)

Predicted flux from an infinite, eternal, static universe - the paradoxical infinite answer, normalised to solar brightness.

Finite-age correction flux119.074W/m²
Expanding-universe flux35.063W/m²
Flux from selected shell0W/m²
Stars in selected shell50,706
Observable horizon radius13.8Gly
Redshift dimming factor0.2945
Sky brightness vs. Sun0.025762
Finite-age flux (W/m²)119.074
Expanding-universe flux (W/m²)35.063
Single-shell flux (W/m²)0
0.025762 (fraction of solar constant)
Dark night sky<0.00001Faintly lit0.00001-0.001Twilight-level0.001-0.1Sun-like0.1+

The actual sky brightness is about 2.58e-2 times the Sun - far too faint to be paradoxical.

  • With your inputs, the finite age of the universe limits the observable horizon to 13.8 billion light-years, cutting the flux from infinite to a finite 1.19e+2 W/m².
  • Cosmic expansion then applies an additional (1+z)⁻⁴ redshift dimming. Only 29.4461% of the finite-age flux survives, leaving 3.51e+1 W/m².
  • Each spherical shell of stars at any distance contributes identical flux (n × L × dr). The paradox arises because summing infinitely many equal contributions should diverge - but the universe is neither old enough nor static enough for that to happen.
  • The Cosmic Microwave Background carries most of the energy that "should" make the sky bright, but it has been redshifted to 2.73 K and is invisible to human eyes.

Next stepTry reducing the universe age to 1 Gyr to see how the finite-age horizon shrinks the flux, or increase H₀ to see how faster expansion amplifies the redshift suppression.

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