How This Olbers' Paradox Calculator Works
Olbers' paradox is the puzzle named after astronomer Heinrich Olbers: if the universe is infinite, populated with stars at a roughly uniform density, and has existed forever, every line of sight from Earth should eventually end on the surface of some star. Adding up the light from every direction should make the entire night sky as bright as the surface of a star like the Sun — yet the sky is dark. This calculator puts numbers on both sides of that argument: the average distance light must travel before hitting a star (the "mean free path"), and the much shorter distance we can actually see given the universe's finite age.
Deriving the mean free path
Treat each star as an opaque disk with cross-sectional area σ = πR², scattered through space with a uniform number density n (stars per unit volume). Along any line of sight, the probability that no star has been intersected within a distance r follows the same exponential law as radioactive decay or light attenuation: P(no hit) = e−nσr. The distance at which this probability falls to 1/e — roughly the average distance to the first star surface — is the mean free path, λ = 1/(nσ). Raising either the stellar density or the stellar radius shrinks λ, because there are more (or bigger) obstacles for a photon's path to run into.
Comparing the mean free path to the observable horizon
Light travels at a finite speed, so we can only see objects whose light has had time to reach us: roughly c × (age of the universe), or about 13.8 billion light-years using the light-travel-time approximation (the true particle horizon is larger, around 46.5 billion light-years, once cosmic expansion is included). This calculator divides the mean free path by that horizon distance to get a ratio, and uses the optical-depth relation 1 − e−horizon/λ to estimate what fraction of the sky would be covered by stellar disks if you could see all the way out to the horizon. Because the mean free path for ordinary stars is many orders of magnitude larger than the horizon, only a tiny fraction of the sky is actually covered — which is why the night sky is dark despite Olbers' argument.
Where this simplified model breaks down
- It ignores cosmic expansion: the true observable-universe radius (~46.5 billion light-years) is larger than the simple c × age estimate used here, and distant starlight is redshifted, which further dims — rather than brightens — the sky.
- It treats all stars as identical: real populations span a huge range of radii and luminosities; using one average radius is fine for an order-of-magnitude estimate but not for precision cosmology.
- It ignores galaxy structure: stars cluster into galaxies separated by enormous voids, so the number density averaged over the whole universe is far lower than the local stellar density near the Sun, which changes the results substantially if you swap one for the other.