Olber's Paradox

Why is the night sky dark, only dotted with shiny stars? Olber's paradox is a titillating problem with a clear and satisfying solution.

Distance before a line of sight hits a star1.565e+8 Gpc1/(nσ) with Sun-sized stars at 0.004 per cubic parsec — beyond this the sky would be covered in stellar surfaces

Recession velocity28,487.07 km/s9.5023% of the speed of light, from the RELATIVISTIC Doppler formula

The naive cz estimate29,979.25 km/s5.238% high — v = cz is only a low-redshift approximation, and it exceeds the speed of light entirely for z > 1

Hubble distance406.9581 Mpcv/H₀ — a first-order estimate that ignores how the expansion rate has changed over cosmic time

Redshift z0.1as entered

Observed wavelength721.908 nma rest wavelength of 656.28 nm arrives 65.628 nm longer

Hubble time1.397e+10 years1/H₀ — an estimate of the age of the universe that assumes a constant expansion rate. It lands close to the true 13.8 billion years by a coincidence of deceleration then acceleration roughly cancelling

Hubble radius4.283 Gpcwhere the recession velocity reaches c

Look-back time, roughly1.327e+9 yearslight travel time over the Hubble distance

Parallax distance10 pc32.6156 light years — the parsec is DEFINED as the distance giving one arcsecond of parallax, so this is exactly a reciprocal and involves no physics at all

Parallax at twice the distance0.05 arcsechalved — which is why parallax runs out: beyond a few thousand parsecs the angle is smaller than anything measurable

Why redshift is not a normal Doppler shiftspace itself expandsthe light is stretched in transit rather than emitted at a shifted frequency, so recession faster than light is allowed at large distances — nothing moves through space faster than c

Against the observable distance3.655e+7×the universe is not old enough for light to reach us from that far, which is why the sky is dark — this is Olbers' paradox and its resolution in one comparison

Time for that light to arrive5.106e+17 yearsagainst a Hubble time of 1.397e+10 years — redshift dims what does arrive, but the finite age is the main answer

The formula

v = H₀d; 1 + z = λ_obs/λ_emit; d(pc) = 1/p(arcsec)

v = cz breaks before you expect

The simple reading of redshift as velocity, v = cz, is a low-redshift approximation. It is already a few percent wrong by z = 0.1 and gives faster than light for z > 1, which is why quasars at z = 6 are not receding at six times the speed of light. The relativistic Doppler formula is used above, and it saturates at c as z grows.

Strictly, cosmological redshift is not a Doppler shift at all. The light is stretched by the expansion of space during its journey rather than emitted at a shifted frequency. That distinction is why recession velocities greater than c are permitted at large distances: nothing is moving through space faster than light, the space between is simply growing.

The parsec is a definition

A parsec is defined as the distance at which a star shows one arcsecond of parallax across Earth's orbit, so distance in parsecs is exactly the reciprocal of parallax in arcseconds. There is no physics in the conversion — it is a definition, which is precisely what makes parallax the one rung of the distance ladder that needs no assumptions.

Why the sky is dark

Olbers' paradox observes that in an infinite, eternal, static universe every line of sight would eventually meet a star, so the whole sky should be as bright as the Sun. The resolution is the finite age of the universe: light from beyond a certain distance simply has not had time to reach us. Redshift dims what does arrive, but the finite age is the main answer.