Olbers' Paradox: Sky Brightness Calculator
If the universe were infinite, eternal, and uniformly filled with stars, every line of sight should end on a stellar surface, making the night sky as bright as the sun. Yet the sky is dark. This calculator works through the maths of Olbers' Paradox step by step: it computes the theoretical flux from an infinite static universe, then shows how the finite age of the cosmos and cosmological redshift each reduce that brightness to the tiny value we actually observe.
Formula
Worked example
Take the default values: n = 0.14 stars/pc³, L = 0.5 L☉, t₀ = 13.8 Gyr, H₀ = 70 km/s/Mpc. Converting: n = 4.9 × 10⁻²³ stars/m³, L = 1.91 × 10²⁶ W. The horizon is c × 13.8 Gyr = 1.3 × 10²⁶ m, giving F_finite = 4.9 × 10⁻²³ × 1.91 × 10²⁶ × 1.3 × 10²⁶ ≈ 1.2 × 10³° W/m². The Hubble radius is c/H₀ ≈ 1.28 × 10²⁶ m and the redshift integral with x ≈ 1.02 gives ≈0.24, so F_expand ≈ 2.8 × 10²⁶ W/m² - about 20 million times fainter than the Sun at 1 AU.
What is Olbers' paradox?
Olbers' paradox is the observation that the night sky is dark, even though in an infinite, eternal, and static universe uniformly populated with stars, it should be blazingly bright. The argument is geometric and elegant: divide the universe into concentric spherical shells centred on Earth. Each shell at distance r contains 4πr² dr × n stars (proportional to r²). Each star appears fainter by 1/r² (the inverse-square law). The two r² factors cancel perfectly, so every shell contributes the same total flux regardless of distance. Sum infinitely many equal contributions and you get an infinitely bright sky. The paradox is named after German astronomer Heinrich Wilhelm Olbers, who described it in 1823, though the problem was stated earlier by Thomas Digges (1576), Johannes Kepler (1610), and Edmond Halley (1720). Edgar Allan Poe even anticipated the resolution in his prose poem Eureka (1848), arguing that the universe must be finite in age.
The shell flux formula and why r cancels
The key equation is dF = n L dr, where n is the number density of stars (stars per cubic metre), L is the mean stellar luminosity in watts, and dr is the shell thickness. This expression contains no r because the 4πr² growth in the number of stars exactly cancels the 1/(4πr²) dimming of each star's flux. The total flux from an infinite static universe is therefore F = n L R, which diverges as R tends to infinity. This is Olbers' paradox in its clearest form. The calculator above lets you choose a shell radius and thickness: notice that the shell flux output stays the same whether you set the radius to 1000 ly or 10,000 ly, illustrating the cancellation directly.
How the finite age of the universe resolves the paradox
The universe is about 13.8 billion years old. Light travels at a finite speed, so Earth can only receive light from stars within a sphere of radius c × t₀ ≈ 1.3 × 10²⁶ m. Beyond that horizon, starlight simply has not had time to reach us yet. The finite-age flux is F_finite = n L c t₀, which is large but finite. With typical Milky Way stellar densities and luminosities this comes out to a value far smaller than the solar constant, confirming that a young, finite universe does not produce a blindingly bright sky. Crucially, even if we lived in a static (non-expanding) universe, the finite age alone would keep the sky dark - the light from distant stars is still on its way.
How cosmic expansion further dims the sky
In our actual universe, the expansion of space stretches the wavelength of photons as they travel. A photon emitted as visible light from a source at redshift z arrives with its energy reduced by a factor of (1+z)⁻¹ (frequency shift) and its arrival rate reduced by a further (1+z)⁻¹ (time dilation), giving a total flux dimming of (1+z)⁻⁴ per photon. Integrating over all shells using the Hubble approximation z ≈ H₀r/c gives the expanding-universe flux F_exp = (n L c / H₀) × integral, where the integral evaluates to (1/3)[1 - (1+x)⁻³] with x = H₀ R_hor / c. For the default parameters this reduces the sky brightness to a tiny fraction of the solar constant, consistent with observations. The Cosmic Microwave Background carries most of the missing energy: it was emitted as bright plasma at 3000 K when the universe was 380,000 years old, but has since been redshifted to 2.725 K microwave radiation, far outside the visible range.
Key cosmological values used in Olbers paradox calculations
| Parameter | Symbol | Value | Source |
|---|---|---|---|
| Speed of light | c | 2.998 × 10⁸ m/s | NIST CODATA |
| Solar luminosity | L☉ | 3.828 × 10²⁶ W | IAU 2015 |
| Solar constant (1 AU) | S☉ | 1361 W/m² | Kopp & Lean 2011 |
| Universe age | t₀ | 13.8 Gyr | Planck 2018 |
| Hubble constant | H₀ | 67.4 km/s/Mpc | Planck 2018 CMB |
| CMB temperature | T₀ | 2.725 K | COBE / WMAP |
| Local stellar density | n | ~0.14 stars/pc³ | Nearby Stars Catalog |
| Mean stellar luminosity | ⟨L⟩ | ~0.5 L☉ | Kroupa IMF estimate |
Standard reference values from Planck 2018 and the IAU. Changing the Hubble constant or universe age in the calculator shifts the expanding-universe flux accordingly.
Frequently asked questions
Why is Olbers' paradox important in cosmology?
Olbers' paradox was one of the first rigorous observational arguments against an infinite, eternal, static universe. Its resolution requires either a finite universe age or cosmic expansion (or both), and historically it nudged cosmologists toward the Big Bang model. Today it is a standard teaching tool for illustrating how simple observation (the dark night sky) constrains fundamental cosmology.
Does the finite age of the universe fully explain the dark night sky?
Yes - a finite age alone is sufficient. If the universe is only 13.8 billion years old, light from stars more than 13.8 billion light-years away has never had time to reach us. The maximum flux is then F = n L c t₀, which with realistic stellar densities is far smaller than the solar constant. Cosmic expansion adds additional dimming on top of this, but even a non-expanding universe of finite age would have a dark night sky.
What role does the Cosmic Microwave Background play?
The CMB is the redshifted relic of the light that filled the universe when atoms first formed, about 380,000 years after the Big Bang. It carries the bulk of the electromagnetic energy that would have made the sky bright in the static model. That energy has been redshifted by a factor of about 1100 (from visible/infrared to microwave wavelengths), so the sky glows in microwaves that are invisible to human eyes. If you could see in microwaves, the CMB would make the sky uniformly bright in every direction.
Why does the shell flux formula not depend on distance?
The number of stars in a shell grows as 4πr² (the shell's surface area times its thickness). Each star's apparent brightness falls as 1/(4πr²) by the inverse-square law. Multiplying these together cancels the r² terms exactly, leaving a flux contribution per unit shell thickness of simply n × L, independent of distance. This perfect cancellation is the geometric heart of the paradox.
What does the Hubble constant have to do with Olbers' paradox?
The Hubble constant H₀ sets the rate at which the universe expands, and therefore how strongly distant sources are redshifted. A larger H₀ means faster expansion and more severe (1+z)⁻⁴ flux suppression from distant shells. It also defines the Hubble radius r_H = c/H₀, which acts as a characteristic scale length in the expanding-universe flux formula. You can see this effect directly by sliding the H₀ input: higher values reduce the expanding-universe flux and the solar brightness ratio.
Could dust absorption solve Olbers' paradox?
Olbers himself proposed that interstellar dust blocks distant starlight. The idea was later shown to fail: in an infinite static universe, dust would absorb so much energy that it would heat up until it reached the same temperature as the stars and re-emit as much light as it absorbed. Dust redistributes the spectrum but cannot remove energy. The real solutions - finite age and cosmic expansion - are thermodynamically consistent.