Olbers' Paradox Calculator — Why Is the Night Sky Dark?
If the universe is infinite and filled with stars, every line of sight should eventually hit a star — making the sky as bright as the Sun everywhere. Yet the night sky is dark. This calculator quantifies exactly how tiny a fraction of the sky stars actually cover, given the finite age of the universe.
Gyr
×10²³
R☉
Parts per trillion of night sky covered by stellar disks — explains why the night sky is dark
- 1
Light-travel horizon radius cT
c × 13.8 Gyr = 130,645,980,000,000,000,000,000,000 mMaximum distance from which light can have reached us since the Big Bang. - 2
Total stars N
2 × 10²³ = 200,000,000,000,000,000,000,000 - 3
Coverage fraction F = 3 N R★² ÷ 4(cT)²
0.000000000004253Fraction of the sky solid angle covered by stellar disks. - 4
Sky coverage in ppt = F × 10¹²
0.000000000004253 × 10¹² = 4.25
How does this calculator work?
The sky coverage fraction F = 3N R★²/(4(cT)²) measures how much of the night sky is covered by stellar disks. For a 13.8 Gyr universe with ~2×10²³ stars, F ≈ 4 parts per trillion — the sky is dark because the finite age limits how many stars' light can reach us. The universe would need to be ~2×10¹¹ times older for every point of the sky to show a star.
Formula
How this is calculated
In an infinite, static, eternal universe with uniform stellar density, integrating the light from shells at every distance gives infinite sky brightness — Olbers' Paradox. The resolution is the finite age of the universe: light can only have travelled a distance cT ≈ 13.8 billion light-years since the Big Bang, so stars beyond that horizon are invisible to us.
The sky coverage fraction F measures what fraction of the total sky solid angle is physically covered by stellar disks. Each star at distance r covers a solid angle πR★²/r², and integrating over all shells from 0 to cT — using stellar density n = N/V_obs — gives F = 3N R★² / (4(cT)²). For the canonical estimate of 2 × 10²³ stars in a 13.8 Gyr universe with Sun-sized stars, F ≈ 4 × 10⁻¹² (about 4 parts per trillion). The sky is effectively all dark between stars.
The calculator also reports T_lit: how long an infinite static universe with the same stellar density would need to be for coverage to reach 100%. The ratio T_lit / T_current equals 1/F — the 'darkness factor'. For our universe this is around 2 × 10¹¹, meaning the universe would need to be 200 billion times its current age for the sky to be fully lit. In practice cosmological redshift and the accelerating expansion add further dimming; finite age is the dominant factor shown here.
Frequently asked questions
Olbers' Paradox asks: if the universe is infinite, uniform and eternal, why is the night sky dark? In such a universe every line of sight would end on a star, making the sky uniformly as bright as the Sun's surface. The resolution is that the universe has a finite age (~13.8 Gyr) and an expanding space, so light from the most distant stars has not yet had time to reach us.
It is the fraction of the sky solid angle physically occupied by stellar disks, expressed in parts per trillion (ppt = 10⁻¹²). At ~4 ppt, only 4 out of every trillion equal-area pixels of the night sky show a star disk. For the sky to be fully covered (F = 1), the universe would need to be hundreds of billions of times its current age.
Yes. The cosmological redshift shifts photons from distant galaxies to lower energies, and the accelerating expansion places most of the universe beyond a causal horizon. This calculator focuses on the finite-age effect, which is the primary and most intuitive resolution of Olbers' Paradox. Cosmological redshift adds further darkening on top.
Also known as
TG we-Calculate Editorial Team. (2026). Olbers' Paradox Calculator — Why Is the Night Sky Dark? [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/olbers-paradox-calculator
TG we-Calculate Editorial Team. "Olbers' Paradox Calculator — Why Is the Night Sky Dark?." TG we-Calculate. 2026. https://we-calculate.com/calculator/olbers-paradox-calculator.
TG we-Calculate Editorial Team, "Olbers' Paradox Calculator — Why Is the Night Sky Dark?," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/olbers-paradox-calculator
@misc{wecalculate_olbers_paradox_calculator, title = {Olbers' Paradox Calculator — Why Is the Night Sky Dark?}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/olbers-paradox-calculator}}, year = {2026}, note = {TG we-Calculate} }
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