• Nocturna Olbers' Paradox pixel-art physics T-shirt, white tee, front view

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Nocturna

Olbers' Paradox Physics

\[dN_{\star} \propto r^{2}\,dr, \qquad F_{\star} \propto \frac{1}{r^{2}} \ \Rightarrow\ dF_{\text{shell}} \propto dr\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

Nocturna wraps herself in the night sky and asks the oldest question in astronomy: why is it dark?

The Physics

If the universe were infinite, eternal and evenly filled with stars, every line of sight would end on a star and the night sky would blaze. A thin shell of radius r holds a number of stars growing as r², while each star's light falls as 1/r², so every shell adds the same brightness. The way out: the universe has a finite age, t₀ ≈ 13.8 billion years, so light from beyond a certain distance has not reached us, and redshift dims what does arrive.

stars in shell ∝ r² dr, light per star ∝ 1/r² ⇒ each shell adds equal flux

  • r: distance to the shell (m)
  • dr: shell thickness (m)
  • t₀: age of the universe, ≈ 13.8 billion years (≈ 4.35 × 10¹⁷ s)

Heinrich Olbers stated it in 1823 and proposed interstellar dust as the answer, which fails; Edgar Allan Poe guessed the real answer in Eureka (1848). Halley and de Chéseaux had raised the puzzle earlier.

Honest caveat: Dust does not work: over time it would heat up and glow.

On the shirt

Front: Nocturna, the pixel-art character, with the title. Back: the shell argument, why dust fails, the finite-age answer and the Poe, Halley and de Chéseaux history. Printed on a unisex tee.

Shipping calculated at checkout by destinationMade to order, ships in 2–5 business days

Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Ter vergelijking: maat M is 20 in (50.80 cm) breed, ongeveer 3.143 × 1034 Plancklengtes (lP = 1.616255 × 10−35 m). Een meetlint is sneller.

Measurements in inches, ±1.5 in tolerance.

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Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

I wrap myself in the night sky and ask the oldest question in astronomy: why is it dark?

Suppose stars go on forever, evenly spread. A thin shell at distance r holds a number of stars growing as r², while the light from each falls as 1/r². Every shell adds the same brightness, so every line of sight should end on a star and the sky should blaze. Heinrich Olbers stated it in 1823 and blamed interstellar dust. That fails: dust soaking up all that light would heat up and glow. Halley and de Chéseaux had worried about it earlier.

The real answer is time. The universe has a finite age, about 13.8 billion years, so starlight reaches us only from a finite volume, and the expansion of space redshifts and dims what does arrive. Edgar Allan Poe guessed it in Eureka in 1848: the light of the most distant stars has not yet had time to reach us.

Equations

\[dN_{\star} \propto r^{2}\,dr, \qquad F_{\star} \propto \frac{1}{r^{2}} \ \Rightarrow\ dF_{\text{shell}} \propto dr\]
\[t_0 \approx 13.8\ \text{billion years}\]
SymbolMeaningUnit
\(r\) distance from the observer to the shell \(\mathrm{m}\)
\(dr\) shell thickness \(\mathrm{m}\)
\(dN_{\star}\) number of stars in the shell \(\mathrm{1}\)
\(F_{\star}\) light received from one star \(\mathrm{W/m²}\)
\(t_0\) age of the universe, about 4.35 × 10¹⁷ s \(\mathrm{s}\)

Sources

  1. Planck Collaboration (2020) Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6
  2. Wesson, Valle & Stabell (1987) The extragalactic background light and a definitive resolution of Olbers's paradox, Astrophys. J. 317, 601
  3. Poe (1848) Eureka: A Prose Poem, Project Gutenberg

Field notes

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