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Thermal Mirage

Heat Refraction Physics

\[n\cos\alpha = \text{constant}\]
$21.99 USD
$21.99 USD
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Meet Thermal Mirage. Hot air over a road is thinner, so light travels slightly faster in it. Skylight skimming the road bends up into your eye, and you see the sky below the horizon: the puddle is sky. On the mug: n cos α = constant. Pixel-art full-wrap black ceramic mug in 11 or 15 oz; equation, symbols and sources in the physics panel below.

Shipping calculated at checkout by destinationMade to order, ships in ~10 days

Details

Size & material11oz · 15oz

Black glossy ceramic mug with a C-handle. Lead- and BPA-free.

  • 11oz: 0.33 l
  • 15oz: 0.44 l
CareDishwasher & microwave safe

Dishwasher safe (top rack) or hand wash. Microwave safe.

ShippingBy destination

Shipping calculated at checkout by destination.

Made to order: about 10 days of production and handling before it ships.

Delivery times & where we ship

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

Hot air near a road has a lower refractive index, so shallow sky rays bend upward and the sky appears below the horizon. With 30 K of heating only rays within about 0.4° of the road turn back.

Thermal Mirage paints a puddle on a dry road, and the puddle is sky.

The refractive index of air depends on its density: n − 1 is proportional to density, the Gladstone–Dale rule, with n − 1 ≈ 2.9×10⁻⁴ at standard conditions. Hot air over a sunlit road is thinner, so its index is slightly lower and light travels slightly faster in it.

In air layered by temperature, a ray obeys n cos α = constant, the formula on this mug, where α is its angle above the horizontal. A ray from the sky heading shallowly down toward the road enters ever lower-index air, flattens out and bends back up into your eye. You see the sky below the horizon, shimmering like water. This is an inferior mirage.

A worked number: if the air at the road is about 30 K hotter than the air at eye height, the index differs by about 2.5×10⁻⁵, and only rays within α_max ≈ √(2Δn) ≈ 0.4° of the road can turn back. That is why the puddle always sits far ahead.

“Come closer,” says Thermal Mirage. “I'll keep moving the water.”

Equations

\[n\cos\alpha = \text{constant}\]
\[n - 1 = K\rho\]
\[\alpha_{max} \approx \sqrt{2\,\Delta n}\]
SymbolMeaningUnit
\(n\) refractive index of air \(\mathrm{dimensionless}\)
\(\alpha\) angle of the ray above the horizontal \(\mathrm{rad}\)
\(K\) Gladstone–Dale constant of air \(\mathrm{m³/kg}\)
\(\rho\) air density \(\mathrm{kg/m³}\)
\(\alpha_{max}\) largest ray angle that can be turned back \(\mathrm{rad}\)
\(\Delta n\) refractive-index difference between cool and hot air \(\mathrm{dimensionless}\)

Sources

  1. Gladstone & Dale (1863) Researches on the refraction, dispersion, and sensitiveness of liquids, Phil. Trans. R. Soc. 153, 317
  2. Ciddor (1996) Refractive index of air: new equations for the visible and near infrared, Appl. Opt. 35, 1566

Source Gladstone, J. H. & Dale, T. P. (1863). Researches on the refraction, dispersion, and sensitiveness of liquids. Philosophical Transactions of the Royal Society of London 153, 317–343.doi.org/10.1098/rstl.1863.0014

Field notes

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