Kelvinor: Kelvin's Age of the Earth
Thermodynamics & Statistical Physics arc, page 60 of 193. Back to The Pixelated Physics Universe
Who they are
Kelvinor raises his thermometer sword over an Earth far older than he thinks.
Kelvinor is a character of the Thermodynamics & Statistical Physics arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
In 1862 William Thomson, later Lord Kelvin, treated the Earth as a body that began molten and has cooled by conduction alone. From the measured rise of temperature with depth, a starting temperature and the rock's diffusivity, the half-space conduction solution gives an age. He bracketed it between 20 and 400 million years. The model left out heat carried by convection inside the Earth (John Perry argued in 1895 that a more conductive interior gives far larger ages) and radioactive heating, discovered after 1896. In 1956 Clair Patterson, using lead isotopes in meteorites, found 4.55 ± 0.07 billion years.
$$ \begin{gathered} t = \frac{T_0^{2}}{\pi\,\kappa\,G^{2}} \qquad (\text{cooling by conduction only}) \\ \text{Kelvin 1862: } 20\text{ to }400\ \text{million years}; \quad \text{Patterson 1956: } 4.55 \pm 0.07\ \text{billion years} \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $t$ | time since the Earth began cooling | s |
| $T_0$ | starting temperature above the surface value | K |
| $\kappa$ | thermal diffusivity of rock (printed k) | m²/s |
| $G$ | temperature gradient at the surface | K/m |
Honest caveat: Radioactivity alone did not fix the estimate; convection mattered at least as much. His conduction mathematics was sound; the model was missing pieces.
Sources
- Thomson (1864) On the Secular Cooling of the Earth, Trans. R. Soc. Edinb. 23, 157
- Perry (1895) On the Age of the Earth, Nature 51, 224
- Patterson (1956) Age of meteorites and the earth, Geochim. Cosmochim. Acta 10, 230
Crossovers
Kelvinor meets Cavenda (Weighing the World 1798). Cavenda weighed the Earth with a torsion balance and found its mean density. Kelvinor tried to date the Earth from its cooling. Cavenda's number held up; Kelvinor's age was far too short, partly because he did not know about radioactive heating and partly because he ignored convection in the mantle.
Kelvinor meets Calorus (Caloric Theory of Heat). Calorus's heat-as-fluid was overturned by work turning into heat. Kelvinor, a founder of the new thermodynamics, used heat conduction to estimate the Earth's age. He got the physics of conduction right, but his age was wrong because he left out heat sources.
Wear the lore
Concept keywords: Kelvin's Age of the Earth, Kelvinor, Thermodynamics & Statistical Physics, physics lore, Pixelated Physics