Deep mines are hot. The deepest gold mines in South Africa reach about 4 km, where the rock is around 60 °C and the tunnels must be cooled. Temperature rising with depth, the geothermal gradient, is the first clue that Earth is a heat engine. In Half-Life and Absolute Age you met radioactive atoms; they are part of the story.
Why it works
Heat flows from hot to cold. Earth's interior is hot for two reasons: leftover heat from the collisions and sinking iron that built the planet, and radioactive decay of uranium, thorium and potassium, which keeps adding heat. That heat must leak out through the crust. Rock conducts heat poorly, so a steady flow needs a steep temperature drop across it, the way a thick blanket keeps a big difference between your skin and the air. The rule for conduction is $q = k\,\dfrac{\Delta T}{\Delta z}$: heat flow $q$ equals the rock's conductivity $k$ times the gradient.
Key idea
Temperature rises about $25^\circ\text{C}$ per km in the upper crust: $T = T_0 + g z$. The gradient is steep only near the surface. Deeper down, the mantle slowly convects and carries heat far better, so the temperature climbs gently, reaching about 5,000 to 6,000 °C at the center.
Worked examples
Worked example: Temperature in a well
The surface is $15^\circ\text{C}$ and the gradient is $25^\circ\text{C/km}$. How hot is the rock at the bottom of a well 3 km deep?
- Warming: $25 \times 3 = 75^\circ\text{C}$.
- $T = 15 + 75 = 90^\circ\text{C}$.
Answer: About $90^\circ\text{C}$.
Worked example: Heat flow through the crust
Granite conducts heat with $k = 3.0\,\text{W/(m·K)}$. The gradient is $25^\circ\text{C/km}$. How much heat flows out through each square meter?
- Write the gradient in SI units: $25^\circ\text{C/km} = 0.025\,\text{K/m}$ (a change of 1 °C is a change of 1 K).
- $q = 3.0 \times 0.025 = 0.075\,\text{W/m}^2 = 75\,\text{mW/m}^2$.
- That is tiny next to sunlight (about $240\,\text{W/m}^2$ absorbed on average), but over all of Earth it adds up to roughly 45 terawatts, and it drives volcanoes, earthquakes and moving plates.
Answer: $75\,\text{mW/m}^2$.
Common mistake
A common slip is to carry the crust's gradient all the way down: $15 + 25 \times 6,371 \approx 160,000^\circ\text{C}$ at the center, hotter than the Sun's surface. The real center is about 5,000 to 6,000 °C. The gradient is steep only where heat must squeeze through rock by conduction. Check any straight-line estimate against what the deep rock is doing: the mantle is solid but flows, and flowing rock carries heat.