A Seveneves Calculation

In the novel Seveneves by Neal Stephenson, the Moon explodes, causing the surface of the Earth to become uninhabitable for 5000 years. There is a colony or two of humans who survive in deep mines. Let's calculate how deep the human hole goes.

The thermal diffusivity of rocks appears to be around 1 mm²/s. 5000 years is around 1.6 × 10¹¹ seconds. You can expect the heat from the “hard rain” to diffuse to a depth of \(\sqrt{1.6\times 10^{11}}\) millimeters or 400 meters. Three times that distance should be safe, so the colony would only have to be \(\frac{3}{4}\) miles down. It might be doable.