Heat conduction. In soil with conductivity k (W/m/K) and volumetric heat capacity C (J/m3/K), temperature obeys dT/dt = alpha d2T/dz2 with diffusivity alpha = k / C. Dry sand has alpha around 2e-7 m2/s, about 1000 times smaller than a metal, which is why sand is such a good insulator.
The exact solution. If the surface follows T(0,t) = Tm + A cos(w t) with w = 2 pi / 24 h, the temperature at depth z is T = Tm + A exp(-z/d) cos(w t - z/d) with the penetration (damping) depth d = sqrt(2 alpha / w). Three consequences: the daily mean is the same at every depth; the swing shrinks by a factor e every d; and the wave is delayed by z/d radians, i.e. (z/d) * 24 h / 2 pi hours. For dry sand d is 8 cm, so at 60 cm the swing is only A e^-7.5 = 0.0006 A and the chamber stays at the mean temperature all day. A yearly cycle penetrates sqrt(365) = 19 times deeper.
Numerical method. The live field is solved by the Crank-Nicolson scheme (second order in time and space, unconditionally stable) on a 4 m column with 1 cm cells and a 60 s step; the surface temperature is the boundary condition and the bottom is insulated. Check 1 starts the solver from a uniform soil and compares it with the closed form above; Check 2 measures d and the delay from the simulated curves alone.
The fox. The fox is shown in the chamber when it is in the burrow window (default 06:00 to 19:00), and walks out otherwise. Water loss uses an illustrative law: 0.5 g/h + 0.35 g/h per C above 30 C of the temperature the animal sits in (air temperature outside, chamber temperature underground). This is not a measured fennec value; it is there to show that the relative saving comes from the temperature the animal experiences, not to predict real water needs. The fox itself does not heat the chamber in this model.
Honest simplifications. The surface is a pure cosine (real sand surface curves are skewed, with a sharp afternoon peak and a flatter night); the soil is uniform and dry or damp with constant k and C (no moisture movement, no evaporation cooling, no vapour transport); the air is a fixed fraction of the surface swing; there is no wind, shade, cloud, or tunnel ventilation, and a fox in the chamber is not a heat source. Real fennec dens are often 1 to 2 m deep with several chambers; typical constants quoted here are textbook-rounded and should be checked against site measurements before use.