An AI-generated pencil sketch of a roast turkey.

The time it takes to properly roast a whole turkey is proportional to its weight to the ⅔ power.

For a spherical turkey of uniform thermal conductivity α\alpha and density ρ\rho, a precise formula has been derived:

t=ln⁡(2(Th−T0)Th−Tf)1π2α(34πρ)2/3m2/3t = \ln\left(\frac{2(T_h-T_0)}{T_h-T_f}\right) \frac{1}{\pi^2 \alpha} \left(\frac{3}{4\pi\rho}\right)^{2/3} m^{2/3}

where the oven is set at ThT_h and the center of the turkey needs to reach a temperature of TfT_f from T0T_0.

The more general ⅔ power law does not depend on unrealistic assumptions about the turkey’s shape or thermodynamic properties; it can be derived from pure dimensional analysis and applied to turkey-shaped meat-based objects by fitting a curve to specific cook times used by chefs.

So whenever I roast a turkey, I find myself digging out my old mathematical modelling textbook to look up the cooking time: 45min/lb2/345 \text{min}/\text{lb}^{2/3} at 350℉.