We model the thermal evolution of the core and mantle using a parametrized convection scheme, and calculate the entropy available to drive the geodynamo as a function of time. The cooling of the core is controlled by the rate at which the mantle can remove heat. Rapid core cooling favours the operation of a geodynamo but creates an inner core that is too large; slower cooling reduces the inner core size but makes a geodynamo less likely to operate. Introducing potassium into the core retards inner core growth and provides an additional source of entropy. For our nominal model parameters, a core containing approximate to 400 ppm potassium satisfies the criteria of present-day inner core size, surface heat flux, mantle temperature and cooling rate, and positive core entropy production.We have identified three possibilities that may allow the criteria to be satisfied without potassium in the core. (1) The core thermal conductivity is less than half the generally accepted value of 50 W m(-1) K-1. (2) The core solidus and adiabat are significantly colder and shallower than results from shock experiments and ab initio simulations indicate. (3) The core heat flux has varied by no more than a factor of 2 over Earth history.\ud All models we examined with the correct present-day inner core radius have an inner core age of < 1.5 Gyr; prior to this time the geodynamo was sustained by cooling and radioactive heat production within a completely liquid core.\ud \u
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