Can mantle convection be self-regulated?

The notion of self-regulating mantle convection, in which heat loss from the surface is constantly adjusted to follow internal radiogenic heat production, has been popular for the past six decades since Urey first advocated the idea. Thanks to its intuitive appeal, this notion has pervaded the solid earth sciences in various forms, but approach to a self-regulating state critically depends on the relation between the thermal adjustment rate and mantle temperature. I show that, if the effect of mantle melting on viscosity is taken into account, the adjustment rate cannot be sufficiently high to achieve self-regulation, regardless of the style of mantle convection. The evolution of terrestrial planets is thus likely to be far from thermal equilibrium and be sensitive to the peculiarities of their formation histories. Chance factors in planetary formation are suggested to become more important for the evolution of planets that are more massive than Earth.