The heating of the earth during its formation

The thermal state of the Earth accumulating from solid bodies is investigated. The conductivity equation is deduced for a growing spherically symmetrical planet which takes into account heating by impacts of bodies, by radioactivity, and by compression of its material. The cooling is produced mainly by impact mixing, which is approximated by extrapolating the parameters from known impact craters to larger sizes. The solution of a more simple conductivity equation for a uniformly heated plane parallel layer with moving boundaries is found. It can be considered as an approximate quasi-stationary solution for the temperature distribution in the outer parts of the growing Earth. The result depends substantially on the sizes of impacting bodies but almost not at all on the time scale of the accumulation. The latter only weakly affects the surface temperature and does not affect the temperature distribution in the layer. For bodies of small radii, r′ < r₁, where the size of the crater is not affected appreciably by gravitation (for the present mass of the Earth r₁ ≈ 1 km), the heating is small. For bodies with r′ > r₁, the heating of the layer is roughly proportional to the ratio $\text{r′}\text{r}{}_1$. Toward the end of the Earth's accumulation the melting point can be reached in the outer layer at r′_M ? 60 km, where r′_M is the radius of the largest body in the power law size spectrum of falling bodies. The estimates of the initial temperature of the Earth can vary within wide limits depending on the mass distribution of large protoplanetary bodies, which at the present time is not known correctly. The initial melting of an upper layer of the Earth a few hundred kilometers thick seems to be possible.