Solutions of the equation of heat flow

The geometry of sunspots has been used to suggest a problem in heat flow. The equation of heat transport is solved for the case of a cylinder with a given thermal conductivity imbedded in an otherwise uniform medium with different conductivity. The surface of this region radiates heat with flux proportional to temperature. At a lower surface, either in heat flux or temperature is held constant. The cylinder can have an anisotropic thermal conductivity. The variations in temperature along the radiating surface have been determined. A simple approximation is noted which has been found to give a general solution with acceptable accuracy. This method may be of some use in other situations requiring the solution of Laplace's equation with a free surface. The analysis is used to set limits on the ratio of diameter to depth for cases which preserve the sharp surface temperature transition across the cylinder.