Considerations of Joule heating and radius reduction in umbral dots

The physical nature of bright features in sunspot umbra such as umbral dots (UDs) is not fully understood yet. Here, by accepting magnetic field reduction in the UDs, the Joule heating mechanism in the UDs is investigated. In this investigation, a UD model is assumed which is characterized by a) a reduced (possibly zero) magnetic field strength in the UD column relative to its surroundings, b) the electric current generated by magnetic field gradient at the boundary of the UD column can produce some heat, and c) when the radius of the UD - because of pressure reduction inside the column - becomes zero, the death of the UD will occur. The rate of radius reduction is estimated to be nearly 100 Ms ^ -1. Without consideration of gas injection from the bottom of the UD column, it is found that the speed of outflow in UDs varies linearly from zero at the bottom to a few 100 Ms ^ -1 at the upper level. It is shown that Joule heating, in some cases, can produce enough heat to explain the brightness of UDs.