A hydrostatic model atmosphere of a typical umbral core is constructed on the basis of both Na I $D_2$ line profile and umbral core-to disk continuum intensity ratio. The observations were obtained by Mullan and Wyller with photoelectric means over the range from $4000{AA};and;6500{AA}$. The computed line profile of Na I $D_2$ and the emergent continuum intensity distribution are compared with the observations. The temperature of umbral cores is found to be lower by $300^{circ}K;to;500^{circ}K$ than that of their ambient umbral region. The detailed physical structure of a working model of umbral cores is proposed.